1 2 , we obtain (25.14').
§ 25.
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
487
We shall need the Lizorkin space of test functions which is adapted to Riesz integra-differentiation. This sp().Ce, unlike its variant for partial and mixed fractional Liouville derivatives (see § may be defined as consisting of functions with Fourier transforms vanishing not on the coordinate planes, but at the origin only. Namely, let
24.9 ),
(25.15) 0, Iii = 0, 1, 2, . . . }. (Compare this with the definition of this space in § 24. 9 , which is considerably more restrictive than (25.15)). The function tP (x ) = exp(- lxl 2 - lxl - 2 ) is an example of a function in the space (25.15). Let us consider now the space � , consisting of 'I = { tJI (x) : tP E S(R") , (vi tP) (O ) =
Fourier transforms of functions in
'I :
� = .1"('1) = { cp (x)
:
E S(R"),
(25.16)
consists of those and only those Schwartzian functions
This allows a simple characterization: the space �
j � cp(x)dx = 0,
Iii =
R•
0, 1, 2, . .
.
(25.17)
[2, 208] - that the Fourier transform
Indeed, it is known - Gel'fand and Shilov p. maps the Schwartzian space S onto itself and
j � cp(x)dx = i- lil j (ix)i cp(x)eiz·O dx = i- lil (vi
R•
R•
(25.15).
according to Thus � is the subspace of Schwartzian functions, for which all the moments equal zero. The space � may be equipped with the topology of the space S(R"), which makes � a complete space. We shall consider generalized functions on � (and on 'I as well) in order to justify some of our operations. We recall that the Fourier transform of a generalized function f E �' ) is defined as a functional j = .1"f, introduced by the rule
(
(j, tJI) = (/, .(/J), tJ1 E 'I .
(25.18)
This definition is correct, since F(w) = � and the Fourier transform is a continuous operation in the topology of the Schwartz space S(R") - Gel'fand and Shilov
(2,
488 p.
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
208]. If
g
is a functional in
w', the relation
(25.18') serves similarly as a definition of the Fourier transform for a functional g E The function lzl-o as an element of the space functional (lzl-o , ,p) = f lzl-o,P(z)dz for all a E
Iii = 0, 1, 2,
n. n.
R"
.
. . However, it is a nonregular functional
w'
w' .
the regular 1C generates since (Vi ,P)(O) = 0, an element of S' or ' if
as
Rea � For such a it will be understood in the sense of regularization achieved by the analytical continuation of the functional (lzl-o, ,P) from the half-plane Rea < The latter is given by means of the equation
(25.19) n =F 0, 2, 4, . . . where cp E S(Rn )), m > Re a - n -1 and Ci = 7rn / 2 2 1 - 2i [i!f(i + n/2)] - 1 • This yields the direct realization of the analytic continuation of the functional (lzl-o, cp) in the half-plane Rea < m + n + 1. We have already dealt with such a regularization in the one-dimensional case - see (5. 6 8). Equation {25.19) is derived in a standard way by subtracting the Taylor sum a-
and taking the relation
into account. The latter can be verified directly by use of the relation
{25.20)
§ 25. In the cases
RIESZ FRACTIONAL INTEGRO-DIFFERENTtATION
489
a - n = 2k, k = 0, 1, 2, . . . , excluded in (25.19), we set (25.21)
by definition.
(25.19) may be represented in a simpler form as i cp)(O)xi cp(x) - li i�[ReL:a)-n J�(V 1 · (25.22) dz (_l z_la ' cp) = R"f lxl a with m = [Rea] - n chosen here (if Re a > n); this, however, has to exclude the values Rea = n,n + 1,n + 2, . . . . Remark 25.1. The functional defined by (25.19) is an analytic function with respect to a for all a E C1 (since it is possible to choose m as large as required; it is evident that the right-hand side of (25.19) does not depend on this choice) except the points a = aA: = n + 2k, k = 0, 1, 2, . . As for the points aA: , the functional (25.19) has a first order pole there and admits the representation 1 (25.23) (� x a ' cp) = (ga , cp) + a - a�c ( A cp)(O), I where ( ga , cp ) is a function analytic in the neighbourhood of the point a�c and 1 lim ( g a , cp) = ( (25.24) 1 1 a., , cp) . The regularization
.
CA:
--
a-a�o
A:
X
25.2. The Riesz potential and its Fourier transform. Invariant Lizorkin space
(25.2)
Arguing formally for the moment we see that operation is to be realized as a convolution (in a generalized meaning) of a function f with the function .r- 1 ( 1 zl -a ). Let us evaluate this function first, treating the Fourier transform in the sense of generalized functions. It is convenient to use the Lizorkin space as a test function space.
(25.16)
Lemma 25.2.
The Fourier transform of the function lxl -a , interpreted according
490
CHAPTER
5.
INTEGRO-DIFFERENTIATION OF MANY VARIABLES
to (25.18'), is given by the relation
(25.25) where 6 = 6(z) is the Dirac delta-function, k = 0, 1, 2, . .. , the constant "Yn (a) being equal to
2k, a =/= -2k, (25.26) -2k, 2k.
a ;/; n + Q= a=n+
Proof.
lz l - a 1)/2 ( ) p a.
is a locally integrable function. We shall use Let Rea < n. Then Bochner's formula to compute the Fourier transform of this function. Let us assume first that ( n + < Rea < n , so that the condition is then satisfied for the function
(25.11)
F(l z l - a ) = lzla - n J,
(25.11)
(25.12)
00
where
J = (27rt/ 2 J P-a+n/2 Jn/ 2- 1 (p )dp. 0
We use the equation
(25.27) (2.52)
(known as Weber integral and obtained by substituting the Poisson integral into the left-hand side of and then interchanging the order of integration). �) , which means that we have proved the first So J = line in in the case ( n + < Rea < n. For the remaining values of a the Fourier transform of the function will be interpreted in the sense of In accordance with we are to show that
(25.27) (27r) nl 2 2 -a+n/2 r ( n;a) /f ( (25.25) 1)/2 lz l-a (25.18' )
1 = ' ;:��� ( lx l� a �) ( lx 1 a ' if>) ' E �, a ;/; n + 2k, a ;/; - 2k.
(25.18' ). (25.28)
§ 25.
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
491
This is true for ( n + 1) /2 < Rea < n since the first line has already been proved for such values. The left-hand side in (25.28) is interpreted1 as (25.19) if Rea � 0. The right-hand side is defined and1 analytic for all a E C since if; E The left-hand side is analytic for all a E C except perhaps the points a = -2k and a = n + 2k, where it has removable singularities: we see that (lz la-n ,
A
Q -
A
A
A
A
(25.30) since (l x l a.,-n ,
= ln2 + Hr' ( l ) + r' (� )1r ( � ) + � �] , 1
cr1
= n+ 2k.
492
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
This may be easily seen from the arguments in (25.29) and (25.30). Thus the operation - 1 lzl-aF is to be realized as the convolution with the function (25.25). In the case Rec:r > 0 this function is locally summable and the corresponding convolution is the lliesz potential (21.1) except for the values c:r = n, n 2 , n 4, . . . In the excluded cases we are to take the logarithmic factor into account according to (25.25). Thus we define the lliesz potential for all c:r, Rec:r > 0, as the convolution .r
+
+
la cp =
J ka (z - y)cp(y)dy,
(25.31)
R"
where
c:r - n :/; 0, 2, 4, 6, . . . ,
er - n = 0, 2, 4, 6, . . . ,
(25.32)
and the normalizing constant is given by (25.26). The function ka (z) is known as the Riesz kernel. Passing to polar coordinates in the integral la cp = ['Yn (c:r)r 1 j IYi a- n cp(x - y)dy, c:r - n :/; 0, 2, 4, . . . , R"
we have (25.33)
I.e. we may interpret the Riesz potential as a result of fractional integration of order c:r in the direction of the vector (see (24.39)) followed by integration with respect to In the notation of (24.39) (25.33) has the form u.
u
( n -a ) 2 ) +r 2IatnT = r (!±..2: 27r(n l)/ 2
s
J (Iatn)(z)du
..
UT
•
(25.34)
-1
The Fourier transform of the Riesz potential Ia cp, Re c:r > 0, is reduced to division by lzla : ·
(25.35)
§ 25.
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
493
for functions
+
Ra
1 J ei� · y <,O( y) dy, y)dy = (2?r)" lYl a
E
cl),
(25.36)
R•
which was proved in Lemma 25.2 for x = 0 - see (25.28) - and follows from (25.28) by the invariance of cl) relative to the translation operator. We conclude from (25.35) that The Lizorkin space cl) is invariant relative to the Riesz potential 1a and Ia (cf)) = cl) with
Theorem 25.1.
0, Re,B > 0.
(25.37)
The invariance of the space cl) is readily seen from (25.35) and (25.36): -a since IYI <,O(y) E \) fora any a and
Remark 25.3. The lliesz potential does not necessarily vanish rapidly at infinity even if 0 for lx l < 1. Then (Ja
(Ia
J dO' J1 p"- 1 (p2 - pO' .
Sa- l
�
� J dO' J1 pn- 1 (p2 Sa- l
Remark 25.4.
0
0
+
X+
lxl 2 ) (a - n )/2 dp
l x 1 2 )( a- n )/ 2 dp � c1 1 x l a- n .
The choice of the Lizorkin space cl) is essential for writing the
494
CHAPTER
5.
INTEGRO-DIFFERENTIATION OF MANY VARIABLES
semigroup property (25.37) for all a and {3 with Rea > 0 and Re {3 > 0. In the case of functions
(25.38)
R•
a > 0, {3 > 0,
a + {3 <
n,
for any unit vector e, l ei = 1. We observe also that (25.39)
Thisiaresult is a paraphrase of the assertion k (a) = la l -a . The lliesz potential of e ·y is a conventionally convergent integralaif 0 < a < (n + 1)/2. In the case a � (n + 1)/2 it is to be interpreted as the analytic continuation in the parameter a.
25.3. Mapping properties of the operator Lp (Rn ) and Lp (Rn ; p)
ra
in the spaces
First of all we remark that the operator Ia is defined for functions
J IYia-n
IYi a- n
l!!l > l
l!!l < l
The existence of the first integral here for almost all z E R:' may be established by showing that it belongs locally to L by using the Minkowsky inequality, while the second one exists for all z, if 1 p
Let 1 � p � oo, 1 � q ::5 oo and a > 0. The operator 1a bounded from Lp (Rn ) into L9(Rn ) if and only if
Theorem 25.2.
0 < a < n, 1 < p < -an , -q1 = p-1 - -an
is
(25.40)
§ 25.
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
495
We omit the proof of this theorem as well as of other statements of this subsection. See the references in § 29.1. A simple way to reduce Theorem 25.2 to the one-dimensional case may be found in § 29.2, note 25.2. The number q = in {25.40) is known as the Sobolev limiting exponent. Later in § 26.7 we shalln��P deal with the Riesz potential of functions
if
o: > 0, 1 < p < oo, 1 < r < oo, o:p- n < 1 < n(p - 1), 1 1 1 J' + n 1 + n ---(25.41) p n < -r -< p ' r = p 0: We observe that Theorem 25.3 contains the "if part" of Theorem 25.2 under the choice 1 = 0, I' = 0 and r =-np(n - o:p)- 1 . We note also a useful particular case 1 = 0, I' = -o:p, r = p in Theorem 25.3: {25.42) I l x l - ap l (Ia
--
--
-
.
R"
which is a generalization of the Hardy inequality (5.45). 1 In the case r = np( n - o:p) there is an extension of Theorem 25.3 to the weights of a general nature. Namely, let p(x) satisfy the so-called Muckenhoupt Wheeden condition
1 I pfiP(x)dx p'' TQi1 I p1 1(1 -P) (x)dx P - 1 $ c < oo, ( IQ I ) ) ( Q
Q
{25.43)
where Q is an n-dimensional cube, IQI being its Lebesgue measure. Theorem 25.4. Let 0 < o: < n, 1 < p < n/o: and q = np/(n - o:p). The operator 1a is bounded from Lp (Rn ;p) into L9(R!';pfiP), if and only if p(x) satisfies the condition {25. 43) with q = np/(n - o:p).
496
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
p
The assertion of the Sobolev theorem does not hold in the case = 1, but a weaker statement is true which is known as a weak type nestimate. Namely, let AJ(t) be the distribution function for a function /(z), z E R : AJ(t) = m { z : 1/(z)l > t } , t > 0.
(25.44)
It is clear that AJ(t) :5 t -P IIfll� with II/II� = J 1/(z) IPdz, the estimate so
R"
(25.45)
is weaker than the estimate liT/11 9 :5 ell/l ip , giving boundedness of the operator T from Lp into L9 • It is in terms of (25.44)-(25.45) that information on the Riesz potential in the Sobolev theorem is changed. Namely, the estimate n q = -n - a,
(25.46)
is valid with 0 < a < n and c not depending on t > 0. We also outline the mapping properties of the Riesz potential operator in the spaces of Holderian functions /(x), x E R,n , where R,n is a compactification of]Rn by the unique point at infinity. To guarantee existence of the Riesz potential 01t.p of a Holderian function it is necessary for t.p( x) to be zero at x = oo and it will be convenient to achieve this by means of a weight. Let where H>. (R' n )
_ _
{ f(x) .. f(x) E C(R' n ), 1/{z + h) - /{x) l :5
( cf. (1.5) and (1.6)).
clh l >.
(1 + l x l )>.(1 + lx + h i ) >. }
Let also
H�CRn ; p) = {/ : I E H\Rn ; p), pflx=o = Pflx =oo = 0}.
The space H>.(Rn ; p) is equipped with the natural norm, which makes it a Banach space.
Let 0 < a < 1 and ;\ + a < 1. The operator ]01 maps the space H>.(R,n ; (1 + lzl 2 ) (n + a)/ 2 ) isomorphically onto the space n>. +a (Rn ; (1 + lxl 2 ) (n -a)/ 2 )
Theorem 25.5.
§ 25.
497
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
and the space H6Ckn ; lz i.B ( + x 2 ) "1 2 ) onto the space H6Ckn ; lz i .B ( if a + A < {3 < n + A and a - {3 < x < n - A - {3.
l
l
+ x 2 ) -a+ "l 2 ),
We omit the proof and note only that it is obtained by mapping the space via the stereographic projection. It is worth observing that the lliesz potential lacp is transformed thereby into a similar potential (up to the weights) over the sphere, i.e. over a compact set in the usual metric, where the Holder property is established by direct estimation - see Vakulov (1, 2). We conclude this subsection by some simple relationships between Riesz potentials and one-dimensional fractional integrals of the Liouville type. For this purpose we need the Poisson and Gauss- Weierstrass integrals R,n onto the unit sphere Sn c nn+ l
(P,cp)(x) =
j P(y, t)cp(x - y)dy,
t > 0,
(25.47)
R•
(W,cp)(x) = j W(y, t)cp(x - y)dy, t > 0,
(25.48)
R"
where (25.49)
is the Poisson kernel and ( 25.50 )
is
the Gauss-Weierstrass kernel.
The Riesz potential l01cp, t.p E Lp(Rn ), 1 < p < nfa, admits of the representations
Theorem 25.6.
Wcp){z) = r (�l t a -' ( P. cp)(z)dt 00
j 0
=
r
00
(l!!)2 j t'l' - 1 (W,cp)(z)dt. 0
(25.51)
498
CHAPTER
5.
INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Moreover, the relationships
(P1la
(25.52) (25.53)
are valid, where the operators I� and 1�/ 2 apply with respect to the variable T. Proof. Substituting the representations (25.47) and (25.48) for P1
=
25 .4. Riesz differentiation (hypersingular integrals)
By Lemma 25.2 Riesz differentiation (- �)a/2 f = .r- 1 lzl- a-a.1'/, Rea > 0, is to be realized as a convolution with the generalizedn function l zl n . Such a convolution, i.e . an integral with the kernel lz - Yl -a- , in contrast to the Riesz potential has n R an order of singularity higher than the dimension of the space and so it will be called a hypersingular integral. Such an integral diverges and our convolution needs to be properly defined. Let at first 0 < a < 1 (or 0 < Rea < 1). We can guarantee convergence of the convolution of the function lzl - n-a (with sufficiently good functions) introducing it as so
(25.56) J IY - zln+a y - j f(x) IYi- nf(x+a - y) y. This integral converges if 0 < a < 1 for bounded differentiable functions and may be considered as a multidimensional analogue of the Marchaud derivative (5.58). An extension to the case a � 1 may be given either in terms of regularization,
f(y) - f(x) d
Rn
_
_
Rn
d
§ 25.
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
499
using Taylor sums as in (25.19) and (25.22), or by taking finite differences. We have already considered both approaches in the one-dimensional case while dealing with the Marchand fractional derivative of the order a 2:: 1 - see (5.68) and (5.80). To realize the Riesz derivative we shall use the approach via finite differences as more preferable although equivalent, generally speaking, to the former (see § 26.5, where the equivalence is shown for good functions in the more general case). Let us define finite differences (��l)(x) of a function l(x) of many variables with a vector step h. Let Th be a translation operator: ( rh l)(x) = l(x - h),
x,
h E R" .
We shall deal both with centered differences ( 25.57)
(with a vector step h and center x) and with non-centered differences (��l)(x) = ( E - Th ) 1 I
=
t.( -1 . (!) /(:• - kh). )
( 25.58 )
To avoid complications in writing we use the notation (��l)(x) for both types of differences, specifically mentioning the choice of difference when it is essential. We consider both types of differences because it will not always be possible to use just one type of difference - see § 26.4 below. Thus the realization of the operation ( -�Y)il 2 , a > 0, is expected to be given in the form of the hypersingular integral (25.59)
where the normalizing constant dn,r(a) will be chosen so that na I would not
500
CHAPTER
5. INTEGRO-DWFERENTIATION OF MANY VARIABLES
depend on l, only if l > a. The construction (25.60)
will be called a truncated hypersingular integral. We find it convenient to have also the designation for the integral (25.59) without normalizing constant: (25.61)
The next section is devoted to a more detailed investigation of the hypersingular integrals (25.59) and of more general constructions. Here we dwell only on the main fact that the hypersingular integrals (25.59) is indeed the Riesz derivative in the sense that (25.62)
under the appropriate choice of the normalizing constant dn (a) in (25.59). ,
Lemma 25.3.
the relation
t
The Fourier transform of the integral T01 f with l > a is given by (25.63)
where
dn ,t(a) = J (1 - e it 1 )1 l t l -n -01 dt
in the case when (�� /)(z)
is
R•
(25.64)
a non-centered difference and
dn , t(a) = J (eiy 1 / 2 - e -iy 1 / 2 )1 IYI - n -01 dy = 21 -01 i 1 J sin1 Y1 IYI - n -01 dy, (25.65) R•
R•
when the difference is centered. Proof.
Let �01f be the integral (25.26) truncated in (25.60). Let the difference as
§ 25. .6.�/
501
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
be centered. Then
i.e.
(1 eix·y )l dy. IY!n +a _
(25.66) J IYI > e It is easily shown that passage to the limit is possible here (in the sense of L2 , for example) , so A
:F(Te01 f) = f(x)
(25.67)
after the dilatation change of variables y = lx l - 1 e. We make also another change variables:
of
(25.68)
where wx('l) is any rotation in Rn , which transforms the first coordinate vector e1 = (1, 0, . . . , 0) into the unit vector xf l x l . It is clear that lei = 1 '7 1 and e � = '1 . e = '71 , so .
1
which transforms (25.67) into (25.63)-(25.64). Similar arguments are to be used in the case of a centered difference. Equation (25.63) yields (25.62) after division by d (a). However, we have to be cautious because of the possibility that the constantn,ldn,l (a) is zero for some a. In the case of the centered difference the question is easy: the normalizing constant dn ,l (a) equals zero identically (for all a) if I is odd and it is certainly different from zero if I is even, this being seen from (25.65). Thus the construction T01 is identically zero: T01 f 0 in the case of a centered difference of an odd order. Consequently, a centered difference is to be taken of even order I = 2, 4, 6, . . . only and then the passage from (25.63) to (25.62) is possible for all a, 0 < a < I. •
=
an
502
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The question regarding the non-vanishing of dn ,l(c.t) in the case of non-centered difference is more difficult, and it will be considered in § 26, where the detailed investigation of hypersingular integrals is discussed. It will be shown, in particular, that dn,I (c.t) as a function of c.t vanishes at the integer points of the interval (0, l). We formulate a corollary of Lemma 25.3. Corollary. The normalizing constant being given by the formulae (25.64) or (25.65) {in the case when it does not vanish}, the hypersingular integral n cr f does not depend on the choice of l, l > c.t. In conclusion of this subsection we give the relations 00
1- j t - 1 -cr ( E - Pt )1 fdt, ncr I = x(c.t, l)
0 < Q < l,
(25.69)
0
ncr/ = X (�1 , l) j t - 1 - !f (E 00
0
W,
t ) 1 /dt 0 < c.t < l,
(25.70)
1
similar to (25.51), where x(c.t, l) is the constant (5.81), while P,f and Wt f are the Poisson and Gauss-Weierstrass integrals (25.47)-(25.48). In the case of "good" functions these relations are derived from (25.52) and (25.53) by inverting the on_e-dimensional operators I� and I�/2 in terms of the Marchand fractional derivative (5.57) with further passage to the limit as t 0. Relations (25.69) and (25.70) may be considered as realization of (5.85) under the concrete choice of a semigroup T, in (5.85). Relation (25.69) is herein a realization of the fractional power (.J=K)cr, while (25.70) gives the fractional power (-A) cr/2 . -+
25.5. Unilateral Riesz potentials
We call integral operators (n/2) Cn- 1 j Y�n tp(x =f y)dy, Cn = r7rn/2 crl.r. tp = r(c.t) ' IYi R"+
R!'
(25.71)
unilateral Riesz potentials. Here x = (x , . . . , x ) E and integration is carried out over the half-space R+ = { x x E1 Xnn > 0}. Since y� I Y I -n I Y i cr-n , the operators (25.71) are similar in a sense to the Riesz potential Icrtp, being in particular bounded from Lp into L9, q = npf(n - c.tp), with 1 < p < nfc.t. The integrals ±tp are immediate generalizations of Liouville fractional integrals (5.4), coincidingI with them if n = 1. That is why we keep to the same notation for these objects. As in the one-dimensional case we shall call potentials If. tp and I� tp left and right-hand sided respectively. We denote x' = (x 1 , . . . , Xn- 1 ) and observe that :
R!' ,
=5
§ 25.
RIESZ FRACTIONAL INTEGRO-DIFFERENTIATION
503
the constructions (25.71) represent "interlacing" of one-dimensional fractional integration and the Poisson integral: an
� J u::-' (P 00
J:;_
where
r( )
••
0
(25.72)
(Py,. (z)dz = J ( le' l i€n ) - 01/J (€)d{, (25.73) R• - •
W
>
c)
W
c).
'f
R"
R"
(25.74)
Proof. We denote Cn - 1 (zn )±a lzl - n e - e Jz ,.J ka± ,£ (z) = r(a ) =
e l n ) a_ ± l e - lz ra l p(z' , l zn I ) , ) r( a_(z
where P(z', lzn l ) is the Poisson kernel (25.49): we have ki,: (e) = (.1"z ,..1"z' k% ,e )(e) = rta; j0 z�- l e- z,.(e+ le'l =fi{,.) dzn . Relation (7 .5) shows that the latter integral equals (e + le' I ien )a under the assumption that arg(e + le' l ien ) = 0 en = 0. Hence By ( 25.54)
=t=
as
=t=
(25.75)
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
504
This yields (25.73)
as
c -+
0, which, in its turn, leads to (25. 74). •
Corollary 1.
Potentials I� possess the semigroup property: I� I� = I±+P .
Corollary 2.
The Riesz potential I01 is representable as a composition of unilateral
potentials:
(25.76) We find now the explicit form of the operators inverse to I�. The desired constructions are to coincide with the Marchaud derivatives D± - see (5.57) in the case 1. We apply formally the Fourier transform Fz' in the variable z' to the relation=(I.+
where the operator I+ is applied with respect to the variable Inverting it by means of the Marchand derivative (5.57), we arrive at . Yn . 1 f(-a)A,(a)
whence after multiplication by e -z.. le'l and application of the inverse Fourier transform F;,1 we finally have IP (z) =
,.7:,�) I r�: ( �� f)(z)dy,
I>
<> ,
R+
where ��/ is the non-centered difference (25.58). So Cn - 1 (I±a ) - 1 / = x(a, l)
I yIYi;;na (�I±y f)( )dy�Da± /, I > a . x
R+
(25.77)
The integrals in (25.77) are hypersingular. It is not difficult to show that for f E S( Rn ) they converge (conventionally) in the sense that D±f = elim -o D± 'e f,
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
where
505
00
D t. f = :(:,'I) j
j j�: (��. f)(z)dy . The constructions D±f are identical with Marchand fractional derivatives if = 1. This was the reason for keeping the same notation as in the one-dimensional case. We also note that <JJit)(e) = (le'l =F ien )a i(e) . £ R•- l
n
§ 26. Hypersingular Integrals and the Space of Riesz Potentials
We now go on to a more detailed consideration of the hypersingular integral (26.1)
defined in § 25. In particular it will be shown that the hypersingular integral generates a "true" inverse to the Riesz potential when the latter is considered within the framework of the spaces Lp. We shall clarify what advantage the centered or non-centered form of a finite difference has in the definition of the/hypersingular integral. Further, hypersingular integrals, more general than , will be considered. They prove to be a natural extension of partial differential operators into the case of fractional orders, since any homogeneous partial differential operator with constant coefficients may be represented as a hypersingular-type integral, as will be demonstrated in § 26.6. A special role in this section is assigned to §n 26.7, where the space Ia (Lp) of Riesz potentials is investigated, the spaces L;,r (R ) , generalizing Sobolev fractional spaces L;(Rn ), aare considered and the relationship between the spaces L;,r (R") and the space I (Lp) is established. We note the relation na
(26.2)
as one of the main results. 26. 1 . Investigation of the normalizing constants functions of the parameter
as
a
dn ,1 (a)
As it was made clear at the end of § 25, we have first of all to answer the question about the zeros of the function dn,l (a) used in (26.1). We find them in this subsection and at the same time we give representations of the integrals (25.64) and (25.65) via elementary functions of the parameter a.
506
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
We introduce the following functions of the parameter a: incentered the casediffoference, the non incentered the casediffoference, the
(26.3)
in terms of which the constants dn,l(a) will be expressed. Here I = 1, 2, . . The former of these functions is familiar to the reader from the one-dimensional case, where it arose in the process of the evaluation of the normalizing constant for the Marchand fractional derivative of order a � 1 see (5.73) and (5.80). We considered there the Marchand derivatives with non-centered differences only. The latter of the functions (26.3) in the case of even I = 2, 4, . . may be represented also as . .
-
.
A:'( a) =
t.(- l) k- 1 G) �� - ki a
(26.4)
The right-hand side here is identically zero in the case of an odd I and so it differs then from AI'( a). The function AI( a), a E R1 , vanishes at the points a = 1, 2, 3, 1 1 and nowhere else. The function A�'(a) has even integers a = 2, 4, 6, . . . , I 2 as zeros in the case of an even I and odd integers a = 1, 3, 1 2 in the case of an odd I. Lemma 26.1. . . . ,
-
-
. . .
,
-
Proof. We consider A, (a) = AI(a) first. Let us use the relation 1 1 f Al(a) = r(1 - a) (I -lne )'(26.5) l e l cw cJe, a < I, obtained § 5 - see (5 .81 ) . It is readily seen from (26.5) that in the case a < I the function AI(a) vanishes only at the poles'of the gamma-function: a = 1, 2, . . , l - 1. If a = I, we have Al(a) #; 0 by (5.74 ). Let a > I. The recurrence relation Al+ 1 (a + 1) = Al+ 1 (a) - Al(a) holds, and this is verified by definition (26.3). It1,!1is then not difficult to show by induction that for a > I, Al(a) > 0 if I is odd and A/(a) < 0 if I is even. Thus AHa) #; 0 for a > I also. Let now A,(a) = Al'(a). Using (26.4) we conclude that if a is an integer and has the same parity as I, then I
in
0
.
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
507
(26.6) - (p-dpd ) a ( pV"P1 ) I Ip=l = A"1 ( a) for an integer a of the same parity as I. Hence the proof of Lemma for A�'(a) follows. Remark 26.1. Lemma 26.1 answers the question about the zeros of the function A1 (a) = Aaa). In some problems related to hypersingular integrals (Samko [31, p.l40]) it is of importance to be sure of the absence of other zeros of the quasipolynomial A1 ( z) in the complex plane except at the points z = 1, 2, . . . , I - 1. This question is open. Theorem 26.1. The normalizing constants dn ,l (a) in (26.1) are analytic functions of the parameter a and are given by the relations dn ,l (a) = .Bn (a) sm. A,(a) (26.7) (a1r/2) , (26.8) Pn (a) = 2ar (1 ?rl +i)n/r2 (�) except for the case of a centered difference of an odd order I, when dn ,l(a) 0. The constant d,. ,, (a) is different from zero for all a > 0 in the case of a centered difference and of an even I, while in the case of a non-centered difference it vanishes for a > 0 if and only if a = 1,3,5, . . . , 2[1/2] - 1. Proof. We begin with the remark that A, (a) = 0 for an even integer a by Lemma 26.1, thus A,(a)/sin(a?r/2) in (26.7), in the case a = 2,4,6, . . . , is to be understood as he-.ma sm. A,(e) (26.9) (e 12) - � ( 1) a/2 .!!._da A1 (a). Evaluating the integral (25.64) in the case of a non-centered difference, we have dn ,,(a) = j (1 - eiY t ) ' dyt j ( IYI2 yn -
--
•
+
=
_
7r
_
7r
+
+
Rl
R•- t
508
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
We find the value of the integral over Rn - l by changing to polar coordinates: J (1 + I TJI 2 ) -(a+n)f2dTJ = I Sn - 2 1 J pn- 2 (1 + p2 ) -(n+a)f2 dp 00
R• - 1
0
� ISn- 2 1 J r
=
0
"
2
Since
2
2
(26.11)
we have
2 ) - (o+n)/ 2 d'l = .-(n - ll/ 2 r ( 1 � '1 1 + 1 J (l
R• - 1
So
"'
) (n ; a) . /f
I
Let 0 < a < 1. Using the relation kL=O( - 1)k (!) = 0 and the equation (26.12)
derived from (7 .6), we transform the remaining integral to 1 k ( ) oo - 1- a 1 L ( - 1) k' J t ( cos kt - 1)dt = � L (-l)k-1 ( k' ) k Joo t -a sin kt dt k=O k =O o 1rA1 (a) _ - 2f ( l + a ) sin ( a 7r / 2) 1
0
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
509
which gives the value of dn ,t (a) presented in (26.7). For other values of a � 1 the required result is easily obtained by analytical continuation in a. In the case of a centered difference and of an even l we obtain (26.13) dn ,l (a) - (-1)1/2 2'- a+ 1 7rr(n(-n1+2)/a2)r (�) joo sint 1 +'at dt similar to (26.10). We make use of the formula 1/2 A: 1 ( l ) 1 1 1 sin t = 2 �( -1) - 112 k ( 1 - cos 2kt) which is simply obtained by expanding sin1 t as a Fourier series in cos kt on the interval [0, 1r]. Using this result and integrating by parts yields 0
_
7r2a- l 1/2 -1 A:- 1 ( l ) ka = r(1 a)sin(a + 7r/2) �( ) l/2 - k in the case 0 < a < 2. It is ea.Sily seen that So
(26.14)
(26.14) is transformed to 00
sm+ t dt - ( 1)112 2a -1- 1 A1 (a) (26.15) r(1 + a) sin(a 7r/2) ' J t1 a Substituting this expression into (26.13) we obtain (26.7) after simple transforma tions. The required result for values a � 1 is obtained by analytic continuation in respect to l The identity dn,l ( ) 0 in the case of a non-centered difference and of an odd is readily seen from (25.65). •
I
_
_
0
a.
a =
510
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
It remains to elucidate the question concerning the vanishing of d ,z(a). The case of centered difference is clear in view of (25.65). So let then difference be non-centered. By (26.7) the question is reduced to that of the zeros of the function Az(a) = A:(a), which were found in Lemma 26.1. Zeros of this function, being a = 1, 2, 3, . . . , I - 1, are prime according to (26.5). So dn,z(a) :/; 0 for a = 1, 2, 3, . . . in correspondence with (26.9). We give also explicit values of dn ,z(a) for even a: a
•
(26.16) (26.17) Q = 2, 4, 6, . . . ,
in the cases of non-centered and centered differences, respectively. Thus, the normalizing constant dn,l(a) being chosen in correspondence with a (25.64), (25.65) and (26.7), (26.1) properly defines the operators n I according to Theorem 26.1, in all cases except the case when the difference is non-centered and a = 1, 3, 5, . . . In this case dn,l(a) = 0 and we come across the phenomenon of "annihilation" of the construction (25 .61): Ta l = 0, a = 1, 3, 5, · · · < I,
(26.18)
if a non-centered difference is used. So a non-centered difference has, on the one hand, an advantage in that its order is not necessarily even. (The requirement to choose I even means that we are in general to take I greater, which is undesirable). On the other hand a non-centered difference leads to the non-desirable effect (26.18) in the case a = 1, 3, 5, . . . We shall consider this special case in § 26.2. .
26.2. Convergence of the hypersingular integral for smooth functions and diminution of order l to l > 2[a/2] in the case of a non-centered difference
Let us show that the hypersingular integral (26.1) converges absolutely for functions l (z), which have bounded partial derivatives of order [a] + 1. For definiteness we consider the case of a non-centered difference.
§ 26.
HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
511
The relation ( AL /)(z) = m
;E b l =m
. 1
I
o
k =O
�� j (1 - u)m - l ;E(- l)m -k km (!)
(26.19)
is valid, where hi = h{1 h�· . We have given its proof in the one-dimensional case - (5.75). In the case of many variables it is proved in a similar fashion by means of the Taylor expansion for functions of many variables with the remainder in the integral form. We conclude from (26.19) that • • •
l(�i f)(z)l � cl h l m
�eR•sup l(vi f)(z)l
lil = m
=
c 1 l h lm , 1 2:: m.
(26.20)
We readily obtain from (26.20) that the integral (26.1) converges absolutely if function f(z) and all its derivatives (Vi f)(x), Iii = [a) + 1, are bounded. We show now that the hypersingular integral (26.1) converges conventionally for I > 2(a/2] in the case of a non-centered difference. In other words, the integer I may be chosen not necessarily greater than a, but as the odd integer which is the nearest to a. We consider I to be odd, since if I is even and I > 2(a/2], then I > a. The following directly verified relation a
(26.21)
is valid, where the function �(T) = !(T - 1)1 (T + l)T- (1+ 1)/ 2 is anti-invariant . relative to the inversion, i.e . P,( T- 1 ) = -�( T) . Equation (26.21) generates the corresponding identity in finite differences ( � /)(z) = ( � /)(z) -
� (A�+ l f)(z + ky) - � (A�+ 1 f) (z + 1 � 1 y) ,
(1 - 1)/ 2
(26.22)
512
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where the function (P� I)(z) is odd in y. Then the limit (l- 1)/ 2 j (Al+ 1 l)(z + k y) d .hm (A'11 l)(z) y y d = � + n J cr IYi IYi n+ cr � k- 1 R" IYI>� � -o
-
11
(26.23)
exists, since l+ 1 > a, which yields the conventional convergence ofthe hypersingular integral ncr 1. We return now to the phenomenon of the annihilation (26.18) of the hypersingular integral of odd order with a non-centered difference. Since l > 2(a/2] is allowed, we may avoid this phenomenon by choosing l = a in the case a = 1, 3, 5, . . . Then At(l) ;/; 0 and the definition of a hypersingular integral by (26.1) with l = a = 1, 3, 5, . . . becomes proper. It should be stressed, however, that conventional convergence in this case is essential even for the very good functions. .
Everywhere below we assume that l > a and l is even in the case of a non-centered difference and l > 2(a/2] in the case of a non-centered difference with the obligatory choice l = a in the case a = 1, 3, 5, . . . . However, in § 26.4 while generalizing the hypersingular integral (26.1) we shall allow odd l in the case of a
centered difference too. The following lemma is close to (25.34).
The Riesz differentiation ( -A) cr/ 2 = n cr can be interpreted as a result of a fractional differentiation in an arbitrary direction u, l u i = 1, with the posterior integration in respect to u: Lemma 26.2.
Da / = - f(- a�:�";�"/2)
J (V;:f)(z)du,
s,._ l
(26.24)
where fJn (a) is the constant (26.8), (V�f)(x) is the fractional derivative (24.38) (24.38') in the direction u and ncr I is taken with a non-centered difference.
The proof is obtained by changing to polar coordinates. 26.3. The hypersingular integral as an inverse of a Riesz potential
The hypersingular integral (26.1) generates an operator mverse to the Riesz potential 1cr cp:
(26.25)
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS cp
513
for sufficiently good functions, e.g. E �(Rn ), which is obvious via Fourier transforms, see (25.35) and (25.62). We shall show that the inversion (26.25) is true on the whole domain of the Riesz potential within the frames of Lp-spaces: cp(x) E Lp(R), 1 � p < nfo:. It is important to stress here that the hypersingular integral na applies to functions 1a which are not "very good" and so it will not be, in general, absolutely convergent. It will be interpreted as conventionally convergent in Lp: (26.26)
where D� f is the truncated hypersingular integral (25.60). We introduce certain auxiliary functions, which are finite differences of the Riesz kernel ka(x). Namely, let us define the function (26.27) where o: > 0, l = 1, 2, 3, . . . and single out the case h = e1 = (1, 0, . . . , 0), setting (26.28) so that (26.29) kz , a (z ) = 'Yn �a) �{-1)1 (!) i z - .l:et la-n if o: - n =F 0, 2, 4, . . and a non-centered difference is chosen. An important role in our consideration will be played by the kernel .
K:
z a { l z l ) = dn ,l (�) l z ln J kz, a ( Y)dy. IYI < I�I ,
(26.30)
Since ka(x) = lx l - a by (25.25), the Fourier transform of the function (26.27) is given by the expression fordifference a non-centered (26.31) for a centered difference. The function �l,a ( h) may be expressed in terms of k1,a (x) by means of the x,
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
514
rotation: (26.32)
where l > a - n in the case a - n = 0, 2, 4, . . . . Here Wz (h) is the rotation (25.68). Equation (26.32) is verified directly by means of the equality lz - ketl = llzlet - kJirl, which yields the relation jy�w;1(h) - ketl = lhl-1lz - khl if we take into account that In lhl kL:=O(- 1)k {!) lz - kh la- n 0 in the case a - n = 0, 2, 4, 6, . . . and l > a - n. l
=
The function kt, a (x) satisfies the estimate
Lemma 26.3.
lkt, a (x) l � c(1 + l x l )a- n - l as lx l � l + 1,
(26.33)
kt, a (x) E Lq (Rn ), 1 - a:fn < 1/q � 1
(26.34)
J kt,a (x)dx = 0.
(26.35)
so that if l > a. Besides R•
Equation (26.35) is valid in the case 2[a/2] < l � a too, if l is odd, the difference in (26.28) is non-centered and the integral in (26.35) is interpreted as lim J kt,a (z)dx. N-oo
lz - !e•I< N
We denote W(s) = ka (x + set ) , s E R1 , so that kt,a (x) = (�� W)(O). By the known relation Proof.
(�� W )(s) =
e
e
J ... J 0
w ( l > (s - St
-
. . . - s,)ds l · · . ds,
(26.36)
0
(J
we obtain kl,a (x) = w<1> ( -0), 0 < < I, whence the estimate (26.33) is easily derived, the inequalities l b 1 and I > k being taken into account in the case n = 0, 2, 4, . . . . Since k, ,a (x) is locally integrable to the power q, 1/q > 1 - a:/n, we see then that (26.33) implies (26.34). Equation a: -
p
cp
k- l
p
§ 26.
HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
{26.35), i. e . the relation k, , a (O) = 0 follows from {26.31) if I > a.
515
Further
Changing the index in the second sum by I - we see that it differs from the first one by the factor (-1 )1 • So v
v
j
lx - !e 1 I
for all N > 0 in the case of an odd I. Lemma 26.4. The estimates
I X:l, a { l x l ) l $ C
kl, a (x)dx = 0
{26.37)
•
{ lxlmin(a-n,O) , In 1 rzr ,
IX:l, a { lxl)l $ clx l a - n - l• '
a =F n, a = n, lxl $ 1 ,
{26.38) {26.39)
are valid, where I* = I in all cases when I > a, and I* = I + 1 in the case when 2[a/2] < I $ a and a non-centered difference is used in {26.38). So X: ,, a ( lxl) E L9{R"), 1 - a/n < 1/q $ 1.
Proof.
The verification of {26.38) is obvious. To prove {26.39) we use the property
{26.35) and have
IK I , a (lz l)l = ��n
k:(a) j
IYI>Ixl
l
kl, a (y)dy ::; lzln:l -a ·
If 2[a/2] < I $ a, so that I is odd, the estimate {26.39) may be obtained because {26.37) enables us to carry out the integration not over the ball I YI < lxl, but over the layer lxl - 1/2 < IYI lxl + 1/2. Namely by {26.37) and {26.33) we have for <
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
516
lzl > I + 1:
L(a) l j lkl,a(Y) i dy ( ) a-1 - (lzl - / ) a- 1 1 <- c n lzl l 2 2 lzl I
I KI , a (lzl)l � lzl n i
lzl - � < IYI
+
__
< _
-
C1 1 X � - n+ a- 1 - 1
-
•
•
We now prove the following theorem which is preparatory for the inversion of the Riesz potential Ia
( D: f)(x) = j K1 ,a (lyl)
(26.40 )
R•
Proof. Applying a finite difference to the Riesz potential, we have (d� /)(z) = f a,, a (e , y)
(26.41 )
where d1 ,a (e, y) is the function {26.27). Hence The interchange of the order of integration is possible in view of the absolute convergence in the case of c > 0. We apply (26.32) and make then the change of variables �wi 1 ( y) = z zE Rn , where w; 1 (y) is the rotation inverse to the rotation (25.68). Then I Y I = le l / l l and kl,a ( z )dz .
The dilatation transformation e -+ ce leads now to (26.40).
•
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
517
The kernel K,,a ( lyl) has an averaging property:
Remark 26.2.
j Kl,a( lyl)dy = 1.
(26.42)
R"
This relation, being a consequence of the choice of the normalizing constants, is easily established indirectly: let ,
4>
R"
Corollary. The Fourier transfonn of the kernel K:, , a ( lzl ) is given by the expression
(26.43) being written for the case of the non-centered difference in (26.27)� E 4> (26.40), Off(z) = k.,, a (Ez)Daf(z).
Indeed, taking / in
we have
E = 1 we obtain then (26.43) by (25.66) and (25.62).
limo D� f D a I = e(L p ) potential within the frames of the spaces Lp (Rn ): Theorem 26.3. The operator
Letting
is the left inverse to the Riesz
(26.44)
The way to prove this theorem is provided by Theorem 26.2. Really, by
(26.42) we have
(D: /)(z) -
(26.45)
R"
from (26.40). By the Minkowsky inequality we obtain li D:/ -
(26.46)
where w�(
518
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
by the Lebesgue dominated convergence theorem, the application of which is justified by the fact K:z,a ( lyl) E Lt(R"), as proved in Lemma 26.4. Let us emphasize that we have demonstrated the Lp-convergence of the truncated Riesz derivative (hypersingular integral) n:I in the range Ia (Lp) of the Riesz potential and have given an estimate of this convergence in (26.46) . One may show that the convergence also holds almost everywhere. We will not dwell here on this and refer the reader to § 29.2, note 26. 1. In conclusion of this subsection we note that besides the hypersingular integrals, considered above, constructions (25.69) and (25.70), defined the previous section, may be used to invert the Riesz potential with E Lp. Namely, the following theorem is true. Theorem 26.31• Let I = la cp, cp E Lp(R"), 0 < a < n, 1 � p < nfa. Then px cp
cp
=
-
00
lim j t - l -a (E - P,)1 fdt ,
1 l) X( a, e -O
l
cp
in
> a,
t
the limit may be taken under the Lp -norm or almost everywhere.
26.4. Hypersingular integrals with homogeneous characteristics
Generalizing the hypersingular integral (26.1), we introduce the construction given by (D na /)(x) = J (Ll�f)(x) IY in+a n(x , y)dy where we shall call the function O(x, y), not depending on l(x) a characteristic. This follows the terminology accepted in the theory of multidimensional singular integrals (see Mihlin [2]). We shall not consider here the constructions in such a general form, but treat the case when O{x, ) does not depend on x and is homogeneous in y: n = O(y'), y' = Y/ I Y I so that y R"
Dna / = dn,l1( a)
J IY in+a
(Ll�f ) (x )
1
O(y ) dy ,
l
>a
(26.47)
R"
We write the normalizing constant as before in spite of the presence of an arbitrary homogeneous characteristic O(y') in order that D0/ be independent of the choice l > a. See Corollary 1 of the Theorem 26.4. As regards more general characteristics, we refer to the references in § 29. 1 (notes to § 26.4) and to § 29.2 (notes 26.3 and 26.4).
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
We note at once that the presence of a characteristic O(y') in the hypersingular integral allows to use this integral with centered differences of an odd order. We shall give a certain classification of hypersingular integrals {26.47) and be concerned with following questions in this section: A) What are the relations giving the Fourier transform of D fi f? B) Under what conditions on O ( y' ) does the convergence in Lp of the hypersingular integral naf with a constant characteristic imply the convergence of the hypersingular integral with the characteristic O(y') and vice versa? In § 26.6 we shall see more clearly what are the hypersingular integrals {26.47). We shall show that the class of such integrals contains, in particular, operators of the form ;:- 1 a:F with a(x) = lxl aa(x'), x' = x/� 1 , being a sufficiently smooth homogeneous function of degree a. In the case of an integer a the class of the operators {26.47) contains all homogeneous differential operators in partial derivatives of order a. First of all we state the following. 519
We shall call the integral {26.47) a hypersingular integral of the neutral type, if it uses a non-centered difference, and a hypersingular integral of the even (or odd) type, if it uses a centered difference of an even order (or odd) respectively. By {26.20) the integral {26.47) converges for l > a in the case of a sufficiently good function f(x) if O(y') E L 1 (Sn - d · For a hypersingular integral of a neutral type and with the even characteristic O( ') = 0( the order of the difference
Definition 26.1.
y
-y')
may be lowered owing to conventional convergence to
(26.48)
l > 2[a/2]
with an obligatory choice l = a in the case a = 1 , 3, 5, . . . . This may be shown following the same lines as in § 26.2. We consider l everywhere below to be chosen as it has been indicated. The neutral type of a hypersingular integral has some advantages in comparison with an even or odd type. This has already revealed itself in the possibility {26.48) to lower the order l. It will prove to be more universal in the problems of inversion of potential type operators. In particular, it makes sense to consider integrals of the even (odd) type for even {odd) characteristics O(y ) only. Namely, if a characteristic is arbitrary: '
n(y') = n+ (Y') + n_ (y') , n± (y') = n(y') 2n(-y') ±
then
D fi/ =: Dfi_ /
{26.49)
520
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
for integrals of the even and odd type respectively. I.e. integrals of the even or odd types are annihilated in the case when the characteristic has an opposite parity. These relations follow for example from {26.69)-{26.71) proved below. On the other hand the hypersingular integrals of neutral type have their own "strangeness" which is reflected in the following remark. Remark 26.3. A hypersingular integral of the neutral type "annihilated" identically: IS
J IYI- n-a (�� f)(x)O(y')dy
Rn
=:
0
for any a = 1, 3, 5, . . . , whatever f and 0 were, if 1 > a. This was established in § 26.1 (see {26.18)), for O(y') = const. For O(y') =F const this annihilation will be obvious in Fourier transforms as seen in {25.68) and {25.69) below. As for the case 1 = a, the hypersingular integral of the neutral type converges conventionally if and only if O(y') is even. This may be seen from the transformations in § 26.2 by substituting the characteristic O(y') into the left-hand side of {26.23). Remark 26.4. In the case of a hypersingular integral of the odd type the normalizing constant dn ,t(a) has not been determined yet. The constant {26.65) equals zero in the case of an odd 1. Starting from {26.7) and taking into account that A�'(a) = 0 for a = 1, 3, 5, . . . by Lemma 26.1, we set by definition dn ,l (a) = f3n (a) cosA"(a) la1l'/2)
{26.50)
for hypersingular integrals of the odd type. We also note that for hypersingular integrals with a homogeneous characteristic, the relation {26.51) D� f = r (- ot�:�"' "/2) J O(u)(V� f)(z ) du Sn - 1
is valid, where (V� /)(x) is the fractional derivative {24.38)-{24.38') in the direction u. This result generalizes {26.24) and is similarly obtained by passage to polar coordinates in {26.47). A) Evaluation of the Fourier transform for D0f. Following the transformations in {25.66)-{25.67), we can easily show that the relation in Fourier transforms F{ Dfi/) = Vn (x) i(x)
{26 . 52)
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
is valid, where the function V0(x) is given by the relations fordifference a non-centered
521
(26.53)
We shall call the function V0(x) the symbol of the hypersingular integral D0 f . This follows the terminology accepted in the theory of multidimensional singular integrals - see Mihlin [2]. It is clear that the symbol vg ( ) is homogeneous of order a: X
(26.54) Theorem 26.4.
symbol are valid:
Let !l (u) E L t (Sn - d · The following representations for the
vg (x) = c:(�=�2) V{l(x) = wn (a)
s
j O(u)(-ix · uNr,
s,._ l
j !l(u)lx
.. - 1
V0(x) = -iwn (a)
·
,t
u l a du,
j !l(u) lx
s,. _ l
n
·
1, 3, 5, . . . ,
(26.55) (26.56)
u l a sign ( x u)du, ·
(26.57)
for hypersingular integrals of the neutral, even and odd type respectively, where Wn (a) = r ((n + a)/2)[211'(n - l )/ 2 r (l/2 + a/2)] - l and the designation ( ie ) a = lel a e - i ...; sign e is used in (26.55). In the case of an even characteristic !l (u) (26.55) and (26.56) coincide and are valid for a hypersingular integral of the neutral type of order a = 1 , 3, 5, . . .
Proof. In the case of a non-centered difference we have (26.58)
As the mner integral is analytic m a, it sufficient to calculate it for IS
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
522
0 < a < 1 . Using the identity
L:l ( - 1 ) 1: (!) = 0 we obtain
1::0
where e = X . lT. Integration by parts in the first of the integrals and using (26.12) give
foo (1 0
p
e iP( ) l
1 +a
. (_ ,e) .
1rAHa) dp = f(1 + a ) sm a 1r
a,
a > 0,
I
> a,
(26.59)
which transforms (26.58) into (26.55) in view of (26.7). In the case of a centered difference we arrive at the inner integra.I
I(e) = j
00
1 P- - a
0
t e
(e i- P( - e- P )
'
dp
e = X · O',
similarly to (26.58), ass uming that 0 < a < 1 again. Applying (26.59) we obtain
( I ) [ - (� - ) ] t ( v -1 ( I )� � _ 1 v=O ( )1 I (I )}
1r � -1t- 1 sin a1rf( 1 + a) �(
1rle l a {
1 2f(1 + a) sin( a 1r /2) +
signe 1 cos(a1r/2) .
� �( - 1 )" _ 1
�
11
_1 1
11
)
2
2
11
1
2-
a
11
a
v
11
a
11 2-
sign .
•
L: { - 1)" {!) I � - v i a = 0 if I is odd and L: { - 1)" (!) I � - v i a x v=O v=O "' AI' ( a ) leiOt ·stgn ( 2' - 11) =: 0 1"f I · even, we see th at J(C) , = 2r(l ) sin ( a "'/2) tOr an even an d +a 1 1r iA" a I(IOt sign ( • (e ) = - 2r 1 + a . ( 26.56 - ( 26.57) . • a1r 2) for an odd I, wluch g1ves Noting that
I
l
l
IS
cos
)
r
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
523
The hypersingular integrals (26.47) do not depend on the order I of the finite difference under the suggested choice of the normalizing constant.
Corollary 1.
The hypersingular integrals of the neutral and even type coincide in the case of an even characteristic. We also conclude from (26.55)-(26.57) that the hypersingular integral D0/
Corollary 2.
of an integer order a is a homogeneous differential operator in partial derivatives under the proper choice of the type of a hypersingular integral. Namely, the following corollary is true. Corollary 3. The with a = 2, 4, 6, . . .
symbol of a hypersingular integral of the neutral or even type or of the odd type and with a = 1 , 3, 5, . . . is a polynomial
Vna (x) = 26{311" ( a) n
� LJ
-:;. x' 1J"1 = a J
n; .
(26.60)
where 6 = 1 for a hypersingular integral of the neutral and even type and 6 = i for a hypersingular integral of the odd type, n; being spherical moments of the function
O(u) :
n; =
J ui n( )du.
(26.61)
(T
s. - 1
To obtain (26.60) from (26.55)-(26.57), the relation
(x · u)m = is to be used. B ) Convergence characteristics.
of
m' � LJ -;fxJ
.
lil = m
J.
hypersingular
.
u3
integrals
with
different
We wish to answer the following question. Let f ( x ) be such a function that its hypersingular integral converges for a certain characteristic 0 1 (y'). Does then the hypersingular integral of the same function converge in the case of another characteristic 0 2 ( y' )? It is convenient to break up this question in the following way: does the convergence of a hypersingular integral with a constant characteristic (i.e. the convergence of the Riesz derivative (26.1)) imply that of such an integral with an arbitrary characteristic, and vice versa? In the first direction this question will be solved positively in the sense that the convergence of n a f implies that of D(i / for an arbitrary bounded characteristic. The boundedness condition may be weakened. Clearly, since the convergence of D (i / with the characteristic O(y') = 0 cannot imply the convergence of n a f, the inverse assertion will require a certain "ellipticity" condition on the characteristic O(y').
524
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
For the function /(z) itself we make an priori assumption that /(z) E L r (Rn ), 1 < r < oo , while the convergence of its hypersingular integral will b e considered in the space Lp(Rn ), 1 < p < oo: a
limo D0 ., /, D0/ = cwhere
(26.62)
n
(A� /)(x) ( )dy. 1 y Doa ,c f = d ,z(a) j n IYI>c I Y in +a '
We do not go into details here and give only the outline of our arguments. The details of the proof may be found in Nogin and Samko [1], [3]. Let f(z) E Lr (Rn ) and let the Riesz derivative n a f exist in L p (Rn ). For such functions one can directly verify that
D aO,c f -- A c D a f
(26.63)
where A, is a convolution operator with the kernel
a,
1 A z a (x, y)O(y') dy (z ) = d ,z(a) J IYin +a n IYI>c ,
(26.64)
Az ,a (z, y) being the function (26.27). This kernel is locally integrable l a,(x) l 5 clzl - n as l z l -+ oo. Using the properties of the function Az, a (z, y)
and (see § 26.3) one can show that the convolution operator with the kernel is bounded in Lp uniformly in €: I IAc ii L ,. -L ,. 5 c, where c does not depend on €. This fact allows us to justify the passage to a limit in (26.63) as € -+ 0 by the Banach-Steinhaus theorem. The limiting operator A = lim A, is an operator of c-o the form A
j ( -ix u)00(u)du I 0, ·
l xl = 1,
(26.65)
SR - 1
provided that hypersingular integrals of the neutral type are used (cf. (26.55)). If the characteristic is sufficiently smooth and satisfies the ellipticity condition
§ 26.
HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
525
(26.65), then the inverse statement can be derived from (26.63), namely that the convergence of Dfi l in Lp implies that of ncr I in Lp, I E Lp. The above arguments are justifiable provided that 1 < p < oo, 1 < r < oo and 1 /p - a/ n 5 1/r 5 1/p. 26.5. Hypersingular integral with a homogeneous characteristic as a convolution with the distribution We shall show that the hypersingular integral (26.47) .may be represented as + a convolution with a generalized function of the form O(z')/lzl n a , i.e. the hypersingular integral coincides with the regularization of a divergent integral. Corresponding with (25.19) we define a regularization of a similar divergent integral with the characteristic 0( z') by the relation p .f.
O(z' ) I = lzl n + a *
j 0( Y') l(z - y)IYi-n(+Ra�- 1 l)(z) dY
IYI< 1
+f
IYI> 1
O(y')l(z - y) dY IYin +a
(26.66)
where
i (R�-1 1)(x) = L (-�) (Vi l)(x) li l � l - 1 J . is the Taylor sum and I > a. Passing to polar coordinates, we easily obtain p.f.
f
IYI
yi O(y') d _ { 0;/
in accordance with the definition moments (26.61). So p .f.
Iii =F a, Iii = a,
5.2. of a p.f.-integral, with 0; being the spherical
O(z') * I = f O( ') l(x - y) - x( y)(�- 1 l)(z ) d y y lzln + a IYin +a R"
+
�'
L..J
(- 1) 1i l o . (VJ. l)(z),
li l � i - 1 J" ! ( 1J" 1 - ;)
I>
a,
(26.67)
526
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where the dash denotes omission of terms with I i i = a in the case of an integer a and x (y) is the characteristic function of the ball IYI < 1. If we choose I t o b e the integer, nearest to a , I > a, i.e . I = [a] + 1, then (26.67) may be written as
p.f.
fl ( z' ) f = f f(z - y) - ( Rf,a1 /)(z) fl (y' )dy, lz jn+a lyn +a R• *
(26.68)
a f:. 1 , 2, 3, . . . , which is directly verifiable. The form (26.68) has an evident advantage in comparison with (26.67) , but it is not suitable for integer values of a. (In the case of an integer a it may make sense at the cost of the restriction J fl (u)du = 0 , which excludes, in particular 11 ( u) Theorem 26.5.
is bounded. Then
=
s._ .
const .)
Let us suppose that 11(u) E L1(Sn _t), while f(z) E C1 (Rn ) and
(D{l/)(z) =
sin(a1r/2) 11( z' ) p. f n + Pn (a) · l xl a
(Dna /)( ...• ) =
sin(a1r/2) fl (x ' ) + 11 ( -x ' ) p . f. Pn (a) 2 lxl n + a
(Dna !)( z ) =
_
*
f
a f:. 2, 4, 6, . . . , (26.69)
'
cos(a ,./2) 11(x') - n( -x ' ) f Pn (a) P· · 2 lxl n + a
../.. a .,..
*
f
*
I, a f:. 2, 4, 6, . . . , (26.71)
'
2 ' 4 ' 6 , . . . , (26 . 70)
for hypersingular integrals of the neutral, even and odd type respectively. In the excluded cases of integer a in (26.69)-(26.71) the hypersingular integral is a differential operator in partial derivatives of order a:
a/ 2 11j (Vj f)(x) (Dan/)(z) - - ( -2{3l )[ ( )],. 1 L '1 J. 1. n J =a Cl'
(a = 2, 4, 6, . . . for a hypersingular integral a = 1, 3, 5, . . . in the case of an odd type).
(26.72)
of the neutral and even type and
Proof. Result (26.72) follows from (26.60) , but can be established directly avoiding
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
527
consideration of Fourier transforms. Let us prove for example (26.69). We have
(DJ!i /)(z) = d.. ,:( a)
j 1�1<:';.: { /(z) + vt- 1 (-1)" G) [/((z - vy) - �� 1 (z)x(vy)) + �( -1)" G) R�� 1 (z)x (vy) } y
R"
d .
Hence after simple transformations
f(x - y) - x( y)�- 1 (x) d a: ) fl ') ( (DnOt /)(x) = - dAH(o:) y y IYi n+ a n ,l R"J ( -1).1li I Ii (Vi /){x), + ;r-L . n ,l ( 0: ) li l� l -1 J 1
(26.73)
where
By direct calculation we find that Ij = ��U�nj , where AHo:)/(o: - l i D is to be replaced by dAH a:)fda: in the case Iii = a: . So ( 26.73 ) turns into (26.69). Similarly (26.70 ) and (26.71) are proved.
26.6. Representation of differential operators in partial derivatives by hypersingular integrals We have shown above that the set of hypersingular operators (26.47) with a homogeneous characteristic contains some homogeneous differential operators, see ( 26.72 ) . Now we prove a stronger assertion, i.e. the inverse statement that all the homogeneous differential operators in partial derivatives of order a: with constant coefficients may be written as a hypersingular integral Dg/ with a homogeneous characteristic. But first we put a more general question , inspired by (26.52) in Fourier transforms. Let a(x/lxl) be a given homogeneous function. Does there exist a characteristic fl (x/ lxl ) such that (26.74) and how can it be constructed for a given a(x/ lxl )? We observe that the expression itself in the right-hand side of ( 26.74 ) may be considered as a generalization of the Liouville differential operation. It contains
528
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
partial or mixed Liouville differentiation as a particular case (see (24.14)-(24.15')) under the choice a(z) lzl-lo:l zf 1 z�· . Such a generalization of fractional differentiation - a convolution with a homogeneous function of order -a - n was considered by Sobolev and Nikolskii [1] . We want thus to show that this generalized differentiation coincides with a hypersingular integral with a certain characteristic. For precision let us deal with hypersingular integral of the neutral type. By (26.55), (26.74) is reduced to
=
• • •
(26.75)
lzl = 1,
a -::F 1, 3, 5, . . . ,
which is an integral equation of the first kind relative to the unknown characteristic O(u). This equation can be solved by means of harmonic analysis on a sphere, which is well presented in Stein [2] and Stein and Weiss [3] - see also Samko [31]. We recall that the Fourier-Laplace series of the function /(z), z E Sn _ 1 , is the series of the form
f(z) - L Ym (f, x) m= O
(26.76)
where Ym (/, z) is a harmonica! polynomial of order m confined to the unit sphere. The harmonic component Ym (/, z) of order m is evaluated by the relation
Ym ( /, z) = dI Sn (m)l J n- 1 S - 1 n
f(u)Pm (X u) du ·
(26.77)
where dn (m) :"t2:_:-22 (m +,: - 2) is a number of linearly independent harmonics of order Pm (t) is the generalized Legendre polynomial:
m.
=
where C� (t ) is the Gegenbauer polynomial and Tm (t) = cos ( m arccos t ) is the Chebyshev polynomial. The series in (26.76) converges to f(x) provided that this function is sufficiently smooth. Expanding the solution and the right-hand side of (26.75) in Fourier-Laplace
§ 26.
HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
529
series, we arrive at the relation
(26.78) The relation
(26.79) is valid for any spherical harmonic Ym (u) if lxl have
= 1; in the case of an integer a we
a - m = 0, 2, 4, . . . , a - m = 1, 3, 5, . . . , a - m = -1, -2, -3, . . . ,
(26.80)
with Cm = 2 l -a 7rn / 2 f(1 + a) [ f ( m +;+ a ) f ( 1 + a-;m ) ] - l and lxl = 1 (see, for example, Samko [28, p. 163], [31, p. 91]). So in the case of a non-integer a from (26.78) we obtain
resulting from the linear independence of spherical harmonics of different orders. So the desired solution !l( x) may be represented by the series
(26.81) a =/= 1, 2, 3, . . . It can be shown that this series converges and is really a solution to the equation (26.75) if the function a(x), l xl = 1, is sufficiently smooth on the sphere. We do not elaborate on the proof of this. We observe also that the "slowly" convergent
530
CHAPTER
series
5.
INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(26.81) may be summed up in terms of the divergent integral:
x = - 27r(n -rl)(/2 fC1 lxl = 1,
!!±.til
O( )
)
+
a:
o:/2) p .f. f;
1
s,. _ .
a( )dcr (ix · u)n+a '
(26.82)
C1
1 , 2, 3, . . .
(see Samko [30, p. 37); the integral on the right-hand side can be made meaningful by regularization or by analytic continuation in respect to a:) . The case of integer a: requires special consideration. We give the corresponding result in the following theorem. Theorem 26.6.
Let a: = 1, 2, 3, . . . and let
aa (V) = L ai vi li l = a be a homogeneous differential operator of order a with constant coefficients ai . There exists a homogeneous polynomial O a ( Y ) of order a: such that (26.83) where a hypersingular integral of the neutral or even type is used if a: = 2, 4, 6, . . . and of the odd type if a: = 1, 3, 5, . . . , the functions f( x) being assumed to be sufficiently good. The characteristic O a ( Y') of the hypersingular integral may be found by a given polynomial a a (x) via the relation
O a (Y) =
1 aa (cr)K(y
s .. _ .
·
cr)dcr, IYI = 1,
(26.84)
where
) [a/ 21 r (n/2) [a/ 2] k (n -1) a - k k !dn (o: - 2k)Pa - 2 �:(t), ( K(t) = r(-1) + r 2 � (�) �) + r (1
-1 < t < 1, dn (o: - 2k) and Pa- 2J:(t) are the same as in (26.77). Proof. For clarity we shall deal with the case of even a: = 2, 4, 6, . . .
By
(26.56)
§ 26.
531
HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
for the Fourier transform of a hypersingular integral, the desired characteristic is to be determined from the equation
lxl = 1 .
(26.85)
We expand a a (x) and !l a ( x ) in spherical harmonic series as above in (26.76) , only these expansions are finite sums now. Substituting the expansions into (26.85), we arrive at the relations
Hence, by (26.80)
In view of (26.80) the summation on the left-hand side here is to be carried out for m = 0, 2, 4, . . . , a only, which agrees with the fact that the polynomial a a (x) contains harmonic components of even orders only. Since spherical harmonics of different orders are linearly independent, we obtain Y.m (n a , x
r r ( 1 + T) Y. ) = (-1) af 2 (�) m (a a J r (1 �) +
x
).
Consequently,
a- ) - Y (aa x) �0 f ( m + 2n + a ) r (1 + 2
a/ 2 a !l a (x) = r(1(-1) + a/2)
m
m
,
which is reduced to (26.84) by means of (26.77). The case a = 1 , 3, 5, . . . is treated similarly. • By straightforward calculations the reader can show that in the case a n when a 2 (x) = aij Xi x; , (26.84) becomes very simple, namely
L:
i ,j = l
=2
532
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where tra 2
= �n aii = ls:_.l J •= 1
s,._ .
a 2 ( u ) du. In particular we see that -
(26.86)
26. 7. The space Ia (Lp ) of Riesz potentials and its characterization in terms of hypersingular integrals. The spaces L;, r (Rn ) We denote by (26.87) the space of Riesz potentials of functions in Lp (Jl!l). It is well defined if 0 < a < n and 1 :5 p < nfa. By Theorem 25.2 and the inequality in (25.42)
la (Lp ) C L9(Rn ), Ia (Lp ) C L9(Rn ; lxl - aP) , 1 < p < nfa,
q
=
np( n - ap) - 1 .
Repeating arguments from § 6.1, we can easily see that I a (Lp) =I L9(�), Ia (Lp) =I L9(Rn ; lzl - aP). I t c an b e shown that Ia (Lp ) does not coincide with any of the spaces L r (Rn ; p), 1 :5 r :5 oo, and thereby needs to be characterized. The space Ia (Lp ) may be defined for p � n/a also, the case a � n being admitted, if we interpret the Riesz potential in the sense of generalized functions on the Lizorkin space (25.16) :
which is proper by the invariance of the space � relative to 1 a . Of course under such a definition this is a space of generalized functions. These functions might be called "quasisingular" in the sense that their finite differences of sufficiently large order I > a are usual functions. (The finite differences of generalized functions are defined in a standard way: (A�/, w) = (/, A� h w)). We have already dealt with such a situation in the one-dimensional case - see Remark 8.2. More precisely, the following lemma is true.
Let f = Ia cp, cp E Lp , a > 0, 1 :5 p < oo, I > a. Then Lp (Rn ) for any h E Rn if p < nfa. If p � n/a, this assertion
Lemma 26.5.
(A�/)(z)
E
§ 26.
HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
533
is replaced by the following: the functional A� / E 4>' is regular and admits an extension from the Lizorkin space 4> to the Schwartz space S:
(A� /, w) = (g,w), w E S,
(26.88)
where g = g( x) = Al ,a (·, h) * cp E Lp (Rn ).
Proof. From (26.32) and (26.33) we derive the estimate (26.89) for lzl 2:: (1 + l) l h l with c not depending on z and h. Thus Al , a (z, h) E Lt(Rn ) for every h, and then it follows from (26.41) that A� / E Lp (Rn ). Let 2:: nfa. Then (26.41) is valid in the sense of generalized functions in 4> : (A� /, w ) = (AI, a (·, h) * cp,w), w E 4>, cp E Lp , which is directly verified. The right-hand side of the latter equation admits an extension to all w E S. Besides this g = A l , a (·, h) * cp E Lp (R") since At, a ( z , h ) E L 1 (Rn ), which was required. • We define the norm in the space [a ( Lp ) by the relation
p
11 /II I"' (L p ) = llcpllp ,
1
� p < oo.
(26.90)
Let us pass to characterization of the space Ia (Lp) · We shall prove this characterization together with a consideration of the spaces
L;,r (Rn ) = {/ : / E Lr(Rn ), Da / E Lp (Rn )}, 11 /II L;,r = 11 / llr + IID a / lip, (X > 0, 1 � p < oo, 1 � r < oo,
(26.91)
which arise naturally as a certain generalization of the space [ a ( Lp). We emphasize that n a f in (26.91 ) is understood as convergent in Lp . see (26.26) . We begin with the following auxiliary theorem. -
Let f(z) E L;,r (Rn ), a > 0, 1 � p < oo, and 1 � r < oo. Then the difference (Ah f)(x) with m > a admits the representation
Theorem 26.7.
( A h) (x) =
j Am,a (x - € , h)(Da /)(€)de
R"
where Am, a (z, h) is the function (26.27).
(26.92)
534
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Let us suppose that na I is constructed by means of a non-centered difference, the difference �h is centered and for simplicity let #; 0, 2, 4, , other variants being considered in a similar fashion. We denote cp� = D� I and Bcp = �m , a ( · , h) * cp. We have
Proof.
a
B
= d.. ,�( a)
{ J t.,.,a(z - y, h)f(y)dy
Lt(Rn) m
-_
Hence after the substitutions (25.68), we obtain
dz
a
in view of (26.89) , the interchange of the
R,.
IYI >e
z = c r , y = c�wz(e ) , where wz(e ) is the rotation
l le l e t - ve/lel l = le - ve l I yields
J �m,a(x - T, h)cpe ( r)dr J (�h f)(x - cr)Ki,a( lrl)dr. =
R,.
. . .
( I ) J f(z)dz J �m,a (Xjyj-nz+a- vy, h) dY
�( _ 1) " V = d , l1(a) � n "-0
which owing to the equality
n
ly- z l >�
R•
Since �m , a (z, h) E if > order of integration is possible. So B cp�
J
-
(26.93)
R,.
Hence (26.92) is obtained as c - 0. Let us justify the passage to a limit in (26.93). Since cp� -+ cp in Lp and �m , a (z, h) E Lt (Rn ) , the left-hand side converges in Lp . On the other hand, since f(x) E Lr and Kt, a (lrl) E L 1 , the right-hand side
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
535
converges to (AI!" /)(z) in L r. The left- and right-hand sides in (26.93) coinciding, their limits have to coincide also (almost everywhere) . This gives (26.92) . • Remark 26.5. The representation (26.92) is an extension of (26.41) to the case
when information
/(z) E Ia(Lp) is replaced by the condition /(z) E L;,r(Rn).
Remark 26.6. Since (26.92) is obtained by the passage to a limit in (26.93), it is clearly true not only for / ( z) E L;,r (Rn ) but also under assumption that (Da /)(z) E Lp(Rn), (AI!" /)(z) E Lp(Rn), > a, 1 5 p < oo. The characterization of the space 1a ( Lp ) is given by the following theorem.
m
Let /(z) be locally integrable and lim /(z) = 0 . Then lxl-oo /(z) E Ia ( Lp), a > 0, 1 < p < oo, if and only if 1) /(z) E L9(Rn), q = npf(n - ap) , and na / E Lp (Jl'l) if 1 < p < n fa ; 2) (Ai/)(z) E Lp(Rn) and there exists lim J lh l - n - a(Ai/)(z)dh if p � . o
Theorem 26.8.
·
n/a,
where
I > 2[a/2]
is chosen as in §
c(Lp) - 1 hI > c 26. 2. In the
na I E L l holds in the "only if" part. Moreover
case
p= 1
the condition (26.94)
Proof. Consider the "only if" part. Let /(z) E Ia(Lp)· If 1 < p < n/a then f E L9 by the Sobolev Theorem 25.2, and if 1 5 p < nfa, then Da/ E Lp by Theorem 26.3. Let p � n fa. Then (Ai/) ( z) E Lp by (26.41) and the analysis of the proof of Theorem 26.2 shows that (26.40) is valid for n: f and therefore there exists lim n: f. We have thereby shown that Ia (Lp ) n Lr C LPa' r (Rn). c-o Consider the "if" part. Let /(z) E L;, r (Jl'l). Then we have (26.92). We observe that the right-hand side in (26.92) is AI!" IaDa
536
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
and then f(z) E /01 by (26.95). As for the case p � n/a , then (26.95) means the same, since the presence of a polynomial agrees with the definition of the space /01(Lp) in the case p � nfa. Therefore, the imbedding L�,r (R") � l01(Lp ) n Lr is proved, which together with the inverse imbedding, proved above, gives (26.94) . The relationship (26.94) yields the sufficiency of the conditions 1), if we take r = npf(n - ap). To obtain the sufficiency of the conditions 2) , we have to repeat arguments given above in the sufficiency part taking Remark 26.6 into account. • Corollary 1.
Let 1 < p < nfa. Then
The spaces L� r(R") do not depend on the type of the finite difference defining D 01 f and on ,order I of this difference under the rule for choice of I, indicated at the end of § 26.2.
Corollary 2.
The spaces L�,r (R") are complete. In view of (26.19) Theorem 28.8 gives a simple sufficiency condition for a function to belong to 101(Lp ), 1 < p < n/a: 1/ (z)l � c/(1 + lxl)><, and I (Vi /)(z) l � c/(1 + lxl)�', Iii = m > a ; A > nfq , JJ > nfp. We present another useful variant of the characterization of the space [01 ( Lp ) which deals with Lp-uniform boundedness of the truncated Riesz derivatives n: / , instead of their convergence. Corollary 3.
Theorem 26.9.
Let 1 < p < nfa. Then f(x) E 101(Lp ), if and only if
f(x) E Lq (R" ), q = npf(n - ap)
or ( 1 + lzl 01 )f(z) E Lp (R")
(26.96)
and there exists a sequence ct --. 0 such that (26.97)
where c does not depend on c�: . The is based on the property of the weak compactness of the space Lp ( R") . It is analogous to the proof of the similar one-dimensional Theorem 6.2 and is therefore omitted, see the proof of Theorem 6.2 which is presented in full; it is necessary to replace the operator (6.22) by the operator dm , a ( · ,
proof
h) *
537
§ 26. HYPERSINGULAR INTEGRALS AND RIESZ POTENTIALS
The space /01 ( Lp) may also be characterized in terms of the convergence in Lp of the hypersingular integrals (25.69) and (25.70), related to the Poisson and Gauss-Weierstrass semigroups. Namely, the following theorem is valid.
Then f(x) E l01(Lp) if and only if
Theorem 26.10. Let 0 < a < n, 1 < p < n/a. E L9, q = np/(n - ap), and
/
00
(L,.)
J t -l - af2 (E - Wt )1 fdt E Lp. 00
J
J� t - 1 - 01(E - Pt ) 1 fdt E Lp
or
lim
�-o
(L,.)
�
�
We also note the relation (26.97')
I E Lq(R"),
1 < p < oo, 1 < q < oo ,
which is valid provided that there exists one of these limits. If q = p this result follows from Theorem 23.4 and Theorem 26.10. Its validity for independent values of p and q may be seen in Samko [34]. We conclude this section by some simple estimates for the integral continuity modulus of the Riesz potential. Let f = [01 a. Then (26.98) and (26.98')
l h l --+ 0.
Here c = ll kt , a l h , where kt , a (x) is the kernel (26.28). The estimate (26.98) is derived from the representation
as
(�� f)(x) = l h l a J kt , a ( Y)
(26.99)
R"
where
wh ( Y) is
the rotation given in (25.68), which is obtained by the following
538
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
transformations
( I ) j I
l _ ""< -1} k (d'h /)(x) _1(a} k "Yn k� -0 =
R"'
the latter coinciding with {26.99}. Since k,, a (x) L t (Rn ) by Lemma 26.3, yields {26.98}. It also gives {26.98'} in view of {26.35).
E
{26.99}
§ 27. Bessel Fractional Integro-differentiation In this section we consider a fractional integr
(E-
E
27. 1 . The Bessel kernel and its properties The Fourier transform of the function {1 of the Bochner relation {25.11). We have r ' [{1 + l z l - "' 2 1 =
+
l .x l 2 } - a/ 2 may be evaluated by means
j 2/�;"';;!�! l l dp.
•/ (2 .-j - •/ 2 lxl ' - • ' 2 p ( 0
Here the integral converges if a > then by {1.86} we obtain
{n + 1}/2. Assuming this condition to be fulfilled,
:r- 1 [{1 + l .x l 2 ) - af 2] =
2(2-n- a)/ 2 K(n - a)/ 2 { 1 xl} �G (.x } a 7rn/ 2 f( a/2} l z l (n- a)/ 2
{27.1}
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
539
where K,(z) is the modified Bessel function ( 1 .85). This expression has a meaning for all a > 0. The function Ga (x) is called the Bessel kernel. We give some of its properties. First of all we shall clear up its behaviour at the origin and at infinity.
The function Ga (x) admitting the estimate
Lemma 27. 1.
Ga (x)
�
�
is
infinitely differentiable beyond the origin
r((n -a)/ 2) I X 1 a- n ' 201r•/'Jr (a/ 2) 2 " lr•.J'Jr(n/ 2) In rxr' r(( /'J n)/ 2) ' r(a/ 2) 2• r n.a-
if 0 < a < n, if a = n , if a >
n,
as lxl --+ 0 and the estimate
l z l (a- n - 1)/ 2 e - lxl Ga (x) "' 2(n+ a1)/ 2 7r(n - 1 )/ 2 f(a/2) as l xl --+ oo.
The statements of the lemma follow immediately from the known properties of the modified Bessel function: K, (z) "' 2" - 1 f(v) z- ", v ::f; 0, K0 (z) "' ln(1/z) as z --+ 0 and K,(z) "' (7r / 2z) 1 1 2 e - z as z --+ oo . - Erdelyi, Magnus, Oberhettinger and Tricomi [2, 7.4.1 (1)]. In this connection we note the relation
/2) 1 / 2 zv e - z J00 e - zt t v - 1/ 2 (1 + t/2) v - 1 / 2 dt K" (z) = ( 7rf(v + 1/2) 0 Corollary.
·
(27.2)
Ga (x) E L t (Rn ), a > 0.
Now, employing properties of the function Ga (x) with firstly its rapid decrease at infinity, it is not difficult to show by means of (25.11) again that
(27.3) for all a > 0, Re a > 0, in fact may be taken. It follows from (27.3) that
Ga G{3 = Ga+f3 , a > 0, {3 > 0, (E - d)Ga = Ga- 2 1 a > 2, *
(27.4) (27.5)
540
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
j G01 (x)dx = 1,
a >
0.
(27.6)
R"
We mention also the relation
(27.7) Its proof may be seen in Stein [2, p. 155]. We now define the Bessel potential by the relation
G01 tp = J G01 (x - y)tp(y)dy.
(27.8)
R"
By Lemma 27.1 it is well defined for example for functions 1 :5 p :5 oo, and represents an operator bounded in Lp ( Rn):
tp(y)
E
Lp(Rn),
(27.9) this inequality being implied by (27.6) and the positiveness of the kernel G01 (x). Equations (27.3) and (27.4) immediately yield the following simple properties of Bessel fractional integration:
(27.10) (the semigroup property), and
(27.11) In contrast to the case of the Riesz potential (27.10) holds for functions in Lp with all 1 :5 p :5 oo, a > 0, (3 > 0. We write below G0 = when it is necessary to admit the case a = 0. The operator G01 may be considered as a constructive realization of a negative power of the operator � . It is of great interest to realize constructively positive powers of this operator as was done in §§ 25 and 26 for ( - � ) 011 2 • Such a realization will be given in § 27.4.
E
E-
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
541
27 2 Connections with Poisson, Gauss-Weierstrass and metaharmonic continuation semigroups .
.
The Riesz potential was seen to be connected with the Poisson integral by (25.52). A similar connection of the Bessel potential with this integral is given in terms of special functions. Thus, the following theorem is valid. Theorem 27.1.
Let
(27. 12) where Pt
The proof of the theorem is obtained by interchanging the order of integration in
the right-hand side of (27.12), which is made possible by Fubini's theorem, and applying then the relation
2(1 - a)/ 2 fi f t ( a- 1 )/ 2 J ( a- l )f 2 (t)P(x, t)dt = Ga (x) r (a/2) 0 00
(27.13)
where P(x, t) is the Poisson kernel (25.49). As for (27.13), it is contained in (1.86) according to ( 27.1). Meantime there exists a modification of the Bessel potential (suggested by Flett ) which is connected with the Poisson integral in a simpler manner. This modification is based on using the function (1 + lxl ) a instead of (1 + lxl 2 ) af 2 in Fourier transforms. We set
f!Ja
{27.14)
R"
where the kernel f!3a (x) is the Fourier original of the function connected with the Poisson kernel P(x, t) by
l!!a {z) = f{�) t a- l - t P(z, t)dt 00
j 0
.
(1 + l xl) -a .
This is
(27.15)
542
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
which may be easily verified by Fourier transforms: 00
(1 + l z l ) - a = r(�) j t a - l e - t P(· , t)dt ,
(27.16)
0
the point denoting the variable in which the Fourier transform is applied. As for (27.16), it follows from (25.54). The kernel � a (z) may be written in the form
(27.17) where Cn is the constant from (25.49), this result being a paraphrase of (27.15). The following properties of the kernel (!S a are easily derived from (27.15) and
(27.17): 1) � a is infinitely differentiable beyond the origin and �a (z) > 0, z I- 0; + l)/ 2)r((n - a)/ 2) l z l a- n as x - 0 if 0 < a < n , �n (z) 2) � a (z) - r((a2r {a)li'("+ I)/l � In � as z - 0 and �a (z) is continuous in the point z = 0, if a > n ; 3) � a (z) - acn l z l - n - l as z - oo , so that �a (z) E L 1 (Rn ), besides this J �a (z)dz = 1. R" It follows from the above properties that the potential �a
Lp (Rn ), 1 � p � oo . 00
f!J"
Th en
f(�) j t"- 1 e -• P,
(27.19)
0
We give also two other integral representations of type (27.12) and (27.15) for the Bessel potential Ga
§ 27. BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
543
integral:
Ga cp r (�/2} f t a/ 2 - l e -t (Wccp)(x)dt. 0 a The second one expresses a cp via metaharmonic continuation 00
=
(27.20}
(27.21} of a function
cp by the formula 00
Ga
=
f(�) J ta- !(M,
(27.22)
0
(compare the representation
(25.51} for the Riesz potential} . The following results
e - '(WcGa cp)(x) = (1�1 \ e - T WT cp)(x))(t } , (MtGa cp)(x) (I� (MT cp)(x) } (t), =
(27.23} (27.24}
are close to relations (27.20} and (27.22} being similar to (25.52} and (25.53). Equations (27.20}-(27.24) become obvious in Fourier transforms if we take into account that
F(e - ' W, cp) = e - t ( l +l � l :l ) t,O(x), F( Mccp) e -t v'l +l� l 2t,O (x), =
(27.25} (27.26}
27 3 The space of Bessel potentials .
.
We shall call the range
(27.27} Q > 0,
1 �p�
00
aa the space of Bessel potentials. Sometimes this space is called the Liouville space offractional smoothness a . This is Banach space relative to the
of the operator
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
544
norm As we shall see below, this space is an extension of the Sobolev spaces L;a ( R!' ) t o the case of fractional order a . S o the spaces defined in (27.27) ar e al so called
Sobolev spaces of fractional order.
Remark 27.1. The spaces Ga (Lp ) and
potential (27. 14), coincide:
� a (Lp ), where � a cp is the modified Bessel
One can prove this by means of the fact that the functions ( 1 + l zl) or 1 and ( 1 + l z l 2 ) or/2 are Fourier transforms of functions integrable over R!' ( cf. the Corollary of Lemma 5.2 in Samko [31, p. 52]). The characterization of the space cor ( Lp ) presented in this subsection, has much in common with that of the space of Riesz potentials given in terms of the Riesz derivative. Among the spaces L;, r (R!') defined in § 26.7, we consider now the important case r = p. We find it convenient to introduce the special designation
Q > 0, 1 :::; p < 00.
(27.28)
The space of Bessel potentials consist of those and only those functions f(z) E Lp (Rn ) which have Riesz derivative of order a in Lp(Rn ), i.e . Theorem 27.3.
We precede the proof of this theorem with the following lemma. Lemma 27 .2.
The set CQ" is dense in
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
545
of the Lemma. 1. We show first that infinitely differentiable functions in Lp (Rn ) form a dense set in L;(Rn ). This is achieved by standard means which are used to show the denseness of "good" functions in various function spaces, i.e. by the identity approximation
Proof
( )
lm (z) = J a( y) l z - ! dy, I E L; , R"
a(y) E ego , I a(y)dy = 1. Then II / - lm l iP --+ 0 as m --+ 00 and it is R" readily seen that D� fm = (D� /) m· So there exists n a fm = lim D� fm and e -o na lm = (D a /)m · Therefore II Da (/m - /)l ip = II (Da /)m - n a / lip --+ 0 as m --+ 0. We note that since a(z) E ego , then Vi lm E Lp (Rn ) for all Ii i = 0, 1, 2, . . . 2. It remains to approximate the function /(z) in the space L;(HR) by COO-functions under assumption that /(z) E C00 (Rn ) and (vi /)(z) E Lp (Rn ), Iii = 0, 1, 2, . . . Let p(z) be any function in ego which has the support in the ball l z l < 2, is identically equal to 1 for l z l $ 1 and such that lp(z)l $ 1. We have to where
show that the ''truncation"
= p(z/N) /(z) e C0 PN (z) /(z) def approximates the function / (z) under the norm II /Il P + un a /l ip as N --+ 00. We denote 11( z) = 1 - p(z) and liN (z) = 11 ( z/N). It is sufficient to verify that IID a ( IIN /) lip --+ 0. We have I
D a (vN I) = VN Da I = d;;J (a) I; k=l
(!) BN,l l
where
BN, J:/ = J IYI - n -a ( �:IIN) (z)( ��- k /)(z - ky)dy, R"
(27.29 )
k = 1, 2, . . . , I, so that the passage II BN,t/ llp --+ 0, N --+ oo has to be justified. Since �:liN = �: JlN , k > 0, and l ( �:p)(z) l $ clyl k /(1 + lyl )'= according to (26.20), by the Minkowsky inequality we obtain
li BN,l lll. 5, cN- k J IYi k - n -a R"
( � r l il A�-· ll i,dy. 1+
(27.30 )
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
546
(26.19) that II Ll�- 1: /l ip � c lyl l- 1: 2: IIVj /l ip = clyll - 1: , Jj J =l Vi f E Lp(Rn) for all Ii i = 0, 1, 2, . . . So from (27.30) we 1:have It follows from
since
Hence simple transformations lead to II BN, A: / IIp � caN - A: + c4 N -a in the case k f; a; if a is an integer and k = a, then II BN, k! IIP � cN -a ln N. • of Theorem 27 .3. Let us prove the imbedding
Proof
(27.31) Since
Ga (x) E L 1 (Rn) by Lemma 27.1, then aa(Lp) C Lp .
Further, we have
1 lxl a . 1 2 2 (1 + lzl )a/ lzl a ( 1 + lzl 2 ) a/ 2
(27.32)
known (Stein [2, p. 157], see also Samko [31, p. 51]) that the function l x l a (l + lxl 2 ) -af 2 - 1 is the Fourier transform of a function in Lt(Rn). So (27. �2) means in Fourier pre-images that
It
is
(27.33) where A is a convolution operator with a summable kernel. By the boundedness of the involved operators (27.33) is extended from the case of functions cp E CC) to cp E Lp(Rn), 1 � p < n fa. The validity of (27.33) for cp E Lp(�) when p � nfa is a corollary of the definition of Ia cp for such cp (see § 26.7). The identity (27.33) implies then that aa (Lp) � Lp n Ia(Lp)· So the imbedding (27.31) is proved in view of (26.94). As well, by (27.33), for I = ca cp we have
II / I lL; = 11/llp + II D a /l ip � ll
Let us prove the inverse imbedding. Let f E L; (Rn). We use the density of CC) in L; (Rn), obtained in Lemma 27.2, and approximate f by functions fm E CC) with + t z , ,r,, respect to the norm 11/llp + II D a /l i p · We have 1 = ( l + l z1l,)""/2 ( l t+l (1 + l x l a ). z Here
(l
c..
i�1�1>: - 1 is the Fourier transform of a summable function (see Stein [2, '2
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
p. 157-158], and also Samko identity in Fourier originals:
(31,
p.
52].
547
The latter relation yields the following
{27.34) where U is a convolution operator with a summable kernel. Since the operators G01 and U are bounded in Lp (Rn ), 1 � p after passage to the limit as we obtain from {27.34) that f E G01(Lp ), i.e. L;(Rn ) � G01(Lp ), as well as 11 /II G"' (L . ) = II (E + U)(f + D 01 /) l iP � c( ll/llp + II D01 / l ip) = ell / IlL; which completes the proof. •
< oo,
m � oo
The operator G01 maps the space L�(Rn ) isomorphically onto the space + n L; P(R ), a � 0, {3 � 0. Corollary.
27 4 The realization of ( E - � ) a/ 2 , hypersingular integrals .
.
a
>
0, in terms of
Since ( E - t1 ) 011 2 f = _r- l ( 1 + lz 1 2 )011 2 .1"f by definition, our discussion is in fact about the inversion of Bessel potentials I = G01cp, Q > 0. The function cp will be first taken to be in the Schwartz space S which is invariant relative to the operator · G01 , and then we consider the problem of inversion for cp E Lp . The operator inverse to G01 will be considered in terms of hypersingular-type integrals, containing the remainder of the Taylor series for the function / (z) (see (26.68)). For f E S and a > 0, a 1= 2, 4, 6, . . . we set
T01 f = L Ca ,; (Vi /)(z} + da J (f(x - y) - (Jlia1 /}(z)] IYI - n - a Aa ( lyl )dy (27.35) R•
j E Aar
where (JlL011 /)(z) is the same as in (26.65), Aa is the set of multiindices with length � (a] and even components, and
d(){ - 7r n/ 2 f(2 (){-a/2) ' - - --:----
w;
=
j uidu,
S n- 1
(27.36) 00
= f e - l + (n+ a)/ 2 e - � - IY I 2 /(4�) de. 0
548
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(n )
We remark that r �W Wj = r (�) . . . r (ia;- 1 ) in the case of even it , . . . , in · We note also that if 0 a 1 , then
< <
() T" I = l(x) - da j l(��i:+� y � a ( i yl)dy.
(27.37)
R•
It is easily verified that the characteristic A a ( lyl) of the hypersingular integral in (27 .35) and (27 .37) stabilizes at the origin and at infinity as a Holderian function; hypersingular integrals with such characteristics were studied in the paper by Samko [20] , the results of which will be used below. Theorem 27.4. Let a > 0, a :f; 2, 4, 6, . . . where Ta is the operator (27.35).
Proof.
S as
and let cp E S. Then Ta Gacp = cp,
We note that the operator inverse to ca may be written for functions in (27.38)
where the Fourier transform .r- 1 ( 1 + IYI 2 ) af 2 is interpreted in the sense of S' distributions. This is easily justified by means of the known Gel'fand-Shilov theorem (Gel'fand and Shilov [1 , p. 179]). So we are to show that (27.39) If Rea
< 0, taking (27.36) into account, we have (r 1 ( 1 +
IYI 2 ) "' 2 , I( x + y)) = da j ����!� I(Y + x )dy . R•
(27 . 40 )
Since the left-hand side is analytic with respect to a in the whole complex plane, we may consider (27 .40) to be valid for Re a > 0 also if the integral in the right-hand side is interpreted as an analytic continuation . Such a continuation is realized by subtracting a Taylor sum of the function f(x) under the integral sign. Carrying out a direct evaluation we arrive at (27.39) . • Remark 27.2. Since ( 1 + l x l 2 ) k =
k E x 2i , then the operator inverse to ca in i (�) l l =O
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
549
the case a = 2, 4, 6, . . . has the form
(27.41) Now, w e shall show that the operator cp E Lp as well, if it is interpreted as
ra inverses the potential I = oa cp for (27.42)
where r: 1 is the construction (27.35) with the integration over nn being replaced by that over IYI > c . Derivatives Vi f of a function f(x) E oa (Lp ) will be understood in the sense of generalized functions. Existence of derivatives Vi I for 1 = oa cp follows from the fact that
Ga (Lp ) C L;:',
m=
0, 1, . . . , [a) .
Let a > 0, a :f:. 2, 4, 6, . . , cp a a r r o cp = cp, where a is the operator (27.42). Theorem 27.5.
Proof.
.
E
Lp , 1
< p < oo.
Then
We are to show that
(27.43) We first prove the uniform estimate
(27.44) with c not depending on c. Its proof will use the idea of Fourier-multipliers in Lp (Stein (2, p. 113]). Since II Vi Ga cp llp :5 cllcp i i P which is easily verified by means of Theorem 3 from Stein [2, p. 114], then (27.44) will readily follow from the estimate
(27.45)
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
550
with c not depending on E, where
D �"'o .•"'., f J f(z - y�y�n+a( Rl,ol/)(z) � a ( ly l)dy. IY I > e =
Let
(27.46)
iz·y - "'""
-- J (e LJ ili lxj !iyi ) �I Yai(ln+ayl ) dy (27.46') l 5[ ) a IYI>e il so that Y:( D �o ;e f ) = ue (z)/(z), f E S. As for (27.45), it is a consequence of the following lemma. Lemma 27.3. The function (1 + l x l 2 ) a f 2 ue (x ) , a > 0, belongs to the space Mp of Fourier-multipliers in Lp , 1 < p < oo {1 � p < oo if O < a < 1}, besides this
(27.47) with c not depending on E.
In the case 0 < a < 1 we refer to Samko [20] where was proved in this case for arbitrary stabilizing characteristics even in a stronger form - in terms of absolute summability of the Fourier integral of the function (1 + uniformly in c. So we give the proof in the case a � 1. We shall use Theorem 3 from Stein [2, p . 114], which states that to show (27.47) it is sufficient to verify the estimates
Proof of the lemma.
(27 .47)
lz l2 )- af2 ue (z)
(27.48) with c not depending on E. By the Leibniz formula it is clear that from the inequalities
(27.48) follows
(27.49) c not depending on c again.
Let us prove
v•O', (z) =
(27.49). If [a] + 1 � 11 � [n/2] + 1, then
j ����r0 �a ( l y l )e'r·• dy IY I > e
(27.50)
since differentiation under the integral sign is possible owing to the rapid decrease which is seen from (27.36). \Ve represent the monomial as of
�a ( l y l) I YI --+ oo
(iy)"
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
551
[lvl/2] (iy)" = E IYI 2m Pj vl - 2m ( Y), where Pj v l - 2m ( Y) are homogeneous harmonic m =O polynomials of orders l v l - 2m (see Stein and Weiss [3, p. 159]). Passing to polar as
coordinates and applying the Funk-Hecke formula, (Erdelyi, Magnus, Oberhettinger and Tricomi [2, 11.4] or Samko [31, p. 43]), and the relation 2.12.2.2 from the hand-book by Prudnikov, Brychkov and Marichev [2], we have
(27.51)
If a - I v i + ( + 1) /2 > 1, then owing to the known behaviour of the Bessel function as e --+ 0 and e --+ we have
n
oo ,
00
IV" u£(z)l � c lzl a - v E e lvl -a- n/ 2 1Ji v l - 2m - l +n / 2 (e)lde = clzl a-v .
J 0
As for the case a - l vl + ( + 1)/2 :5 1, the estimates in view of the uniform boundedness of the integrals
n
(27.49) follow from (27.51)
which is obtained owing to monotonicity of the function A a (e). Let now I v i If a -::f 1, 3, 5, . . . , we have
IV" u£ (z)l :5c
J
<
[a]+ 1.
IYI Iv l -a- n l( z . Y) l [a]+ l - l v l dy
1!11 < 1�1-1
+ c E lzl li l li i S [a] - v IY> I�I - 1
j
If a
= 1, 3, 5, . . . , by the Taylor formula with the remainder i n the integral form,
552
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
we have
" L...J
-nv Ue ( ) _ ·I vi v
X -1
(a + 1 'I J·
Ivi) ! ; X
li l=o+l- l vl f ( l - e) o- l vl de f y" +i I Y I - n -o Ao (lyl)ei(z·y dy. lr l > e Transforming the integral over I Y I > in the same way as the integral in (27.50), 1
X
0
c
we obtain (27.49). • The uniform estimate (27.44) being thus obtained, it remains by the Banach Steinhaus theorem to verify (27.43) on the Schwartz space S which is dense in L,. Since T0G0cp = cp for cp E S by Theorem 27.4, it can be shown that 0 by means of the Parseval equality and the Lebesgue II Te0 G0 cp -- cpl12 ---+ e -o dominated convergence theorem - employ the representation (27 .46') at first. Let us choose r > 1 so that p was between 2 and r and let fJ = 2(r - p)/p(r - 2). Then we derive from the interpolation inequality 11/11, 5 11111: - 6 11/11� and the uniform estimate (27.44) that (27.43) holds for cp E S. •
= 2, 4, 6, . . . the operator (27.41 ) is inverse to the potential G0cp within the frames of £,-spaces in the weak sense: Remark 27.3. It is easily verified that in the case a
In conclusion we give three other ways to effectively construct the operator We shall limit ourselves to giving very brief information, the corresponding references may be found in § 29. 1 . The first way is connected with the transformation of the operator (27.35) to the form
(E - �)01 2 , a > 0.
(27.52) where
h0 (y) E
L1 (Rn ) and D0/ is the hypersingular integral ( ( no = j �� nf+) x ) n( y)dy IYi o n
(27.53)
R,.
the characteristic fl(y) of which is a polynomial of degree less than a explicit expression for h 0 (y) and fl(y) in Nogin [7] .
-
see the
§ 27.
BESSEL FRACTIONAL INTEGRO-DIFFERENTIATION
Let a > 0, a :/:; 2, 4, 6, . . . , 1 < p < Ta Gacp = cp where a Tal = ha * I + lim n O, £ I (L , )
Theorem 27.6.
oo
553
and tp E Lp(Rn). Then
e-o
D n,e l being the "truncated" (lyl > e) integral, corresponding to (27.53).
We remark that a certain advantage of the construction (27.52) in comparison with (27.35) consists in the fact that it contains a hypersingular integral of a simpler form, constructed in terms of translations of the function l (z). The second way is related to (27.20) and is similar in a sense to the statements for the Riesz potentials in (25.70) and in Theorems 26.3' and 26.10. Let us denote 00
17'\�eal = x(a1/2, I) j t -l-af2 (E - e -t w, )1 1dt 0 < a < 21 '
e
where
x(a/2, I) is the constant (5.81).
Theorem 27. 7. (L p )
lim e-o
��Ga tp =
tp
Let tp E Lp(R"). Then
if 1 � p < oo .
lim
(p .p .) e-o
(27.54)
'
��Ga cp =
tp
if 1
�p�
oo
and
Theorem 27.7 remains valid if we set
1 �ae / - x(a,-I) j t -l-a ( E - 1\f )1 1dt 0 < a < I, 00
'
e
'
{27.55)
where M, is the semigroup (27.21). Finally, the third way is based on the idea of introducing hypersingular integrals with the "weighted" differences
p) �1a f = _1_ J (A� I)(z, dn ,l (a) I Y in+a dy, R"
( �� f)(z, p) =
� (!)<-l)ip(l:, y)/(z - ky).
(27.56)
For distinctness we consider the case of non-centered differences. We remark that the "weighted" difference with the weight p(k, y) = e - k y has already occurred in consideration of the Bessel potentials in § 18.4, as in (18.77). We define the weight
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
554
p(k, y) in (27.56) by the relation p( lc , y) = 2 1 - (n +a)/ •
( ( � a )) - 1 (lc lyl)
n
If f E S, the integrals (27.56) with this weight have the following properties: a) the integral (27.56) converges absolutely if l > a and conventionally if 2[a/2] l � a; b) if a 1, 3, 5, . . . and I > a, then T,a f = 0; c) :F(T,a f) dn ,l (a)(1 + l zl 2 ) a/ 2 /(z), where the constant dn , l (a) is the same as in § 26. The statement of Theorem 27.7 remains true if we replace l)� f by the truncated integral (27.56) with integration over IYI > c (Rubin (24, 25]).
<
=
=
§ 28. Other Forms of Multidimensional Integro·
differentiation
Fractional integro-differentiation of functions of many variables defined in the whole space Rn , which was considered in §§ 25-27, was a realization of the fractional power ( - � ) - a/ 2 or (E - �) - a/ 2 , � being Laplacian. The direct generalization of this approach is to consider fractional powers of an arbitrary differential operator in partial derivatives. We do not consider this question in such a generality, but focus only on fractional powers of the simplest differential operators (28. 1) (ultra-hyperbolic case) (28.2) (parabolic case). Fractional integro-differentiation considered in §§ 25-27 pertains, thereby, to the elliptic case. The realization of a negative fractional power of the former operator will lead to Riesz hyperbolic potentials with Lorentz distance and of the latter to parabolic potentials. These potentials are first considered, then the realization of a positive fractional power of the parabolic operator (28.2) is given. In the conclusion of this section we study the fractional integration which adjoins in a sense to constructions in § 24 and differs from them by the fact that integration in the corresponding fractional integrals is carried out not over the octant or a parallelepiped with the opposite vertices z ( x 1 1 . , Z n ) and a ( a 1 , . . . , a n ) as in § 24, but over a pyramid with the vertex at the point z and
=
=
. .
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
555
a base being a constant hyperplane not depending on x. We call such a fractional integration 'pyramidal'.
28. 1 . Riesz potentials with Lorentz distance (hyperbolic Riesz potentials) Similarly to the definition of Riesz fractional integro-differentiation in the elliptic case in § 25, the idea of introducing fractional powers is obvious, at least formally, in Fourier transforms. In fact, since the application of the operator (28.1) is reduced by Fourier transforms to multiplication by the quadratic form:
:F( -D,cp) = P(x):Fcp, P(x) = x� +
· ·
·
+ x; - x; + t -
·
· ·
-
x� ,
{28.3)
it is natural to introduce the fractional powers ( - Dp )A as the operators, which are defined by Fourier transforms by means of multiplication by the fractional power of the quadratic form P(x). We note, however, that in contrast to the elliptic case, the quadratic form P( x) has no definite sign. Of course, one might avoid raising negative values to a power by considering IP(x)IA or jP{x) jAsign P(x). But the fractional powers obtained in this way do not contain the usual integer powers of the operator - Dp , the former for odd ..\ and the latter for even ones. So we shall use the standard way of raising to a power with the choice of the "principal" value. We introduce the standard notations:
p� =
{ �(z) l'
if P(x) > 0, if P(x) :5 0,
p_A =
{ 0IP(x)IA
if P(x) if P(x)
2: 0,
< 0,
and
P(x) > 0, P(x) 0.
<
So
{28.4) Starting from {28.3) we introduce two following forms of the fractional power of the D 'A lembertian:
{28.5)
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
556
(cf. (7 .4) in the one-dimensional case ) . The formula of the Fourier transform of fractional powers of quadratic forms is known in the theory of generalized functions. We give it in the form (28.6) convenient for our purposes, where "Yn ( a ) is the familiar normalizing constant (25.26) used for the elliptic lliesz potential. The proof of (28.6) may be found in Gel'fand and Shilov [2, p. 349]. By (28.6) the operation (28.5) with A = -a/2 is to be realized as a convolution with the function e± !!jZ wi i
n ( a)
(P ± iO) -r = or - A
1
--
"Yn (a)
(e± !!.jL ra P-�+ - + e± T- r• P -�- ) . n-
.
or - A
or -
•
or - A
-
(28.7)
We denote this convolution by
so that
( - Dp) ±a/ 2 = IP ±io ·
By
(28.7) this operator has the form (28.8)
where and
K±
r (y) = .JP[Y) =
JX� + · · · + X� - x;+l -
· · ·
- X�
denote the following cones:
K+ = {x : x E Rn , P(x) � 0}, K_ = {x : x E R" , P(x) � O}. The function r(y) is called the Lorentz distance and the cone K+ is known as the light cone or characteristic cone. The integrals in (28.8) converge in the case of sufficiently good functions if a > n - 2. One can make sure that this is so by representing r" -a ( y) as r" - a ( y) = I Iy' 12 - I y" 1 2 1)n - a)/ 2 where y' -- ( Yl , . . . , Yp ) E RP and YII -- ( Yp+ 1 , . . . , Yn ) E Rn -p ,
§ 28. OTHER FORMS OF INTEGRO-DIFFERENTIATION
557
by passing to repeated integration over RP and Jl'l -P and introducing polar p coordinates in the integral over Rn- , which will give the condition (n - a)/2 1. In the case a � n - 2 the construction {28.8) is not determined, but it admits the continuation, analytical in respect to a which is well suited for a � n - 2. We do not elaborate on such a continuation, see the references in § 29.1. It follows directly from {28.5) that <
a+f3 a 1{3P ± iO -- JP±iO IP±iO
{28.9)
·
We put the following question: is it possible to introduce an operator of hyperbolic fractional integration of the type {28.8) with the integration over only one of the cones K+ and K_ and such that the semigroup property {28.9) remains true? It is clear that for this purpose we have to start not from the function (P iO) >. , but from the functions P� or P� . In view of {28.4), {28.6) after simple transformations yields relations ±
( . ( 1n (a)a) 1r12 (
. p - a 1r p -a / 2 + Sill. p1r p - a/ 2 :F( p(+a- n ) / 2 ) -- 1n (a) Sill . \ I n - a ) 1r /2 Sill 2 + 2 :F( p(_a- n ) / 2 ) -
_
Sill n -
)
'
{28.10)
n - P p - a/ 2 . n - P - a 1r p_- a/ 2 . Sill -7r + + Sill
2
2
).
Hence {28.11)
if and
(
)
p=
2 ' 4, 6, . . .
:F p!_o; "> = (-1) � - l 'Yn (a) sin aTr/2 P� � sill { n - a)Tr/2 .
if
n-p=
{28. 12)
2, 4, 6, . . .
Therefore, the potentials p = 2, 4, 6, . . . ,
{28.13)
558
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(-1) �- l cp(x - y)dy n - p = 2, 4, 6, . . . , lp_ 'P = ....;. H....;...._ �---:.. (-a:--)- J l rn - a (y)l ' K_ with
a > n - 2 and
a?r/2 Hn, (a) = 'Yn (a) sm. sin ( n - a ) 1r/2
(28.14)
(28.15)
admit the semigroup property
(28.16)
in view of (28.11) and (28.12), provided that p = 2, 4, 6, . . . in the case of the sign + and p = n - 2, n - 4, n - 6, . . . in the case of the sign - . We single out the case p = 1. The form (28.13) may not b e applied i n this case, while (28.14) may be used for odd
n:
(n + l)/ 2 Ipa_ 'P - ( -1) H�(a)
cp(x - y)
J rn -a(y)l dy, K_ l
p = 1, n = 3, 5, 7, . . .
(28.17)
However, there is another variant m the case p = 1, namely that which was introduced by Riesz. In the light cone K+ we consider the positive half-cone
(28.18) and designate X - y) dy, a > n - 2, l0a cp - H 1(a) J It'( n r n x+ - a (y) -
--
(28.19)
+
emphasizing the relation of the operator (28.19) to the D'Alembertian 82 82 - p - -/l.r . The normalizing constant Hn (a) is chosen as P x2 xl
-
• •
•
0
=
n
Hn (a) = 2 sin(n'Yn-(a)a) ?r /2 = 2"- t .- - t+n / 2 r
which becomes clear below.
( i) r ( "' � - n ) +
(28.20)
§ 28.
559
OTHER FORMS OF INTEGRO-DIFFERENTIATION
This is the potential (28.19) that is usually called It may be also rewritten as l()(
1 D
( )
the hyperbolic Riesz potential.
rn - 01 ( x - y)
(28.21)
where
is the negative light half-cone with the vertex shifted to the point x. Let us show that the operator /0 admits the semigroup property
(28.22) provided that the normalizing constant is given by (28.20). To prove this we first calculate the Riesz potential of the exponential function ex 1 • In this case we have
and if we choose
Hn (a) =
j e-Y 1 r01 - n (y)dy
K+ +
(28.23)
we obtain
(28.24) Calculating the integral 00
(28.23) we have
Hn (a) = J e - 111 dy1 o
J (y� - le i2 )(0t- n)/2 � , e = (y2 , . . . , Yn ) E Rn - l .
1ei < Y1
560
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Hence
Hn (a) = f yf- 1 e -Y tdy1 f { 1 - l7712 )(a -n )/2 df] 1771<1 1 2 (a n) 2 " 2 1 = r(a) I Sn - 2 f{ 1 - p ) - / p - dp, 00
0
0
which coincides with {28.20) after simple transformations. We observe that a more general equality
ea·x. . . 2 ' loa (ea·x ) = -(a-=i,._.. a-=-�----- a��,...)a--:.- /�
{28.25)
may be derived from {28.24) ( cf. (25.39) ) . This is the result (28.24) which allows us to give a direct proof of the semigroup property . {28.22 ) . Taking > n - 2 and {3 > n - 2, we have
a
1/j lg tp = Hn (<>)lHn (PJ K f(x) r"- n (x - y)dyK_;f(y) ,.P-n (y - e)tp(e)de. _;
K+(y) K+(x ) for y E I<+ (x) .
C It is clear that integration we obtain
So, interchanging the order of
IQigtp = Hn (<>)lHn (P) j 'l'(e)de j r" - n (x - y),.P - n (y - e) dy D(x,O K_; (x) where
K+(y)
(28.26 )
D(x,e) = K+(x) Kt(e) (introduce the characteristic function of the cone n
while interchanging the order of integration ) . By means of the shift along the surface of the cone and dilatation we see that the inner integral is equal to
ra+.B- n (x - e)Bn (a, {3) , Bn (a , {3) = f
D(O,e1 )
e 1 = ( 1 , 0, . . . , 0) ,
ra - n (y)r,B - n (e 1 - y)dy ,
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
561
where the integral Bn (o:, {3) is clearly a constant. So we conclude from (28.26) that
+ o
o o iP-
n (o:, /3) Hn ( o: {3)1a+f3 IP · l a ]P - HB(o:)H n n ( /3) Taking here v; = e x • , we see by (28.24) that the relation Bn (o:, /3) = Hn (o:)Hn (/3)/Hn (o:
+
(28.27)
{3)
must hold (cf. (25.38)), which immediately transforms (28.27) into the former of the relations (28.22). As for the latter, it is obtained by direct checking after differentiating under the integral sign. Remark 28.1. The Fourier transform of the hyperbolic Riesz potential
given by the result
J0tp is (28.28)
where q = e 2fisign x 1 if r2 ( x ) > 0 and q = 1 if r2 0, r2 ( x ) = x � - x � - · · · - x � . We conclude this subsection by consideration of the Riesz hyperbolic potential (28.21) in the two-dimensional case:
<
or
( x - y)dy1 dy2 (y� _ y� ) l - a/ 2 '
v;
=
(28.29)
=
The change Yl + Y2 2e 1 , Yl - Y2 26 of variables transforms this potential to Liouville fractional integration in each variable, which was considered in § 24:
(28.30)
562
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
where Z t = (zt + Z 2 )/2, z2 = (zt - Z2 ) / 2. Equation (28.30) can be represented in the operator form:
Ta - A - 1a/ 2, a/ 2Arp Jo'P _
1
(28.31)
++
where A is the operator of the linear change of variables: (Arp)(z) = rp(zt + z 2 , Zt z2 ) and 1:'.;· a/ 2 is the operator of the Liouville fractional integration (24.20). The following theorem is an immediate consequence of (28.31) and Theorem 24.1 on the Liouville fractional integration 1:�2 , a/ 2 .
Let n = 2, 1 � p < oo, and 1 � < oo. The operator 10 is bounded from Lp (R2 ) into Lr(R2 ), if and only if
Theorem 28. 1.
r
0 < a < 2, 1 < p < 2/a,
r = 2p/(2 - ap).
28.2. Parabolic potentials We consider negative fractional powers of the heat equation operator (28.2) in this section. As usual, we find it convenient to single out the ''time" variable by redenoting it as t and the space variables as x 1 , . . . , X n , i.e. to carry out the investigation in the ( + ! )-dimensional Euclidean space Jl!1 +1 of the points ( x , t), where z E Jl!1 and t E R1 . We shall deal with the negative fractional power of the operator
n
T
=
-.6. +
8 X 8t
(28.32)
where .6.x is the Laplacian with respect to variables Xt , . . . , X n . For the operator (28.32) we have in Fourier transforms:
Trp = ( lxl 2 - it)�(x, t)
(28.33)
where the Fourier transform is applied in Rn + 1 :
(Frp)(x, t) = �(x, t) =
j eix·�+itT rp(e, r)dedr.
(28.34)
Rn+ 1
So a negative fractional power r- a/ 2 , a > 0 , may be naturally introduced in Fourier transforms via the function ( l x l 2 - it) -a/ 2 with the choice of the principal
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
563
value: arg ( l x l 2 - it) E ( - 7r/2, 7r/2). This function will be represented as the Fourier transform of a function defined in terms of the Gauss-Weierstrass kernel
(28.35) Let
{
t - l +af 2 W(x, t), t > 0, 1 h a (x, t) - f(a/2) t < 0. 0, _
Lemma 28.1.
(28.36)
The Fourier transform (28.34) of the function ha (x, t) is equal to (28.37)
The proof of the lemma is obtained by direct calculation:
IX·'-
. o-.. - - 1 e 1·t T dT I e - � + . t: de. h a (e T ) eu:·'-t: +1·t T tJ.edT = {4 7r)n / 21f( a/2) I T -2 0 R" 00
4�
'
The inner integral here is the product of the easily evaluated one-dimensional integrals so that further verification of a (28.37) is easy in the case of small a > 0, the general case being achieved by the usual analytic continuation with respect to
a. •
Lemma 28.1 allows us to introduce the negative fractional power of the heat equation operator T as the convolution
r-af 2 cp = Ha cp';;!
I ha (e, T)cp(x - e , t - T)dedT
R"+ 1
or in the explicit form
(28.38)
564
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
is the half-space {(z, t) : z where t 0}. That is the operator which is called the parabolic potential operator. If we use the definition
Rf-+1
(28.39)
(WT
E Rn, >
(47rT)-nf 2 J e - lel2 /(4T )
=
R•
for the Gauss-Weierstrass operator, we can rewrite the operator from in the form
(28.38),
(28.40)
Ha, as it appears
r(�/2) J0 r"1 2- 1 ( WT
( H "
=
(28.41)
for the lliesz potential) . 0 on a rather large set of functions, The operator H a is defined for any for example on functions
(25.51)
a>
> a/2. Lp(Rn+l) Theorem 28.2. The integral H 0 0 with cp E Lp(Rn+l) converges absolutely almost everywhere if 0 a n + 2 and 1 � p (n + 2)/a and the operator Ha is bounded from Lp(Rn+ l ) into L9(Rn+ l ) , where 1 p ( n + 2)/a and (n + 2)p/(n + 2 - ap). We do not give the proof of this theorem, but see the references in § 29.1. Similarly t o the spaces l0(Lv) of Riesz potentials or the spaces G0(Lv) of <
<
<
<
<
q =
Bessel potentials, the spaces
(28.42) of parabolic potentials may be introduced . These spaces are well defined by Theorem in the case � p < + In the next subsection we give a characterization of the spaces Ha in terms of the convergence of certain hypersingular-type integrals. We note that the spaces of type may be also considered in the case of functions f(z) defined in the half-space Rf- 1 only. Finally, we remark that similarly to what was stated above, we can consider a negative fractional power of the differential operator E - d.x + 8f8t , where E is the identity operator. In Fourier transforms we have then to deal with the function
28.2
1
( n 2)/a. ( Lp) ( 28.42) +
(28.43)
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
565
so that the fractional power
(E - tJ. 0 ) -a/2 x+ 8t
(28.44)
defined in this way, that is the corresponding potential, relates to parabolic potential, H01cp, already considered above in the same manner as the Bessel potential G01cp relates to the Riesz potential l01cp. The fractional power is realized as the convolution with the original of the Fourier transform and has the form
(28.44) (28.43)
- 1 j T- 1-a-/2 00
- r(a/2) (compare
0
e - T ( WT
(27.20) with the Bessel potential). operator (28.45) is also called a
(28.45)
The parabolic potential. Owing to the T and, factor e- its domain in essentially larger than that of the potential in particular, it is bounded in any space L,(Rn + 1 ), � p < oo. For further information about the operator and references see notes
(28.45)
1
§ 29.2,
28.3. The realization of the fractional powers and
(
E
-
�x +
:t ) a/2 ,
a
>
(28.39) 28.2-28.3.
( -�x
+
:t ) 0112
0, in terms of a hypersingular integral
In this subsection we construct effectively the hypersingular integrals T01 I and '.t01 I, a > which are inverse to the parabolic potentials I = H01cp and I = 1l a cp defined by and So it is natural to call them the parabolic hypersingular integrals. They will contain the non-standard finite differences, which reflect the different behaviour of potentials with respect to the space variable z and the time variable t .
0, (28.39)
(28.45).
We shall present first the scheme for the formal construction of the operators and '.ta , a > inverse to the potentials H01 and 1[01 respectively. , and then justify the fact that they indeed invert the corresponding potentials. We shall use the Gauss-Weierstrass kernel below and we recall that it has the following
T01
0,
(28.35)
566
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
properties (Stein and Weiss
[3, p. 16, 24]) :
J W(x, t)dx = 1,
(28.46)
J W(y, t)W(x - y, r)dy = W(x,t + r),
(28.47)
R" R"
(28.48)
j
where
(F�
(28.49)
R"
is the Fourier transform in the space variables x. Let t) = (H or
f( x,
(
)
F�
to both parts of this
(28.50) where 1:/ 2 is the operator of the one-dimensional fractional integration ( 5.4 ) applied in respect to the variable q . Inverting the operator 1: /2 in ( 28.50 ) by means of the Marchaud fractional derivative (5.80) we arrive at the relation e tlel:l (F�
00 1 j - x(o:/2, /) "' (1) k (- 1
1
_
LJ
o
k :O
) k ( :F� f)(e, t - k q)e (t -k,)lel:l
Now we multiply by e - tl { l l and then apply the operator
dq . l q +or/2
Fi 1 , which gives
(28.51 )
'P(z, t) = x(a�2, /) j [t(z, t)+ :p - 1 ) k (!) j f(x-y, t-k71)W(y, k'l)dy] 71 ,:�/ 2 . oo
I
k-1
0
Hence after the substitution
y
-+
R"
../ky and simple transformations we have
t) j (��q.l,+f)(x, 2 / or
R.f+t
W ( y , q )dydTJ
":Jlrorf
( 28.52)
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
567
�(-l)k (!) /(z - ../ky, t - k'l)·
(28.53)
where
(A� ,, /)(z, t) =
= (Ha ) - 1 in the form of the
Thus we have constructed the inverse operator ra hypersingular integral Similar arguments applied to the equation 1l a
(28.52).
1
j
R;+ •
= f(x, t) yield the following
(A111 '11 /)(z ' t '· e - ") , 7] d d7] d�J('!a )( t) W( y ) y / z, ' 7J l + a/ 2
where the weighted difference of the type
(28.54)
(28.53):
(A�,, I)(z,t; e - " ) = t (!) (-l) k e - k" l(x - Vky, t - k71) k :O
(28.55)
is used. Weighted differences have already occurred earlier in the elliptic case, as in It is not difficult to show that the integrals in and converge conventionally for f E at any point (x, t), provided that they are understood as
(27.56).
(28.52)
S(Rn+ l )
(28.54)
(28.56) where r: 1 and cr: 1 are the truncated integrals and with the integration over the shifted half-space R��1 {(y, 71) y E 7J > c }. We show now that the hypersingular integrals ra I and era I ' interpreted as suggested in are real inverses to the corresponding potentials f H a
=
(28.56),
Theorem 28.3.
{28.52) (28.54) : Rn,
= (Rn+ l )
= S = S(�+1)
Let I = H a 0 and . Then ra I = £-0 lim Tea I =
Proof. We shall base this on the representation T£a f =
j
R+•+ t
W(y, 7J) /Ct, af 2 (7J )
(28.57)
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
568
where have
K 1 , 01 (TJ) is the kernel (6.7') . We remark at once that by (6.8') and (28.46) we W(y, TJ) K1 ,at(TJ) E L 1 (R" + 1 ),
j W(y, TJ)K1,at(TJ)dydTJ 1 .
(28.58)
=
R"+l
Let us prove (28.57). Substituting the expression for the finite difference
(A� ,, f)(x, t) =
t( - l )l (!) j ha (z - ../ky, ( - k'IJ)'I'(x - z, k= O
R"+l
t - ( )dzd(,
h 01 (x, t) being the kernel (28.36) , into the integral T:X I and interchanging the order of integration, we obtain T�.. l
1
J f (Ot/2)x( Ot/2,l) R"+l x
[
1
00 ) (( - kTJ) :/ 2- 1 dTJ / k( 1 at
�(- ) k-0
'
k
E
TJ1 + / 2
j W(y, 'I)W(z - ../ky, ( - k'IJ)dy] 'l'(x - z,
R"
t - ( )dzd( .
Carrying out the change of the variable y ---+ yf-/k, k :f= and observing that W(y/Vk, TJ) k" 1 2 W(y, kTJ), by the property (28.47) of the Gauss-Weierstrass kernel we have
0,
=
T�.. l
1
J f(Ot/2)x(Ot/2, l) R"+l x
[
1
00 ) / (( - kTJ) :/ 2- 1 dTJ] k( 1 at
�(- ) k-0
'
k
E
TJ1 + / 2
W(z, ()cp(x - z, t - ( )dzd( .
Hence, after the substitutions z ---+ cz, TJ ---+ c TJ and simple calculations we arrive at (28.57) . The properties (28.58) allow the possibility to pass to a limit as c ---+ under the integral sign in (28.57) . As a result we obtain lim Te01 I =
0
Theorem
28.4. Let I = 1l 01 0, and
The proof of this theorem is similar to that of Theorem 28.3,
and is obtained
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
569
from the representation
c:r: I =
j W(y, 7J)X:,, � (71)e-e"cp(x - .fiy, t - c71)dyd71
R+.+ •
( cf.
(28.57)), which is proved analogously to (28.57). The hypersingular integrals T01 I and '!'01 I are inverse to the potentials I = H01
of the corresponding truncated integrals. Theorem 28.5.
Then
Let 0 < a < ( n + 2)/p,
1
oo,
and I = H01
(28.59) Equation valid for
Proof.
Theorem 28.6.
(28.57), Lp, (28.57) Lp L9, n 2)p/( n 2 - ap). 28.2 (28.57) Lp (28.59). Let 1 � p < a > 0 and let I = 11.01
The proof is similar (see also references in § 29.1). 28.4. Pyramidal analogues of mixed fractional integrals and derivatives In we introduced mixed integrals and derivatives of fractional order. The domain of integration for such operators is a rectangular parallelepiped with opposite vertices x = (x�, . . . , xn ) and a = (a� , . . . , an) · In particular, it may be an octant with the vertex x. The kernels of these operators have singularities on those faces of parallelepiped which pass through the point x. Now the domain of integration will be chosen to be a certain pyramid with a vertex x and with a basis situated on the fixed hyperplane ( which does not depend on x). As regards the kernel, we assume it to have singularities on the hyperplane passing through the point x. As a result the so-called pyramidal analogues of mixed fractional integrals
§ 24
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
570
and derivatives arise which, as it will be shown below, have essential distinctions in comparison with mixed fractional integral and derivatives considered in § 24 and which are not reducible to them. See for comparison the domains of integration for mixed fractional integrals and derivatives and for their pyramidal analogues in the two-dimensional case in Figures 4 and
5.
�
�
-
�
- +-------�
-------. x(x 1,�)
Figure 4. The domain of integration of the integral
/�+
1
Figure 5. The domain of integration of the integral
1�4c
Later on we shall require some auxiliary terms and notations. Let A = ll a k ll (aj k E R1 ) be a matrix of order n x n with determinant !AI = det A =Fi 0, aj = {aj l ! . . . , aj n } be its line vectors and aj k be the elements of the inverse matrix A- 1 . Without loss of generality we put l A I = 1 . Let also A · x = (a 1 · x, . . . , an · x), (A · x) a = (a t · x) a 1 (an · x) a,. where a = ( at , . . . , an } and b = {b t , . . . , bn ) E ·Rn , c = { c 1 , . . . , en ) E Rn , -oo < bj , Cj < oo. We denote by •
•
•
Ac (b)
= { t E Rn
:
A
· { b - t) � 0, c · t � 0}
{28.60)
the n-dimensional bounded pyramid in RR with the vertex at the point b, with a basis on the hyperplane c · t = 0 and with lateral faces situated on the hyperplanes aj · (b - t) = 0, j = 1, . . . , n. In particular, if A = E = ll6j k ll is a unit matrix and c = 1 = {1, . . . , 1} then {28.60} is the simplest model pyramid of the form
E1 {b) = {t E Rn : b � t ,
t 1 ·
� 0}.
{28.61)
If n = 2 it coincides with a triangular bounded by the lines t 1 = b t , t2 = b2 and t 1 + t2 = 0 {see Figure 6). The Abel type integral equation ( see (28.65)) which will correspond to the above mentioned pyramidal analogues of mixed fractional integrals is a generalization of the equation considered by Mihlin [1, p. 48] . The next proposition contains conditions of non-emptiness and boundedness of the pyramid {28.60) and the expression for interchanging the order of integration, similar to the Dirichlet formula ( 1 .32) .
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
Figure 6. The domain of integration of the integral
571
181 cp
The pyramid Ac (b) given by (28.60) is non-empty {bounded) in Rn if and only if A- 1 c · b > 0 {A - 1 c > 0, respectively). The proof of this Lemma follows from the fact that the linear transform t - A- 1 t, b - A- 1 b maps the pyramid Ac (b) into the pyramid
Lemma 28.2.
(28.62) Lemma 28.3. If a function f(t, T) given on Ac (b) x Ac (b) is measurable then the following expression for interchanging the order of integration
J dt J j(t, T)dT = J dT J Ac(b)
is true, where u
Ac(b)
Ac(t)
(b, T)
j(t, T)dt,
(28.63)
u(b,T)
= {t E Rn : A · T � A · t � A · b},
under the assumption that one of the repeated integrals in convergent.
(28.64)
(28.63)
The proof is obtained directly by Fubini's Theorem 1 . 1 .
Now we consider the Abel-type integral equation on the pyramid 1
J
cp(t)dt (A . (x - t)p- a = f(x),
is absolutely Ac (b)
X E A c (b),
(28.65)
where < a < 1 (which means < Q l < 1 , . . . , 0 < O:n < 1), r (a: ) and x < b (z 1 < b l J . . . , zn < bn ) ·
= r(at ) . . . r(a:t)
r(a)
0
Ac(z)
0
572
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
In particular if Ac(b)
= Et (b) (28.65) has the form
1
f(a)
J (zcp(t)dt - t) l - cr = /(z),
z E Et(b),
(28.66)
E 1 (z)
which does not coincide with the Abel multi-dimensional equation (24.1). Therefore (24. 1) is not a particular case of (28.65). However the method which was used in § 2.1 may be applied for inverting (28.65). Following this approach we replace t by r and by t, respectively in (28.65) , multiply both parts of the resulting relation by (A · t)) - cr and integrate over the pyramid Ac(z). Applying Lemma 28.3 we have
z (z -
l f( <>)
j tp(r)dr j (A · (z - t))- a (A · (t - r))0- 1 dt = j (A · (z - t)) -a f(t)dt
Ac(z)
Ac(z)
e1(z,r)
(28.67) where the domain r) is given by (28.64). To evaluate the inner integral in (28.67) we introduce the new variables s; = [a; · t)]/[a; · r)]. Taking the equalities 1 - s; = [a; · (t - r)]/[a; · ( x - r)] and (1.68) and ( 1.69) into account we obtain
u(z,
(z -
(z -
n 1 sj 0; (1 - s;) a;- l ds; j (A · (z - t)) -a(A · (t - r))a- l dt = JgJ 0
e1(z ,r)
(28.68)
= f(a)f( l - a). Hence (28.67) may be rewritten in the form
J cp(t)dt =
Ac(z)
J (A · _ a)
1 f(1
( z - t))
_ 0 f(t)dt def := ft
Ac(z)
- cr (z).
(28.69)
Here we change the variables
(
)
1 . ! , X = X l ' . . . ' Xn ' At = A - 1 . �� = X d d d dt dn where
d = (dt , . . . , dn ) is given by (28.62).
(28.70)
Then in accordance with Lemma 28.2
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
573
(28.69) is equivalent to
I
.,P( r)dr = g (y) ,
(28.71)
E1(y )
where E1 (y)
is the pyramid (28.61) and
(
.,P ( r) = tp A - 1
·
�) ,
( · �) IIn d�c . k= 1
g (y) = h -a A - 1
(28.72)
To invert (28.71) we rewrite it in the form
.,P ( r) dr
= g(y).
(28.73)
Successive differentiation of this expression with respect to Yn , Yn - 1 , . . . , Y1 yields the relation
.,P(y) Here we change the variable z f)
=
= A- 1 n
-1
f) ·
Y1
{)R · ·
·
f)
Yn
g (y).
� similarly to (28.70) so that
f)
aj k -- "' {) LJ dk {)Xj , k = 1 , 2, . . . , n . Yk i= 1
(28.74)
Then finally we obtain the following inversion relation for (28.65) :
(
n n {) 1 (z) = cp f( 1 - a) II � aj k fJxJ. k=1 =1 J
)
I (A . (x - t))-a f(t)dt.
Ac(x)
= E1 (b) then (28.75) has the form
In particular, if A c (b)
tp
(x)
= f(1 1_ a) fJax I (x - t) -af(t)dt E 1 (x)
where
:x
=
8 1 ••• 8
:
(28.75)
!
•
.
(28.76)
574
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Now we give a justification of the solvability of (28.65) in the space Lt(Ae(b)). For this goal we introduce the notation
IA. (L ) = { ,
g :
g(:r ) =
I Ac
(z )
h (t)dt, h (t) E L , (A. (b)) }
(28.77)
and observe that if g then there exist partial derivatives of g up to order almost everywhere on Ac ( b) and
E lAJLt)
n
( [!n {;n
ii; t
0
)
O:r; g( :r ) = h (x).
(28.78)
The following statement is an analogue of Theorem 2. 1 .
The Abel-type equation Lt (Ae(b)) if and only if
Theorem 28.7 .
11 -a (:r ) � r ( l � a)
(28.65)
with
0<
a < 1 is solvable in
(A . (:r - t)) - Q f(t) dt E IA.(L l ), I Ac ·
(z )
n ft -a (z) lc·z= O = L: a-jn Oa · ft - a (z ) lc · z=O = XJ j:l 0 ) n (n = IT L Ctjk 8 x · fi - a (x) lc · z=O = 0. J k =2 j= l
(28.79)
· · ·
(28.80)
These conditions being satisfied (28.65) has a unique solution given by (28.75) .
=
Proof. In the model case Ac (b) E1(b) the theorem follows from (28.71) and (28.73). In the case of an arbitrary pyramid A c (b) it is obtained from (28.71) and (28.73) after the change of variables (28.70) with the property (28.74) being taken into account. • Corollary.
Equation (28.66) is solvable in L1 (E1(b)) if and only if 11 -a (:r ) = r ( l �
a) I (x - t) - Q f(t)dt E LE, (L l ), Et (z )
(28.81)
§ 28.
8
575
OTHER FORMS OF INTEGRO-DIFFERENTIATION
i l·z=O = Xn lt - a (x) l l ·z=O =
ft - a (z) h-z=O = 8X ft - a (x) n = aaX
2
...
a
·
·
·
a
(28.82)
0.
These conditions being satisfied (28.66) has a unique solution given by (28.76). Theorem 28.7 gives the criterion of solvability for the Abel-type equation (28.65) in terms of the auxiliary function fl - a (x). Simple sufficient solvability conditions in terms of the function f(x) itself are given in Theorem 28.8 and its
Corollary.
Theorem 28.8.
order n and
Let the function f(x) have continuous partial derivatives up to v11 f(x) l c ·z=O = 0 ,
(28.83) Then the Abel-type equation (28.65) is solvable in Lt(A c (b)) and its unique solution is representable in the form
a
j (A . (x - t))-a k=IIl ( J?== l ai k 7Jt.J ) f(t)dt.
1
f( l - a )
0 $ I.BI $ n - 1.
"
"
(28.84)
Ac(z)
Proof. First we consider the model equation represent fl - a (x) as
ft - a (x) = r(l �
o
)
Similarly to
(28 .73)
we
J (x - w af(t)dt
E 1 (z)
j
Zn-1 X
- (z 1 +· · ·+zn-�+ t n ) X
(28.66).
(X n - 1 - t n_ I ) -a"- 1 dtn - 1
. . •
7
- (t�+· · ·+t .. )
Carrying out the successive integration by parts and taking
(28.83)
into account
576
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
we have
1 !t -a (x) = f(1 - a )
J�
Et ( )
(x - t) 1 - cr of(t) dt. (1 - a) ot
(28.85)
The function (28.85) satisfies (28.81) and (28.82). Indeed, the former is clear, while the latter is checked by successive differentiation of (28.85) with respect to Xn , Xn - 1 , , x2 . Hence according to the Corollary to Theorem 28.7 we obtain the statement of the theorem for the model equation (28.66). The general equation (28.65) is reduced to the model one by means of the change of variable (28.70). The theorem is proved. • . . •
Corollary.
and
If the function f(x) has a continuous partial derivatives up to order n v13 f(x) h-�= 0 = 0,
0�
l.Bl � n - 1
(28.86)
then (28.66) has a unique solution given by the relation
J
(x - t)
-a
0
Et(�)
0 dt. 1 t . . . tn
an f(t)
(28.87)
Remark 28.2. Equation (28.71) is an example of an integral equation of the first kind arising in the problems of integral geometry where the unknown function is to be restored by its known integrals over some sets, see for example, Gel'fand, Graev and Vilenkin [1, p. 111]. Proceeding from (28.65) and (28.75) we introduce the operators
( f.t
J (A . (x - t)) l - cr ( n n (VAc f)(x ) = f(1 1- a ) lJ ( � a_ ; k ox; ) J k - 1 J- 1 A c:(� )
0
0
I
A c: (�)
a > 0,
f(t)dt (A · (x - t)) cr '
(28.88) (28.89)
0 < a < 1,
which we shall call pyramidal analogues
of the Riemann-Liouville mixed fractional integrals and derivatives of order a = ( a 1 , . . . , an ) , cf. (24.5) and (24.9). Expressions (28.88) and (28.89) are defined for functions given on A c (b). Theorems 28.7 and 28.8 contain conditions for the existence of the mixed fractional derivative (28.89). In particular, if a function f has continuous partial
§ 28.
OTHER FORMS OF INTEGRO-DIFFERENTIATION
577
n
derivatives up to order and (28.83) hold, then the mixed fractional derivative (28.89) is representable in the form (V�J) ( x ) = r(l
� a) j
Ac(r)
(A · ( x -
t))- a ft c t= a;k O�J- ) !(t) dt . k=l J l
(28.90)
With the aid of (28.67) one may check the semigroup property
(28.91)
fJ1 ,
. . . , an + f3n ) · for any function
a
1
0
(1JE1 f)(x) = f ( 1 - a) o x
f(t) dt J (x - t)a '
E1(r)
We shall call them the
a > 0,
(28.92)
O < a < l.
(28.93)
model pyramidal analogues of the Riemann-Liouville mixed fractional integrals and derivatives of order a. In the case a k > 1 , we introduce the model pyramidal analogues of mixed fractional derivatives similarly to (24 . 10 ) by the relation
(28.94) where
(!_) [a] (_!____ ) [at] . ( .!_ [a,.] OXn ) OX 1 OX . .
=
Below only the operators Example 28.1. Let
(28.92)-(28 .94) will be considered.
(t) = (t 1 +
· · ·
t
+ n ).B 1 , a > 0. Then -
l i +.B 1 (IE1a
1
{3 > 0,
(28.95)
578
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
f a 1 /)( X ) - f (P) ( X t + · · · + Xn ) P - lai - t , ("" vE (P I a I} _
(28.96}
_
where l a l = a 1 + · · · + an . This example shows that the model pyramidal analogues of mixed fractional integrals and derivatives (28.92} and (28.94} keep power behaviour invariant - i.e. transform power functions into power functions - on the hyperplane x 1 +· · · + x n = 0 which is the basis of the pyramid E1 (x) . Unlike this, mixed fractional integrals and derivatives (24.5} and (24.9} keep power behaviour invariant on the hyperplanes x1c = a1c , k = 1 , . . . , n , since
f a (t - a)/3 - l )(x) = f {,B} (x - a) a +/3 - l , a > 0 , (Ia+ ( a + P)
�(�)a) (
(V�+ (t - a) P - 1 )( z ) = f (
z -
a fJ
> 0,
{28.97}
a)P -a- 1 , a > 0, {J > a,
(28.98)
Similarly to (24.6} and (24.9} we may introduce the model pyramidal analogues of mix�d fractional integrals and derivatives with respect to some of the variables. Let
x = (x', x"), x' = {xt , . . . , Xm ), x" = (xm+b . . . , Xn }, a' = (at , . . . , am , O, . . . , O), E� ( x) = {t
E Rn
:
t' � x', t" = x" , t' · 1 + x' · 1 � 0}.
(28.99}
Then we put
1 ( J_E 1
f (x' - t')a - 1 cp(t', x")dt', I
( )
a' > 0,
{28.100}
E� (x)
{) [a 1] + 1 � 1 ('D£1 /}(x} = o (1�:- [a ] - 1 /}(x}, a' > 0, x
(28. 101}
where [a'] = ([at], . . . , [am], 0, . . . , 0). In particular, if x" = (x2, . . . , xn ) then (28.100} and (28.101} coincide with the one-dimensional fractional integrals (2.17}
§ 28. and derivatives
OTHER FORMS OF INTEGRO-DIFFERENTIATION
579
(2.29):
(IE: cp)(x) = (I�(x� +··+x ,. )
(28.102) (28.103)
if x" is fixed. The above arguments show that pyramidal analogues of mixed fractional integrals and derivatives are specific constructions different from mixed fractional integrals and derivatives. In particular, they may not be represented as a tensor product of one-dimensional fractional integrals. However, some properties of one-dimensional fractional .integrals and derivatives may be transferred to such operators. For example, the index law (28.91) is valid. An analogue of Hardy Littlewood Theorem 5.3 is also true. To formulate it we consider the case of two variables just as was done in § 24.4. We introduce the space Lp 1 ,p� (Et(b)) of functions ( x 1 , x ) equipped with the norm
f
2
(28.104) The following theorem is a corollary of Theorem 24.1.
If 1 < Pi < 1/o:j and 1/qi = 1/Pi - O:j , j = 1, 2, then the operator of the pyramidal analogue (28.92) of the mixed fractional integral is bounded from Lp 1 ,p� (Et(b)) into L91 , 9� (E1 (b)).
Theorem 28.9.
cp(t)
cp(t 1 , t2 ) t t 2 {( t, 2) R
Proof. Let given on the E Lp 1 ,p � (Et(b)). We define the function triangle E1 (b) (see Figure 3) to be zero on the rectangle ll(b) = E : < bt , -b } . Let = < b x E Et(b); < < t 0, x E ll(b) \ Et(b)} -� {
t2
XE1 (b)
IIIE.
The theorem is proved.
•
580
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
§ 29. Bibliographical Remarks and Additional Information to Chapter 5 29. 1 . Historical notes Notes to § 24.1. The solution of the multidimensional Abel equation {24.1) was probably known long ago, even though it was not found in the early publications. It is natural to suppose that it was already known to Holmgren [1] {1865-1866) - see the paragraph below. The Laplace-transform method of solving {24.1) may be found in Vasilache [1] {1953) for n = 2 and in Delerue [1] {1953) for an arbitrary n. Notes to § 24.2. There are many papers where partial or mixed fractional integra differentiation, in each variable or in a part of them, is "introduced" . It is worth emphasizing that these concepts arose long ago, and the first person who indeed introduced Riemann-Liouville fractional integra-differentiation for functions of two variables was Holmgren [1 , p. 14] (18651866)). These ideas were essentially used in problems of the function theory by Montel [1 , p. 172] (1918) yet. Notes to § 24 .4. Details about the spaces Lp with mixed norm may be found in the books by Besov, D'in and Nikol'skii [1 , ch. 1] (1975) and in the paper by Benedek and Panzone [1] (1961). The idea of considering fractional integration in the spaces Lp is contained in the latter paper where it was realized for the Riesz potential operators. The boundedness from Lp of mixed fractional integration, into L q , qi = Pi (1 - c::tj pi )- 1 , directly for the operators 1: 1 ® and similarly for partial fractional integration, was noted by Skorikov [3] (1977), the case of an arbitrary number of variables was observed by Magaril-ll'yaev [1] (1979), although it was probably known earlier. Notes to § 24.5. The existence of relations between different fractional integration operators via the bisingular operator was first noted by Pilidi [1] (1968). In particular, he gave (24.30) for cp e Lp with 1 p min(a j 1 , a ; 1 ). Relations (24.30)-(24.32) for cp L;; were given by Skorikov [3] (1977). Notes to § 24.6. Partial and mixed Marchaud fractional derivatives (24.33)-(24.35) in the case of two variables appeared in Marchaud [1] (1927). Fractional integra-differentiation in a direction for functions of many variables was first introduced by Kipriyanov [1]-[6] (195�1967) in a form different from (24.38) and (24.39) (refer also to § 29.2, note 24.3). Notes to § 24.7. Theorem 24.4 in a sli ghtly weaker form was proved by Skorikov (3] (1977). Notes to § 24.8. The relations of this subsection are well known and are easily derived from the corresponding one-dimensional results. There are a number of papers which contain detailed proofs of these results, mostly in the case of two variables, as for example, Jain [1] {1970) where {24.56) is contained. The expressions in (24.52) and (24.53) with 2 may be found in Raina and Kiryakova [1] (1983). Notes to § 24.9. The space � . discussed here, was introduced and investigated by Lizorkin [1] (1963). It was considered also by Yoshinaga [1] (1964). Notes to § 24 .10. Weyl-type partial and mixed fractional derivatives appeared first in Bessonov [1] (1964). There, for functions J in Lp the author considered whether there exist in Lp certain partial or mixed derivatives, if there exist in Lp some other such derivatives (see § 29.2, note 24.5). The detailed and complete investigation of fractional differentiability of periodic functions of many variables was undertaken by Lizorkin and Nikol'skii [1] (1965) on the basis of Lp-spaces. Refer also to Nikol'skaya [1] (1974), where the problem of the existence of mixed Weyl fractional derivatives is related to the rate of convergence of partial sums of multiple Fourier series. Theorem 24.7 is a slight modification of theorem 4 from Lizorkin and Nikol'skii [1]. Notes to § 24.1 1 . The definitions (24 .65) and (24.66) seem not to have been introduced elsewhere. Theorem 24.8 may be considered as new.
1:2
E
< <
n=
§ 29. ADDITIONAL INFORMATION TO CHAPTER 5
581
JCOt
Notes to § 24. 12. The polypotentiaiS of type were introduced by Okikiolu [5) (1969), in a more general form with power weights, though. The connection between the polypotentials and·1{0t in the form {24.69) not been noted earlier. Notes to § 25.1 and 25 .2. Relation (25.1 1) is usually connected with the name of Bochner [1, p. 263 and 315] , whereas (25.14) and {25.14') for the Fourier transform of radial functions in terms of fractional integration were given by Leray [1] (1953). The space � defined in (25.16} firstly appeared in Semyanistyi [1] (1960}. Afterwards Lizorkin [1] {1963}, and also [5) {1969}, [8] (1972), gave a thorough investigation of spaces of such a kind, together with application in the theory of functional spaces with fractional smoothness. Since then such spaces have become objects of current research and are often used in mathematical literature. We remark that this particular space appeared also in Helgason [1 , p. 162] (1965). In the book of Helgason [2, p. 20, 59, 62] {1983), who it appears was unaware of Lizorkin's papers, the invariance of the space � relative to the Riesz potential operator was proved in connection with its applications in the theory of the Radon transform. Relation (25.25} is fomally known long ago, its justification in the sense of distributions on was given by Schwarts [1, t. 2, p. 1 14] (1951) . The result obtained there was different from (25.25) by the polynomial summand in the second line, which is orthogonal to � (see Remark 25.2). The interpretation of the Fourier transform of the function in the sense of distributions on � was suggested by Semyanistyi [1] (1960), and refer also to Lizorkin [8, p. 242] (1972). A potential with the kernel first appeared in the thesis of Frostman [1] (1935} devoted to the problem of the existence of the unique equilibrium potential of a compact set in This potential was introduced by Riesz, who was Frostman's teacher - Riesz [2] (1936} and [4) (1938}, and also [5] (1939} and [6] (1949}. (In pointing out the role of this outstanding mathematician we would like to mention here his obituary written by Girding [2] and the paper by Mikolas [9] with brief remarks on Riesz's works). We do not concern ourselves here with the connections of Riesz potentials with superharmonic functions but refer to Landkof [1] . The invariance of the space � defined in (25.16} relative to the Riesz potential was noted by Semyanistyi [1] (1960) and Lizorkin [1] (1963). Notes to § 25.3. Theorem 25.2 is due to Sobolev [1] (1938). The proof of necessity part follows Stein [2, p. 140]. The proof of Theorem 25.2 based on convexity theorems was later suggested by Thorin [1], 1948. An elementary argument immediately reducing the Sobolev theorem to the case n = 1 , was suggested by du Plessis [2] (1955} (see § 29.2, note 25.2). Muckenhoupt and Stein [1] ( 1965} gave the proof based on the interpolation of linear operators. This proof is given in Stein's book (2]. A rather simple proof which is well-suited for all n 2:: 1 was given by Hedberg [1] (1972). We refer also a development of Hedberg's idea in a more general situation in Meda [1]. We also mention Yoshikawa [1], who obtained the statement of the Sobolev theorem as a corollary of a general similar result for fractional powers of the form
JCOt
has
S(Rn )
lx l - a
l x l a -n
Rn .
A -a = r(a) - 1 J ta-l Tt /1 Tt lli=ln lx1p• (lx'l2 lx"I2 PI 2 x' = (xt, ... ,xm ) x" (xm+ t t ... ,xn ) 00
f
0
being a semigroup of operators satisfying certain assumptions . .
Theorem 25.3 was proved by Stein and Weiss [1] (1958). Similar theorems with weight or
+
with
and
=
were proved by
Nilrolaev [1], [2] (1973). Theorem 25.4 i s due t o Muckenhoupt and Wheeden [2] (1974). A simpler proof of this theorem can be found in Weiland [3). A simple proof and even in a more general case of the so called anisotropic Riesz potentials was given by Kolcilashvili and Gabidzashvili [1] (1985} - see also Kokilashvili [2, p. 36-54) (1985). There are also other generalizations (the so-called two-weighted ones and others) : see § 29.2, notes 25.7 and 25.8. The weak type estimate (25.46} for Riesz potentials was revealed by Zygmund [4) (1956). Theorem 25.5 was obtained by Valrulov [1,2] (1986). Theorem 25.6 was proved by Stein and Weiss [2, p. 57] (1960) in the case of the Poisson integral and by Johnson [1] (1973) in the case of the Gauss-Weierstrass integral. r-Jotes to § 25.4. The realization of Riesz differentiation in the form of the hypersingular integral (25.62) appeared first in Stein [1] {1961) in the case 0 2. The general case > 0 was considered by Lizorlcin [6] (1970) who introduced the hypersingular integrals
a
F-1 1elaF'P< a <
582
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(25.61) with a centered difference. Lizorkin [6) indeed defined an entity more general than (25.61 ) , which corresponds t o the �called anisotropic case. H e characterized the space of anisotropic Bessel potentials in terms of such anisotropic hypersingular integrals. The Fourier transform of a hypersingular integral, that is (25.63), was studied by Lizorkin (6, p. 82] in the case of centered differences, and by Samko (17, p. 1170) (1976) in the case of non-centered differences. Expressions (25.69) and (25.70) for inverting Riesz potentials f = [01 2(a/2] in the case of a non-centered difference was noted by Samko (17) (1976). The case 0 < a < 2 when one may take I = 1 , was known earlier - Stein [1] (1961) and Lizorkin [6, p. 85] (1970) . The way to avoid the phenomenon of the annihilation (26.18) by means of the choice I = a in the case of a non-centered difference was suggested by Samko (17) (1976) . Fractional differentiation in a given direction was considered by Wilmes [1 , 2], who defined it in Fourier transforms without realization in untransformed functions and gave (26.24). Notes to § 26.3. The �ults of this subsection, except Theorem 26.31, were obtained by Samko (17) (1976). The latter theorem was proved by Rubin [26] (1987). Notes to § 26.4. Hypersingular integrals with homogeneous characteristics appeared in the papers of Wheeden [1]-(4] (1967-1969). The characteristics can be found in Fisher [7) (1973). However, hypersingular integrals with a � 2 were introduced in all these papers via regularization (subtracting the Taylor sum), and not by taking finite differences. In the form (26.47) hypersingular integrals were considered in Samko (21] (1977), (26] (1978) and [28] (1980) in the case of homogeneous characteristics, and in Samko (18] (1976) and [20] (1977) in the case of a non-homogeneous characteristics. It needs to be said that Wheeden and Fisher investigated hypersingular integrals in the context of the theory of Bessel potentials, i.e. the hypersingular integral D01 J was considered to be in Lp together with j. The approach taken in the cited papers by Samko and in this book allows one to take f Lr and D01 f E Lp with different r and p which includes both the "Bessel" and "Riesz" situation. The classification of hypersingular integrals with a homogeneous characteristics both in type and in order of the used finite difference was suggested in Samko [26, p. 235] (1978) and [28, p. 198] (1980). Relations (26.55)-(26.57) were obtained by Samko in the same papers, the first of these expressions in the case 0 < a < 1 being proved by Radzhabov [1] {1974). The question about the simultaneous convergence of hypersingular integrals with different characteristics was considered by Nogin and Samko [1] (1981) and (3] (1982). The same question in the case of non-homogeneous characteristics was investigated by Nogin [1] (1980). The papers by Horvath [1] (1978) and Ortner [2] (1985) concerning problems of analytic continuation and compositions of convolutions with ''pseudo-functions" k ..\ E R1 , are also connected in a sense with the context of § 26.4. Notes to §§ 26.5 and 26.6. Theorem 26.5 was proved in Samko [26] (1978). The answer to question (26.74) in the form (26.81) and (26.82) and Theorem 26.6 was obtained in Samko [28, p. 205-207) (1980) . Notes to § 26.7. The investigation of the space /01(Lp) presented here follows the papers by Samko (17), [18], (1976), [20, 23] (1977) and [31] (1984) and include consideration in the sense of distributions for p � n/a. The space I01(Lp) had arisen earlier, but without constructive characterization in terms of convergence of hypersingular integrals. We refer to, for example, Maz'ya and Havin [1] (1972), where the authors showed that the closure of C0(Rn ) in the norm IIF- 1 I(I01.1"
O(x, y)
E
( 'Jiy) lxl� ,
§ 29. ADDITIONAL INFORMATION TO CHAPTER 5
583
of order less than a, which are acquired Wlder this closure. In particular, in Samko's papers there are obtained both Lemma 26.5 {[31 , p. 78]), the characterization of the space Ia (Lp) presented in Theorem 26.8 ([17), [18], (1976) and [31 , § 13], (1984)}, Theorem 26.9 ([23] (1977)), and the estimates (26.98}-(26.99) ([17) (1976)). The spaces Lp,r (Rn ) arose as a natural generalization of the spaces of Riesz and Bessel potentials coinciding with the former if r = np/(n - ap), and with the latter if r = p. They were investigated in Samko [17] (1976) and [20) (1977). Theorem 26.10 was proved in Rubin [23] , [26], (198�1987), (26.971 ) with different p and q was proved in Samko [34]-[35] (1990) and the statements (26.98) and (26.981 } were given in Samko [17) (1976). , Notes to § 27.1. The fractional powers (E - /i) -a/2 became named Bessel potentials after the papen by Aronzajn and Smith [1) (1961), Calderon [1] (1961), Aronzajn, Mulla and Szeptycki [1) (1963), Aronzajn [1) (1965) and Adams, Aronzajn and Smith [1] (1967). The Bessel kernel as the Fourier original (in the sense of distributions) of the fWlction (1 + lxl) -a/ 2 was considered in L. Schwartz [1 , vol. 2, p. 116] (1951 ). Notes to § 27.2. Theorem 27.1 and the modification � a n - 2 admits an analytic continuation to the whole complex plane except for a finite number of poles, and this was shown by Riesz. Another proof of analytic continuation was given by Fremberg [1) (1945), and also [2] (1946). We further point out the method of an analytic continuation given in the book by Baker and Copson [1 , p. 60] (1950) in the case n = 3. Nozaki [1] (1964) considered the Riesz hyperbolic potential (28.19) in the general case (28.3} with an arbitrary p = 1, 2, . . . , n instead of p = 1. He evaluated the nonnalizing constant (28.20) coJTesponding to an arbitrary p and verified the semigroup properties (28.22) and the analytic continuability in a. We refer to the case p ':# 1 in the paper by Trione [2] (1987) or its reprint [3) (1988) . Relation (28.28) was established by Schwartz (1 , t. 2, p. 120) (1961) . In connection with the calculation of the Fourier (Laplace) transform of the power of the Lorentz distance, we mention the book by Vladimirov [1 , § 30) (1963), and the book by Trione (1] (1980), the latter .
t
t
D
D is
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
584
being specially devoted to the Fourier-Laplace transfonn of functions depending on the Lorentz distance. The statement of Theorem 28.1 should be considered as known, but the authors cannot find the corresponding reference. The realization of fractional powers of differential operators is closely connected with the investigation of "generalized" functions IP(x)l\ where P(x) is a polynomial. In this connection we can refer to the following papers: Gel'fand and Graev [1] (1955), Fedoryuk (1] (1959), Bresters [1] (1969), Bernstein and Gel'fand [1] (1969), Atiyah [1] (1970) and Palamodov [1] (1980), and also the book by Gel'fand and Shilov [2, ch. Ill, § 4) , which contains some results for arbitrary functions raised to the power �. Notes to § 28.2. The parabolic potentials H011p and 1{011(), - (28.39) and (28.45) - were introduced by .Jones [1] (1968) in connection with investigations of the heat-type equations. The further study of these potentials and of the spaces H01(Lp) and 1f.01(Lp) was undertaken by Sampson [1] (1968), Bagby (1] (1971) and (2] (1974), Gopala Rao [1] (1977) and (2] (1978), Chanillo [1] (1981), Nogin [2] (1981) and (5) (1982) and Nogin and Rubin [1], [2] (1985) and (4] , (5] (1986). The proof of Theorem 28.2 can be found in Gopala Rao (1] (1977). Notes to § 28.3. We followed here Nogin (2] (1981) and Nogin and Rubin [1] (1985) and [4] (1986). Notes to § 28.4. The Abel-type integral equation (28.65) in the particular case n = 2,
I � �II
and = "2 = 1/2 was first solved by Mihlin in 1940 in connection with the 1 problems arising in investigations of wave reflection from a rectilinear boundary. For this we refer to Mihlin [1, p. 48) and references in Preobrazhenskii (1 , p. 9], and also applications of equations of such a type to problems of supersonic flow over space angles in Fedosov [1]). The solution of (28.65) in the case of an arbitrary natural number n and c =
A=
"1
k -1
� (0, . . . , 0, 1 , 0, . . . , 0) was obtained in Kilbas and Vu Kim Tuan (1] (1982). The other results of this subsection, in particular, the definition and the properties of the pyramidal analogues of mixed functional integrals and derivatives are published here for the first time.
29.2. Survey of other results (relating to § § 24-28} .. _ 1(), at = Raina (3] dealt with the evaluation of the mixed fractional integrals , atn ) , of functions of the form IP(t) = exp(-Epiti)P(Eti), P being a polynomial, in terms of some special functions. This is an extension of the one-dimensional result of H.M. Srivastava [4], (see § 9.2, note 5.5 in this connection). His result was corrected and further developed by R. Srivastava [1]. We observe also that the mixed fractional integrals (24.6) were used by Kosdunieder [1],
1� .
24.1.
(atb
• • •
F�n)
[2] to derive some properties of the Lauricella hypergeometric function of n variables Prudnikov, Brychlrov and Marichev [2, p. 745) . . 24.2. Erdelyi-Kober-type modifications of fractional integration in the case of two variables l: ® were given by Verma [1] and Mourya (1]. Namely, the operators Ig · l l: ® Ig+ · l '1 ' + .� , .� , ,, (in the notation of §§ 18.1 and 24.3) and some others were introduced. Mourya [1] established the main properties of these operators, and gave applications to some special functions. We observe that there is a misprint on p. 173 and errors on p. 175 of this paper. Verma (1] considered the two-dimensional Mellin transform of mixed Erdelyi-Kober fractional integrals. Various compositions of two operators of such a type were studied by Raina (2). Mixed Erdelyi-Kober fractional integrals were used in Verma [2], [3] to study properties of the two-dimensional integral transform generalizing the two-dimensional Hankel transfonn. Kaul (1] introduced modifications of generalized fractional integration of the Saxena-type (23.5) and (23.6) for the case of two variables, and gave the expression for their Mellin transform J
J
1� . 1 J
1� . 1
§ 29. ADDITIONAL INFORMATION TO CHAPTER
5
585
and the expression of fractional integration by parts. These results were extended to the case of more general multidimensional operators by Saxena and Modi [1] and Mathur and Krishna [1].
24.3. A number of papers by Kipriyanov [1)-[5) dealt with the fractional derivative /�a) (z) , < a < 1, of a function / ( z) at the point z nn in the direction from the point a nn . The initial definition of such a derivative by Kipriyanov says that /�a) (z) is a function satisfying the
E
0
E
relation
(29.1) where e is the unit vector in the direction from
Kipriyanov
a to i.e. e = (z - a)/lz - a!. As was shown by [2]-[3] this definition yields the following relation for /�a) (z) :
fa(a) (z) - r(1 a- a)
(cf.
lz -a l
J 0
z,
(
)
/ (z) - f+(z - ee) 1 - e -. n- 1 + r(n ) /(z) - /(a) (29.2) el a l al r (n - a) l z - al 1 +a -n _
z -
(24.3811)). In the papers [1)-[5] Kipriyanov studied various properties of such fractional derivatives and investigated spaces of functions in the domain n c nn having fractional derivatives in all directions. ReJations of these spaces to the Sobolev spaces wt(n) were shown. We mention some of these results, e.g. [3].
/�a) (z) exists, is continuous and bounded for ( a) E 0 a) J sup 1/� (z) ld:c < oo and a > 1/p, then /(z) E H; -1 /P , p > 1. o aen 1.
If
x,
2. If
0 < a < ,\ � 1 and continuous in respect to ( z, a).
x
0, then
/ (z) E Ha ( O) ; if
/ (x) E H� (O) , then Ji )( ) exists for all (x,a) E 0 a
x
x
0
and is
c(a) (O) be the space of functions continuous on 0 , such that /�a) (z) is continuous with respect to (z, a) E 0 0 and let 11/ll c (O = 11 / ll + (z,amax eOxO 1/�a) (z) l. Then ) ) 11/ llc (O) � All /l l w�(O) if l > nfp and 0 < a < min (l , l - n/p), wt being Sobolev space of an integer order I. 4. A similar result holds for the spaces L�a) (O) with the nonn II/IIL9 (n + 11 /�a) II L.,(OxO · ) ) Moreover, imbedding of the space wt into c< a) and L�a) is compact under the usual assumptions 3.
Let
c
X
- Kipriyanov (Sa]. We note that the operator
also
� aP /(z)
=
p
J (p2 - -r2)-a /(a + -rw)-rn-l d-r w, lwl p (29.3)l a l a
1 d pn -1-a r ( 1 - a) dp
0
of fractional differentiation in a given direction = 1, with was investigated by Kipriyanov (7]. He showed that the operator + �pa /(z) = 2p1 a
r(1 - a )
P
J 0
=
x -
,
can
being a fixed point, be transfonned to
( p ) n -1 d-r r(n-r(n)a) /p(z)a
J(x) - /(a + -rw) !:.
(p2 - -r2 ) 1 +a
(29.3)
+
'
p
=
lz - al,
586
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
(cf. (29.2) and (24.3811) ) and the operator inverse to (29.3)
can
be given in the form
Moreover, the operator �� was used in this paper to construct spaces of the L a ) (O)-type of fractionally differentiable functions, which are compactly imbedded into the Sobolev spaces wt (O) under the appropriate assumptions on p, l, q and a. 24:.4:. Liouville fractional partial and mixed derivatives (24.15) , treated in the sense of distributions: (D +. . J, tp) = (f, D� . _ IP) , tp � with � being the space (25.461), were ' introduced by Lizorkin [ . Lizorkin and Nikol'skii [1] introduced the spaces s; L(Rn ) = s; (Rn ) with dominating mixed derivatives. The space s; (Rn), a = (ab . . . , an ) , consists of functions J Lp (R") with the mixed derivative V+. .. . + J Lp (Rn ) and with all support derivatives in
�
.
E
.
.
�
E E
.
Lp (Rn ). The •upport deri'llati'lle for + J is the derivative ... + ! where a is obtained from a = (a t , . . . , an ) by replacing some of a1c by 0. It was shown in this paper that
Vt
..
s; = Ga(Lp ),
Vi
a > 0, 1 < p < oo ,
(29.4)
where aa (Lp ) is the space (24.74) of polypotentials (24.72). In the paper by Nikol'ska.ya (1] the character of the approximation of a function J s; (Rn ) by its partial Fourier sums was investigated. Brychkov (1] considered the spaces s; for all a Rn, defined as distributions, and proved (29.4) under the appropriate interpretation of the left- and right-hand sides. The concept of smoothness order of the generalized functions in s; with respect to a part of variables was introduced in this paper (see also Brychkov [2] in this connection) , and the relation of this order to the existence of the trace of a generalized function on hyperplane was shown. 24:.5. Let f(x) be a 211"-periodic function of many variables and let v ( a ) J, a = (a t ' . . . , a n ) be the Weyl fractional derivative (24.60) , (24.62). The following statement was proved by Bessonov [1]. Let J Lp (�), � = {x : 0 � x; < 21r} , 1 < p < oo. H there exist v ( a ) J Lp (�) and v (f1) J Lp (�), ai � 0, f3i � O, then Lp(�) for 'Y = Oa + ( 1 - 0){3, 0 5 0 5 1. A certain generalization of this statement was obtained by Nikol'skii [5], p. 238. .. _ J, {3 = 24:.6. A similar question on the existence of some fractional derivatives i (f3t , . . . , f3n ) when there exist some other such derivatives of certain orders a = ( a , . . . , a� ) was considered in a general form by Magaril-D'yaev [1] in the non-periodic case in the general context of the spaces Lp(Rn ) with mixed norm. This is the so-called problem of the intermediate derivative. This question is connected with the multiplicative inequalities for fractional derivatives, the validity of which was completely investigated in this paper. The development of the results for the problem of intermediate derivative may be seen in the paper [2] by the same author, where it was given in certain generalized terms. ' 24:.7. Fractional differentiation on the manifolds M = rn X Ir"" i.e. on the product of a certain number of circles and axes, was considered by Magaril-D'yaev and Tikhomirov [2] in the presentation of some questions of the hannonic analysis on such manifolds. It was defined by using the analogue of the Lizorkin-type space � adapted to circles and axes forming M. The problem of the intermediate derivative, Bernstein-Nikol'skii and Favard inequalities for fractional derivatives and other questions were considered in the mixed norm spaces Lp(M). 24:.8. The Bernstein-type theorem for mixed fractional derivatives of a function f(x, y) oftwo variables was already given by Montel [1, p. 187], in the following form: if lf(x, y) - Pn , m (x, y)l 5 + Bn -6 , lxl < 1, IYI < 1, 'Y > 0, 6 > 0, where Pn , m (x, y) is an algebraic polynomial, then f(x, y) has all mixed fractional derivatives of order (a, {3) such that a/"'f + {3/6 < 1, a > 0, {3 > 0. 24:.9 . The connection between the existence in Lp of the mixed Weyl fractional derivative
E E
E
E
E
V�'YJj E
V�
.
i
Am-'Y
V�.,t 'a2) J of a function f(xl , x2 ) of two variables, and the behaviour of its partial Fourier sums and mixed continuity modulus was considered by Esmaganbetov [2].
§ 29. ADDITIONAL INFORMATION TO CHAPTER
5
587
24.10. Let a function f(x) , considered in the domain 0 C Rn , be the restriction onto 0 of a function g(x) which in the whole space Rn belongs to the Bessel potential space Lp (� ) (§ 27). The characterization of such functions f(x) in tenns of partial Riemann-Liouville fractional derivatives, or more generally, Chen derivatives {18.18), was noted by Skorikov [2] (see § 23.2, note 18.14 in the case n = 1). Some considerations in similar situations in the case n = 2 were developed by Biacino and Miserendino [1]-[3], Biachino, Di Giorgio and Miserendino [1] and Biachino [1]. We also mention Miserendino [1], [2] where for functions in L�(O) with an integer a and a parallelepiped 0 in Rn the author investigated the problem of the existence of mixed fractional derivatives at the points of the boundary 80 and their belonging to the space L 2 (80). 24.11. The definition of Griinwald-Letnikov differentiation given in § 24.11 for functions defined in the whole space Rn , can be adapted to functions defined in a region. This is realized in a similar fashion to § 20.4, where Griinwald-Letnikov fractional differentiation on the finite interval was defined. In the case of two variables the definition, generalizing {20.42), has the fonn
{29.5) where a = (a 2 , a 2 ), a = (a t , a 2 ) and x 1 � a lt x2 � a2. Such a definition was used by Krasnov [1], who gave some properties of the fractional derivatives {29.5). The development of this definition for the case of analytic functions given in the region G C � may be found in Krasnov and Foht [1]. Such an approach to fractional derivatives of functions of many variables was considered in this paper in connection with the derivation of certain integral estimates for solutions of elliptic differential equations. This was earlier studied in Foht and Krasnov [1] in the case of the Laplace equation. In the latter paper the definition of fractional differentiation of analytic functions was used, reduced to fractional differentiation of monomials xi , j = (it , . . . ,jn ), but the authors discarded all the monomials with ik < a if only for one k = 1, . . . , n. a 24.12. Mixed fractional differentiation !:5\ !,..01 .., 5". BZ•"+l1 with the fractional power
u;
B::::
u..,l
• • •
u
-Grs t/v,
+
of the Bessel differential.operator By = + k > 0, was used by Brodskii to define certain function spaces with fractional s�oothnes� in the half-space. 24.13. The estimate
O
[1], [2]
211" , p
-
for fractional differentiation {24.48' ) in the given direction w and the function /(x) of the exponential type p was given by Wilmes [1], [2] (see also § 23.2, note 20.5). The case r = 1r /p, 0 < a < 1, yields the Bernstein inequality with the constant 21 - a . 25.1 . The Lizorkin space {25.16) consists of Schwartzian functions J S with the Fourier transform vanishing at the origin. The similar space � , adapted to partial fractional derivatives and integrals (see § 24.9) is related to the vanishing of the Fourier transfonns on the coordinate hyperplanes. In some other problems a necessity arises to deal with the Lizorkin-type space �V of Schwartzian functions such that (l> k cp)(x) = 0, lkl = 0, 1, 2, . . . , x V, where V is a given closed set in � . Such spaces were investigated by Samko [25], [29], who gave their characterization in the case when V is a cone in Rn and proved the denseness of �V in Lp(Rn ) if m(V) = 0. This denseness was obtained for an arbitrary such set V if p � 2, and for a certain class of such sets V, which were called quasibroken in [29), if 1 < p < 2. Later on it was shown that V with m(V) = 0 may be arbitrary in the case 1 < p < 2 also (Samko [36); see also the case of mixed nonn spaces
E
E
588
CHAPTER
5.
INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The denseness of �V in Lp(Rn ), 1 < p < oo, for V an origin or a union of the coordinate hyperplanes was earlier established by Lizorkin [5], [8]. The space � � corresponding to the case V = {0}, is dense in Lp(R1 ; lxP') provided that 1 < p < oo and 'Y > - 1 or p = 1 and 'Y > - 1, but is not an integer, we refer to Muckenhoupt, Wheeden and Wo-Sang Young [1, Theorem 6.10), where the space Soo was considered. This consists of Schwartzian functions with Fourier transfonns identically equal to zero near the origin, which is somewhat narrower than �{0} · We observe that in Samko [29], [36) the denseness of similar spaces narrower than �V was indeed shown. 25.2. du Plessis (2) observed that the Sobolev theorem is reduced to the case n = 1
L;;(Rn ) in [36]).
immediately, since
n n lxln � II lx; I , or more precisely, lxln � n n/ 2 II lx; I · j =l j =l
This follows from the
fact that the geometric mean is dominated by the arithmetic one. So
,r.p(y),dy < n I lxr.p(y)dy - Y in -a - I II lx · y · l l -a/n
R•
R"
j =l
J
_
J
an d then the one-dimensional Hardy-Littlewood theorem 5.3, being applied to each variable, gives the desired estimate for Ilia 'PIIq , = - i" . We note that the Sobolev inequality Ilia fllq � cll fllp was shown by Lieb [1] to admit a maximizing function f such that equality holds. A similar function does not exist for the Bessel potential - see Lieb [1, p. 352]. The sharp constant c was explicitly evaluated by Lieb (1] in the case q = p 1 or p = 2 or q = 2. 25.3. In the limiting case p = n f a the operator Ia is not defined as an absolutely convergent integral on the whole space Lp(Rn ), but IIIar.pi iBMO � cllr.pllp on the set which is dense in Lp , where BMO is the space of functions with bounded mean oscillation. We refer to § 4.2 (note 3.3) and § 17.2 (note 13.1) in the one-dimensional case, where the references to the multidimensional case are also given. Harboure, Macias and Segovia (2] obtained weighted estimates for the Riesz potential in the case p = n f a . The paper by Mizuta (2] is also relevant. It concerns the so called total differentiability of Riesz potentials of functions in Lp, p = n / a with an integer a . We mention the paper by Stricb.artz [2) as well, who studied the Riesz ( and Bessel) potentials of functions in BMO. Chanillo (2) considered commutators b(x)(Ia f)(x) - Ia(bf)(x), f E Lp, with b E BMO. In this connection see also Komori (1] for the estimate
t ;
1
1 1 q r
Ol
- = - - - - n p
In the case and only if
[2).
n = 1 the commutator bDa - nab was shown by Murray (1] to be bounded in L2 if nab E BMO . The discussion of such iterated commutators may be found in Murray
25.4. The analogue of the Sobolev Theorem 25.2 for the mixed norm space L;;(Rn ) as in the definition in §§ 24.4 and 24.12, was given by Benedek and Panzone [1]. Ia i1 bounded from L;;(Rn ) into Lq-(R'l ), p = (pl , . . , Pn ), q = (qb . . . , qn ), if 1 < Pi < n /a a.nd 1 /qi = 1 /Pi - afn . A more general assertion in this direction was obtained by Lizorkin [7), who considered the wide class of conv@lution operators, including in parti�ular the a.ni1otropic Rieu potential• .
I [p(x - y)]a-n r.p(y)dy
(29.6)
R•
where p(x) is a non-negative a-homogeneous function of degree 1 (i.e. p(t4l Xl J , t4"xn ) = tp(x) for t > 0 with a; > 0, L a; = n ) , non-vanishing as lxl = 1. Lizorkin proved the boundedness of • • •
589 1 < Pi � qi < a = i=2:n l ai (1/Pi - 1/qi ), [1 , 32] 112 p(x) = (Jt=l lxj 12a; )
§ 29. ADDITIONAL INFORMATION TO CHAPTER the potential (29.6) from
':#
Pn qn ;
Lp(R")
into
Lq(R")
with
The book by Besov, D'in and Nikol'skii
5
oo,
is relevant here. A similar statement
p.
in a more general
for the anisotropic Riesz potential with the kernel
situation of the weighted mixed norm spaces and on subspaces of different dimensions was proved by Kokilashvili In this connection we observe that the Sobolev type different dimensions theorems for Riesz potentials were first obtained by D'in Besov, D'in and Nikol'skii p. give such a theorem in the general anisotropic case. We mention also Adams and Bagby where some cases of estimates for the usual (isotropic) Riesz potential operators from Lp into Lf are specified. 25.5. The continuity of potential type integrals
(1].
(2].
[1],
[1, 34]
Ig�P = 1 lx -tla-n �P(t)dt, E Lp(O), E R" p > n/a [2, 48). (3, 256]), if < a - nfp < 1, if a - nfp = 1, (29.7) if a - n/p > 1, 1/p + 1/p' 1. ,i-n HJ\ case 1 1.6). a A(x, 11) x -111 x, l [2,E Lt196]. = (29.7).p = fP (R") n .Lp(R"), 1 < p < n/p < a < 1 + nfp n/p (R") = {g : g(x + h) - g(x) o(lh.la -n/p ) a l �P E h. � x lh.l - (1] [1]). (9]. (29. 7 ) Ig�P case a - n/p > 1 Ig�P E Ha-n/P(O) if a - nfp '/: 1, 2, Ig�P E Ha-n/p ,l/p' if a - n/pl(i�P= 1, 2, [12]). 25.6. In a l lx l ) 1 �P( 1 f d I I y 1 1 (I: �P)(x) � �,n ("'...) lx - JIYin-a = "Y_n (a)_ Pa- u ( ) fP(p)dp O < a � , IYI< a U(..\) = i(n"'j;) -Fl (T, 1 - y; ;.; ..\2 ) (17]). 25.3 has R" Ba
n
in the case of a bounded region 0 Moreover (Sobolev p.
and
was first noted by Sobolev
p.
0
The definition of the space (0) in the one-dimensional is in § (Definition The case of the kernel with the first factor Holderian in but under the particular value oo, was treated by Mihlin p. There is a similar result for 0 R" , close to The Riesz potential la�P with oo, and =
is a Holderian function: = uniformly in as 0} (du Plessis This result was extended to certain convolution operators by Cotlar and Panzone and to more general operators by Kilbas The estimate for the integrals in the is specified as follows: . • . •
. . . and (0) is a corollary of the generalized Holderness of the potentials more general than the c ase of radial functions the expression
'
p
o
This
(Kilbas
oo
holds, where (Rubin Theorem the essential specification for radial functions. Let be the ball in centered at the origin and with radius a, 0 oo . The following theorem holds (Karapetyants and Rubin and Rubin in the general case ) .
Let1/p�P(, Iqzl=) Ep(1Lp(-Ba;mp)lxl" ),The1 n< IpI <�PI < n(p -�1)cl, lfPnI � 1, < a' <withn, m < m a, . : L.(B.; I�I.,o ) L.(B. ;I�I.,) = q(mn- a + vfp) if > ap - n a nd = q(mn + np) if � ap - n, > O, e�cep t the ca•e � ap - n if a =
Theorem 29.1. 0 � �
v
vo
v
0
oo , v
oo.
vo
e-
v
e
590
CHAPTER
5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
The JWOO/ of this theorem is based on the representation which is of interest itself of the Riesz potential with a radial density in terms of one-dimensional fractional integration in the radial variable: namely
1�.{.2 and t:!�
are the Riemann-Liouville fractional integration operators given by (2.17) Rubin (17, 22) is relevant where the inversion of potentials with a radial density and a ch81"acterization of the range were given. 25.7. In the papers by Kokilashvili and Gabidzashvili [1) and Gabidzashvili [4] and also
where
(2.18).
�� b:�:::::�: :=::.:):i=:ti(.b�;.,:)�;.-:: 4
j = 1, . . . , n ) . Applications to imbedding theorems for general weighted Liouville spaces were also given. The limiting case (p') - 1 {2: cri ) = n - a may be found in Gabidzashvili [5). Gabidzashvili [1)-[3] extended Theorem 25.4 to certain potential type operators T01 generalizing (29.6) in the apace of homogeneous type, and proved the theorem on two-weighted estimate for T01• He also gave Koosis-type theorems which allow one to have the weight by a given weight and vice versa. A more general situation into so that T01 is bounded from of homogeneous measure apace can be found in Kokilashvili and Kufner [1] and [2], where the conditions on the measure were in p81"ticul81" found for the boundedness within �the frames of Lebesgue, Lorentz and Orlicz spaces. 25.8. The weighted Theorems 25.3 and 25.4 were extended to more general cases involving P2 ), including the weak type estimates. the two-weighted estimates from P l ) into These questions have recently been developed extensively and essentially (f81" promoted). The two-weight problem of weak type inequalities for the Riesz potential was first solved in Sawyer [1], and more effectively by Gabidzashvili [1]-[3]. As for norm inequalities, the two-weight problem was completely solved by Sawyer [2], and in other terms by Gabidzashvili and Kokilashvili [10)-[11]. For various other investigations and generalizations we refer to the survey by Dyn'kin and Osilenker [1], the book by Kokilashvili [2] and his papers [3]-[9] and the papers by Gatto, Gutierres and Wheeden [1], H81"boure, Macias and Segovia [1], [2], Stromberg and Wheeden [1], [2], Heinig [1], [2], Andersen [2], Chanillo and Wheeden [1], Ruiz and Torrea [1], Gabidzashvili [4)-[6], Kokilashvili and Gabidzashvili [2], Gabidzashvili and Kokilashvili [1], Gabidzashvili, Genebashvili and Kokilashvili [1] and Fofana [1]. The reader can also find other references in these papers. The Muckenhoupt-Wheeden weight condition (25.41) is not always easily checked, so Abdullaev [1) gave the weighted boundedness conditions in the case when the set of singularities of the weight functions Pl and P2 satisfied certain assumptions and P l and P2 were functions of the distance from this set. 25.9. Mapping properties of the operator within the frames of the Hardy spaces (nn ) , in the case 1 < p < oo up t o the equivalence of 0 < p < + oo , which coincide with norma, were investigated by Stein and Weiss [2]. Their results were developed by Krantz [1], for operators more general than The case of certain generalized H81"dy spaces can be found in Taibleson and Weiaa [1] and Youngsheng Hang [1]. Two-weighted estimates of the Riesz potentials in the weighted Hardy space were given by Stromberg and Wheeden [1], [2] and Gatto, Gutierres and Wheeden [2]. 25.10. Sobolev's theorem 25.2 was extended to the generalized Riesz potentials
J =l
-n
p,
X
Lp(X;p) Lq (X;r)
Lq (Rn ;
Lp(R'I;
101•
Lp(Rn )
101
HP(R'I; p)
Ot = loll'
r
O(y) VJ(:.r: - y) dy I IYina
R•
HP
§ 29. ADDITIONAL INFORMATION TO CHAPTER 5
E
O(y) = O(y/IYI) Ln/ (n -cr ) (Sn - d by Muckenhoupt [1]. (see § 4.2, note 3.2) was extended to the n-dimensional case:
with
In
591
this paper the limiting cases
p = n /01 and p = 1 were also considered. In particular, the one-dimensional Zygmund-Flett result
R".
where B is a ball in Another proof for 0 :: 1 in the cases p = n/01 and p = 1 was given by Hedberg [1], who also proved the multiplicative inequality
where 0 < 01 < n, 0 < 8 < 1, 1 < p < oo, p < q � oo, 1 /r = (1 - 8)/p + 8 / q . A more general case of O(y) Lr(Sn - d with r � n/(n - Ot) was considered by Muckenhoupt and Wheeden [1], who proved the estimate ll lxi�'(I{l (R" ) and of fractional order 01 = 01 1 , , Otn ), 01j � 0, generalizing the Liouville spaces L�a ) and consisting of functions J which have partial Liouville derivatives of order Otj in each variable belonging to respectively), were investigated by Chuvenkov [1). 25.13. Connections between Besov spaces (their definition may be found in Nikol'skii (6] or Besov, D'in and Nikol'skii (1]) and fractional differentiation was revealed by Lizorkin [4), (9] who had shown that Bessel fractional integro-differentiation acr - ( 27.8 ) - realizes the isomorphism between Besov spaces: Ga(B; , s> = B;J r , 1 < p < oo , 1 � 8 � oo , 1 � r < oo. Herz [1, p. 315] showed that a similar statement is true for the Riesz fractional integration: JCW(Ar 9) = A cr� r r .., is defined as the completion under the appropriate interpretation of the Besov space. That is Ap,u in the Besov norm of the space of infinitely differentiable functions, the Fourier images of which have a compact support not containing the origin. The obvious relation to Lizorkin space � is evidently seen here. 25.14. The Riesz potential arises naturally in the theory of Radon transform - Helgason [2, p. 20, 29 etc]. We also observe that Bredimas [6) used one-dimensional fractional integration in the problem of inverting the spherical Radon transform. 25.15. The modification of the unilateral Riesz potentials (25.71) adapted to the case of a
E
( E LM (EM), LM (EM • • •
(
E�) (R") (R")
p,
p,
592
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
half-space, namely
x E Rf., [01cp,= Ecp LE Lp(R"), p Lp(Rf_)
and coinciding with the Riemann-Liouville integral (5.1) if n 1, was considered by Rubin [20]. In particular, it was clarified whether the Riesz potential coincides with the unilateral potential t/J in the half-space 0, where t/J also. This is the case, if < 1/p, and then one can explicitly construct the operator A bounded in such that 0. H 1/p, in order that such an equality is satisfied, it is that 8; J(x' ,O ) - 0 1 necessal")' and suffiaent 1 /p] under the · J - 0, 1, . . . [ 8xJ,. appropriate interpretation of the trace.
18+ a(la cp)(x) = (10 Acp)(x), + Xn >
Xn >
a>
, x - (x1, ... , xn ) , •
,a-
SimilM modifications
R"
of unilateral Riesz potentials, connected with a sphere in were introduced in Rubin [19], [22], where the following results were obtained: i) the semigroup property; ii) inversion formulae given below in the case 0 < < 1 for simplicity, namely
a acn-1 (B+a ) - 1 I = r(n/2) l(x) r(n/2 - a) lx l 2 01 r(1 - a) ( B� ) -1 I = aen - 1 f r( 1 - a) +
f ( lxl2 l-(x)IYI2-)01l(ylx)- Y l" y, d
IY I < Ix l
IY I > Ixl
iii) the representation of the Riesz potential /01
d.
as
a composition
(25.76), and also Rubin [28).
The radial limits of the Riesz potential were studied by Mizuta [1], who showed that with � and + {3 < n, then there exists a Borel set C such that lim (l01cp)( r u) 0, with the Riesz capacity 0, for each u r-oo This question in a more general setting is discussed in Kurokawa and Mizuta [1].
if
cp E25.16. Lp(R"; lxl-11) c p(E) {3 0, p > 1 ap = 0 a
=
a>
E ESn-1Sn\E.-1
25.17. Kurokawa [1] proved that the Riesz (and Bessel) potentials approximate the identity operator as - +O both almost everywhere and in the Lp-norm. Namely, i) let 1 � p < oo converges to at each Lebesgue point of I; ii) let 1 < q < p and and then then n converges to I in the -norm.
a E Lp(R"), I I E Lp Lq , 1a I 101I
I Lp
25.18. The periodic analogue of the Riesz potential, i.e. the operator generated by the
§ 29. ADDITIONAL INFORMATION TO CHAPTER 5
593
multiple Fourier expansion of the form rx c.p -
L l.pJ: I k l -a eib
Jk J;tO
k (kb ... , kn ),
,n,
where = 0t > 0, 0 � Xi < 2'11' , i = 1, . . . was considered by Cheng Min-teh and Chen Yung-ho (1] (1956), [2] (1957) and Wainger (1] (1965). The one-dimensional case may be seen in § 19.3. Certain developments can be also found in Cheng Min-teh and Deng Dong-gao [1]
(1979).
25.19. The Riesz-type generalization
.7°'1' �=
=
hn (at}] -1 I c.p(x Y)l�la-n dy,
(ln Yl t ··· • ln Yn ),
R"+
Rf. =
o
{y
E
R" :
Yl > 0, . . . , yn
>
0}
of Hadamard's constructions (18.42)-(18.44) was considered by Emgusheva and Nogin (1] together with the corresponding Riesz-type fractional derivatives in Rf. which are invariant relative to dilatations. 25.20. Let be locally Lebesgue integrable in R" and let be the Riesz = [0 potential, 0 < Ot < n. The inequality
f(x)
1 h"
g(x) J I lg(x) -g(xo)ldx � ch I IX - XQ In-IJa(x)l(hdx IX - xo I) +
Rn
1� -�ol< h
holds, which is obtained by direct estimations, c depending only on Ot and n (Samko). This easily yields the convergence of the left-hand side to zero as h - +O nnder the sole assumption that is finite. This proves Love's conjecture (see § 17.2, note 12.6). Love himself developed proofs for the case = 2, but they were lengthy and he noted the fact that this conjecture can also be proved by means of Theorem 1.11 from Landkof [1]. 26.1. In connection with Theorem 26.3 we observe that the truncated Riesz derivative D� J of the function J 1
(Ia iJI) (xo)
n E I0(Lp ),
n/at,
which yields the almost everywhere convergence. Here
�, (x)
and Ci (i = bypersingular
=
t- n
Lp (R"),
1•
I lc.p(x- y) - c.p(x)ldy, Wt (x)
l!l l
is the same as in (26.39), 1
=
E l • -a
I to-l· - 1 �t (x)dt, £
1, 2,3,4) are absolute constant. The almost everywhere convergence
integrals D{l/ with a characteristic investigated by Nogin [8], [11].
O(y) not necessarily homogeneous wasthe of
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
594
(D )(x) lxi) -N1 , I('Di )(x)
O(y/IYI)
26.2. The hypersingular integral {} J with a homogeneous characteristic of functions sufficiently smooth and vanishing at infinity, admits the following estimates. Let � c( 1 + for i = l with J l � c(1 + l > a and N1 > a, N'}. > n. Then
f(x) E C1(Rn ),
if
, 1/(x) l
lxi) -N� , I i
O(u) E Ll (Sn_l ), and
I(D{lf)(x) l � c(1 lxl) - min(a+N. ,N� ,n+a) if O(u) is bounded Samko [26], [31, p. 89]). The continuity modulus, of fractional order in general, w-y 5) = sup I A'Y J l x , lhl < cS X = Lp(Rn ), 1 � < or X = BC(Rn ), of the hypersingular integral D{l/ is estimated as follows: UJN(Dcr / 6) <- cb(5) w-ys(Ja, S) + c f wt>,l(f,t) +a a(t)dt' +
(
26.3.
p
(!,
oo,
- I
where
0
a(t) = J O(tu)du, t S•- •
6
t
>
0
0, b(5) =
J1 t-1 -aa(6t)dt, 00
-y
>
A
0, 0 < a < � l. Here l is the
a(t)
order of the difference, defining D 0 J. In the case 0 :: const both and b(5) are constants and may be chosen any such that a (Samko and Yakubov (4]; the case of integer -y and A and X = can be found in paper [2] by the same authors). 26.4. In the case of the characteristic the equal to the spherical harmonic hypersingular integral admits the simple expression in terms of n
A C
A>
D{l/
O(y') Dy"' J = AYm ('D)Da-m J if a � Dy,J = AYm (V)Ia-m J if a �
m,
a-m f:
Ym (y')
m,
A
where is the constant depending on a, n, m and on the type of the hypersingular integral { Samko [26], [31, p. 90]). Compare these formula with the representation (26.83) in the case
a = m.
The paper by Horvath, Ortner and Wagner (1] is also in a sense relevant. In it there were the power xi being considered distributional convolutions with the kernels taken in the certain graded algebra associated with the La�l�e operator, so that the components of xi with respect to the canonical basis of the algebra are harmonic polynomials See also Ortner [2]. 26.5. There exists a characterization of the space 1 < p < n a, not in of Strichartz's constructions terms of na J as in Theorem 26.8 and 26.9, but in t
en a j � lxl-j +a-n ,
ermsIa(Lp),
(
Yj (x). /
2 d ( A: ) f)(x) dt. This characterization is given in a more general context of y l I y 0 1!11 < 1 the Lorentz spaces Lp , q - Bagby [3]. Added in proofs. The characterization of the range JO! (Hp ) in these terms, Hp being Hardy spaces, 0 < < 1, was given by Strichartz [3]. we discard the condition J E Lq(Rn ) in the characterization of the space Ia (Lp), (see Theorems 26.8 and 26.9), we in a situation when J(x) "contains" a polynomial, because the function J (x) determined by the given seminorm I D a /l ip only. The thorough investigation of the question how does the function J(x) "turns into" a polynomial at infinity with dependence of the defining seminorm, was undertaken by Lizorkin [10], [11]. In this connection we observe j t-2a -l J 26.6. H p
is
are
§ 29.
ADDITIONAL INFORMATION TO CHAPTER 5
595
/Pa-n/Ipa Lp
that the space ( ) with 1 < p < oo, p � n/a may be treated as a subspace of the quotient space S' where S' is the Schwartz space of temperate distributions and is the subspace in S' consisting of polynomials of a degree not exceeding ex - nfp. For p 2 we refer to Pryde [1] and to Davtyan [1], [2] in the case 1 < p < oo. Davtyan considered a more general case of the non-isotropic Riesz potentials. This assertion for isotropic potentials was known earlier, but the authors cannot find the corresponding reference. The paper by Kurokava [2] is also relevant. There the author dealt with the case of an integer a and investigated the representation of the function in the "Beppo Live spaces" {! e 'D' : 1)i e , a } in the form where is a polynomial = +
p= a-n/p
J Lp = P(x) If
P(x) If.
J Lr
Lp .
=
f(x) IaJl(Llp),r c i DIPIJII c Daflip , l r 2I Lp,r (R") J Lr
Iii
=
Ia (Lp ) = {! : J e 1)i J e Iii = ex}, 1 p nfex, = np/(n - exp), ex = 1, . . . , n - 1. 1a ), nfp, f(x) e Ia (Lp) E Lq , 1)i J e Lr , Iii = ex, na-(a)'Dj J e Lp. 1/q = 1/p - a/n, 1/r = 1/p - (a - [ex])/n. 26.10. Ia (Lp) f(x) E Ia (Lp) Xi , i = 1, . . . , n Lq ,
<
<
Wf. J E Lp Rn r
Lp ,
2,
q
In the same papers the characterization of the space the higher derivatives: if and only if
(Lp
0<
ex
<
was given in terms of
J
where
The following coordinate characterization of the space on the behaviour of function in each variable [3]). Let
2,
involves information (Nogin and Rubin
1 (x- te ) (I�
ex
Rt
-
be the partial Riesz potential in the variable
Xi and let
"--v-'
i-1
CHAPTER
596
Theorem 29.2. Let
5.
INTEGRO-DIFFERENTIATION OF MANY VARIABLES
1 < p < oo a nil 0 < a < 1/p. Then
101 (Lp ) =
n
n
n If (Lp ) anil th e norm•
i=1
11 / IIJar(L, ) anll E 11/ IIJ�(L , ) are equivalent.
i=1 An extension of this theorem to the case of anisotropic Riesz potentials of the type (29.6} was given by Emgusheva and Nogin [2]. In fact, they dealt with a more general situation, considering the anisotropic spaces •
a
= (a 1 , , an) � 0 and ch81"acterizing it in terms of Riesz derivatives both in the whole space Rn and in each variable separately, or in v81"ious unifications of variables. . • •
The space Ll,r was in fact considered earlier by Davtyan [2], [3], who used the anisotropic hypersingular D01 J , introduced by Lizorkin [6], to define this space as L:,r = {/ : I e Lr , na I e Lp }. The latter is an evident generalization of the space (26.91). Davtyan showed that Ll,r = Lr n Ia ( Lp ) , where Ia ( Lp) is the range of t�e anisotropic Riesz potential of the type (26.9). He also proved the inversion theorem D01 Ia J = J, J e Lp. 1 � p � nfa•,
1/a• = n- 1 E aj 1 .
26.1 1 . The ch81"acterization of the weighted space l01[Lp (Rn ; p)) of fractional integrals with weight p (x) satisfying the Muckenhoupt-Wheeden condition (25.43) was given by Nogin and Samko (4). Thus /(x) l01[Lp(Rn ; p)), 1 < p < n/ a, if and only if /{x) e Lq(pfiiP) with 1/q = 1/p-a/n and either there exists limo D� J in the norm of Lp (Rn ; p) or l i D� fi lL • (R• ·p) � c . cIn the case n = 1 this statement was obtained by Andersen [1]. A more general situation for the spaces L;,r (P1 1 P2 ) generalizing the spaces (26.91), P l and P2 being Muckenhoupt-type weights was studied by Nogin [9], [12]. The case of power weights was previously considered by Nogin [6]. 26.12. The following theorem on the simultaneous approximation of functions and their Riesz derivatives holds. Eve ry f-nction f(x) with the finite norm
E
•
E Ar
E Ap
(29.8}
can be app rozimatell by C0(Rn)-.function• with reapect to thia no rm, 1 < p < oo, 1 < r < oo anll a > 0. We refer to Samko [17, § 5) if 1 < p < nfa, Nogin and Samko [2] if 1/p - a/n � 1/r � 1/p
and Nogin [5, p. 28] in the remaining case. 26.13. A function J.&(x) is called a multiplier in the space X if I' I e X and 11 1'/llx � cll f llx for all J In order that a bounded function J.&(x) to be a multiplier in the space l01(Lp ) of fractional integrals, 1 < p < n/a, it is necessary and sufficient that the operators I
E X.
Nc cp =
J Nc (x, y)cp(y)dy,
R•
l
Nc (x, y) = L
v =l
(�)
J ltl-n -01 (6f �-&)(x)6,_11, 01 (x - y -
ltl> c
vt, t)dt ,
l
> a,
where 6m, Ot (x, h) is the kernel (26.32), should be b�unded in Lp ( Rn ) uniformly in e. The conditions which follow are sufficient: 1 ) I ltl -n -01 11 6� �-&ll n j 01 dt < oo, 2) for each i = 1, . . . , l - 1 R•
there exists a number ai such that 0 < Oi < min( a, I - i) and
I
R•
ltl -n - Ot+Ot i l l6�1'11n /(n - Ot i ) dt <
§ 29.
ADDITIONAL INFORMATION TO CHAPTER 5
597
c>.. (.Rn ),{
A > at, are multipliers in la(Lp) (Samko [24]}. l'(x) E 1, X E 0 The discontinuous function xo(x) = , where 0 is the half-space or a ball, is a 0, X fl 0 multiplier in Ia ( Lp) if 1 < p < 1/ at, 0 < at < 1. The case of the region 0 satisfying the �called Strichartz condition may be fom1d in Strichartz [1] for Bessel potentials, and in Nogin and Rubin [3] for ruesz potentials. 26.14. The operator of multiplication by functions of the fonn c(x)(1 + lxl 2 ) -v/ 2 , > 0, is a compact operator from the space Ia (Lp) into the space 1a -e ( Lp) provided that 0 < e < at and II > e, c(x) E A > at - e (Umarkhadzhiev [1]). 26.15. The ruesz derivative na J as the limit (26.26) was treated by Emgusheva and Nogin [1] in a more general setting with integration in (25.60) over \ Gc instead of {t : It I > e}, where oo. In particular, fnnctions
11
c>.. (Rn ),
Rn
Ge is an arbitrary neighbourhood of the origin with the only assumption that m (K n Gc ) - 0 as e - 0 for every compact set K in We observe that Gc may thereby be unbounded. In particular, it was shown that existence of Da J does not depend on the choice of Gc .
nn .
26.16.
The generalized Marchaud-type differences
1
L: cif(x - kiy) (§ 9.2, note 5.1 1) may i =l
be used in multidimensional ruesz differentiation na J as well. This was done by Kuvshinnikova [1] together with an application to inversion of some multidimensional potential-type operators. 26.17. In connection with Theorem 26.10 and the assertion (26.971) we mention the following statement of Samko [34]. Let J(x) E Lr ( R ) , 1 < r < oo . Then lim D� J exists in
Lp(Rn),
operator
1
oo, if and only if II(E H the limit exists, then
< p <
(25.47).
Pt )a /l ip
n
�
eta, t E R�,
c-o
where
Pc is
the Poisson
II(E - p,a )JIIp � Ata iiDa fl iP and
ll�i: /l ip � clhi01IID01 /l ip, h e
nn .
26.1 8. We note that the apparatus of hypersingular integrals Dfi/ with homogeneous characteristics and the technique developed in § 26 were essentially used in a recent paper by Kochubei [1] in constructing and investigating the fundamental solution of the Cauchy problem for the periodic pseud�differential equation
t E (0, T] uc (x, t) + E
nn '
are
can
2'7.1. The following Bernstein-type inequality
II(E - A) a/ 29llp � c(1 + a2 )a/ 2 119 llp holds for Bessel differentiation (E - �)a1 2g = F-1 (1 + lel 2 )ai 2 Fg in the case of fm1ctions g E Lp(R" ) with Fourier transforms supported in the cube lej l � a, j = 1 , . . . , n. Also, the Bohr-Favard-type inequality
II(E - �) -a/ 29llp � c(1 + a 2 ) -a/ 2 119llp is valid, where G01 = (E - �) -a/ 2 is the Bessel fractional integration operator and the functions
598 g
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
E
Lp (Rn ) have Fowier transforms vanishing in the same does not depend on g(z) (Lizorkin (5]).
GaVJ = where Ga (x)
.ia
J Ga (z - Y)VJ(y)dy,
cube. The constant c in both cases
a = (a l t . . . , an ),
R•
the kernel with the Fowier transfonn
n
p(x) being the positive solution of the equation L xjp-- 24; (x) = 1, aj = a./aj · The space j= 1 Ga(Lp) of functions representable by the anisotropic Bessel potentials aaVJ, VJ e Lp(Rn ), coincides with the Sobolev space L�a) (Rn ). This consists of functions f Lp(Rn ) with Liouville fractional partial derivatives aa; J /8zj; also belonging to Lp(Rn ). Moreover, the space Ga(Lp)
E
is characterized in tenns of the corresponding anisotropic The case a 1 = · · · = an w as presented in § 27.3. 27.3 . We indicate here the integral operators potentials, but have a unilateral character similar to
hypersingular integral (Lizorkin
[8]) .
which are similar by their nature to Bessel the potentials {25.71). We denote
where dn - 1 = 2 1 -n / 2 1r-n / 2 , a > 0 - the unilateral Bessel potentials, ien ) -a. Similarly to (27.22) the equality
g"f<() = ( .j1 + lel2 - e� =F
r�) J ti�-1 (MYra VJ(·, zn =F Yn )) (z' )dtln 00
G%VJ =
0
holds and similarly to
{25.76) the representation aa
=
a%12 a�12 is valid. Here
The operators inverse to G% are realized as hypersingular integrals with weighted differences as at the end of § 27.4), thus
(G%) - 1 f =
xt:, J f�= (6�1.,J)(x)dy, I)
Ri-
I>
a,
§ 29. ADDITIONAL INFORMATION TO CHAPTER 5
599
where
(_&� J)(x) J(x)e"' IYin /2Kn/2 (1YI) =
Theorem 29.3. Let a > 0 and IIIPIIP are equivalent. Let
Then I E G% (Lp), 1 � then
�Pf- = p
1
<
p
l
+
L {�) (-1) k ( kl y l )n f 2 Kn/ 2 (kl y i )J(x- ky).
k=l
< oo. Then G% (Lp ) = Lp and the norma iiG%1PIIL ; and
00
j j IYI-n y;a(.&�11 J)(x)dy, lim0 cpf- e e R"
� oo, if and only if
e-
(L,.)
l > a.
Lp(R" ). If I e G% (Lp), 1 <
p
� oo,
lim f- eziat. a/molt everywhere alao. e-0 !p Besides the operators G% , one may consider unilateral potentials of the form
Cn- 1 1fn/2 Their action in Fourier images is reduced to multiplication by (1 le'l ien ) -a In the case these potentials coincide with the operators V) -a consideredwithin =
� =
§
n =
1). 18.4.
+
1
r (n/2)
(29.9)
•
=t=
(E ±
(d. (25.75)
Operators G% defined via Fourier images are found in the book by Triebel [1 , p. 282], where the mapping property G� : L: - L;+P was proved for them. In explicit form the potentials G%1P were given in Rubin [20], [24], where in particular, the inversion of these potentials and Theorem 29.3 were proved. The potentials (29 .9} were considered in Eskin (1]. 27 .4. We note the characterization of the space (Lp) of Bessel potentials in terms of "truncated" integrals (27.46), (27.54) and (27.55).
aa .
Theorem 29.4. Let 1 < p < oo, a > o, a #: 2, 4, 6, . . . Then e Ga (Lp) if and only if vi f E Lp , U l � lal, aftd Ofte of the following COftdition.t i• aa.tiafied: i ) the aequence D �A0 ,e f of i•tegr4l• (27.46) converg e• in Lp -norm; ii) sup l iD � e f l lp < oo . e>O We observe that hypersingular integrals f have a certain advantage in comparison with used in Theorem 27.3. That is, they contain the characteristic the Riesz derivatives
oo
na exponentially vanishing at infinity. aa )
J(x)
D�o
�a (lyl)
Theorem 29.5. The apace ( Lp , Ot > 0, 1 � p < oo, conai.tt• of thoae and OftiJ tho•e j.nction• f (x) E Lp for which the aequence !D? f converge• in Lp -norm, wh ere !D: i• one of the i•tegral• (27.54) aftd (27.55), or the truncation of the integral (27.56). The former of these theorems was established by Nogin (9], the latter by Rubin (23], (25]. 27.5. The presentation of the Bessel potentials theory from the point of view of fractional powers of operators can be found in Fisher (3)-[6]. We mention also the paper by Fujiwara [2), where the domain of the fractional power (E in the half-space (under the third boundary condition) was characterized in terms of the interpolation spaces [H •P (� , LP(R�p]. The
J
-/1)a/2
2 )
600
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
[1) {(x, 31) ,R'&+1 :[1x) Rn 31
previous paper Fujiwara is also relevant. Purely imaginary fractional powers of an elliptic-type differential operator of the second order were treated by Fujiwara 27.6 . Ginzburg studied trace problems in certain weighted Bessel potential spaces in the strip 0 < < b }, defined via Bessel fractional differentiation in x and E E normal differentiation of the first order in 31· 28. 1 . FUnctions
[3).
,
p(e) Je� -e� - ... -e�,
c.p[P(e)) Ki {e : p2 (e) x i31, E Rn , 31 K+ {31 : p2 (31) 311 [1]). [1)
> o,e 1 > o} and is supported in the light cone = There the function z = > O, < 0} and K11 (z) is the McDonald function, + x = E and it is assumed that the integral in the left-hand side converges absolutely (Domingues and Trione In this connection the book by Leray is relevant, where the expression for the Fourier-Laplace transform of Lorentz-invariant functions was given in terms of a composition of the one-dimensional Fourier-Laplace transform with fractional integration, this being similar to (25.14). 28.1 1 • Kipriyanov and Ivanov introduced the hyperbolic Riesz potential of type (28.19) within the frames of a general Lorentz space X defined as a smooth manifold in equipped with the Lorentz structure and the metric This was defined by the equation • · · ch with = this also being known as = Lobachevskii space. In fact the authors dealt with the following modification
[1) 2rzy [x, 31)2 [x, x)-1 [31, 31] -1 [x, 31) X1 311 - X2 31r2zy-. Xn31n ,
Rn
(IaJ)(x) Hn ( ) J f(31)sh a-1rzyd31 Dz 101 =
1
a
D:e
of the Riesz hyperbolic potentials, being the corresponding analogue of the cone (28.18). They proved Sobolev-type theorem on the boundedness of from Lp (X) into Lq (X) with 1 < p < q < oo, 1/q = 1/p - a/n, together with the weak-type estimate in the case p = 1. 28.1 11 • Riesz introduced surface hyperbolic single- and double-layer potentials. The with the Lorentz distance, and S(x) a part of a former was given by
[7)J g(u)r01-n (u - x)dsu S( z ) S K:;(x) h(x), � g(x), x E S.
given surface which is cut off by the cone (cf. (28.21)). Ivanov and Kipriyanov [1] applied such potentials with a = 2 to the Cauchy problem for the wave equation with the initial = data u(x) = 28.2. The spaces
a > 0,
1 � p < oo ,
1�
r
< oo ,
L;,r {! : l fl r + +, r c; H01c.p c.p Lp(�+1 ): r c�.r Lr nH01(Lp ), r
(compare with the spaces = u.r- 1 1e i 01.1"J II, < oo } , see (26.91)) were introduced and investigated in Nogin and Rubin [1], [5]. They generalize the spaces and and coincide with the former if 1 < p < oo, 0 < a < 1 n/2, r = (n 2)p/(n 2 - ap) and � with the latter if a > 0, 1 < p = < oo. The spaces consist of functions E representable as the potential with E =
1�
p,
< oo, a > 0.
+ Ha (Lpf+) Lr1t(R01n(Lp) +1 )
§ 29.
ADDITIONAL INFORMATION TO CHAPTER 5
601
They are characterized by the relations
Cp1r
=
{! :
c,aIr = {! : J e Lr ' where T01 is the operator
J E Lr , T01 f E Lp }, 1 5 p, r < oo, sup IIT:r fl iP <
�>0
00
(28.52) and T� f is
},
a > O,
1 < p < oo, 1 5 r < oo ,
the truncated integral in
Ot >
o,
(28.52).
The spaces
CJ1p = 1t01(Lp ) admit a simil8l- characterization in tenns of the hypersinsular integral
(28.54).
H 0 < 0t < 2, the characterization of the space 1t01(Lp ) in tenns of hypersingular integrals with first onler differences was also obtained by Chanillo [1] and Nogin (5). In the case 0 < a < 1 Bagby [1] used other means - Strichartz..type constructions - for the same purpose. There is the characterization of the space 1t01(Lp ) with a � 1 in non-constructive terms - via reduction as in Gopala Rao [1, 2]. Nogin and Rubin (1), (5) obtained the "separated" characterization of the spaces Cp1r , which reveals different behaviour of functions j (x , t) € £p1 r in the space variable X and in the time variable t, thus
c;1r
where
{f : J e Lr , D � J e Lp , n:/2 f e L, , },
1 < p < oo, 1 5 r < oo, a > 0 and D� J =
=
1 �- +O dn l ( a ) I lim
(L. (R•+ 1 ))
-
l
1: ( �)f(x - kf1,t)lfll-n - a dfl J ""(-1) LJ
lrl > � 1: =0
�
is the "partial" Riesz derivative in X and n:/2 j is the partial Riesz derivative in t. Instead of the space derivative D� f the set of partial coordinate derivatives D�i f, i = 1, 2, , n, may be used to give the similar "coordinate" characterization of the spaces Cp1 r . These characterizations were applied by Nop and Rubin (1 ), (5) to investigate multipliers in Cp1r and establish a connection of c;lr with Sobolev spaces in the case of integer a. Some sufficient tests for multipliers in tenns of the spaces cA(Rn) were, in particular, given in these papers. The wide class of regions in nn +l , containing the class of so caRed Strichartz regions was also defined, the characteristic functions of which are multipliers in Cp,r · Multipliers in 1t01 (Lp ) were studied by Bagby [1]. Chanillo [1] showed that functions in c0 (nn+1 ) are multipliers in 1t01 (Lp ). Nogin and Rubin (1], [5) showed that the space C�'J, m = 1, 2, , consists of functions f(x, t) E Lp , which have the partial derivatives a2m f /8xJm , j = 1, , n, and am f /8tm in • • •
. • •
.Lp(,R'I+ l ). In the case of odd a = 2m - 1,
• • .
m = 1, 2, . . . , a similar assertion was obtained, the only difference being that smoothness order m - 1/2 with respect to t is fractional. Thus the :-----
- 2 partial Riesz denvative n,m 1/ was used. In the case r = p statements similar to above were proved by Gopala Rao [1], but his results contain superfluous information on mixed derivatives in the sufficiency part. Nogin and Rubin managed to avoid this excess information. Bagby [1] gave interpolation theorems for the spaces 1t01 (Lp )· We also obeerve that the spaces 1t01(Lp ) coincide up to the equivalence of the norms with the anisotropic spaces of functions in Lp which have the generalized Liouville derivatives of order a in x;, j = 1, , n, and of order a/2 in t, all belonging to Lp . This is proved by means of the Lp -multipliera technique. Such spaces are particular cases of the general anisotropic spaces investigated by Lizorkin (5), (6], [8] . 28.3. As it was shown by Nop and Rubin [2], the hypersingular integrals TO f and '!'01 f (28.52) and (28.54) - which invert the potentiaJa J = H01cp and J = 1t01cp, cp Lp , respectively, may be interpreted as the almoet everywhere limits of the corresponding "truncated" integrals. .
•
•
•
.
.
• • •
E
·
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
602
28.-4. Fractional powers of the hyperbolic operator Err - i 4' were investigated by Sprinkhuisen-Kuyper (1), (2). 28.5. One more variant of fractional integration is defined by the relation
3irr
in Fourier images. This was investigated by Takano (2], who showed that the operators Ta J form a semigroup of the class (Co) in the space Lp(R" ; p) , with a Muckenhoupt-type weight p (x) for every a > 0. The generator of this semigroup is the hypersingulM operator ncr. This was interpreted by Takano as the closure of the operator F- 1 1x i 01F in the space Lp(R" ; p). The case p(x) = (1 + tzl 2 )-m /n , m > n, was considered eMlier in Takano [1]. This semigroup in the one-dimensional case Mo&e earlier in some probability problems (see Kat [1] and Elliott [1] in the case 0 < a < 1 and S . Watanabe [1] in the case 0 < a < 2). 28.6. In connection with the various forms of fractional integro-differentiation of functions of many variables, we mention the case of functions given on the unit sphere in R". There are many papers devoted to fractional powers of the Beltrami-Laplace operator on the sphere and their generalizations and analogues. We refer the reader to the review paper by Samko [30], where other references can be found. We also cite Samko [22) and Pavlov and Samko [1], where the direct analogues of Riesz integro-differentiation on the sphere were considered, including spherical hypersingular integrals. See also Colzani (1] and Vakulov and Samko [1]. 28. 'T. There is a generalization of fractional integro-differentiation to the case of many variables whim involves integration over a cone with the vertex at the point x E R". instead of the half-axis ( -oo,x) in the one-dimensional case. This is discussed in the book by Vladimirov, Drozhzhinov and Zav'yalov [1]. The definition is as follows. Below r is a closed convex acute solid cone in R"; r• = {( E R" : ( ·x � 0 Vz E r} is a cone conjugate to r ; C = int r• is the set of inner points of r• ; is the convolution algebra of temperate generalized functions with support in r ; H( C) is the algebra of functions holomorphic in the tube region TC = {z = x+i11 : z R", 11 E C} which are Laplace transforms L(g] = (g((), e*z·e ) of generalized functions g E The function K:c(z) = J eiz·(cJ.e is called the Cauchy kernel of the tube region TC . The open convex acute
Sf.
Sf.. E
cone C is called regular if [K:c(z)] - 1 E H(C). I'
The operation of fractional integro-differentiation is introduced as follows. Let C = int r• Let Dr(() = L -l [K:0(z)](() be a regular cone. Then K:0(z) = [K:c(z)]a E H(C) for any 01 E be the Laplace original of the power K:0(z) defined as the chMacteristic function of the cone r, if a = 0. The operator of fractional integro-differentiation is defined as the convolution with the distribution Or((). These operators form an abelian group with respect to 01 . It can be shown that the tensor product of the one-dimensional Liouville fractional integrals of the same order, and Riesz potentials with the Lorentz metric are particular cases of the above operators. 28.8. Another approach to fractional integro-differentiation connected with cones in R" was eMlier developed by Gindikin (1], [2] (see Vainberg and Gindikin (1]). In contrast to note 28.7 the order of fractional integration is "multidimensional" here. Let V be a convex cone in R" , not containing straight lines, such that there exists a group G(V) of linear transforms preserving V, with the following property. For all points z and 11 E V there exists a unique transformation 1/J E G(V) for which 1/J(z) = 11 · H we fix a point e E V, then we can transfer the multiplicative structure of the group G(V) to V: zy = g(x)y, where g(z)e = z, g(z) E G(V). A function satisfying the condition J(xy) = /(z) f (y) is called a complex power function. Every
Rl .
also
I = 1, has the form /(z) = n (xk (z))PJr tl::.fxP, k= l fractionally p = (P l t . . . , PI ) E C1 • The number I is called the rank of the cone V, Xk (x) rational functions of coordinates of the point The cone V is defined by the inequalities Xk (x) > 0, k = 1, . . . , l. Let d�-i be the measure invariant with respect to the group G(V). It is connected with euclidian one by the relation d"' = zddx, where d = (d t . . . . , d1 ) E R1 • Further, let
such function /, nonnalized by the condition /(e)
are
x.
§ 29.
ADDITIONAL INFORMATION TO CHAPTER 5
603
rv (p) be the Siegel integral of the second kind, or r-function of the cone V. It may be represented as a product of usual f-functions. The Riemann-Liouville operator of order p = (Pl , . . . , P l ) E C 1 is defined as
('Pv cp} (x) =
1 f v (p)
f cp(y}(y - x)P+ddy 00
where integration is carried out over the cone (x, oo ) consisting of points y such that y - x E V. In Gindikin [1] a detailed investigation of the operators Pv was undertaken, and applications to constructing fund.amental solutions of certain differential equations and solving some problems of integral geometry were given. The Abel equation Pv cp = f was considered. Analysis similar in a sense to the latter was developed earlier by Gi.rding [1] for the case of cones of symmetric and Hermitian matrices. Faraut [1] also considered the case where the Riesz-type potential was introduced for a symmetric cone in a Jordan algebra and its applications to the Cauchy problem were given. The Rie -Liouville operators connected with homogeneous cones were also dealt with in the paper by Watanabe [1]. We observe that the Riemann-Liouville operators 'Pv were used by V.S. Rabinovich [1] in investigating multidimensional integral equations of convolution type, the symbol of which has a singularity at infinity of the type of complex power function related to some cone. In this paper the analogues of the Sobolev-Slobodetskii function spaces were constructed in terms of the operators 'Pv . This was applied to the problem of Noether properties for convolution type equations. 28.9. Let Ac(b) be the pyramid (28.60), and x E R!' , k = (kl , . . . ,kn) be a vector-function. Let us denote k(x) = kl (x) . . . kn(xn) and consider the Abel-type equation :c
mann
J k(A (x - t))cp(t)dt ·
=
f(x), x E Ac(b).
(29.10)
A c(:c )
which is more general than (28.65). Let the vector-function l (x) exist for the given kernel k(x) such that
J (1(:c ,r)
l(A · (x - t))k(A · (t - r) )dt = 1
(29.11)
where u(x, r) is the region (28.64). Then (29.10) is solved by the same method that was applied to (28.65), and the unique solution of (29.10} is
op(z) =
ii ( t •i,;' a:J J l(A · a=l
J =l
(z - t))J(t)dt
Ac(:c )
(Kilbas). Equation (29.10} is a pyralil.idal an�ogue of the Sonine equation (4.1), while (29.11 ) is such an analogue of the condition (4.2 11 ) . 28.10. Beautroux and Burbea [1) considered the "fractional radial differentiation" operator for holomorphic functions f(z) = L: a k z k in the unit ball in en defined as na J = L: 0. They proved imbedding theorems with respect to the parameters p, q , a , and gave
604
CHAPTER 5. INTEGRO-DIFFERENTIATION OF MANY VARIABLES
Lipshib,.type estimates for functions in
A� , a ·
Similar results in the context of more general
Hardy-Bergman-type spaces H�·9 ,/3 can be found in Jevtic [2). 28.1 1. Ricci and Stein [1] defined fractional integration in the context of nilpotent groups by considering convolutions with kernels supported on an analytic homogeneous manifold V in the group. Typical examples of these convolutions are fractional integration along a homogeneous curve 11 = R�, k > 0, or the case when V is the n-dimensional forward light cone in nn+ 1 • They investigated the problem of Lp - Lq boundedness of such convolutions.
zk , z E
Chapter 6 . Applicat ions to Integral Equat ions of t he First Kind wit h Power and Power- Logarithmic Kernels In this chapter we present applications of fractional integrals and derivatives to integral equations of the first kind
Mtp ::
f k(z, t)tp(t)dt = l(z),
0
z E 0,
the kernels k(z, t) of which have a singularity of the type lz - t la- l , 0 < a < 1 , or, more generally lz - t l a - 1 lnm lz!,1 . Fractional integrals arise naturally in the former case, the latter leading to their generalization considered in § 21. The kernels k(z, t) are ass umed to be real-valued for simplicity, while the right-hand side and the unknown function may be assumed to be complex-valued. In the case k(z, t) = c(z, t)lz - t la-l the investigation and solution of the above equation is based on the idea to single out the fractional integration operator explicitly. Namely, the equation Mtp = I generally speaking can be represented in the form Ia N tp = I, N = v a M, where [a and 'D0 are suitable forms of fractional integra-differentiation. The equation N cp = g, g = 'D0 I, obtained after inverting the Abel equation proves to be an equation of the second kind under rather weak assumptions on c(z, t). In the case when c(z, t) is continuous at t = z this is a Fredholm equation of the second kind. If the function c(z, t) is allowed to have a jump at t = z , this case being of special interest and often occurring in applications, then the situation proves to be more difficult. The equation of the second kind which results is of a singular type. The theory of such equations is well known and we ass ume the reader to be familiar with the simplest ideas and facts of this theory. An excellent presentation may be found in the well known books of Gahov (1] and Muskhelishvili (1]. Nevertheless , we display the main results of this theory as required in the present chapter in the preliminary subsection § 30.1. In certain
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
606
simple cases for example in § 30.1 , we shall manage with minimal knowledge of the singular integrals considered in § 11. Some types of singular equations ( the "dominant" ones, in particular) are solvable in closed form, i.e. via quadratures. This will allow us to obtain an explicit solution - in closed form - of a certain class of the equations of the first kind considered. As regards the most general case, we shall obtain results which have a qualitative nature, and characterize the solvability of the equation. Among these the reader will find the criterion of normal solvability and the formula for the index (see definitions for these notions in § 31.1). We call attention to subtlety connected with the representation of the operator M in the form M = 1a N, which will always be our particular concern in the investigation of the equations. This representation obviously assumes the identity M = Java M to hold, but in general la va I #; I see for example (2.60). We can be sure of the validity of the relation Ja v a I = I by taking functions I representable by the fractional integral of order a as in (2.59). So we must check the fact that the range of the operator M is embedded into the range of the fractional integration operator 1a . We shall always keep this checking in mind. As for the equations of the first kind with a kernel involving a power logarithmic singularity, the scheme of investigation is the same. Fractional integration is replaced by the operators 1a ,m considered in § 21, and fractional differentiation by the operators inverse to 1a , m realized as a convolution with the Volterra function. -
§
30.
The Generalized Abel Integral Equation
We begin with a consideration of the equation which has become known as the generalized Abel equation. The case of the equation on the whole axis will be easier in a sense. In the first subsection we give necessary preliminaries concerning the solution of singular integral equations for the reader's convenience.
30. 1 . The dominant singular integral equation The equation at (z)
a2(z) 1r
f
-oo
=
l(x ).
(30.1)
is known as the dominant singular integral equation on the whole axis. We assume the coefficients a 1 ( z ) and a2 (z} to be real-valued for simplicity, while the right-hand side and the unknown function may be assumed to be complex valued. Let
(30.2)
§ 30.
THE GENERALIZED ABEL INTEGRAL EQUATION
607
We denote
O(x) = arg G(x) .
(30.3)
The integer
x=
00
2
� j d arg G(x) -
(30.4)
oo
is called the index of (30.1). Equation (30.1) is solvable in closed form and its solvability is characterized by the following theorem.
Let assumptions (30.2) be satisfied. If x � 0, then (30.1) is unconditionally solvable in Lp(R1 ), whatever the right-hand side f(x ) E Lp(R1 ), 1 < p < oo , is and its general solution is given by
Theorem 30.1.
c�: a2 (x+)Z( x) + at(z) f( x)
•
A:=l
(x
A(x)
i) A:
_
a2 (x)Z(x) f00 f(t )dt A(x) 1r Z(t)(t - x) - oo
(30.5)
where CA: are arbitrary constants, which may be complex the sum being omitted if x = 0, A(x) = a�(x) + a�(x ) and
•/ Z(x) = ( : � ;. ) - 2 y'A{x)e -r(z) , 1(x) _
If x
<
1 ! 2 .
00 ln
11"1
- oo
[ ( !:;: ) • G(t)] dt. t-X
0, (30.1) is solvable if and only if
f00 f(x)dx
- oo
Z(x)(x + i)A: = O,
k = 1, . . . , l xl .
(30.6)
We remark also that
Z(x), Z(1x) E H (R. 1 ) , .\
(30.7)
608
CHAPTER 6. INTEGRAL EQUATIONS OF THE Fffi.ST KIND
The singular equation of the form 00
a t (z) ,P(z) + !.7r j a 2t(t)-,Pz(t) dt = g(z) -oo is also solvable by quadratures, unconditionally solvable for x � general solution H
c�: Z(z) + a t (z) g (z) Z(z) 1/J(z) = � L::' 1r (z + i)A: A(z) 1:- 1 _
(30.8) 0 and has
the
00
a 2 (t)g(t)dt f Z(t)A(t)(t - z)
- oo
(30.9)
where A(z) and Z(z) are the same as in (30.5). The necessary and sufficient condition for solvability of (30.8) in the case x < 0 is
f Za(z2 (z)g(z) )(z + i)A: d 00
...
...
-oo
=
O,
k = 1, . . . , lxJ.
The characteristic singular equation on an inten;al cp(t)dt = !( z ) , a < z < b, a 1 (z) cp( z) + a27r(z) f --t-z 0 b
--
(30.10)
has a more complicated solvability picture. Let as before
a t (z), a 2 (z) E H.\ ([a, b]); a�(z) + a � (z) "I 0, z E [a, b].
(30.11)
Solutions of (30.10) may be looked for in the space H* = H*(a, b) of Holderian functions with integrable singularities at the end-points of the interval ( cf. Definition 13.1). This is caused by the fact that the solutions of (30.10) have in general singularities at the end-points. Sometimes it is desirable to find solutions bounded at one end or at both. We denote
H = H((a, b]) = U H"([a, b]) = H* n C([a, b]),
(30.12)
n; = H* n C([a, b)), n; = H* n C((a, b]).
(30.13)
1' >0
§ 30.
609
THE GENERALIZED ABEL INTEGRAL EQUATION
We shall use one of the spaces H* , n: , n; or H as a space in which solutions are to be found. In contrast to the case of the whole axis the index of (30.10) is known to depend on the space of solutions. Let G(x) = e i8 (� ) be the function (30.3). Let us choose the value of arg G( x) so that
(30.14)
0 � O(a) < 21r. Let
Ra
{ O,1,
=
if we look for solutions bounded as x --+ a, if we admit for solutions unbounded as x --+ a
and n6 is defined similarly. By X we denote any of the spaces H* , We set K
= Kx =
[O��]
n: , n; or H .
+ R 0 + R& - 1 for the space H* , for the space n: or for the space H
(30.15 )
n; '
and
JJa =
1 - Ra - O(a) 21r ,
where [O{b)/(21r)] denotes the entire part of the number. We emphasize that JJa = JJa( X ) and JJ6 = JJ,(X.) and -1 < JJa < 1, -1 < JJ6 < 1. Theorem 30.2.
Let conditions (30.11) and (30.14) be satisfied and f(x) = (x - a) l -"/.(x) • (b - x) l - v, ' /.(x) E H.
Then (30.10) is unconditionally solvable in the space X if lla 0 and its general solution is
x�
( )
U'J • r ...
> JJa ,
v,
> JJ6
and
a 2 (z)Zo(z) (z a ) "'• (b z) "'" P• (z ) + a 1 (z)/(z) -1 A(z) A(z) _
_
_
a2 (z)Zo(z) 1rA(z)
( z - a ) "' • ( b - z ) "' " f(t)dt J t - a b - t Zo(t)(t - z) ' ,
a
( 30.16 )
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
610
where A(z ) is the same as in (30.5), P.Jt-t (x ) is a polynomial of degree x - 1 with arbitrary coefficients ( P.Jt - t (z) = 0 if x 0} and =
Zo ( z)
=
� [i �(�� + /J(a) ln(z - a) - /J(b) ln( b - z)] E H.
exp 2
(30.16')
If x < 0, (30.10) is solvable if and only if
J b
Zo
/�z)(z - a)�- l dz x ( x - a) �'•
b - z ) �'•
=
k=
0,
1, . . . ' lxl.
This condition being satisfied, (30.10) has the unique solution given by (30.16) with P.Jt- t :: 0. 30.2. The generalized Abel equation on the whole axis In this subsection, using Liouville fractional integra-differentiation following equations of the first kind
M
Q
J z
-oo
_Ma ,p =-
J z
00
_
z
(30.17)
00
u(t) ,P(t)dt + v(t) ,P(t)dt g ( ) - X• ( z - t) l -a J (t - x) l -a _
- oo
J� we solve
(30.18)
effectively in closed form. Equations (30.17) and (30.18) are known as the generalized Abel equations with, exterior and interior coefficients, respectively. These equations can also be written as
1;2�"'11�� (
J
00
M a
Ct (z ) +
-oo
- a
z - t)
J c1 (t) +IXc2-(t)tsigI I-an (x - t)
00
- oo
=
u(x) +2 v(x)
'
C2 ( X ) =
4'· o/
(t )dt
=
(30.17')
- g (X) ,
(30.18')
_
u(x) -2 v(x) .
(30.19)
§ 30.
THE GENERALIZED ABEL INTEGRAL EQUATION
Equations (30.17) and fractional integration:
611
(30.18) are evidently rewritten in terms of the operators of
Ma tp = u(z)r(a)I+ tp + v(z)f(a)l� tp = /(z),
(30.20)
(30.21) Ma .,p :: f(a)I+ (u .,P) + f(a)I� (v .,P) = g(z), where I± are the fractional integration operators (5.2) and (5.3). We shall seek solutions of (30.17) and (30.18) in the space L,(R1 ) , 1 < p < 1/a. A principal question for (30.17) and (30.18) - as well as equation of the first kind
- is the following. How can we characterize admissible right-hand side functions f(z) and g(z) of (30.17) and (30.18) for a given space of solutions, L,(R1 ), in the present case? We shall see that the ranges M a (L,) and M a (L,) coincide with the space Ia ( L, ) of fractional integrals, which was studied in detail in § 6, or perhaps differ from 1a ( Lp ) by a finite-dimensional space. As for the coefficients u(z) and v(z), we shall assume that
u(z), v(z) E H>. CR1 ) ,
.\ > a,
(30.22)
u(z), v(z) E H>.CR1 ),
.\ > 0,
(30.23)
in the case of (30.17) and
in the case (30.18). Here R} is the axis, completed by the unique infinite point and H>.Ck1 ) is the Holder space defined by (1.6). Clarifying condition (30.22), we remark that it is caused by the requirement that multiplication by u(z) and v(z) preserves the space of functions representable by fractional integrals of order a (see Theorem 6. 7). We suppose that the coefficients u ( z) and v( z) do not vanish simultaneously:
(30.24) The key point in solving (30.17) and (30.18) is the connection between the fractional integration operators I± and the singular operator which was established in § 11.
S,
A. Solution of the generalized Abel equation with exterior coefficients.
Applying
(11.10) to (30.20), we have
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
612
i.e. we represent the operator Ma as a composition:
(30.25) where a 2 (z) No� = a 1 � + a 2 S� = a 1 (z)�(z) + 1r
00
j
-
oo
�(t )dt , t-z
(30.26)
a 1 (z) = f(a)[u(z) + v(z) cos c:t7r], a2 (z) = f(c:t) sin c:t1rv(z). (30.27) It is clear now that the solutions of (30.17) will be given in closed form: we are to solve successively the singular equation
(30.28)
No� = !
of the form (30.1) using known results and then invert the Abel equation. While solving these equations in the way mentioned we have to guarantee the solvability of the Abel equation I+ cp = � . Namely, we seek solutions cp Lp and suppose that f Ia(Lp)· But inverting this Abel equation we do not know, generally speaking, that � Ia(Lp)· So we have to make sure that any solution �(z) of the dominant singular equation (30.28) is representable by the fractional integral of order c:t, when the right-hand side is. This will be shown in Lemma 30.1 below. We solve now (30.28) by using the result in (30.5). As was noted above we are to solve this equation in the space 1a ( Lp). We shall find all the solutions of this equation in the space L9, q = p/( 1 - c:tp), (taking into account that Ia(Lp) C L9 by Theorem 5.3) and then show that these solutions belong to Ia(Lp), if f Ia(Lp)· It follows from (30.24) and (30.22) that the assumptions (30.2) are satisfied. The index x of (30.28) (see (30.4)) in terms of functions defined in (30.19) has the form
E
E
E
E
� J d arg [ct(z) + ic2 (z)tg a21r ] . 00
x=
(30.29)
-oo
The number x is integer-valued: x = 0, ±1, ±2, . . . . If x � 0, the general solution of the singular equation according to (30.5), by the relation � ....
(z) _ - � c�: _
(30.28)
is given,
v(z)Z(z) u(z) + v(z) cos c:t?r + /( z ) r( Q)A(z) ( z + i)l:
v(z)Z(z) sin c:t?r A(z) 1rf(c:t)
00
f
- oo
f(t)dt Z(t)(t - z) '
(30.30)
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
613
where Ci are arbitrary (complex) constants,
(30.31) (
and the function Z z) is equal to Z( z) =
(: � �)
· - N/ 2
e-r<�>,
�
(30.32) If x 0, the necessary and sufficient conditions of solvability for (30.26) appear according to (30.6): the right-hand side is to be orthogonal to a finite - oo
<
number of linearly independent functions.
Let at(z), a2(z) E H>.. CR1 ) , ..\ > If f(z) E 1°(£,), then all the solutions E £9, q = p/(1 -ap), of the equation a 1 C) + a2SCJ = I belong to l0(Lp) as well. Proof. Since I E L9(R1 ) , all the solutions in L9(R1) are given by (30.5). Then the second and the third terms in (30.5) belong to p:r(L,) by (30.7), Theorem 6.7 and Corollary 1 of Theorem 11.4 . Further, by (5.28) we have ( z + i)- t = li.V't , where V' t (z) = const ( z i) � a E L,(R1) for every p � 1, k = 1, 2, . . . . So ( z i)- i E 1°(£,) and then also Z(z)v( z) ( z + i)- t E 1°(£,) by Theorem 6. 7 . In view of Lemma 30.1 solutions V'(z) of the starting equation (30.17) or (30.25) may by found by the relation C)
Lemma 30.1.
ct.
+
+
- :-
V'( z ) = Da+ A;...... , n a+ A;...... -
•
Q
()()
� ( z) - ct ( z -
t) dt,
(30.33) r(1 - a) J0 where ct(z) is the function (30.30). The solution V' E L,(R1) of the Abel equation was found in accordance with Theorem 6.1. Thus, taking Lemma 30.1 and Theorem 30.1 into account, we see that the initial equation (30.25) is solvable for any right-hand side f E Ia(L,) if x � 0. This means that 1°(£,) � Ma(L,) if x � 0. On the other hand, M0(L,) � Ia(L,), which follows directly from (30.17) in view of (6.1) and Theorem 6.7. So t l +a
(30.34)
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
614
In the case x < 0 (30.34) is valid "up to a finite-dimensional subspace" , which is produced by the finite number of linearly independent solvability conditions
(30.6).
Thus we have proved the following theorem. 30.3. Let assumptions (30.22) and (30.24) be satisfied and let x be the index (30.29). Equation (30.17) is unconditionally solvable in L,(R1 ), 1 < p < 1/o:, for any right-hand side f(z ) E P�(L,), if x � 0, and its general solution is given by (30.33) and (30.30). If x < 0, then (30.17) is solvable for those and only those right-hand sides f(x) E la (L,), which satisfy the conditions
Theorem
00
J Z(x)(x + i)k = O, f(x)dx
k=
1, . . . , 1 x l,
(30.35)
-oo
If these conditions are satisfied, the equation has the unique solution given by (30.33) and (30.24) with CJ: = 0. B.
Solution of the generalized Abel equation with interior coefficients.
Applying (11.10) to into account, we obtain
(30.21)
and taking the commutation property of (11.12)
Ma cp = r(o: )l+[(u + V COS 0:7r)cp + sin o:7rS(vcp)).= g , i.e. we represent the operator if a
as
a composition
(30.36) where
J a2 (t)cp(t)dt t-X , 00
No cp = a 1 cp + S(a2 cp) = a 1 (x)cp(x) + .!_ 7r
- oo
(30.37)
and a 1 (x) and a 2 (x) are coefficients {30.27). Now we are to solve successively the Abel integral equation and then the singular integral equation
(30.38)
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
615
of the form (30.8). Here D+a 9 - f(1 a- a) J g (z) -t t+g (za - t) dt E L 00
_
P
0
provided that E Ia (Lp) (we have used Theorem 6.1 here). It remains to solve (30.38) in the space Lp. In contrast to the previous case of exterior coefficients, when we solved the singular equation first and the Abel equation afterwards, we deal with the inverse order for solving now. Hence there arises no need to represent the solution of the singular equation by fractional integral as before. Solving (30.38) using (30.9), we have g
_ sina 7r Z(z) j v(t)(D+g)(t) dt ( 30.39) 1rf(o) Z(t)A(t)(t - z) ' provided that x � 0, with x the same as in (30.29) . Here A(z) and Z(z) are functions (30.31 ) and (30.32) . In the case x 0 there arise necessary and sufficient solvability conditions for 00
- oo
<
(30.38):
v(z) (D+ g)(z) dz = 0 ' J Z(z) (z + i)k 00
k=
1, . . . , 1xl.
- oo
Us.ing (5. 17) for fractional integration by parts, we can transform these orthogonality conditions to the form 00
j g(z)tP,, (z)dz = 0,
k=
1, . . . , lxl,
(30.40)
-oo
where Toa justify the applicability of (5.17) we have to show that v(t)(t + i)-k /Z(t) E I (Lp) · This fact was established in the proof of Lemma 30.1. As a result, we arrive at the following theorem. in
616
CHAPTER 6. INTEGRAL EQUATIONS OF THE Fffi.ST KIND
Let assumptions (30.23) and (30.24) be satisfied. Equation (30.18) is unconditionally solvable in Lp(R1 ), 1 < p < 1/o:, for any right-hand side g(z) E Ja (Lp), if x � 0 and its general solution is given by (30.39). If x < 0, (30.18) is solvable for those right-hand sides g(z) E Ia (Lp) which satisfy orthogonality conditions (30.40), then in this case the unique solution is given by (30.39) with CJ: = 0.
Theorem 30.4.
30.3.
The generalized Abel equation on an interval
We consider now the equations of the above type in the case of an interval:
J
J 6
�
cp(t)dt u(z) (zcp-(t)dt p - a + v(z) (t - z) l - a = f(z), t a �
J �
u(t)cp(t)dt (z - t) l-a +
J (tv(t)cp(t)dt - z) l- a = g (z) '
a<
z < b,
(30.41)
6
a
b,
(30.42)
where 0 < a < 1. This case is of especial interest because of the various applications where such equations arise - see references in § 34.2, note 30.8. In contrast to § 30.2, the behaviour of the coefficients at the end-points of the interval will have an essential influence on the existence of solutions, their quantity and the nature of the singularities. We shall give the complete solution of (30.41) and (30.42) in closed form and clarify the influence of the end-points on the solutions of the equation. We shall seek solutions of (30.41) and (30.42) in the space H* of Holderian functions with integrable singularities at the end-points (see Definition 13.1). We recall that it consists of functions of the form
(30.43) where et > 0, e2 > 0 and cp* (z) is Holderian on [a, b) . As for the right-hand sides f(z) and g(z), we assume that
f(z), g(x) E H� ,
(30.44)
where H� is the space defined in (13.53) or (13.59). This is justified by Theorem 13.14, stating that fractional integration of order a maps the space H* onto H� one-to-one. Comparing the setting of the question with that of the case of the whole axis, we remark that the difference in choice of the space for the solution is caused by
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
617
the fact, that Holder spaces are not convenient for the whole axis . This is because of specificity connected with the infinite point (although they may be used, too), while the spaces Lp on the interval [a, b] prove to be less convenient than H* since it is not easy then to characterize singularities on the right-hand sides. As regards the coefficients u(x) and v(x), we assume that
u(x ) , v ( x) E H�([a, b]), � > o: ; u2 (x ) + v2 (x ) ¢ 0, x E [a, b].
(30.45)
As in § 30.2, the method used to solve the considered equation will be based on the connection between fractional integrals via a singular integral. In this case (11.16)-(11.19) will be used. Solution of the generalized A bel equation with exterior coefficients. A.
We rewrite
(30.41) in terms of the fractional integration (2.17)-(2.18):
(30.46) One can eliminate 16_cp or 1:+ cp using (11.18) or (11.19), respectively. Both ways are equivalent and give the solutions which differ only in form. Substituting the fractional integral 16_cp from (11.18) into (30.46) , we arrive at the singular equation
a l (x ) � (x ) + a27r(x )
b
j t�(t) dt a
-X
=
f (x) , (b - X ) a
(30.47)
where
a 1 (x) = u(x) + v (x) cos o: 1r, a2 ( x) v ( x) sin a1r =
and
�
x
( )
z:
=
1 cp(t)dt . (b - x)a J (x - t) l-a a
(30.48) (30.49)
Thus we have to solve the singular equation (30.47) and then invert the Abel � equation (30.49). In this process we must take care to find such solutions of (30.47), for which (30.49) is solvable. We use well known results (Gahov [1] and Muskhelishvili [1]) from the theory of singular equations on an open curve, the finite interval in our case. Let
(30.50)
618
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
in correspondence with §
30.1, where the value of O(z) is fixed by the condition 0 � O(a) < 211" ,
{30.51)
and let x be the integer (30.15). We solve (30.47) basing ourselves on Theorem 30.2 and taking into account the fact that the right-hand side of (30.47) is in H* , if I E n:. In the case X 2: 0 we obtain z
I (z - t) 1 - a = cp(t)dt
a
a 2 (z) Zo (z) a (z) (z a) P• (b - z ) a+ P•px-1 ( z) + 1 l(z) A(z) A(z) _
_
a 2 (z) �o (z) 1rA(z)
fb ( zt -- aa ) P• ( bb -- zt ) a+p, a
where A(z) and Zo (z) are the functions
/J a
= 1 - na -
l(t)dt Zo (t)(t - z) (30.52)
(30.31) and (30.16' ),
O(a)
2 11" ,
(30.53)
and Px - 1 { z) is a polynomial of degree x - 1 with arbitrary coefficients ( Px -1 ( z) = 0 in the case x = 0). It remains to invert {30.52) as the Abel equation relative to cp(t). The principal fact that we are to investigate is to discover whether the right-hand side of (30.52) allows the solution of this equation. Since we need to find any solution cp E H* , we need to clarify, by Theorem 13.14, whether the right-hand side belongs to the space n:, if l(z) E n:. We begin with the following lemma. Lemma 30.2.
The weighted singular operator
b(
) ( )
b - z P " -1 f z - a P • -l(t) dt sP • , p , I - b-t t-z 1r t-a _
a
--
maps the space n: onto itself provided that a - 1 < IJ a � a and a - 1 < /J b � a . Proof. Let Ht'(et , e2 ) b e the space maps Ht'(e 1 , e 2 ) onto itself, if
(13.51).
By Theorem
11.1 the operator Sp.,p,
619
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
Since H: is a union {13.53) of the spaces H6(t 1 , t2 ), the operator s,... ,,... then preserves this union invariant, if Pa � a < 1 + and which � a < 1+ coincides with the assumptions of the lemma. • We turn now to the right-hand side of (30.52) and recall that /(t) E H:. By Theorem 11.1 we have Zo(z) E H6([a, b]). Since multiplication by a function belonging to H>.([a, b]) ,\ > a preserves the space H�, the right-hand side of (30.52) belongs to H: by Lemma 30.2, provided that
Jla
Jlb
a - 1 < Pa � a, a - 1 < Jib � a. By
Jib
(30.54)
(30.53), in order to satisfy the latter of these conditions we need to choose (30.55)
As regards the former condition, we have
nna = 0,1, a=
if if
O(a)/(211") < 1 - a , O(a)/(211") � 1 - a,
(30.56)
and then the index (30.15) takes the form x=
(we recall that
{ 1, [ O(b) 2 11" ] + 0,
if O(a) if O(a)
(1 - a)21r, � (1 - a)21r
<
(30.57)
0 � O(a) < 211" ).
O(a) = (1 - a)21r corresponds to the case u(a) = 0. Thus the integer n a and nb being chosen by (30.55) and (30.56), the right-hand side of (30.52) belongs to H:, and therefore (30.52) is solvable relative to cp by Theorem 13.14. Inverting this Abel equation, we obtain the general solution of the generalized Abel equation (30.41) in the case x � 0: Remark 30.1. The relation
sin a1r d j� u(t) +v(t) cos a1r f(t)dt (x) = � + CJ:
cp
1:-1
_
a
b v(t)Zo(t) dtj ( x - a ) "'• ( b - z ) "'• + a f(s)ds ( sin1ra1r ) 2 .!!_dx J� A(t)(x - t) a s-a b-s Zo(s)(s - t) ' a
a
(30.58)
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
620
where Ci are arbitrary constants and
(30 .59)
G
with
Z0 (t) and A(t) functions (30.16') and (30.31) and
{ -8(a)/(21r),
if 0 5 8(a)/(21r) < 1 - a, 1 - 8(a)/(21r), if 1 - a 5 8(a)/(21r) < 1, l' b = 8(b)/(21r) - [8(b)/(21r)] - 1.
l' o
=
B y interchanging the order of integration
X
L...J Ck <J'k (z) +
-
sin a1r 1r
(30.60)
(30.58) is transformed to d 1 /C(z, t)f t)dt, ( dz b
(30.61)
a
where
/C(z, t ) _-
u(t) + v(t) cos a1r (z t ) + A(t) _
-a
We can transform also (30.58) to another form not containing integrals in the principal value sense. In view of (11.16) we have •
sm a1r
--
1r
Replacing here
s - a) 1 ( -t-a b
a
a
= - cos a1r
v
( ) + + Ib - <1' · -n a a
a
a by 1 - a, we obtain the representation of the singular integral in
621
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
the form sin a?r -1r
,
o
)
t , cJe d (-t - a t -a
f
sin a 1r
8 -
€
( 30.62) Applying this representation to the singular integral in the last term in ( 30.58 ) , we arrive at sin cx1r d
-1
�
j A(t)u(t)f(t) (z - t) a dt a
d j v(t)Z(t)dt d j dr j
2
t
�
a
,
T
a
( 30.63) where
Z(t) = (t - a) l-a+ l'• (b - t) a +�£• Zo(t). case x < 0, the singular equation ( 30.47 )
As regards the Theorem 30.2 if and only if
�t){:t } at' d) + 0 , J Zo(z z - a �'• b - z a �£•
is solvable by
,
=
k = 1, . . . , l xl .
( 30.64)
a
If these conditions are satisfied, ( 30.47) has a unique solution which is given by the same resulting relation ( 30.52) with the term containing P. _ 1 (z), being omitted. The final result of this investigation is stated in the following theorem. Theorem 30.5.
Let the assumptions in ( 30.45) be satisfied, let + e -a w i v(z) O(z) = arg u(z) u (z ) + e a"'•v. (z ) ,
0 ::; O(a) < 21r,
and let x be the integer ( 30.57) . The generalized Abel equation ( 30.41 ) is unconditionally solvable in the space n• for each right-hand side f(z) E H!, if x 2:: 0, in particular, if O(b) 2:: 0. Its general solution is given by ( 30.58 ) or ( 30.63) . If x < 0, the equation is solvable if and only if the conditions in (30.64) are
622
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
satisfied and then it has the unique solution given by (30.58) or (30.63) under the choice crc = 0. B. Solution of the generalized Abel equation with the interior coefficients.
As in the case of the whole axis, the equation with interior coefficients is easier because we are to solve the Abel equation first and afterwards the singular equation. Equation (30.42), that is
is transformed by means of (11.16) to
Inverting the Abel equation relative to the expression in the square brackets, we obtain the singular equation
a1 (x),P(x) + .!_I a2t(t),P(t) - x dt = g1 (z) , b
1r
(30.65)
iJ
where
a1 (z) and a2 (x) are the same as in (30.48) and ,P(z) = ( z - a)a
sin a1r 1r
(z
(30.66)
(t)dt a)a .!!. dx._ I (xg - t)a . �
_
iJ
It remains to solve (30.65). Its solvability is determined by the same integer :x as above and the expression for its solutions is known - Gahov [1] or Muskhelishvili [1]. The derivation of the final result is left to the reader.
30.4. The case of constant coefficients We consider especially the generalized Abel equations (30.42) in the case when the coefficients are constant.
(30.17), (30.18) and (30.41),
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
623
A. Equation on the whole axis.
If the coefficients u and v are constant, the solution of (30.17) may be obtained avoiding an application of the general theory of singular integral equations with variable coefficients (but the connection with the singular integral, as before) . Let us consider these equations given in the form (30.17'):
M a cp = _1_ r(a)
00
j Ct + c2sign (z - t) cp(t)dt = /(z) ,
z E R1 ,
lz - tp-Ot
- oo
(30.67)
where 0 < a < 1, Ct and c2 are constants, ci + c� #; 0. The factor 1 /f(a) is taken for symmetry with the inverse operator. We note that the integral M01 cp is the Feller potential considered in § 12. The following theorem presents the explicit form for the inverse operator (M01) - 1 . We stress that the integrals in (30.68) are interpreted as conventionally convergent in the norm of the space Lp (R1 ) . They converge also almost everywhere.
Equation (30.67) is solvable in Lp (R1 ) 1 < p < 1 /a for any right-hand side /(z) E l01(Lp ) and has the unique solution
Theorem 30.6.
,
a
cp(z) = Af(1 - a) a
= Af{1 - a)
,
00
.
f Ct + c2signl(z+ - t) [/(z)
-oo
lz - tl a
00
_
j 2ct /(z) - (ct + c2)/(zl+-at) - (ct - c2)/(z t
0
(30.68)
f(t)]dt +
t) dt
'
where A = 4[c� cos2 (a1f/2) + c� sin(a1f/2)]. Proof. We have
(cf. (12.10)) . So the connection the singular operator yields.
(11.10) between the fractional integration I±_
and
(30.69) with a 1
= u + v cos a1r and a2 = v sin a1r.
Since S2 cp :: -cp by ( 11 . 2 ) , the singular
624
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
operator N with constant coefficients is invertible and
Inverting the Abel equation in operator, we obtain
(30.69)
by Theorem
6.1
and then the singular
(30.70) where D+ is the Marchaud fractional derivative from (11.11') into (30.70), we obtain
(5.57), (5.60). Substituting SD+f (30.71)
which gives (30.68). • We single out the important particular cases of (30.67): 00
f -oo
00
f sign (x - t) ,P (t)dt = g(x),
l()(t)dt lx - ti l-a = f(x),
lx - ti l-a
-oo
l()(t), ,P(t) E Lp(R1 ), 1 < p < 1/o:, and f(x), g(x) E Ia (Lp)· (30.68) solutions of these equations are given by the relations
where
(30.72) According to
00
1r f(x) - f(t) dt tg 0: j l()( X) - � 2 11' 2 lx - tl l +a ' - oo _
(30.73)
00
o: o:1r g(x) - g(t) ,P (x) = 2 11' ctg 2 j lx _ tl l +a st. gn (x - t)dt. - oo
(30.74)
Thus we have obtained the inversion of the potentials 1a and H a see (12.1), (12.2), (12.1') and (12.2'). In the case of sufficiently good functions /(x) and g(x), e.g. differentiable ones and those such that 1 /(x)l � c(1 + lxl) - ", v > 1 - o:, these relations may also be written in the form
l()(X ) =
_.!._ tg o:1r .!!._
21r
2 dX
00
f sign (x - t) f(t)dt,
- oo
Ix - t Ia
(30.75)
625
§ 30. THE GENERALIZED ABEL INTEGRAL EQUATION
1 a?r d ctg .,P(x) = 21r 2 dx
00
g(t)dt lx - tier ·
f
-
oo
(30.76)
We note that a simple condition on the right-hand side /(x), sufficient for the solvability of (30.67), is that it may be represented in the form (30.77) Then f(x) E Jer ( Lp ) , 1 < p < 1/a, by Theorem 6.6 and so (30.67) is solvable in Lp . For functions f(x) of the form (30.77), the inversion given in (30.68) may be also written in the form
d
00
j c2 + clx1s-igntier(x - t) f(t)dt.
(30.78)
- oo
Equation on the interval.
As regards the equation
� f u
b
a<x<
b,
(30.79)
with 0 < a < 1 and constant u and v, we shall obtain its solution from the general relation (30.58). The function O(x) from (30.50) is now constant: 0( x) = 0 = arg
( it is clear that 0 = 0
u + e - er 1ri v , u + eer ..•v .
0 < 0 < 21r
(30.80)
if and only if v = 0).
The inequalities 0 < 0 < (1 - a)21r and ( 1 - a)21r < 0 < 21r are equivalent to the inequalities uv < 0 and uv > 0, respectively. <•+2•ar)i Proof. From (30.80) we have ; = 1 - �,1 _ 1 e -er1r a . Since 1 - eu = - 2 ie /2 x
Lemma 30.3.
·
sin( z /2 ) , we obtain
! _ sin ([O - (1 - a)21r]/2)
v-
which easily yields what is required. •
sin(0/2)
•
•
u
626
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
We derive from Lemma 30.3 that the index x, defined in (30.57) , of (30.79) equals 1 if the coefficients u and v are of the same signs and equals 0, if they have different signs. We conclude then from (30.58) that the generalized Abel equation (30.79) with the coefficients u and v of different signs has a non-trivial solution in the case f(x ) = 0:
�
J -a - 9/(211') (b - t)a- 1+9/(211')dt <po (z) = .!!._ dx (x - t ) (t - a)
1 a
1 d ( b z ) a-1+9/(2r) ds = - s -a (l - s ) - 9/(2r) s + -� x-a _
0
( b - a ) ( 1 - a - (} /( ( x - a) 2 _
21r)) f1 s-a (1 0
_
( ) a- 2+9/(211') ds. s ) - 9/{211') s + xb -- Xa
Applying ( 13.18), we obtain const <po (z ) = (x - a )a+9/(211') (b - z) 1 - 9/{211') ,
0 < (} < (1 - a ) 2 .
7r
(30.81)
Taking (30.63) and (30.81) into account we state the resulting conclusion in the following theorem.
Equation (30.79) is solvable in the space H* whatever the right-hand side f(x) E H; was. The solution is unique, if uv > 0, and contains one arbitrary constant, if uv < 0. All the solutions are given by the fonnula
Theorem 30. 7.
�
c u sin a1r d f(t ) dt cp (x) (x - a)a+ 9/(2"") (b - z) 1 - 9/(211') + A -1r- dx j (x - t)a a
-
v A
( sin n1r-) 2 d f� Z(t)dt)a d f dr 1 f b
t
1r
dx
a
(x - t dt
a
(t - r) -a
T
f(s)ds Z(s ) (s - r) { ao .82)
where c = 0, if uv > 0, and c is arbitrary, if uv < 0, and A = u2 + 2uv cos a1r + v 2 ; Z(t ) = (t - a) 2 -a- 9/(2,.. ) (b - t ) a - 1+ 9/(2"") , if uv > 0, and Z(t) = [(t - a) /(b t )p -a- 9/(211') , if UV < 0. Let us consider the following important particular cases. The Carleman
§ 30.
THE GENERALIZED ABEL INTEGRAL EQUATION
627
equation
J lxcp-(t)dttp- a b
= f(x)
(30.83)
a
which attracted the attention of many authors (see references in § 34.1) is contained in (30.79) under the choice u = v = 1 . The result in (30.82) gives its unique solution if we choose c = 0 and A = 4 cos2
( )
a/2 Z(t) = tb -- at
a1r , 2
By simple calculations it may be shown that (30.83) with the right-hand sides
f(x) = (x - a) n ,
n
= 0, 1, 2, . . . ,
and
f(x) = [(x - a)(b - x)]
has the solution
respectively. There exists another inversion formula for the Carleman equation (30.83):
cp (x)
tg (a1r/2) d = dx 21r
b
j sign (x -a t) f(t)dt lx - tl
a
sin2 (a1r /2) d + 21r2 dx
j M(x ' t)[(t - a)(b - t)] - a/2f(t)dt ' b
a
where
J b
- a)(b - y) M(x,t) = J(t - a)(b - t)t -- J(y y a
-
-
dy �-=---� ( l - a)/2 lx - Y":""-:"-:-----:� la [(y - a)(b - y)]
X -:--
(30.84)
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
628
Compare this with (30.75) . Let us prove this result. Equation (30.52) in application to the considered particular case {30.83) has the form
:t: (t)dt f (z
_
=
a
On the other hand, since (30.83), we have
_!_ tg 211"
a1r 16 [ (z - a)(b - z) ] a/2 f(t)dt. 2
(t - a)(b - t) t - z
a
{30.85)
6 2 ! /(z) + _!_ tg a1r f (z - a)(b - z) 01/ f(t) dt . f6 (t
{30.86)
a
:t:
Inverting (30.85) and (30.86) as Abel equations and summing up the result, we obtain the relation 2 1-as
2
=
2B
(
where ( 12.44), (12.45) and (12.48) are used for brevity. By simple transformations we represent the second term as
X
f6 It -dtzl01 f6 ([ r(r(t)T) ] 01/2 [ r(r(t)T) ] (l - a)/ 2) f(TT-)dtT ' _
a
r(t) (t -
a
t) .
= where a)(b - We apply ( 12.47) to the first term here, after which (30.84) is obtained by simple calculations. As regards another particular case
f6 sign lz -ti(z l--at) 'f/J(t)dt g(z) =
(30.87)
a
this differs from (30.83) by the fact that the corresponding homogeneous equation has a non-trivial solution. The result in (30.82) yields the general solution of (30.87):
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
629
:&
c 1 a1r d f g(t)dt ,P (z) (z - a)(l +a)/ 2 (6 - z) (l+ a)/ 2 + 21r ctg 2 dz (z - t) a
z: 2 c T d f (t - a ) ( 1 - a)/ 2 dt + 1r2 dz 6 - t (z - t) a
Cl
os
Cl
( )
t b d f dT f 6 - s ( 1 -a )/2 g(s)ds (s - T) a · (t - T) l -a T s - a X dt Cl
(30.88)
The relation
b
cos2 � 2
d j N(z, t)(t - a)(l - a)/ 2 (6 - t)(l - a)/ 2 g(t)dt, - 21r2 dz Cl
(30.89)
is also valid, where b
N(z, t) = f .j(t - a)(6 - t) - .j(y - a)(6 - y) t-y
Cl
X
sign ( z - y)dy lz - Yl a [(y - a)(6 - y)](l - a)/2 .
Its derivation is similar to that of (30.84) .
§ 3 1 . The Noether Nature of Equations of the First Kind with Power-Type Kernels In this section we shall investigate the problem of normal solvability and the index ( Noether nature) , properly interpreted, for equations of the first kind:
M VJ = f lzc(z,t) - ti l - a VJ(t)dt = /(z), 0
z E 0, 0 = [a, 6], -oo :5 a < 6 :5 oo.
(31 .1)
630
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
The generalized Abel equations, considered in §30, are their particular cases. We shall reduce (31.1) to an equivalent integral equation of the second kind. The latter will be in general singular. We give first the necessary preliminaries from the theory of N oether operators and from the theory of singular integral equations. Then we shall consider the case of the whole axis n = R1 , and in conclusion deal with the case of a finite interval n = [a, b], -00 < a < b < 00.
3 1 . 1 . Preliminaries on Noether operators We give here the bare minimum concerning Noether operators, which is necessary for the presentation of questions on the solvability of (31.1). These questions are developed in §§ 31.2 and 31.3. The theorems about the Noether properties of operators in abstract Banach spaces given below are known as the Nikol 'skii Atkinson-Gohberg theorems. More details on the theory of N oether operators can be found in the books of Gohberg and Krupnik [4] and S.G. Krein [1]; see also references in § 34.1 . Let X and Y b e Banach spaces. The ring of all linear operators bounded from X into Y will be denoted by [X --+ Y]. Definition 31.1.
The set Zx(A) = {
of all zeros of the operator A in the space X is called the kernel A E [X --+ Y] . For the adjoint operator A* E [Y* --+ X*] we denote similarly
of the operator
Zy· (A* ) = { ,P : A*,P = 0, ,P E Y* }. This subspace in Y* is called
the co-kernel of the operator A.
The operator A E [X Y] is called normally solvable in X if its range A(X) consists of elements orthogonal to the co-kernel:
Definition 31.2.
/ = A
--+
,P E Zy· (A*).
The operator A E [X --+ Y] is normally solvable in X if and only if its range A( X) is closed in the space Y .
Theorem 31.1.
We introduce the following notation for the dimensions of the kernel and co-kernel:
n = nA = dim Zx (A),
m
= rnA = dim Zy· (A* ).
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
631
The number rnA is called the deficiency number of the operator A. Sometimes both numbers rnA and nA are called the deficiency numbers, the former of them being also known as the nullity of the operator.
The operator A E [X -+ Y] is called the Noether or Noetherian operator, if it is normally solvable in X and its kernel and co-kernel are finite-dimensional: n < oo, m < oo. The ordered pair ( n, m) is called the dimensional characteristic or d characteristic of the operator A . The difference x = n - m is called the index of the operator. We shall use the designation x = xx -y (A) and write xx (A) in the case X = Y. It is known that the index of the operator is stable Definition 31.3.
relative to small perturbations. For Noether operators the following statements hold. Theorem 31.2. If A E [X -+ Y] the operator BA E [X -+ Z] is also xx - z (B A)
and B E [Y -+ Z] are Noether operators, then Noetherian, and
= xx -y (A) + Xy-z (B) .
If the operator A E [X -+ Y] is Noetherian and T is a completely continuous operator from X into Y , then the operator A + T is also Noetherian, and xx-y (A + T) = xx-y(A) .
Theorem 31.3.
The operator A E [X -+ Y] is Noetherian if and only if it admits the left and right regularizers R, E [Y -+ X] and R,. E [X -+ Y]: R,A = E + T1 and AR,. = E + T2 , where Tt and T2 are operators completely continuous in X and Y, respectively. Theorem 31.4.
The usual examples of N oether operators are the singular integral operators and (30.10). We recall known results concerning the Noether properties of these operators which will be needed below.
(30.1), (30.8)
Theorem 31.5.
The singular integral operator a (x) No tp = a t (x)tp(x) + 2 1r
00
j tpt (-t)dxt
-oo
=
f (x)
(31.2)
where a 1 (x), a 2 (x) E C(R1 ) and are real-valued, is Noetherian in the complex valued space L,(R1 ), 1 < p < oo, if and only if a�(x) + a�(x) :/ 0, x E R1 . The operator No has the d-characteristic (x, 0) if x � 0 and (0, l xl ) if x :5 0, where x is the integer (30.4).
632
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
A similar operator on a finite interval will be considered with a "regular" term: ,
,
N
(31 .3)
Cl
In the case of an interval the conditions of normal solvability and the relation giving the index are already dependent on the space of the solutions. We shall give two known assertions. The first will concern the "classical" setting of the question of normal solvability, when one considers the orthogonality to solutions not of the adjoint equation N* ,P = 0, but of the homogeneous tmnsposed equation: ,
,
a 1 (z) ,P (z) - .!.1r j a2t(t)¢(t) - z dt + J K(t, z) ,P (t)dt = 0. Cl
(31 .4)
Cl
and solutions of (31 .3) are sought in the space of functions, Holderian in (a, b) and having integrable singularities at the end-points with non-fixed orders. The second assertion will deal with the case when the Noether nature is interpreted in the sense of Definition 3 1 .3 and the operator in (31.3) is considered in the space Lp (p) with the weight function p(z) = (z - a)�'(b - z)". We recall that the coefficients a 1 (z) and a2 (z) in (31.3) and (31.4) are assumed to be real-valued. As regards the kernel K(z, t) we assume that
gn(z - t) , "'""( z, t ) = A(z, t) + lzB-(z,tlt)si l -e
0 < e � 1,
where A(z, t) and B (z, t) are functions, Holderian in both variables. Let H*, n: , H6 and H be the function spaces considered in § 30. 1. As previously in § 30. 1 , G(z) denotes
Since a 1 (z) and a 2 (z) are real-valued, we have G(z) = e il (z) . We consider the value of 0( z) = arg G( z) to be chosen by the condition 0 � 0( a) < 21r with 0( z) extended by continuity to other points z E (a, b). Theorem 31.6. Let a 1 (z), a2 (z) E H, ai{z) + a�(z) # 0, z E [a, b), and O(a) #; 0, O(b) #; 21rk, k = 0, ±1, ±2, . . . . Then (31.3) with the right-hand side /(z) satisfying the assumptions of Theorem 30.2, is solvable in the spaces H*, n: , H6 or H if
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
633
and only if b
j /(z)t/Jj (z)dz =
0,
(31.5)
a
where {tPj } is the complete system of solutions of (31.4) in the spaces H, H& , n: or n• , respectively. The difference of the numbers of linearly independent solutions of the homogeneous equation {31 .3) and of the solvability conditions given in (31 .5) is equal to the index x given by (30. 15) . Let a 1 (z), a 2 (z) E C([a, b]). The operator (31 .3) is Noetherian in the space Lp(p), 1 < p < oo, p(z) = (z - a)�'(b - z)", - 1 < I' < p - 1 , - 1 < v < p - 1, if and only if I) a�(z) + a�(z) -:# 0, a :5 z :5 b; II) 6(a) -:# 21r 1 !" , 6(b) -:# -21r � ( mod 21r). These conditions being satisfied, the index of the operator {91.9) is equal to
Theorem 31. 7.
[ 21r
"' = 6(b) + 1 + v p
] [ +
1 + I'
p
_ 6(a) 21r
]
(cf. {90. 15)). In the case tt(z, t) = 0 the operator given by d-characteristic (x, 0) if x � 0 and (0, lxl) if x :5 0.
(91.9}
has the
In the following subsections we consider the question of the Noether nature of the operator in (31.1). Equation {31. 1) as an equation of the first kind is not normally solvable generally speaking, so the question of its "Noetherness" requires the proper setting. The operator M will not be in general Noetherian as an operator acting from certain space X into the same space X . In the case X = Lp(R1) it is not even bounded from X into X - see the Hardy-Littlewood Theorem 5.3. If mesO < oo, the operator M is bounded in Lp(O), but M cannot be Noetherian as an operator from Lp(O) into Lp(O). Indeed, if M E [Lp(O) -+ Lp ( O )] is Noetherian, then by Theorem 31.4 there exists the bounded operator R such that RM = E + T, where T is completely continuous in X = Lp(O), mesO < oo , (under rather weak assumptions on c(z, t)), we see then that the identity operator E = RM - T is completely continuous in X, which is impossible. A natural question arises: may one construct, for the operator M non Noetherian from X into X , such a space Y that the operator M is Noetherian as an operator from X into Y? The construction of such a space for achieving Noether properties is sometimes called the normalization of the operator. By the Hardy-Littlewood Theorem 5.3 the operator M with a bounded function c(z, t) is bounded from Lp(O), 1 < p < 1/at, into £9(0), q = p/(1 - atp) . It is not, however, a Noetherian operator from Lp into L9 • Let us show this for c(z, t) = 1 . Correspondingly with Theorem 3 1 . 1 we have t o show that the range M(Lp) i s not closed in L9 • For c(z,t) = 1 by (30.34) we have M(Lp) = Ia (Lp) C L9 , where I a (Lp) is the space of fractional integrals. We recall that Ia (Lp) -:F L9 - see (6.5).
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
634
Since infinitely differentiable finite functions belong to Ia (Lp) as for example, by Theorem 6.5, the space Ia (Lp) is dense in L 9 and consequently is not closed in L 9 • In the general case under rather weak assumptions on c( z, t) we shall show that M ( Lp ) � Ia ( Lp) and that the range M ( Lp) is closed in I a ( Lp). In other words, we shall treat the operator M as Noetherian from Lp into Ia (Lp), the latter being considered as a Banach space relative to the norm (6.17).
31 .2. The equation on the axis The equations to be considered are 00
(M
f
-oo
c(z, t) lz _ ti l -a
(31.6)
We allow the function c(z, t) to be discontinuous on the diagonal t = z:
{
t < z, c( z, t) = u(z,t), v(z, t), t > z,
{31.7)
so that
(M
z
00
f v(z, t)
- oo
(z - t) -a
(t - z) -a
z
(31.6')
In the case of continuity at the diagonal (u(z, z - 0) = v(z, z + 0)) the operator M will always be of the Fredholm type i.e. with the index equal to zero, in the corresponding setting. It will be of the Fredholm type also, if, e.g. v(z, t) = 0. We define now the class of admissible functions u(z, t) and v(z, t) in (31.6). Let R� be the half-plane {{t, z) : t < z} and = {(t, z) : t > z } .
R2_
junction u(z, t) defined in the closed half-plane R� belongs to the class H;(R�) if i) u(z, z) E C(R1 ) ; ii) u(x, t) is Holderian in z of order A, 0 < A < 1, unifonnly with respect to t:
Definition 31.4. A
{31.8)
H;(R2_) is defined similarly. Writing c(z, t) E H;(Ri) will denote that u(z, t) E Hz (R+ ), v(z, t) E, Hz (R_ ). We observe that functions in H; ( .Ri) are bounded, but are not necessarily continuous in t. For example u(z, t) = z{l + z ) - 1 sign t E H;(.R�), A = 1. The class
�
-2
�
-2
2
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
M(Lp) C pr(Lp), 1 < p < 1/o:. We represent the operator M
A. Imbedding as
635
u(t, t)cp(t) dt + v(t, t) cp(t) dt + K+
00
(31.9)
_
_
00
where z
Ku+
u(z, t) - u(t , t)
J (x - t)l -a
(31.10)
00
v
_
(t - x -a
z
By means of (11.10) we obtain
(31.11) where
No
[u(z, z) + v(x, z) cos o:1r)
Theorem 31.8. If c(x, t) E
sin o:1r r
00
j v(t, t)
- oo
t-x
(31.12)
n;(Rl) with .\ > o:, then M(Lp) � Ia (Lp ) and (31.13)
Proof. By (31.11) and the boundedness of the singular operator in Lp( R1 ) (see Theorem 11.2) it is sufficient to prove the statement of the theorem for Kt
0
+ t a J[u(z, x-ts)- u(x-ts, z-ts))k(s)cp(z-ts)ds, 00
0
(31.14)
636
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where k(s) is the function
(6.10). Let us prove (31.14). For /(z) = K!cp we have J(z) = J [u(z, z - s) - u(z - s, z - s)]sa- 1 cp(z - s)ds and so CX>
0
CX>
f(z)-f(z-t)=t a j (u(z, z-t-st)-u(z-t-st, z-t-st)](s+ 1) a- 1 cp(z-t-st)ds -1 CX>
-ta j (u(z-t, z-t-st)-u(z-t-st, z-t-st)]sa- 1 cp(z-t-st)ds 0
CX>
= j [u(z, z-t-s)-u(z-t, z-t-s)]sa- 1 cp(z-t-s)ds 0
CX>
+ta j k(s+1)[u(z, z-t-ts)-u(z-t-ts, z-t-ts)]cp(z-t-ts)ds, -1 which gives
(31.14).
By this result
_:!!__ u(z, z-t -s)-u(z-t, z-t -s) a tn (z-t-s)ds D a+ ,£ Ku+ cp f(1-a) JE t 1+a J0 s 1 -a CX>
CX>
r
+ r( l':.cr) J � J [u(z, z-ts) -u(z-ts, z-ts)]k(s)
CX>
According to (31.8)
c
CX>
CX>
g(z - t)dt I AE (z)l � (1 + lzl )>. j t 1 +a - >.(1 + l z - tl)>- ' g (z) _- j lcp(zS 1-- as) l ds . (31.16) 0 0
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
637
Here g(z) E L9(R1 ), q = p/( 1 - ap) , by Theorem 5.3, so
According to (6.32) the inner integral is dominated by
cll
c(1 + t) - .\f a , so II Ae llp �
0
B,(z) = r(l � a) J k (s)ds J u(z, z - t) - ;(z - t, z - t) cp(z - t)dl 0 00
00
u
or by ( 6. 11 )
B. (z) = - r(l� a) J k(s)ds J u(z, z - t) - ;(z - t, z - t) cp(z - t)dt e•
oo
0
0
dt j k(s)[u(z, z - ts) - u(z - ts, z - ts)]
Q
Hence
oo
dt j s.\ k(s)
oo
0
0
and applying the Minkowsky inequality, we obtain
I f e
oo
dt l i B. II, � c t t -A s .\ l k(s) l ds The inequality
{
_l
oo
l
} 1/p
a > 0, - oo
( 3 1 . 18)
( 3 1 . 19)
638
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
is valid. Indeed let h > 0 since the case h < 0 is reduced to the former one by the change z = -y. Denoting the left-hand side of (31.19) by J (h), we have
f
00
J (h ) 5
0
11/1 ( -z)ldz (1 + z)•(l + z + h)• +
(f f) h/2 +
0
00
h /2
11/l (z) l dz (1 + z)•(l + lz - h i )•
h /2 00 � (1 + h) - 4111/l l h + j (1 + h - z) - 411/l (z) ldz + j (1 + z) - 4 11/l(z)l dz 0 h/2 -a � (1 + h) - 4111/llh + ( 1 + 2h ) 111/llh , which yields
(31.19).
By
(31.19) we obtain 00
£
IIBe llp � cll.- 1 dt j s>. lk(s)l(1 + ts) ->. ds 0
0
from (31.18). The inner- integral is estimated as follows: 00
J(t) = j s>. lk(s)l(1 + ts) - >. ds � et -a
as
t --+ 0.
(31.20)
0
In fact since
lk(s)l � csa- 2 for s � 1, we have 00
J(t) � c + c j s>. +a - 2 (1 + ts) - >. ds 1 1 =c + c J e -a (e + t) ->. t:Ie 0
1/t 1 J a ->.e - a (1 + e) ->. de � et - a . =c + et 0
By
(31.20) (31.21)
639
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
and then, according to (31. 15) , II D+ , .:Kj
M
of the potential type by the
Theorem 6.1 allows one to invert the fractional integration to the right by fractional differentiation, provided that we deal with the range p:x(Lp ), i.e. If.Di./ = /, f E Ior (Lp ) · Hence the imbedding M(Lp ) � Ior (Lp) obtained in Theorem 31.8 allows to write
M
N = Di.M,
(31 .22)
1 < p < 1 /a .
The following lemma presents an explicit expression for the operator N. 31.1. Under the assumptions of Theorem 31.8 the operator N = D+M, with M the operator given by (31.6), has the form
Lemma
(31 .23 )
N = r (a) No + T,
where No is the singular operator given by (31 .12) and
( 31.24)
where a 7':!" - f( 1 a) u
00
X
(31.25)
0
0
S
d+t j v(x, x+t+s)-v(x+t, x+t+s)
T.-
00
!!__ f u(x, x-t -s)- u(x-t, x-t-s) f __!
00
The operators TJ and T"- can be also represented as �
T,}" tp = f( l � a) f T,}" (:r, T)tp(T)dT, -oo
T.- '1' = f l � ) ( a
J T.- (:r, T)tp( T)dT, 00
�
(31 .27)
640
where
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
T) - u(t, T) T.+ (z ' T) = 1 (z u(z, 1+a (t - T) l -a dt t) T �
u
1 1 _ 1 u(z, T) - u(T + s(z - T), T] ds - z-T ' s 1 -a (l - s) l + a 0
_ _
_
Tv
(
(31.28)
T) - v(z, t) dt z, T) = 1 (t -v(z, 1+ z) a (T - t) 1 -a T
�
1 1 _ 1 v(z, T) - v[z , z + s(T - z)] ds . - T-z s 1+a (l - s) 1 -a 0 _ _
Proof. From (31.11) we have D+M
D+Kt
Di./ = cos a1rD� / + sin a1rSD� f
so
(31.29)
for f E Ia (L,). Since K;
dt 1 u(z, z - t - s) - u(z - t, z - t - s) t" (z - t - s)ds 1 1+ s 1 -a r(1 - a t a 00
a
)
00
T
0
0
+ r(1 - a) 1 k(s)ds 1 u(z, z - t) - tu(z - t, z - t) cp(z a
00
0
00
0
_
t)dt.
(31.30)
The pass age to the limit here is easily justified if it is in the norm of Lp . It is not difficult to show that the integrals in (31.23) exist not only in the sense of convergence in the £,-norm, but in the usual sense also for almost all z. In (31.30) the second term is identically zero in view of (6.11). Expression (31.26) can be deduced by obvious symmetry arguments. The representation (31.27) and (31.29) are obtained by easy transformations. • Remark 31.1. It follows from Theorem 31.8 that the operator T in (31.23) is bounded in L,(R1 ) at least, if 1 < p < 1/a. It will be shown below that the
§ 31. EQUATIONS WITH POWER-TYPE KERNELS
641
operator is completely continuous in Lp(R1 ). Now we observe that the kernels (31.28) admit the estimates
c(x - ) >.- l IT_i (x, r) l � (1 + rlxl) >- , which follow from (31.28) by (31.8), c being a constant. C. The Noether nature of the operator M.
The previous consideration paved the way for the result on the Noether nature of (31.6). Let c(x, t) be of the form (31.7). Theorem 31.9. Let c(x, t) E H; ( .R�J with � > a. The operator M is Noetherian from the space Lp(R1), 1 < p < 1/a, into the space Ia (Lp) if and only if
(31.31) Under this condition the index of the operator M is equal to XLp-Jor(Lp)( M )
=
� j d {c1 (x) 00
arg
+ ic2 (x)tg
-oo
where c1 (x) = u (x, x) + v(x, x), c2 (x) = u(x, x) - v(x, x).
a11" 2}
(31.32)
Proof. Since � > a the imbedding M (Lp) � 1a (Lp) and the representation in (31.22) are valid by Theorem 31.8. Elementary arguments show that the property
of the operator M to be Noetherian from Lp into /a ( Lp) is equivalent to that of the operator N from Lp into Lp . The condition (31 .31) is necessary and sufficient by Theorem 31.5 for the operator No to be Noetherian and under this condition the index of the operator No is equal to the integer (31.25). So in correspondence with Theorem 31.3, it is sufficient to show that the operator T is completely continuous in Lp(R1 ) . Since a composition of a bounded and continuous operator is a completely continuous operator again, we observe by (31.24) and the similarity of the operators TJ" and T; that it remains to prove the complete continuity of the operator TJ only. We give this statement separately in the following lemma.
If u(x, t) E H; (Ji�), � > a, then the operator TJ" is completely continuous in Lp (R1), 1 < p < 1/a.
Lemma 31.2.
Proof.
We check the Riesz criterion
(31.33)
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
642
(J
� I> N
)
l(ti" cp)(z) IP dz .
l/P � q(N) II 'I' II, , 'I (N) N-::oo 0,
(31 .34)
which guarantees (see Dunford and Schwartz [1, p.324]) the complete continuity of the operator Tj . We observe that the representation of the operator Tj given in (31.27) and (31.28) ''works well" for checking (31.33), while (31.25) is well-matched to (31.34). We denote F(z) = (Tjcp)(z). By (31 .27) we have �+ 6
F(z + 6) - /(z) =
�
j Tj(z + 6, T)
�
The estimate for A1 is simple in view of (31.8):
�c
f6 (6 - T).\- l dTII
Further,
We represent Tj(z + T + 6, z) - Tj(z + T, z) �+ T A1 =
f �
A2
_
-
u(z + T + 6, z) - u(z + T, z) dt (z + T + 6 - t) l+ a (t - z) l -a
� + T+ 6
f
�+ T
as
A1
'
u(z + T + 6 , z) - u(t, z) (z + T + 6 - t) l+ a (t - z) l -a dt
'
+ A2 + A3 where
(31 .35)
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
643
z+ T u(z + T, z) - u(t, z) dt. Aa = f [(z + T + 6 - t) - 1 -a - (z + T - t) - 1 -a ) (t - z) l -a . z Let us estimate A 1 first:
where
(13.18) is used. Thus >. -a !Al l :::; cl (1 + lz6+ rl)>-rl -a .
The estimate for A2 is obtained similarly and coincides with applying the mean value theorem to A3, we have
where 0
where
< e < 6/r,
(31.36) (31.36).
so
c 1 = c1 (e) and 0 < e < .\ - a.
By the estimates for A� , A2 and Aa we have
a(r)dr IIA2 IIP <- c6e f rl ->.+e + c6>.- a f a(r)dr rl -a '
a(r) =
Further,
{_[ (1
00
00
0
0
+ lz + ri) - >-P i fP(z) IPdz
}
�p
(31.37)
0 :::; a(r) :::; llfP IIP and it is easily shown that llall9 :::; KllfP IIp for q > max(p, 1/ .\ ), where K = K(q). The where
.
It is clear that
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
644
T 1 (31.37) T_ , T)dT cl shows that the integrals in 00 are finite and admit the estimate f a( with 11 = c � 0
c
so
00
00
s
s.
I(ti . f [( 1 + lz J) ( 1 � lz -t I )] f o �I>N �I>N a d d t 1 z :5 Ct i i �I P (N + 1)• f tl+ a ->. f (1 + l z J )( A- clf a(1 + lz t J ) Af a �I>N Here > 0; we choose c < a and c < � - then the inner integral is estimated according to (6.32), so that u
0
as
)
(
(
00
o
c
(
)
0
1 /p
sl - a as
(
>.fa
_
a;
)
)
00
N which completes the proof of the lemma and, thereby, that of Theorem 31. 9 . Thus, in view of (31. 2 2), equation (31. 6 ) is reduced to the equation of the u
0
•
second kind
00
a1r v(t, t) (t)dt + T
sin
- oo
§ 31. EQUATIONS WITH POWER-TYPE KERNELS
645
where T is the completely continuous operator. Theorem 31 .6 yields the following corollary.
Let u2 (z, z)+v 2 (z, z) =f. 0, z E R1 • Equation (31.6) with the right-hand side f(z) in Ia (Lp) is solvable in Lp , 1 < p < 1 /o:, if and only if
Corollary.
J 1/J; (z)(D+ J)(z)dz = 0, 00
-
oo
j = 1 , . . . , m,
where 1/J; (z) is a complete system of linearly independent solutions of the homogeneous singular equation N* ,P = 0, adjoint to (31.38). D . Regularization of operators of potential type.
In the conclusion of this subsection we consider the problem of a regularization of (31.6), i.e. reducing it to a Fredholm equation, not a singular one. Our goal will be an effective construction - in the explicit form - of the regularizer for the operator M, treated in correspondence with Theorem 31.4. This regularizer will be the operator o: sin O: ?r
Rf = ?rA(z)
00
j lzco-(z,t 11t)+a [/(z) - f(t)]dt,
(31.39)
-oo
where
A(z) = u2 (z, z) + 2u(z, z)v(z, z) cos o:?r + v2 (z, z), u(z, z), t < z, co ( z, t ) = v(z, z), t > z.
{
Let c(z, t) E n;(Ji�J and let u(z, z), v(z, z) E H>.. ( R1), A > o: . If u2 (z, z) + v 2 (z, z) =f. 0, z E R1 , then the open.dor (31.39) is a regularizer of the operotor (31.6) : (31 .40 ) RMtp = tp + T'tp, MR/ = I + T''J,
Theorem 31.10.
where T' and T" are operotors, completely continuous an respectively. Proof. First of all we remark that the operator
Lp
and
1a ( Lp),
R is bounded from Ia (Lp) into
646
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Lp. To see this, we rewrite the operator R in terms of fractional differentiation: z) a v(z, z) a RI = ru(z, (a)A(z) (D+ /)(z) + r (a)A(z) (D_f)( z) ' after which the estimate II R/ IIp � cii / III.. (L. ) follows immediately from Theorem 6.1 and ( 1 1 . 1 1"). It follows from Lemma 31.2 that the operator z
1_ u(z, z) - u(t,t) cp(t )dt (u I+a I+a u) cp - r (a) J (z - t) l- a -oo _
_
_
is completely continuous from Lp into Ia (Lp), if u(z) E H >. ('.k1 ), � > a. So, while considering the composition RM and M R, we may interchange the operation of multiplication by a Holderian function with the operators I+ and I�. According to (31 .9) we have M = M0 +T1 7 where M0cp = r (a)I+(ucp) + r (a) J�(vcp) and T1 is an operator completely continuous from Lp into Ia (Lp) (see Lemma 31.2) . Therefore,
RMcp = A1 (uD� + vD�)(ui+ + vl�)cp + T2 cp = 1 [(u2 + v 2 )cp + uD�I�vcp + vD � I+ucp] + T3 cp, A where T2 and Ta are operators completely continuous in Lp (R1 ) . Since D+I�cp = cos a1rcp + sin a1rScp and D� I+cp = cos a1rcp - sin a1rScp by (11.10) and ( 1 1 .1 1) , we have
RMcp = ! [(u2 + v2 + 2uv cos a1r)cp + sin a1r(uSv - vSu)cp] + T3cp.
The operator uSv - vSu is completely continuous in Lp(R1 ) , 1 < p < oo , since Su - uS is - see for example Gohberg and Krupnik [4, p. 33] . So RMcp = cp + T'cp, where T' is completely continuous in Lp . The validity of the second part ( 31 .40 ) is verified similarly. •
31.3. Equations on a finite interval Let us consider the operators of a potential type in the form
Mcp -= u(z)
b
cp(t)dt cp(t)dt J (z t)l -a + v(z) J (t _ z)l -a + J T(z, t)cp(t)dt _
a
= /(z)
b
z
z
a
( 31 .41 )
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
647
and investigate them from the point of view of the proper setting of the problem of normal solvability. In the end of this subsection we shall obtain the statement on the N oether nature of the operator M as an operator from Lp into JC�(Lp) = Ia[Lp(a, b)]. In such a setting the adjoint operator M* is interpreted as an operator from the space Ia (Lp)* = n:+ (Lp ) of generalized functions into Lp' · In view of wide applications of such equations on a finite interval, we shall be also interested in the "classical" setting of the problem, when · instead of the adjoint operator M* one deals with the transposed operator · M T '· _
b
�
b
v(t) fjJ(t)dt _ ' + J (tu(t)- fjJz)(t)dt l -a + J T(t, z)·.,.,·(t )dt - g (z ) .,., _ J (z - t) l - a � G
(31 .42)
G
and the spaces of solutions tp and 1/J are characterized in simple terms, being in general topological spaces and not Banach ones. A.
Reduction to the equation with the Cauchy kernel
Applying (11.18) and ( 1 1 .16) to (31.41) and (31 .42), respectively, we reduce them after elementary transformations to the equations with the Cauchy kernel:
a1 (z
a 1 (z
) Ai. ( '�if
) ·'·(
.,.,
a z) + a2 (z)(b - �)
7r
z) _
b
�(t)dt JG (b - t)a (t - z) + K
Ai. '�if
=
/ { z) ,
b
a2 (t)(b - t) a fjJ (t )dt K* ·'· _ (z) , 1r (b - z ) a J t - z . + .,., - 9t 1
(31.43 )
(31 .44 )
IJ
where
a t (z) and a2 (z) are functions (30.48) and b
�
�(z) = J tp(t)(z - t)a - l dt, 9t (z) = G
K-t =
r(�{D�+/,
.
SID
�c:t7r) ddZ J g(t)(t - z)a- l dt,
K· ., =
�
r( )
o:_r.,,
(31.45)
(31.46)
where T is the integral operator with the kernel T(z, t) and r is its transposed operator. It is necessary to remark that the "right-hand" representation r = If_ Df_ TT used in the pass age to (31 .44) is valid for functions 1/J which are considered below ' since we shall show that the function r 1/J is representable by the fractional integral of order a.
CHAPTER 6. INTEGRAL EQUATIONS OF THE Fffi.ST KIND
648 B.
The class of admissible perturbations
T.
It is natural to consider the perturbing operator T to have the kernel T(x, t) with a singularity of order weaker than 1 - a , in order that the operator K in (31.4�) should have a weak singularity. By means of the expression for fractional integration by parts we arrive at the requirement that the fractional derivative of T(x, t) with respect to t of order a
6 ,.,..( _- 1 d ex ) - ex l i' (x, t) - r ( a ) dt J T(x, s)(s - t) ds
(31.47)
t
must have a weak singularity. The following lemma presents a simple sufficient condition for the admissible kernel T(x, t). Lemma 31.3.
Let
{
(x, t)(x - t) f11 - 1 , t < x, T(x,t ) = cCt (x, 2 t)(t - x)"Ill2 - 1 , t > x,
(31.48)
where a < f3i :5 1 and Ci (x, t) are functions, bounded in [a, b] x [a, b], differentiable in respect to t as t :/; x and such that 18ci /8tl :5 Ai lx - tl - 1 , i = 1, 2. Then the kernel rf-ex > (x, t) is representable as the sum of a Volterra degenerate kernel and the kernel with a weak singularity: r}t ex) (x t) = T,0 (x t) + '
'
{ cc<�,l -ex6)) (b - X)f12- l (b - t) -ex r
0,
.
'
t > x, t < x,
(31.49) (31.50)
Proof.
Let
t < x at first. We have
In the first integral we change the variable: s = t + w(x - t) and then differentiate under the integral sign, while in the latter integral we differentiate first and then
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
change the variable:
649
s = z + w (z - t). We obtain:
1
+ (z
+
_
t).Bt - a
8c1 (z, s) ( 1 - w).Bt dw J0 8s w0
a
(z - t)l+a - ,8:�
j
(6 - t)/(� - 4) 0
c2 (z, z + w(z - t)) dw w l - .8, ( 1 + w)l+a .
The integrals in the first and in the third terms are bounded since bounded. For the second term we have
ci (z, t)
are
Let t > z. Integrating by parts in (31.47) and then differentiating under the integral sign with respect to t, we obtain
The first term here is a degenerate kernel, others admit the estimates
c -< (t - z)l+a-p, .'
650
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
,
c I (s - z)c2 (z,2-,8s)ds � (s - t) a · � (t - z) l + a- ,8�
·
t
Gathering all the estimates we obtain
(31.49) and (31.50).
C. Noether theorems in the "classical" version.
Since equations (31.41) and (31.42) are reduced to the singular equations _ theory of the latter equations. The (31.43) and (31.44), we can use the Noether main work here is the proper choice of the space for the solutions cp and ,P in the situation of the "classical" approach, based on Theorem 31.6. The problem of such a choice did not arise when we considered the operator M in (31.41) as Noetherian from Lp (a, b) into Ia [Lp (a, b)] in the setting prescribed by Definition 31.3. In the theorems below, the value + v(a)e -awi O(a) = arg u(a) u(a) + v(a)eawi is chosen within the bounds
0 � 0 (a) < 211" .
Let u(z), v(z) E H >. ([a , b]), A > a, and f(z) E H� , let the kernel T(z, t) satisfy the assumptions of Lemma 31.3 and let u2 (z) + v 2 (z) =I 0, O(a) =I 0, O(a) =I (1 - a)211" , O(b) ::j; 2 11"k, k = 0, ±1, . . . . Equation (31.41) is solvable , in the space H* if and only if J/(z),P;(z)dz = 0, where {1/J; } is a complete system a of solutions of the homogeneous equation (31.42) (with g(z) = 0), which have the form (31.51) ,P(z) = (z - a) - a (b - z) -a ,P. (z) with a function ,P. (z), Holderian on [a, b]. The difference between numbers of linearly independent solutions of (31.41) and (31.42) is evaluated by (30.57). Theorem 31.11.
The proof of this theorem is actually set up by the reduction to (31.43) and (31.44), and follows as we shall show from the known facts for such equations. Let us rewrite (31.43) and the homogeneous equation corresponding (31.44) in the form , a (z) <J, (t) dt + K, <J, = j, (z), 2 a l (z) <J, (z) + 11" -t-z a
(31.52)
a2 (t) 1/J, (t) dt .!:_I t-z 11"
(31.53}
--
I
,
a l (z),P, (z) -
a
+ K 6'f/J6
= 0.
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
651
�, (z) = (b - z) -a �(z), /b (z) = (b - z) -a /(z), t!J6(z) = (b - z)0 1/J (z), K , = (b - z) - °K(b - t)0 and K& = (K, )*. Equations (31.52) and (31.53) are singular integral equations with the Noether nature presented in Theorems 31.6 and 31.7. The problem to be solved is the following. If we seek solutions 2?r(1 - a) . Then solutions t!J6 (z) of (31.52) are sought, by Theorem 31.6, in the space of functions bounded at both end-points, i.e. in the space H. Therefore 1/J ( z) is to be taken in the class of functions of the form Here
�
1/J (z) = 1/J, (z)/(b - z) 0 , 1/J, (z) E H.
(31.54)
The functions 1/J (z) may be sought, indeed, in a wider space (31.51). The simplest way to see this is the following. We might reduce (31.41) and (31.42) to the singular equations of the type (31.43) and (31.44) or (31.52) and (31.53) with the weight function (z - a)0 instead of (b - z) 0 • To obtain this we have to apply (11.17) and (11.19) instead of (11.18) and (11.16). By this method we would arrive at the equation of the type (31.52) with the right-hand side (z - a) -a / (z) and the equation of the type (31.53) with the solution 1/Ja (z) = (z - a)0 1/J (z). Arguments repeating those given above show that the choice of the class of the same solutions 1/J(z) must be the following: 1/J(z) = 1/Ja (z)/(z - a)0 with 1/Ja (z) E H. Comparing this with (31.54) we attain (31.51). Applying Theorem 31.6 to (31.52) we see that (31.43) is solvable in H* if and , only if J f(z)t!JJ(z)dz = 0, where {1/Ji } is a complete system of solutions of the
homogeneous equation corresponding to (31.44) in the space (31.51). If (31.43) is solvable in the space H* for the right-hand side / (z) E H!(C H*), then all its solutions �(z) belong to the subspace H! too. We omit the proof of this assertion, but give some elucidations. For a singular equation, as in (31.52), with the kernel K( z, t) which is Holderian in both variables, it is known that if its right-hand side has integrable singularities at the end-points of the power type with orders not exceeding a given number, and the Holder property of the kernel is of sufficiently large order, then the solution, which is a priori considered in H* , has similar singularities. It is known that this can be proved by the method of Carleman-Vekua-regularization (see Gahov [1]) which is based on the inversion
a
-
652
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
of the dominant part of the equation. After this the required information about solutions is obtained from the theory of Fredholm equations. In our case, that is under our assumptions on the kernel T(z, t), the difference is only in the fact that we arrive at the Fredholm equation with a kernel having a weak singularity (see Remark 31.3 below), and the statement is true as well. This requires detailed estimates which occupy too much space and therefore are omitted. Since (31.43) is solvable and all its solutions belong to the space n: , then {31 .41) is solvable in H* . It remains to observe that the last statement of the theorem follows from its validity for (31 .52) and (31.53). • Remark 31.2. The choice (31.51) is that of the space, "adjoint" to the space H* for solutions of (31.41) and (31.42). This choice of adjoint spaces of (weighted) Holderian functions for (31.41) which is an equation of the first kind enables us to obtain the statement of the Noether theorems-type, i.e. Theorem 31.1 1 . We would like t o stress, that the main idea underlying the choice of the adjoint
spaces proves to be the requirement that
6
J(z - a) a (b - z)a lcp(z),P(z)lds < oo. a
As
regards the theory of singular equations, the rule for the choice of adjoint spaces is
6
J lcp(z),P(z)l dz < oo . a
Remark 31.3. One can investigate the regularizer (interpreted in the sense of
Theorem 31.4), construct it explicitly and show that the kernel of the regularized equation, i.e. the Fredholm-type equation of the second kind, has a weak singularity. Namely, if u(z), v(z) E H >.. , � > a, and the kernel T(z, t) satisfies the assumptions of Lemma 31 .3, then the kernel of the regularized equation is dominated by lz - tl" - 1 with I' = min(� , Pt , P2 ) - a. D . The N oether nature of potential type operators from
Ia (Lp) ·
Lp
into
Let us consider the equation of type (31.41) in the form
j u(z, t)cp(t)dt + j6 v(z, t)cp(t)dt /( M
=
a
(z - t) l - a
(t - z) - a
_
(31.55)
We suppose that the functions u(z, t) and v(z, t) satisfy the assumptions 1) they are Holderian of order � > a in z uniformly in t: (31 .56) where on t;
Zt � t, z2 � t and Zt � t, z2 � t, respectively, and A and A do not depend
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
653
2 ) u(z, z) = u(z, z - 0) E C([a, 6]), v (z, z) = v(z, z + 0) E C([a, b]). Following the same lines as in §31.1 ( see (31.9)-(31.12)) an d applying (11.16), we reduce (31.55) to the form (31.57) where
(31.58) tJ
and a 1 (z) = u(z, z) + v(z, z) cos a1r,
a 2 (z) = v(z, z) sin a1r, and
�
y) T1 cp = j u(z,(zy)- -y)u(y, 1 - o cp(Y)dy,
(31.59)
tJ
1!
6
v(z, y) - v(y, y) 2 cp = j (y - Z ) 1 - o cp( Y)dy.
(31.60)
z
We shall prove that the operators T1 · and T2 are completely continuous from Lp into l0 (L,), 1 < p < 1/a. Firstly we show that the operators of the form (31.55) are bounded from L, into l0 (L,). We denote �
M1 cp = f tJ
u(z,t) cp(t)dt, (z t) 1 _ 0 _
6 f M2 cp = (t v(z,t)1 _ _
�
z) 0 cp(t)dt.
Let -oo < a < b < oo, 0 < < 1 , 1 5 p < oo. The operutor M1 with the function u(z, t) satis!Jing the assumptions 1} and e) is bounded from L,(a, b) into I:+ [L,(a, b)]. If u(z, z) #; 0, a 5 z 5 6, then Mt [L,(a , b)) = I:+ [L,(a, b)].
Lemma 31.4.
a
Proof. Let p > 1 . We have
MlC/) - j u(t, t)cp(t)o :e
tJ
(z - t) l -
clt
+ TilCfJ
654
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
T1 is the operator (31.59). Let us show that T1 is bounded from Lp (a, b) into I:+ [Lp(a, b)]. First of all, using Theorem 13.3, we shall prove that T1 (Lp) C I:+ [Lp(a, b)]. In accordance with this theorem let
where
..7.
z-c f(z) - f(t) dt , ) 'l"c ( z - j 1+a (z - t)
- oo
where f(t) = (T1
< b,
a < z < b,
and it is assumed that
as
s < a.
z-c
z -e s) - u(t, s) dt ) j ( 1/Jc ( z) =
a
•
z c•
f f + r (o:) k (s)ds oo
u(z , t) - u(t, t)
0
(31.61)
where k(s) is the function (6.10) . Hence elementary manipulations which are left to the reader, give the uniform estimate ll .,j;c liP � c with c not depending on £ . So T1
(z) = naa+ T1
By (31.61) and (6.11) we arrive at
V1
z
z
f(l a:
a:
f u(z, s) - u(t, s) ) f
(31 .62)
•
The kernel of this integral operator is dominated by const ( z - s )�- 1 in view of (31.56) and so the operator V1 is bounded in Lp. This is equivalent to boundedness of the operator T1 from Lp into 1!\[Lp (a, b)]. Further, we have M1 1, it evidently holds on a set dense in L 1 . Since the operators M1 , 1:+ and V1 are bounded in L 1 , this result is extended by continuity to the whole space L 1 . •
§ 31 . EQUATIONS WITH POWER-TYPE KERNELS
655
Corollary 1. Let u(z, t) and v(z, t) satisfy assumptions 1) and 2). The operators M1 and M2 are bounded from Lp(a, b) into pr[Lp(a, b)], -oo < a < b < oo , 0 < < 1, 1 < p < 1/o:. Indeed, by symmetry we state that the assertion of Lemma 31.4 holds for the operator M2 as an operator from Lp(a, b) into If_ [Lp(a, b)]. It remains then to recall the coincidence of the ranges: I�\[Lp( a, b)] = If_ [Lp (a, b)] in the case 1 < p < 1/o:. Q
Corollary 2. Let u(z, t) and v(z, t) satisfy assumption 1). The operators T1 and T2 are completely continuous from Lp (a, b) into Ia [Lp(a, b)], - oo < a < b < oo , 0 < < 1, 1 < p < 1/o:. Indeed, as in Corollary 1 it is sufficient to consider the operator T1 only. Its complete continuity from Lp into Ia (Lp) is equivalent to that of (31.62) in the space Lp (a, b). As it was observed in the proof of Lemma 31.4, the kernel of the operator V1 is dominated by the kernel (z - s)�- l with a weak singularity. Then the operator V1 is completely continuous in Lp see for example Krasnosel 'skii, Zabreiko, Pustyl'nik and Sobolevskii [1, p. 97]). We are ready now to give the result on the Noether nature of the operator M in (31.55) in the following theorem in which 0:
-
G( z )
u(z, z) v(z, z)e ia - u(z, z)++ v(z, z)e-ia •• eil(z) · _
_
-
Theorem 31.12. Let the functions u(z, t) and v ( z, t) satisfy assumptions 1) and 2). Then the operator M in (31.55) is Noetherian from the space Lp, 1 < p < 1/o:,
into the space Ia (Lp) if and only if
z :5 b; 211' (mod 211'). ii) O(a) ":f 211' 1 -pap (mod 211'), O(b) -:f ""# a :5
(31.63) (31.64)
These conditions being satisfied, the index of the operator M is given by (31.65) Proof.
Representation (31.57) and Theorems 31.2 and 31.3 show that the property of the operator M to be Noetherian from Lp into Ia ( Lp) is equivalent to that of
656
CHAPTER 6. INTEGRAL EQUATIONS OF THE Fffi.ST KIND
Lp into Lp .
the operator N01 from operator
The latter is equivalent to the property of the b
a2 (y) ,P(y)dy a1 (z),P(z) + .!.j 1r y-z
(31.66)
a
to be Noetherian in the space Lp(p) with the weight p(z) = (z - a) -ap . The singular operator and the operator of multiplication by a continuous function are permutable up to an operator completely continuous in the space Lp (p) see Gohberg and Krupnik [3], [4], so the criterion for (31.66) to be Noetherian is contained in Theorem 31.12.
....
Remark 31.4. Theorem 31.12 may be extended to the case of the space Lp(p) with the weight p( z ) = (z - a)�'(b - z) " , -1 < p < p - 1, -1 < v < p - 1. In correspondence with Theorem 31.7 the conditions in (31.64) are to be replaced by the conditions �(a) :f. 21r 1t" - a and O(b) =/:= -21r 7 ( mod 2 1r ), the conditions in (31.65) requiring similar changes. The literary citations in §34.1 should also be referred to.
(
)
Remark 31.5. A theorem about the Noether nature - in the sense of Definition
31.3 - of the operator M holds in the case of the spaces H6(p) similarly to Theorem 31.12. 1/p < a < 1 the statement of a type such as Theorem 31.12 holds. This gives the Noether nature of the operator M from Lp into a special space ( see §34.2, note 31.1). E. The case v(z, t) = 0. We now specially consider an important. particular case of (31.55), namely Remark 31.6. In the case
z
u(z,tl)
J (z a
_
t) -a cp(t)dt = /(z),
0
< a < 1.
(31.67)
It is reducible to a Volterr� integral equation of the second kind if u(z, z) =/:= 0. Indeed (31.57) takes the form f(a)I:+ [u(t, t) cp(t)] + T1 cp = f . Inverting here the fractional operator 1:+ , we arrive at the equation·
u(z, z)cp(z) + . - K (x, s) cp(s)ds = n :+ /, Slll Q1r Q1r
X
J a
(31. 68)
§ 31. EQUATIONS WITH POWER-TYPE KERNELS
657
where
K, (z ' s)
z:
=
u(z, s) - u(t, s)
J {t - s) 1- a (z - t ) 1+a dt 1 1 _ u(z, s) - u(s + e(z - s), s) ds. =_ z-sJ e 1 -a (1 - e) l+a Here I K-(z, s) l � c(z - s) - 1 if u(z, s) satisfies {31.56). •
0
according to {31.62). 'Thereby, in view of Lemma 31.4 the following result is proved.
Let u(z, t) satisfy {31.56) and let u{z, z) 6e continuous on (a, b] � z � b. For every /{z) e i:+ (Lp), p 2:: 1, {31.67) is equivalent to the following Volterra equation of the second kind:
Theorem 31.13. and u (z, z) I 0, a
z:
( 3 1 .69)
(J
where g ( z) = u(;,z) ( D�+ /){z) and the kernel A(z, s) = ai::�u :,z has a weak singularity. Therefore, {31.67) is unconditionally and uniquely solvable in Lp( a, b) for any /(z) E I:+(Lp) · The case a 2:: 1 may be similarly considered, provided that u(z, t) is differentiable up to order (a] and (olal u(z, t ) ) /oz [a] satisfies (31.56) and the requirement of continuity at t z. z
=
31 .4. On the stability of solutions Solutions of integral equations of the first kind are not stable in general and the problem of inversion of such equations is known as an ill-posed problem. The simplest equation of the first kind - the Abel equation: z:
is not stable __. n- 0
<J'n (2
•
a, (z J - t) 1-a
{ 31.70)
4
for example in the 'space C((a, b]). In fact let us choose / (z) = = [(z a)/(b - a))" so that -+ 0 as n -+ oo. The solution by of {31.70) IS equal to ( z) = 6-a (z) = . . . + a There IS no , 26) S0 II II C = <• r < t > -+ <"-tr lD VIeW 0f (1 66) •
-a 1 / nl l c n . <J'n r(1a{D(Ja+ /n r n <J'n narr(-a r(n+1- a) r a
n rxn+l)- o) . ( 1:-(J ) n- a nar<•-�� r(arf(n+l •
.
•
658
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
stability in the case of the space Lp(a, b), 1 � p � oo also, which can be justified by the same example modified in the following way, namely
(31.71) Then
(31.72) as n --+- oo. The instability of solution of the equations of the first kind is a reflection of the simple fact that the operator generated by the left-hand side of the equation, maps the considered space of solutions onto its proper subspace, the space itself being essentially wider than this subspace. Therefore the inverse operator is not bounded. This means that for equations of the first kind it is natural to estimate the proximity of solutions and that of the right-hand sides in different metrics. Namely, the proximity of right-hand sides is to be considered in a stronger metric, so that when dealing for example with (31.70) in a certain space X we have to take another space Y for the right-hand sides providing the boundedness of the inverse operator from Y into X. Such an approach can naturally give stability of the solutions. The spaces X = Lp and Y = L 9 are not suited for this goal whatever p E (1, oo] and q E (1, oo] were, since the inverse operator is never bounded from L 9 into Lp . We can achieve stability by taking Y = I:+ (X) , i.e. requiring that the class of right-hand sides consisted of functions representable by the fractional integral of a function in the given space. (Compare this with similar arguments in connection with the normal solvability of integral equations of the first kind in the end of §31.1.) In other words, it is natural to define stability for example for (31.67) in such a way that not the condition 11 /l lx --+- 0, but uv:+ fllx --+- 0 would imply llcpllx --+- 0. This means that under such a definition of the stability we are interested in the a priori estimate
(31.73) Of course, the idea formulated above is trivial for the simplest equation such as (31.70), but it proves to be substantial and useful in a more general situation of (31.67) and of (31.55) or (31.41) all the more so. This idea is effective under the choice, for example, of X = Lp or X = H:>t. , X = H:>t.(p) , since the spaces 1:+ (X) are well studied and characterized in the case of such X - see §13. We recall also that the space I:+ (Lp ) was shown to be coincident with the Sobolev-type space n a ,p - see § 18.4. We confine ourselves to the illustration of the afore mentioned idea by the example of (31.67).
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
659
Let u(z, t) satisfy the assumptions of Theorem 31.13. The solution of (31.67) exists, is unique in any of the spaces Lp (a, b) , 1 � p < oo, and admits the estimate (31.74) whatever the right-hand side f was, the constant c depending only on a and the kernel function u(z, t).
Theorem 31.14.
Proof. By Theorem 31.13 it is sufficient to prove the estimate ll
where wp (/, t) = ll f(z + t) - /(z)llp · Here, evidently, a simple estimate ll
equations. We do not elaborate on these considerations. We also do not concern ourselves here with the problems connected with the regularization in the sense of Tihonov of the ill-posed problem of solving the integral equation of the first kind. There are many investigations devoted to this goal. We refer, for example, to the paper by Zheludev (1], directly concerning the Abel equation (see also (2]), where numerical methods of solutions were suggested based on the method of Tihonov 's regularization. The papers by Gerlach and Wolfersdorf (1], Hai and Ang [1], Gorenflo (6] and Ang, Gorenflo and Hai (1] are also relevant. We also note the paper by Savelova (1] which deals with some questions of stability of fractional differentiation, including the multi-dimensional case.
§ 32. Equations with Power-Logarithmic Kernels The present section deals with the solution of integral equations of the first kind with power-logarithmic kernels and a variable limit of integration
660
CHAPTER 6. INTEGRAL EQUATIONS
rta) f z
a
( Z' - t)( Z' - t)"' - ' Jn�
c
a
Z'
OF
T HE
FffiST
KIND
� t 'P (t)dt = f(z ) ,
(32.1)
b,
where [a, b] is a finite interval of the real axis , a 0, -oo < {J < oo, 1 > particular, the inversion formula for convolution operators of the form
>
b - a�
In
z
(I':f'P)( z) =
r(�) f( z - t)"'-' m� � t 'P(t)dt = /(z) , Z'
a
(32.2)
a < z < b,
is given. The solution of the latter equation unlike the · case {J = 0 considered in § 2 requires the application of special functions of the Volterra type. This is necessary for the following reason. In the case of a natural power of a logarithm ({J = m = 1 , 2, . . . ) the function za - 1 lnm z ( which is naturally considered instead of z a - 1 lnm ( 1I z)) is obtained by ordinary differentiation of a power function za - 1 with respect to a m times: z a - 1 lnm z = c:rn za - 1 1dam . Therefore it is natural to expect that the power-logarithmic function z a - 1 ln� (11z ) with an arbitrary real exponent {J can be obtained by fractional differentiation (1I z ) a - 1 with · respect However after doing this we obtain instead of the function za - l ln� (1 /z ) to itself a certain special function known as a Volterra function, whose main term of asymptotic expansion is equal to za - qn� ( 1/z) - see (31 . 1 1) and (32.37) below. For this reason we first give · some properties of the special Volterra functions and prove some identities for them. We then solve in closed form the integral equations of the form
a.
(32.3)
a < z < b,
a > 0,
with constant coefficients Am k and integer nonnegative powers of logarithms. In conclusion we obtain the criterion for solvability of (32.1) and , in particular, of (3;2.2). We note that unlike {32.3) the unknown . function
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
661
32. 1 . Special Volterra functions and some of their properties We consider the function
{32.4) called the Volterra function as given by Erdelyi, Magnus, Oberhettinger and Tricomi [3, 18.3] and M.M. Dzherbashyan [2, Ch. 5, § 11, p. 261]. We recall some its properties. The function p( z , u, a ) is defined for Re u > -1 and is an analytic function of z with branch points z = 0 and z = oo and has no other singularities. It is an entire function with respect to a. It can be extended to an entire function with respect to u by the relation
(32 .5 ) Reu > - n -
1,
and hence, in particular,
n = 1, 2, . . .
{32.6)
From {32.4) the differentiation formula
tF
dzn p( z , u, a ) = p( z , u, a - n)
{32.7)
follows. The asymptotic expansion of the Volterra function at zero is given by the relation
)
- 0' - 1 1 p(z , IT, <> ) = z " In ;
(
N[ ];.-1 {-1)"{; 1) ( ( �n �rN) ]. +
n
p(l, -n - l , a)
(
1 In
;
)
-n
+0
Re o: > -1,
l (;) I arg ln
<11" ,
z --+ 0,
{32.8)
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
662
where ( u + 1)n is the Pochhammer symbol given in (1.45). In Lemma 32.1 below we prove two integral identities which are used in the sequel. They involve certain special functions. We denote for simplicity
(l�g)(o:) =
{ (l�g)(o:),g)(o:) ev: fl
if p > if p <
0, 0,
(32.9)
where l�g and v: fl g are the right-sided fractional integral (5.3) and fractional derivative (5.8) considered on the half-axis (0, +oo). We denote by l�[g( z)](t) an application of the operator I� to a function g( x) with respect to at the point t. We introduce the functions
T, T
T,
v(z) = p(x, 0, 0), vh (z) = (d/dx)v(xe h ), 1 z T - l (t), -00 < p < oo, l't ,p (x) = I� f(T) :y
[ () ]
(32.10) (32.11)
and we shall call these Volterra functions also. The following lemma is true.
The following integral identities
Lemma 32. 1.
J ta- l [ln t + h - .,P(o:)]vh(X - t)dt = -za- l , �
o: > 0,
(32.12)
0
J l'a ,p (t)l' l -a, -f3(X - t)dt = �
"'(
, 0<
0:
<
1,
-00 < /3 <
oo,
(32.13)
0
are valid for the Volterra functions (32.10) and (32.11). Proof.
Let
h E R1 . We consider the relation 0:
>
0,
0'
>
0, (32.14)
which is directly checked by (1.68) and (1.69). We first apply the integration operator J! with respect to u to (32.14) and then the differentiation operator 1; 1
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
with respect to
a.
663
Then in view of Stirling's formula {1.63} we obtain the relation
t a- 1 e h a ln t + h - f'{a) dt f (z - ty - 1 e h (T - 1 ) dr f r(a) [ r(a) ] z
00
0
0
Setting here u =
r(r)
1 and using (1.67), (32.4} and (32.6), we have
z
j ta- 1 �nt + h - 1/l(a)]v((z - t)eh )dt = za /a,
(32.15)
a > 0.
0
Differentiating this result with respect to z and applying (32.7) and the estimate v(z) = O(ln(1/z) - 1 ) as z -+ +0, following from (32.8), we derive as a result the first identity (32.12) in view of (32.10). Let now -oo < {J < oo. Similarly to the previous case {J = 1, where ordinary operators · of the integration and differentiation of the integer order were applied to (32.14), we transform this relation first applying the operator I� with respect to a to (32.14) and then the operator J;P with respect to u see (32.9). Setting u = 1 - a (0 < a < 1) and h = - In -y, "Y > b - a, in the relation thus obtained and taking (32.11) into account we obtain another identity {32.13). The lemma is proved. • -
Remark 32.1. By
(32.8)
the function defined by
(32.11)
has the asymptotic
behavior
( ) In- 1 ;"Y + (In ;"Y ) ] ,
z a- 1 (In "Y ) - P 1 + {J d 1 P a ,p (z) = :y ; r(a) da r(a) z -+ +0,
()
[
0
-2
(32.16)
at zero. Therefore (32.13) can be continuously extended to the cases a = 0, {J > 0 and a = 1, {J < 0. If we set a = 1, {J = 1 in (32.16), then according to (32.7) it is not difficult to obtain the following asymptotic result at zero:
(32.17) for the function
vh (z) defined by (32.10).
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
664
32.2. The solution of equations with integer non-negative powers of logarithms First we consider the simplest equation of the form
(32.3) in the case m = 1:
f(�) J(z - t)"- 1 [ln(z - t) + A)10(t)dt = /(z), :r:
a>
0.
(32.18)
(I
Without loss of generality we suppose that operator
A 11 = 1
and
A1 0 = A.
We apply the
( J!+ g)(x) = J vh (x - t)g(t)dt :r:
(32.19)
where Vh has the form (32.10) to both sides of this equation and set h - .,P(o:) = A. Then from (32.12) it follows that (32.18) is solvable together with the equation
t
ra
) J (z - t)" - 1 \0(t)dt = - J vh (z - t)f(t). :r:
:r:
(32.20)
According to § 2, the unique solution of the latter equation, and therefore of (32.18) is given by the formula
(32.21) where 1)�+ is the Riemann-Liouville fractional derivative given in Now we consider (32.3) with m = 2:
t
r a)
:r:
j(z - t)"- 1 [In2 (z - t) (I
+ A2 1 In ( z - t) + A 20)10( t)dt = !( z),
(2.30). a>
0. (32.22)
Again without loss of generality we suppose that A22 = 1. Setting h = h 2 in (32.12) and differentiating it with respect to o: arid adding (32.12) multiplied by h1 - .,P( o:) to · the relation thus obtained we find
J ta- 1 {ln2 t + [h2 + ht - 2.,P(o:)) In t :r:
0
=
+
[h2 - .,P (o:)][h t . - .,P(o:)] - .,P'(o:)}vh � (x - t)dt
-za- t �n z + ht - .,P(a)), o: > 0.
(32.23)
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
665
Let us suppose now that the constants ht and h 2 are connected with the constants A2 o and A 2 1 by the relations (32.24) and J!+ and J!f. are the operators (32. 19) with h = h 1 and h = h2 respectively. Applying first the operator J!f. and then the operator J!+ to the both sides of (32.22) and taking (32.23) and (32.24) into account we find
r(�) j �
tl
( :e - w - ' �(t)dt = ( - 1) 2 ( J!V!U (t ) )( z ) .
(32.25)
Hence it follows that (32.22) is solvable together with the latter equation and its unique solution has the form (32.26) Analyzing the solution of (32.18) and (32.22) we see that the relations in (32. 12) and (32.23) played the main. role in constructing the solutions given in (32.21) and (32.26) respectively. Similarly, for solving (32.3) with an arbitrary m the leading role is to be played by the identity which generalizes (32.12) and (32.23) and is given by the following statement.
Let m = 1, 2, . . . and the constants Am; = Am; (m, a) be defined by the recuJTent relati.ons
Lemma 32.2.
Amm (m, a) = 1 ,
m ;::: 1;
Ato(l, a) = h t - tP(a), . . . , Ato (m, a ) = hm - tjJ (a);
Am ,m -t ( m, a) = Am - l ,m� 2 ( m, a) + At,o(1, a), Am; (m, a) = A m - t ,; (m, a)Ato( 1 , a) + j=
m ;::: 2;
.
. d Am - tJ (m, a) + Am - l ,j - t (m, a), da
l, . . . , m - 2, if m ;::: 3;
Am o (m, a) = Am - t,o(m, a)Ato(1, a) +
(32.27)
d A t,o(m, a) , da m -
m ;::: 2.
666
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Then the identity
j a:
ta- l
O
m
m- 1
J =O
J =O
?: Amj (m, a) lnj t Vh ,. (z - t)dt = -z a- l ?: Am - l ,j (m - 1, a) lnj z
is valid.
(32.28)
If m = 1 , then in view of (32.27) we have Au ( 1 , a) = 1 , A1o(1 , a) = h1 - .,P(a). If m = 2, then A22(2, a) = 1, A21(2, a) and A2o(2, a) are given by (32.24). Therefore (32.28) coincides with (32.12) for m = 1 and with (32.23) for m = 2. Changing h1 to h2 and h2 to ha in (32.23), differentiating this relation with respect to a and summing it with (32.23) itself (with h1 = h2 and h2 = h3) multiplied by A10(1, a) = h1 - .,P(a), we obtain (32.28) with m = 3. Continuing
Proof.
this process and using the method of mathematical induction we prove that (32.28) is valid for an arbitrary natural m, and this completes the proof. • Let now the constants h1 , . . . , hm be connected with the constants Am,m- 1 , . . . , Amo(Amm = 1) by means of (32.27) and let J!.f. , . . . , J!+ be the operators given in (32.19) with h = h1 , . . . , hm respectively. Applying the operators J!.f. , . . . , J!+ successively to both sides of (32.3), using Lemma 32.2 and the assertions of § 2 we obtain the following theorem.
Let the constants h1 , . . . , hm be connected with the constants Amm = 1 , Am,m - 1 , . . . , Amo by (32.27) and let J!.f. , . . . , J!+ be the operators given in (32.19). Then (32.3) is solvable together with the equation
Theorem 32.1.
(32.29) and its unique solution has the form (32.30) where I:+ is the fractional integral (2.17) and v:+ is the fractional derivative (2.30). In particular, if a = 1, then the unique solution of (32.3) with the pure logarithmic kernel has the form (32.31)
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
667
Now we investigate (32.27) which reflects connections between the constants Amm , Am,m - 1 , . . . , Amo involved in (32.3) and the constants h t , . . . , hm contained in its solution (32.30). If m = 2, then (32.27) coincides with (32.24) which by the Vieta theorem is the same as the numbers h t - tjl(a) and h2 - 1/J (a) which are roots of the quadratic equation
(32.32) If m = 3, then (32.27) has the form Ass(3, a) = 1, As2 (3, a) = h t + h2 + hs - 31/J (a), Ast (3, a) == [h t - ,P(a)][h2 - 1/J (a)] + [h t - 1/J (a)][hs - 1/J (a)] + [h 2 - 1/J(a)][hs - 1/J (a)] - 3 1/J '(a), Aso(3, a) = [h t - ,P(a)][h2 - 1/J (a)][hs - 1/J (a)] - As, 2 (3, a) ,P'(a) - 1/J "(a). By the Vieta theorem this is equivalent to the fact that the numbers h2 - ,P( a) and hs - ,P( a) are roots of the cubic equation
h t - 1/J (a), (32.33)
Similarly, in the general case of an arbitrary natural m the numbers h 1 ,P (a), . . . , hm - ,P (a) are roots of a certain algebraic equation of power m, namely
(32.34) with the coefficients am - b . . . , a 1 , ao expressed via the constants A m,m - t , . . . , Am o (Amm = 1) involved in (32.2), for example, am - 1 = Am,m- 1 , etc.
32.3. The solution of equations with real powers of logarithms We pass on to solving statement. Lemma 32.3.
(32.1).
We note at once that
(32.13) leadsto the following
The solvability of the equation
J JJa,p (z - �)
a
(32.35)
CHAPTER 6.
668
INTEGRAL EQUATIONS OF THE FIRST KIND
in L(a, b) is equivalent to that of the equation
J
(I
·� J 1'1 -a,-p (z - t)f(t)dt z;
(I
(32.36 )
in L(a, b) where J'a ,p (z) and 1' 1 - a, - p (z) are the functions (32.11). We introduce the notation
and observe that function (32.11):
a>
0, -oo < {3 < oo,
a=
0, 0 < {3 < oo,
(32.37)
Ua ,p (z) is the main term of asymptotic expansion of the Volterra J'a ,p (z) = Ua ,p (z)[l + 0 ( 1 ))
z -. +0.
as
In the following the ensuing lemma plays a fundamental role. Lemma 32.4.
relations
Let c(z) be an absolutely continuous function on (O, b - a) and the
c(z)u a ,p (z) - co . z ( ) co = z:lim c(z), = r o l'a ,{J ( ) be valid. Then for any function r.p(z) E L(a, b) the relation Z
(32.38)
J c(z - t)ua,p (z - t)
z;
0
(I
t
+ J J'a,p (z - t ) dt J t/J (t - r) r.p(r)dr z;
(32.39)
(I
(I
is true where ,P (z) =
- � J r'(y)dy J J'a,p (t) !1'1 -a,-p (z - t)dt y
z;
0
0
E L( O, b - a) .
(32.40 )
669
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
,P(x)
Proof. First we show that E L(O, b - a). Let a > entire part of {3. In view of (32.16) we have
0, {3 > 0 and [,8] be the
b-a b-a 11 dt b- aln-� l. ln� J 11/J(x) l dx :=;c J l r'(y) l dy J t 1 _0 J11 (x � t) l+:-t dx b-a 11 dt b-a t =c t 1 - a J11 [ln Z - t + ln rt ] �] J l r'(y) ldy f _ R
0
0
R
J_
0
__
0
0
Applying the binomial formula and carrying out the change
b-a ] � j lf/>(z) jdz $c L (� ) j lr'(y) jdy j t'(!_a
b-a
1
•=0
0
x = t r we obtain
7J
0
0
b- a 11 dt (b- • l ln &.r- 1.81+1 + l in &. I dr ( ) j lr'(y) ldy j t 11j/ (r - 1 ) 1 + o t b- a (b-a)/11 oo b a ] � $ c 0 (�) j l r' ( y) j dy ( j ln r + j In � ) 1 (b- a)/11 l ln T -1 1 �P-�]+i + lin T -1 1 li dr l)l +o
LB1 i
• -0
X
:=;c
•lt
0
'
0
0
_
_
(r -
��] ( ) �b-al r'(y) ldy Joo (I T: 1 I{J-�]+i l)l+I o T:1 li) 1 i
In
LB1
•=0
0
+ In
(r
_
< + oo, The cases a =
0, {3 > 0 and a > 0, {3 < 0 are considered similarly.
In
T
dr
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
670
We use Lemma 32.2 now. Setting
/(z)
:1:
=
taking (32.38) into account we have
� J#Jl- a,-p (z - t)f(t)dt
f c(z - t)ua,p (z - t)
:1:
a
� J #Jl - a,-p (z - t)dt J[coiJa,p (t - r) + r(t - T)Pa,p (t - r)]
:1:
=
a
a
t J co J
:1:
=
a
a
(32.41)
a
One may check directly that
� J Pl-a,-p (z - t)dt J r(t - T)IJa,p (t - r)
:1:
a
a
J dt J .,P(t - r)
:1:
=
a
a
Then from (32.41) it follows that
� J #Jl-a,-p (z - t)f(t)dt co J
:1:
t
:1:
=
a
a
a
a
But according to Lemma 32.3 the relation
J
J t
:1:
/(z) = Pa ,p (z - t)[co
a
is valid. Hence (32.39) follows. The lemma is thus proved. Further we consider the operator
J
•
:1:
(T.p
(32.42)
§ 32. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
671
It is completely continuous in Lp(a, b), 1 :::; p :::; oo, and has zero spectral radius, Zabreiko [1). Therefore the operator coE + T.p is invertible in Lp {a, b) and the following statement giving the criterion of solvability of {32.1) in Lp (a, b) follows from Lemma 32.4.
Let a function c(z), c(O) #; 0, satisfy conditions of Lemma 32.4 and let u a ,� (z) have the form {32.37). The equation
Theorem 32.2.
j c(z - t)ua,� (z - t)cp(t)dt = /{z) :1:
{32.43)
(I
is solvable in Lp (a, b), 1 :5 p :5 oo if and only if the free term /{z) is representable in the form
f :1:
/{z) = JJa ,�(z - t)x(t)dt, x(t) E Lp{a, b).
{32.44)
(I
This condition being satisfied the equation has the unique solution given by the expression
d� ( � i Pt -a , - p (z - t)f(t)dt)
{32.45)
where (c0 E + T.p) - 1 is the operator inverse to the operator :1:
(coE + T.p) cp(z) = co cp(z) + j ,P(z - t) cp(t)dt
{32.46)
(I
and the function ,P(z) is defined by {32.40), {32.11) and {32.38). Corollary. Equation {32.2) is solvable in Lp (a, b), 1 :5 p :5 oo, if and only if the free term f(x) is representable in the form {32.44). This condition being satisfied the equation has the unique solution cp given by the expression {32.47) A characterization of the space of the right-hand side f of {32.43) via a convolution with the Volterra function � 1' 1 - a , - � is contained in Lemma 32.3 which is a generalization of Theorem 2.1 for equations of the form {32.44).
672
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Remark 32.2. It is easily shown that the relation r:f cp = c,
cp E Lp , is impossible in general if c #; 0 in the case p � 1/a, 0 < a � 1. Namely, if p < 1/a, fJ E R1 or p = 1/a, fJ � -1 + a then cp = c = 0.
§ 33. The Noether Nature of Equations of the First Kind with Power-Logarithmic Kernels The present section deals with the investigation of the Noether nature of integral equations of the first kind with power-logarithmic kernels ,
f Kcp = _
a
c(x, t) fJ -y _ lx - ti l -a In lx - t ! cp(t)dt - f(x), a < x < b,
(33.1 )
where [a , b] is a finite interval of the real axis, -oo < fJ < oo, 0 � a < 1 ({J < - 1 if a = 0) , -y > b - a , and the function c(x, t) has a jump of the first kind at the diagonal x = t:
{
c(x, t) = u(x, t), t < x, (33.2) v(x, t), t > x. Equation (31.1) considered in § 31 is a particular case of (33.1 ) for 0 < a < 1, fJ = 0. We shall study the Noether nature of the operator K in the corresponding proper setting, namely from the space Lp = Lp( a , b), 1 < p < oo, into a special Banach space X. We shall show that the appearance of the factor lnfJ l:r:! tl weakening or strengthening a singularity of the kernels at the diagonal x = t (depending on the sign of {J) and in the kernel of the potential type operator (33.1) changes only the range of the operator K. As for the Noether nature of (33.1), it remains the same in the sense that the Noether properties of K from Lp into X
are equivalent to that of a certain singular integral operator in Lp which does not depend on fJ. The scheme of investigation of the Noether properties of (33.1) is the same as for (31.1) with power kernels in § 31. We note only that the presence of the logarithmic factor with power unity considerably complicates the proof of the imbedding theorems for the ranges of the operators and I�! , with power-logarithmic kernels (see (21.1)) playing an important role, and the identities giving a connection between these operators with singular integral operators. To derive the former we use properties of the Volterra function (32.1 1) obtained in § 32, and to obtain the latter we use the method of differentiation of arbitrary order with respect to a parameter in the relation connecting fractional integrals with singular operator. The results obtained are used in the investigation of the property of the operator (33.1) as to whether it is Noetherian.
1:/
§ 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
673
33. 1 . Imbedding theorems for the ranges of the operators P;f a ,fJ and lbLet 0 5 a <
1, -oo < {J < oo. We recall the notations: z
1 f InfJ ..:L. z- t - f(et) ( z - t) l- a
Jaa ,{J
(J
a ,
a
b
InfJ ..:L.
(33.3)
1 f t -z f(et) z (t - z) l- a
= 0, 0 < a < 1, we shall use the symbols 1:+
(33.4) (33.5) We obtain the imbedding theorems for such spaces as a corollary of a more general assertion. We introduce the Banach space
I!+ (L,) =
{1 :
j g(z z
/( z ) =
a
t)
g(x) E L(O, b - a), more general than result of § 32.
(33.4)
and
(33.5).
1 5 p 5 oo.
11 / llr!+ (L.l = llv>IIL.
}•
(33.6)
The following theorem is derived from the
Let JJa ,fJ (x) be the Volterra function (32.11) and ua ,p(x) be its main part (32.37) near the origin and c(x) be an absolutely continuous function on (0, b - a] and c0 = z-o lim c( z ) . Then the following statements are true: 1) If c0 :/; 0, then (33.7)
Theorem 33.1.
674
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
2) If co = 0, then the imbedding
(33.8) is valid and the operator of this imbedding is completely continuous. Proof. The assertions of theorem follow from Lemma 32.4 of the previous section: the former in view of the boundedness of (32.46) in Lp(a, b) and the latter in view of the complete continuity of the Volterra operator (32.46) in Lp(a, b). • Corollary.
Let
.1 � p < oo, o � a < 1, o < 6 < a
if a > o, -oo < Pt < fJ2 < oo.
Then the imbedding (33.9) (33.10) are valid and the operators of these imbeddings are completeiy continuous. 33.2. Connection between the operators with power-logarithmic kernels and the singular operator Earlier in § 11, the relations in (11.16) and (11.17) connecting the Riemann Liouville fractional integration operators I:+ and Ib'_ with the singular operator were found. They played an important role in the investigation of the Noether nature of the potential type operators (31.55) with the power-type kernel on a finite interval [a, b]. It is natural to expect that a connection between the operators I:f and I:! with power-logarithmic kernels and the singular operator are to play similar role for the potential type operators (33.1) with the power-logarithmic kernel. Proceeding from (11.16) and (11.17) we shall find such connections by applying the method of differentiation of an arbitrary order with respect to the parameter a to these equations. It will allow as to pick out a natural power of logarithm under differentiation of an integer order, while a modification of this method will help to cover the case of an arbitrary order.
S
S
§ 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
675
We introduce singular integrals with power-logarithmic kernels
( a)
a ln m t- a � a t - x- cp(t ) dt,
_
a lnm .!=!. � a cp t dt, b_; ( )
1 j" t - a
(Sa1a1m ct')( x) = 1r
__
x
a
b
1f
(Sb1a1mct')(x) = ;
Lemma 33. 1. If m = 0,
a
-
( bb - t ) X
(33.11)
1, 2, . . . and
1 < p < oo, -1 + 1/p < Q < 1/p,
(33.12)
then the operators Sa1a1m and S,1a1m are bounded in the space Lp(a, b) . Proof. The assertion of the lemma follows from Theorem Theorem
1.5
if m =
1, 2, .
.
..
11.3 if m = 0 and from
Remark 33.1. It is known that the conditions given by (33.12) are not only sufficient but also necessary for boundedness of the operators Sa1a and S6 1a in the space Lp(a, b), Gohberg and Krupnik [5]. Remark 33.2. Let li Sa a m il = max liSa a m ilL be the norm of the operator p I I I I II'PII L , = l Sa 1 a1m . Then Theorem 1.5 yields the estimate m < .!:. f ta- 1/p l tln_ 1t l dt , li Sa1a1m II _ 1 I 00
1r
m=
1, 2, . . .
(33.13)
0
The similar estimate for the operator S6 1a1m is also valid. We rewrite (11.16) and (11.17) in the form
(33.14) (33.15)
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
676
interpreting them in the limiting ease a = 0 as tp = tp . Similar identities are also true for the operators r:f and I�! with power-logarithmic kernels. Namely, the following theorem holds. Theorem 33.2.
the relations
Let tp E Lp( a , b), 1 < p < oo, 0 � a < 1/p, -oo < {J < oo. Then I�! cp = r:f [cos( ar)cp + sin( ar) Sa 1a f{) + Tt tp],
(33.16)
I:f tp = I�! [eos(ar)tp - sin(ar)S& 1a f{) + T2 tp] ,
(33.17)
are valid where T1 and T2 are operators completely continuous in Lp (a , b). Proof. We have to prove (33.16). If {3 = 0 it coincides with (33.14). If {J =f. 0 then,
as was said above, to obtain the required result we apply differentiation of order {3 with respect to a to (33.14). We begin with the ease of natural {3 = 1, 2, . . . . Multiplying (33.14) by f(a) and differentiating the relation thus obtained {3 times with respect to a and dividing by r( a) we have
(33.18) where
i ar) N!X -_ di (eos E+ L · da:J .� •
i=O
(i) · I
{Ji- i sin(ar) •
•
A -• ua:J
.
8o a a· I
I
Hence we obtain (33.16) for natural {J by the Corollary of Theorem 33.1 and Lemma 33.1. Let now {3 be an arbitrary real number. It would natural to think that we have now to apply differentiation (or integration) of an arbitrary order {J to (33.14). However we can not use the operators of Liouville fractional integra-differentiation since the operator Sa 1 a of the form (33.11) does not preserve the space Lp( a , b) if 1/p < a < 1, and (33.14) is not true if a � 1. It turns out that in this situation the truncated operator
{
1/f(a), Ca = 1, is well-suited for our purpose, where
{
a > O, a = O,
(33.19)
Pt,e is the projection
t � [a, a + c] 0 � a < a + c < 1/p. ( p+a 1E f)(t) = /0,(t ) ' te[ a, a + c] '
(33.20)
§ 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
We replace Q by t in (33.14) and then apply (33.19) with respect to t. A. First we transform the left-hand side of the relation obtained. For ( if Q = 0 we take {3 > 1) by (33.19) we have
(I+ I6t - 'P)(z) = Ca"(a r(1,B) -{J
b
= Ca
7£ a
677
{3 > 0
b
1 V'(y)dy -y' (t - o ) l- P r(t) 1 (y - z) l - t r(t)dt
£
z
( )
T"' -1 y - z dT. f (/3) "'( ) 1 1z (yV'(y)dy 1a -z 0
R
'T
Using the result
(P > 0), (33.21) we find
(33.22) Here
(33.23) according to the relation
j u
as
e - • t a - l dt = ,.a - l e - u
[1 + c�l) ] 0
as
given by Erd«Hyi, Magnus, Oberhettinger and Tricomi In view of Theorem 33.1
l u i -+ ()()
(33.24)
(1, 6.9.2(21), 6.13.1(1)) .
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
678
where Vt is a completely continuous Volterra operator in L,(a, b). Everywhere in the following we shall denote by V2 , V3 , Volterra operators completely continuous in L,(a, b). Hence according to (33.14) and the Corollary to Theorem 33.1 we have • • •
,
z) (t )dt = Ia+� N +e v; = Ia - .8 TVT N +e v; t.p, Ca f (t A_p (tZ)-1 -aa+ a 1 t.p a+ 2 a 1 � t.p �
where N, cp = cos(t1r)cp + sin(t7r)S0 , ,cp. Substituting this expression into obtain the relation
(33.22) we
T a - ,8 Tvacp a - ,8 'P - 1a+ 1-+,8 I,t _ cp = I,_:
where Va = V2 Na+ � V1 . Let now {J < 0, a > and (33.23) we have
0.
Then, using
]+ ]+.8 I'b- l/'J r =ca -v1 a (-1)[- ,8 1
x
Taking
r-z
and
(33.23)
and
(33.21)
0,
then by
d[- .61 + 1 1 1 +,8+[- ,8) -p r(t) + ' r 1, a,e Ibda[- ,8)+ 1 t
[I - 1 -,8-[-,8) n
(33.18)-(33.20)
(33.25)
tn
_l_ T-�
-
)]
" 1 +,8+[- ,8] ( T - z ( r)dr (r - z) 1 - e t.p \
(33.23) into account, by Theorem 33.1 and (33.14) we find
Then from (33.26) we obtain (33.25) with Va = VsNa lf.&. B . Now we transform the expression i! cos(t1r)I! + cp.
If {J >
§ 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
679
(33.18)-(33.20) we have I-+p cos(t1r) Iat + cp -
_
( )
a+e cp(y)dy 1 cos(t7r) z --y t dt. a 'Y Ca J z - y r(,B) J (t - a) l - fJ 'Y a z
-
a
(33.27)
Integrating the inner integral by part and taking (33.21) into account we obtain
( z --y ) t dt ( z --y ) a Ja (t a) l-fJ 'Y - 'Y
+e 1 a f(P)
c
os(t r )
_
_
-
-
In - fJ
'Y
_
z y [cos(a ) - .\ (z - y)], r
_
where
Hence according to Theorem 33.1
(33.27) takes the form (33.28)
For p < 0 a similar result is found in the same way as in case A. C. Finally we consider the expression i�J! + sin (t r) a t cp. Let P > 0. Carrying out integration by parts as in case B, then according to (33.11) we have
S
,
(33.29) where
�(u, T) =
In .1.
r(P) J T in ; E
. •-·
,tJ- • .m
(()I +
. m- •
'!J 1rds --+ 0
as
u --+ 0.
The expression similar to the first term on the right-hand side of (33.29) (with sine replaced by cosine) was considered in case B. Therefore using Theorem 33.1
680
we reduce
CHAPTER 6. INTEGRAL EQUATIONS OF THE Fffi.ST KIND
(33.29) to the form
Here the operator-function Vi is completely continuous in Lp(a, b) for r > 0. If r = 0, then complete continuity does not hold in view of the first statement of Theorem 33.1. If we return t o the proof of Lemma 32.4 of the previous section we c an check that the operator-function VT is continuous with respect to r in the operator topology on (0, e] and uniformly bounded with respect to r on [0, e], Also in view of (33.13) we have
� j tor+T-l/p lt 00
II ST +or, l ll � c =
-
1 l - 1 l ln t 1dt.
0
�
Therefore the operator J VT ST + or,1 dT is completely continuous in we obtain the result
0
Lp(a, b),
and so
(33.31) from (33.30). In the case {J < 0 an analogous result is found by arguments similar to those above. Collecting (33.25), (33.28) and (33.31) obtained in the cases A, B and C we arrive at (33.16). Relation (33.17) is derived from (33.16) by applying the operator A : f(z) -+ f(a + b - z) to the left and to the right. The theorem is proved. • From (33.16) and (33.17) the coincidence of the space given in (33.4) and (33.5) follows. Namely, the following assertion is true.
If 0 � a < 1/p, then the Banach spaces r;f(Lp) and I�!(Lp) coincide up to the equivalence of norms:
Corollary.
(33.32)
§ 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
681
33.3. The Noether nature of equations (33.1) We consider
(33.1) which according to the supposition (33.2) has the form
t) lnP -1j v(x, t) lnP -1K cp :: j (xu(x, - t) 1 -a x - t cp(t)dt + � (t - x) l- a t - x cp(t)dt. b
�
(33.33)
a
where - oo < {3 < oo, 0 ::; a < 1 ({3 < -1 if a = 0). We assume the functions u(x, t) and v(x, t) to satisfy the following conditions similar to conditions 1) and 2) for (31.55) in the case 0 < a < 1: thus 1) u(x, x) = u(x, x - 0) E C( [a , b] ), v(x, x) = v(x, x + 0) E C( (a , b]) ; 2) in the case a < 1 the functions u(x, t) and v(x, t) are Holderian of order A > a with respect to x and uniformly in t; 3) in the case a = 1 the functions u(x, y) and v(x, t) are differentiable with respect to x and the inequalities
l ouox l ::; (x -ct)1 l -et ' l -oxov I -< (t - cx)2 l-et '
e1 >
0'
lu(x, t) - u(t, t)l ::; ca(x - t y � , lv(x, t) - v(t, t)l < c4 (t - x)e� , e2 > 0,
hold where Ct , c2 , C3 and c4 are some positive constants. Let us consider the case 0 ::; a < 1/p ({3 < -1 if a = 0). By the Corollary of Theorem 33.2 we shall study the mapping properties of the operator K from the space L,(a, b) = L,, 1 < p < oo , into the space Ia ,P (L,) defined by (33.32). We represent the operator given in (33.33) as
(33.34) where
(33.35) �
1_ u(x, t) - u(t, t) lnp _1_ cp(t)dt (r.r1 ,P cp )( X) = _ , r(a) J (x - tp- a x-t a
1_ j v(x, t) - v(t, t) 1nP _1_cp(t)dt. (r:2 ,P cp)(x) = _ r(a) (t - x)l- a t-x b
�
(33.36)
682
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
Lemma 33.2. If 1 < p < oo, 0 � a < 1/p and u(z, t) and v(z, t) are Holderian functions of order ..\ > a with respect to z uniformly in t, then Tf'f3 and r;·f3 are completely continuous operators from Lp(a, b) into JCJt,f3 (Lp) and
,.,a,(3 .L 1
a ,f3 v;-b - /a+
T.2a ,(3
_
- 16a,-f3 irv2 , _
(33.37}
where the operators v1 and v2 are completely continuous in Lp (a, b). Proof. If 0 < a < 1, fJ = 0, then the assertion of thus lemma coincides with LeJ"ma 31.4 and ,.,a,O = /a v;(33.38) a+ 1 ' T.2a,O = /a6- TT2 ' .L 1 V
where the operators a sin a1r v;1
2
a sin a1r 1r'
Ja (s)ds J (t -u(z,s)ls)-
:r:
(33.39)
'
J (s)ds f' (s -v(z,tps) -(tv(t,- s))1+ dt 6
:r:
.
-
:r:
a
z
a
(33.40)
are completely continuous in Lp(a, b). We transform (33.38) by using the operator given in (33.19), following the same lines as for (33.14) and (33.15) in the proof of Theorem 33.2. After some transformations similar to those above with Theorems 33.1 and 33.2 being taking into account, we have
where the operators Ts and T4 are completely continuous in Lp(a, b). Hence the truth of the theorem follows in view of the complete continuity of the operators V1 and V2 in Lp (a, b). • From (33.27) it follows that the Noether nature of the operator given in (33.3) is equivalent to that of the model operator given in (33.35). The latter in view of Theorem 33.2 is representable in the form
(33.41) where Na is the singular integral operator given by continuous operator in Lp(a, b).
(31.58) and T5 is a completely
§ 33. EQUATIONS WITH POWER-LOGARITHMIC KERNELS
We conclude from
(33.41)
683
that the Noether nature of the operator K from
L,(a, b) into l01•fJ(L,) is equivalent to that of the singular integral operator N01 lp = [u(x, x)+v(x, x) cos(aw)] ll'(x)+
sin (a w)
1r
j ( -t -a) 6
a
Ot
x-a
v(t ' t)l,O(t)dt (33.42) t-x
in L, (a, b). Using here Theorem 31.7 about the Noether nature of the operator N01 we obtain the following statement similar to Theorem 31.12.
Let 1 < p < oo, 0 � a < 1/p and -oo < f3 < oo if a > 0 and f3 < - 1 if a = 0 and let the functions u(x, t) and v( x, t) satisfy conditions 1) and !) indicated above. The operator given by (33.33) is Noetherian from L,(a , b) into [01•{1 ( L,) if and only if Theorem 33.3.
1)
u2 (x, x) + 2u(x, x)v(x, x) cos aw + v 2 (x, x) #= 0, a � x � b;
2)
8(a) #= 2w( -a + 1/p), 8(b) #= 21r/p'
( mod
21r),
(33.43) (33.44)
where x) + v(x, x)e - ia11' e il (%) - u(x, u(x, x) + v(x, x)eia11' '
0 � 8(a)
<
2r.
These conditions being satisfied the index of the operator K is equal to ifO � 8(a) < 2r( -a + 1/p), if21r( -a + 1/p) < 8(a) < 2 w ,
(33.45)
(33.46)
where k is the integer number defined by the condition 8(b) - 2rk E (-21rfp, 21r /p').
(33.47)
In particular, if a = 0 and f3 < -1, then the operator given by (33.33) Noetherian if and only if u(x, x) + v(x, x) #= 0, a � x � b,
is
(33.48)
and the index is equal to x = "'L .-Io·' (L. ) = 0 . Comparing Theorem 31.12 with Theorem 33.3 we obtain an interesting fact.
684
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
The appearance of the factor potential type operator Theorem 33.4.
_
1
tp = f( ) cr M
,
j a
lnP l !tl �
in the kernel of the
c( z , t) (t t lz - t il- er tp )d
which weakens or strengthens (depending on the sign of {3) a singularity of the kernel on the diagonal z = t , changes the range of the operator but does not influence its Noether properties in the sense that if O < a < 1/p, then the Noether nature of the operator K from Lp(a, b) into l0 •P(Lp) is equivalent to that of the singular integral operator given by (33.42) in Lp. Remark 33. 3. Theorem 33.3 may be extended to the case of the space Lp([a, b], p)
with the general power weight
n
Zk l"" . The theorem about the k l from the Holder weighted space H6(P) into the
p{z)
=
n lz =
Noether nature of the operator K generalized Holder weighted space n; +a ,p(p) is also valid - §
33.2.
34.2, notes 33.1 and
33.3 presents the Noether nature of the operator given by (33.33) in the case 0 � a < 1/p. A similar assertion is also true in the case 1/p < a � 1 (see § 34.2, note 33.3). In particular, if a = 1 and {3 > 0 the Noether nature of the operators giv�n in (33.1) with pure logarithmic kernels is valid. More Remark 33.4. Theorem
general investigations of the N oether nature of operators with such kernels have also been carried out (see § 34.2, note 33.4).
§ 34. Bibliographical Remarks and Additional Information to Chapter 6 34.1. Historical notes Notes to § 30. 1. Here we confined ourselves to the case of equations on the axis and on an interval of the axis. The reader may meet the theory of singular integral equations in more detail in the books by Gahov [1] and Muskhelishvili [1], and also Gohberg and Krupnik [4], where these equations are considered on curves in the complex plane. Notes to §§ 30.2 and 30.3. The generalized Abel equation, both in the case of inner and external coefficients, on a finite interval and on a half-axis first arose in Zeilon [1] (1924). In connection with this paper, which was unknown to researchers who dealt with generalized Abel equations between 1960 and 1970, one should refer also to § 17.1, Notes to §§ 11.2 and 11 .3. In this paper by Zeilon a certain fonnalism was proposed of reducing the generalized Abel equation to the characteristic singular equation, or to the Hilbert boundary value problem for analytic functions connected with the latter. This paper also contained the first attempt to consider systems of generalized Abel integral equations with constant coefficients. Zeilon suggested a method of reduction to a singular equation, which is surprisingly original for the year 1924. He
§ 34. ADDITIONAL INFORMATION TO CHAPTER 6
685
stated that the solution of the generalized Abel equation could be reduced to the successive solutions of a singular equation and the usual Abel equation. However, he did not give either a correct solution of the singular equation or an investigation of its solvability at all. Such a solution was known to be obtained much later by Muskhelishvili [1] and by Gahov [1]. The solution of the generalized Abel equation with a sufficiently complete investigation of solvability was first given in this way by Sakalyuk (1] (1960) and (4] (1965) in the case of a finite interval. Sakalyuk used Carleman's method of analytic continuation. The restricted assumptions made in these papers were essentially weakened by Wolfersdorf [2] (1965) and Samko [1] (1967) and (5] ( 1968). In the papers by Samko the solution was based on another method presented in § 30. It is based on the direct connection with the singular operator which allows one not only to clarify the connection between admissible solutions and right-hand sides of the generalized Abel equation, but to investiga�e general equations of the first kind with a power-type kernel as well. The generalized Abel equations (30.17) and (30.18) on the whole axis were solved by Samko (4) { 1968), [7) {1969) and (9) (1970). The presentation in §§ 30.2 and 30.3 follows Samko (27, §§ 3 and 9] (1978), see also a survey by Samko [32]. Notes to § 30.4:. The explicit solution (30.67) was given by Samko (10) (1970), (14] (1973). Refer also (13] (1971) where the sOlution of this equation was obtained in the form {30.63). The cases in (30.72) were noted by Samko [5) (1968). These cases were in general known earlier. We refer to (30.73) in Stein and Zygmund [2] (1965) for functions IP E L 2 (R1 ) with a compact support and (30.74) in Heywood (1967). Heywood (2] (1971) showed that the integrals in {30.73) and (30.74) are conditionally convergent for almost all z if f(z) E JOt(Lp), 1 < p < 1/a. The solution of (30.72) in certain spaces of generalized functions was investigated by Jones (2] (1970). The generalized Abel equation (30. 79) with constant coefficients on a finite interval was specially considered by Wolfersdorf (3) {1969), who gave its solution in tenns of integrAls with the hypergeometric kernel. The solution in the form (30.82) was given Samko (27, § 9] {1978) and it was presented in the book by Gahov (1] (1977). The assertion of Lemma 30.3 was noted by Wolfersdorf [3] (1969) and repeated by Ganeev (1] (1979), [2] ( 1982). In the latter paper the solution of the generalized Abel equation with constant coefficients on a finite interval was also given via the hypergeometric function, but it differed from the result of Wolfersdorf (3]. The solution of (30.83) was first given by Carleman [1] (1922). His solution may be seen in § 34.2 below. Another form of this solution was obtained by Ahiezer and Shcherbina (1] (1957) (see also § 34.2). Willi&IDS [1] (1963), who considered {30.67) in connection with some problem of electrostatics, &JTived at the same result. Krein (1] {1955) &JTived at a certain method of solution of integral equations in his investigations connected with the inverse Stunn-Liouville problem. In particular, he obtained the solution of the Carleman equation (30.83) . In this connection we refer also to Mhitaryan [1] (1968). The form (30.84) for the solution of the Carleman equation was indicated by Samko (3] ( 1968) who also obtained (30.89). Notes to § 31.1. Theorems 31.1-31.4 are well-known theorems in the theory of Noetherian operators. In abstract Banach spaces they were first obtained by Nikol'skii (3] ( 1943) in the Fredholm case when the index is equal to zero, and they were extended by Atkinson (1] and Gohberg (1] ( 1951 ) to the case of non-zero index. One may use Gohberg and Krein (1) (1957) and Kato (1] (1958) together with the books cited in the beginning of § 31 .1 to make an acquaintance with the theory of Naetherian operators in Banach spaces in more detail. Theorems of the type 31.5-31.7 are well-known in a more general case of an arbitrary smooth curve in the complex plane. We gave formulations only for the case of the axis or any interval required. This case was considered especially by Widom (1] (1960). Theorem 31.5 is well-known, as for example in Gohberg and Krupnik (4). Theorem 31.6 is a classical version of statement on the validity of Noether theorems for singular integral equations on an open contour - see the books by Gahov [1] and Muskhelishvili [1]. However, the assumptions on the kernel K:(z, t) in these books were less general. Theorem 31.7 in a more general form was obtained by Gohberg and Krupnik [1-3] (1968-1971). We used the expression given by Karapetyants and Samko [2] (1975) for the recalculation of the index when writing the index in an explicit form. Notes to § 31 .2. The proper setting of the Noether nature for equations of the first kind with a power-type kernel developed here was suggested by Samko [6] (1968). He investigated the
686
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
Noether properties of {31.6) in such a setting in the papers [12] (1971) and (14) (1978). The presentation in § 31 .2 follows the latter paper in the main. Theorem 31 .10 was proved by Rubin [2] (1972). Notes to § 31 .3. The results of this subsection were obtained by Samko [1] (1967), [6] {1968) and (27) (1978) except for Theorem 31 .12 concerning the Noether nature in Lp on a finite interval, which was given by Rubin [3) (1973). The result for Lp(P) noted in Remark 31.4 was obtained by Rubin [1) {1972) including the case a = -oo or b = oo. The Noether nature in Lp(P) in a more general weighted case was investigated by Rubin [2] (1972), [8] {1975). The criterion of the Noether nature for the operator {31.55) from H�(p) into H;+a (p) mentioned in Remark 31.5 was obtained by Rubin (7) (1974). We remark that the decisive point here is the application of Theorem 13.13 on the isomorphism between the spaces HS'(p)
and H;+a(p), realized by the Riemann-Liouville operators (2.17) and (2.18). It allows one to characterize the class of the right-hand sides f of {31.55) in the same simple terms as the solution
m
j ta-l lc=O L Bm lc lnlc t 0
v[(x - t) eh )dt = -
m -1
: L Bm - l , lc lnlc x,
x
k=O
a>
O,
(34.1)
where v(x) has the form (32.10), cf. the relation in (32.28). We also observe that the solution of the simplest equation (32.18) is contained in the books by M.M. Dzherbashyan (2, Ch. 5, § 1.1, p. 264] and Volterra [3, p. 102). Lemma 32.2 and Theorem 32.1 obtained by Kilbas were not mentioned earlier, as well as equivalence of {32.27) in Lemma 32.2 to the fact that the numbers h1 , h2, . . . , hm in terms of which the solutions (32.20) of {32.3) is expressed, are connected with the roots of the algebraic equation (32.34). Notes to § 32.3. The results of this subsection were obtained by Rubin [10] (1977). Notes to § 33.1. Theorem 33.1 was proved by Rubin [10] (1977). Earlier he obtained the imbeddings in (33.10) in the cases of natural and nonnegative real powers of logarithm in the space Lp (a , b) - see (4] (1973) and [9] (1976), respectively. Notes to § 33.2. The results of this subsection were given by Rubin [4] {1973) and [10] (1977). Notes to § 33.3. The Noether nature of the potential type operators {33.1) with the power-logarithmic kernel from the space Lp (a , b), 1 < p < oo, into the spaces (33.32) was
§ 34. ADDITIONAL INFORMATION TO CHAPTER 6
687
investigated by Rubin for natural, nonnegative real and arbitracy powers of logaritlun in the papers [4) (1973), [9] (1977), respectively.
34.2. Survey of other results (relating to § § 30-33) 30.1 . Ca.rlema.n (1] obtained the solution of the equation
1 rp(t)dt J0 lx - tj 1 -a = /(x), < x < 0
(34.2)
1,
in tenns of the contour integral
rp(x) =
i sin
¥
21r 2
1 t(t- 1) a/2 t d _ �dx � (t-x)a ! [ ] a - t
J(a) da
r�
(34.3)
a(a - 1)
o
where r� is an arbitrary closed contour intersecting the positive real half axis R point 30.2. The solution of the Ca.rlema.n type integral equation
� only in the
x.
a
rp( )d J0 jx2 yy2jy1 -a = f(x), 0 < x < a, _
which is reducible to (34.2) by changes of variable is known in the form
This differs from that derived from (30.82) or (30.84) similar result for (34.2) was obtained by Williams (1]. 30.3. The integral equation of the first kind
-
see Ahiezer and Shcherbina [1]. Later a
c(x-t) Joo lx - tj 1 -a rp(t)dt = /(x), 00
-oo
-
<x<
(34.4)
oo,
with a difference kernel, where
. c(x) = { u(x), v(x), x < O, , u(x),v(-x) E H 1 �
X > 0,
(R+ ),
-' > a
,
is reduced to an integral equation of the second kind with difference singular kernel N'{J :
a 1 rp(x)
sin a1r + 1r
00
Joo
-
v(-t-It-xl ) "P(t)dt X
00
+
Joo T(x-t)"P(t)dt = g(x)
-
688
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
a1 = g(x) = rfaJD+ J. T(x), u(x) tJIT(x)(x)l clxlcr-1 x 1 Nh (x), h (x) E L (R1 )Ncp = >..cp p.Scp N his1 ITcp(x) S(hc(12 cp)lxl) -cr- >.. xN-p.1 1 2 1 (a = = c cp(x) { / f(t)dt - tJ f(t)dt } - >.. 1 P(x,t)f(t)dt. .!!_ dx (x - t)cr J (t - x)cr J c= cis utJ A = u2 tJ2 , = tJsin2 v sA11'in�:Ja11' r(r(1 -1 9/-cr-(211'9')r(2(11'-cr)) ) UtJ ' UtJ x- tt ) UtJ P(x, t) = (x(1 - x)] -1 -cr2F1 ( P(x, t) = -Jz { x - x)) -cr2F1 ( /,r (-: ) } UtJ u = tJ u = -tJ.
where u(O) + t�(O) cos cr1r, The kernel which is evaluated via and explicitly is a continuous function everywhere except for the origin, and has the estimates � as - 0 and � + as - oo. The operator • + • where and are constants, and has the structure + + . H the operator invertible, then the operator has the same form - Samko (16], (27, § 5). 30.4. The generalized Abel equation (30.79) with constant coefficients on a finite interval was solved by Wolfersdorf [3] in the form 0, b 1): z:
sin cr1r + A1r
Here
0
0 if utJ > 0 and
arbitrary if
..\
"f
1
> 0 and
z:
u
_
1
< 0,
0
+ 2utJ cos cr1r +
a1r r (2 - 6j(21r})r(1 - a) A1r2 (a + 1) r (1 - a - 6/(21r)) "f
\. 1\
1
<0
6 2 - -, 1 + a; 2 + a; --21r X(1 - )
if
> 01
, a; 1 + a; z: 1 t) if < 0 where 6 is given in (30.80). Compare this with (30.84) and (30.89) in the cases and Another form of solution was given by Ganeev (2] also in terms of hypergeometric functions. 30.5. The generalized Abel equation (30.41) is also solved in closed form in the case when it is considered on a smooth curve in the complex plane. This case was investigated by Sakalyuk [3] and Peters (2]. The latter author also solved the generalized Abel equation (30.42) on a smooth curve and the equation and
[ (1
1-
f (z cp(tt))dt1-cr k f (t cp(tz))dt1 -cr = f(z), z E C, C, Caz Cza k =. k = e211' i cr +
(34.5)
in the case of a closed contour and being two its constituent arcs. Here k "# 1 and The cases 1 and which imply the degenerate cases of the corresponding k "# Riemann boundary value problem were considered by Chumakov and Vasil'ev [1]. We note that by reduction to (34.5) on a closed contour Peters [2] solved, for example, the equation ••
c. .
e211'icr .
c
f11' 0
xt)- 1t)-cr cp(t)dt = f(x), l
sign ( l sin(x -
0<
x
< 1r.
A modification of (30.41) which corresponds to the case of system of intervals was considered by Chumakov [2]. 30.6. The "exceptional" cases of the generalized Abel equation (30.41), when and vanish simultaneously at a finite numbers of points of the interval [a, b), was investigated by
tJ(x)
u(x)
§ 34. ADDITIONAL INFORMATION TO CHAPTER 6
Vasil'ev (2, 5). The solution of (30.41) in the case when u (x) and v(x) may be infinite at the points was gi�n by Orton (1] in some spaces of generalized functions. 30.7. The algorithm for solving the equation z
u(x)
J a
P� - � (x t) l -a cp(t)dt v(x)
b
+ J (tP�x)-l -�a cp(t)dt
_
_
=
689 a
and
b
f(x)
z
more general than (30.41), where P(x) is a polynomial, was indicated by Sakalyuk [2). Peters (2] considered such an equation in the case of an open smooth curve. We refer also to the case v(x) :: 0, f(x) :: 1 in Neunzert and Wick [1]. We also observe that some generalizations of (30.41) with the Gauss hypergeometric function (1.72) in the kernel on a finite interval and a closed contour was considered by Chumakov (1], and Peschanskii (1], respectively. 30.8. The generalized Abel equation (30.41) on a finite interval has many applications. We note some of them. First of all there are boundary value problems for differential equations of mixed type as for example, WoJfersdorf [2]. We also note the paper by Bzhikhatlov [1] where an equation similar to the generalized Abel equation (30.41) was obtained while solving the boundary value problem for equations of degenerate type. WoJfersdorf (4) gave applications of the generalized Abel equation to problems of game theory. Applications of the particular cases, when u :: v :: 1 or u :: -v :: 1, to problems of magnetobydrodynamics have been indicated by Lundgren and Chiang [1]. Applications of {30.41) with constant coefficients to contact problems of the theory of creep and plasticity were given by Arutyunyan [1, 2] and Arutyunyan and Manukyan (1]. Similar applications in a more general case of a variable coefficient of friction were given by Rubin (15). We also observe that the special case = f of the generalized Abel equation arose in an application to a non-linear Hilbert boundary value problem - Hatcher [1]. 30.9. Although the first attempt to solve systems of generalized Abel integral equations was undertaken by Zeilon (1], their real investigation was begun comparatively recently. More serious approaches were made by Lowengrub and Walton (1] and Walton (1]. In the former paper a system of two equations of the type (30.41) - considered in a non-general form - was reduced to the Riemann boundary problem for two pairs of functions by the method of analytic continuation. In the latter paper the system of two equations more general than those of the form (30.41) was reduced to a complete singular equation. Investigations in the papers were influenced by applications to mixed problems of the theory of elasticity. We refer also to Lowengrub [1] in this conn�tion. Interesting results for system of generalized Abel equation were obtained by Vasil'ev [1, 3, 4). After constructing some Hermitian forms by the given system he obtained the criterion of uniqueness of the solution and the criterion of absolute solvability of the system in terms of sign definiteness of these forms. In certain cases the numbers of solutions and of solvability conditions were expressed via the rank and the signature of these forms. In particular, Vasil'ev (4) reduced systems of the generalized Abel equation U = V f, where and f are vector-functions and U and V are matrix-functions, to a system of singular (U (V = f. integral equations. He gave a similar investigation for systems of the form Complete results on the solvability of systems i n terms of the method of Hermitian forms were given in the case of constant coefficients, for example, when U and V are number matrices. In the case of systems of the form
cosa1f'1�+ cp - 1f_ cp
1:+ c,o+ 16_ cp 1:+ cp) + 16_ cp)
cp
C + C (x t) J t Ix - t11 -a cp(t)dt - f( ) 00
- oo
2 sign
-
_
X ,
690
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
where Ct and C2 are number matrices Vasil'ev [1, 3] obtained explicit relations for inversion of the type (30.68). Vasil'ev [6] considered systems of generalized Abel equations on the whole axis containing a vector-function rp( -x) together with a vector-function rp(x). We also note the papers by Penzel [1, 2] where systems of generalized Abel equations were studied on the half-axis R� in the spaces Lp(R� ; p) with weight p = x"Y. 30o10o The integr�differentiation equation .\
tp
J 1
( x) =
0
sign (x - t)
lx - tla
tp
' (t )dt,
0
1,
appean in problems of approximation of function, under the application of the variational method to the approximation of functions by positive polynomial operators - Kogan [1, 2]. In these papers, by inversion of the right-hand side, this equation was reduced to a homogeneous Fredholm equation of the second kind with a positive defined operator. 30o1 1 o Some results on the solvability of the non-linear integr�differential equation
J F(x, t, rp(t), rp'(t))(x - t)a-1 dt z
=
J(x), x > 0,
0
1,
0
are contained in the paper by Sadowska [1]. 3 1 o 1 o The generalized Abel equation (30.41 ) , i.e. the equation Mrp = u(x)I:+ rp+v(x)If_IP = j, was solved in § 30 in the case 0 < a < 1. It may be considered in the case a � (0, 1) as well. H a < 0, then this relation can be interpreted as a differential equation of fractional order. A comprehensive investigation of the equation Mrp f with an arbitrary a E R 1 was given by Rubin [11]. In pal"ticular, for every a, he constructed a pair of Banach spaces X and connected with the spaces Lp(a, b) such that the operator M is Noetherian from X into 3 1 o2 o The Noether nature of integral equation of the first kind
=
Y.
Mrp :=
J c(x, y)lx - Yla-l rp(y)dy
n
=
Y
J(x), x E 0,
where the function c(x, y) is discontinuous at the diagonal x = y was investigated by Rubin [2, 5) in the case when 0 is a union of intervals of the whole axis and M is considered as an operator from Lp(O; p ) into Ia[Lp(O ; p)) . Rubin [6] also investigated the Noether properties of the operator M in the case when 0 is a sufficiently smooth curve in the complex plane. 3 1 o3 o Rubin [8], [13] investigated the Noether nature of the operator M of potential type (31.1) in a more general case, when c(x, t) may have jumps not only for t = x but for t = a k , k = 1, . m, x = bj , j = 1, . . . , n as well, the case ak = bj being especially important. This generalization is interesting, particularly for the reason that it allows one to consider the equations o o ,
t J ck (x, t)lz - tla-l
k=l
=
/(x)
a ,.
"with a finite number of potential type kernels" . 3 1 o4o The Noether nature of potential type operators stated in Theorem 31.9, can be demonstrated under weaker assumptions about the behavior of u(x, t) and v(x, t) at infinity.
691
§ 34. ADDITIONAL INFORMATION TO CHAPTER 6 That is we restrict ourselves to the case of degenerate functions
u(x, t)
m
=
n 1:=1
L a�:(x)b�:(t),
Lp l01(Lp)
v(x, t) � L c�: (x)d�:(t), and if we consider the Noether property not from 1:=1 from L� into L;+ a . Here � = L�(Rl ) is the Sobolev space of fractional potential type operator being considered as closed one - Skorikov [1].
L
into
but
smoothness, the
31.5. The N oether nature of the equation
- t) f mlx(x,x - t i l -a �P(t)dt 00
=
f(x)
- oo
more general than {34.3), with an "almost difference" kernel was considered by Skorikov (4]. He obtained the criterion of the Noether property and the index formula in the case when m (x, y) has a jump of the first kind with respect to y if y = 0. The investigation was based on the reduction to a certain convolution-type singular equation. 31.6. Samko and Vasil'ev [1] singled out some classes of equations more general than the generalized Abel equation which are reduced to complete singular equations with a meromorphic kernel, and therefore can be solved in closed form. �
Iij+ IP + Jk(x,t)�P(t)dt to the fractional 0 integration operator 10+ in Lp{O, a ) were considered by Kalisch (2] and Malamud (1], (2]. We 31.7.
Problems of similarity of the operator
recall that the operators A and B are named similar in the space X if there exists a linear operator in X invertable in X such that = CB.
C
AC
3 1 .8 . Atkinson (1] considered the equation
f (xPu(x,:- tt)p )a �P(t)dt �
=
f(x),
0<
x < b,
0<
at < 1,
0
which coincides with (31.67) if p = 1. He showed that the unique solution IP of this equation has a smoothness of the form IP(x) = xPa -l + P 1P(x), 1/1 cn (o, b), if the additional smoothness assumptions on the kernel u(x, t) cn +2 {(x, t) R2 : 0 � t � X � b} and the free term + l n f(x) = xP g(x), g(x) e c [o,b] hold and the inequality pat + {J > o is satisfied. The method of investigation is simil81' to the consideration of {31.67) on reduction to a Volterra integral equation of the second kind, and on applying the method of successive approximation to the latter in a special fractional space.
E
E E
31.9 . In connection with the stability problems for the Volterra equation (31.67) of the first kind, Vessel& [1] proved the estimate
provided that u(x, t) and Ut {x, t) are continuous. The estimates of such a kind are inspired by the fact that some information on the derivative of the unknown solution is known in certain applications. A more general norm of the fractional Sobolev space JlP ·• was also dealt with. A generalization of such an estimate presented in the context of the ideas of regularization was given by Ang, Gorenflo and Hai (1].
CHAPTER 6. INTEGRAL EQUATIONS OF THE FIRST KIND
692
32.1. The equation of the fonn
z:
J c(x - t)k(x - t)
(a , b),
(34.6)
0
which is more general than (32.1) was investigated by Rubin and Volodarskaya [1] and Rubin [13], [14). Here the function c(x) satisfies certain conditions connected with the absolute continuity on [0, b - a), and the kernel k(x) the fonn
has
( ;) ,
k(x) = xa- l g tn
a � O,
(34.7)
"Y > b - a.
Here the factor g {In ;) introduces such a singularity (or zero) into the kernel which is weaker than that of the power function. For example g(x) may have the fonn
;
n
g (x) = IJ In�" , Ink = �, k= l k times
-oo
< f3k < oo.
In the cited papers the weakening or strengthening of the singularity of the kernel is obtained by the application of a convolution operator with respect to the variable a, the kernel of this convolution being, in general, a distribution. 32.2. Rubin [12] obtained the characterization and inversion relations for the convolutions I:f.1
E
00
t J'(x) = - f r(t x_ -a a+ 0
1)
exp
' (a) dt. (t rr(a) )
Then the following statement is true. Theorem 34.1. A n exi•tence of the limit
(Bf)(x)
= elim -o
z:-e
J [f(x) - f(y))#'1(x - y)dy a
where we take f(x) := 0 6eyond the intenuJ.l [a, b), in Lp (a, b) i. the nece••ary condition for the repre•enta6ility of a function f Lp (a, b) in the form f = r:_f.1 1P with 0 < a < 1,
E
E
§ 34. ADDITIONAL INFORMATION TO CHAPTER 6
693
32.5. Volterra and Peres [1] considered the equation
(34.8) a(x, t) being a certain given function, which is more general than (32.3). By means of the relation in (34.�) this equation was reduced to a simpler Volterra equation of the first kind in which logarithmic factors were absent. In particular, if 01 = 1 and m = 1 (34.8) was reduced to a Volterra integral equation of the second kind under some smoothness assumptions on a(x, t) with respect to x. 32.6. Veber [4] obtained the inversion formula for the simplest integral equation (32.2) on the axis:
r�) 1 (x - t)0- 1 In(x - t)
= J(x)
- co
in a certain space of generalized functions which is invariant with respect to Liouville fractional integr
6
x y !. 1 I x - y I
1n
+
=
f(x),
0 � a < x < b < +oo,
(34.8')
Cl
with a = 0 in the form
��
where v 2 and vU� are the Riemann-Liouville fractional differentiation operators (2.22) and (2.23), was first obtained by Ahiezer and Shcherbina [1] (1957) as a particuJar case of solution of more general than (34.8') integral equation (39.10'), with Gauss hypergeometric function in the kernel. A closed-form solution and the solvability conditions of this equation in the case of any a � 0 were investigated by Vasilets [1] in the weighted spaces Lp ([a, b]; p) with the power weight p(x) = (x - a)l' (b - x)". 33. 1 . Theorem 33.3 in the case of the natural powers of logarithm was extended by Rubin [4) to the weighted spaces Lp ([a, b) : p) '!here
n
p(x) = IJ lx - xk l"• , lc= l
a=
x1 <
· · ·
< Xn = b.
(34.9)
Ru� [9) indicated that this result was true for real nonnegative powers of logarithm. 33.2. Kilbas [6) investigated the Noether properties of the operator (33.1) with power logarithmic kernels with natural powers of logarithm from the weighted Holder space HS(P) into
694
CHAPTER 6. INTEGRAL EQUATIONS OF THE FffiST KIND
H�+a,f' (p), < � < < � + a < = ,p H�(p) H�+a,f' (p) 1:f Lp ::: Lp(a,b), < a < (34.10) r;! I:f [cos(cur) + ( a?r) Sta , a - 1 + Tacp] + c1 (cp), (34.11) r:f I:![cos(a?r) -sin(a?r)Sb , a - 1 + T4cp] + c2 ( cp), Lp c1 (cp) c2 (cp) T4 Ta [10} r;f (Lp) R 1/p < a < 1 (33.1) Lp 33.3-33.4. 32.2. Sta,astJ, S&and,a s., (32.16) (34.1)(32.17)(34.11) case1/p -a1/p1=<1/p,
the generalized weighted Holder space where 0 1, 0 1, {J 1 , 2, . . . is the weight in (34.9). We note that the main role here is played by the theorem on the isomorphism between the spaces and realized by the operators and r;! with power-logarithmic kernel if {J is natural. Its proof uses the properties of the convolution operator (32.19) with the special Volterra function (see § 32.7), together with the results of § 21. 33.3. H cp e 1/p 1, then the relations similar to (33.16) and (33.17) are true
sin
and are some are completely continuous operators in and and where functionals. These relations were proved by Rubin who applied them to the investigation into of the Noether properties of the operator from EB if remark He also proved statement similar to Theorems This method of investigation does not work in the exceptional since the involved in and are bounded if and operators and the operators involved in and are bounded if Therefore the problem of the investigation of the Noether properties of the operator in the remains open. 33.-i. The Noether nature of the potential type operators with pure logarithmic kernels (a = > acting from the space was investigated by Rubin in the case of the first power of the logarithm, and by Kilbas in the case of natural and powers of logarithm. Kilbas [3], [5], [7) investigated the Noether properties of the operator of the fonn
'6
Kj
-oo <
a
oo.
more general than the operator with pure logarithmic kernel > He considered this operator as acting from the space oo, where is the weight into a special space. Here fJj , m ) , 'Y > and among the functions Kj there may be both continuous and piecewise continuous ones with jumps at the diagonal Other results connected with such equations may be seen in the papers by Kilbas 33.5. Rubin and Volodarskaya [1] and Rubin [13] investigated imbedding of the ranges of the convolution operators (34.6) which are more general than the imbedding (33.10). The introduction of the �called generalized Volterra functions and the investigation of their asymptotics by means of properties of the Laplace transfonn lie at the base of these papers. 33.6. The Noether nature of potential type operators with a more general kind (34. 7) of singularity was considered by Rubin and Volodarsk:aya [1] and Rubin [14].
Chapter 7. Integral Equat ions of the First Kind wit h Special Funct ions as Kernels Abstract . In this chapter we shall be concerned with applications of fractional integra-differentiation to the investigation of one-dimensional integral equations of the first kind b
j K(x, t)f(t)dt = g(z),
-oo � a < x < b � +oo,
{1)
a
with the kernels K{z, t) containing special functions. Such equations are closely connected with integral transforms - see § 1.4. Many problems from other fields of mathematics such as differential equations discussed in Chapter 8, function theory and others, also problems in physics, mechanics and other natural sciences are reduced to them. The so-called dual and triple integral equations - see typical examples in the last section of this chapter - can be written in the form {1). Equations of the form {1) are equations of the first kind and therefore the problem of their inversion is an ill-posed problem. Many methods of solution are known for such equations, which depend on the type of the kernel. The convolution equations, i.e. the equations of the form {1) with K(z,t) = k(x - t), have been most studied - Gahov and Cherskii (1], Titchmarsh (1]. Hirshman and Widder (1], H.M. Srivastava and Buschman (3]. We observe that the latter book contains many equations with special functions as kernels and their solutions, as well as a large bibliography of the papers devoted to solution in closed form of convolution equations with variable upper or lower limits of integration. Solutions of equations such as { 1), which are generalizations or modifications of the Abel integral equation {2.1), and also solutions of the composition type equations connected with the Abel equation are constructed in this chapter. It turns out that the most effective method is the factorization method, also called the composition expansion method, which gives a representation of some classes of
696
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
the operator ( 1) as the compositions of fractional integro-differentiation operators together with other operators which have known inversion formulae. Many . mathematicians throughout the world have made considerable contributions to the development of this method, which was started for the equations considered in the early 1960s - see § 39.1 . However, we note that long before this the implicit idea of the factorization method was used by the Russian mathematician Sonine [4, 5] at the end of the 19th century and the method itself was in fact applied by Lebedev [1] in 1948 - see § 39.1 and § 39.2 (notes 37.3 and 35.7) .
§ 35. Some Equations with Homogeneous Kernels Involving Gauss and Legendre Functions In this section we consider the Mellin convolution integral equations with the Gauss hypergeometric function or the Legendre function as kernels which are important in applications. We shall show that these equations can be inverted in terms of compositions of two fractional integro-differentiation operators with power weights, or in terms of operators involving Gauss or Legendre functions and usual differentiation in a quite symmetrical way. We shall use the results of § 10.1 in these investigations.
35. 1 . Equations with the Gauss function We deal with the following four integral equations involving the Gauss hypergeomet ric function as a kernel (35.1)
t: (x - r)e- 1 2 F1 (a, b; c; l - ;T ) cp(r)dr = g(x), I f(c)
(35.2)
e
(35.3)
/ (r - z)e-1 2 F1 (a, b; c; l - ;T ) cp(r)dr = g(x), d
f(c)
(35.4)
considered on an interval 0 � e < x < d � oo. In accordance with Remark 10.3 we denote the operators on the left-hand sides of (35.1)-(35.4) by 1 /�+ (a, b)cp, 2 /�+ (a, b)cp and 3/:f_ (a, b)cp, 4/:f_ ( a, b)cp if d < oo or 3 /: (a, b)cp, 4 /: (a, b) cp if d = oo.
§ 35. KERNELS INVOLVING GAUSS AND LEGENDRE FUNCTIONS
697
By the relations given in {10.22)-{10.29) the operators j lc(a, b) are the compositions of two one-sided fractional integrals or derivatives with power weights. The conditions for such a representability are given by Theorem 10.4 and Remark 10.3, which show that if tp{z) belongs to the space Lp or some of its subspace, then under corresponding conditions the operators i Ic(a, b) map a certain subspace of Lp and some or other of the relations given in {10.22)-{10.29) are valid. This means that if the right-hand side, i.e. a function g(z), is taken from the space Dj indicated in Table 10.2 which is mapped by the operator j lc(a, b), then the corresponding equation i Ic( a, b )tp = g is uniquely solved by successive inversion of two integro-differentiation operators composing jlc(a, b). Realizing these inversions we obtain from {10.22)-(10.29) the following representations of solutions of {35.1)-{35.4):
tp{z) = z- a I;:xa I!+ cg(z ),
tp{z) = z - b I!; c xc- a I;:x a+b -cg(z), tp{z) = I!; c x a I;:x - 4g(z ),
tp{z) = za+b - c I;;xe-a I!; e x - " g(z), tp{z) = x - " I�: e xe- a li�z a+b - eg(z), tp{z) = z -a li� Xa J�: eg(z),
tp{z) = za+b - e li�ze- a I�: e x - "g(z).
(35.5) {35.6) (35.7) (35.8) (35.9) (35.10) (35.11) {35.12)
To formulate the corresponding theorem we denote by Ej a column of numbers which is obtained after replacing (10.22)-(10.29) by (35.5)-(35.12) respectively in column Ej of Table 10.2. Then the following statement is true.
We consider (35.1) and {35.2) with e = 0 on the interval {0, d) and {35.3) and {35.4) with d = oo on the interval {e, oo) provided that Rec > 0. If the conditions Aj from Table 10. 2 are satisfied for the operators Bj and a given function g(x) E Dj , 1 � p < oo, then the corresponding equation Bjtp = g of the forms (35.1)-(35.4) has a unique solution given by the expression Ej where e = 0 and d = oo. The statement of the theorem remains t"'e when e > 0 and d < oo correspondingly, and in these cases the assumptions in the conditions Aj involving p are omitted. Theorem 35.1.
Proof. In fact, all the conclusions of the theorem were obtained in the proof of Theorems 10.2 and 10.4 and Remark 10.3. We additionally note that, for example, if e > 0 then the singular point = 0 of the kernel of (35.1) lies beyond the interval
T
698
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
of integration. This fact allows us to remove the conditions containing p from the assumptions At and A2 , since singularities of the function
( )
d m X a X - a 1m-b Xa 1(m-c)-(m-b)g {X )
or
( � r g(z),
m=
1, 2, . . .
'
It should be emphasized that the latter relation is not equivalent to {35.5) since the function g( x) here is assumed to belong to the subspace of functions in Lp ( e, d) representable by the form g = I� x , x E Lp(e, d), i.e. having the derivative g (m ) (x) and g(e) = g'(e) = · · · = g ( m - l ) (e) = 0. Comparing two last expressions with {10.24) and taking {10.19) into account we arrive at the following representations for the solution of {35.1):
r
{
dzm
za
�
! (xr-(mr)-m-c)c- 1 (-a, m - b., m - . 1 - ;T ) g(r)dr} , 2 F1
c,
� (x m- c- 1 !
m
c)
{35.13) {35.14)
0 < Re c < m , g E cm ((e, d]) g(e) = g' (e) = · · · = g <m - l ) (e). It is clear that Lp , .(e, d) = L,(e, d) if e > 0 as is seen in transform other relations given in {35.6)-(35.12).
{10.2).
Similarly we can
§ 35. KERNELS INVOLVING GAUSS AND LEGENDRE FUNCTIONS
699
35 .2. Equations with the Legendre function Different particular cases of equations (35.1)-(35.4) with the kernel involving Chebyshev, Legendre, Gegenbauer, Jacobi, etc. polynomials arise in applications to differential equations as for example in § 40.2. Here we consider equations that will be most useful below, i.e. equations with the Legendre function given in (1.79), (1.80) as a kernel:
J(z2 - t2)-P/2Pt (i) f(t)dt :1:
=
g(z) ,
(35.15)
=
g(:z),
(35.16)
j(t2 - z2 ) -p /2p: (i) f(t)dt = g(x),
(35.17)
e
f(:z2 - t2) -pf2Pf: G) /(t)dt :1:
e
d
:1:
j (t2 - z2) -Pf2Pf: ( ;) f(t)dt = g(:z), d
{35.18)
:1:
provided that 0 ::; e < x < d ::; oo and Re p < 1 in all cases and that the integrals indicated above converge at the variable end point. We shall obtain solutions of these equations by the Mellin transform. We consider the first equation in more detail. In (35.15) we change the variables and the functions by the substitutions
z2
=
y, y/T = '1,
(71 - 1) -P /2 H(71 - 1) P: (Vii'J = h(71), g( VY) = 9t ( Y), where H(e) is the step function, namely H(e) = Then this equation has the form
f h(y/T)cp(T)T- 1 dT
1 if e > 0 and H(e) = 0 if e < 0.
00
0
=
(35.19)
9t (y).
700
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Applying the Mellin transform defined in (1.112) to the latter and 11.14 (1) from the book by Marichev [10] and taking the convolution theorem given in (1.115) into account we obtain the relation - s) 2 �J f((1 + p +f(1v)/2s)f(1/2 - s)f((p - v)/2 - s) 91 (s), Rep < 1, Re(2s - p - v) < 1, Re (2s + v - p) < 0.
'P ( s) _
•
-
•
(35.20)
The inverse path from (35.20) to its original tp(t), and hence f(t), can be carried out by various methods. We shall indicate three of them here, which lead to different forms of solutions. A. Using the representation given in (10.35) and grouping the gamma functions by different ways (vertically and crosswise) we obtain the following two representations for the function tp( y): 'P( Y)
=
2 - IJ ( y( 1 - �J -11)/2 J��+v- 1 )/2 ) { y 1 +( 11 -�J)/2 J��- 11- 1 )/2 y - 1 /2 )g1 { y) , {35.21)
tp( y) 2 _ #J (y( 1 -�J-11)/2 J��+ 11)/2 y - 1 /2 ) {y 1 +( 11- �J)/2 J��-v)/2 - 1 )g1 ( y) . (35.22) =
These representations coincide for sufficiently good functions g 1 ( y) (for which the operators in the parenthesize are commutative) since they differ from each other by replacing v by - v - 1 which is not essential according to the property Pf(z) = P� 11 _1 (z). Making the inverse change in (35.21) by the expressions given in (35.19) we arrive at the following representation for the solution of (35.15):
11- 1 )/2 X - 1 g ( X ) /( X ) - 2 1 - IJ X -vJe(+iJ +;zv� 1 )/2 X 2+ 11- IJ ]e(�J+ ;z � _
(35.23)
where 1:+ ;z � is the operator defined in § 18.2, see (18.41). B . Another representation for the solution can be obtained if we multiply the numerator and denominator of (35.20) by f(1 + (p + v)/2 - s) and then apply the duplication formula given in (1.61) for the gamma-function together with (10.35) and its analogue in the form
xf1 /2 rJt-{j x - a/2 f(x) O + ; v'i"
=
_1_ 2 7r i
21 <
-y + ioo
J
-y - ioo
f{1 - a - 2s) /* (s)x -' ds, f{1 - {J - 2s)
1 - Rea.
{ 35 . 24)
§ 35. KERNELS INVOLVING GAUSS AND LEGENDRE FUNCTIONS
701
Then
cp
*
(
+ (JJ + v)/2 - s) (s ) , s ) = 2" r(1r(1+ - +2s)r(1 I' v - 2s)r((JJ - v)/2 - s) 9 1 *
cp( y) = 2 " (y- (JJ +v)/2 ��:t'�)( y1 +(v-p) /2 J(}_; - 1y(p + v)/2 )g1 ( y) , and finally the desired solution of the equation
(35.15) can b e written in the form (35.25)
C. In many cases the third form of representation for the solution of (35.15) via an expression almost symmetric with (35.16) is used. To obtain it firstly we note that the difference between parameters of gamma-functions in the numerator and the denominator in (10.35) is equal to {3 - a and if Re ({3 - a ) < 0, then the integral on the right-hand side of (10.35) corresponds to the fractional integral xf3 Ig;P x- a , and if Re ({3 - a) > 0 then it corresponds to such a fractional derivative. Analogously, the difference of parameters in (35.20) is equal to 1 - I' with Re(1 - p) > 0, i.e. the right-hand side of (35.20) corresponds to the fractional derivative of a function g1( y) . So in terms of originals it is convenient to write this fractional derivative as a composition of the operators including the usual differentiation operators. To return to originals, we use the fact that according to ( 10.35) multiplication by (1 - {3 - s) n in terms of the Mellin transform corresponds to the operator n (n n z - fJ ) /2 xf3 ( /z) xn - fJ when a - {3 = -n and that the correspondence zP/2 Multiplying and dividing the right-hand +-+ (1 - {3 - 2s) n follows from (35.24). side of (35.20) by (1 - {3 - s) n or (1 - {3 - 2s)n with the corresponding numbers {3 = 1 + (v - p )/2 or {3 = n we write the right-hand side of (35.20) in the form
( 4J:)
(�-'-
2- P r(1 - s)r(1/2 - s) v ) r((1 + I' + v)/2 - s)r((JJ - v)/2 + n - s) -2- - n 91 (s), 2-p- n r ((1 - n)/2 - s)r(1 - n/2 - s) r(( 1 + I' + v)/2 - s)r ((JJ - v)/2 - s) (1 - n - 2s)n g1 (s). •
8
•
If 1 - Re JJ - n < 0, then the first gamma-functions correspond to the kernel of ''fractional integral" of the form (q - 1)- Pt /2 H(TJ - 1)Pt'11 (TJ- 112) with specially selected parameters 1'1 and v1 ; see Marichev [10, 11.13(4)] . The second multipliers correspond to the above operators of usual differentiation with power multipliers which can lie both outside and inside of the integral. In the latter case additional assumptions on the function 91 ( Y) are needed. Passing to the originals in the given
702
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
expressions after the changes given in (35. 19) we finally obtain the following four representations for the solution of (35.15):
X
J(z2 - r2)<"+�J)/2- 1 P�;:-�J ( ;) g( r)dr, :r:
e
/( z) =z -n
J (z 2 :r:
_
r2 )
e
(35.26)
( ;) Tl - �J -11 (35.27)
J(z2 - T2)(n+•)/2- 1 T-n p;-n-" (;) g(T)dT } • :r:
e
J :r:
/(z) = ( z 2
_
r2 ) (n+�J)/2- 1 p; - n - �J
e
( ; ) g(n) ( r)dr.
{35.28) (35.29)
The latter relation contains an operator in the form (35. 16). Thus, replacing here g
(35.31)
/{z) =z -n
J (z2 :r:
_
t2 ) (n+�J)/2 - 1 p;-n- �J
e
(�) t�J+n - 1 (35.32)
cf. (35.25) with (35.30) and (35.28) with (35.32).
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
703
Replacing in (35.15) and (35.16) t 2 - P f(t) by /(t - 1 ), z " g(z) by g(z - 1 ), z by and e by d- 1 we arrive at (35.18) and (35.17). This property allows us to use the solutions obtained for finding corresponding solutions of (35.17) and (35.18). In particular, the solutions
z- 1
/(z) = (z/2) " + 1 I:: "- 2 z 1 _ , _ , I�!;�2 zP -"- 3g(z), (35.33) 1 1 + 11 p + " 1 (35.34) /(z) = 2 ,+ 1 zP +,+ 1 Id-"- ;z-2 z - Id"- z -"- g(z) follow from (35.30) and (35.25). We shall not write analogues of (35.26)-(35.29) which can be obtained in the indicated way. We only note that solutions of (35.17) and (35.18) are representable by expressions different from (35.28)-(35.29) and (35.31)-(35.32) respectively, only by changing the intervals of integration of (e, z) to (z, d) and (z 2 - t 2 ) (n + p)/ 2 - 1 to ( -1) n (t 2 - z2 ) (n +p)/2- 1 . Using the method which led us to the solution given in (35.25) one can see by direct evaluation that besides (35.33) and (35.34) other forms for the solution of (35.17) and (35.18) which are admissible are (35.35) (35.36) respectively. On the basis of results from § 10.1 and using Theorem 18.1, Lemma 31.4, Remark 10.3 and the symmetry property of the Legendre function Pf (z) = P�11 _ 1 ( z), by which we can assume Rev � - 1/2 without loss of generality, the results obtained can be stated as the following theorem.
Let Rep < 1, Re v � -1/2 and 0 < e < d < oo. Equations (35.15), (35.16) and (35.17), (35.18) are solvable in L,(e, d), 1 � p < oo, if and only if g E I::; �' (L,(e, d)) and g E I�: " (L,(e, d)) respectively. These conditions being satisfied, each of the equations has the unique solution given by (35.25), (35.30) and (35.35), (35.36). The cases e = 0 and d = 0 in (35.15), (35.16) and (35.17), (35.18) respectively are more complicated and therefore they require further investigations (see § 39.2, note 35.3).
Theorem 35.2.
§ 36. Fractional Integrals and Derivatives as Integral Transforms When introducing the fractional integro-differentiation operators I:+ and I6_ in § 2, we in fact considered separately three cases namely integration if Re a > 0,
704
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Rea < 0
Rea = 0,
differentiation if and integro-differentiation of imaginary order if -:/; However, there is a way of defining these operators for all values of c:r at the same time if we use Fourier or Mellin transforms. Thus we may write down the equations
a 0.
-y +ioo 1 f f{1 - a - s) /* {s + a)x- • ds, 10a+ / ( x ) = i {36.1) 21r -y-ioo f{1 - s) Re(s + c:r) < 1, -y +ioo {36.2) et(z) = 2�i-y -fioo r(��)a/•(s + a)z- • ds , "'( = Res > 0, see {10. 3 5) and {10. 3 6), which follows from (7.17) and (7.18) after applying the inverse Mellin transform defined in {1.113). Here / *{s) denotes the Mellin transform of a function /{x) given in {1.112). The integrands on the right-hand sides of {36.1) and {36.2) contain the
ratio of two gamma-functions. This stimulates us to consider more complicated constructions of such a kind involving the ratio of arbitrary products of gamma functions as for example constructions similar to the integrand in defining the Meijer G-function. Realization of the latter approach in terms of originals will lead to an integral transform of type with the Meijer G-function as a n kernel where k(z, t) c� Its particular cases were indicated in §
{1.95)
=
(� I��:� ).
{1.44)
1.4 .
However the language of originals proves to be inconvenient for constructing the theory of integral convolution transforms. There are two reasons for this. First of all, as in the case of 10+ and J� , a definition of the transform can prove to depend essentially on parameters of the G-function though the G-function itself depends on its parameters analytically. Secondly, the G-function exists not for all values of m, n, p, q. For example, it does not exist if p q or m n p, q � Therefore it is more convenient to use the language of the Mellin transforms in order to define integral convolution-type transforms by using the Parseval relation given in
= =0
= = 0,
0.
{1.116).
36. 1 . Definition of the G-transform. The spaces 9Jt;:;(L) and L�e ,-y ) and their characterization
h*{s) {1.116)
Taking in to be a ratio of products of gamma-functions as in we arrive at the following concept.
{1.95)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
Definition 36.1.
705
The G-transfonn of a function / (z) is defined by the integral
( I��? I f(t)) (x) 1 j (bm ) + s, 1 - (an ) - s _ = -_ 21ri r [ (a; + l ) + s, 1 - (b� + l ) - s ] 1• (s ) z • ds,
(G/)(x) :: a;.n
(36.3)
u
where r (a( b;m+)1 )++s,s, 1 1--(b(a�n+)1 )--8 8 - r a b t ++s,s, .. .. .. ,, bam, ++ 8,s, 1 1--b a1 --s,s, .. .. .. ,, 11 -- ab,n -- 8s m+ l n+ l m n n f(b; + s) n r(1 - a; - s) j=- l,--j= - -= --,--l-- -- -- -- ---- -- -- n r(a; + s) n r(1 - b; - s) j=m+ l j=n+ l
[
]
[
] (36.4)
f•(s) is the Mellin transfonn - see (1.112) - of a function /(z) on the line +1 ) = u = {s, Res = 1/2} = {1/2 - ioo, 1/2 + ioo} ; (an ) = a 1 , a 2 , . . . , a n; (a; an+ b an+2 , . . , a,; (bm ) = bt , . . . , bm; (b� +1 ) = bm+b · · · , bq; and the components of p- and q-dimensional vectors (a,) and (b9 ) are complex numbers which satisfy the conditions
.
Rea; =F 1/2 + 1, j = 1, . . . , n; Reb; =F -1/2 - 1, j = 1, . . . , m.
(36.5)
It is clear that if m = n = p = q = 0, then
(do8 1 : 1/(t))(z) = /(z).
(36.6)
It is natural to care about the convergence of the integral in (36.3). Since in accordance with (1.65) the gamma-function has a power-exponential asymptotic expansion as l lm s l oo then the space of functions is to be characterized by j•(8) with power-exponential weight at infinity. This justifies the necessity to introduce the fs:>llowing three definitions.
-+
706
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Definition 36.2.
The ordered pair ( c•, -y•) where
+-q c* = m + n - p2 ,
(36.7)
is called a characteristic of the G-transform (36.3), the value '1 = 2sign c* + sign -y * is called an index of the G-transform and the function bm ) + s, 1 - (an )+- s H ( s ) = f [ ((a; + l ) + s, 1 - (b� l ) - s
(36.8)
]
(36.9)
and the number p + q are called an image of the kernel and an index of the complexity of the G-transform, respectively. Definition 36.3. Let c, -y E R1 and 2signc + sign-y � 0. We denote by rot;,�( L ) the space of functions f(x), 0 < x < oo, representable in the form
j
f(x) = 2�i /*(s)x - •ds,
(36.10)
(T
(36.11) where F(s) E L(u). We denote rotO,� ( L) as rot- 1 (£) for brevity. We denote by L�c,-y ) the space of functions f(x) , 0 < x < oo, satisfying (36.10) and (36.11) where F(s) E L 2 (u), 2signc + sign-y � 0, and the integral along u is assumed to be mean square convergent. The pair (c, 1) is called a characteristic of the space rot;,�(L) and L�c,-y ). I t follows from (1.65) that the asymptotic relation
Definition 36.4.
(36.12) holds and hence the integral
j
h(s) = 2�i H (s)x - •ds (T
(36.13)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
707
is convergent everywhere in the case c• > 0 except at the point x = 1 if TJ = 0, 0 < 'Y * � 1, p = q, and it is divergent everywhere if TJ = 0. However, if 11 = 0 and p :/; q, then the function h( x) may be defined via the convergent integral of the form given in {36.13) with the contour u coinciding with the line Re s = 1 /2 + c;sign (q - p), £ > 0. Therefore the case 11 > 0 is connected with the usual direct G-transform realized by the Meijer G-function. The cases 1J < 0 are singular and, in particular, they correspond to the inverse transform of Laplace, Stieltjes and Meijer-type transforms if c• < 0 and of a fractional differentiation if c• = 0, 'Y• < 0. As for the cases 11 = 0 they correspond to Watson type transforms, i.e., in particular, to the Narain transform - see Marichev [10, Sec. 8.3], the Hankel transform, and Y- and H-transforms considered below in § 36.7 when p :/; q, or to transforms of the type of integrals of imaginary order when p = q. The given definitions of the spaces rot��(L) and L�c ,-y) take the behavior of H(s) at infinity into account. Namely, if c + c• > 0 or c + c• = 0 and 'Y + 'Y* � 0 (which can be briefly written as the inequality 2sign ( c + c•) + sign ('Y + 'Y * ) � 0) then the integral in {36.3) is absolutely convergent in the case of rot;,�(L) or in square mean in the case of L�c ,-y ) , respectively. We give some properties of these spaces. 1) The relation L�o,o) = L 2 {0, oo) is true. 2) x - 1 f(x - 1 ) E rot��{L) or L�c ,-y) if and only if f(x) E rot��(L) or L�c ,-y) , respectively. 3) The sets of spaces rot��{L) and L�c,-y) are well-ordered in the sense that
{36.14) if 2sign ( c' - c) + sign ("'(1 - "'() > 0. 4) The spaces rot;,�(L) and L�c,-y) with the norms ll f ll rat- 1 = (c •r)
j
I F( s) ds l ,
11/II L 2 = IIFII L2 (u)
(36.15)
(1
and the usual operations of addition and multiplication by scalar are Banach spaces which are isometric to L( -oo, oo) and L 2 ( -oo, oo ) , respectively. 5) The following theorem gives a characterization of spaces rot��(L) and L�c,-y) in terms of spaces rot- 1 (L) and L 2 (0, oo) , respectively.
a) The spaces rotO,�{L) and L�o,-y) consist of functions f(x) representable in the fonn f(x) = x-"Y J6+ 0 consist of functions f(x) for
Theorem 36.1. oo ,
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
708
which there exist the constants Mj depending only on f(z) such that
m > c + "'f, n = 1 , 2, 3, . . . ,
(36.16)
and the norm an (36.16) is evaluated in the space rot- 1 (L) and L2 (0, oo), respectively.
The proof will be given only for the space L�c,'"f), since it is similar for the case of
rot;,�(L).
a) Let
c = 0.
Then by definition
/(z) E L�o,'Y) if and only if
f(z) = 2�i j s -'Y F(s)z - • ds, F(s) E L2(u),
(36.17)
(7
where the integral converges in square mean. We transform (36.17) to the form
1 r(1 - s) F (s)z ds, /(z) = 21ri J r(1 + 'Y s) 1 _
-·
(36.18)
(7
According to (1.66) F1 (s) belongs to L2 (u) if and only if F(s) E L 2 (u). Since 1 � 0 then f(1 - s)/f(2 + 1 - s) E L 2 (u) and in accordance with the Parseval relation (1.116) in the space L2 - see , for example, the book by Titchmarsh [1, p. 127, Theorem 73 and p. 71, the formula 2.1.17] - we have
1 r(1 - s) F (s)z ds = 1 (1 - t)'Y
(7
where
F1 (s).
Then from (36.18)
f(1 - s) 1 d d 16+ l +'"f - .S TY+ 1
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
709
Since F1 (s) E L2(u) then 0. Hence it follows that d� IJt 1 0. A function /{z) belongs to L�c , -y) if and only if
� j s-'Ye-wcllm •I F(s)z-' ds,
/{z) = 2 i
F(z) E L2(u).
q
It follows from {1.65) that the functions F(s) and
m > 1 + c, belong or do not belong to the space L2 ( u) simultaneously. The function smr- 1 (1/2 - c - "Y + m + 2cs) satisfies the conditions of the theorem in the book by M.M Dzherbashyan [2 , Subsection 2.3.2]. Therefore L�c,-y ) coincides with the space Lf of Dzherbashyan [2, p. 90] where
4>(s) = sm /f (1/2 - c - "Y + m + 2cs ) ,
m > "Y + c.
Hence the product
sme- 2c• ln n
ft ( 1 + k + m - 2csc - "Y - 1/2 ) '
1:= 1
s E u,
converges boundedly to the above function 4>( s) as n � oo. The latter implies that sme- 2c• ln n fi 1 + l:+ -��,. _ 1 72 /4>{s) is uniformly bounded with respect to s
1:= 1
(
m
)
and n. Then by a statement due to M.M. Dzherbashyan [2, p. 90] /(z) belongs to Lf = L�c ,-y ) if and only if the estimate in (36.16) is true. The theorem is thus proved. We just note that in the case of rot;,�(L) the corresponding result by Vu Kim Than [4) for the space rot+ 1 (L) instead of the latter cited statement should be used. •
36.2. Existence, mapping properties and representations of the G-transform We now consider the question of the existence and ma�ping properties of the operator of the G-transform in the spaces rot;,�(L) and L2c,-y ) , and also obtain its
710
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
representations in the usual form via the integrals containing Meijer G-functions in the kernel along the ray (0, oo) . The following statement answers the questions concerning existence and mapping properties.
The G-transform defined in (36.3) with the characteristic (c* , -y*) exists in the spaces rot;:� (L) and L�e,� ) if and only if
Theorem 36.2.
2sign (c + c*) + sign (-y + -y *)
;::: 0.
(36. 19)
Under this condition the G-transfonn isomorphically maps the spaces rot;:� (L) and L�e ,� ) onto the spaces rot;�e• �+� · (L) and L�e+e • .�+�· ) , respectively. . Proof. Let / (z) E rot;:� (L) or L�c.� ) . Then it follows from (36.10) and (36.11) that f*(s) = s -�e- 1r ellm •l p(s) , F(s) E L(u) or F(s) E L2 (u) respectively. Since ( c* , -y*) is the characteristic of the G-transform, then the asymptotic relation given in (36. 12) holds with the values defined in (36.7) . Hence taking s E u into account we obtain the representation
(36.20) where F1 (s) E L(u) or F1 (s) E L2 (u) . Hence the integral in (36.3) defining the . . . . . ( • 1 � I" f It eXISts, IS a functiOn m the space nnG-transtorm, J""e+e • .�+�· (L) or L 2e+e '�+�· ) . We find now the conditions for the existence of this integral. Since lz -' 1 = z - 1 1 2 for s E u and F1 (s) E L(u) or L2 (u) , then only the power-exponential weight s -�- -r · e- r ( e+e• ) llm •l influences on its existence. If c + c* > 0, then this weight decreases exponentially as IIm sl ---+- oo for any value of "Y + -y* . If c + c* = 0, then this weight decreases only for "Y + -y* > 0, or it is bounded for "Y + -y* = 0. These can be unified into the same condition of the form (36. 19). If these conditions are not satisfied, then the weight increases at infinity and therefore the integral (36.3) is divergent. Isomorphism of the map is clear from the above arguments. • Theorem 36.3.
Let the inequality 4sign c* + 2sign -y* + sign IP - ql > 0
(36.21)
and the conditions Re bj > - 1/2, j = 1, 2, . . . , m;
Reaj
<
1/2, j = 1 , 2, . . . , n,
(36.22)
hold. Then the G-transform defined in (36.3) exists in the space vn- 1 (L) and may
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
711
be represented as the following Mellin convolution integral 00
(G/){z) = J �n pq 0
(� I (b,)a, ) y
y y
( ) f ( y) d
{36.23)
containing the Meijer G-function. condition in (3 6. 21) holds, then the G-function
Proof. If the
a;'."
(• I ��:� )
exists and is integrable on any interval [c, E], 0 < c < E < oo, everywhere including the singular point z = 1 provided that c* = 0, p = q. For 0 < 'Y * < 1 this function has a singularity of order 0{{1 - zp* - 1) as observed from 8.2.1.48 in Prudnikov, Brychkov and Marichev [3]. Consider firstly p � q. Then this G-function has asymptotic estimates near the singular points z = 0 and z = oo which may be written in the form (see Marichev [10, Sec. 8.3] or [12]):
O(lzla -1 ) + clz iP cos[(q - p)z1/(q-p) + 6], z a> - 1max � k � n Reak and c = 0, c5 = const or 0, c #= 0, q � p + 2, (q - p)p = {1 + p - q)/2 ---+-
* c =
oo ,
{36.24)
* 'Y .
Hence it follows that
{36.25) for c• > 0 or c* = 0 and 'Y * > 0, q = p. Therefore according to {36.22) the left-hand integral converges boundedly if s E u. The latter means that there exists a constant c > 0 such that for any c > 0, E > 0 and t E R1 the estimate E
I
j K(z)zit-1/2dzl £
�
c,
(I )
K{z) -- �n z (a, Pf (b,)) z • -1 '
holds - see Vu Kim Tuan [4]. Further, the inequality q � p + 2 following from the one q > p for c• = 0 is to be taken into account. Then from (36.21) and (36.24) it
712
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
follows that the additional term of the form 00
c
• - 1 xl2( q -p)) - 1 - 1 1 2 cos[(q - p)x 1 f(q-p) + 6]dx x J1 00
j t
= c (q - p) 9 = Im s ,
cos[(q - p)t + 6]dt,
(36.26)
1
s E u,
arises on the right-hand side of (36.25) in the remaining case c* = "'' * = 0, q ;:::: p+ 2. The latter integral in (36.26) converges boundedly also. So the conditions given in (36.21) and (36.22) ensure a bounded convergence of the integral in (36.25) for s E u . Hence by the Parseval relation - see (1.116) - in the space rot- 1 (£) the integral in (36.23) is convergent and is equal to (36.3) - see Vu Kim Than [1]. Now let p ;:::: q. Applying the reflection relation ( 1.36) and the translation relation ( 1.97) for the G-transform to the right-hand side of (36.23) and making the change of variables x = 1/x' and y = 1/y' we have
-(b,) ) ]:_ ( _!_) ( _!_) Joo qp ( � -(ap)
_!_ (G/) x' x'
G':.m
=
0
x' y'
y'
f
y'
dy' . y'
(36.27)
Using now property 2) of the space rot- 1 (£) given in § 36.1, we easily arrive at the case p � q. •
( I�::� )
Remark 36.1. If c• = -y• =
0 and p = q, then the G-function c;-.n :r has 1 a non-integrable singularity of order 0((1 - x) i� - ), Im.,P = 0, at the point x = 1 and hence the integral in (36.23) is convergent in such case only for f(x) = 0. The condition in (36.21) excludes this case.
Let 2sign c* + sign "''* ;:::: 0 and let the conditions in {36. 22) be satisfied. Then the G-transform exists in the space L2 (0, oo) and can be represented in the form
Theorem 36.4.
: J a;:'+���. ( � I �b�i� -n 00
(Gf)( :r ) =
.,
0
)
!(y)dy
(36.28)
where (ap )+1 = a 1 + 1 , a2 +1, . . . , ap+l. If additionally 2sign c* +sign ("Y* - 1/2) > 0, then the G-transform defined in (36.3) exists in the space L2 (0, oo) and may be represented as in (36.23) .
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
713
Proof. If 2sign c* + sign-y• � 0 then the function given in {36.9) is bounded on the line u in view of the asymptotic estimate in {36.12). Therefore H(s)/{1 - s) L 2 (u) and {36.3) may be written in the form
E
(Gf)( z) =
J H(s) f* (s)z1 -• ds. �� 21r1 dz 1 - s
(36.29)
(T
Applying now Parseval's relation, given in {1.116), in the space L 2 (0, oo) to the right-hand side of {36.29), and the translation relation {1 .97) for the G-function, in particular, leading to the relation {36.30) we arrive at the representation in (36.28) without difficulty. If 2signc• + sign (-y* - 1/2) > 0, then not only H(s)/(1 - s) L 2 (u) but also H(s) E L 2 (u) since H(s) = O(lsl --r* ). Therefore Parseval's relation can be applied to {36.3) immediately which yields (36.23). •
E
Remark 36.2.
Since, for example,
( I � I f(t)) (z) = G8�
(GA� I � I l(t)) (z) = L HI m ; z} and
L _, H - I m ; ., } - see (1.119)-( 1.121) - and the Meijer
G-function of the form (36.23) or (36.30) does not exist in the case m = n = 0 of the G-transform, then not every G-transform may be represented by (36.23) or (36.28). The cases 2signc* + sign -y• > 0 and 2signc• + sign -y• < 0 of the G-transform correspond to the direct and inverse classical integral convolution transforms and the cases 2signc* + sign-y• = 0, i.e., c* = -y• = 0, correspond to the Watson transform for q -::/; p ( see the books by Titchmarsh [1, Chapter 8] and M.M. Dzherbashyan [2, Chapter 2, Section 1]) or to the fractional-type integrals of imaginary order Ib� and Ii! for q = p. This can be seen in more detail in § 36.1. Thus the general approach to the theory of integral convolution transforms with homogeneous kernels is realized via the representation in the form {36.3).
36.3. Factorization of the G-transform We introduce the following definition.
We call any representation of the operator in (36.3) by a composition of other G-transforms with less indices of complexity - see Definition 36.2 - a factorization of the G-transform.
Definition 36.5.
The most simple G-transforms except the trivial one in {36.3) are those which have indices of complexity equal to 1 . They are connected with the Laplace transforms given in { 1 . 119)-{ 1.121) and are defined in the following way.
714
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Definition 36.6.
The following transforms of a function f(z) 00
t f a = (I) e - (z: f t )± t f(t ) � ,
(36.31)
+ioo 1 -y f*(s)z - • ds, = 21ri j f(±a ± s) ioo
(36.32)
0
-y-
Re(s + a) � 0, with L {
Let G1 , . . . , G, be G-transforms such that transforms of their kernels H1 (s), . . . , H,( s) satisfy the condition
Theorem 36.5.
H1 (s)H2 (s) . . . H,(s) = H(s)
(36.33)
where H(s) has the form (36.9). Let also the transforms Gt , . . . , G, have indices of complexity less then G and let them have characteristics (ci , -yi ) , . . . , (ci , "Yi), respectively. Then if the condition in (36.19) holds the G-transform - see (36.3) in the space rot;,�(L) and L�c,-y) can be factorized via the transforms G� , . . . , G1 arranged in some order -
(36.34)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
715
if and only if (it , . . . , i,) is a rearrangement of the numbers (1, 2, . . . , 1) such that the inequalities
(
2sign c +
t ci; ) + sign ( + J=tt 'Yi; ) � 0, 'Y
J= t
(36.35)
k = 1, 2, . . . , 1,
hold. For any group of transformations Gt , . . . , G, of the above form there always exists at least one rearrangement satisfying (36.35). Proof. The existence of G-transforms satisfying the condition in (36.3) is clear. For example, this condition holds if we set I = p + q and take f (bi + s), i = 1, 2, . . . , m , f (1 - ai - s), i = 1, 2, . . . , n, r- t (ai + s), i = n + 1, . . . , p, r- t ( 1 - bi - s), i = m + 1, . . . , q as the transforms of the kernels Ht (s), . . . , Hp+9(s) respectively, which yields the factorization via the direct and inverse modified Laplace transforms defined in (36.1) and (36.2) - see Remark 36.2. It is clear from (36.33) that the relations � ci. L..J 1
i= t
= c. ,
(36.36)
are true. Applying Theorem 36.3 I times in succession we easily obtain necessary and sufficient conditions given in (36.35) which guarantee the existence of the whole compositions ( Gi,. . . . Gi 2Gs1f)(z), k = 1, 2, . . . , I. These compositions belong to ( 1 ') the spaces rot;, \, ( L) or L c I"Y where c' = c + L ci . , 1' = 1 + L 1i. . Hence
I
2
k
j=t
J
k
j =t
J
+
+
by (36.36) we have that (G/)(z) E rot;+t · �"Y +"Y• (L) or (G/)(z) E Lc2 c • I"Y "Y • , c respectively for k = I. We show now the existence of the rearrangement (i t , . . . , i,) for which the conditions, given in (36.35), hold. For this purpose we choose the indices it , . . . , i, in the following way. Let it be the index of the largest number among ci . If there are several such ci , then we take it as the index of the largest number among the numbers 1i in the pairs ( c; , 1i ) where ci are the largest numbers and it is an arbitrary index i from this set if there are several such numbers 1i . Withdrawing the pair ( c; , 1i) with such indices we consider the rest of the pairs and choose the next index i2 and the corresponding pair among them in a similar way. As a result after such a choice we have the inequalities
j = 1, 2, . . . , I - 1.
(36.37)
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
716
We show that (36.35) follows from (36.37). We assume on the contrary that (36.35) is not true, for example, for k = 1 - 1: 2sign
(
) (
1- 1 l- 1 c + � ci; + sign 'Y + � 'Yi;
J =1
That is, the inequality
c + c;1 +
· · ·
c + c! +
· · ·
J =l
)
< 0.
{36.38)
+ ci1 _ 1 < 0
(36.39)
or ..
+ c!· · - · = 0 '
(36 .40)
'Y + ,.,; + . . . + 'Y�- 1 < 0 .
is true. According to (36.36) from (36. 19) we have that 2sign (c + c!It +
· · ·
+ c!11 1 ) + sign ( -vI + -v! lit +
-
· · ·
> 0' + -v� 111- 1 ) -
I.e.
c + c! + ..
or
c + ci1 + "'VI
· · ·
· · ·
+ c!· · - · + c!. , > 0
(36.41)
+ c;, _ 1 + ci, = 0,
(36.42)
. . . + "'V� + "'V� 111 > 111-1 + "'V� l it + -0
hold. If {36.39) and {36.41) or {36.39) and (36.42) or {36.40) and {36.41) hold simultaneously, then c;, > 0 and since c;1 � c;� � � c;, by construction then c + c;1 + + c;, _ 1 > 0 which contradicts (36.39) . Now we suppose that the system of conditions in (36.40) and (36.42) holds. Then c;, = 0 and 'Y� > 0. So it follows from (36.19) and (36.40) and the inequalities c;1 � c;� � � c;, = 0 that c;1 = ci� = = c;, = 0, c = 0, from (36.37) that 'Yi1 � � 'Yi, > 0 and from the condition 2sign c + sign 'Y � 0 that 'Y � 0. The latter contradicts (36.40) and therefore (36.38) is not true. Similarly one may prove that the inequalities in (36.35) hold for the values k = 1 - 2, . . . , 1. • · · ·
·
· ·
· ·
· · ·
· ·
·
·
36.4. Inversion of the G-transform Theorem 36.6.
G-transform
Let g (x)
E rot;�c• ,-y+-y• (L)
or g (x)
E L�+c* ,-y+-y* .
Then the
(36.43)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
717
is the inverse of the G-transfonn defined in (36.3) with the notation (G/)(z) = g(z). If additionally the conditions of Theorem 36.5 are satisfied, then the inverse operator in (36.43) is factorized by the relation (36.44)
+
+
Since g rot;.t • 'l' +'l' (L�c c* ,-y -y* )), then by Theorem 36.2 the G-transform c, * in (36.43) exists and maps a function g into a function f in the space rot��(L�c,-y)). Applying now the G-transform in (36.3) to (36.43), evaluating the left composition directly, reducing all gamma-functions and using (36.3) we obtain the function g(z) on the left-hand side. The inverse G-transforms G-;:J 1 involved in (36.44) have the characteristics ( - ci; , --y;; ), j = 1, 2, . . . , I. Therefore the conditions in (36.35) with respect to (36.44) have the form
Proof.
E
I
I
i =k
i=k
• 2sign (c + c• - L ciJ + sign ("Y + -y - L "Yi;) � 0,
(36.45)
k = 1 , 2, . . . , I. It is clear that (36.45) and (36.35) coincide and hence Theorem 36.5 enables us to factorize the inverse operator (36.43) by (36.44) for the above rearrangement ( it , . . . , il ) · • Remark 36.3. If the characteristic (c, -y) of the space
the condition
I
rot�� (£) or L�c,-y) satisfies
I
(36.46) 2sign (c - E lei) + sign ("Y - L hi ) � 0 i= l i= l then the restrictions in (36.35) are true for all rearrangements and therefore the order of application of th-e operators Gi; in (36.34) and G� 1 in (36.44) may be arbitrary, i.e. these G-transforms are commutative in such a case. We consider the important case when all the Gi , i = 1, 2, . . . , I, are modified Laplace transforms as defined in (36.31) and (36.32). Applying the Mellin transform - see (1.112) - to these relations we see that the transforms of the kernels in (36.9) containing only one gamma-function in the numerator or in the denominator
r(±a ± s) r-·1 (±a ± s)
+-+
za A±z- a ,
+-+
za A±1 z- a ,
Re (a + s) � 0, Re (a + s) � 0,
(36.47) (36.48)
718
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
correspond to such operators. Hence each of the above Laplace transforms corresponds to one of the gamma-function in the G-transform (36.3). That is, this general G-transform can be factorized via the compositions of the operators 6 6 ,.. ; A+ z - ; ,
.._
j = 1, 2, . . . , m ;
z a; - 1 A _ z 1 - a; , - 1 ,.. ,..a; A+
.._
.._
-a; ,
j = 1, 2, . . . , n; . J = n + 1 , . . . , p ,·
z 6; - 1 A: 1 z 1 - 6; ,
(36.49)
j = m + 1 , . . . , q.
under certain conditions. By the symbols
we denote these operators respectively taken in indicated order. Their characteristics being evaluated by ( 36. 7) are equal to ( 1/2, -Re bi), j = 1 , 2, . . . , m , ( 1/2, Reaj ), j = 1 , 2, . . . , n, (- 1/2, Reaj } , j = n + 1, . . . , p, (-1/2, -Rebj ) , j = m + 1 , . . . , q respectively. We denote them by { fh ,6), . . . , (Op+ q , ep +q) in the indicated order. Theorems 36.5 and 36.6 can be formulated with respect to such a factorization in the following way. Theorem 36. 7.
Let the conditions in (36.19) and (36.22) and the inequalities
j = n + 1 , . . . ,p; (36.50) Rebi < 1/2, j = m + 1, . . . , q, hold. Then the G-transform defined in (36.3) can be factorized in the spaces rot��(L) and L�c,-y ) via a composition of modified Laplace transforms with power multipliers (36.49) applied in same order �eai > -1/2,
(36.51)
if and only if (i1 , , ip+q ) is a rearrangement of the numbers (1, 2, . . . ,p + q) such that the inequalities • • •
k
k
J=1
i=1
2sign (c + L (Ji; ) + sign (-y + E ei; ) � 0, k = 1, 2, . . . ,p + q,
hold.
(36.52)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
719
rot;..t • ,"Y +"Y ( L) or L�+c* ,-y +-y* and the conditions c in (36.19), (36.22), (36.50) and (36.52) be* satisfied for a certain ream�ngement (it , . . . , ip+q) of the numbers (1, 2, . . . , p + q) . Then the G-transform defined in (36.43), inverse to (36.3), can be factorized via the operators A 1 , , Ap+f with the characteristics ((Jl , e1 ), . . . , ((Jp+q , eP+f) by the relation
Theorem 36.8.
Let g (z)
E
. • .
f(x) = (G- 1 g)(z) = ( A� 1 Ai./ . . . A��9g)(z).
( 36.53)
Remark 36.4. In the case of operators in (36.49) the transforms of the kernels
Ht (s), . . . , Hp+q(s) are single gamma-functions of the form r(bj + s), r(1 - Oj - s), r- 1 (aj + s) or r- 1 (1 - bi - s) respectively. We can combine these gammcrfunctions
into various groups by different methods, for example in pairs. As a result we can obtain many variants of composition expansions for the G-transform by using (36.34) , for example, via fractional integrals and derivatives, Hankel, and Stieltjes and Meijer transforms and their inversions. Remark 36.5. If the conditions in (36.22) or (36 .50) are violated for some indices j provided that (36.5) hold, then instead of the corresponding operators in (36.49) there arise operators such that in the kernels of the direct Laplace transforms r f _ ,,. the function e - z will be changed to e - z - L: �· For such indices j the
k=O
gamma-function corresponding to the image of the kernel can be changed by three gamma-functions by the formula
+ r(a + s) = ( - 1) i r(a k + s)r(1 - a - k - s) , f(1 - a - s)
(36.54)
-k < Re (a + s) < 1 - k. Under a suitable choice of k the condition violated for the latter will be now satisfied. But such an operation leads to an increase of the index of complexity of the G-transform. From what has been said above we conclude that the following statement holds which is important in the theory of integral transforms.
The G-transform and, in particular, the classic direct and inverse integral transforms of the convolution type is the composition of a certain number of direct and inverse Laplace transforms of the form (36.31) and (36.32), whose admissible order of application depends on the space of functions f and the parameters of the transforms. For sufficiently good function spaces of the type rot;,� (£) or L�c,-y) , with the condition, for example, 2c - p - q > 0, the operators
Theorem 36.9.
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
720
A1 ,
. • •
, Ap+q are commutative and their composition forms the G-transform defined
in (36.3).
36.5. The mapping properties, factorization and inversion of fractional integrals in the spaces rot�_; (L ) and L �c ,-r )
The fractional integro-differentiation operators z- a ( J�+ /)(z) and z -a (I�/)(z) of' an arbitrary complex order a are both defined in the spaces rota,� ,(L) and L�o ,-y ) with 1' = max(O, -Rea). These operators map the spaces rot;,�(L) and L�c ,-y) with 2sign c + sign (')' - i) � 0 isomorphically onto the spaces rot;,� '+ Re a (L) and L�c ,-r' + Re a) , respectively and they can be factorized via the composition of the. operators given in (36.31) and (36.32) by the following relations under the corresponding conditions indicated below Theorem 36.10.
c
•
(36.55)
if /(z) E rolO,� (L) or L �o,-r) and Re a > - 1/2; (36.56)
if /(z) E rol�/� ,- Re a (L) or L �l / 2 ,- Re a) and Re a > - 1/2; (36.57)
if /(z) E rolO,�(L) or L�O,-y) and k > -Rea - 1/2; (36.58)
if /(z) E rolO,�(L) or L�o ,-y) and Re a < 1/2; ( 36.59)
if /(z) E rot�/2 ,.., (L) or £�1 / 2,-y) and Re a < 1/2; ( 36.60 )
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
721
if f(x) E rota,;(L) or L�o.� ) and k - 1/2 < Rea < k + 1/2 .
Proof�
Let n be an integer such that Rea + n > 0. Then we have
� � J f*(s)IgJn x-' ds
= x-a d n i 2
q
d" a +n j [ 1-s - ' ds 21ri r 1 + a + n - s f* (s)-x dxn
1 = x -a _
]
q
=
f [ 1-s 21ri q r 1 + a - s I (s) X ds. 1
]
*
_,
( 36.61 )
All interchanges of the order of integration and differentiation that have been made in ( 36.61 ) are valid because of the absolute convergence of the integrals given above. The latter is true by l si - � F(s) E L(u) - see ( 36. 11 ) - or by Parseval's relation in the space L 2 (u) and l si -� F(s) E L2 (u). If now the transform of the kernel is written as the product r - 1 (1 + a - s) r ( 1 - s), then the factorizations in (36.55) and (36.56) with the corresponding conditions on the spaces and Re a are easily obtained as particular cases of Theorem 36.7. Let now a be an arbitrary complex number and k be a positive integer such that Rea + k + 1 / 2 > 0 . Then by (1.47) the product r - 1 ( 1 + a - s) r (1 - s) can also be written in the form
Now by the arguments given after Remark 36.3 and the relations 8, 1 4 and 12 in Brychkov, Glaeske and Marichev [ 1, p. 24], and also by the choice of k, the images ( 1 + a - s) k , r- 1 (1 + a + k - s) and r (1 - s ) correspond to the operators x -a (J!:) k x a+k , x - 1 - a - k A: 1 x1+a+k and x - 1 A_ z 1 the characteristics of which are (0, -k) , ( - 1 /2, Rea + k ) and ( 1 /2, 0) , respectively. Hence the factorization of the form ( 36.57) follows. The remaining relations are proved in the same way. The first statements of this theorem are the direct corollaries of Theorem 36.2. • Corollary 1.
Any function f(x) in the space rot;,�(L) or L�c.�) can be represented
722
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
in the form f(x) = x - -r IJ+ cp(x) where cp(x) E rot;,J(L) or cp(x) E L�c ,o), respectively. from the fact that x- -r Jti+ is the G-transform with the ) characteristic (0, -y and therefore it is an isomorphic mapping of the space rot;,�(L) or L�c,o), respectively onto the space rot;:�(L) or L�c ,-y ) provided that 2signc + sign 1 � 0.
The proof follows
If a = ifJ is a pure imaginary number then the operator I�� as an automorphism in the spaces rot;;�(L) and L�c,-y) . The proof follows directly from the fact that the operator z- iB I�� the
Corollary 2.
IS
G-transform with the characteristic (0, 0).
36.6. Other examples of factorizations This subsection deals with examples of factorizations in terms of fractional integrals of Hankel and Bessel transforms, generalized Laplace transforms and potential type integrals. For convenience we shall denote by { h(x) }cp the Mellin convolution in 00
( 1.114) and set {h(l/z)}cp = J h(tfx) cp(t)t - 1 dt. 0
The following statements are valid. Theorem 36.11.
The Hankel transform of modified form Re v > -1,
( 36.62 )
maps the spaces rot;;�(L) and L�c,-y) isomorphically onto itself and it can be. factorized as the composition ( 36.63 )
If additionally f(x) E rotl/� Re v/ 2 (L) or f(x) E L�/ 2 ,Re " / 2 (L), then the composition , in another order ( 36.64)
can be also admitted.
The proof follows from the fact that if Rev > - 1 , then the operator in ( 36.62)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
723
admits the representation
f(
(
) (
{ Jv 2 Vx) }f = 2�i r s + v/2 r - 1 (T
1 + v/ 2 - s
)f* ( )x-• d
( 36.65)
s
s
according to the sixth relation given in ( 1 .118) and the Parseval relation - see So it is factorized via two operators x" / 2 A + x-"1 2 and x- l - v/ 2 A: 1 x 1 + " / 2 as seen in the arguments after Remark 36.3. Their characteristics are equal to ( 1 /2, Rev/2) and ( -1 /2, -Rev/2) , respectively. So applying Theorem 36.5 we obtain the conditions guaranteeing the existence of the compositions of these operators in one or another order as well as ( 36.63) and ( 36.64) themselves. •
( 1 . 1 16) .
Theorem 36.12.
The Bessel Y -transform of the modified form
j¥. (2/f) t(y)�, 00
{Y.(2v'x}}/ =
( 36.66)
IRe v l < 1 ,
0
()
where Yv z is the Bessel function of the second kind {Erdilyi, Magnus, Oberhettinger and Tricomi {e, 7.!.1]} maps the spaces rot;,�(£) and L�c,"'f) isomorphically onto itself and it can be factorized via the composition of the Hankel operators Jv = { Jv 2y'X)} and the fractional integrals or derivatives of the form K = x-"1 2 I: 1 '2 x( v +l)/ 2 and I = x - ( v +l)/ 2 I��2 x"l 2 • If c = 0 and v < 1 /2, then the operator Jv can be in any position and the operator K must be applied after I:
If 2sign c + sign
arbitrary.
(
(
( 36.67)
Yv = KIJ, = KJv i = J,KI. -y
- 1 /2)
� 0,
then the order of application of the operators
is
Proof.
In accordance with ( 1 . 1 16) and 9.4 ( 1 ) in the book by Marichev [ 10] we form the G-transform as
( GY f)( x) =
1 21r i
jr [s - ( (T
8 + v/ 2, 8 - v 1 2 v + 1 ) /2, ( 3 + v)/ 2 - s
] f* ( )x-•d 8
8"
( 36.68)
Its characteristic is equal to ( 0, 0 ) and so by Theorem 36.2 it exists in any spaces
rot;,�(£) and L�c,..,) and is an isomorphic mapping of this space onto itself. If additionally the conditions in ( 36.22) which have the form -1 < Re v < 1 are satisfied, then according to Theorem 36.3 the integral in ( 36.68) is transformed
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
724
into the integral in (36.6). Now we write the transform of the kernel in the form r
r
r
+1-s [ v/2s ++v1/2- s ] [ s -s(v- +v/21)/2 ] [ (3v/2+ v)/2 - s] ·
Using the second relation in ( 1 . 117) it is not difficult to establish the following Mellin correspondences
r r
[ db - s ] [ ad + s ] -
_
S
+-+
8
+-+
Z 1 - dzdO+- b Z b- 1 '
Re(a + s) > 0 ,
(36.69)
Re ( b
(36.70)
-
s) > 0 ,
(d- a - 1 )/2 z(a -d+l)/2 { Ja +d- 1 (2 V- fZ\} ., ) ... ""
'
( 36.71 )
Re(a + d), Re (a + s) > 0 , from (36.1), (36.2) and (36.65). Applying these relations to the above image of the kernel I it is easy to obtain the whole factorizations indicated in this theorem. The conditions on the characteristics of the spaces are deduced from Theorem 36.5. • Theorem 36. 13.
The Bessel H-transform of the modified form (36.72)
where H 11 (z) is the Struve function {Erdelyi, Magnus, Oberhettinger and Thcomi maps the spaces rot�;(L) and L�c,-y ) isomorphically onto itself and it can be factorized as the following compositions
[2, 7.5.4]}
11 {H11 (2 ..fi)} I = ( J11 )(z ( 11+ 1 )/2 r:. 11 2 I�!2 z- ( + 1 )/ 2 )/(z) , {H11 (2 y'Z) } I = z(ll+l)/2 1: 1 /2 z - 111 2 ( J11 )z ll/2 1!' 2 z- ( 11+ 1 )/2 /(z) ,
(36.73) (36.74) (36.75)
where /(z)
E rot- 1 (L)
rot0, � 1 2 ( L) or £ �· 11 2
or /(z) E L 2 (0 , oo) in the first two cases, and /(z) E in the third case. Moreover, the operators on the right-hand
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
725
sides of {36.73) and {36. 74) are commutative and the operator (J,) in {36.75) is commutative with the operators z( " + 1)/ 2 I: 1 '2 z - " /2 and z" /2 I!.' 2 z -( "+1 )/ 2 .
The proof is similar to that of Theorems 36.11 and 36.12 and is based on the representation of the transform {H,(2.Ji)} / in the form s v/2 s v { H (2VZ} }/ =� 21r1 /r [ 1 + +/2 - ] r [ + (+ +/21)/2 ] {36.76) 11
II
(7
S
S
II
provided that -2 < Rev < 0, which follows from 9.5(1) in the book of Marichev (10) . •
Theorem 36.14.
The generalized Laplace transform
{36.77) Rev < 1, where D,(z) is the parabolic cylinder function (Errlilyi, Magnus, Oberhettinger and Tricomi (2, 8.2]) maps the spaces rot;:� ( L) and L�e,"'f) isomorphically onto the spaces ftl) - 1 . the 1o� 11owang +1 2 Re 2 · 1y an d a·t a dmats . :JJ'-e+1 / 2 ,"'(-Re v / 2 (L) an d Le2 / ·"'�- "/ , respec tave composition expansions D,{f(y); z
} = z 1 /2I: "/2 z(v- 1 )/2A+ f(z),
D, {f(y); z } = A+ z 1 1 2 I: "'2 z<" -1)/ 2 f(z) ,
{36.78) {36.79)
where Rev < 0 and 2signc + sign('Y + {1 - Rev)/2) � 0 in the latter case.
Proof. In accordance with 8.30{1) in the book by Marichev (10) we consider the G-transform
(Gn /)(z) = 2�i j f [ � tl::. �)�2 ] r(s) :e- • ds. 8
{36.80)
(7
It has the characteristic {1/2, -Rev/2) and therefore exists in any space rot;:� (L)
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
726
,) , · or L (c ;,J"c-+l l/ 2,-y - Re " / 2 (L) or L(c2 + l/ 2 -y - Re "/ 2) by Theorem . . 2 -y and maps 1"t mto nn The reducibility of to follows from Theorems and and the factorization relations and follow from Theorem It is easy to formally write these relations on the basis of the correspondence
(36.80) (36.77) (36.78) (36.79)
D, r [ s s, (1s - 1/2v)/2 ] +-+
+
+
36 2
36.3 36.4 36.4.
=
r[s]r [ s +s(1+ -1/2v)/2 ]
(36.81)
The theorem is proved. • In the conclusion of this subsection we illustrate another proof, although formal, of the decomposition for the potential type integral in
(12.39)
(12.34)
00
1
la
X>
0
For any function
.,-"( I�
2.5(1)
0.
(36.82)
f E rot;,�(L) (L�c,-y) ) we have the representation
/)(z) = 2�i f r [ .- ;/;,o:i ! :i2-. ] f* (s)z -• ds,
(36.83)
tT
see in the book by Marichev obtain the following factorization
[10]. Using now (36.69) and (36.70) we easily (36.84)
where an arbitrary order can be admitted. The transform of the kernel may be decomposed into two fractions by another way which leads to the factorization via Hankel transform.
36 .7. Mapping properties of the G-transform on fractional integrals and derivatives Theorem 36.15.
Let the conditions 2sign ( c +
c* ) + sign ('r + i* + Re a) � 0, 2signc + sign ( i + Rea) � 0
(36.85)
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
727
hold. Then there exists the G-transform defined in (36.3) of the operator x- a (I�+ f)(x) in the spaces rot;,�(L) and L�c ,"'f) and it is evaluated from the relation (36.86) The
Gi 1 /
proof follows =
x- a l�+ f
from Theorem 36.5 where we have to set l = 2, and Gi1 / = Gf in (36.3) and take the G-transform
(a;:'+��� I (::) �l l I(t)) I
,
( " ) instead of that in ( 36.3).
Let the conditions of Theorem 36.15 be satisfied. Then there exists the G-transform defined in (36.3) of the function x- a (I�f)(x) in the spaces rot;,�(L) and L�c ,"'f) and it is evaluated from the relation Theorem 36.16.
(36.87) The
proof is similar to that of Theorem 36.15.
36.8. Index laws for fractional integrals and derivatives For fractional integrals and derivatives there are known the so-called index laws given by ( 10.4), ( 10.5), ( 10.42) and (10.43), see also Theorem 10.7. In this subsection we prove the theorem characterizing the conditions under which index law as well as the analogue of Theorem 10.7 for the operators I�+ and I� in the spaces rotO,; ( L) and L�o ,..., ) hold.
Let (a t , . . . ,an ) and ({Jt , . . . , fJn ) be two sets of arbitrary complex numbers. Then the operators zPi /�+-Pi x- a i are commutative in the space {f(x) : f(x) = x6 g(x), g(x) E rotO,; (L) or g(x) E L�o ,..., ) } with
Theorem 36.17.
1
i
= max { O, max Re L:
Moreover, if an addition
j=l
- CXi; ), { it , . . . , ii } C { 1 , 2, . . . , n } } ,
({Jt , . . . , Pn )
(36.88) (36.89)
is a certain rearrangement of the set
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
728
(nt, . . . , nn), then the composition of all these operators is an identity operator (36.90) If the conditions in (36.88) or (36.89) are not satisfied, then the above operators are not commutative.
f(x) = x6g(x) where g(x) E rot0,�(L) or L�O,-y ) be
Proof.
Let the representation true. Then by (36.10) we have
zPi1 ];�1 -Pi1 z-ai1 f(x) = zPi1 ];�1 -Pi1 2�i I g* (s)x6 -a i1 -• ds. tT
Since
Re(6 - ni1 - s) > -1 according (36.89), then for m > Re(,Bi1 - ni 1 ) we find
1 I r [ -1 .B-i1 -i1 s-+s 6++6 m ] g* (s) --x ern /3 Il =x�'� 1 1 211"i dxm 6 - i1 +m- ds Q
Ai
tT
-s+6 = 2?r1 i I f [ 11 -- ni1 ,Bi1 + 6 ] g* ( ) 6 _11 ds = cp( ) - S
X .
S X
tT
All transpositions of operators that have been carried out are valid since all integrals we have used converge absolutely or (according to Parseval 's relation) in L 2 , in view of the condition E L 2 (u) respectively. The constructed E L(u) or function belongs to the space E rot0 � , (L) or , ' o , > L� -r , where = t ) } . The conditions in (36.88) and (36.89) rewritten in the form
cp(x)
g*(s)s"Y
g*(s)s"Y {/(x) : f(x) = x6g(x), g(x)
1' 1 + Re(ni1 - ,Bi k
1* = max{ O, max Re L(,Bi; - ni;), {it , . . . , ik } C {1, 2 , . . j= t
6 > t
.
, n} \ i t } ,
§ 36. FRACTIONAL INTEGRALS AS INTEGRAL TRANSFORMS
729
with respect to the parameter 1 ' are also satisfied. So we can apply operators of the type with i2 E 1 , . . . , n} \ i 1 to again. Continuing this procedure we obtain by Theorem 36.5 that the composition exists for any rearrangement (i 1 , . . . , in and moreover it is representable in the form of the G-transform
xP•� I;�� -P·� x- 01 ·� { (xP•,. I;�,. -Pi,. x-a ,,. ) . . . (zP•• I;�· - 13'• z- 01'• )f(x) ) �
J r [ 11 -- aPii,. ++ 66 -- s,s, .. .. .. ', 11 -- Paii.l ++ 66 -- ss ] g* (s)xcS- 6 ds � r [ 1 - a t + 6 - s, . . . ' 1 - an + 6 - s l * (s)x 6- s ds, = 21r1 J 1 - Pt + 6 - s, . . . , 1 - Pn + 6 - g 21rz
,.
u
(36.91)
S
u
see (10.48), and therefore it does not depend on other applications of composing operators. The change of this order leads to the change of the order of the gamma-multipliers in {36.91). If in addition . . . , Pn } = a 1 , . . . , an } , then all gamma-functions in (36.91) are cancelled and the right-hand side has the value which yields {36.90). Let now the conditions in (36.89) be broken, i.e. let there exists a number j such that 6 � Rea; - 1/2. At first let 6 < Re a; - 1/2. Then the operator E rotO,�(L) does not exist for the function = or L�o .� ) if e < Re a; - 1/2 - 6. Similarly, in the case 6 = Re a; - 1/2 the function can be found in the space indicated in the formulation of this theorem such that the above operator does not exist on this function. Finally we suppose that the condition in {36.88) is not satisfied for a certain
{Pl !
{
x6g(x)
f(x) x- 112+Ee - z
xPii;,t-P;x - 01if(x) f(x)
set (il l . . . , ik ) · Then by Theorem 36.5 the composition does not exist in the space rot0,�(L) or £�0·�>.
•
-P• j=lll (xP•; I;� ; x- 01'i )f(x)
Theorem 36.18. Let the conditions of Theorem 36.17 be satisfied except {36.89)
instead of which we assume the opposite inequality
6 < max Re ai - 1/2
l �j�n
{36.92)
to be hold. Then the statements of Theorem 36.17 with Io+ replaced by I_ are true.
The proof is similar to that of Theorem 36.17.
Corollary 1. Let 1 = max{ O, -Rea, -ReP, - Re a + and 6 > max( - Rea , - ReP) - 1/2, or 1 = max{ O, -Rea, -Re {a + P)) and 6 > -Rea - 1/2. Then the first and the second relations in {10.4) hold in the spaces = E rotO,�(L) } and = E L�o.�) } , respectively.
g(x)
{f(x) : f(x) x6g(x), g(x)
( P)) {f(x) : f(x) x6 g(x), .
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
730
The proof is obtained on the basis of the representation Z
-a- fJ zO+a lO+fJ !(Z ) = 21r1 i j r [ +p +p +1 -1 -s ·] r [ p +1 -1 -s ] /• ( ) -• d S
Q
(1
S
S Z
S.
1
Let = max(O,Ren,ReP,Re(n+P)) and 6 > max(Ren,ReP)-1/2. Then the relation
Corollary 2.
(36.93) holds in the spaceso {/(z) : /(z) = x6g(x), g(x) E VJtO,� (L) } and {/(z) : f(z) = x6g(x), g(x) E L� ,-y) } . The proof follows from Theorem 36.17 and Corollary 1. Corollary 3 . Let = max{ O , -Rea, Re(n + P)} and 6 > max( O , -Rep, -Re(n + P))-1/2. Then (10.42) holds in the spaces {/(z) : /(z) = x6g(x), g(x) E VJtO,�(L) } , o ) , and {/(z) : /(z) = x6g(x), g(x) E L� -y } . Proof. Taking the result n + P + = 0 into account we rewrite the left-hand side of (10.42) in the form 10..: - fJ xa +fJ x- fJ 1g+ x -a- {J 10+ xfJ f(x). This composition exists and has the value 1
1
�j r ( 1 + n + P - s] r (
211"1
(1
1-s
1-s ] 1+P- s 1 + p - s r ( 1 + + p s ] !• (s)x - 'ds = f(x) Q
-
provided that the conditions of the Corollary are satisfied. This completes the proof, cf. the proof of Theorem •
10.6.
§ 37. Equations with Non-Homogeneous Kernels In this section we consider some classes of integral equations of the first kind solvable in closed form, the kernels of which are not directly representable in the form These are equations in which the kernel is of the type and which takes the form 0), (J = y r , after the substitutions z = ln y and r, certain non-convolution equations with Bessel functions, equations of composition type decomposed into more simple invertible equations, and 4) also equations connected with the integration of special functions with respect to a parameter - Kontorovich-Lebedev and Mehler-Fock type transforms.
K(xft). t = ln (2)
(1) K(ln
f
K(x - t),
(3)
(
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
731
37. 1 . Equations with difference kernels
We investigate the following seven left-sided Abel-type integral operators
(A�+' /)(z) =
1� (z - t)a- 1 J-(a- 1)/2(l(z - t))f(t)dt, 1� (z - t)a- 1 -
(37.2)
r (a)
(I
a � /)(z) = ( Ba-t
(37.1)
Ja/2 - 1 ( l (z - t))f(t)dt,
r (a)
(37.3)
� 1 a ,� )( z) - (z - t) a - 1 1-a- 1 ( lv z - t)f(t)dt, ( Ca+ � 1 (z - t) a - 1 J- ( (z - t))f(t)dt, a (Da+� /)(z) al 1� (z (I
I
(I
=
(s:+ /)(z) =
(I
(37.5)
�(�;-• la- t (J,..j(z - t)(z - t + 7)/(t)dt, l arg "YI <
1� ( (I
(37.4)
r (a)
(I
(E';+A.� l)(z) =
�
r (a)
71',
)
(37.6)
z - t a- 1 f(t) dt
2sh -2
( 37.7)
r (a)
and the corresponding right-sided operators Ia,fJ ,- � Aa,� Ba,� ca,� Da,� ' E::..�,., and Sf obtained from (37.1)-(37.7) by replacing z - t by t - z and [a, z] by [z, b], b � _oo. In the case when b = +oo we shall use the index - instead of the symbol oo- for thea ( above operators - see, for example, ( 10.56). In the expressions given above Re a > 0), p, l and "Y are some complex parameters, 1F1 (a; c; z) is the confluent hypergeometric function defined in ( 1 .81) and J., (z) is the Bessel-Clifford function defined by /4) " J-., (z) = J.,- (t.z) = r( + 1) ( 2z ) _., J., (z) = � (( -z+21)�ck! (37.8) ' The function l., (z) will occur below as well. 6-
-
v
6- '
'
oo
v
6- ,
6-
'
6-
732
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
According to the obvious relation J,(O) = J,(O) = 1 we obtain
- Aaa+,O - Baa+,O - Daa+,O - caa+,O - JaOt+ · - Eaa+,o,,. laa+,{J ,O -
(37.9)
This allows us to consider the operators given in (37.1)-(37.6) and (37.7) taking the asymptotic estimate 2sh ( T/2) .- T as r --+ 0 into account - as certain generalizations of the fractional integration operators 1:+ . Proceeding from the representations of kernels of the above operators via series, and using Legendre's duplication formula given in (1.61) it is not difficult to write the following relations characterizing the structure of the operators in (37.1)-(37.4):
=
A aa+, >. -
_
1:+ (E - .\ 1�+ ) - P,
(37. 10)
� (a/2}1: ( - .\ 2 ) k 1aa++ 2k LJ k= k! O
(37.11)
(37. 12)
(37.13) where E is the identity operator. The sums in the middle parts of these relations are called the Neumann generalized series. Since the operators I!+ are bounded in Lp(a, b), p � 1, b < oo - see the proof of Theorem 2.6 - then these series may be summed for l.\1 < II I�+ II i !{ a b ) " After evaluating the sums one may omit , this restriction on .\ and on the sums in (37.10)-(37. 13), since the operators in (37. 1)-(37.4) are analytic functions with respect to .\. A similar approach can be applied to the operators in (37.5)-(37.7) and to the corresponding right-sided operators, but in the case when b = +oo we must be careful while using Neumann generalized series to take into account the asymptotic estimates of the special functions in the kernels at infinity.
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
733
We also note that from (37.10) and (10.58) the interesting operator relation (37. 14) follows where e± >.z denotes the operator of multiplication by the function e ± >.z see the relation after (18.77). On the basis of (37.10)-(37. 13) and the semigroup property in (2.65) it is easy to write the relations
-
(37. 15)
and similar expressions for the right-sided operators. From the connection of the operators in (37 .1 )-(37 .7) with fractional integrals noted in (37.9) we can conclude that the operators in (37.1)-(37.7) have the same range in L, ( a , b) as I:+ , i.e. the following statement is true.
Let -oo < a < b < oo, 0 < a < 1, 1 � p < oo. Then the operators in (37.1)-{37.7) are bounded from L, ( a , b) onto I:+ [L,(a, b)] C L, (a, b).
Theorem 37.1.
proof follows directly from the properties 1F1 (,B; a; O) = J., (O) = 1 , 2sh (T/2) T as T --+- 0 and Lemma 31.4. Comparing (37.1)-(37.7) with (1.122) it is easy to see that, if a = 0 or a > 0 and a function f(t) is defined to be zero on the interval 0 < t < a, then the operators in (37.1)-(37.7) can be written as the Laplace convolutions in (1.122). Then after evaluating the Laplace transforms of the kernels by the corresponding relation 6.10(5) in Erdelyi, Magnus, Oberhettinger and Tricomi [1) and 2.2. 12.8 (cases I; + 1 , I; +2 , I�), 2.12.9.3 if n = 0, 2. 12.11.5 and 2.4.10.4 in Prudnikov, Brychkov and Marichev [2) and [1] respectively, it is not difficult to obtain the following Laplace transforms of (37.1)-(37.7) by the convolution statement in ( 1 . 123):
The
,_
(LI�f· >. f)(p) = p-a (l - >.p- 1 ) - P(Lf)(p), (LA�+>. f)(p) = (p2 + >. 2 ) -a f2 (L/)(p),
Re >. > 0, Rep > 0;
(37.19)
Rep > lim >.!;
(37.20)
734
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
(LB;:._>"· /)(p) = p(p2 + A 2 ) -(a +1)/2 (Lf)(p), Rep > jlmAj; (37.21) (LC�f.>. /)(p) = p-a exp [-A2 /(4p)](Lf)(p), Rep > 0; (37.22) a (Lf)(p), Rep > IIm.\ l ; (LD�+A /)(p) = (37.23) + ,\ 2 + a- 1 e ((p - .jp2 + A 2 )-y/2] 2 xp , >. (LE�f. -y f)(p) = (L/)(p), 2 2 .jp2 + A2 p + .jp + A
C .j� ) ( )
+ (1 - o:)/2) (LS0a+ f)(p) = f{p f(p + (1 + o:)/2) (L/)(p),
Rep > jlmAI ;
(37.24)
2Rep > Reo: - 1.
(37.25)
-
Comparing the right-hand sides here with those of (37.10)-(37. 13) we observe that the first four relations are obtained from (37.10)-(37.13) by replacing I�+ by p- 1 see (7.14) . As before, from (37.19)-(37.25) we easily obtain the operator relations in the form (37.26) (37.27) (37.28)
Eaa,+>. ,-yD{Ja+, >. - Eaa++{J ,-y ·
(37.29)
in addition to (37.15)-(37. 18). We consider now the inversion problem for the operators in (37. 1)-(37.7). (37.19)-(37.25) show that the Laplace transforms (Lh)(p) of the kernels h(x) - see ( 1 . 1 19) - of (37.1)-(37.7) and the corresponding inverse values ((Lh)(p)] - 1 have the same form and differ from each other only by values of the parameters. In the simplest case of the fractional integration operator /�+ ' to which (Lh)(p) = p- a with the condition Reo: > 0 corresponds, this condition for the Laplace transform p0 of the kernel of the inverse operator Dg+ obliges us to represent p0 in the form p0 = pnp- (n -na) , where Re (n - o:) > 0, with the value pn corresponding to the operator {�) = D�+ - see ( 1 .124). Similar operations are to he carried out when inverting the operators in (37.1)-(37.7). Taking the above arguments into account we shall construct the solution of the equation (37.30)
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
735
Using (37.10), { 10.58), {10.59) and 6.3(7) in Erdelyi, Magnus, Oberhettinger and 'I'ricomi (1] we formally arrive at the following representations for the solution of {37.30):
()
() ( )1 ()
- .!!._ l ]l+ m -a , -fJ , 'JI. .!!._ m ( z ) - dz o+ dz 9 d l 2: (z - t)l+m -a- 1 F ( -P; l + m - a ; �(z - t))g(m ) (t)dt = dz f{ l + m _ a) 1 1 (I
.!!__ 1 - dz R(z) '
(37.31)
/(z) = e"-2: I;_(e - "-2: I:+ a g(z),
(37.32)
/(z) = I�+ a e"-2: 1;_fe - "-2: g (z),
(37.33)
()
()
.!!_ m (e - "-2: g (z)). -a fJ-a,-'JI. __ ' J'+m , /(z) = e "-s .!!_ dz dz o+
(37.34)
Proceeding analogously from (37.20)-{37.25) we can formally write the following representations for the operators inverse to (37.2)-(37.7) :
/(z) ={ (A:.t"') - 1 g } (z) = (A;_;• "-g)(z) = (I;.i + � 2 ) 1 A!'J 2m -a, "-( r;.; + � 2 ) mg(z)
(
)1
' 2: (z - t)21+2m -a- 1 d2 (�(z - t)) (37.35) J = dz2 + �2 f{2l + 2m - a) l+m -(a+1 )/ 2 X
(: r
-
(I
.
+ �·
g(t)dt .
{ (B:l) -1 g } (z) = I�+ (A;;_; - 1 • "-g)(z),
( )1
(37.36)
d l 2: ( t)l+m-a- 1 )g(m ) (t)dt, l (� { ( C:.f.'J1. ) - 1 g } (z) = f(l + m a ) l+ m -a- 1 vz=1 dz (37 .37) 0
-
Z -
-
736
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
{ ( n:.;.� ) - 1 g}(z) = (n;;-; •� g} (z) =
2•
[ t. G ) n!:; a .�
I •
]
A;;-V !:; g (z),
k - 1 =::; Re a < k, k = 1, 2, . . . ,
(37.38)
{ (E:.f..\ '"Y ) - 1 g} (x) = (E!+ a , .\ , -"Y A;�·.\ g)(x) = (E!_; cr+1 , .\ , - ')' D;!·.\ A;�·.\g)(x),
k - 1 =::; Re a < k + 1, k = 1, 2, . . . , =
(37.39)
a+1 [San+- cr (Ia+ + ( 1-a 2 - - k) k Ia-+1 + -2-) k g] ( ) -1
X '
2k-- 2 =::; Re a < 2k, k = 1, 2, . . .
(37.40)
To justify these inversion relations we denote by R( x) the integral operator next after the signs ( d�) 1 in and and after the signs
(37.3 1), (37.34) (37.37) ( � + ,\2) 1 in (37.35) and (37.36). Then conditions for the invertibility of the operators in (37.1)-(37. 7) can be collected in the following general theorem. Theorem 37.2. Let 0 =::; a < x < b < oo in the equation [h * l](x) - see (1.122), let h be one of the operators defined in (37.1)�37.7), Re a > 0, let g be a given function on [a, b] and let I be an unknown function. We assume that l(x) = g(x) = 0 for 0 < x < a in the case a > 0. Then the solution I = h- 1 g of this equation exists and is unique in the space I E Lp (a, b), b < oo, if g(x) E I:+ (Lp(a ,b)) , p � 1. In the case when p = 1 this solution can be represented by the corresponding relations given in (37.3 1)�37.40) provided that the following additional conditions are satisfied: 1) 0 < Re a < I + m, l,m = 1,2, . . . , g E ACm ([a, b]), g(a) = g'(a) = · · · = m g ( - 1 ) (a) = 0, R E AC1([a, b]), and R(a) = R'(a) = · · · = R(1 - 1 >(a) = 0 for (37.31), only if g E ACm or R E AC1, then I = 0 or m = 0, for (37.31}; 2) 1 p < oo and Re/3 > 0 or Re/3 < 0 but g(x) is representable in the fonn g (x) = (I:+ p ,P)(x), ,P(x) E Lp (a, b), for (37.32); 3) 1 =::; p < oo and Re.(a - /3) > 0 or Re (a - {3) < 0 but g(x) = (1�+ 1/J)(x), ,P(x) E Lp (a, b), for (37.33); 4) the conditions similar kto+2those of 1} for (37.34)�37.37); k+ 5) 0 < Re a < k, g E AC ([a, b]) and g(a) = g'(a) = = g( 1 >(a) = 0 for (37.38); 6) 0 < Re a < k + 1, g E Ack+5([a, b]) and g(a) = g'(a) = . . · = g(k+4>(a) = 0 for (37.39); 7) 0 < Re a < 2k, g E AC2k ([a, b]) and g(a) = g'(a) = · · · = g(2k- 1) (a) = 0 for (37.40). =::;
·· ·
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
737
The follows from the existence, uniqueness and coincidence of the corresponding Laplace transforms of the equations and their inversions, and also from the existence of all given operators in the indicated spaces of functions and /. The condition E is obtained on the basis of Theorem 31.13 and the conditions 2) and 3) follow from Theorem 10.9 if we take into account the fact that for the existence of the corresponding fractional derivatives the condition E in the last subcases in 2) and 1) must be changed by the condition E or E The conditions of the = form = ··· = = 0 ensure the vanishing the terms outside of integrals when using ( 1 . 124). •
proof
g
g I:+ (L,(a,b))
g I:+ (L,(a, b)) g I:_;P [L,(a,b)] g J�+ [L,(a,b)]. g(a) g'(a) g(m- l ) (a)
Remark 37.1. According to (37.8) the replacement
A by iA in all relations of this
subsection for the operators defined in (37.2)-(37.6) is equivalent to replacing the functions I, and J, by each other. Remark 37.2. Making the reflection operation in all relations of this subsection, i.e. replacing x by x, and the corresponding changes in the functions f and
a+bthe corresponding relations and results concerning the g,. it is. not difficult to write . a,p, ). Aa, ). Ba, ). Ca , ). Da, ). Ea, ). ,-y , and sa m the case nght-s1ded operators I when b < oo. If b = oo, then such results are in general still valid under stronger 6_
6_ ,
,
6_ ,
6_ ,
6_ ,
6_
6_
conditions.
37.2. Generalized operators of Hankel and Erdelyi-Kober transforms The generalized operators of Hankel and Erdelyi-Kober transforms considered here will be used later in § 38 when considering certain types of dual integral equations. Generalized Erdelyi-Kober operators differ from the operators
J(x2 - t2 )"12J,(> ..jx2 - t2)
,P(x),
Rev > -1,
(37.41)
j (t2 - x2 )"12J,(A../t2 - x2)
Rev > - 1 ,
(37.42)
•
=
0
CX)
�
only by the weight factors. The operators in (37.41)-(37.42), in their turn are obtained from the operators C�+). and C�· ). - see (37.4) and the text below - by the changes a = v 1 and 2" + 1 tf(t 2 ) = A"
+
738
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
relations for (37.41) and (37.42) follow in the form
�
� J t(x2 - t2) -("+1)/ 2/_,_ 1 (,\ vfz2 - t2),P(t)dt
(37.43)
0
and
(37.44) 0, -1 < Re 11 < 0 and 1/J E AC([O, b]), and we replace the ,P(z) = O (e - �� z" - 1 / 2- e ) as z -+ oo and > 0 for (37.42). 00
If
c
symbols I, and J, with each other, which is equivalent to replacing ,\ by i,\ in (37.41)-(37.44), then (37.41) and (37.43) are valid under the same conditions. However the conditions for the validity of (37.42) and (37.44) are changed since the integral in (37.42) must converge and the function I,(z) in the kernel bas an exponential growth at infinity. On the basis of (37.41) and (37.42) we introduce the following
generalized
Erdelyi-Kober operators
J� (TJ, o:)f(z) = 2 a ,\ 1 -a z - 2a- 2'1
f� t2'1+ 1 (z2 - t2)(a- 1)/2 Ja- 1 (,\ vfz2 - t2 )f(t)dt,
(37.45)
0
00
R� (TJ, o:)f(z) = 2 a ,\ 1 -a z 2'1
f t 1 - 2a-2'1 (t2 - z2 )(a- 1)/2 Ja- 1 (,\ vft2 - z2)f(t)dt,
(37.46)
�
0,
where o: > 1J > - 1/2. We also define the operators Ji� ( 7J , cr) and Ru(7J, o:) via the right-hand sides of (37.45) and (37.46) by replacing Ja- 1 by Ia- 1 · is defined by the relation
The generalized operator of Hankel transform X
00
f T1 -a- 2<7 (r2 - y2)<7 J2'l+a ( vf(z2 - a2 )(r2 - b2 ))f(r)dr. y
(37.47)
It is obvious that this operator is connected with Sf1 ,a, 2 given in ( 18 .19) by means of the relation s
s
( 0, o:,0, 0 ) = ( 0, o:,0, 00 ) = 1],
U
1],
s
f1 ,a, 2 .
(37.48)
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
739
We give formally, without indications of the space of functions some properties of the operators introduced above. As usual these properties can be checked by direct calculations for sufficiently good functions and then can be extended to functions in Lp . 1 . It is obvious that if A --+ 0, then the operators in (37.45) and (37.46) coincide with the Erdelyi-Kober operators ( 18.8):
Jo (TJ, a) = I, ,a , Ro (TJ, a) = K, ,a .
(37.49)
2 . On the basis of (18.1 1) we have
Jo (TJ, 0) = E, Ro (TJ, 0) = E.
(37.50)
3. Translation relations
J>t (TJ, a)z 2{j /(z) = z 2{j J>t (TJ + {3, a)/(z), R>t (TJ, a)z2{j /(z) = z 2{j R>t (TJ - {J , a)/(z),
(37.51) (37.52)
follow from (37.45) and (37.46) . 4. Evaluating the corresponding repeated integrals, using 2.15.35.2 with 2 6 + c2 = 1 in Prudnikov, Brychkov and Marichev [2], it is not difficult to prove the results (37.53)
Ru (TJ, a)R>t (TJ + a, {3) = R>t (TJ, a)Ri >t (TJ + a, {J) = K,,a+fj
(37.54)
provided that a > 0, {3 > 0 - see (37. 18). 5. Taking the last relations into account and using (37.50) we can define the operators J>t (TJ, a) and R>t (TJ, a) for a < 0 via solving the corresponding integral equations (see § 18.1) that is . (37.55) (37.56)
740
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
where -n < a < 0, n = 1, 2, . . . . Hence we deduce the relations
Ji).1 (TJ , a) = J.x (TJ + a , -a) , JA" 1 (TJ , a) = Ji>. ( TJ + a , -a). Ri.x1 (TJ , a) = R.x (TJ + a , -a) , RA" 1 (TJ , a) = Ri>. (TJ + a , -a).
(37.57) (37.58)
6. For the operators in (37.45) and (37.46) the following analogue of the formula for integration by parts 00
00
J xf(x)J>.(TJ, a)g(x)dx = J xg(x)R>. (TJ, a)f(x)dx
(37.59)
0
0
is valid . 7. By calculating the compositions of the operators in terms of the relations 2. 12.35.2 and 2.12.35.6 in Prudnikov, Brychkov and Marichev [2] the following operator relations
Jo,('J + a , fJ)S
( O,
( 1'
0, ,\ , a , (T ,.,
) = s ( 0,, ,.,
) ( A , ,\ {3, ) s ("·, a , U - TJ - a ) = h (q, a + {J) , ) s ( '1 +,\,a , {3, 'I + ) = R;� ('J, + •>+
0, ,\ R� ('J, a)S '1 a {3, = s 0, a +A , {3, , ,., ,
( s ( o, ,
s '1 +o , a , ,.,
t ) ) 'I + tOt +
A, a + /3, .,. - 'I - Ot + fJ)/2 ,(37.60)
(T
,.,
(T
0, 0
0
0,
0,
a, 0
fJ)/2
/2
0 + fJ/2 CJt
CJt
, (37.61) (37.62)
fJ). (37.63)
can be proved, which connect the operators in (37.45)-(37.47) with each other. 8. Using (37.62) with a + f3 = 0 we can deduce that the equation
(
)
S 0TJ ' ay , uy f(x) = g(x) , , has the solution
(
0, f(x) = s TJ +y, a -a , ,
� ) 9 (")
(37.64)
(37.65)
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
741
from whence the operator relation s- 1
( o,
y, Y
a,
TJ ,
follows.
u
)=
s
( +y,a, fJ
0,
-a,
�)
(37.66)
37.3. Non-convolution operators with Bessel functions in kernels In this subsection we consider two following non-convolution operators with Bessel functions in kernels: z: -+ a- 1 ( Ja ,>. f)(x) = J (x -r (t)a) Ja- 1 ( >-. .jt(x - t))f(t)dt,
0
(I-a-, >. f)(x) =
z:
(x -r t) a - 1 I- ( .jx(x - t))f(t)dt, J (a) a- 1 >-. 0
(37.67) (37.68) (37.8). (37.68)
where l11(z) and l11 ( z ) are the Bessel-Clifford functions defined in We shall also deal later with the operators differing from and by the replacement of the symbols J and l by l and J, respectively. We shall denote these operators by l;t >. and J;, >. . It is obvious that
(37.67)
(37.69) All the above operators coincide with the fractional integral
a ja+,O - ja+,O - j-a ,O - j-a ,O - zO+
(37.70)
in the case ).. = while for other values of ).. they are also closely connected with the fractional integral, being compositions of the operators Jt>. or It>. with 1g; 1 in one or another order. These properties are consequences of the foilowing statement.
0,
Let Rea > 0 and f E L,(O , b) , b > oo, p � 1 . Then the operators introduced in (37.67) and (37.68) are defined and are bounded from L,(O, b) onto
Theorem 37.3.
742
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
the space I0+ [Lp(O, b)]. If also Re ,B > 0, then the following composition relations (37.71) (37.72) hold. The first statement follows immediately from Lemma 31.4 . (37. 71) and (37.72) are proved by direct evaluation, which will be shown, for example, for (37.72):
Proof.
J z
0
(z - t) a - 1 r (a) Ia - 1 ( >.. .jz(z -
t
-1 t))dt J (t -r(r)P ,B) f(r)dr 0
- J f(r)dr J(z - t)a- 1 (t - r)P- 1 l-a_1 (J.. Vz(z - t))dt . z
-
0
z
f (a) r (,B)
r
(37.8) 2] (37.72). (37.71) (37.72) (37.55) (37.56).
2.15.2.6
Changing the variable t = z + (r - z)02 and taking and the relation in Prudnikov, Brychkov and Marichev [ into account we easily obtain the value of the inner integral and thereby the right-hand side of • It is important to note that and enable us to define the operators 1:, >. and I; >. for negative values of a by the method which has already , been applied in § and in the definitions in and Namely, we set
18.1
= ( � r , ,
(37.73)
( i;,>./)(z) = (J;+n, >.J )(z),
5 1 - n, n = 1, 2, . . . We also note that (37.71) and (37.72) yield the representations -n
< Re a
(Ja+, >.f)(z) = Ioa+- 1 (J-1+,>./)(z), ( I;, >./)(z) = ( il,>. I�_; 1 /)(z),
(37.74) (37.75) {37.76)
which characterize an important role of the operators J-1+, >. and Il, >. in the
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
743
investigation of (37.67) and (37.68) with an arbitrary a. The following statements contain the conditions for the validity of (37.73)-(37.76).
Let f E AC(n- 1 ) ((0, b]), n = 1 + (-Rea] and Rea < 0. Then the constructions in (37.73) and (37.74) exist. Theorem 37.5. Let f E Lp(O, b), b < oo, and p � 1 . Then (37.75) as true fora Re a > 0 while (37.76) is valid for Re a > 1 or for 0 < Re a � 1 if f E I�_; [Lp (O, b)].
Theorem 37.4.
of these theorems are obvious and follow from the connection of the operators in (37.67) and (37.68) with the fractional integrals mentioned above in (37.71) and (37.72). Now we consider the inversion problem for the operators in (37.67) and (37.68). As it follows from (37.75) and (37.76) this problem is reduced to the inversion problem for the operators J{>. and 11>. . To solve the latter we prove the following auxiliary statement.
The proof
'
Lemma 37 .1.
'
The identity (37.77)
is valid. We denote by A the left-hand side of (37.77) and expand the Bessel functions in the integrand in series with summation over k and I, respectively. Differentiating these series with respect to T and replacing I by k 0 � � k, we obtain
Proof.
m,
A _ _ � � ( -1) - L:' �
l:- O m - 1
m m(�2t/4)m (k - m)(�2 z/4)l:-m fz: (m!)2 ({/c m)!)2 _
( _
T T
m
t )m - 1 ( _ ) l: -m- 1 d Z
T
T.
f
We evaluate the inner integral by the relation 2.2.6.11 in the hand-book by Prudnikov, Brychkov and Marichev (1] which yields the value
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
744
It follows from 4.2.3.1 and 4.2.3. 18 in the cited hand-book that the finite sum is equal to -tkzl: - kz( -t) i . Hence we derive the right-hand side of (37.77). • Theorem 37.6.
!
Let F(z) E AC([O, b]). Then the equation
( Jt>. f)(z) = f(z) +
j f(t) :z Jo ("A.jt(z - t))dt = F(z) :r:
(37.78)
0
has a unique solution of the form
f(z) = z - 1 li:>- (zF(z))' = F(z) - z - 1
J0 F(r)r:T I0("A .jz(z - r))dr. :r:
(37.79)
Proof. Replacing z by T in (37.78) , multiplying both sides of these equation by
and then integrating with respect to T from 0 to z we have
J0 f(r)r:T lo ("A .jz(z - r))dr J0 T:T lo ("A.jz(z - r))dr :r:
:r:
+
x
j f(t) :T Jo (>.,/t(r - t))dt j F( r)r:T 10( >.,/:t(:t - r))dr. T
:r:
=
0
0
To evaluate the inner integral we interchange the order of integration in the second term on the left-hand side and use Lemma 37.1. As a result we arrive at the relation
J0 f(t) :z Jo ("A.jt(z - t))dt J0 F(r)r:T Io ("A.jx(z - r))dr. :r:
:r:
z
(37.80)
=
Now it is easy to obtain (37.79) by subtracting (37.80) from (37.78).
•
Let 0 < Re a < 1, g E llf [L (0, b)] and g(O) = 0 . The operator inverse to (37.67) exists on such functions+g(z)1 and can be represented in both of
Theorem 37. 7.
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
745
the following forms: :1:
{ Jt. � ) - 1 g(x) = x - 1
j Io (>.. y'x(x - t))d[t{J0+g)(t)],
{37.81)
0
(Jt, � ) -l g(x) = x - l (f1-a ,�
(37.81)
(37.82)
(1 - a)g(x). {37.75), (37.78)
Proof. follows directly from the relations and applied to the equation { Jt � /)(x) = g(x). To obtain the representation in ' we introduce the notation
(37.79) (37.82) {37.83)
(37.76) (37.81). (37.82)
(37.81) {1 {37.83)
and apply to Then the right-hand side of has the 1 form x - (11 a �
(10.12)
{37.84) =
{1 - a)g(x) + xg'(x)
provided that g(x) is sufficiently smooth. The latter relation can be obviously extended to the function g(x) E •
Jg+ (L 1 {0,b)). Theorem 37.8. Let 0 < Re a < 1, x - 1 h(x) E £ 1 {0, b) and h(x) E Jg+ (L t { O , b)). Then the operator inverse to {37.68) exists on such functions h(x) and can be represented in the form {37.85) {37.67), {37.68), {37.81)
37.7
proof follows directly from and Theorem after replacing xa (x 1 -a g(x)))' by f(x), xf(x) by h(x) and a by 1 - a. The conditions for
The
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
746
h(z) ensure the existence of the operator inverse to (37.68) and its representation by the form (37.85). • 37.4. Equations of compositional type In a series of problems of mathematical physics one sometimes meets integral operators, generating integral equations of the first kind with Volterra kernels, such that they may be represented as compositions of more simple invertible operators. We call these equations of a compositional type. Some typical examples of such equations are considered in this subsection. A.
If we
set z = y and denote u(y, y) = cp(2y) and change the variables (1 + 8)y = t,
2y = z, r(t)t#'+P - 1 = f(t),
( 2 ) 1 - #£ - 2p cp(z) 13 ; 1
= g(z )r(p)
(37.86)
in the solution (40. 2 8 ) of the hyperbolic equation in (40.19), then we arrive at the integral relation
z - t .\ t(z - t) ) f(t)dt = g(z) , I (z -r(t)Pp) - .:.....2 (JJ , 1 - p; p; � 4 �
1
2
(37.87)
0
involving the Humbert function defined in (40.2 5 ) in the kernel. The inversion formula for this relation can be obtained by two methods. One is by considering the solution of other boundary value problem for the corresponding hyperbolic equation ( 40.19). The other is by studying the structure of the operator in (37.87). We shall illustrate both ways. To use the first we consider the second Cauchy-Goursat boundary value problem with the conditions
u(z, z) = cp(2z), u11(z, O) = 0
(37.88)
for (40.19). We stress that the function in ( 40. 2 8 ) satisfies the condition u11 (z, O) = 0 provided that a constant is given by ( 40.31). According to the papers by Kapilevich [1; (5.1 ) , (5.3)] and Gordeev [1] the solution of the second Cauchy-Goursat problem has the form
747
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
u(z, y) =
�-
y
J [
+
(37.89)
�+ 11
[
where fl and R are the Green-Hadamard and Riemann functions respectively for (40.19). Their explicit representations via triple series were given by Kapilevich [3, p. 148 1]. Setting y = 0 in (37.89) and taking the condition u(z, 0) = T (z) into account we arrive at the relation y
�
T(z) =
j [
(37.90)
0
The kernel of this equation can be easily obtained from (5.31a) in Kapilevich [1] :
�R R H(x, 0, 71, 71) = x2 2P z - P 7JP +2P R1 2P B2 (JJ, 1 - JJ; 1 - p; - � ' -T), 2z 2
ic
2 2
1 - 2p �= = f 2 f -li ), p < 1. ( 1 - p) (p + 1/2) ' R z(z - 2 7]
(37.91)
Substituting {37.91) into (37.90) and using the changes in {37.86) we obtain the relation �
j
/(z) = 22P- 1 laicf (p)z - 1 [(g(t)t 1 - 2P- P)� + (1-' + p)t - 1 g(t)t 1 - P - 2P] 0
Hence taking the values ic and 13 from (40.31) into account we find finally the inversion formula for {37.87) in the form 1
/(z) = ;
t - z �2 z (t - z)) d(t 1 _P g(t)), j tP (fx(-l -t)p)-P .....::...2 (JJ, 1 - JJ; 1 - p; 2t '4 �
0
O < p < l.
(37.92)
The functions g and f in these relations must naturally be such that the integrals
748
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
t2-�'g(t) --+-
t�' t
tf(t) --+-
here exist. For this the conditions + 1 g ( ) --+- 0, 0 and 0 as 1 0 and the condition g E AC ([0, b]) are sufficient. The fact that (37.92) really inverts (37.87) follows from the uniqueness theorems for the solution of the initial Cauchy problem and the second Cauchy Goursat problem for ( 40.19) with � = ib, b > 0. The result obtained coincides with the known inversion relations in special cases, below. 1 . If � = 0, then (37.87) and (37.92) according to (40.25) and ( 1.80) have the form
t --+-
z
j(x2 - t 2 )(p- t )J• p�;• ( ;) f(t) dt = g(x),
(37.93)
0
f(x) � J tP (x2 -t2 ) -PI2 z
=
0
P�,
(-i ) d(t1 -Pg(t)).
(37.94)
The latter relation can be also rewritten by taking out the derivative of the integral
(37.95) z
=
:x j(x2 - t2) -p/2 P�, (T) g(t)dt. 0
(37.93) and (37.95) are the special cases of (35. 16) and (35.31) for e = 0 and n = 1, respectively. 2. If � = JJ = 0, then (37.87) and {37.92) become the reciprocal relations and = (Jb+ /) { z) = 3. If JJ = 0, then in accordance with (40.25) and (37.8), (37.87) and (37.92) can be rewritten in the form of the relations and = /(z) = where = + {1 cf. (37.67}, (37.82} and (37.69}. We consider the structure of the operator in (37.87}. For this purpose taking
g(x) /(z) d� I�+P g (x).
z- 1 (J1-p,>..
cpt (z) xg'(x)
(J:,>.. /)(z) g (x) p)g(z),
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
749
(18.41) into account we evaluate the composition
X
11
J (y -r(a)t) a- 1 l_a- 1 (). Jt(y - t))f(t)dt
[ ( ).2 ) i 2 t& f(t)tk dt ] I r(p)r( ) 4 (a: 0
00
=
{;
hl:!
X
J(z2 tl:
_
a:
0
y2 )p - 1 (y t) k +a - 1 dy _
t
=[E . . . ](z t)P +i + or - 1 (z + t)P- 1 _
=[E . . . ](z - t)P+k + or - 1 (z + t)P - 1 f(p)f(k + a)
r(p + a + k)
(
-z x 2 F1 1 - p, a + k;p + a + k; tt + z
--
)
)
(
z-t x 2 F1 1 - p, p; p + a + k; 2;"" dt =2(2z)P - 1
tl:
J
(z - t)P +a- 1
r(p + a)
0
z - t ).2 x 82 p, 1 - p;p + a; 2;"" 4 t(z - t) f(t)dt
(
'
)
(37.96) provided that
Rep > 0.
Thus we arrive at the operator which differs from (37.87)
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
750
by a change of variable only. Hence (37 .87)
in
the form
z (x - t)P - 1 ,..... ( t ..\2 p, 1 - p; p; � , 4 t(x - t) ) f(t)dt .::. 2 J f(p) X-
0
(37.97)
..\ =
follows. In particular, in the case 0 we obtain from (37 .97) the following compositional representation for the operator in (37.93)
J (x2 - t2 )(p- l)/• G) f(t)dt = (2x)1 - P I�+ ;•' (2x) - 1 IK+P f(x). :1:
P��·
0
(37.98)
)
This relation conforms to (35. 16) and (35.30 if we take the operator equality into account. The following statement is a consequence of the above arguments.
10;;z'l!J+ = (2x) - 1
Let Rep > 0, g(x) E AC1 ([0, b]), b < oo, and t�' + 1 g(t) � 0 and t 2 - �'g(t) as t � 0. Then (37.87) is invertible by (37.92) in the space of functions f(x) E AC((O,b)), b < oo, such that tf(t) � 0 as t � 0. Theorem 37.9. �0
We also note that under the appropriate conditions by using (37 .82) the solution of (37.87) can be represented in the compositional form
f(x) = x - 1 (11+, -p ,>. cp)(x), cp(x) = 2�' xP - I' dxd [x2+1' -P I�.r':z2x�' - 1 g(x)].
(37.99)
B . Now we consider the integral equation
z (x - t)'Y - 1 ...... ( t r( "Y) .::. 1 a, a , ,B; ...,. ; 1 - ; , .A (x - t)) f(t)dt = g(x), J 1
0
(37. 100)
where " 'k '
(a)"Y;)( a ) (.B); xi yA: , x )= � L...J j, A: =O ( J· + kJ· .
.::. 1 (a, a1 , ,B; "'( ; , y
......
1
t
lxl < 1 ,
(37. 101)
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
751
is one of the Humbert functions (see Erdelyi, Magnus, Oberhettinger and Tricomi We shall investigate this equation in the space of functions = {I : t O, , < oo } . The following statement is true.
[1; 5.7(25)]). q Qq l(x)x E L ( b) b Theorem 37.10. The integral equation in (37.100) has the solution I E Qq, q < min(O, Re (-y - a - /3)), and q � 0 when Re(-y - a - /3) > 0, if and only if g(x) E Iri+ (Qq ). This condition being satisfied the equation has a unique solution. If also min(Rea', Re(-y - a), Re (-y - ,8)) > 0, then the solution of (37.100) can be represented in the form (37.102) (37.101) [1]
2.10(1)
2.10(12)
Proof. It follows from and and in Erdelyi, Magnus, Oberhettinger and Tricomi that the function defined in has the following asymptotic expansions
x,
Bt(a, a', /3; -y; y) =
{ 0((1O( (1- xpx)),-a-P), ln -
0(1),
(37.101)
Re(-y - a - f3) < "Y - a - ,8 = 0, Re(-y - a - ,8) >
0, 0,
(37.103)
x --+ 1. 9Hence we obtainq that the integrand in (37 .100) can be rewritten as qt l(t)[O(t - ) + O(fY -a-P - )] for t --+ 0 which guarantees the existence of this integral. Applying now Lemma 31.4 in this case when p = 1 relative to the space Qq we obtain that g E Iri+ (Q q) if I E Qq. The inverse assertion follows from Lemma 31.4 as well. Let now E Iri+ ( Q q) and let the conditions of the second part of theorem ' be satisfied. Then by Theorem 10.9 the operator IJ+ a, a , >.. defined in (10.55) ' transforms the function I E Q9 into (Ici+ a, a , >.. l)(x) E Iri+ 0(Q9). This means that ' x-P(Ici+ a,a ,>.. l)(x) E Lt(O, b), and so the operator IgJ", Rea + n > 0, can be ' , a a , applied to the latter function. Considering IgJ"x P(Ici+ >..l)(x) as a repeated as
g
integral, changing the order of integration and using the integral representation
x,
Bt(a, a', f3; -y; y) X
tFt(a'; -y - a; y(1 - u))du,
Rea > 0, Re (-y - a) > O, x ft [1, oo),
752
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
for the function 8 1 we obtain
:r: (z_.. -__t)__...; "Y+_n- 1 + ',� laO+ n z-P (f!-a, O+ a J)(z) z-P J ..:. r(-y + n) _ 0 8 1 (o + n, a', {J ; -y + n ; 1 - t/z, �(z - t))f(t)dt. =
x
(37.104)
On the basis of (37.103) and (37.104) we can conclude that and hence E E Therefore both sides in (37. 104) can be differentiated n times which yields the relation
zP I�J" z-P I�J" z -P(IJ+ a,a' .� /)(z) IJ:n -P (Lt ).
(IJ+ a ,a' ,� /)(z) IJ:"(Q9)
:r: (z - tp- 1 � ' a laO+ z -P ( f!-a . P , O+ J)(z) z J r(-y) 0 =
(37.105 ) Finally, taking ( 10.58 ) into account we arrive at the following compositional representation for (37.100 ) :
:r: (z - tp- 1 .... ( J r(-y) .:.1 o , o', {J; -y ; 1 - ;t , �(z - t)) f(t)dt 0 zp 10+a z-P 10TY+-a� a' e�:r: lao+' e - �:r: J(z) g (z) . =
=
Inverting now each operator in this composition we obtain ( 37.102 ) .
•
( 37.106 )
37.5. The W-transform and its inversion
W
In this subsection we consider some properties of the s
The W-transfonn of a function f(z) is defined by the integral
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
=
2�i j f[v - iz - s, v + iz - s] 1 - (an ) - s r [ (,Bm ) + s, ( a; + 1 ) + s, 1 - ( ,a;- + 1 ) - s ] I* ( 1 - s) ds ,
753
q
X
(37.107)
where v (Rev > 1/2) and components of the vectors (ap ) and (,89) - see explanation to (36.3) - are complex parameters satisfying the conditions in (36.5), f*(s) is the Mellin transform of a function f(z) and is the contour = {s , Re s = 1/2 } . u
It is obvious that the following relation
u
(37.108) with fl (y) = y- 1 /(y- 1 ) which connects G- and W-transforms is valid - see (36.3). Definition 37.2.
The transforms 00
(37.109)
Ki�{f(t) } = j Ki� (t)f(t)dt, 0
00
Ki-;1 {g (t) } = --i--z j t sh 7rtKit (z)g(t)dt, 1r
(37.110)
0
where K (t) is the Mcdonald function - see (1.85) - with an imaginary index v = iz arei� called direct and inverse Kontorovich-Lebedev transforms, respectively. The formulations and proofs of the following theorems use the terminology of § 36. Theorem 37.11. The W -transform defined in (37. 1 07) with the characteristic (c* + 1, -y• - 2Rev + 2) where Rev > 1/2 exists on functions in the space rot;;�(L) if and only if
c 1) + sign ( + -y• - 2Rev + 2) � 0.
2sign(c* + +
36.2 (c* 1, -y• - 2Rev 2).
'Y
(37.108) 1
(37.111)
follows from Theorem since according to the W-transform can be considering as the G-transform evaluated at the point and having the characteristic + + •
The proof
754
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Let the conditions in {36.22) and the inequality
Theorem 37.12.
{37.112) 4sign(c* + 1) + 2sign(-y* - 2Rev + 2) + sign 1 2 + p - q l > 0 hold. Then the W-transform defined in {37.107) exists on functions in the space rot- 1 (L) and can be represented in the form (W/)(z) = I c;'+�!2 (t � 1 - " + iz , �P�)" - iz, (a, ) ) t(t)dt. 00
0
{37.113)
37.11 into account, we see 36.3. Theorem 37.13. Let the conditions in (36.19) and in (37.111) hold. Then the W transform can be represented on functions in the space rot;,�(L) as compositions of the G-transform, defined in {36.3) with the direct Kontorovich-Lebedev transform, defined in (37 .109) by the relation
If we take the arguments in the proof of Theorem that the proof follows from Theorem
{37.114) - see ft (Y) in {37.108). 37.11
Proof. It follows from Theorem that the W-transform exists on functions in the space under the conditions of this theorem. We take in Marichev into account and rewrite it in the form
[10]
rot;,�(L)
9.3(1)
00
�r[v - ix - s, v + ix - s] = I K2iz(2-JY)y" - ' - 1 dy. 0
{37.115)
{37.107) to s- ] I* { 1 - s) ds ( an +) + l ) + s, 1 - (,a;n l ( Wf)( X) = 21r1 i I r [ (a;(,Bm) ) s + s, 1 -
This enables us to transform
q
00
X
2 I K2iz(2-JY)y"- ' - 1 dy. 0
(37.116)
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
755
The condition in (36.19) allows us to interchange the order of integration in {37. 116) by Fubini's theorem - see Theorem 1.1. This easily yields the representation {37.114) after changing the variable 2,fY = t and using {36.3). •
Let f(x) E rot- 1 (L). Then the Kontorovich-Lebedev transform, defined in {37.109) has the following compositional representation in terms of two direct modified Laplace transforms and the inverse Laplace transforms, defined in {36.31) and {36.32) and {1.1 19), respectively
Theorem 37.14.
{37. 117)
Proof. The integral representation for the Mcdonald function
Ki� (t) =
00
j e-tchu
cos uxdu
{37.1 18)
0
- see Prudnikov, Brychkov and Marichev [1; 2.4.18.4] - enables us to write {37.109) in the form
Ki� {f(t) } =
00
00
0
0
j f(t)dt j e-tch u
Since /{t) E rot- 1 (L), then the estimate
00
cos uxdu.
{37. 1 19)
00
� j dt j F(s)t -' ds j e-tch u cos uxdu
I Ka�{/(t)} l = 2
0
(1
00 00
0
� j IF(s) l ds j j e -tch ut - 1 1 2dtdu 00 -J-idu 1 1 = F( s) I ds JcliU < +oo 211' J I �
2
(1
0 0
(1
0
follows from {37.119). This allows us to change the order of integration in {37.119):
Ka�{ f(t) } =
00
j 0
00 cos uxdu j e- tch u f(t)dt. 0
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
756
Making the change of variable u 2 .../11 and replacing by ../i and using the factorization of the cosine Fourier transform which is obtained from (36.62) 2 we arrive at if we take the relation and (36.63) when into account. • 2/(rz) cos z J
= /
J_ 1 /2(z) =
z
v = -1/
Theorem 37.15.
(37.117)
Let f E rot;,�(L), 1 /2 < Rev < 3/4 and the inequality 2sign (c
+ c* ) + sign (-y + -y - 1/4) � 0 •
(37. 120)
hold. Then the inverse W-transform operator ((W/)(z) = g(z)) defined in (37.107) has the form
1
+ 1) , -(an ) �Y-3/2+v K - 1 {g ( �) }) (z). (37.121) ; /(z) = !2 (cp,q-qm,p- n � --(a ';' 2i..Jli; 2 (/3 + ), -(/3m ) If also q-m-n > m + n - p, Re /3j < 1/4, j = m+ 1, . . . ,q, ze2"'z g(z) E L 1 (0, oo) , and the conditions in (36.22) hold, then (37.121) can be represented in the form /(z)= -i- J t sh 2rt 00
1r Z
X
0
n+ 1 ), 1-(an ) ) g (t)dt . 1-( 1+v-it a cpq-+m2,q,p-n+2 (z 1 1+v+it, 1-( , 0 1 1-'amq + ) ' 1- I-'m ) (37.122) a
K��{g( T/2) } to the W-transform (W/)(z/2) =
Proof. Applying the operator
g(z/2) we obtain the relations
K�p{9 (�) } = 1r!z J TshrTKiT (2vz)(Wf) ( �) dT 00
0
= ,}vz J rsh rrK , (2y'i)dr 00
,
iT ) iT ) i J r (v - - 2 r (v - s + 2 r [ (/3m ) + ( n+ 1 ) + 1 1--(a(mltn+)1 ) - ] I ( 1 - )ds.(37. 123) 0
s
1
x 2r X
D
0P
S,
8'
,.,,
- S
•
8
s
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
(37.120) [1] 1/2 < Rev < 3/4.
757
The condition in enables us to use the results by Vu Kim Than and Yakubovich which justify the interchange of the order of integration in provided that Evaluating now the inner integral using from the cited paper
(37.123) (19)
00 j Tsh7rTKiT(2y'X)r(v - iT/2)f (v - + iT/2)dT = 21r2x" - a s-
s
0
we can rewrite
(37.124)
(37.123) in the form
(37.125) To obtain now the representation in (37.121) it is sufficient to use Theorem 36.8 relative to the G-transform given in (37.125) and the reflection and translation relations ( 1. 9 6) and ( 1.9 7) for the G-function. Now we shall prove that the solution can be represented in the form (37.122). Since q - m - n > m + n - p and the conditions in (36. 22) hold, then by Theorem 36.3 (37.121) can be rewritten 00 m n ( -(a +l ), -(an ) ) 11 1 f(x) = -21r-2 ! ap,q-q ,p- xy 1 -(,8;'; +1 ), -(Pm ) y- dy 00 (37.126) J TSh 1rTKiT(2y'Y)(Wf)(T/2) dT. as
0
X
0
(37.122)
To obtain now it is sufficient to interchange the order of integration in and then apply the relation
(37.126)
00 ) ) n Jy- "KI·T (2 h.\y Gp,q -qm ,p-n (xy � -(-(a!J;n;+l+l ),-(a ) ), -(fJm dy ! t m ,p n 2 l v - iT,-(v +fJ;'iT+l, -),(a-(;+lfJm),) -(an ) ) - 2 0p-+ 2, q - + ( and the property of G-function given in ( 1 . 9 7). The interchange of the order is possible according to the conditions rsh 7rTg( T/2) E L 1 (0, oo ) and &{Jj < 1/4, V HI
0
_
X
758
j= m+
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
1, . . . , q, and the estimate
f aqp,q-m,p-n (xy � --((!J;'a;+t )),, --((fJamn )) ) Y KiT (2vr.:\YJdY +l
00
0
_,
00
< const
j y-Re"(zy)- 114K0(2Vi)dy < +oo. 0
The theorem is thus proved.
•
37.6. Application of fractional integrals to the inversion of the W-transform
37.13 (36.32)
As was shown in Theorem the W-transform is representable as a composition involving the G-transform, defined in The latter can be factorized via the operators in and which may lead to fractional integra-differentiation operators after pairing according to Theorem To give details we investigate an important special case of the W-transform defined in which in its turn generalizes the known Mehler-Fock transform. Namely, we consider the transform defined by
(36.31)
(36.3).
36.10.
(37.107)
(Fa/)(z) = (W�§ 1 "\h)) I /(t)) (z) - iz - s)r(v + iz - s)r(1 - lkt - s) /* (1 - s)ds (37.127) =- _2?ri1_ f r(v r(1 - Pt - s)r(1 - fJ2 - s)r(1 - f33 - s) where Rev > 1/2 and the parameters at, fJt , f32 and {33 satisfy the conditions in (36.5). We shall call it the F3-transform since the kernel of the integral in (37.127) has the form (10.4 7) and is connected with the Gorn function Fa given in (10.45). Picking out the functions r( v - iz - s) and r( v + iz - s) from the kernel we calculate the characteristic (36.7) for the four remaining gamma-functions: (I
(37.128) Now it is easy to formulate the four following theorems which are special cases of Theorems
37.11 - 37.15.
The F3-transjonn defined in (37.127) with the characteristic (0, -y• - 2Rev + 2) where Rev > 1/2 exists on functions in the space rot;,�(L) i/
Theorem 37.16.
§ 37. EQUATIONS WITH NON-HOMOGENEOUS KERNELS
759
and only if 2signc
+ sign('Y + 'Y• - 2R.ev + 2) � 0.
(37.129)
Let the conditions in (36.22) with respect to the parameters fJ1 , fJ2 and f3a be satisfied and (37.130) 'Y• - 2R.ev + 2 > 0. Then the Fa-transform defined in (37.127) exists on functions in the space rot- 1 (L)
Theorem 37.17.
a1 ,
and can be represented in the form
00
(Fa/)(z) r(2 - 2111 - /3) j
where
Fa
( l - v - P - iz,a', l - v - P+ iz, b'; 2 - 2v - P; 1 - t, l - I) !(t)dt,
(37.131) (37.132)
Theorem 37.18.
Let /(z) E rot;,�(L),
1/2 < Rev < 3/4 and 2sign (c - 1) + sign('Y - Re{J- 1/4) � 0.
(37.133)
If the conditions
(37.134) R.e/3 < 1/4, Rea' < 1/4, Reb' :< 1/4 � 2 hold and ze • g(z) E L(O, oo), then we have the following inversion relation for the Fa-transform (g( -z) i.e=/g(z)): + it)f(1 - b1 + it)f(1 - {J + it) ( _F.a- 1 g)(z) = z211-r 1 J f(1 - a1f(1 - a1 - + it)f(2it) aF2 (1 - a' + it, 1 - II - b' + it, 1 - {3 + it; (37.135) 1 - v - a' - b' + it,2it + 1; 1/z)z1'g(t)dt. 00
II -
- oo
X
II -
b'
II -
II -
II -
II -
760
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Theorem 37.19. Let Re v > 1/2 and the parameters a = Pt + {3 , p�, f32 and Pa = {J satisfy the conditions in (36.22) for m = 0, n = p1 = 1, q =23 and let the
inequality
(37.136) 2sign (c - 1) + sign("Y - Re{J) � 0 hold. Then the following factorization of the F3 -transform in the space rot;,� (L) is valid via the Kontorovich-Lebedev transform (37.109), the modified Laplace transforms (36.1) and (36.2) and the fractional integro-differentiation operators: (Fa!)(z) = 22- 2" K2iz { [t2" - 1 t6 A: 1 t' A: 1 td It+t�• G I G )) ] c: ) } , (37 .137)
(Fa!)(.,) = 22 2" K2iz {[t2 - l t6A: 1 t' It+ tdA: 1 z�· G I ( n) ] c:) } . (37.138) -
•
{Fa/)(z) = 22- 2" K2iz { [t2" - 1 t6 It+t' A: 1 tdA: 1 t�• G I G) ) ] c: )} . {37.139)
The values of the parameters b, e, d, 6 and "Yl with the condition Re6 > 0 for each of these relations are given in Tables 37.1 - 37.3, respectively. Table b a1 - 1 a' - 1 {3 - 1
e
b' - a' {3 - a' b' - a'
d
5
1 - b' + {3 1 + b' - {3 1 + a' - b'
a' + b' - {3 a' b'
Table b a' - 1 {3 - 1 a' - 1
e
1 + b1 - a' 1 + b' - {3 1 + {3 - a'
e
{3 - a' - b' - 1 {3 - a' - b' - 1 -a' - 1
"Yl
-a' - b' -a' - b' -a' - b'
37.2
d
5
{3 - a' - b' - 1 b' - 1 a' - 1
a' a' a' + b' - {3
Table b b' b' {3
37.1
"Yl
1 - {3 1 - a' 1 - b'
37.3
d
5
a' - {3 {3 - a' a' - b'
a' a' a' + b' - {3
"Yl
1 - a' 1 - b' 1 - a'
§ 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
761
The proof follows from Theorem 36.5 if we take the arguments in the proof of
37.11
Theorem into account. As has already been noted, the F3-transform includes the Olevskii and Mehler-Fock transforms which are obtained from when a' = and a' = respectively, by using namely: {J =
1/2 - v
(1.79),
(37.107)
0
0,
00
1 ,B (37.140) r(2 - 2v - {J) j1 (t _ 1 ) 1 - 2v2 F1 (1 - v - fJ - ix , 1 - v- fJ + ix; 2 - 2v - {J; 1 - t)f(t)dt, g ( x ) = J t 1 / 4 -vf 2 (t 1) 1 / 4-v / 2 p��/f�iz (2t 1) f (t ) dt . (37.141) 1 These expressions differ from the representations (8.55) and ( 8 .42 ) in Marichev [10) by some changes of variables and functions. The conditions in (37.130) for the validity of (37.140) and (37.141) have the form 2 - 2Rev - Re{J > 0 and 3/2 - Rev > 0 respectively, and it is satisfied owing to the assumption 1/2 < Rev < 3/4 of Theorem 37.15. As regards the Mehler-Fock transform of the form (37.141), the following statement which is derived from Theorem 37.19 immediately is valid. Theorem 37.20. Let 1/2 < Rev < 3/4 and let the inequality (37.142) 2sign ( c - 1) + sign (-y - Rev - 1/2) � 0 hold. Then the transform defined in (37.141) can be factorized in the space rot;,� ( L) x
00
_
_
by the relation
(37.143) where the parameters have the values: b = -1, e = 1/2 - v and d = 1/2 + v or b = -1/2 - v, e = v - 1/2 and d = 1. § 3 8 . Applications of Fractional Integro differentiation to the Investigation of Dual Integral Equations Solutions of many applied problems in mechanics with mixed conditions are reduced to the so-called dual and triple integral equations. It is typical of such equations
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
762
that an unknown function satisfies different integral relations on different intervals. One may find the fundamentals of the theory of such equations in detail, and including surveys of various methods of their solution, for example, in the books by Sneddon 7 or Uflyand and Virchenko - see also the survey by Popov We consider here some typical examples of the applications of fractional integro-differentiation to reducing dual and triple equations to a Fredholm integral equation of the second order, the theory of which is well known.
[1].
[3,
6],
[1]
[3]
38. 1 . Dual equations Example 38.1. In investigations of mixed boundary value problems in mathematical physics using the Hankel transform, one often meets dual integral equations with the Bessel function given in in kernels of the form
(1.83)
J t - 2a [1 + R(t)]lP(t)J,(xt)dt = F(x), 00
0
J t -2fl lJ(t) Ju (xt)dt 00
0
=
0<
x < 1,
G(x), 1 < x < oo.
(38.1)
R(t), F(x) and G(x) are the given functions and lP(t) is an unknown one. To (38.1) in the form which is suitable for using the Kober operators defined (18.5) (18.6) we make the following changes of variables
Here represent in and
Then by using
t/J(y) = y- 1/2 \J(2..fii) , /(x) = 22a x -a F(-/Z), g(x) = 22fl x -fl G(..fi), k(y) = R(2.../i) .
(38.2)
(18.19) these equations are rewritten in the form
(38.3) s,/2-a,2a, t (1 + k)t/J = /, Su/2-{J ,2{J, l tP = g, where the functions f and g are given on the intervals It = (0, 1) and /2 = (1, oo ) , respectively. We first consider the case k ( y) 0. Let =
v)/2 + {J - a. (38.4) Applying the operators in (18. 5) and (18.6) respectively to (38.3) and using (18.21) A = (p +
§ 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
763
we arrive at the relations
{38.5) Let a function
h be defined by the expression
{38.6) Then
{38.5) can be rewritten as a single equation {38.7)
for which the inversion relation has the form
{38.8) {18.20). {38.2) {38.1) 0. All {38.1) =
{38.6)
according to Substituting here the value from and making the inverse changes given in we obtain the explicit solution �( t ) of the dual equation for R(t) :: the operations given above are valid under appropriate conditions on parameters and functions. The solution of can be obtained by another method. To obtain it we represent the function I as I It + 12 where ,.
and similarly to this we write 9
={�
{38.9)
= 91 + 92 · Then by {38.5) we have {38.10) {38.10) can be considered as
Since the functions It and 92 are known, then the equation relative to unknowns l2 and 9 1 in the form
{38.11)
764
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
The first term on the left-hand side vanishes on the interval It and we obtain the relation from which the function 9t can be found by the second result in (18.17). Hence the function 9 = 9 + 9 will be obtained. Finding now a function t/J via 9 from the second result int (38.3)2 and using the inversion relation given in ( 18.20) we finally arrive at the representation of the solution in the form (38.12) (38.11) can be considered on the interval I2 . Then the second term on the left-hand side vanishes and after similar arguments we may arrive at the representation of the solution in another form (38.13) t/J = s,../2+a,-2a,tl· One can show that the constructed solutions in (38.8), (38.12) and (38. 13) are equivalent to each other. Now we consider the case of (38.1) with an arbitrary function R(t). We shall use the above solution given in (38.8) in the special case R(t) 0 as a "trivial" solution containing some unknown function h = h + h of the form (38.9). Substituting it into (38.3) we arrive at the system t 2 =
(38.14)
Inverting the second of these equations by the second relation in (18.17) we find the function h on the interval I2 : h2 = K;12_a,,->.92 · Taking the expression h = h t + h2 and the first relation in (18.22) into account we can write the first of these equations on the interval It in the form I:/2+/3,,.. - >. h t + S,.. / 2- a ,2a,tkS,/2+/3, 11->.-2 a ,t h t = I - s,.. / 2- a ,2a ,tkS,/2+/3,1J->.-2a ,t h 2.
(38.15)
Inverting now the operator I:/2+/3,,..- >. by the first equation in (18.17) and using the first result in (18.21) we finally arrive at the relation h t + S,../ 2- a ,>.-11+2a ,tkS,/2+/3,11->.-2a ,t h t
= H,
H = I:/2+a ,>.-,.. f - S,.. /2- a ,>.-11+2a ,t kS,/2+P,11->.-2a,t h2
§ 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
765
on the interval It . Under appropriate conditions on the given functions this will Its be a Fredholm integral equation of the second kind relative to the function kernel contains the function and the right-hand side is a known function since the function has already been obtained. Constructing the solution of this equation we can find the function and then the solution of ( ) by ( ) and
h1 .
k
h2
h
38.1
38.8
(38.2 ).
Example 38.2. Here we consider the dual integral equation of the form (X)
j t -" - "(t2 - k2).8 J11 (zt) 'll(t)dt = F(z ), 0 < z < 1, i
(X)
j J.,(zt)'l!(t)dt = G(z), 1 < z < oo, i
(
38.16)
k � 0. As in the previous example we use notations /1 = (0, 1 ) and 12 1, ) After the substitutions 'lt(t) = t"+ 1 1/J(t), F(z) = (z/2)11- 2,8 /(z), G(z) = (2/z)" g(z) (38.17) and the use of the notation in ( 37.47) , (38.16) has the form
where =(
oo ,
S
(g: s
I'
( 0'11,
�2p,
O,
�) ,P(z) k ) 1/J(z) o
=
/(z), z E It ;
( 38.18 ) g(z), z E /2 . Applying the operators in ( 37.45 ) and (37.46 ) respectively to ( 38.18) and using the properties in (37.60 ) and ( 37.61 ) we arrive at the equation
where
h = h1 + h2
-
see (
-
11
,
=
(
38.19)
(
38.20)
38.9) - and h1 (z) = J;t (JJ - {3, {3 - p)/(z), h2 (z) = Rt (/3, 11 - {3)g(z).
766
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Using now the inversion relation given in {37.65) we find {38.21) whence it is not difficult to obtain an explicit solution of {38. 16) after the inverse change {38. 17) . Example 38.3. Now we consider the dual integral equation of another type
PIJ { H(T) 1/J (T); x} = f(x) , PIJ { 1P(T); x} = g(�),
0 :5 x :5 a;
a < x < oo,
IP I < 1/2,
{38.22)
where P�J { 1P(T); x} =
00
J P�t'/2+iT (chx)1/J(T)dT,
0 :5 x < oo ,
{38.23)
0
is the inverse generalized Mehler-Fock integral transform with the associated Legendre function p:=r/2 + i 7-{z) defined in {1 .79) in the kernel. We note here that if we replace 1P(T) by 7r - 1 Tsh T1rf{1 - v + iT)f{1 - v - iT)g(T) and g(x) by 2v - 3/2 sh 11 2 - v xf( ch 2 (x/2)) in the second relation in {38.22) and put Jl = 1/2 - v , then by {8.42) and {8.43) in Marichev [10] we obtain the inversion of the direct modified Mehler-Fock transform defined in {37.141). The function H( T) in {38.22) is assumed to be known. We introduce analogues of the Erdelyi-Kober operators defined by the following fractional integrals of one function by another one namely ch x (see § 18.2)
x
z
J (sht){ l +�J±l ±�J)/2 {chx - cht) -112=FIJ
IP I < 1/2,
{38.24)
1 ) K±�J {
00
J (sht){l - �J=f l =f�J)/2(cht - ch x)- 112=F �J
lll l < 1/2.
{38.25)
§ 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
767
Without writing down the operators obtained by inverting (38.24) and (38.25), we denote them as I±! and K±! respectively. Then by direct evaluation one may prove two following compositional relations:
(38.26) X
-w E Lt (O, oo), where IJJI < and :Fe and
(1.109).
(38.27)
1/2, w(t) = r (1/2 + JJ + it) r (1/2 + JJ - it)[r (1/2 + it) r (1/2 - it)]- 1 , :F, are the sine and cosine Fourier transforms defined in (1.108) and
We introduce the notation
0 :5 z :5 a, a < z < oo,
(38.28)
where G2 (z) = �K; 1 { g(t) ; z}, a < z < oo, and G1 (z) is a certain as yet unknown function. Then applying the operators I:! and K; 1 respectively to (38.22) and taking (38.26) and (38.27) into account we obtain the relation
:Fc { H (T) 1/J (T)i s} = F(z ), 0 :5 z :5 a; a < z < oo, where F(z) = �I:! {f(t); z}, using (1.111) we find the value
0 :5 z :5 a.
(38.29)
Inverting the second of them and
(38.30) where the symbol
(! )
(.1", G)(T) here and below means that the integration in the
corresponding integral in (1.109) is carried out only on the interval ( a, b) instead of (O, oo) . To find the unknown function G1 we substitute (38.30) into the first equation in (38.29) and as a result we obtain the Fredholm integral equation of the second kind. After solving it and finding the function G, defined in (38.28), the required function 1/J ( T) can be obtained by (38.30).
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
768
38.2. Triple equations Example 38.4. We consider triple integral equations of the form 00
I t - 213 J, (xt)'l!(t)dt = F1 (z), 0 < z < a, 0
I t -201 J, (xt)'l!(t)dt 00
0
=
G2 (z),
00
I t - 213 J, (xt)'l!(t)dt = F3(z), 0
F1 , G2
where and substitutions
a<
z < b,
(38.31)
b < z < oo ,
'I!
F3 are given functions and is an unknown one. We make the ,P (t) = t - 1 \l! (t), f(x) = (2/z)213 F(z), (38.32) g(x) = (2/z?01G(z),
setting
z _ Ij, F(x) = j� Fj(x); Fj(x) = { 0F(z), zE (38.33) lj, =l It = (0, a), J2 = (a, b), J3 = (b, ) and defining similarly the function G(z), the functions Gb G3 and F2 being still L...t
e
oo
unknown. Then by (18.19) the system of equations in (38.31) has the form
S,/2- /3,2/3,2 1/J(z) = /(z), s,/2 - cr,2cr,2 1/J (z) = g(z).
(38.34)
From the latter equation we obtain (38.35) 1/J(z) = s,/2+cr, - 2cr,29(z), by using (18.20) where g(z) = 9 t (z) + 92 (z) + 93(z), i.e. g contains the unknown
§ 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
functions
169
g1 and g3. To find them we introduce the notations (38.36)
I- 1 K
I
inverse to to the first relation in (38.24). Then and apply the operator we apply the operator to the second result in (38.34) and use the relations in (18.17) and ( 18.21) . This leads us to the result
I- 1 I = Kg:
I- 1 I = I- 1 S.,P = I,/2+fJ ,(p -11)/2+a -fJ S,/2-fJ,2fJ ,2 .,P(x) = S,/2-fJ,(p-11)/2+a+{J,2 1/J(x), Kg = K,/2-fJ,(p-11)/2+fJ- a Sp/2- a,2a,2 1/J(x) (38.37) = S,/2-fJ,(p-11)/2+a+{J,2 1/J(x). In the same way after analogous applications of the operators K - 1 and I it is not difficult to obtain a second relation in the same form, which together with the first one gives the system of equations
I- 1 l(x) = Kg(x), K- 1 l(x) = Ig(x).
(38.38)
Equation (38.38) can be rewritten in the form
X
It
E
I3.
I3,
g1
on the intervals and respectively. Inverting these relative to and respectively we arrive at the following results for these unknown functions:
'('�) K9,(.,) + ( : ) 1 ( � ) - ( : ) r 1 (!) Kg2 (z),
91 (.,) = - ( : )
r
r
r
g3
1 11 (.,) (38.41)
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
770
ga(z) ( :)r 1 ( � ) /g1 (z) + (: )r 1 ( � ) r 1 /a(z) - ( :)r 1 ( ! ) /g2(z). =-
Excluding
(38.42)
93 from here we finally arrive at the relation
91 (.,) = ( : ) r 1 ( � ) K (:) r 1 ( � ) 191 (.,)
+ (: ) r 1 { ( � )r 1 t. (z) - (! )Kg2 (z) }
- (: ) r 1 ( � )K ( :) r 1 { ( � ) r 1 /a(z) - (! ) Jg2 (z) } .
(38.43) Evaluating the compositions of operators given in (38.43) one can see that this relation, under appropriate assumptions on the functions, is a Fredholm integral Considering this equation to be solvable equation of the second kind relative to and using (38.42) , (38.35) and (38.32) we can find the solution of the system of equations (38.31). At the end of this section we shall prove that (38.31) in the special case = 0 is equivalent to the system of equations from the following example.
91 ·
F1 = F3
Example 38.5. We consider the triple integral equations
r(e + s/6) cp (s);x } = 0, E It u /3; 1 rot { r(e + f3 + s/6) (38.44) u ) r(1 + sf rot- 1 { r(1 + + sju) cp (s)j } = 92 (x), where Ij (j = 1, 2, 3) are the intervals in (38.33), while {3, e, 6 > 0, u > 0 are real parameters and the unknown function cp*(s) = rot{cp(x); s} is the Mellin transform - see (1.1 12) - of a certain function cp(x). •
7J
7J -
Q-
X
•
X
a,
7J ,
Using the notation of (38.33) and (23.1) and (23.2) we reduce the system of equations in (38.44) to the form (38.45) or
§ 38. APPLICATIONS TO DUAL INTEGRAL EQUATIONS
771
( : ) I� ;,,( 'l'l (z )+ ( !) I� ; ,,( 'l'2 (z) + ( � ) I� ; ,,( rpa(z) = 0, z E I,, ( � ) r.f+;•,o 'l'l (z) + ( : ) r.f+ ;•,o 'l'2 (z) = g(z), x E Ia, on the corresponding intervals. Using (18. 17) we invert the third, the second and the first of the equations (38.46) relative to the functions and and obtain the relations
cpa, cp2 cp1
cpa (x) = 0, (38.47)
Substituting the value of 'Pl from the third relation into the second one we finally arrive at the Fredholm equation of the second kind
'l'2(z) = ( : ) IO+�"·•+a ( � ) I.f+;•,• ( : ) r::�.HP (! ) r� ;,,( 'l'2 (z) (38.48) + ( : ) rn.t;. ,. +a Y• (z ). relative to cp2 • After inverting this equation it is not difficult to restore the solution
of the system of equations in (38.44). In conclusion we note that if we set
F1 = Fa = 0, =e+ = (eto - .Bo + ,a = < - e - 1) 12, 7] ,
v
Q
7] -
e - 1)/2,
(38.49)
7]
in (38.31), then according to ( 18.19) , ( 18.22), (23. 1) and (23.2) this system of equations has the form (38.44) with o: being replaced by o:0 and ,8 by ,8o and
772
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
6 = u = 2. Obtaining the solution of this system of equations we find the solution of the system of equations given in {38.41) , {38.49}:
w(t) = s,,{Jo +e - ,cp(t ) .
§ 39. B ibliographical Remarks and Additional Information to Chapter 7 39. 1 . Historical notes
d
Notes to § 35.1. The equation of the fonn (35.4) with = 1 and variable lower limit was first considered by Higgins [3] {1964). He obtained its solutions in the form {35.1 1) and (35.14) in the space of sufficiently smooth functions. We note that earlier special cases of this equation with Chebyshev and Legendre polynomials in the kernel were studied by Ta Li [1] {1960), [2] {1961) and Buschman [1] {1962) respectively (see § 39.2, note 35.1 below). Methods of fractional integra-differentiation were first applied by Love [2,3] {1967) to studying (35.1 ) - {35.4) in the spaces Qq and Rr given in § 17.1, Notes to § 10.1 (see § 39.2, note 35.2 below). Theorem 35.1 was not published previously, but the case p = 1 is contained in the papers by Love mentioned above. Notes to § 35.2. Equation (35.16) was first obtained and inverted by Copson [5, p. 353] {1958) when solving the Dirichlet problem for the hyperbolic equation (40.19) with ..\ = 0 in the first quadrant. However, this result was not specially noted in the above paper. Therefore it is generally accepted that the paper by Buschman [3] {1963) was the first where the inversion relations for (35.16) and {35.18) were obtained. Using methods of fractional integra-differentiation and the theory of generalized functions Erdelyi [9] {1964), [12] {1967) found the solution of (35.15) and {35.17) in the form (35.25) and {35.35) (see § 39.2, note 35.3 below). Notes to §§ 36.1 and 36.2 . The idea of defining fractional integrals and derivatives via the inverse Mellin transform operators given in (36.1) and (36.2) was suggested in the papers by Kober [1] {1940) and Erdelyi [4] {1940), though previously it was suggested by Zeilon [1 , p. 4] {1924). The definition of the G-transform via the Mellin-Barnes integral in (36.3) was first introduced by Vu Kim Than, Marichev and Yakubovich [1] {1986) in the form which slightly differs from {36.3). Earlier the special case of such a transform in the form (36.23) was considered by Narain Roop [1] {1959), and this case was investigated more carefully in Narain Roop [2-4] (1962-1963) and Fox [2] {1961). The space !m0 �(L) was introduced by Vu Kim Tuan [1] {1985) who denoted
L - 1 . The more general s�e !JR� 1 (L) was introduced by Vu Kim Tuan [4) (1986). The space L�c ;y ) is a special case of the space L� which was introduced by Akopyan [1, p. 6] {1960)
it by
see also M.M. Dzherbashyan [2]. Theorems 36.1-36.4 were proved in the papers by Vu Kim Tuan [5) {1986) and Vu Kim Than, Marichev and Yakubovich [1] {1986). Notes to § 36 .3. The idea of the factorization of integral transforms, i.e. their representation as compositions of other "more simple" integral transforms, probably first appeared in the expansion expression of the Stieltjes transform represented as the composition of two direct Laplace transforms - Widder [1] {1938), [2] {1946) and also Erdelyi, Magnus, Oberhettinger and Tricomi [4, Ch. 14). Tricomi [2] {1935) pointed out the expansion formula of the Hankel transform via the composition of direct and inverse Laplace transforms. Compositions of such a kind were systematically applied by Hirschman and Widder [1) (1958) to studying convolution
§ 39. ADDITIONAL INFORMATION TO CHAPTER 7
773
type transforms. As far as the G-transfonn of the fonn (36.23) is concerned, the idea of its factorization was developed in Fox (3] (1963) , [6] (1971) and Rooney (5] (1983) (see § 39.2, notes 36.1-36.4- below). The technique of factorization of the G-transform of the fonn (36.3) via simpler G-transforms of such a kind, together with special tables and notations were first developed formally, i.e. without characterization of all conditions and spaces of functions, by Brycbkov, Glaeske and Marichev (1) (1983), and also (2] (1986). There were prior investigations by Higgins (4,5) (1965) and Marichev [3) (1973) , [6] (1976). The factorization problem in the space !Dt�� (L) was solved in the paper by Vu Kim Than, Marichev and_Yakubovich [1] (1986) where Theorem 36.5 was proved. Notes to §§ 36.4-36.8. Theorems 36.6-36.8 and 36.1�36.13 were proved by Vu Kim Tuan, Marichev and Yakubovich (1] (1986) and by Marichev and Vu Kim Tuan (2] (1985), [3) (1986) respectively. The other results in these subsections were obtained by Vu Kim Than and have not been published previously. Notes to § 37.1. The solution of (37.1) with � = 1 was obtained by K.N. Srivastava (5] (1964) in a more cumbrous form that given in (37.31)-(37.34). The solution of (37.1) in the form (37.31) with I = 0 and a = 0 was given by Wimp [2] (1965) in terms of the Laplace transform, we refer to Rusia (4,6) . The most perl'ect investigation of the equation (37.1) was carried out by Prabhabr [1] (1969) (the case � = 0, a = 0), [2] (1970) who used methods of fractional inte�differentiation with Re a > 0 and found the solutions (37.32) and (37.33) when Re,B > 0 and Re ,B < Re a, respectively. Equation (37.2) with a = 0, a = 2n + 1 and � = i, i.e. the equation
j (x - t)n In (x - t)f(t)dt I&
= g(x)
0
where In (z) is the modified Bessel function given in (1 .84), was first considered by Tedone (1] (1914) for integer non-negative n. In the case of natural n he reduced this equation to the simplest form with n = 0 and obtained its fonnal solution. The solution of (37.2) with a = 0 and a = 1 was found by Elrod [1] (1958) by using (37.20). A formal solution of (37.41) which differs from (37.4) only by change of variables was obtained by Burlak [1] (1962) and Sneddon [2] (1962). Sufficient conditions for its solvability were indicated by Srivastav [1] (1966) . We note only that (37.4) in another form wa8 already solved by Sonine in 1884. For this see § 39.2, note 37.3 below and also the Brief Historical Outline at the start of this book. The solution of (37.5)-(37.7) was given by Marichev [8] (1978). The presentation in this subsection follows this paper. Notes to § 37.2. The solutions (37.43) and (37.44) of (37.41) and (37.42) were formally obtained by Burlak (1] (1962) and Sneddon [2] (1962) ; see also the papers by Srivastav [1] and Rusia (3]. The necessary conditions for the existence of the solution of (37.41) in the space of continuous functions were given by Bharatiya [1] (1965), and the criterion of solvability for this equation in the space L2(0, oo) was given by Soni [1] (1971). It is generally considered that the solution of (37.41) and hence (37.4) was first given by Bur1ak and Sneddon. However, in actuality the equation which was solved by Sonine (4) (1884), see also [6) (1954), is reduced in the special case to (37.41) by quadratic change of variables (see § 39.2, note 37.3 below). We also note that Sneddon, who did not know of this paper by Sonine, called the operator in (37.41) the Sonine operator on the basis of the fact that Sonine had calculated two special integrals of such a kind of Bessel functions, which are known in literature as Sonine integrals. For these we refer to (2.54) and the handbook by Prudnikov, Brycbkov and Marichev [2, equations 2.12.4.6 and 2.12.35.12]. We also observe that the inversion formula for (37.41) with 11 = 0 in the space L2(0, oo) was indicated in the paper by Rozet [1] (1947) (see § 39.2, note 37.4 below).
774
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
The generalized Erdelyi-Kober operators defined in (37.45) and (37.46) and generalized operator of the Hankel transform given in (37.47) were introduced by Lowndes [1] (1970), who had proved (37.48)-(37.66) and other results. Notes to § 37.3. The operator inverse to the operator (37.67) in the form (37.81) was first constructed by Bakievich [1] (1963) for 0 < a < 1. However, in the special case a = 1 the inversion of this operator, in the form of the relations in Theorem 37.6, was obtained earlier in the paper by Vekua [1] (1945), and also his book (3, p. 69, 70]. In this paper the expressions of the form (37.78) and (37.79) and more general relations proved in 1942 and connecting the solutions of the Laplace equation �u = 0 and the Helmholtz equation �u + �2 u = 0 were indicated. In this connection, § 40.2 and § 43.2, note 40.1 should be seen. A special case of (37.71) written in another form was indicated in the paper by Henrici [1 , p. 256] (1957). Other results of this subsection are the development of investigations given in the paper by Lowndes [9] (1985) where a modified form of the operator (37.68) was indicated and studied. Theorem 37.7 was mentioned in the paper by Marichev [1] (1972) as a special case of a more general statement. Notes to § 37.4. The results of this subsection were given by Marichev [1] (1972). Notes to § 37 .5. Special cases of the so-called integral transforms with respect to an index, or with respect to a parameter of special functions were discovered and investigated comparatively recently. The book by Lebedev [4] presents the fundamentals of the theory of Kontorovich-Lebedev transforms defined in {37.109), {37.110), and Mehler-Fock transforms given in (37.141), (38.23). In the paper by Wimp [1] published in 1964 the considerably more general transform (37.113) with the Meijer G-function in the kernel was introduced. Its inversion formula was obtained, and five special cases were considered. For this the book by Marichev [10, Section 8.4] should be seen. A more compact form for the inversion relation given in {37.122) was found by Yalrubovich (1] (1985), who also investigated the compositional stJUcture of this transform. The presentation in this subsection follows the paper by Vu Kim Than, Marichev and Yakubovich [1] {1986). We also note that different representatives of the above transforms with G-functions and H-functions in the kernel were considered in the papers by Kalla [1] {1970) and Shah [1] (1972). Notes to § 37.6. The Fs-transfonn in the form (37.131) was introduced in the paper by Brychkov, Marichev and Yalrubovich (1] (1986). Theorems 37.16-37.20 were proved by Yakubovich and have not been published previously. Notes to §§ 38.1 and 38.2. The solution of the dual equations {38.1) in the case = G = #-£ = v = 0, a - {3 = 1/2, and F 1 was first obtained by Beltrami [1] (1881) . We also note that earlier, Weber (1] (1872) had formulated problems for partial differential equations reducible to dual equations. Different special cases of (38.1 ) with = G = #-' = v = 0 in the main were studied much later in the papers by King (1] (1935), Busbridge [1] (1938), Tranter [1-3] (1950-1954), Gordon (1] (1�54) and Noble [1] (1955), and also Titchmarsh [1] (1937, 1948). Various methods which mainly differ from methods of fractional integro-dift'erentiation were used in the above papers. Fractional integration of order 1 /2 was first applied by Copson [3] {1947) to solve dual equations of the form (38.1). In this context see also Noble (2] (1958) and Sneddon [1] (1960). This method was developed in the papers by Copson [6] (1961) , Peters [1] {1961) and Lowengrub and Sneddon [1] (1962), [2] (1963) in order to solve (38.1 ) in the case #-' = v, 0. Fractional integration with Sonine operators (see § 39.2, note 37.3) was first applied by Ahiezer (1] (1954) to solving dual equations of the form (38.16) with F = #-' = v = 0 and modified order of conditions on :c. In his paper (2] (1957) this result was generalized to the case v = -1-' · The papers by Peters (1] (1961), Burlak [1] (1962), Lowndes (1] (1970) , etc. were devoted to the investigation of (38.16) with arbitrary #-' and v . However, all these articles contained an improper setting of the problem. Instead of integration over (k, oo ) in the first equation in {38.16) , integration over (0, oo ) was taken. But this is not correct for arbitrary {3 if the values of the function (t2 - k2 ){j on 0 < t < k have not been specified. It should be noted that in the papers by Erdelyi and Sneddon [1] (1962) and Sneddon [2] (1962), the systematical application of the Erdelyi-Kober type operators given in (18.1)-(18.6) to
R
::
R
R ::
§ 39. ADDITIONAL INFORMATION TO CHAPTER 7
775
the theory of dual, triple, etc. equations was begun. At the same time, the investigations in these and many following papers were carried out formally without studying the spaces of the functions, and the conditions on the pMameters. Subsections §§ 38.1 and 38.2 were written using material from the papers by Erdelyi and Sneddon [1] (1962) and Sneddon [2] (1962) Example 38.1, Lowndes [1] (1970) Example 38.2, Virchenko and PonomMenko (1] (1979) - Example 38.3, Virchenko and Makarenko [2] (1975) Example 38.4, Lowndes (2] (1971) and Sneddon (5) (1975), [6] (1979) Example 38.5, with some modifications. We also note that the solution in (38.8) was obtained by Titchma.rsh [1, p. 334] (1937), when � = 11, and by Peters (1] (1961), while the solutions in (38.12) and (38.13) were found by Noble (1] (1965) and the method of "trial" solution (Example 38.1) was used by Gordon (1] (1954) and Copson (6] (1961). -
-
-
-
39.2. Survey of other results (relating to §§ 35-38) 3 5 .1. The integral equation
J (t -rx)c(c) -1 #'1 (a, P; c; l - ;t ) f(t)dt 1
=
g(x),
0 < XQ � X � 1,
(39.1)
was considered by Higgins (3] who obtained two fonns for its solution
f (x) X
(
#'1 m , -P; a
- c
+ m; 1
- �) dt,
under the appropriate assumptions for a BUfticiently smooth right hand side g (x). Here If_ is the fractional integro-differentiation operator given in (2.18) or (2.34). We note that the above paper by Higgins was the first one in which the technique of factorization of the �1-transform by using fractional integro-differentiation operators was applied. The solution of the equation
f(x) =
m 1 t x)m- - 1 r��1� c) j e - e X
X
�1
(- a, -P; m - c; 1 - ;) g(m) (t)dt,
m = 1 , 2, . . . , m > Rec > 0, which differs from (39.1) by replacing 1 - tfx by 1 - x/t was obtained by Wimp (2] by using the
Laplace transform. Earlier special first kind Tn (x) in the kernel
cases
of this equation with the Chebyshev polynomial of the
J1 � f(t)dt t2 - x2
= g(z),
X
and with the Legendre polynomial
Pn (x) in the kernel
0 < zo � "' � 1,
(39.2)
776
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
1
J Pn (t/x)f(t)dt = g(x),
0 < xo $
x $ 1,
(39.3)
z
were considered in the papers by Ta Li [1, 2] and Buschman [1], respectively. The classical scheme for the solution of the Abel equation in (2.1) (see § 2.1) was used in these papers. In the paper by Widder [4) {39.2) and (39.3) were solved by the method based on the Laplace transform. We also observe that in the paper by Erdelyi [7) the equation of the form (39.3) was solved by using Rodrigues formula. The solutions of the equations of the form (39.3) in which Pn (x) is replaced by the generalized Legendre polynomials Q mk (x) and by the generalized Rice polynomials H�a, fJ ) (x) and by the hypergeometric function ¥"3(-n, a + f3 + n + 1, e, e ' ;p, p1, 1 + a; x), were obtained by R.P. Singh [1] and Dixit [1, 2], respectively. The solution of the integral equation
1
j(t - x)a P�� ,/3) (� - 1) f(t)dt = g(x),
0<
xo $ x $ 1,
(39.4)
z
with the Jacobi polynomials
P�a,/J) (x) = (a + 1)n [n!] - 1 �'t (-n, n + a + f3 + 1; a + 1; (1 - x)/2), which is a special case of (39.1), was first obtained by Higgins [2] - see also K .N. Srivastava [4] and Rusia [5) . In the papers by Bhonsle [1, 2], Rusia [2, 7), C. Singh [3] and K.N. Srivastava [3, 6-8] solutions of the equations of homogeneous type containing Jacobi polynomials in the kemel
were obtained. 35.2. We denote by Qq (a, b), 0 $ a < b $ oo, the space of functions r.p(x) given on (a, b) almost everywhere and such that the function xfr.p(x) is locally integrable on (a, b). Love [2] obtained necessal'y and also sufficient conditions for the existence and uniqueness of the solution of (35.1) and (35.2) in Q9(o, d), 0 < d < oo, and their explicit solutions in the form (35.5)-(35.8). These investigations were continued in the paper by Love [3] where (35 9) (35 . 12) for the solutions of (35.3)-(35.4) in the space Q q (d, oo), 0 < d < oo, were proved. We note that the representations for the solution of the equation of the form (35.2) as the compositions of two Erdelyi-Kober-type operators given in (18.1) and (18.2) were formally done by Buschman [5). Miiller and Richberg [1] obtained the inversion formulae in the special case of ' (35.3) with Chebyshev polynomials of the first kind in the kern.el
.-
f(t)dt = g(x), f� t2 - x2 00
x > 0,
z
which is an analogy of (39.2) . R.K . Saxena and Kumbhat
[3] deduced the inversion relations for the operators
r�:) f c'l-"Y (t - x)"Y- 1 :#'1 (a, {3; 1 - �) f(t)dt 00
z
-y;
(39.5)
§
777
39. ADDITIONAL INFORMATION TO CHAPTER 7
for f(x) E Lp(O, oo), 1 � p � 2 ; under the appropriate assumptions on the parameters. Goyal and Jain [1] found the inversion relations for the operators more general than {39.5) with Wright hypergeometric function in the kernel. Braaksma and Schuitman (1] obtained the inversion relations for the operator of the form
(35.4)
; f (1 - � ) c- 1 �1 (a, b; c; 1 - ; ) f(t) �t A' A 23.2, 18.3).
Af
00
( )(x) = r c)
:t:
in the space of test functions T(�, ll) and for the operator conjugate to in the space of generalized functions T' (�, Jl) (see § note Using methods by Love [2, 3], Prabhakar (4) constructed the solution of the equation
x > 0,
Re c
> 0.
{39.6)
McBride [1] obtained invel'Sion relations for the equation H�J = g (see § 23.2, note 18.5) more general than (35.1), and also for similar generalizations of (35.2)-(35.4). The investigations were given in the spaces of test Fpp and generalized Fp p function considered in § 8.4.
Saigo [1, 2] found the inversion relations for the operators T;/·" and 1::!·" , -oo � a < . b � oo (see § 23.2, note 18.6). In the paper by Smirnov (5) the inversion relation for the equation of the first kind with
Gauss hypergeometric function in the kernel, which arose in solving the Trikomi problem for equations of mixed type with two degenerate lines, was obtained by reduction to an equation of the form (35.1). 35.3. Using methods of fractional integro-differentiation Erdelyi (9] obtained necessary and also sufficient conditions for the existence of the solution, integrable on a finite interval, of (35.15) for Re ll < 1 and Re v � -1/2, and he constructed the explicit solution of this equation in the form (35.25). Erdelyi also indicated the form (35.35) for the solution of (35.17) with = oo, which was investigated earlier by K.N. Srivastava [1]. There is a mistake in his inversion expression see also Habibullah [2]. In the paper by Erdelyi [12], (35.15) with arbitrary real Jl and " was studied in the space of generalized functions with support on (O,oo) and conditions for the solution to be an usual function were found. Smirnov [4, 6) indicated sufficient conditions for the invertibility of two equation of the form (35.15) and (35.16) written in another way and gave their inversion relations. Din Khoang Anh (1) obtained the conditions for the existence and uniqueness of the solution of (35.15)-(35.16) with e = 0 and of (35.17)-(35.18) with = oo in the spaces Lp((O, a), x'"Y) and Lp((a, oo), x'"Y), 0 < a < oo, respectively, and constructed the inversion relations for them. We note that a series of pages by Buschman [2], Higgins [1] and K .N. Srivastava [2] were devoted to solving equations of the form (35.17) with Gegenbauer ultraspherical polynomials C�(x) in the kernel instead of P,Nx). 35.4. In the paper by Sneddon (4) the general method based on the Mellin transform defined in (1.112) and the convolution theorem in (1.115) was suggested for solving the equation
d
d
j ( �) f(t) �t a
k
:t:
As examples, solutions of (35.4),
= g(x),
0 � x � a, a � oo.
(39.7)
(35.15), (35.17) with d = oo, (39.1)-(39.3), etc. were given.
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
778
35.5. Kalla and Saxena [2] obtained the inversion relations for the operators
�x-'1 - 1 r(1 _ a)
f rf1 ( z:
0
a , {J + m; -y; -;;A
)
atP t'1 J(t)dt,
{39.8)
{39.9) more general than {23.5) and {23.6) for provided that the conditions
E
E
f(x) Lp(O,oo), 1 � p � 2, or f(x) rolp(O, oo), p > 2,
Re ("Y - a - {J) > m - 1 ,
-y :f:. 0, - 1 , -2, . . . ,
m = 1 , 2, . . . , f.' > O,
1 /p + 1/p1
= 1, I arg(1 - a)l < hold. Here !Dlp(O, oo) means the subspace of Lp(O , oo), p > 2, consisting of functions which the inverses of the Mellin transfonn of functions in Lp'(-ioo, ioo) - see also § 23.2, note 18.1. Re77 > max(1/p, 1/p' ),
1r
are
In the paper by Parashar [1] equations more general than (23.5) and {23.6) with the Meijer G-function in the kernel were considered, and the uniqueness of their solutions in the space L1 (0, oo) was proved provided that certain conditions hold. The inversion relations for such equations in Lp(O, oo), 1 � p � 2, or in !Utp(O, oo), p > 2, were obtained in Kalla [3] by using the Mellin transfonn. These results were extended by Kalla [12], Kalla and Kiryakova [1] and Kalla [11] to more general equations with the Fox H-function and an arbitrary function, respectively in the kernels. One may find other generalizations of (23.5) and (23.6) in § 23.2, note 18.4. 35.6 . Love [6] obtained necessary and also sufficient conditions for the solvability of the integral equation 00
J rf1 (a, b; c; -x/t)t"f(t)dt = g(x),
0<x<
oo.
0
He constructed the solution of this equation on the basis of the representation of the left-hand side as the composition of the Rie -Liouville fraction� integration operator in (5.1) and the Stieltjes transform (see § 9.2, note 7.3). Another fonn for the solution of this equation via the inverse Mellin transfonn was given earlier by Swaroop [1] . These investigations were continued in the papers by Prabhakar and Kashyap [1] and Love, Prabhakar and Kashyap [1] where the integral equations
mann
f tcr(c)-1 rf1 (a, b; c; - ;t ) f(t)dt = g(x), f tc-r(c)1 1F1 (a; c; -xt)f(t)dt = g(x) 00
00
0
0
were solved on the basic of the representation for the left sides as the compositions of the Rie Liouville fractional integration operator given in (5.3) and Stieltjes and Laplace transfonns, respectively. 35.7. Lebedev [1] proved that the integral equation
mann-
(TJ)(x) = ! JK ( t x ) j (t)dt = g(x), t+x + a
1r
0
2../iX
0�
x � a,
{39.10)
§ 39. ADDITIONAL INFORMATION TO CHAPTER 7 where K(u) = (7r/2)'}/i't{1/2, 1/2; 1; u 2 )
is
779
the elliptic integral of the first kind has the solution
a �dx f Jt2tdt x2 dtd f Jt-rg(2-r)d.,.2-r . t
f(x) =
_
! 1r
_
X
_
0
This result is based on the representation for the operator T as the composition of the left-sided and right-sided fractional integration operators by the function x 2 :
(TJ)(x) = ! 7r
x
f 0
fa
dt f (-r )d-r . 2 2 Jx t t J.,.2 t2 _
_
This method was simultaneously applied by Copson [3] to solving a two-dimensional equation in electrostatics. It should be noted that the papers by Lebedev and Copson were the first publications where integral equation of the first kind with special functions in the kernels were solved by means of their representations as the simpler equations of Abel type, where solutions were already known. In other words, the method of factorization for integral operators with special functions in the kernels was used in these papers for the first time. We also note that Lebedev [1] applied his result to the equation
fa 0
(
)
1 1 4xt (xt)m (x + t) 2m+1 �1 m + 2 ' m + 2 ; 2m + 1 ; (x + t) 2 f(t)dt = g(x),
and in the paper by Kalla [10] a similar equation with other parameters
fa (x
+ t) - 2 a�1
0
0$
(
1 a, 2 ; 1 ;
x $ a,
)
4xt (x + t)2 f(t)dt = g(x), 0 < a < 1,
was solved by method of Lebedev. In the paper by Ahiezer and Shcherbina [1] of the equation
fa (x2 0
1 + t2 )P
:l"l (�
2'
p + 1 . !. 2 ' 2'
)
4x2 t2 (x2 + t2 ) 2 ! (t) dt g (x),
was given where 0 < a $ +oo, 0 < 2p < q < 2p + 2. case of (39.10') ; see also Williams [1]. a
If
0<
x
a
< a,
formal solution
(39.10')
p = 1 /2, q = 2, then (39.10) is a special
35.8. By using Erdelyi-Kober type operators given in (18.1)-(18.4) Lowndes [4, 6] obtained solution of the integral equation of the first kind
f6 �(x, t)cp(t)dt a
=
f (x),
a<
x < b,
(39.11 )
780
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
in the following cases:
- 1 t6 1J K:(x, t) = r(1u6xCT'1 - a)r(1 - /3) a=
00
I
max(z,t)
0, b = +oo, 0 < a , /3 < 1 , u > 0, 6 > 0, -oo < J,£, '1 < oo ;
and
-u( cr+'l ) t6( 1 -p -J3 ) - 1 K: 1 (x, t) = u6x r(a)r(/3)
min(z,t)
I a
b < +oo, u > 0, 6 > 0,
a>
0, {3 > 0, - oo < J,£, v < oo,
where 1/l(t) is a certain function. The paper by Williams (1] is also relevant. As examples the equation a
I 0
K� (plx - t l ) -, = l x - t l "'
rp(t)dt
f (x), 0 < x < a,
Rep > 0, 1�1 < 1/2,
with the modified Bessel function K� (z), and the equation of the form (39.10') were considered. Ahner and Lowndess (1] found the solution of (39.11 ) with the kernel K:(x, t) more general than the above equations, and applied their results to solving dual and triple integral equations of the form (38.16), (38 .44). 36.1 . Fox (3, 5, 6] developed a method of solving the integral equation 00
Kf(x) :::
I k(xt)f(t)dt
= g(x),
x > 0,
(39.12)
0
provided that the Mellin transform k* (s) = rol{k(x); s}, - see (1.112) - of the kernel k satisfied certain conditions. On the basis of the convolution theorem in (1 .116) for the Mellin transform the method of solving (39.12) in L2 (0, oo) was developed in the paper by Fox [3] provided that the function k* (s) satisfied the functional equation
k* (s)k* (1 - s) =
j )r(a; + ('lj + s)/m; ) IJn r(a; r+(('l('lj ++ 11 -- s)/m . s)/m )r(('lj + s)/m )
j=1
j
;
;
(39.13)
The simplest case when k* (s)k* (1 - s) = 1 is well-known, as for example, in the book by Titchma.rsh (1, p. 40 1) . In the paper by Fox [3] the solution of (39.12) with the kernel k(x) satisfying (39.13) was expressed via the Erdelyi-Kober type operators defined in (18.1)-(18.4) . For example, if (39.13) holds with a 1 = a, '11 = 'I and m 1 = m, then the formal solution of (39.12)
§
39. ADDITIONAL INFORMATION TO CHAPTER 7
781
has the form 00
f(x) =
I (I�+;m;f1/mk)(xt)(I�+;m;f1/mg)(t)dt 0
where the operator IO+; tT, f1 is given in (18.1). As an example, the solution of (39.12) with k(x) = �xOt cos(x - onr/2) was given also. In the paper by Fox [5, 6] on the basis of direct and inverse Laplace transforms defined in (1.119) and (1.120) a method was developed by which a formal solution of (39.12) with the kernel k( x) such that {39.14) can be obtained. As examples, the solutions of (39.12) were obtained with k{x) = sin x, k(x) = foJ11 (x) and k{x) = x11 K11(x) where J11(x) and K11(x) are the Bessel function of the first kind defined in {1.83) and the Macdonald function given in {1.85), respectively. This method was applied by Fox [1] and by R.U. Verma [8] to finding the inversion relations for the Vanna transform defined in (9.6) and for the transform of the form {39.12) with the Meijer G-function G� : � (x) - (1.95) - in the kernel, respectively. 36.2. It is said that k(x) and h(x) form a pair of Fourier kernel• if they satisfy reciprocal inverse relations
I
00
I
00
k(xt)f(t)dt = g(x),
0
h(xt)g(t)dt = f(x),
(39.15)
0
and each of them can be considered as the inversion expression of one another. H h(t) :# k(t) and h(t) = k(t) then the Fourier kernels are called non-aymmetrical and aymmetrical Fourier kernels, respectively. Often more general relations than (39.15)
1
00
0
where kl (x) =
�
� 1 g(t)dt, 1 h1 (xt)g(t) � 1 f(t)dt,
k 1 (xt)j(t) t
�
00
=
0
�
0
J k(t)dt, h1 (x) = J h(t)dt are considered. 0
0
t
�
=
(39.16)
0
They are reduced to {39.15) provided
that differentiation of the integrand is possible. Investigation of the Meijer G-function as a symmetric Fourier kernel was first given by Narain Roop [1]. These investigations were developed in the paper by Fox (2] where the new general function of hypergeometric type called later the Fox H-function was first introduced and studied. The theory of Fox H-function may be found in the paper by Braaksma (1] and in the book by Mathai and Saxena [1] and in the handbook by Prudnikov, Brycbkov and Marichev [3]. Problems connected with symmetrical Fourier kernels were also considered by Marichev [10] and Verma R.U. [4). Using a method based on the direct and inverse Laplace transforms R.U. Verma (5, 6] reduced equations of the form {39.12) with G- and H-functions, respectively in the kernels, to the equations with symmetrical Fourier kernels, and constructed the solutions of such· equations in closed form. In the paper by Kesarvani [2-5) G- and H-functions were studied as non-symmetrical Fourier kernels. We also note that Kesarvani (7) found necessary and sufficient conditions for functions
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
782
f(x), g(x) E
L2 (0, oo ) to satisfy the dual equations (39.15) with
�:.
where G 2 is the Meijer G-function defined in (1.95). 9 In the case when h(t) = in (39.15) or in (39.16), functions and are called k-tran&jorm& of each other. Mainra [1] and B. Singh [1) studied the classes of k-transforms (39.15) in which Fourier kernels are respectively the generalized Watson functions w�!:::·.:�': expressed via multiple integrals of products of Bessel functions, and via the function formed by the application the Erdelyi-Kober-type operators given in (18.1) and (18.3) to the Watson function. Soni [2] proved that if L , oo ) , Re a > 0 and Re 71 > -1/2, then and are and are k-transforms if and only if the functions where the Kober operators defined in (18.5), (18.6) are the same k-transforms. 36.3. A series of papers by Fox [2], Kesarwani (5], R.K. Saxena [2, 4), K.C. Gupta and Mittal (1], Rattan Singh (1], Kalla (9), Buschman and Srivastava (1], Kumbhat [1], Nasim [1] are concerned with finding inversion relations in Lt (0, oo ) or in L2(0, oo ) of equation of the form (39.12) with the Fox H-function or its special cases in the kernel. Solutions include the fractional integration operators of Erdelyi-Kober-type given in (18.1) and (18.3) in some special cases R.K.Saxena (2], Kalla [9], Buschman and Srivastava [1]. As examples the inversion relations were obtained for the Varma transform given in (9.6), and for the Hankel and the Meijer transforms (see § 1 .4). In the papers by V.P. Saxena (1], Bhise and Madhavi Dinge (1] and Madhavi Dinge (1] integral operators of the form (39.12) with the Fox H-function in the kernel were represented as compositions of the Erdelyi-Kober-type operators defined in (18.1) and (18.3), and operators of the form (39.12) with the Fox H-function of less order. R.U. Verma [7) formally constructed the solution of a two-dimensional integral equation with a kernel which is a product of two Fox H-functions of one variable. We note that Raina and Koul [1,2] and Raina [1] proved that the fractional integrals defined in (5.1) and (5.3) of Fox H-functions are H-functions also but of greater order. 36.4. Rooney (5] investigated the mapping properties on the integral transform
k(t)
f(x) g(x)
/(x), g(x) E 2 (o It, f(x) K;, f(x) a a
g(x)
(Kj)(x) from the weighted
space
0
Cp,r
It,a , K;,/(x)a
J (xt I at , · · ' , ap ) f(t)dt, x { f : l f(x } 00
=
=
G�"
JxP
)J• .,- t dx <
(39.17)
> 0,
bb . . . , bq
oo,
1�r<
oo
(x)
-
see
Rooney [3]
-
onto £1 - JJ ,• • By using the Mellin transforms he obtained a characterization of the range of the operator in terms of the ranges of the Erdelyi-Kober-type operators given in (18.1) and (18.3), and of the modified Hankel and Laplace transforms:
K/
00
(Ht,11 f)(x) J(xt) 1/k-l/2J, (Ikl(xt)11k )f(t)dt, =
Re71 > - 1 ,
0
(Lt ,af)(x) J(xt) -a e-lk J(zt)l/A: f(t)dt, k 00
=
0
=
±1, ±2, . . .
,x
> 0,
under the appropriate assumptions on the parameters of Meijer G-function in the kernel.
In
§ 39. ADDITIONAL INFORMATION TO CHAPTER 7
783
the case p + q = 2m + 2n, the conditions and the expression for inversion of the equation = were given. Let
(Kj)(x) g(x)
f(x) x-'Y-112H1 ,-y [t'Y+1 12f(t)](x), £a+-y+ 1/2+1/p,p £-y+1/2+1/p,p' 'H-y
=
V�J is a fractional derivative in (5.8). Gasper and Trebels (4] proved the bowtdedness of the operator V�'H-y from into and deduced necessary conditions for 'H-y-multipliers of type (p, p). 36.5. A series of papers is concerned with the application of the Erdelyi-Kober-type operators (18.2) and (18.4) to finding inversion relations for the integral transforms (39.12) with special Macdonald fwtction and Whittaker fwtction in the kernels. Saksena [1] , Fox [5], Okikioulu [3,6] and Manandhar [1] used the Erdelyi-Kober-type operators defined in {18.1 )-(18.3) and the Mellin transform given in {1.112) to finding inversion relations in Lp 0, oo) for the transform
K11 (x)
Wt,p (x)
(
00
x'Y j(xt)a-112K11 _1,2 (xt)f(t)dt g(x), x =
(39.18)
> 0,
0
in the cases p = 2, "( = 0, a = 11 > 1/2; p > 1, 11 > 1/p, 'Y � 1 - 2/p > 'Y - 11 , a + 1 - 1 /p > 11 > -a + 1/p and 1 � p < oo, 11 E a > l11 - 1/21 + 1 /2 + 1/p' 1/p + 1/p' = 1) if v > 0 and a > l11l - 1 /p' if 11 < 0, respectively. Okikioulu [3) showed that the operator in {39.18) can be represented as the composition of the modified Laplace transform and the Erdelyi-Kober-type transform defined in (18.3), and it is bounded from Lp(O, oo) into Lq (O, oo), 1 /q = 1 - "( - 1 /p > 0. He also proved that integral transforms of the form {39.18) with replacing by
R1 ,
y1/2-11Y1/2_11 (y) y1/2-11 H1/2_11 (y)
(
ya-1/2K11_1,2 (y) y112-11 J11_1,2 (y), Jp(y), Yp(y), Hp(Y)
and where are Bessel and Struve functions - as given in {1.83), and Erdelyi, Magnus, Oberhettinger and Tricomi [2, 7.2.1, 7.5.4) are representable as compositions of the Erdelyi-Kober-type fractional integrals defined in (18.1), (18.3) and the cosine- and sine- Fourier transforms given in {1.108), {1 .109) . The Kober operators in { 18.6) were applied by Saksena (1] and R.K. Saxena (1] to obtaining inversion relations for the Varma transform given in (9.6). We also note that the inversion of the Vanna transform in terms of the direct L and inverse L Laplace transform operators was fowtd by Fox [7). H .M. Srivastava [1] obtained the inversion relation for the transform
-1
00
j (xt)11- 1 12e-xtf2 wk+ 1/2 ,p (xt)f(t)dt g(x), x =
{39.19)
> 0,
0
which coincides with the Varma transform in the special case 1.1. = 11. H.M. Srivastava proved that if E is the solution o£ (39.19) and - v � k < v + 1/2, then = , is the inverse Laplace transform operator and where L is the Erdelyi-Kob r operator
-1 k 11J-+,1k,11-110 f(x) L I�'i� v -kg(x)
f(x) L2 (0,oo) L-1
given in {18.2). Other representations for the solution of (39.19) were given by Pathak [1). We also note that McKellar, Box and Love [1] and Love [8] applied fractional integro differentiation to finding inversion relations for the integral transform of the form {39.12) with the Struve function in the kernel - Erdelyi, Magnus, Oberhettinger and Tricomi [2, 7.5.4). 36.6. K .J. Srivastava [1-3] applied the Kober operators defined in (18.5) and {18.6) to the investigation of the generalized Mellin transform given in (9.7) and acting from L O, oo) into 00 Lp ' (O, oo) {1 � p � 2, 1/p + 1/p' = 1), and to studying in L2 (0, oo) the transform
H11 (z)
p( J wp,ll (xt)f(t)dt 0
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
784
with the Watson function
wp,11 (x) = x112 J t-1 J11 (t)Jp(xft)dt 00
0
in the kernel, where
Bessel function given in (1.83) K.J. Srivastava [1] and [2], [3] , respectively. 36.7. Marichev [1,2] considered the equation -
f f(t) (x-r(c)t)e- 1 Fa ( z:
1
Fa
is the
x t ) dt =g(x), c > O,
a, a , ,B, ,B' ; c; 1 - t , 1 - ;
a
0�
J11 (x)
a x b <
<
�
oo ,
Re
involving the Appel function of two variables defined in {10.45) in the kernel. He obtained the solution of this equation via the composition of three fractional integro-difl'erentiation operators or via the operator with the function in the kernel (see § 10.3). Similar equations with the Appel function in the kernel were studied by Marichev and Vu Kim Than [1). Vu Kim Tuan [6] proved necessary and sufficient conditions for the existence and uniqueness of the solution of the Abel-type equation
Ft
Fa
a
z:
f (x � t)e-1 ( ;t ) p(x, t)f(t)dt = g(x), (c) p(x, x) x) E Qq p(x,t) = 1Ft(a;b;�(x - t)), i"t(a,b;c;1 - xft), Fa(a,a';b,b';c;1 -
being an analytic function, ":# 0, in the space of function / { (see § 17.2, notes to § 10.1). He extended these results to equations containing Kummer, Gauss and Appel hypergeometric functions: 0
l'(x, t) xft.1 -tfx).
36.8 . H.M. Srivastava and Buschman [1] considered operators of the fonn
x-"f -a J(x at)0 -1 t"� f(t)dt, x6 J(t bx)P-1 c6 -P f(t)dt, a= b= Ft 2F1 . 00
z:
+
+
0
z:
from which the Kober operators in {18.5) and (18.6) are derived if - 1. They - 1 and proved that the composition of these operators is the integral operator with a homogeneous kernel involving the Appel function and, in particular, the Gauss hypergeometric function 36.9. Habibullah [1] .investigated the integral operators
AJ(x) = x� J(xt)b-l i"t (a, b; c; - xt)f(t)dt, 00
0
00
BJ(x) = x� /(xt) a-l tFt(a,c; - xt)f(t)dt, 0
C J(x) = x� J(xt)a-l t� (a, c;xt)f(t)dt, 00
0
�1
tFt
where is the Gauss hypergeometric function defined in (1 .72), is the confluent hypergeometric function given in (1 .81) and tJ is the Tricomi function - Erdelyi, Magnus,
§ 39. ADDITIONAL INFORMATION TO CHAPTER
7
785
Oberhettinger and Thicomi [1,6.5]. He proved that the operators A and B can be represented as compositions of the Erdelyi-Kober-type operators given in {18.1) and the generalized Stieltjes and Laplace transforms, respectively, and the operator C can be represented as a composition of the generalized Stieltjes and Laplace transfonns (see § 1.4 and § 9.2, notes 7.3, 7.8). On the basis of these results Habibullah proved the boundedness of the operators A, B and C from Lp(O, oo), p � 1, into Lq (O, oo), 1/q = 1 - 1/p - � � 0, and found their inversion relations under the appropriate assumption on parameters. We note that this type of result in another space of functions was previously obtained by Swaroop [1). We refer also to Marichev [10, Sect. 8.2) with respect to the operators A and B, and to Love [6] with respect to the operator A. 36.10. Let It, be the Kober operator defined in {18.5), let af'1 be the Gauss a hypei'geometric function given in {1.72) and let Tn be the integral operator (Tn f )(x) = s
{-1)"xf(x) + J k(t/x)f(t)dt. Erdelyi [5) proved that if f(x) E L2 (0, oo), k(x) = [B(n, £1 + 0 1)] -1 x / 2 af't (1 - n, £1 + n - 1 ; �-« + 1 ; x), £1 > -1, then Tn be represented in the form can
"
36.1 1. By using the fractional integrals xfJ Ig+ x"Y and xfJ J�x"Y Marichev [6] proved relations of compositional type which are special cases of (36.34). Analogues of (11.27)-(11.30) which realize the connection between certain pairs of the integral operators with special functions in the kernel and the singular operator are more interesting. We indicate two pairs of such a type, namely
{39.20) where IRe £11 <
00
1/2, {J, (2.fi)}rp = J J11{2v;Ji)rp(t)t-1 dt, and 0
2J8+ ( 1J, b)rp(x) =
sin IJ1r sin b1r sin( c - (J - b)1r '
,
"
_
sin(c - 1J)1r sin(c - b)1r ' sin(c - a - b)1r
_
(Srp)(x) = !. 1r
f rp(t)dt t-x .
{39.21)
c ¢ IJ + b,
00
These relations were previously obtained in Marichev (3]. All expressions of this kind hold for sufficiently good functions, and can be proved by applying the Mellin transform defined in (1.112) to both sides of the equation. We note in addition that by using relations of such a type and the Mellin transform, Marichev [7) indicated the class of complete singular equations with power-logarithmic kernels solved by quadratures. 36.12. H .M. Srivastava [1] employed the Riemann-Liouville fractional integro-differentiation in identities involving infinite series. He showed how such exercises cuhninate in linear and bilinear generating functions for a wide variety of special functions. 0
786
CHAPTER
7.
INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
Saigo and Raina [1] and Saigo [8] evaluated the generalized fractional integrals and derivatives given in § 23.2, note 18.6 of a general class of polynomials with essentially arbitrary coefficients of several elementary functions, and of the Bessel and the modified Bessel functions given in (1.83) and (1.84). A similar problem for the two fractional calculus operator introduced by R.K. Saxena and Kumbhat [2] and involving the Fox H-function was studied by Saigo, Kant and Koul [1]. 37.1 . A series of papers by Widder [3], Buschman [4], Khandekar [1], K .N. Srivastava [9,10], Rusia [1,5], C. Singh [1,2,4], B.K. Joshi [1], H.L. Gupta and Rusia [1] were concerned with the solution of special cases of (37.1) with the generalized Laguerre polynomials or with the Whittaker functions M1c in the kernel. Methods based on the Laplace transform or on reducing such equations to the Abel integral equation (2.1) were used. By using the Laplace transform S.D. Gupta [1] solved the equation
Lg(x)
11 (x)
z z J ( -t)(x - t)aL� ,, [(x- t)�']J(t)dt g(x), ( z) ea
0<
=
0
involving the generalized Laguerre polynomial L� , , Laplace transform by
a > 0, cf.
x < a,
in the kernel which is defined via the
Re(p - a ) >
0,
(37.19).
By using the Laplace transform H.M. Srivastava [3] obtained the inversion relation for the integral equation more general than (37.1) with the confluent hypergeometric function of several variables 00
L
� 2 ( ab · · · , ar ; b; zl t · · · , zr ) =
m t , . . . ,mr=O
( t)
( ar )mr z;n1 a m1 ( b)m1 + · · ·+mr m 1 ! • • •
He also indicated that this method can be applied to solving the general equation (4.2) in the case when the Laplace transform of the kernel Lk{p) is representable in the form Lk(p) = [(p - a) n (Lkt )(p)] - 1 where k1 ( is a certain function. 37.2. By using the Laplace transform Kalla [2] and T.N. Srivastava and Y.P. Singh [1) obtained the solution of the equations - see (37.2) - namely
x)
fz (x2 - t2 )" J, (.\(x2 - t2))e-b(z,-t ,)f(t)dt g(x), =
a
z
J t)" Jff {.\(x- t)�')J(t)dt (z -
=
g(x),
0
J11 {x)
J/:(x)
is the Bessel function defined in (1.83) and is the Bessel-Maintland function where (see § 23.2, note 18.2). The method of fractional integro-dift'erentiation was applied by Prabhakar [3] to solving the
§
39. ADDITIONAL INFORMATION TO CHAPTER 7
integral equation
z:
j (x - t),8-tE:,,8 [�(x - t)]f(t)dt
= g(x),
787
Re P > 0,
a
and a similar equation with variable lower limit where
E: ,,8 (z) =
f: k!f.(�{�p), Re a > 0, is
k=O
the generalized Mittag-Lefler function. A method based on the Laplace transform was used by H.M. Srivastava and Buschman to find the solution of the most general convolution equation z:
J 0
I
]
Ap) f(t)dt = g(x) (x - t)a-t H;'' q" x - t (a(bt ,! ABt ),), .· ·. ·. ', (ap, ( b9 , B9) t
[
l
[2]
{39.22)
with the Fox H-function in the kernel. As examples, special cases of this equation with the generalized Wright function p\P 9 - 1 , a special case of the Meijer G-function of the form G ( x) and the generalized hypergeometric function pFp -1 were considered. In this connection we also point out the paper by Nair [1], who considered convolution equations with the generalized Wright function p'iPq and the generalized Bessel-Mailtand function JC(x). In the paper by H.M. Srivastava [2] a solution of the equation similar to (39.22) with variable lower and infinite upper limits was obtain by using the Mellin transform. Also, as special cases, the equations with the G-function of the form a;:t <x), with the generalized hypergeometric function of the form pF9 (x) , with the Whittaker function W>., (x) and with Bessel functions Jv(x), Kv(x) and Yv(x) l' were considered. 37.3 . Sonine (4, 5] (1884) and also [6, p. 151], proved that the operators in (4.2) and (4.21 ) with the kernels k( x )
_
- (2x
) P J-p(2i.fli} , -
� ;�
P -t Jp -t (2-.fii) )P l(x) = (2x) (2../XY -t ,
(2iy'ii) -P and their special cases with p = 1/2 of the form k(
X
-
Vii
) cos(2iJii) _
and also the operators with the kernels
'
< p < 1,
..j1rX
l(x) = cos(2Jii)
-
have the property given in
0
(39.23)
(39.24)
oo
(4.211) together with the fractional integro-dift'erentiation operators
k(x) = xa -t /r(a), l(x) = x-a fr(1 - a),
0
< a < 1.
After the quadratic changes of variables and functions
x = a + b2 - ( 2 , t = a + b2 - T 2 , 2i.J'Y = �� 2/(t) = ..J;F(T) , g (x) = G(()
7. (4.2) (4.2')
CHAPTER
788
the operators in
INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
and
with the kernels
{39.23)
have the fonn
0 which coincide with fonn
{37.41), (37.43) ! b
e
and the operators i n
cos(.\
(4.2')
1, {39.24)
with the kernels
have the
�) -rF(-r)d-r = G(e),
�
v •- -
c; 2
[1]
The authorship of the latter expression is attributed to D.S. Jones in some papers. 37.4 . Rozet obtained the inversion relation for the integral equation in under the appropriate assumptions on the kernel As an example he found relations for the solutions of the equations
[1]
{39.12) L2 (0,oo) K(.x, t). � Jn � J1 (2 ) ! � x t)n f(t)dt = (.x) t j(t)dt = g(z), ! ((v�) (37.68) = 0, (J+, = 2i. f)(.x) = g(.x) (37.67) =1 [4) 39.2, 37.11 (37.[1].67) (5.1) [1] f(.x) L2 (0, oo) .x"/2 JJ11 (2.../Xt)t-1112f(t)dt L2 (0, oo) [9,10]0. 0
X -
0
the first of which is the limiting ·case of with o ,\ The inversion relation for the non-convolution equation given in in the case o was obtained by Mackie example of more general result (see § note below). By using the non-convolution operators given in differentiation operators defined in and (5.8), Soni characterization of the space of functions
when v > Lowndes investigated properties of the operators
where
m
is the positive integer for which
0
� o, ,\ � <
O,A
such that
E
0
o
9
+
'
with the operator The paper by Soni is an and the fractional integro solved the problem of the 00
E
0
0,
+m �
1,
and of the operators li). ('1, o ) defined
§ 39. ADDITIONAL INFORMATION TO CHAPTER
(1.83) J(1.01_814), (37.68). [1101] _1 ,
by the above relations with being replaced by functions given in and cf. 37�5. In the paper by Pollard and Widder applied to finding the solution of the equation
J k(x - t)f(t)dt �
=
where
7
J11 (z) 111 (z) and
789 are the Bessel
the method of operational calculus
was
g(x), x > 0,
(u > 0) of which is connected with the solution of the heat the kernel k(u) = Aiu exp ( u of this equation is expressed in tenns of the Riemann-Liouville fractional equation. The solution integro-differentiation operators of order 37.6. Let and be the operators in and and let Lp , v :: Lp([O, oo) ; x"P - 1 ) be the weighted space. H � oo, then the operators and respectively. H a.re bounded in Lp , v when 11 + and 11 > then 0 + and a.re bounded in Lp , v if 11 + + and 11 > respectively - Heywood and Rooney and Heywood 37.7. Following the papers by Prabhaka.r and using q-fractional integrals (see § note Prabhalcar and Chakrabarty obtained the inversion relations for the q-integral analogue of with 1F1 being replaced by the basic confluent hypergeometric function 1� 1 in the kernel (see Slater 37.8. Pinney considered the integral equation 0
- fu-) J>. ('l,a) R>. ('l,a)1/2. a 1/2, 1 <(37.p 4<5) (37.46) J>.('l,a) <, a)2 2'1R>. ('l,a) - 27}, 2a), << min(1, a < 1/2,2/p)R2/(1>. ('l,a) 2a) < p < 2/(1 J>.('l max(1,2/p) - 2a - 27}, [1] (32a]. 2'1 23.2, 18.1 5) (37.1) [1] [1,2] [1][1]).
J p(t)rp( .jx2 t2)dt f(x), 00
+
=
x > 0,
(39.25)
0
and found its solution
rp(x) �-!.x �dx J f( .jx2 t2)q(t)dt r/2 p(x) q(x) J p( rp (4.2") (4.2). also 00
+
0
provided that for a function
there exists the function
such that
r cos
)q( r sin
rp)drp 1
for any r > 0. He also obtained sufficient conditions for the truth of the latter relation, which is an analogue of for the Sonine equation He considered the following special case: 0
=
where
is the generalized Legendre function - the Legendre polynomial L;11(z) for the natural 11 particular, - tl -2p as 11 - 0, and
p(t)
where if I' =
J1/2. -I' (z)
0
p(t) = t1 -I'J_11(at), < Rei' < 1, p(t) t1 -21', < Rei' < 1, (1.83); (39.25) p(t)
= n.
In
at
i s the Bessel function of the first kind given i n i n particular, = cos In the case = 0 the connection of with the Abel
CHAPTER 7. INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS
790
equation ( 2.1 ) was shown and when
J
00
37.9 . Soni
[3]
cos
p(t) at (39.25) � f(t)dt g(x), x > t-x = cos
was reduced to the equation
=
0.
found the integral representation
J(x- t)"12J, [2Vk(x- t)]f(t)dt X
/2 ] <x-t) " [ d Jt Jo[2 Vk(t- T)]j(T)dT dt k" r(v + ) dt 0
z
=
]
-
1
for the operators of the fonn (37.4) , and proved the necessary and sufficient conditions of solvability in for the equation 0
0
L2 (-oo,oo)
d� J Jo[2 Vk(x-t)]f(t)dt g(x), -oo x oo, z
<
=
- oo
<
and obtained its solution in the fonn
f(x) .!!._dx J Jo[2 Vk(t-x))g(t)dt. 00
=-
Soni
[3]
also investigated the conditions for the uniqueness of the solution of the equation
d� J Jo[2 Vk(x- t))f(t)dt g(x), a > -oo, L2 (a,+oo),L (a, +oo). (g ::: 2 [ ] [1] K:(x, t) J(T2 - b2 )r 7"-m-n-l Jm (xT)Jn (tT)dT. z
=
in because the corresponding homogeneous equation 0) can have a non-trivial solution in Similar investigations were done for the operators with variable lower limit - see also Bouwkamp 1 . By using the method of representation via the composition of two equations of the fonn (37.4) , Williams solved (39.11 ) with the kernel a
00
=
b
J, (x)
We note here that other integral equations with the Bessel function in the kernel considered on the whole line arise in applications. Two such equations can be obtained from 40.26) and 40.27) if we carry out differentiation with respect to y for y = 0 and take 40.45 ) and (40.44) into account in the fonner relation and if we set y = 0 and take (40.32 ) and (40.47) into account in the latter relation. Then for 1J. = 0 after the substitution = with IPI < 1/ , ¢. 0, we obtain the relations
( p
(
T(x) l2g(x)
(
2
39. 7 791 00 ( j It � �2P J-p (.\. lt - xl)dt g(x), x oo 00 sign(t -x) d v(x) - -211" J It- 1 1 2p dt [g(t)Jp-1 ( .\.lt -xl)]dt, J,(z) (37.8). g(x). Kv(z) (1.85) and [Yv(z) 2, 7.2.1]. 21 9. 1 [ 1 ] Jv is Kv Yv:(37.4) j(t- x)(v - 1)/2Kv(2Vf=X)J(t)dt g(x), 00 j(t-x)(v - 1)12Yv(�../t -x)f(t)dt g(x). §
ADDITIONAL INFORMATION TO CHAPTER
=
-oo
<
<
oo ,
-
ctgrnr
=
X
- oo
-
where is the Bessel-Klifford function _given in One may check that the second relation inverts the first under appropriate assumptions on the function 37.10. Let be the Macdonald function defined in let be the Bessel function of the second kind - Erdelyi, Magnus, Oberhettinger and Tricomi By using the Mellin transform and results and 20 from Table V.K. Vanna found the solution of equations similar to with variable lower and an infinite upper limits, in which replaced by and by 00
=
�
=
�
37.1 1. By using the method of reducing to a boundary value problem for the partial differential equations
a2r.p c(x,11)r.p 0, c(x,11) c(11,x), r.p(x,O) f(x), x 0, -axa11 (39.26) r.p(0,11) -!(11), 11 0, x 0, 11 (1] a a J K:(x, t)g(t)dt f(x), g = !.2 ( ax"" - a11r.p ) I (39.26) 40.2. K:(x,t) R(x,x;t,O) (37.67) R(x,xo;t,to) 1 +
=
=
>
=
in the first quadrant
>
> 0 Mackie
>
=
obtained a solution of the integral equation
�
�= '11
=
0
where = As examples
and with a =
is the Riemann function of and the equations
�
Definition
�
j Pn (x/t)r.p(t)dt g(x), j Pn (t/x)r.p(t)dt = g(x) Pn (z) (5] =
0
in the kernels were solved, cf. the equation considered the equations 0
with the Legendre polynomials 37.1 2. Prabh.akar �
J a
(x �(c)t)c-1
�t (a, b;
c; 1 -xft, .\.(x- t))j(t)dt = g(x),
(39.3).
a < x < {3,
(39.27)
7. fz (x �(c)t)c-1 �1 (a, c; 1 - tfx, -X(x- t))f(t)dt g(x), x (39.28) �0,1 (a, c(39.x, 2)7) (39.28) (10.64), 0,(35.1) (35.2), 0 (39.27) (39.28) (37.1). 1 f(x) e�zx-61;:;.e-�zxbI!+cg(x), f(x) I!+ce�zxb I;;:;.e-�zx-6g(x) � (2.17). [6) (39.27) (39.28}
792
CHAPTER
INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS =
b;
0t
<
< (3,
a
b ; ; y is the Humbert function given in where 0t > (3 < oo . In particular, if ,\ = then and coincide with the Love equations and and if b = they then coincide with the equations of the form Prabhakar found the criterion of solvability for and in L (at, (3) and obtained their solutions in the form
=
=
respectively. Here ] + is the Riemann-Liouville fractional integro-difl'erentiation operator defined in Similar results were proved by Prabhakar for the equation of the form and with variable lower and finite upper limits and for more general equations. For example, for the equation
(h(t) ��!;)]c-1 �1 (a, c; 1 - ��;:, -X(h(x) -h(t))) /(t)dh(t) g(x), Jz m' 0 h(t) h(t) e C00(at, {3], h'(t) (39.0.6) t 0 h(t) (37.1). (1] I-oooo k(x- t)f(t)dt g(x) k(x) "2k -iIioooo (l(t)] -1 ezt dt l(t) (2.1), (1] (39.1) (39.4) (1] (37.60} (37.61), (37. 4 7) (23.5} (23.6). (1] z (39.29) J(x- t)n Im (x-t)f(t)dt g(x), 0, 0, (1.84) Im (z) (39.29). (39.239.9) 1 , 36.0,1 1, 2, ... (39.29)0,1,2, ... - 1, - 2, . . . 0. (1] 37. 1 5 � (37.107). (37.109) (1,2] (37. 1 41} [1-3]. fJ
=
b;
Ot < X < (3,
where > In particular we note that if ,\ = and = then the latter equation is reduced to and if b = and = t, then it is reduced to 37.1 3. Tanno investigated the inversion problem for the convolution integral transforms =
where
=
and
is a meromorphic function
with real zeroes and poles. He also considered such a problem for more general transforms. The book by Hirschman and Widder should be consulted in this connection. As examples, solutions of the Abel equation the Higgins equation and the equation of the form were obtained. 37.14. R.K. Saxena and Sethi found the operational relations similar to and and connected the generalized Hankel transform operator given in with the modifications of the operators in and 37.15. Tedone considered the integral equation
=
m
�
m
+n �
a
with the modified Bessel function given in in the kernel. On the basis of properties of the function he proved some difference-recurrence relations for the operator in This was used to reduce the solution of with m, n = to the successive solutions of simpler equations with m = n = 0 - § Note to - and the differential equation. Also was reduced to the solution of the Volterra integral equation with a difference kernel if m = , m +n > ,n= 37.1 6. Yakubovich, Vu Kim Than, Marichev and Kalla proved a theorem similar to Theorem in the space L2(R ) and considered the special cases of the W-transform given in The Kontorovich-Lebedev transform defined in and the Mehler-Fock transform given in of generalized functions were investigated by Glaeske and Glaeske and Hess
§ 39. ADDITIONAL INFORMATION TO CHAPTER 38.1. When investigating the special case of equation
(38.1 )
Lebedev
[2]
7
793
obtained the Shlemil'ch
�/2 J� r2 -t2 : - j cp(r cos O)dO g(r) 1/2 [1]. 2 18.2. 0, [3] (38.1) (1.83), ak · [1] 0 (38.1) 0. r
=
0
0
containing the fractional integral of order by the function x - § The inversion relation of this equation was used by Lebedev and UO.yand 38.2. In the case G = p = v Tranter represented the solution of as a series 00
ak Jv+2k+n [1)
(x) of Bessel functions given in and as a result he reduced this problem to k :O solving the infinite system of linear equations with respect to Cook used the integral analogue of such a method and reduced this system to Fredholm's integral equation of the second kind. Nasim and Aggarwala suggested a method for solving based on its decomposition into two systems of more simple dual equations with F(x) = and G(x) = studied the problem of the uniqueness for the solution of the dual 38.3. Walton equation in the space of generalized functions. 38.4. Buschman and Kesarvani indicated that by using the Erdelyi-Kober-type fractional integration operators given in and and the Mellin transform defined in the dual integral equation with = and the dual integral equation with Meijer G-functions in the kernel, respectively can be reduced to a single equations of the same form given on the half-line oo ) . By using the Mellin transform Nasim and Sneddon investigated dual equations with kernels of general form. As an example, dual equations of the form with different Bessel functions and trigonometric functions in the kernels were considered. 38.5. By using two-dimensional analogues of Erdelyi-Kober operators - § (note - Makarenko studied dual and triple integral equations with Bessel functions in the kernels. Using the method in Example Virchenko and Gamaleya considered the system of dual equations of the form 38.6. By using fractional integra-differentiation Sethi and Banerji reduced equations more general than to the Fredholm equation of the second kind. 38.7. The system of dual equations associated with the Mehler-Fock transform and reduced to when p and H(-r) = th-r1r/-r was first investigated by Grinchenko and Ulitko Using fractional integrals of order by the function chx, they reduced such a system to the Fredholm integral equation of the second kind. Two systems with the same kernels and two systems with trigonometric kernels were solved by Babloyan by using a similar method. A system of the form with the integer 1J was first investigated by Ruhovets and UO.yand These results were developed by Virchenko and Makarenko We also note the paper by Lebedev and Skal'skaya who studied the system with p and H(-r) of a special form and obtained the equation with the Gauss hypergeometric function
L:
(1.1 12)
(38.1) [1,2[]6] [0,
[1]
[1].(38.22) =0
[61] ) (18.7) (18. (38.1) R(t) 0
[1]
(38.16). 38.2 (38.16) g= =0 1/2
[1] (1].
(38.1)
(1]
[1].
29.2
24.2)
[1]
(38.22)
[1]
j ;<� t =l'l (a,-a;�;1 - �) dt=g(x) 1 [2] 35.1)
(38.22)
�
instead of fractional integrals. They used the result by Love (§ for the inversion of the last equation. Investigating three dual equations with the generalized associated Legendre function in the kernel Virchenko used the inversion relation for the more general equation with the Gauss hypergeometric function
[1]
7. (a b chx-cht ) dt = g(x), tp(t)(chx cht)c chx+d J mare [2] = j
794
CHAPTER z:
INTEGRAL EQUATIONS WITH SPECIAL FUNCTIONS -1
�1
, ; c;
c
> 0.
a
S.P. Pono nko considered the system of dual equation of matrix type P#' (Aj (r)'IP(r)) j and using she reduced this system to a system of Fredholm equations of the second kind. Together with the equations similar equations associated with the inverse Kontoro vich-Lebedev transform were considered in some papers. Dual and triple integral equations of such a type were investigated by Lowndes and the former equation was studied as the limiting case of the latter equation. One case of dual equations was solved in closed form, and other ones were reduced to the Fredholm integral equation of the second kind. 38.8. Equation generalize the systems from the papers by Cook and Borodachev and intersect with the system from the paper by S.P. Ponomcu-enko All these systems are reduced to Fredholm equations of the second kind by using Erdelyi-Kober-type operators. 38.9. Virchenko and Makarenko and Makarenko considered dual and triple integral equations with kernels more general than namely the Watson function - § (note above. These equations were reduced to Fredholm equations of the second kind. 38.10. The method given in § (Example was used by Yirchenko to find the inversion relations for two dual equations with the functions �1 and in the kernels, -by their reduction to the equation A f g where A is the operator in § (note 38.1 1. K .N . Srivastava and Virchenko and Romashchenko considered triple integral equations of the form with 0 and arbitrcu-y respectively. The result from the paper by Pathak was generalized in the latter paper. By using the Erdelyi-Kober operators given in and Lowengrub and Walton and Walton reduced the system of triple equations of the form to a system of generalized Abel integral equations - § (note Lowndes and Srivastava reduced a certain class of triple series equations involving the generalized Laguerre polynomials. to the triple integral equations of the form and obtained an exact solution of these triple series equations. 38.12. Quadruple and n-tuple integral equations were considered in some papers. Using the in method of Cooke Ahmad investigated the 4-tuple equations with the functions the kernels. The method used in § when solving the system was applied by Dwivedi and Trivedi to similar triple and quadruple equations. A further generalization was made by Vu Kim Tuan who considered n-tuple equations with the Meijer G-function in the kernels of the form
[1]
[2],
[1] 38.1
[5]
.:!_dx J xaPm+·'li·++ll ,q; +m+l (xy I (bmo, ()a, ;)<4-, (ep), -) 1 ) J(y)dy - ·(x) x .:!_dx J GPm;;+,ll++ll,q+m;+l (xy 1 (b1,tn(a,, )),, (dC4J q ),O ) f(y)dy - g, (x), •
00
9 a
•
0
Jv(z)
(38.16) '
00
_
.
0
[0,
where 0 < a o < a 1 < · · · < an - 1 < an = oo is the disjunction of the half-line oo) into intervals, 9i ( are the given functions and the remaining parameters satisfy the appropriate assumptions. Vu Kim Tuan proved that this system is solvable in the space f( E L2 (0, oo) if and only if 9i E L2( ai- l , ai i = 2, . . , n. In this case a unique solution was found in closed form. Vu Kim Tuan used the methods from the papers by Fox and R.K. Saxena where solutions of dual equations with functions more general than the Fox H-function in the kernels were formally constructed, and the Erdelyi-Kober operators were applied.
x)
), 1,
.
[4)
x)
[3,4]
Chapter 8 . Applicat ions t o D ifferent ial Equations In this chapter we shall be concerned with the applications of fractional integro differentiation to the investigation of partial differential equations of the Euler Poisson-Darboux-type, and of ordinary differential equations of fractional and integer order. In particular, the Cauchy, Dirichlet and Neumann singular initial value problems are solved in closed form, and the existence and uniqueness theorems for certain types of differential equations of fractional order are proved. The resulting solutions have wide applications to 'fiicomi-type boundary value problems for partial differential equations of the mixed type.
§ 40. Integral Representations for the Solution of Partial Differential Equations of Second Order via Analytic Functions and Their Applications to Boundary Value Problems The present section deals with certain aspects of the general theory of elliptic equations with anaJytic coefficients as developed in the book by Vekua [3] ( 1948), and their applications to the so-called generalized Helmholtz two-axially symmetric equation. We prove that if the Erdelyi-Kober operator I, , a defined in ( 18.8), and the generalized Erdelyi-Kober operator J). ( fJ, o:) defined in (37.45) are applied to analytic functions in each variable, then we obtain the solutions of the above Helmholtz equation. These results are applied to the solution of certain boundary value problems.
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
796
40. 1 . Preliminaries Let there be given the following general homogeneous partial differential equations of the second order with two independent variables and with analytic coefficients
Ht u u�� - Uyy + A t (x , y)u� + Bt (x , y)uy + Ct (x , y)u = 0 , Ku u�� + Uyy + A(x , y)u� + B (x , y) uy + C(x , y)u = 0, =
=
(40.1) (40.2) (40.3)
and equations of the form
(40.4) H* v ve, - (a(e, 71)v)e - (b(e, 71)v), + c(e, 71)V = 0, (40.5) K* v v�� + Vyy - (A(x , y) v)� - (B (x , y) v)y + C(x , y)v = 0, conjugate to (40.1) and (40.3). Generally speaking we assume that the complex variables x , y , e and 71 are connected with each other by the relations =
=
e = X + iy, 71 = X - iy, x = <e + 71)/2 , y = <e - 71)/2i. We denote by ( the number conjugate to e and note that ( = case when and are real..
x
y
(40.6) 71 holds only in the
We introduce the differential operators
(40.7) In particular, the following formal relation is obvious namely
(40.8) where d 2 is two-dimensional
Laplacian.
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
If the coefficients of
797
(40.1)-(40.3) are connected with each other by the relations
4a(e, TJ) } = A (e + '1 , e 4b(e, '1) 2 2i
'�
)
±
'.B ( e +2 '1 , e 2i-
'�
), (40.9)
A t (z, y) = A(z, -iy), B t (z, y) = iB(z, -iy),
(40.10)
Ct (z, y) = C(z, -iy),
(40.1)-(40.3) (40.3).
(40.4)-(40.5), (40.6) e, TJ, (40.1), (40.4)
then and respectively can be transformed into each other by the changes given in and replacing y by iy, while becomes Nevertheless, for real z and y the above equations are classified in the following way. Equations and are called hyperbolic-type equations in the first and second canonical forms, respectively, and and are called elliptic-type equations. Let = z + iy be a point lying in a certain simply connected domain i) in the complex plane C, z and y themselves may be complex. Then we denote by f> a domain containing the point '1 connected with by ( Further we denote by (i), f>) a (Cartesian) product of i) and f> called a cylindrical domain. We introduce the following definition:
(40.5)
(40.2)
(40.2)
(40.3)
e
40.6).
e
A simply connected domain i) in the complex plane C is called a basic domain of (.lO. l} and {40. 9} if their coefficients a, b, are analytic with respect to variables (e, '1) E (i), f>), the functions A, B, C are analytic in (z, y) E i) and the relations in (.l0. 9} hold.
Definition 40.1.
c
We indicate the most important representation relations via analytic functions for the solution of ( which will widely used below in special cases. First such a relation uses the concept of a Riemann function.
40.3),
solution v = R(e, '1) = R(e, TJ;eo, TJo) of the Goursat boundary value problem for the conjugate equation (40.4) with the conditions
Definition 40.2. A
,
R le=eo = exp j a(eo, t)dt, flo
(
R l,=,o = exp j b( Eo
r, '10
)dr
(
40.11)
798
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
is called the Riemann function of the operator H given in (40. 1} corresponding to the point ({o , 7Jo ) E �.
We note the following properties of the Riemann function: a) if a, b E C 1 (�), c E C(�), then the Riemann function R of the operator H in the domain � exists and it is an unique; b) the relations R,({o, 7J) = a({o , 7J)R({o , 7J) and Re ({, f}o) = b({, 7Jo )R({, 7Jo) hold on the characteristics e = {o and 7J = f}o ; c) the normalization condition R({o , f}o ; {o , 7Jo ) = 1 holds; d) the reciprocity condition R* ({, 7Ji {o , 7Jo ) = R({o, f}o ; {, 7J) is valid, i.e. the function R({, 7J j {o, 7Jo ) with respect to the last variables {o and 7Jo is the Riemann function R* of the conjugate equation (40.4) corresponding to the point {, '1 · More detailed information about the Riemann function and its tables in different special cases of equations of the form (40.1) may be found in the books by Babich, Kapilevich, Mihlin, Natanson and others [1), Kosljakov, Gliner and Smirnov [1] and in the paper by Andreev, Volkodavov and Shevchenko [1]. Other references are also cited there. The following statement is true - Vekua [3, p. 31).
Let � be a certain basic domain of {40.9). Then all solutions of � - that is, those expanded in the Taylor's series in V can be represented in the fonn
Theorem 40.1. (40. 3) regular in
-
e
j ()(t)R(t, 7Jo i {, ()dt j ()*(r)R({o , r; {, ()dr, e
u(x, y) = o:o R({o, 7Jo ; {, () +
+
eo
q
({o , 7Jo ) E ( � , i>).
o
(40.12)
Here ao is a certain constant and ()({) and ()* ( 7J) are certain functions analytic in and � respectively, and uniquely defined by means of u(x, y) provided that the relations in (4 0. 6) hold.
�
The following statement gives several similar integral representations for the solution of (40.3) by means of functions satisfying conditions simpler than those for the Riemann function.
Let -y({, 7J , 9) be a function analytic in the cylinder (V, D, D) where D is a certain basic domain of (40. 1}, and let this function satisfy {40. 1} with respect to { and '1· Let in addition
u(x, y) =
j -y({, (, 9)
L
(40.13)
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
799
along any closed contour L lying in � satisfies {40. 9} provided that the conditions in {40. 6) and {40. 9} hold. If in addition the relation�
[ a811 -r(e, Ji� [ :e 1 <e .
lim B-e
7J ,
0) +
o )] H(e . .,h<e . .,, 11>]
.,, II)
a(e, 71)-r(e,
7J ,
=
0,
{40.14)
=
o,
{40.15)
respectively hold, then the integrals of the form e
j -r(e, (, O)
u(x, y) =
{40.16)
eo
(
u(x, y) =
j -r(e, (, O)
{40.17)
(o
respectively, where eo lies on the boundary of �' are regular solutions of (40. 9} in �.
40.2 with the conditions e = X + y, e = X - y, and the conditions in {40.10) instead of those in {40.6) and {40.9) is also true with respect to {40.2). In this case the conditions for the functions -r {e,(,O ) and
Remark 40.1. Theorem
analytic may by weakened up to the existence of their continuous derivatives of the second order. Remark 40.2. The following functions
e
R(O, e, 71), 7Jo E � ; f R(t, e, 71)(t - O) a- t dt, 71o ;
71o ;
9
-r {e,
0,
0) satisfying the conditions in Theorem
40.2. If A, B and C are analytic in �, then any solution of {40.3)
are two examples of a function
7J ,
regular in � is also analytic in � with respect to x and y .
Theorem 40.3.
Re o >
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
800
40.2. The representation of solutions of generalized Helmholtz two-axially symmetric equation In this subsection we obtain different integral representations for the solutions of two mutually connected cases of and namely the generalized
(40.3}
(40.2},
Helmholtz two-axially symmetric equation
HP>. ,.u =•r
Uzz
+ Uyy +
2p
-Uz
Z
+
2p + � 2 u = -u 11 Y
0,
p, p, � - const ,
(40.18}
and the corresponding hyperbolic equation
(40.19} (40.18}
obtained from after replacing y by -iy. We also consider some special inversion relations for above equations. In the special case J.l = � = is the so-called equation of axially symmetric potential theory. This is discussed in more detail in § and § Notes to §§ and Here we only note that itself arises, with the special values z = X, y = p = m - and = n - if we need to find the monochromatic solution of the form
0 (40.18}
41.1
41.2.
Y, 2
1
Y,
(40.18) 2p 1,
41.1
43.1,
Y}e±i>.t,
u = u(X , t ) = u 1 (X, X 2 = z� + z� + . . . + z� , Y 2 = y� + y� + . . . + y� for the D 'Alambert wave equation
(40.19'} }
in the space with co-ordinates ( z 1 , z 2 , . . . , Zm , Yt , Y2 , . . . Yn and time t. The possibility of an effective construction of the solution of is enabled by two remarkable properties of the operator H;, 11 • There are "the relations of conformity"
(40.18}
(40.20} Hp>.,p (yt - 2p u) = yt - 2p H1>.-p,p u, and a compositional representation for the solution of (40.18} by means of the Erdelyi-Kober operators, defined in (18. 8 } and (37.45}, of harmonic functions even
§ 40. INTEGRAL -REPRESENTATIONS FOR THE SOLUTION
801
x and y. This representation follows from the relation J(z)/ /J J(Y )/ p J(z){ O 0) .u.A / = Hp,..x IJ I(z)/ /J J(Y)1/ p J(z){O 0) / (40.21) - 1 2, - 1 2, - 1 2, - 2 , where the indices x and y in the above operators imply a variable to which these operators are applied. According to (40.20) each solution of (40.18) corresponds to another solution of this equation obtained from by replacing I' by 1 I', or p by 1 p, and by multiplying by x1 - 2�-' ,or y1 2P respectively. It follows from (40.21) that if f is harmonic, i.e. satisfies the Laplace equation �/ = 0, then the function u = n;,"' I�{1 2 . "' I�{1 2 ,p Jiz>(o, 0) / is the solution of ( 40.18). Similar properties are also true for {40.19). Equations {40.20) is easily verified by direct evaluation. Equation {40. 2 1) can be proved as a direct corollary of Lemmas 40.1-40.2 given below. These lemmas in
.X
'
'
.X
u
-
-
'
u
-
,
characterize the application of the Erdelyi-Kober-type operators to the differential operator
{40.22) 40.3
These lemmas as well as Lemma below were proved in the papers by Lowndes and Notes to and and 5, - see also Erdelyi note
[7, 9) 40.1.
Lemma 40.1.
[8, 10)
§ 43.1,
§§ 40.2
40.3
§ 43.2 ,
Let f(x) E C2 {0, b), b > 0, and /{0) = /"{0) = 0. Then
Let a > 0, f E C2 {0, b), b > 0, x2" + 1 f(x) is integrable at zero, and x2"+ 1 /'(x) -+ 0 as x -+ 0. Then
Lemma 40.2.
and, in particular, if A = 0, then I, , a L�z)f(x) = L�1.a i, ,a f(x).
Let a > 0, f(x) E C2 (0, b), b > 0, and z" + k f(k)(z) with k = 0, 1, 2 are integrable at zero. Then
Lemma 40.3.
802
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
where la , >. is the operator obtained from the operator in (37.68) by replacing i by
J.
[9] proved that by using Lemma 40.3 we can find the full system of (40.18) in a neighbourhood of the origin in the form
Lowndes solutions of
z = r cos O,
y = r sin O,
An - const , n =
0, 1, 2, . . . ,
(40.23)
where J,(z) is the Bessel function given in and P�a , b ) (z) is the Jacobi polynomial - Erdelyi, Magnus, Oberhettinger and Tricomi Lowndes used the known result for with � = obtained by the method of division of variables in polar coordinates. In particular, we also note that in the case of the system in has the form Anr-PJn+p(�r)C�(cos O) and An r"C�(cos O), where C� (z) are Gegenbauer polynomials if I' = and I' = � = respectively. However, to find integral representations for the solutions of ( it is more suitable to use another approach based on Theorem and We denote by y, a special solution of connected with the solution 1(e, fJ , t) from Theorem by means of Using the relation in and its special cases when I'P� = we can conclude that this solution should be found in the form
{1.83),
(40.18)
0
(40.23)
f(z, t) 40.2 0,
[2, 10.8].
(40.18)
0
40.2 (40.21)
(40.18) (40.6).
0, 40.18) (40.21).
(40.24) where x = -pf ( y = -�2 p/ , p = - t) 2 + y2 , and a , b, c must be defined. Substituting into and carrying out some operations we arrive at an equation in partial derivatives of second order with respect to O(x, y) . This equation is a linear combination of two equations of the form from Erdelyi, Magnus, Oberhettinger and Tricomi with (J = 32 and the coefficients z - 2 and � 2 in two cases only connected by means of namely + {J = 1 = p; = o: c = o:{J b = a= p p b = c = -p, o:{J = a= o: + {J = 1, 1 = - p. So we conclude that O(z, y) = 32 (o:, {J; 1; x, y) where
4zt), (40.24)
1) 2)
,
-p , 1 - 2 , -p , 0,
4 (40.18)
(z
[1], (40.20): 1, p(1 - p), p(1 - p),
5.9(29)
1, 1
l z l < 1,
(40.25)
is one of the Humbert confluent hypergeometric functions with the parameters o: = I' and {J = I' · Integral representations for such a function in terms of the Legendre function P;(z) = P,(z) given in and the Bessel-Clifford function
1-
(1.79),
803
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
given in
(37.8) were obtained by Kapilevich [1 , p. 1243] in the form
82 (1!, 1 -
1
Jl ; "'( ;
x, y) =('Y - 1) f( 1 - tp - 2 P_ 11 ( 1 - 2xt) 0
X
J..,. - 2(2y'(l -
t)y)dt,
(40.25')
Re"'( > 1.
can It should be noted that if < 11 ;/; then the variable x = be on a cut - a singular line - of the Humbert function. To exclude this we multiply the special solution in with the parameters indicated in 1 ) by the piecewise constant multiplier 1 ) lsign 11 l , where 0° �1 , 11 is a constant. Then we integrate the expression obtained respect to the parameter in accordance with As a result we find the function
xt 0, 0 (40.24) ltr(t) lt l" (signxt +
-p/(4xt)
t
(40.13).
U
t + 1 ) 1sign l'l (z, y) _1 IX 1 _ , y1 _ 2p /00 r(t)((zlt l" -(signx t)2 + y2]l-P -
1
-oo
Similarly on the basis of
(40.26)
2) we obtain the second function
u(x, y) =lt lzl
_ 11
+ 1)1s•gn. l'l f00 v(t) lt l "(signxt [(x - t)2 + y2]P
-oo
(40.18)
(40.27)
which is obviously the solution of as well as the first one. We note that if and p = - 1 , . . . and p = 1, . respectively, then the functions in are not defined, and a special approach is required to constructing solutions in these cases. One may satisfy oneself by direct verification that the constructed solutions of the form satisfy the corresponding condition in Therefore on the basis of and Remark after replacing by and the interval -oo < < oo by < < < = in we obtain the function
0, -2, (40.27)
2, . .
(40.24) (40.14). (40.16) 40 . 1 y -iy t 0 x-y t x+y, t x+Oy (40.26)
(40.26)
804
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
U
1 (x,y) =I3X- p j (1T(x- ()+2)O1y)-P (X + y)P -1 (}
(40.28) (40.19) when p 0. y iy in (40.28) we arrive at the relation u(x, y) = - i(2i) 1-2P /3x- P J T(u) u1A (e - e - 1 ) 2P- 1
which satisfy the hyperbolic equation Making the inverse replacement by
>
L
u = x + iycostp, which after the change (}
= y cos
tp
(40.29)
can be also rewritten in the form
u(x, y) =13x - P yl y i -2P j T(x + iO)(x + i0)1A (y2 - 02 )P- 1 y
-y
(40.26)-(40.30)
In order that the above integrals in may exist , the functions and 11 must satisfy certain conditions given below and in the theorems of the next subsection. It should be noted that the integral operators in these expressions are compositions of the form I�z{1 2 , p iC:{1 2 .p Jfe\O , O f which will be observed below while investigating their boundary values. From the above arguments we arrive at the following statement.
T
) - (40.2 1) -
p
Let p 0 and � 0 and let T(z) be an analytic function of z = x + iy in the neighbourhood of z = 0 and let it be an even function of x and y, i.e. T(z) = T(z), T( -z) = T(z), ReT is an even function of x and y, ImT is an odd function of x and y, and ImT = 0 when xy = 0. Then (40.29) and (40.30) give all the classic solutions u E C2 of (40.18) in the neighbourhood of (0, 0) even of x and y for which u(x, ±0) is bounded. When 0 < p < 1/2 and 0 < < 1/2 respectively, then the relations u,(x, 0) = 0, or more exactly u,(x, y) = O(y), and z ( , y) = 0, Theorem 40.4.
>
p
U
O
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
or more exactly ux (x, y) = O(x), respectively, are true. If in addition
805 ·
(40.31) [ (p 1/2)] - 1 = r(p + 1/2)[y'?rr(p)] - 1 , then the functions given in (40.28)-{40.30) satisfy the Dirichlet condition 13 = B ,
u(x, 0) = r(x)
(40.32) and u(x, y) = r(x) + O(y2 ) as y -... 0. If r(z) has singularities at the points z = z1 and z = -z1 , then ±zt, ±z1 are singular points for the solution u(x, y) given in (40.29), (40.30) too. proof of the basic conclusions of this theorem can be found in the book by Gilbert The condition in can be verified directly by the passage to the in limit as and Theorem shows that the operator in maps analytic functions into solutions of In the general case its inverse operator was constructed as a cumbersome series by Gilbert p. As far as the special case JJ = A = is concerned, such an operator can be found explicitly in a relatively simple way Gilbert Next we prove that other special cases of the inverse operator - when can be constructed in closed form. JJ A = Case A. Let JJ = > 0. We assume that = Then with the condition in has the form
The
[2]. y -... 0 (40.28) 40.4 (40.18).
(40.32) (40.29).
(40.29)
[2, 205-206].
0
(1]. 0-
0, p r(z) r(z). (40.30) (40.31) 1 - 2p 11 u(., , y) = B(p, 1/2) 1 T(., + iy)(y2 - 0 2 y>- 1 Jp - t (>..jy• - o•)dO -y
1r- 1 /2 f(p + 1/2)Ji'> ( -1/2,p)Rer(x + iy) = 1r- 1 / 2 r(p + 1/2)y 1 - 2p ( qt /)( w), (40.33) /(t) = Rer(x + h/i)t - 1 1 2 , w = �, where J, (z) is the Bessel-clifford function given in (37.8), Ji' > (fJ, a) and c�:: are the operators in (37.45) and (37.4) applied with respect to the second variable. Using the inversion relations of these operators given in (37.57) and (37.37) we =
806
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
obtain the following representation for the operator inverse to ( 40.33): Re
T(z + iy) = J;rr- 1 (p + 1/2)JJf>(p- 1/2 -p) ,
X
_
lm -p - t ( � Jy2 t 2 ) -
( 1 dtd ) m (t2P- 1 u(z, t))tdt. t
( 40.34)
Now the corresponding statement follows from Theorem 37.2.
Let g(y) = u(z, V"Y)yP- 1 12 E ACm ([O, b]), b < oo for all z and g(O) = g'(O) = = g(m - l ) (O) = 0 and 0 < p < m. Then each solution u(z, y) of (40. 18) with Jl = 0 corresponds by {40.94) to a certain hannonic function Re T (z + iy). If u(z, y) is an even function of z and y, then ReT ( z + i y) is also an even function of z and y and the relation T(i) = T( z) holds for the corresponding analytic function T( z) . Case B. Let � = 0, p > O.We assume that an even function u is given on the boundary r of the unit disk l z l � 1:
Theorem 40.5.
· · ·
u(z, y) lr = /(
r = {z = cos so ,
!( so) = !(- so ) = !(1r - so) , By substituting = z + i y = e i �P, z + i (J = t in at the following integral equation:
z
- �-� , y , 2 - 2p
y = sin so , I�PI � ?r }, It'
:F ±1rj2.
(40.35)
(40.30) and using (40.35) we arrive
z
� ayB(p, 1/2) J T(t)t�-' [(z - t)(z - t)]P - l ( ( z - t)(t - i) ) dt = /{ so) , x 2 F1 JJ, 1 - JJ;p ; z
2t(z + z)
( 40.36)
i
Integration in ( 40.36) is taken along the vertical interval joining the points and This interval can be replaced by a segment of a circle r of the form t = ei a , l a l � so , since T(t) is analytic in the unit disk and other integrands are analytic everywhere except the points t = 0, oo lying on the boundary or beyond the corresponding segment of a circle. According to the Cauchy theorem, the value
z.
z, i,
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
807
of the integral is unchanged for such a replacement of the contour of integration. In order to choose a single-valued branch of (40.36) with respect to the variable z we make a cut along the half-line y = 0, z =:; 0 and set I 'P I < 1r. Then [(z - t)(i - t)]P - 1 = [2t( cos a - cos
'P T(eia)eia(p+p) F (p, 1 - p,. p,. -1 (1 - -cos a )) da J- tp (cos a - cos
f( cos a) l sin a l = Re (T(t)tP+P ) , 71 = cos a, z = cos cp, d = 1, 0 =5
a<
I 2,
7r
0 =5
(40.38)
-
g( z) - f(p2 +P.J-i ( 1 - z2 )p - 1/2 z P f( arccos z ) . 1/2) _
Using Theorem 35.2 concerning the inversion of this equation we can formulate the following statement.
Let p > 0, p < 1/2, let f( 0, and let (If_ h)(z) be the fractional integral defined in (2. 18) . Then the solution of (40.36) on the circle l z l = 1, zy =/; 0, in the space of even functions T(f) = T( -f) = T(t) is reduced to the solution of the following Hilbert boundary value problem - Gahov [1] . The Hilbert problem 40.1 . I t is required t o find a function T(z) = e(z, y) + i77(z, y) analytic in a quarter of the disk lzl < 1, Z > 0, y > 0, continuous Theorem 40.6.
right up to its boundary I and such that limit values of the real and imaginary parts of T( z) satisfy on l the linear relation
a(t)e(t) + b(t)71(t) = c(t), t E I,
(40.39)
where a = c = 0, b(t) = 1 on the axes z = 0 and y = 0 and a(t) = cos a(p + p ) , b(t) = - sin a(p + p) , c(t) = f(cos a) sin a, 0 � a < 1r /2, on a quarter of the circle.
808
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
(40.35)
Under the circumstances f is expressed via the given function f(cp) in by the relations ( The solution should be found in the space of functions T( z) which are bounded near z = and have an integrable singularity at the point z = i.
(35.34), 40.38). 1/2,
1
1
40.6
1/2
Remark 40.3. If p > and the analogue of Theorem then - p < obtained by using the property in ( is true. Then t 1 - JJ must be substituted instead of tl' and p must be replaced by - p in
40.20) 1
(35.4). Proof of Theorem 40.6. After inversion of (35.18) we arrive at the problem of finding the limit value T(t) of the function T(z) which is even and analytic in the right half-disk, if its boundary value in (40.38) of the form Re [T(t)tP + JJ] = f(cos a)j sin al is known. As was noted in Theorem 40.4 the function T(z) must have the property lmT(z) = 0 on the coordinate axes, i.e. when zy = 0. These three conditions may . by joined to the one given in (40.39) with the indicated coefficients. We evaluate the index x and the number of solutions of the Hilbert problem 40.1. Following the book by Gahov [1, § 30] we rewrite the relation + Re [T(t)tP JJ] = c(t) in the form
Hence we deduce the boundary condition of the corresponding Riemann problem (
1
40.40)
Here T+ (t) is the limit value on the circle lzl = of a function T(z) analytic in a < 1r and T - (t) is the limit value on lzl = of lz l < as z -+ t = e ior , a certain function T (z), lzl > 1 , which is not necessarily analytic. By a similar method the relation T+ = T- reflecting the analyticity of T on the axes is reduced. Since T(t) is even, then the homogeneous condition T+ (t) = -t - 2P - 2JJT - (t) can be continued to the whole circle except the singular point t = lying on the 2 2 cut. After a circuit around arg t P JJ has a jump which equal to - 7r (p + p) - Gahov [1], So the index x is equal to [-2p - p] + in the space of solutions integrable at the point t = and the value lxl characterizes the number of solutions of the homogeneous Hilbert problem when x � or the number of solvability conditions of the corresponding inhomogeneous problem when x < The solution of this problem itself can be constructed from the expressions given in the book by Gahov [1 , Substituting the above function T(z) into ( with � = we obtain the solution u ( z, y) of the following Dirichlet problem. The Dirichlet problem It is required to find a real-valued function everywhere except u ( z, y) even in z and y and continuous in the disk l zl perhaps the line z = This function must satisfy with � =
1
0�
§ 43.2].
/2,
-1
-1
1
-1,
0,
§ 46]. 40.2.
0
0.
1
(40.18)
�1
4
0. 40.30)
0, 0
1/2,
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
;/; 0
809
40.35) on the circle 0 and uy ( z, 0) = 0;
in the disk when zy and with initial data indicated in ( l z l = and it must be continuous and bounded on the line y = this function is bounded on the axis z = if only p <
1
0
1/2.
40.3. Boundary value problems for the generalized Helmholtz two-axially symmetric equation
40.26) - 40.28)
In this subsection on the basis of ( we construct solutions of ( the Dirichlet and Neumann problems in the half-plane for ( and of the half-homogeneous Cauchy problem in the characteristic triangle for We also indicate the character of the behavior of the solutions near singular lines, and the conditions for functions which guarantee the existence of integrals. At first we clarify the behavior of as z --+ It follows from that the representations
(40.26)
00
40.18)
(40.19).
0.
(40.25)
y'
.:.....2 (a, /3; "'{; z, y) = E ( ) 1 1 2 Ft (a, /3; "'{ + l; z) • 1 :0 "'f I -
(
40.41)
are true. Substituting an expansion of the Gauss hypergeometric function of the form into we obtain the following leading term of as z --+ oo , I arg( -z) l < 1r :
(10.13)
(40.4 1)
(40.41)
[;: �:::: :] o F1 (7 - a ; y)(-z)- " + f [P: �:::: � ] oF1 (7 - ,8; y) { -z) - 17 , a '1,8;
22 (a, ,8; 7; z, y) - r
(
40.42)
B(a, a ; "'{; z, y) - r a, "Y- a (-z) - ln( z , f3 = a. "'{
[
]
a
-
)
2.10(7) (40.42) (40.26) 1/2,
The latter relation follows from in Erdelyi, Magnus, Oberhettinger and Tricomi Applying to we find that if J' < then the value u(O, y) exists; if p > then the limit lim lzl 2�' - 1 u(z , y) exists; and if J' = s--o then the limit lim ln - 1 l z l u(z, y) exists. These properties show that the solution z-o given in and a has a power singularity of order O (z 1 - 2�' when p > logarithmic singularity of order O (ln lzl ) when p = on the line z = Now we use the as�ptotic expansion 0F1 (v ; -y) = O y< 1 2 ")/ 4 cos( 2 ..JY +
[1].
(40.26)
1/2,
1/2
)
[
-
1/2,
1/2 0.
810
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
1r(1 - 2v)/4)] as y -+ oo - Marichev [10, (6.21)]. Then from (40.42) we find that a f; {J,
(40.43)
z, y, -+ oo.
(40.26) 0 (40.26)
Therefore the integrand in has the following estimate as t -+ oo, namely Hence the O T t lti�' +P - 31 2 if � f; and 0[T(t)lti 2P - 2 (1 + 1tl 211 - 1 )] if � = improper integral in is convergent if the functions T(t)lti P +P - 31 2 and T t 1t 1 2P - 2 + 1tl 211 - 1 ) , are integrable at infinity. The above investigations lead to the following two statements.
] [ () ( ) (1
0.
Let T(t) be a continuous function bounded on the axis ( -oo, oo) and satisfying the following conditions at infinity: I T(t)l < Clt 1 1 1 2- p -p - E when J' > 0, � f; 0 or I T(t)l < C lt1 2- 2�' - 2P -£ when I' � 1/2, � = 0 and I T(t)l < Clti 1 - 2P - E when problem 0 < I' < 1/2, � = 0 where C is a constant and € > 0. Then the Dirichlet in the half-plane y > 0 which consists of finding the solution u E C2 (y > 0, z f; 0) of (40.18) with the initial data in (40.32) at the points z, -oo < z < oo, z f; 0, is solvable for p < 1/2, p f; 0, -1, -2, . . . and its solution is expressed by (40.26) with the coefficient
Theorem 40.7.
= [B(1/2, 1/2 - P)t 1 = f (1 - p)[y'i"f (1/2 -p)] - 1 .
(40.44) The behavior of the derivative u11 as y -+ 0 is characterized in Table 41.1 when q = 0. The behavior of u and as z 0 is also given in Table 41.1 but with q = 0 and replacing p by I' and y by z. The solution given in (40.26) has the It
Uz
-+
following behavior as y -+ oo:
Let v(t) be a continuous function on the axis ( -oo , oo) and let v(t) satisfy the following conditions at infinity: lv(t)l < CltiP - P - 1 1 2 -£ when � f; 0 and lv(t) l < Clti 2P - 1 -£ (1 + 1 tl 1 - 211) when � = 0 where C is a constant and € > 0. Then the weighted Neumann problem in the half-plane y > 0 which consists of finding the solution u E C2 (y > 0, z f; 0) of (40.18) with initial data Theorem 40.8.
(40.45)
§ 40. INTEGRAL REPRESENTATIONS FOR THE SOLUTION
811
is solvable for p > -1/2, p f; 0, 1, 2, . . . , and its solution is expressed by the relation ( 40.46)
Here u2 (x, y) has the form (40.27) with the coefficient 12 =
_
B(p, 1/2)
( 40.47)
21r
and the functions
are singular solutions of (40.18) satisfying the homogeneous condition of the form (40.45). The behavior of the solution u and its derivatives as x --. 0 and y --. 0 are characterized in Table 41.1 in the same way as was indicated in Theorem 40.7, and the solution's behavior as y --. oo is obtained from Theorem 40.7 on the basis of (40.20). Remark 40.4.
(40.46) shows that the solutions of problems with data on singular
lines can not be unique without additional restrictions on the space. In the case = 1/2 one more solution involving the function ln(p/y) in the kernel can be added to ( 40.46). For more details we refer to § 41.4.
p
Remark 40.5. Expanding the function in (40.27) with the constant ( 40.47) in the neighbourhood of the point p: u 2 = p-1 u 20 + u2 + O (p) we write the second term of this expansion
u2
x -JJ (x, y) = l i 21r
X
j (t) lt iJJ (sign xt 00
-
v
oo
l: (-.L_) 4xt
(
_
�2
_!!_
4
)
'
[
+
1) 1sign JJ I
� ��l: f: (1'(k) �:+( 1l).k.l.
l: ,l=O l:+ l 1 ln p + c - E -; i=l 1
] dt,
(40.48) where C is the Euler-Mascheroni constant. This function is the solution of the problem given in Theorem 40.8 for the singular case p = 0. If � = I' = 0 and the
812
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
condition of the form
I v(t)dt 00
=
0
(40.49)
-oo
necessary for solving the Neumann problem in the half-plane holds, then ( 40.48) leads to the known Dini relation for the solution of the Neumann problem in the half-plane y > 0 for the Laplace equation
u(z, y) = 2�
I v(t) ln[(z - t)2 00
+ y2 ]dt.
(40.50)
-oo
If p ;:::: 1/2,
then (40.27) represents also the solutions of the following weighted Dirichlet problems Remark 40.6.
1 v(z), p > 1/2, p f:. 1,2, 3, . . . , 1 - 2p lim ln - 1 y u(z, y) = v(z), p = 1/2, Y -+O
limo y2P- 1 u(z, y) =
y-+
and
-
(40.51) ( 40.52)
(40.48) is the solution of the problem in (40.52) also. In conclusion we note that on the basis (40.28) we can similarly obtain the
solution of the following half-homogeneous Cauchy problem for the hyperbolic equation (40.19)
u (z, 0) = r(z),
lim
y-O+
y2P u11 (z, y) = 0, 0 < z < 1,
(40.53)
and make the corresponding statement about its solvability.
If T E C2 ([0, 1]), 0 < p < 1/2, and 13 be given in (40.31), then the Cauchy problem in (40.53) in the triangle D = {0 < z - y < z + y < 1} has the classic solution u E C2 (D) of the fonn (40.28).
Theorem 40.9.
§ 41 . The Euler-Poisson-Darboux Equation The present section deals with integral representation for the solutions of the Euler-Poisson-Darboux equation in the elliptic and hyperbolic cases, and their applications to the construction of the solution of the Dirichlet, Neumann and Cauchy boundary value problems. As in § 40 we show by using the Erdelyi-Kober
§ 41 . THE EULER-POISSON-DARBOUX EQUATION
813
operators defined in (18.8) that solutions of the simplest equations with constant coefficients are reduced to solving the Euler-Poisson-Darboux equation.
41 . 1 . Representations for solutions of the Euler-Poisson-Darboux equation
The generalized Euler-Poisson-Darboux equation E(/3, {3* ) u = U('l - f3*ue( _- {3u11 ,
=
0, {3* , {3 -
(41.1)
const ,
is of especial importance while solving problems of axially symmetric potential theory. This equation corresponds to (40.1) with a(e, = p< • /(e - fl), b(e , = {3/(e - and c(e, = 0. After the substitutions given in (40.6) it has the form
'7)
'7)
'7)
-
'7)
>
2p = 0, 2q + -u, E+ U = Uzozo + Uyy + -Uzy y
(41.2)
f3 = p + iq, {3* = p - iq.
(41.3)
We note that the Laplace equation in Ira
(41.3') the solution of which is sought in the form
with the axis of symmetry r = 0, is reduced to (41.2) with q = 0, 2p = n 2 and y = r. Such a solution is called a 2p + 2-dimensional axially symmetric potential and satisfies the equation -
(41.3") Since the operator in (41.3") is even with respect to r, then each solution of (41.3") is an even function of r, and if p = 0 then the 2-dimensional axially symmetric potential is a harmonic function. A further two cases will be different. We shall assume f3 and /3* to be real or complex and conjugate numbers with �{3 = "&{3* = p while investigating (41.1) or ( 41 .2), respectively. By direct evaluation we prove the following statements.
814
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
A function u is the solution of the equation E({J, {J*)u = 0 if and only if the function
Lemma 41.1.
(41.4) satisfies the equation E(1 - p•, 1 - {J)v = 0. Lemma 41.2.
If u 1 (e, q) satisfies (41.1), then the function (41.5)
where a, b, c and d are arbitrary constants such that ad - be "# 0 is also a solution of (41.1). Let {3 > 0, {3* > 0, and let r(O) be an arbitrary function continuously differentiable twice. Then the integral
Lemma 41.3.
1
J1 (fJ, {J*) = J r[e + (71 - e)t]tP •- 1 (1 - t)P - 1 dt
(41.6)
0
satisfies (41.1) and it also satisfies- (41.2) in the case p > 0 provided that the conditions in (40.6) hold. We note that Lemma 41.3 follows from Theorem 40.2 if we set i (e, f1 , 0) = C(e - o) - P ('1 - 0) -P • , C -
const .
(41.7)
Then the integral of the form (40.17), i.e. ,
J2 ({3, {J*) = j i (e, 71, O)v(O)dO,
'1o
= e,
(41.8)
'lo
where v(O) is an arbitrary function continuously differentiable twice and {J < 1, {3* < 1, will satisfy (41.1) and by using Lemma 41.1 (41.8) transforms into (41.6). It is obvious that (41.6) and (41.8) are linearly independent solutions if {J "# 1/2, p• "# 1/2, p "# 1/2. Theorem 41.1.
If 0 < {J < 1, 0 < {3* < 1 and either {J "# 1/2 or {J* 'f; 1/2, then
§ 41 . THE EULER-POISSON-DARBOUX EQUATION
815
the general solution of (41.1) is given by 1
u(e, '7) = j r(8)t.B * - 1 (1 - t).B - 1 dt 0
1 * 1 + (TJ - e) .B .B j v(8)t - .B(1 - t) - .8 * dt 0
= J1 ({J, {J*) + J2 (f3, /3* ) = J (/3, /3*), 8 = e + (TJ - e)t,
(41.9)
where r(8) and v(8) are arbitrary functions continuously differentiable twice. If 0 < p < 1, p #; 1/2 then the general solution of (41.2) is given by (41.9) where r(8) and v( 8) are arbitrary continuous functions. In the case {J = {J* = p = 1/2 such a solution has the fonn 1
u(e, '7) = j r(8)( t - t 2 ) - 1 12 dt 0
1
+ j v(8)(t - t 2 ) - 1 1 2 ln [(t - t 2 )(TJ - e)]dt.
( 41.10)
0
Remark 41.1. In particular, if r(8) and v(8) are analytic functions in a certain domain 1J, i.e. for (e, TJ) E 1J, then (41.9) and (41.10) present all solutions of (41.1) analytic in 1J. We characterize briefly the process for finding the general solution of ( 41.1) for other values of the parameters f3 and {J* . Let -k < /3* < 1 - k , - 1 < {3 < 1 - l, k , l = 1, 2, 3, . . . , {J + {J* #; -1, -2, -3, . . . . Then according to the inequalities 0 < 1 - k - {J* < 1 and 0 < 1 - l - f3 < 1 the function J1 ( 1 - k - {J* , 1 - l - /3) given in (41.6) is the solution of the equation E(1 - k - {J* , 1 - 1 - {J)u = 0. We evaluate the derivative
(41.11) provided that r( 8) is sufficiently smooth. It is obvious that the integral on the right-hand side of (41.11) satisfies the equation E(1 - {J* , 1 - f3)u = 0. Therefore
816
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
multiplying (41 .11) by ( 7J - e) t - P - P* and using Lemma 41.1 we obtain the function
( 7J - e ) t - p - p ·
J T(k+l> (e)t-P (1 - t)-p· dt, t
e = e + (7J - e)t,
0
which is the solution of (41 .1). After similar transforms with the second solution J2 we arrive at the general solution of (41 .1) in the form u (e, 7J)
ak+l - (7J - e ) t - {j - {j * k J ( 1 - k - f3* , 1 - 1 - f3) , ae 871,
where J({3, {3*) is given in (41.9). Using the inequalities {3* + k the general solution
( 41.12)
> 0 and f3 + I > 0 we can obtain another form of
( 41 .13) If, for example, {3 = {3* = 1/2 - k, k = 0, 1, 2, . . . , then Jt and J2 will become linear dependent solutions and it should be substitute the right-hand side in ( 41 .10) instead of J into (41.12) and (41 .13). If one of {3 and {3* is an integer, then (41.1) can be integrated in quadratures by the cascade Laplace method - Babich, Kapilevich, Mihlin, N atanson and others [1 ,p.43]. It should be noted that we can arrive at the representation in (41.9) by using the Riemann method and the Riemann function of (41.1):
RE (e, 7]; eo , 7Jo) = ( 7J - e)P+P * ( 7J - eo) - {j ( 7]o - e) - p • 2 Ft ( /3 , {3* ; 1; u ) , = (e - eo )( 7J - 7Jo ) , ( 41 . 14) (e - 7]o} (7] - eo ) 0"
where 2 Ft is the Gauss hypergeometric function given in ( 1.73) - Babich, Kapilevich, Mihlin, Natanson and others (1 , p. 48] . It is not difficult to transfer the above results to the case of the elliptic equation (41 .2). Thus from (41 .7), (41.3), (41 .6) with Im = 0 we obtain the special solution of (41.2):
t
1 (e, 7J, t) =c t (e - t) - P (t - () - P = ct exp[-(p+iq) ln(e -t)-(p- iq) ln(t-()]
§ 41 . THE EULER-POISSON-DARBOUX EQUATION
817
=ct exp[-2p In 1e -tl- (p+iq)i arg(e -t)-(p-iq)i arg(t- ()] =ctle -t i -2P exp [-(p+iq)it/1 -(p-iq)i(r - t/1 )]
( 41 .15)
where
le - tl = J(z - t)2 + y2 , .P = arg(e - t) = arccos[(z - t )/rt ] , Yl � 0.
(41. 16)
rl =
Setting
c2
= 11 T(t) or c2 = l2v(t) in (41.15),
( 41. 17)
and integrating along the axis
-oo < t < oo and using Lemma 41.1 with respect to (41.2) (see also (40.20)), we arrive at the following two solutions of (41.2):
u(z, y) = lt y1 - 2p
00 f u(z, y) = /2 - oo
2t.P j00 [(z -T(t)e 2 t) + y2) 1 -P dt,
( 41 . 18)
- oo
v(t)e2f.P dt. [(z - t) 2 + y2]P
(41 . 19)
These representations are analogues of the solutions given in (40.26) and ( 40.27) 0 for I' .\ 0. ( 41.18) and ( 41.19) for (40.18) coinciding with ( 41.12) when are also analogues of J1 (/3, /3*) and J2 (/3, /3*). The analogy of the solution given in (41. 10) and containing the logarithmic function is the solution of (41 .2) with 1/2 of the form:
q=
p=
00 1 u(z, y) = /3 - oo
= =
T(t)e 2t.P dt, C +C y'(z t) 2 + y2 1 2 ln (z - ty) 2 + y2 _
[
]
( 41 .20)
where C1 and C2 are arbitrary constants. Carrying out the changes given in ( 40.6) and the substitution ( 1 - 2t)y and taking ( 41 .3) into account we obtain the following analogue of ( 40.30) :
= (}
( 41.21)
818
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
In particular, if q = and case of with � =
0 T(z) 0:
(40.33),
u(z, y) 2 =
13 y1 - 2P
( ) then from
= T z ,
(41.2 1) we obtain the special
y
j ReT(x + i8)(y2 - fP )P- 1 d8 0
(41.22) where I��l is the Erdelyi-Kober operator defined in and applied with respect to y. The following analogue of Theorem is valid for and
40.4
(18.8)
(41.2 1) (41.22).
Let p > 0 and let T( z ) be analytic function in the neighbourhood of z = 0. Then (41.2 1) presents all classical solutions u E C2 of (41. 2 ) in the neighbourhood of (0, 0). If in addition T(z) is an even function of y and satisfies the condition T(i) = T(z ) , then (41.22) presents all classical solutions u E C2 of (41.2) with q = 0 even with respect to y in the neighbourhood of (0, 0). Under the circumstances the value u(z, ±0) is bounded and if 0 < p < 1/2, then u11(z, 0) = 0 and u11(x, y) = O(y) as y --. 0. If in addition (40.3 1) holds, then the functions u( z, y) given in (41.21) and (41.22) satisfy (40.32). The relation inverse to (41.22) is obtained from (40.34) when � = 0, and hence lm -p - t (�Jy2 - t 2 ) = 1, provided that the conditions in Theorem 40. 5 hold.
Theorem 41.2.
In conclusion of this subsection we note that on the basis of the relations in
17.2 , note 16.2 we can deduce the connection between the asymptotics of T( ) u(z, y) as y --. +oo. Indeed, if
§ and
z
00
+ iy) """ k=O (41.23) L d�; (z)y- 21: as y --. +oo, then from (41.22) and (17.17) we obtain the following asymptotic expansion of the solution u( z, y) of the boundary value problem defined in (40.32) and (40.31) for (41.2) with q 0, namely 1} + i2yk+);2k+ u(z, Y) "'laf( > [ � ( -1) kVJtk!(rY)({g(z t y k) p k=O (41.24) r(1/2 k) + k=O L d�: ( z) r (1/2 + p - k) y21: ] as Y ___. +oo, where VJt(Y) {g(z + iy) ; 2k - 1} is the Mellin transform defined in (1.112) of the function g( z + iy) with respect to y at the point 2k + 1. ReT(z
=
P
oo
L
§ 41 . THE EULER-POISSON-DARBOUX EQUATION
819
41 .2. Classical and generalized solutions of the Cauchy problem After replacing y by iy the elliptic equation equation of the form E
(41.2) is transformed into a hyperbolic
- u = u�� - u, - -2yq u� - -2py u, = 0.
(41.25)
The last equation is characterized by the Cauchy problem considered in the characteristic triangle n- = { 0 < -y < :r: < + y} with initial conditions
1
u(z, 0) = r(z), 0 :5 :r: :5 1,
lim
y- - o
( -y) 2P u,(z, y) = v(:r:), 0 < :r: < 1.
(41.26)
The solution of this problem (with exactness to coefficients ) can be obtained from if we set and other relations for the general solution in Subsection
(41 .9)
41.1
e = x + y, TJ = :r: - y, P* = v + q, f3 = p - q
(41.27)
in the above expression. Thus the following statement is true.
If T E C2 ([0, 1]), v E C2 ((0, 1)), 0 < {3* < 1, 0 < {3 < 1 and 1, then the Cauchy problem giv2en in (41.26) for (41.25) in the domain is properly set and its solution u E C (D -) is given by
Theorem 41.3. {3 + {3* < n-
1 u(z, y) = - A 1 (-y) 1- 2P j v(:r: + y( 1 - 2t))(1 - t) -,8• t - P dt 0
1
+ B (P� fl") J r(:r + y(l - 2t) )(l - t)P - t tp •- t dt,
(41.28)
0
where A1 = [( 1 - 2p)B(1 - {3* , 1 - {3)] -1 • Corollary. If the condition
:5 1/2, (41.29) holds on the characteristic y = -:r:, then the following representation of v(:r:) via u(z, -:r:) = fb(z), 0 :5
:r:
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
820
T( z) and 1/J( z) is valid:
1 - 2 - {3) {J .!!._ p • (�) z dz 10 + 1/J 2 v(z) _- _ 2 r(1Pf-(12p) 1 - 2Pf(2p)f(1 - {3) d 2p + 2 f(1 - 2p)f({3* ) dz 10+ T(z),
( 41.30)
where IC+ is the fractional integration operator defined in (2.17). Proof. Substituting y =
'7
and
2z by z we obtain
-z into (41.28) and using (41.29) after replacing 2zt by
1/J (i) = - A t 22P - 1 f(1 - P* )I�f. p• z - P v(z) +
Applying the operator
Using now
f(2p) zt - 2p zP zP•- t T(z) . o+
r(p•)
IC�- t to this relation we find
(10.12) we evaluate the composition in the last term:
- P c 12p- t Z 1- 2p 1P z - P )Z 2p- t T(Z ) p· -t Z 1- 2p 1O+ fJ zp •- t T(Z) - 1O+ lO+ O+ O+ _ 1 - P 1P - P 12p- t t- 2p 2p- t - O+ c O+ z O+ Z ) Z T(Z) _
which yields
(41.30). •
Remark 41.2. The relation m ( 41.30) is also true provided the conditions 0 < {3 + {3* < 1, {3* > 0, {3 < 1 less restrictive than those in Theorem 41.3 hold.
(41.30) is widely used while investigating equations of mixed type - see, for example, the books by Tricomi [1], Bitsadze [1-3] and Smirnov [2, 7] . However, (41.30) is usually proved by more a complicated method which does not use (10.12).
Remark 41.3.
§ 41 . THE EULER-POISSON-DARBOUX EQUATION
821
In many cases, especially while solving boundary value problems for equations of mixed type, the conditions r, v, u E C2 from Theorem 41.3 lead to restrictions on the given functions, which are verified with difficulty. This problem is solved by considering formal solutions of the type ( 41.28) which are not sufficiently smooth and can be approximate by solutions in the space C2 • Now we consider a class R 1 of such a generalized solutions of the Cauchy problem introduced by K.l. Babenko
[1, 2].
The function in (41.28) with {3 = {3* = p is called a generalized solution of (41.25) with q = 0 in a class Rt in the domain n- = {0 < -y < X < 1 + y } if 0 < p < 1 and Definition 41.1.
( 41.31) where H >. ([O, 1)) is the space of Holderian functions defined in §
1.1.
The following statement is true.
Let generalized solution given in (41.28) with q = 0 of the Cauchy problem defined in (41.25), (41.26) be in the class Rt . Then u� , u11 E C(D-), the function u11 satisfies the second condition in (41.26) and there exists a sequence {un }� 1 , un E C2 (D- ), of classical solutions of (41.25) such that n-oo lim Un = u in any closed triangle D; = {e < -y < x < 1 + y}. Theorem 41.4.
Proof. Taking the coordinates (41.27) in (41.28) and using Lemma 13.2 according (41.31) we arrive at the representations
to
I (J
r (O) = r(O) + (0 - s) -p+e
v( O)
(J
=
v(O) + I (0 - s)P- l +e .p(s)ds,
(41.32)
0
where e > 0 is sufficiently small and
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
822
(41.33)
= + e = (8 - q)t + '1 (1.73). u (e, 71) = A l 22P - 1 B(1 - p, 1 - p)( q - e) 1 - 2P v(O) + r(O)
Using now the substitutions (} (71 - e)t inner integrals by Then we have
or (}
we evaluate the
-
- 22p - l A 1 B(1 - p,p + t:)(q - e) -P
j t/>(s)(q - s)' 2 F1 (p, 1 -p; l + e; : = ; ) ds 1 -p +t:) ('1 - e) -P + B(p,B(p,p) j �(s)(q - s)' 2 F1 (p, 1 -p; l + e; : = ;) d• ,
x
e
,
x
e
+ j �(s)(q - s)-•+• 2 F1 (p,p- e; 2p; : = : ) ds e
0
X
2 Fl
1( -p, 1 -p - 2 - 2p; ee -- '18 ) d8 . cj
(41.34)
All the above integrals are proper integrals and have continuous derivatives with respect to e and '1 in the domain V. The derivative of the last integral in ( with respect to e also exists since by (e - 8)P- l +�: 2 F1 as except the first and the last ones have 8 -+ e . Derivatives of all terms in order q - e) -P ] as '1 -+ e. Applying the operator ( q - e) 2P , to
0[(
(10.13) (41.34)
41.34) ( i=;) = 0(1) ( : - � ) (41.34)
§ 41. THE EULER-POISSON-DARBOUX EQUATION
and passing to the limit as
,.,
__...
823
e we find
x
[v(O) + j ,P(s)(e - s)"- 1 +c ds] {
0
and this passage is carried out uniformly for 0 � e � (} < 1. Hence after the substitutions given in (41.27) we obtain the second relation in (41.26). Since cp, 1/J E C([O, 1)), then by Weierstrass 's theorem there exist sequences of functions {cpn (s)}� 1 , { 1/Jn (s)}�= l E C2 ([0, 1)) tending to cp and 1/J respectively, uniformly on any interval [O, Oo], Oo < 1. By (41.32) they correspond to the functions Tn , Vn , n = 1, 2, . . . , which belong to C2 ([0, 1)) also. But then the corresponding solutions in (41.28) denoted by Un belong to C2 (D - ) and Un __... u too. •
41 .3. The half-homogeneous Cauchy problem in multidimensional half-space We consider the Cauchy problem in the half-space (x 1 , . . . , Xn ) E Rn , y > 0} for the hyperbolic equation
Rf.+ 1
=
{(x, y) x
=
(41.35) with the half-homogeneous initial conditions
u(x, 0) = T(x), uy (x, 0) = 0,
(41.36)
where T(x) E C2 (Rn ) is the given function. It is known, (Courant and Hilbert [1, p. 466]) that if 2p = n - 1, then the solution of the problem given in (41.35) and (41.36) is unique, and it can be represented as the spherical mean Mn (x, y; T) of the function T in the space Rn by
u(z, y) = Mn (z, y; T) = I S l l J T(:t + yt)dcr. n - t s..
(41.37)
-1
Here t = (t t , . . . , t n ) lies on the surface of the unit sphere Bn - 1 i.e. I t I = 1, du is an element of the unit sphere surface area and I Sn - 1 1 = 21rn/2 /f(n/2) is its surface
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
824
area. Its value is evaluated by
(25.9) when f = 1.
·
In particular, the relations
( 41.38) ( 41.39) follow from ( 41.37). We apply the Erdelyi-Kober operator I, , o o defined in (18.8) and denoted by I��l below, with respect to y to the solution in (41.37). Then using Lemma 40.2 with A = 0 we select the parameters of I��l so that the operator in ( 41.35) with 2p = n - 1 will be transformed into the operator in ( 41.35) with arbitrary 2p. In this way we construct the function
+ 1/2) I u(z, y) = r(pr(n/2) n / 2- t,p +( t - n)/2 Mn (z, y ; r) .
(41.40)
According to Lemma 40.2 this function is the solution of (41.35) and the constant coefficient r�1!H)) is chosen so that the conditions in ( 41.36) hold in view of
(41.38).
We note that if we use the definition of the Erdelyi-Kober operator I, ,a for considered in subsection 18.1, then (41.40) remains the solution of the Cauchy problem given in (41.35) and (41.36) in the case p < (n - 1)/2. However, Bresters (2) proved that the solution given in (41.40) is not a unique in the case a
<0
p < o.
By using (23.1)
(41.40) can be represented in the form
+ 1/2) u(z, y) = r(pr(n/2)
I
"Y+ioo
")' - ioo
r((n - s)/2) rot(y ) {M (z, y r) s} y ds, n ; ; r((1 - s)/2 + p) -·
0 < 1 < 1,
( 41.41)
where rot 0 and r (t) has the simplest asymptotic expansion of the form 00
r(t)
�
L akt - k , t -+ oo. k:O
( 41.42)
§ 41 . THE EULER-POISSON-DARBOUX EQUATION
825
Then, as was done by Berger and Handelsman [1], 00
M1 (z, y; r)
�
L ck(z)y- 21: ,
(41.43)
k =O
where co (z) = ao ,
k = 1 , 2, . . . is a polynomial of degree 2k - 1. Hence by using ( 17.18) we obtain the asymptotic expansion of the solution of (41.25) with q = 0:
u ( z, y)
�
f(p + 1/2) Vi +
2(- 1)1: rot(r > {M1 (z, y; r); 2k + 1} [� f:'o k!f(p - k)y2 + 1 1:
l
�
f(1/2 - k) 2 f:'o f(p + 1/2 - k) c�;(z)y- 1:
·
(41.44)
In conclusion of this subsection we note that Lemma 40.2 in the case of arbitrary �, enables us to construct the solutions of the Cauchy problem for arbitrary homogeneous linear hyperbolic equations of second order with constant coefficients, on the basis of the solution given in (41 .37). Indeed, the above equations by changes of variables can be reduced to the equation n
L v�. � . - v, - �2 v = 0. 1:= 1
(41 .45)
According to Lemma 40.2 the solution v(z, y) of this equation can be transformed to the solution u(z, y) of (41.35) with 2p = n - 1 by the relation
u(z, y) =
(
)
r(n/2) (y) 1 n- 1 2 - v(z, y) , v'i Ji >.. - 2 , -
(41.46)
where Jf' > (f1, a) is the generalized Erdelyi-Kober operator J>.. ( f1, a ) defined in (37.45) and applied with respect to y. Then (41.36) is true for the solution v(z, y)
826
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
also, and the inverse transform is given by the expression
(
)
1 - n M (x, y,. T). ,fi (y ) n u(x, y) -_ f(n/2) J). 2 - 1, 2- n If on the basis of the solution
( 41.47)
v(x, y) we construct the function y
v(x, y) = J v(x, t)dt,
( 41.48)
0
then it is also the solution of conditions
(41.45)
satisfying the half-homogeneous initial
( 41.49) v (x, 0) = 0, Vy {z, 0) = T(x) symmetric with (41.36). Replacing T(x) by v(x) in (41.48) and adding it with the function given in (41.47) we arrive at the solution of the Cauchy problem of the general form ( 41.26) with p = 0 for (41.45). 41 .4. The weighted Dirichlet and Neumann problems in a half-plane In this subsection we apply the results given in ( 41.18)-(41.20) to solving the Dirichlet and Neumann problems in the half-plane y > 0 for (41.2) with any parameters p and q . The following analogues of Theorem 40.7 and 40.8 are true.
Let T be a continuous function bounded on the real line ( -oo, oo) . Then the Dirichlet problem in the half-plane y > 0 which consists of finding the solution u(x, y) E C2 (y > 0) of (41.2) in the space of functions bounded at infinity under initial conditions in ( 40.32) is properly set for any real q only in the case p < 1/2. The solution is given by (41.18) with Theorem 41.5.
_
I1 Theorem 41.6. Let 11 be a line ( - oo , oo) and if p � 0,
(1 - 2p)B(1 - ,8, 1 - ,8) 22Pref""
·
(41.50)
continuous function absolutely integrable on the real then in addition lv(t)l < C jt j 2p - l -e , c > 0, It I oo , C - const. Let also the necessary condition in (40.49) be satisfied. Then the weighted Neumann problem in the half-plane y > 0 which consists of finding the solution u E C2 (y > 0) of (41.2) vanishing at infinity and satisfying the boundary condition in (40.45) on the axis - oo < x < oo is properly set only in three cases: --+
§ 41. THE EULER-POISSON-DARBOUX EQUATION
827
1) p > q, q is an arbitrary real number; 2) -1/2 < p < 0, q = 0 and 3) p = q = 0. In the first two cases the solution of the above problem is given by (41.19) with ( 41.51) and in the third case the solution is given by (40.50) and is continuously connected with (41.19) and (41.51) only for q = 0, p -+ 0. These solutions contain integrals which converge absolutely and uniformly in the set { y � 0, lzl < R} and these solution vanish at infinity of order u = O(p- 2P - l) and u� , u11 = O(p- 2P - 2 ) as p = ..jz 2 + y2 -+ oo . ) Let p = 1/2 and r be a function continuous on the real line and lr(t)l < Cltl - e , e > 0, lt l -+ oo, C - const. Then the weighted Dirichlet and Neumann problems in the half-plane y > 0 which consists of finding the solution u ( z , y) E C2 (y > 0) of (41.2) vanishing at the infinity and satisfying the conditions
Theorem 41. 7.
lim ln - 1 yu(z, y) = lim yu11 (z, y) = r( z ) ,
11 - + o
11 - + o
-oo
< z < oo,
(41.52)
respectively, are solvable and their solutions are given by (41.20) where 1/J has the form (41.17) and
or by the equivalent relation (41.54) where C, C1 and C2 are arbitrary constants and 1/J (z) is defined in (1.67). (41.18) with respect to y, and taking the limit as y -+ +0 in the equation obtained and in (41.19), provided that p < 1/2 and ( 40.32) and ( 40.45) hold, and further, making the changes of variables Remark 41.4. Differentiating
Ct = ch qr,
c2 = -sh qr,
a=
1 - 2p,
we obtain the relations in (12.11) and (30.78) which reflect connection between the direct and inverse Feller transforms.
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
828
and
The following boundary value problems are solvable for other parameters p q.
A. If p > 1/2 and q is any real number, then the weighted Dirichlet problem (40.51), -oo < x < oo , is solvable and its solution is given by (41.19), (41.51). B. If p = 0, q =F 0, then the weighted Neumann problem lim ln - 1 yu11 (x, y) = v(x)
(41.55)
Y -+O
is solvable and its solution is given by ( 41.19) with p = 0, C. If p < 0, q =f: 0, then the Neumann problem
u11 (x, 0) = v(x) ,
-oo <
/2 = [4qe 9'��' sh q ] - 1 . 1r
x < oo,
(41.56)
is solvable in the space of functions bounded at infinity and its solution is given by (41.18) with
r(t) = -pq - 1
J v(8)d8 t
+ C,
C-
const ,
- oo
where C = 0 if u vanishes at infinity. D. If q = 0, p = -1/2, then the weighted Neumann problem lim o(yln y) - 1 u11 (x, y) = v(x) , 11-+
-oo <
x < oo,
(41.57)
is solvable and its solution is given by ( 41.19) with q = 0, 12 = -1/2. E. If q = O , p < -1/2, then the weighted Neumann problem
(41.58) under the conditions ( 40.49) and lv(t)l < Clt l - 2 - e:, c > 0, I t I --+ oo, is uniquely solvable in the space of functions vanishing at infinity together with their derivatives of the first order and its solution is given by (41.18) with q = 0 and
J (t - 8)v(8)d8, t
r(t) =
- oo
It =
2f(1 - p) . y'ir( -1/2 - p)
In order for the above problems to become uniquely solvable we must assume additional conditions on the solutions, and remove the singular solutions of (41.2) of p the form y 1 - 2P r12P - 2 e 29tP , y 1 - 2P ! rl- 2 e 29 tP , 1 and In y! r}- 1 ln(yr1- 2 )e 29tP if p - 1/2 .
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
829
Table 41.1
p q
u=O
Uy = 0
< -l 0 1
y
-l 0 1
-l < p < O
y ln y
0 1
y - 2p
0 0 1 1
#:- 0 1 1
v
1
,- 2p
l
v
>l v
ln y yl - 2p 1 y- 2p ,-
Table 41.1 reflects the dependence of the orders of the solution u of (41.2) and its derivative uy as y --+ 0 on the parameters p and q. This table also show which weights correspond to properly setting Neumann and Dirichlet problems with conditions on the singular line y = 0.
§ 42. Ordinary D ifferential Equations of Fractional Order Equations in which an unknown function y( x) is contained under the sign of a derivative of fractional order, i.e. equations of the form
F(x, y(x), v:: w 1 (x)y(x), v::w2 (x)y(x), . . . , v:: wn (x)y(x)) = g(x),
(42.1)
where v:J = v::+ or v::_ are called ordinary differential equations of By analogy with the classical theory of differential equations, differential equations of fractional order are divided into linear, homogeneous and inhomogeneous equations with constant and variable coefficients. Differential equations of fractional order are studied both in the space of regular functions, i.e. functions summable to a certain power and continuous and differentiable up to certain order in a classical sense, and in various spaces of generalized functions. Analogues of the Cauchy and Dirichlet problems for differential equations of fractional order often arise in applications. Thus if we would like to find the solution y(x) of (42.1) with initial conditions
fractional order.
then we say that we are dealing with the solution of the Cauchy-type problem for If the values of the unknown function or of its derivatives of integer or fractional order are given at the end points of a certain interval (z0, z 1 ] then we say that we are dealing with the Dirichlet-type problem for (42.1). In this section we shall be concerned with the proper setting of certain problems for differential equations of fractional order. Thus we shall study questions concerning the solvability of these equations in certain spaces of functions.
(42.1).
830
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Further, the applications of the theory of fractional integra-differentiation and the theory of differential equations of fractional order to integration of certain classes of differential equations of integer order will be considered.
42. 1 . The Cauchy-type problem for differential equations and systems of differential equations of fractional order of general form We need to find a function
y(z) satisfying the equation
da y(z} = /(z, y), dza n
n-
1 < a 5 n, n = 1,2, . . .
,
(42.2}
0/
= -[-a], where here and below tz 01 = vg+ , with initial conditions k = 1, 2, . . . , n ,
(42.3 }
where / (z, y) is the given function and a, b t , . . . , bn are given constants. We shall consider a series of theorems on the existence and uniqueness of the solution of the above problem. We denote by Rn the following set of points (z, y) in a domain D lying in R x R:
(42.4} where a, h and b 0 are certain constants.
Let f(x, y) be a real-valued function continuous zn Lipschitzian with respect to y:
Theorem 42.1.
1/(z, Yt ) - f(x, Y2 ) 1 5 A IYt - Y2 l
D
and
(42.4'}
and let the condition sup(z ,y )ED 1/(z, y)l = bo < oo hold. Then there exists a unique continuous solution of the Cauchy-type problem given in (42.2}, (42.3} for n = 1 in the domain Rt C D. For the case n = 1 we refer to (42.4}.
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
831
Let f(x, y) satisfy the conditions in Theorem 42.1. Then there exists a unique continuous solution of the Cauchy-type problem given in (42.2), (42.3) for n = 1, 2, . . . in the domain Rn C D.
Theorem 42.2.
Let !A:(x, y� , . . , Ym), k = 1, 2, . . , m, be real-valued functions continuous in a domain Dm C R X R!" and satisfying the conditions
Theorem 42.3.
.
.
m 1 /k (x, Yl , . . . , Ym) - b: (x, Z t , . . . , Zm)l � A E IYi - Zi l , k = 1, 2, . . . , m, i= l
and
sup
(z-,yl , · · · ·!f,. ) ED.,.
k= 1 , 2 ... m
1/k (z, Yb . . . , Ym)l = M <
oo.
Then there exists a unzque
, , continuous solution of the Cauchy-type problem
k = 1, 2, . . . , m, 0 < a � 1,
in the domain
where a > Mh/f(a + 1).
1 x1 - aYk (x) - f(n)bk I � a,
}
k = l, 2, . . . , m ,
Let f(x , Yl , . . . , Ym ) satisfy the conditions in theorem 42.3. Then there exists a unique continuous solution of the Cauchy-type problem
Theorem 42.4.
da
dx a Yk (x) = l�c (x, yt , . . . , ym ),
-
k = 1, 2, . . , m, .
in the domain
j = 1, 2,
. . . , n,
n-
1 < a � n,
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
832
fl..
==
{ (z,
!ft . . .
where
. , !lm )
E Dm
:
0<
Z
:5 h,
,.,n-a!li (z) - r(a �: + l ) l :5 a, J: =
}
1 , 2, . . . , m .
.
n - 1 hn -j h . a>E . f(a . .:_ 1 ) , bi ,o = M, k = 1 , 2, . . , m. J =O
_
J
Let Pi (z), k = 0, 1 , . . . , m, and /(z) be functions continuous on an interval (O, h). Then the Cauchy-type problem
Theorem 42.5.
k = 1, 2, . . . , m, has a unique solution continuous on (O, h) . Theorem 42.6.
Under the conditions in Theorem 42.5 the Cauchy-type problem m
cJ(m - i )a
l: Pt(z) dz(m - i)a Yi (z) = /(z), k=O
n - 1 < a � n,
.
k = 1 , 2, . . , m, j = 1, 2, . . . , n ,
has a unique solution continuous on (0, h).
The proof of Theorems 42.1-42.6 differs little from the proofs of the corresponding
theorems for differential equations of integer order. Therefore we prove Theorem 42.1 . Integrating (42.2 ) where fz-_0101 = I�+ we have
833
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
and hence according to the property in (2.61) and the condition in (42.3) we obtain
a- 1 + f (x - t) a- 1 /(t, y)dt. y(x) = b1 xf(a) f(a) :r:
(42.5)
0
So, the problem in (42.2) and (42.3) is reduced to (42.5). We show now that if the continuous function satisfies (42.5), then it satisfies ( 42.2) and ( 42.3). Indeed, applying the operator d�OJ to (42.5) we have
/(x, y)
dOl d:r: OJ
y(x) - f(x, y). The condition in (42.3) for n = k = 1 can be obtained without difficulty if we
and, hence,
apply the operator a�01-_\ to
(42.5):
:r:
=b1 + rt1) j f(t, y(t))dt, 0
and then set
x = 0.
It follows from our arguments that ( 42.5) is equivalent in the above sense to (42.2) with initial conditions given in (42.3). We accomplish the rest of the proof by the method of successive approximations, although the method of compressed mappings - Kolmogorov and Fomin [1, p. 73) - may be also used. Let
a- 1 + f (x - t)a- 1 f(t , Y (t))dt , n = 1, 2, . Yn (x) = b1 xf(a) n- 1 f(a) :r:
0
First of all we require the points
.
..
(42.6)
(x , Yn (x)) to be lie in R1 for 0 < x � h.
The
834
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
estimate
I.,'-"Yn(Z) - �� 1 ;��; 10 (z - t)"- ',(t, 7/n- t (t))dt :r:
r
)
=
< bo x -< boh r(o: + 1) r (o: + 1)
(42.7)
follows from the condition sup I /( z, y) I = b0 • If the condition b0 h /f( o: + 1) < a (:r: ,y )eD
holds, then (z, Yn (z)) E R1 for 0 < z � h. Now we estimate the difference Yn (z) - Yn - l (z). By
(42.7) we have
IY1 (z) - Yo(z)l � bo z a /f(o: + 1) � bo h a /f(o: + 1). By using the Lipschitz condition given in with n = 1 we find that
l!12 (z) - y, (z)l
l = r at ( )
(42.4') and the last estimate from (42.6)
:r:
j< z - t)"- 1 [/(t, y, (t)) - f(t, I/O (t))dt 0
$ r�) 1 (z - t)" - ' lut (t) - 7/o (t)ldt :r:
0
2 1( a- 1 bo ta dt � rAbo h a � rA z t ) ( o: ) r (o: + 1) (2o: + 1} " 0 :r:
Repeating such
an
estimate many times we finally arrive at the inequality
It follows from here that the sequence Yn ( z) tends to a certain limiting function y(z) uniformly with respect to z (0 < z � h). This limiting function is continuous in (0 < z � h) and satisfies the inequality
which is obtained if we pass to the limit in
(42.7) as n -+ oo.
Now carrying out
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
835
the passage to the limit in (42.6) as n --+ oo we obtain ( 42.5) according to the continuity of /(z, y). We prove that the solution y( z) is unique for sufficiently small h . Let A h a /f (a + 1 ) < 1 and we suppose that there exist two solutions y(z) and Y(z) of the problem considered. Substituting them into (42.5) and subtracting one from the other we obtain
ly(z) - Y (z)l =
z
J 0
(z t) a- 1 �(a) [/(t, y(t)) - /(t, Y(t))]dt
� J (z - t)a- 1 i y(t) - Y(t) idt . z
�r )
0
We suppose that the difference I Y( z) - Y ( z) I admits a maximal value 6 at a certain point z = e lying on the interval 0 < z � h. Then for z = e from the last inequality we have that 6 � A r- 1 (a)6h a a - 1 or 1 � A h a / f (a + 1) which contradicts the assumption. This completes the proof of Theorem 42.1. Theorem 42.2 is proved similarly to Theorem 42.1, only in this case we set
Theorems 42.3 and 42.4 are extensions of Theorems 42.1 and 42.2 to the case of systems of differential equations of fractional order. Theorems 42.5 and 42.6 are special cases of Theorems 42. 1 and 42.2. • In conclusion of this subsection we consider two examples in which Theorems 42.1 and 42.2 are used. Example 42.1. We solve the following Cauchy-type problem
n - 1 < a � n,
k = 1 , 2, . . . , n. Using the proof of Theorem 42.1 we have
836
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
n za- k Yo (z) = L blc r (a - k + 1) ' lc= 1
� J(z - t)"- 1 Ym- t (t)dt.
Ym (z) = Yo (z) + f )
z:
0
Hence for m = 1 , 2, . . . we find
Y1 (z) = Yo( z) + .\
n
2 lc
[; b�c f(2az -a-k + 1) , ...,
which in the general case yields n m+ 1 . 1 z. aj - lc , m = 1 , 2, . . '""' = b (z) Ym L..J _\JL..J k '""' f (aJ - k + 1) k= 1 i = 1
.
After we pass to a limit as m -+ oo we obtain the following representation for the solution: oo n . 1 zaj - k y(z) = '""' L..J _\JL..J b�c '""' . 1) r(aJ - k + lc= 1 i = 1 .
00 = L bk za- lc Ea, 1 + a- 1c(.\z a ) ,
lc= 1
where Ea, p (z) is the Mittag-Leffler function defined in (1.91). In particular, if a = n = 1, then 00 zj - 1 Y(z) = b1 '""' .) = b 1 e�z: � _\i - 1 _r_ ( J = J 1
and we have "a joining" of the known solution of the Cauchy problem for the equation of first order and of the above problem for the equation of order a. Example 42.2. Now we construct the solution of the Cauchy-type problem for
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
837
the inhomogeneous differential equation
da dz a y(x) - �y(x) = h(x),
n-
1 < a :5 n ,
with initial conditions
k = 1, 2, . . . , n . Similar to the previous example we have
Ym (x) =Yo (z) +. r a )
+ r (1a)
t
I(x - t)a- 1 Ym- 1 (t)dt :r:
0
I (x - t)a- 1 h(t)dt, :r:
0
and hence
n m+1 . X aj - J: """ (x) Ym = L., b��: """ L., �J- 1 f (aJ. - k + 1) J:= 1 j= 1
I
m �J- 1 + r(aj ) (z - t) aj - 1 h(t)dt. J-1 0 Passing to a limit
as m -+ oo
�
•
:r:
we find the solution of the Cauchy-type problem
n
y(x) = L b��: xa- J: Ea , 1 +a- J: (�xa ) J:= 1
I :r:
+ (x - t)a- 1 Ea, a [�(x - t) a]h(t)dt. 0
42.2. The Cauchy-type problem for linear differential equation of fractional order We consider the linear differential equation of fractional order
n- 1
�"· y(x) + L P��: (z)�"·-•- 1 y(z) + Pn (x)y(x) = /(z), J:= O
(42.8)
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
838
where
l'l'\O'o -- vOcr+o - 1 ' k = 1, 2, . 1: u1: = LO ai - 1, k = 0, 1, . . . 0 < ai � 1, j = 0, 1, . . , n,
AJ
...,n
, n,
j=
k = 1, 2, .
a1: = u1: -
ao = uo 1 -
()
(42.9) ()
- obviously, UA: - 1 , , n, and PA: x and f x are + given functions. We require to find the solution y( x) of this equation satisfying the initial conditions
D O",. y(x ) jx = O
.
.
= 61:, k = 0, 1,
...,n-
1.
(42.10)
We begin the investigation of this Cauchy-type problem for the case
Pk ( x) =
0, k 0, 1, . =
. .
, n,
i.e. we consider the equation
(42.10).
(42.11)
with initial conditions given in The following statement is true. Theorem 42.7.
Let a function f(x) E L 1 (0,a) be represented in the form
(42.12) where j( x ) E L 1 (0, a). Then there exists a unique solution of the Cauchy-type problem defined in (42.11), (42.10) represented in the form
Proof. It is obviously follows from (42.9) that
(42.13)
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
839
Hence, (42.11) may be rewritten as
or (42.14) Thus the problem defined in (42.1 1) and (42.10) is reduced to the problem given in (42.14) with initial conditions in (42.10) where k = 0, 1, . . . , n - 2. Applying again (42. 13) to (42.14) similar to the above arguments we obtain (42 .15) Now the problem defined in (42.11) and (42.10) is reduced to the problem given in (42. 15) with initial conditions (42.10) where k = 0, 1 , . . . , n - 3. Continuing this process we reduce (42.1 1) to the equivalent equation
which has the form (42.16) if we take into account the relation
We prove that the above function y( z) satisfies the initial conditions in (42.10). For this we apply the operator 1)C7o given by
to (42. 16). Setting here z = 0 we obtain the condition in (42.10) for k = 0. Applying the operator 1)C7 1 to ( 42. 16) and setting then z = 0 we arrive at the condition in (42.10) for k = 1 . Continuing this process we verify that all conditions
840
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
in (42.10) are satisfied. The uniqueness of the solution of the above Cauchy-type problem follows from (42.16) too. • Now we formulate the main theorem of this subsection.
Let the functions Pi (z), k = 0, 1, . . . , n, be Lipschitzian, i.e. they satisfy the Holder condition of order � = 1 - Subsection 1 . 1 on an interval [0, a), and let /(z) be continuous function in [0, a) which can be represented in the form
Theorem 42.8.
-
(42. 17)
where i(z) E L 1 (0, a). If ao > 1 - an , then the Cauchy-type problem defined in and (42.10) has an unique solution continuous on [O, a].
(42.8)
We set l)C7• y(z) = �(z). Then (42.8) h as the form of the Volterra integral equation of the second order
Proof.
�(z) = w(z) +
z:
j W(z, t)�(t)dt,
( 42.18)
0
where
W(z, t) = - Pn (z) (z -f t)u. (un )
-1 (42.19)
( 42.20) (42.19) shows that the kernel W(z, t) has a weak singularity at t = z. Applying the method of successive approximation to (42.18) we find that this equation admits not more than one solution �(z) E L 1 (0, a) continuous on [0, a]. Hence, by Theorem 42.7 we deduce the uniqueness of the solution of the problem defined in ( 42.8) and (42.10). The proof of the existence of the solution of this problem
841
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
according to Theorem 42.7 is also reduced to the proof of the representation
da ,. - 1 <J (x) = dx a ,. - 1 <J (x), i (x) L 1 (0, a) . _
E
(42.21)
Indeed, since (42.18) admits not more than one solution <J (x) = l) CT • y(x), then by Theorem 42.7 it is sufficient to prove that the function
is the solution of the problem given in (42.8) and (42.10). Thus we must to verify the condition given in (42.21) and in Theorem 42.7. We shall now do this verification. Taking into account the conditions a 0 > 1 - c:tn , c:tn > 0 and the inequalities
in accordance with
(2.44) we write the relations
da ,. - 1 z CTk +at ,. - 1 k = 0, 1, . . . , n - 1, r(1 + u�:) dx 01 · - 1 r (u�: + c:tn ) ' da .. - 1 z CT,. - CTk + a.. - 1 X CT,. - CTk = r(l + O'm - O'A: ) dx01 · - 1 r (um - O'A: + c:tn ) ' m = k, k + 1, . . . , n - 1, zCTk
which together with
(42.17) allow us to write (42.20) in the form
w(x)
(42.22) Now we apply the result by Dzherbashyan and Nersesyan [6, p. 17], who proved that for any functions g(x) L 1 (0, a) and p�:(x) C([O, a]) there exists a unique function G(x) L 1 (0, a) satisfying the relation
E
E
E
(42.23)
842
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
Similar statements may be found in Lemmas 3.2 and 10.1. On the basis of the above result, ( 42.22) can be rewritten as
a,. - 1 w(z) = ddza,. - 1 w(z), where w(z) is a certain function in have
L 1 (0, a).
(42.24)
Further from (42.18) and (42. 19) we
and hence according to (42.23) we obtain
d- · v(z), v(z) E £ (0, a), �(z) = w(z) + dz1 •
(42.25)
where Hence it follows from {42.24) and (42.25) that
-1 d- • v(z). + �(z) = dcr· w(z) 1 -• dzzOtra If now x � 1 -an , then the representation in (42.21) has been proved. If x < 1 - an , then according to (42.23) there exists p E N such that (p - 1)x < 1 - ern < px and
da,. - 1 w(z) + d-P• v (z), v(z) E L (0, a). �(z) = dz 1 a ,. - t dz -P •
This completes the proof. • In conclusion of this subsection we indicate that the simplest Cauchy problems for differential equations of fractional order (42.8') and (42.8") VC:.. y - >..y = 0, z > 0, y(O) = 1, 0 < a < 1, have the solutions y = z01 - 1 Ea,a (>.. z 01) and y = exp ( ->.. 1 101z), respectively. Here the fractional derivatives Vg+ and V� are defined in (5.9) and the Mittag-Leffler function Ea,p (z) in {1.91).
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
843
42.3. The Dirichlet-type problem for differential equation of fractional order We consider the differential equation of second order with fractional derivatives in the form
Ly :y" (x) + a o (x)y'(x) m + E a�:(x)Vg.f. (w�: (x)y(x)) + a m+ 1 (x)y(x) = f(x) 1:= 1
( 42.26)
on the internal [0, 1], where 0 < a 1: < 1 and functions a0 (x), a m+ 1 (x), a�:(x), w�:(x), k = 1, 2 , . . . , m, f(x) are continuous on [0, 1]. In the theory of boundary value problems for the equations of the form ( 42.26) the following two theorems are of importance. The first of them is an analogue of the Hopf principle - Bitsadze [3, p. 25, 26].
Let w�: (x), k = 1 , 2, . . . , m , be non-decreasing positive functions on [0, 1] satisfying the Holder condition with exponents KJ: > a�:, 0 < a�: < 1, k = 1 , 2 , . . . , m, and a�:(x) E C[0, 1], a 1: � 0, 0 < x < 1, k = 1, 2 , . . . , m + 1. If y E C2 (0, 1) is the solution of (42.26) which differs from a constant, then the positive maximum and negative minimum of y(x) can be only at the end points x = 0 or x = 1.
Theorem 42.9.
Proof. We suppose the opposite, i.e. there exists x0 , 0 < x0 < 1, such that max y(x) = y(xo ) > 0. We note that if cp(t) is continuous on [0, x] and satisfies o<�< 1 the -Holder condition of order x > a at the point t = x and has a maximum at this point then (13.1) yields (Dg+ cp)(x) > 0. Therefore, since functions w�:(x) are positive and non-decreasing on [0, 1], then WJ: fl has a positive maximum at the point x0 • Hence there exists 6 > 0 such that the inequalities
0 < y(x) < y(xo ), a�:(x)(Dg.f.w�:y)(x) � 0, k = 1 , 2, . . . , m, hold for any x E [x0 - 6, x0] . From the equation Ly = 0 we have
y" + ao (x)y' =
m
-
E a�:(x)Dg.f.w�: (x)y - am+ 1 (x)y. 1: = 1
Hence
y" + ao (x)y' 2:: 0, x E [xo - 6, x o] .
(42.27)
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
844
On the interval
[zo - 6, zo] we introduce the preliminary function z(z) = y(z) + cg(z),
where g(z) = exp ( -pz) - exp ( -pzo), p > 0, 0 < c < [y(zo) - y(zo - 6))/g(zo - 6). Substituting y(z) = z(z) - cg(z) into (42.27) we have
z" + ao(z)z' � cp exp ( -pz) [p - ao(z)). If we choose p such that p > a0 (z) for z E
[z0 - 6, z0), then we obtain
z" + ao(z)z' > 0.
(42.28)
z(z) has a maximum on the open interval (z o - 6, zo), then the conditions and z" $ 0 hold at the corresponding point. This contradicts (42.28). Therefore z( z) can have maximum only at the point zo since
If only
z'
=
0
o - 6) z(z0 - 6)
z(zo ), z'(zo) � 0.
But 0 $ z'(zo) = y'(zo) + cg'(zo) = y'(zo) - cpexp (-pzo) and hence y'(zo) � cp exp( -pz0) > 0 which contradicts the necessary condition of the extremum y'(z 0) = 0. Thus the extremum of y(z) can not be at the inner point of the interval [0, 1) . • The second theorem is an analogue of the Zaremba-Giraud principle - Bitsadze
[3, p. 26).
Let the conditions in Theorem 42.9 be satisfied. If y E C[O, 1] u C1 (0, 1) u C2 (0, 1) is the solution of the equation Ly = 0 given in (42.26) and
Theorem 42.10.
max y(x) = y(1) > 0 y• = O�z� l
(y. = O�z min y(x) = y(l) < 0), �l then y'(1) > 0 (y'(1) < 0). If y• = y(O) > 0 (y. = y(O) < 0), then y'(O) < 0 (y'(O) > 0) provided that the additional conditions hold, namely y(z) E C1 [0, 1), Wk (z) E C1 [0, co], Wk (O) =I= 0, k = 1, 2, . . . ' m, where co is a small positive number.
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
845
The proof follows directly from Theorem 42.9. Definition 42.1.
boundary condition
We say the problem of finding the solution of (42.26) with the y(O) = y(1) = 0
( 42.29)
is the Dirichlet-type problem for this equation. Let the coefficients of (42.26) satisfy the conditions in Theorem and ao (z) = 0, at (z), Wt (z) E C1 [0 , 1), k = 1 , 2, . . . , m. Then the Dirichlet type problem defined in (42.26) and (42.29) is solvable unconditionally and uniquely in the space of functions C[O, 1] n C2 (0 , 1).
Theorem 42. 11.
42.9
Proof. It is not difficult to verify the following equalities
tp(z) = ztp(z) - (z - 1)tp(z)
1
= :X[! ttp(t)dt + f (t - 1)tp(t)dt] z
0
z
� [/0 ttp(t)dt + z(z - 1)tp(z) + J (t - 1)tp(t)dt - z(z - 1)tp(z)] 1
z
=d
z
�
1
= dz'l [/ (z - t)ttp(t)dt + J z(t - 1)tp(t)dt] 0 z 1 ' = dz2 J G(z, t)tp(t)dt, z
0
where
G(z, t) =
{ t(zz(t -- 1),1),
t � z,
t > z.
Therefore in the c ase a 0 (z) = 0 we can rewrite (42.26) �
dz'l
�(z) = 0,
(42.30)
as
(42.31 )
846
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
where
t
�{ z) =y(z) +
J0 G(z, t)am+ t (t)y(t)dt
m t
t
+ L: J G(z, t )a�; (t)(Vg+ w�: y)(t)dt - j G(z, t) f(t )dt. l: = t o
o
{42.32)
From the immediate verification of this by using (42.29) we can prove that �{0) = �{1) = 0. Hence, from {42.31) we derive the relation
�{ z) = 0.
{42.33)
Hence we have proved that the Dirichlet-type problem given in (42.26) and (42.29) is equivalent to the integral equation (42.32) and ( 42.33). The immediate verification shows that this equation is a Fredholm equation of the second kind. In view of Theorems 42.9 and 42.10 and the boundary conditions in {42.29) the homogeneous integral equation {42.32) and {42.33) with /{ z) = 0 is equivalent to the Dirichlet-type homogeneous problem, and therefore it has only the trivial solution y = 0. From here we deduce that the Fredholm inhomogeneous equation is solvable unconditionally ·and uniquely, and hence the Dirichlet-type problem is also solvable unconditionally and uniquely. •
42.4. Solution of the linear differential equation of fractional order with constant coefficients in the space of generalized functions We consider the linear differential equation of fractional order n
L a; ]0i y(z) = /{ z),
j=t
{42.34)
where here and below Ja; = 1;+ = V�; , and with constant complex non-vanishing coefficients a t , a2 , . . . , a n and different real exponents at , a 2 , . . . , an . This equation generalizes the Abel integral equations of the first and second kind, and the usual linear differential equations of integer order with constant coefficients. We shall find the solution y( z ) of ( 42.34) in the space Sf. of tempered distributions with support in [0, oo ). One may obtain more detailed information about this and other terms and notation of this subsection in the book by Vladimirov [2] .
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
Let /(z) E S.f. . We write (42.34)
as
847
the convolution
k(z) * y(z) + /(z)
(42.35)
where n
k(z) = L: a; /a; (z), j =l and a generalized function Ia ( z) E s� is /a (z) =
(42.36)
{f
z+- l /f(a), a > 0, (N) a+ N (z), a :5 0, a + N > 0, N = [ a] + 1 . -
-
( 42.37)
We apply the Fourier-Laplace integral transform to both sides of ( 42.35). For /( z) E S� this transform is defined by
F(z) = L[/(z)](z) = L[/](z + iy) = V[f(e)e- (Y ,O](z),
( 42.38)
where V [g (e)](z) is the Fourier transform of a generalized function g(e) E defined in the usual way - Vladimirov [2 , p. 105]. We introduce the notation n
· if(a; / 2
aJ e Y(z) = L[y(z)](z), K(z) = L[k(z)](z) = � . L...J za, i =l
S�
( 42.39)
where the branches of the power functions are given by the condition z a; > 0 for z = z > 0, j = 1, 2, . . . , n. Since /(z), y(z), k(z) E S� , then F(z), Y(z), K(z) is analytic in the upper half-plane c + = {z : Imz > 0 } in the complex plane C . We denote by H the set of functions G(z) of complex variable z = z + iy, analytic in c + and satisfying the estimate I G(z)l :5 M(1 + l z i 2 )Pf 2 (1 + y-"), z E c + , for certain real non-negative constants M , p and q which do not depend on z. The set H called the Vladimirov algebra, is a multiplicative algebra relative to addition and multiplication of analytic functions, and multiplication of a function by a complex number. The following assertion is true - Vladimirov [2, p. 173] .
The algebras S� and H are isomorphic algebraically and topologically. The Fourier-Laplace transform is an isomorphism of S� onto H and the relation L[k(z) * y(z)](z) = K(z)Y(z) holds for any generalized functions k(z), y(z) E S� . Theorem 42.12.
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
848
Thus ( 42.34) has a solution in the space S.+ if and only if the solution Y(z) of the algebraic equation K(z)Y(z) = F(z) is in the Vladimirov algebra H. Further if Y E H , then a unique solution of (42.34) is given by the relation
y(z) = L - 1 [F(z)/K(z)].
( 42.40)
Here L - 1 is the inverse Fourier-Laplace transform which maps H onto Si. . If a function K(z) of the form (42.39) does not have zeros in c + , then 1/K(z) E H and hence Y(z) = F(z) /K(z) E H. In this case (42.34) is solvable in Si. for each f(z) E S.+ . Now we assume that the function K(z) has I zeros z = z; E c + , j = 1, 2, . . . , I. This set can not he infinite since then K(z) = 0. Therefore the function Y(z) = F(z)/K(z) is in H provided that the conditions F(z;) = L[f(z)](z;) = 0, j = 1 , 2, . . . , I, hold. We denote by N and N0 the sums of orders of zeros with positive and zero imaginary parts respectively, of the function K(z). Also if K(z) vanishes at origin of coordinates, then we shall not take this zero into account. Then the following expression is true
1
1
N = - [arg K(z)].y - - (N0 - 1m�n a; ) . �J� 21r 2
(42.41)
This follows from the generalized principle of argument - Gahov [1, p. 100]. Here [wJ.r means a change of w after a circuit in a positive direction along a closed contour ; consisting of an upper half-circle surrounding all zeros of K ( z) and of an interval of the real axis joining this half-circle. Thus we obtain the following statement.
Let N � 0 defined in (42.41) be the sum of orders of zeros with positive imaginary parts of a function K(z) of the form ( 42.39) analytic in c + and being the Fourier-Laplace of the generalized function k(z) given in (42.36) . If N = 0, then (42.34) is solvable in the space Si. for each f(z) E 84- . If N > 0 and Zt , z2 , . . . , z, are all zeros of K(z) with orders rt , r2, . . . , r,, r1 + r2 + + r, = N, such that Im z; > 0, j = 1, 2, . . . , I, then (42.34) is solvable in Si. if and only if the function F(z) = L[f(z)](z) has zeros z1 , z2 , . . . , z, with orders more or equal to r1 , r2 , , r, , respectively. If the solution exists, then it is unique and is given by (42.40). In the case N = 0 the solution of (42.34) can be also represented in the form y(z) = f(z) go(z) where uo(z) = L - 1 [1/K(z)] is the fundamental solution of the operator k( z) i.e. a generalized function with support in [0, +oo) satisfying the equation k(z) uo(z) = 6(z) where 6(z) is the Dirac delta-function. Let now the right-hand side f(z) in (42.34) he in the space D'+ consisting of generalized functions with support in [0, oo). This is obviously wider than S.+ . We construct the solution y( z) which also is in D'+.
Theorem 42. 13.
· · ·
• . •
*
*,
*
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
849
Let c be a real non-negative constant such that c > max l zj I · 1Si 9 1/ K(z + ic) E H and the generalized function
Then
( 42.42) is, in general, in the space D'+ . We note that relation g(z) = go(z) holds in the case N = 0. We consider the following convolution
g(z) does not depend on c and the
k(z) * g(z) = L - 1 [K(z)] * e c:�: L - 1 [1/ K(z + ic)] = e c:�:(L- 1 [K(z + i c)] * L- 1 [1/K(z + ic)]) = e c:�: 6(z) = c5(z), where c5(z) is the delta-function. From this we obtain that the function g(z) is the fundamental solution of the operator k( z)• in the space D'+ and
( 42.43)
y(z) = /(z) * g(z). Hence we arrive at the following statement.
If f(z) E D'+, then a unique solution of (42.34) in the space D'+ is given by (42.43) where g(z) is the fundamental solution of the operator k(z) • in the form (42.42).
Theorem 42.14.
g(z) in the space D'+ can be in a narrower space. So if g(z) and / (z) are continuous on [O, oo), then y(z) in (42.43) is continuous on [0, oo) and (42.43) has the form
Remark 42.1. A generalized function
:1:
y(z) = j f(t)g(z - t)dt, z > 0. 0
This representation is also true in the case when
L 2 (a, b)Va, b E R1 }.
g(z), /(z) E L�oc
=
{
42.5. The application of fractional differentiation to differential equations of integer order We consider two examples in which the theory of fractional differentiation will be applied to the integration of ordinary differential equations of the second and nth order.
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
850
Example 42.3. We consider the following equation of second order
( 42 . 44) We shall find its solution as the fractional derivative y = V�0z(z) with 1>�0 = V� o + order p of which must be defined. Using the Leibniz relation given in (15.11) for fractional derivative in the form
zV�t 1 z(z) =V�t 1 (zz(z)) - (p + 1)V�0z(z) , zV�t2 z(z) =V�t 2 (z 2 z(z)) - 2(p + 2)V�t 1 (zz(z ))
+ (p + 1)(p + 2)V�0z(z), from ( 42.44) we obtain
1)�: 2 ([a 2 + b2 z + C2 z2]z(z)) + V�t 1 ([a t + bt z - 2c2 (P + 2)z - b2 (P + 2)]z(z ))
+ V�0 ([ao - bt (P + 1) + (p + 1)(p + 2)c2]z(z)) = 0. We shall find the solution of this equation in the space of functions z( z) integrable on any finite interval and satisfying the conditions z(z o ) = z'(z o ) = 0 2 which we need in order that the operator relations 1>�0 -Jz = -Jz V�0 and 1>�0 -J;.r = 2
� V�0 be valid when p < . 0. Then the previous equation can be written in the
form
�.
{ d� [:., (a2 + b,x + c2x2) + a1 + hJ x - 2c2 (p + 2)x - b2 (p + 2)] +ao - b , (p + 1) + c2 (p + l)(p + 2) } z(x) = ( 42 . 45) 0.
We define a parameter p as one of solutions of the quadratic equation
ao - b t (P + 1 ) + c2 (P + 1 ) (p + 2) = 0. Then from (42.45) we obtain the simple differential equation
( 42 . 46)
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
851
the solution of which is easily given by the method of separation of variables, namely
( 42.47) Here the parameters a, b and c must satisfy appropriate conditions which ensure that the relations z(x0) = z'(x0) = 0 hold. Thus the solution of {42.44) has the form
(42.48) with the parameter p being defined in (42.46). The integral in {42.47) can be evaluated in different forms in accordance with the relations between its parameters. The detailed investigation of all its values and the corresponding representations of the function z{ x) was accomplished by Holmgren [2] (1967). Here we give only the results concerning one of these cases. Let c2 I 0, b� - 4a 2 c2 I 0. Then a2 + b2 x + c2 x 2 = c2 (x - a )(x - [3) , and after simple evaluation, the function z(x) from {42.47) h as the form � z(x) = �2+ 1 (x - a)P - 9+ 1 (x - fJa) p - r + 1 where q = � c �(a - fJ ) , r = - c�(a�fJ) and the conditions Re (p - q) > 0 or Re(p - r) > 0 when x0 = a or x0 = {3 respectively must be satisfied. The special case of (42.44) is the hypergeometric equation - Erdelyi, Magnus, Oberhettinger and Tricomi [1, 2.1(1)] - for which a2 = 0, b2 = 1, c2 = -1, a 1 = c, b 1 = -(a + b + 1), ao = -ab. Then the corresponding equation (42.46) yields the values Pt = a - 1 and P2 = b - 1. From (42.47) and (42.48), for the first of these parameters P1 = a - 1, we obtain the following representation for the solution of the hypergeometric equation: 1 a- c { 1 ) c-b - 1 . v0 -X y(X) - -na+ X _
(42.49)
This is a modification of the Euler integral representation given in {1.73), and also the expression 3 in Table 9.1. Evaluating the integral in (42.49) we arrive at the representation of the solution via the hypergeometric function in the form
y(x) = f (af (2- -c +c) 1) x 1 _c 2 F1 (1 + a - c, 1 + b - c ; 2 - c; x). Example 42.4. We consider the following ordinary differential equation of order
852 n
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
with binomial coefficients
(42.50) Using the integro-differentiation operators of arbitrary ord.er we construct one of the solutions of (42.50) in the form of ( n - I)-dimensional integral. For this we first introduce the polynomials
n
n 1: = IJ L b�: z bn (z - ,\k), 1: = 1 1: =0
(42.51)
where we denote by ,\ 1 , ,\ 2 , . . . , ,\n the roots of .,P(z) such that ,\j k = 1 , 2, . . . , n . Making the substitution
#; ,\1: , j #; k,
k= O
.,P(z) =
n
j,
y = exp(,\ 1 z)Y(z) ,
,\ 1
#; 0,
( 42.52)
and using the identity
we rewrite (42.50) in the form II'
(�t + � ) + (�t + � ) Y(z)
zt/1
We shall find the solution of this equation
as
Y(z) = 0.
( 42.53)
a derivative of order p, thus
dP
-
Y(z) = d Y1 (z) zP where the function conditions
Y1 (z)
is integrable on any interval
(0, a )
Y1 (0) = y�(O) = · · · = y�p - 1 ) (0) = 0.
and satisfies the
( 42.54)
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
853
Then ( 42.53) is reduced to the following equation
(42.55) We introduce the polynomials (42.56) Since ,P(A1) = 0 then ,P1 (z) has order n - 1. By setting
(42.57)
we obtain that z1 (:.,) 111 (z) = 0 follows from the homogeneous Abel equation of order -p - 1 given in ( 42.55) and �1
written in terms of the notation in (42.56). Hence under the conditions in (42.54) and a1 < 0 (42.50) is reduced to the equation (42.58) of order n - 1 . Continuing by analogy the above procedure of lowering the order, we arrive at the following system of differential equations of orders a; 1: -
( 42.59)
Here Yn-t (z) = exp(At,n-tz)(an equation of first order
+ bnz)- a• is the solution of the last simple (42.60)
854
and
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
z = .\ 1 ,m is the root of the equation 1/Jm (z) = 0 and 1/Jm( z) = z - 1 1/Jm - 1 (.\ 1 ,m - 1 + z), m = 1, 2, . . , n - 1, .
( 42.61 )
<po (z) = ,P(z), lrm = <J'm - 1 (.\ 1 ,m - 1 )/,P:,._ 1 (.\ 1 ,m - 1 ),
m = 1 , 2 , . . . , n.
( 42.62) (42.63)
We prove that .\ 1 ,m = .\m +1 - .\ m , m = 1 , 2, . . . , n - 1. From (42.56) , (42.51 ) and (42.61 ) we have the relations
n 1/J2 (z) = z - 1 1/J1(.\1,1 + z) = z- 1 bn II (z + .\ 1 + .\ 1 , 1 - .\t ). i= 2 Since z = .\ 1 , 1 is a root of the equation 1/J 1 (z) = 0, then we can set .\ 1 , 1 = .\2 - .\ 1 . Continuing such arguments for .\ 1 ,m , m = 2, 3, . . . , n - 1 we obtain .\ 1 ,m = .\m +1 - .\m , 1/Jm(z) = bn m = 1 , 2,
. .
n
II
k=m+1
(z + .\m - .\t ),
. , n - 1.
(42.64)
The conditions in (42.54 ) and a 1 < 0 are concerned with the function Similar conditions for other functions Yi ( z) have the form
Yt(O) = y�(O) = · · · = y�n - t) (O) = 0, k = 1, 2, . . . , n - 1, at < 0 , k = 1 , 2, . . , n, an = 0, .
Y1 (z).
( 42.65 )
the last condition an = 0 follows from Yn - 1 (0 ) = 0. Under these assumptions after convoluting the system in (42.59) we obtain the following representation for one of solutions of (42.50 ) :
(42.66)
§ 42. ORDINARY DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER
855
One may prove that (42.65) yields the following conditions for at :
ct 1 < 0, ctn < - 1 , «l'n - 1 < -1, ctn - 2 < -2, . . . , a-2 < - n + 2.
(42.67)
They can be extended if we use analytic continuation of the Abel integrals in (42.66). The special case of (42.50) for n = 2, a2 = bo = 0, 62 = -61 = 1 is the Kummer degenerate hypergeometric equation
zy" + (
c -
z) y' - ay = 0 .
(42.68)
For this equation
0,
c -
a, a-2 = a ,
a-2 < - 1 . The solutio:O: given in ( 42.66) for
f(1 - a ) 1 -c �: z e 1 F1 ( 1 - a ., 2 - c,. -z ) , f(2 _ c)
(42.69)
where 1 F1 is the Kummer degenerate hypergeometric function defined in ( 1 .81 ) , an d give also by formula 9 in Table 9.1. Here the conditions c - a < 0, a < - 1 c an b e extended t o the ones c - a < 1 , a < 1 which ensure the convergence of the integral.
856
CHAPTER
8.
APPLICATIONS TO DIFFERENTIAL EQUATIONS
§ 43. Bibliographical Remarks and Additional Information to Chapter 8 43. 1 . Historical notes
[3; § 6, 10] (1948). § §§ [2-5) (1952-1955).(40.19) [5) (1955) 0 40.2 [7) (1960), 0). [8] (1963),�) [10, 11] (1965) [14] (1970) (40.22). 0 [8] (1963) L� [10] (1965)K, ,o 40. 2 (18.8) - § 43.2, (40.140.8) 1. (40.19) 0 [5] (1979), [7] (1981), [9] (1985) 40.1-40.3 [1] (40. 2 3). (1823). (40.2[1]0) {1915). [3] (1953) §40.21)-{40. 2-40.3 23) [4] {1976), [9] {1978) (40. 1) 1sign #J I {40.26), {40.27) §§ (40.48). (41.1)(1772) [1 , 177, 426-432] {41.1) [1] 0(1823),{3 1/2. (41.25) 0 (41.22) [1, 40, 381-395] (1860). (41.1) {3 (41.2) 1. O, (41.31 ), 0 1948(41.14). [ 1 ] (1881) [1], 1915, [1] {41.25) 0, 0 1 (41.1 ), (41.25) {41.26) 1923. {41.1), (41.2) (41.25) 0, 1/6[1] yu�� [1-3]o, [1-3,7]. [3] (1953), [8] (1965). [ 3 ] {41.31) {41.31 ), {41.22) 41.1 {41.2) 0, 0 [1] (1957). 1/2. (41.18) {41.19) [5) {1976). 41.2
Note to
40.1. The presentation in this subsection follows the book by Vekua
Notes to 40.2 and 40.3. The method of fractional integro-difl'erentiation for the differential equations in generalized axially symmetric potential theory was first considered by Weinstein In the paper and also he proved relations connecting the solutions of with I' = ..\ = with eaCh other for different values of the parameter p by means of the fractional integral (Lemma with ..\ = This idea was developed by Erdelyi and who investigated the properties
given in In particular, in the papers and of the differential operator he proved Lemma in the case ..\ = and its analogue for the Erdelyi-Kober operator defined in note On the basis of these statements Erdelyi found relations connecting the solutions of and with I' = ..\ = with each other for different values of p by means of the Erdelyi-Kober operators. The results of Erdelyi were generalized by Lowndes who proved Lemmas and the relations in We also note that such an idea embryo was in fact suggested in Poisson "Expressions of conformity" of the form were indicated by Weinstein though such results had already appeared in Darboux The presentation in follows the papers by Marichev with in certain modifications in and with inserting the regulator (signxt + and Notes to 41.1 and 4:1.2. Equation as a special case of a more general equation was first obtained by Euler p. in connection with his investigations concerned with the motion of air in pipes of different section, and with the vibration of strings of variable thickness. Euler found the solution of with < = p• < Such an equation of the form with q = was solved by Poisson who obtained the hyperbolic analogue of the representation for the solution given in called the Poisson representation. The general solution of for p• = was given by Riemann p. He constructed the solution of the Cauchy problem by using a certain auxiliary function, and the method was called by his name later The fundamental solution of with q = i .e. the solution of was first indicated by Beltrami in the case 2p = This result was extended to p > only in by Weinstein who found two representations for such a solution. We note that in much later than Euler and Poisson, with q =
§ 43. ADDITIONAL INFORMATION TO CHAPTER
::/:- 0
8
857
(41.23) (41.24) (1] (1968).(1-3] [ 1 -3], (1] (1951)(41.35) 41.4 - [2)[[.12]] (1823) (1985). R1 3, 1. (41.35) (41.36) [2] (1953) +1 , uP y1 -2pu1 -p , P (41.40). u uP u� u y (41.35) (40.20). 41.3 (8] (1983). (1] (1975) (41.46)-(41.49) (41.2) 0 41.5 [2] (1947). 41. 5 41.1/2.4 (5) (1976) D112y y/x (1] (1918)[1] (1919). 'D��2y y/x 2 nl} y (1y] fx,(1925), [3, 99] (1982), 1 J F('D�\ y (x); x)dx.
q was proved by Marichev. The relations in and were fmmd by Berger and Handelsman Theorem was proved by Gordeev In the case {3* = {3 this result in other notation was known earlier - Bitsadze Smimov and Gilbert The space was introduced by K.I. Babenko who proved Theorem see also Notes to .§ 41 .3. Equation was first considered by Poisson in the case n = p :::: The Cauchy problem given in and was investigated by Weinstein for different values of the parameter p. He obtained the solution of this problem in the form Weinstein indicated and used the relations = connecting the solutions = of the equation of the fonn with each other for different values of p The presentation in subsection follows the paper by Berger and Handelsman except for obtained by Lowndes Notes to § 41 .4. The Dirichlet problem for with q = was first considered by Velma In particular, he proved Theorem in this case. Theorem for any real q as well as other results in subsection were obtained by Marichev with certain more accurate definitions in the case p = Notes to § 42.1. The paper by O'Shaughnessy was probably the first where the methods for solving the equation = were discussed. Two solutions of such an equation were suggested by O'Shaughnessy and discussed by Post They were essential
[1]41.(1975). 3
=
different because really they were the solutions of two different equations, namely =
and Mandelbrojt
=
but this was not taken into account by O'Shaughnessy and Post. Later see also Volterra p. arrived at a differential equation of
fractional order in investigating the extremum problem for the functional
0
Mandelbrojt had assumed that the corresponding variations are equal to zero and obtained the differential equation of fractional order = with Cauchy conditions. The paper a > by M. Fujiwara where, in particular, the equation = containing the Hadamard fractional differentiation operator defined in was considered, may also be regarded as a prehistory of the theory of differential equations of fractional order. The first serious step in this theory was made by Pitcher and Sewell who proved Theorems and on the existence and uniqueness of the solution of the Cauchy-type problem for the equation under other conditions than those in subsection = Barrett obtained the solution of provided that the conditions in hold. These results were generalized later in the papers by Al-Bassam Al-Abedeen and Al-Abedeen and Arora They proved certain theorems similar to the corresponding theorems from the theory of linear ordinary differential equations. The presentation in § is based on the simplified results from the above papers. Notes to § 42.2. The main presentation in this subsection follows the paper by M.M. Dzherbashyan and Nersesyan The solution of is given in M.M . Dzherbashyan and the solution of is well known, for example, Titchmarsh Notes to § 42.3. The Dirichlet problem for was investigated by M.M. Dzherbashyan Nahushev and Aleroev The presentation in § follows the above papers by Nahushev. Note to § 42.4. The presentation in this subsection follows the papers by Didenko Kochura and Didenko Notes to § 42.5. The idea of applying fractional integro-differentiation to constructing solutions of ordinary differenti� equations was first suggested by Liouville who considered Using the idea of Liouville this equation was studied by Holmgren Sohncke and Letnikov part Ill] A more complete and detail investigation was given by Letnikov.
Fa ('D�+y(x); x] 0 1 )ay(x), 0, (!Di)(x) (ax y 'Di(18.54) (1] (1938) 42.1 42.2 ('D� y)(x) f(x, y) 42.1 . (1] (1954) + {42.11) {42.10) ( 4 ) (1965), [ 8 ] (1982), (1] (1976) [1] (1978). 42.1 [6(42.] (1968). (42.81) (1]. (7), 8"(42.) 26) [8] (1970), (4) (1976), (5) (1977) [1 , 2] (1982). 42.3 (173) (1984}, [1] (1985). (3] (1832), (42. 4 4). [2] (1867), [1] (1867) (4, (1874). [1] (1933)
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
858
Equation (42.50) was also investigated by many authon. The methods of fractional calculus were used by Letnikov (9] (1888) , Nekrasov (2] (1888}, (4] (1891) and Karasev (1] (1957), and also Alekseevskii (1] (1884). The presentation in. § 42.5 follows the papen by Letnikov (4] (1874), [9] (1888) and Holmgren [2] (1867).
43.2. Survey of other results (relating § § 4G-42) 4:0.1 Erdelyi [8, 10, 14) proved the statement for the second Erdelyi-Kober operator in (18.8) similar to Lemma 40.2 in the case = 0.
A a > o, I E 2 (o, ) x2'1- 1 /(x) x2'1 /'(x) L,<�>K-,,aJ(x) = K-,,a L,<�_>01 /(x) L��) K,,a L��> cx-2'1 f(x)) = x-2'1L(_:J f(x) I0a+ ;�2 L(�), l = L(�) 1001+ ;�2 I' 1-01;�2 L,(�) l = L(�) 1-01;�2 I
Lemma 4:3.1 . Let Then
where
antl
oo tuul
C
antl
be integrable at infinitr.
are giv en in (40.22) antl (18.8), re•pectiv elr.
It was noted that by using the relation 'l-Ot
the relations
'1-0l
follow from Lemmas 40.2 and 43.1 under the appropriate assumptions indicated by Erdelyi [8). These relations were generalized by Lowndes [5], thus
J>.01L'1(�) / - (L'1(�)-a A2 )J>.01 /' R>.01 L'1(�) / - (L'1(�)-a - A2)R>.01/ J>. (f1,a), R>. (f1 ,a) Jf R� Jf = x2a+2'1 J>.(fJ,a)x-2'1 /, R�J = x-2'1 R>.(f1,a)x2a+2'1 f. � J>.f(x) = J�f(x) = Jl f'(x) = j Jo(AJx2 0 /(0) = +
where the operators
and are connected with the generalized Erdelyi-Kober operators defined in (37.45) and (37.46) by the expressions
In particular (Lowndes [7]), the following representations hold
t 2 ) f ' (t)tlt
0;
00
R>.f(x) = R�f(x) = -Rl!'(x) = j Jo(A Jt2 � = A j pJl (AJt2 � - 2 f( ) J>. R>. x2 )J ' (t)dt
00
f(x) -
provided that t 1/
t 2 - x2
x2 )J(t)dt,
t - 0 as t - oo and the operators inverse to
and
are evaluated from
§ 43. ADDITIONAL INFORMATION TO CHAPTER
8
859
{J� ) -1 f(x) Ji� f(x), (R� ) -1 f(x) Ruf(x) 3,37.57 37.58) . 7 40.1. f E C2 (b, ) b > 0, x-1/2f(x) -+ 0, x1/2 /'(x) -+ 0, x-1/2 f"(x) -+ x -+ R� f"(x) = ( � - ..X2) R� f(x), 0. o, 5 > f E C2 (b, ) b > 0, j(A:) (x) O(e-6� ) 2 f(x). k 1, Ri� f"(x) ( � + ..X ) �� mma 43.2 r � 1 e (..X r ) 11 r 11 K o (� - _x2)11 0 (�3 - .,X2 )11 0 u r, r J�aux2 + y22, + uyyu +r-1 , r 0 Jx2 + y2 + z2 ,11 R u �2u u�� + uyy 0 Ip(fl, a) 39.2 37.� 4.), 2 "'Y( ) x1 -"'Y -x dxd 1+"'Y-dxd x2 dxd 2 + 1 + -y)x-.d +m m 0, [x10'1+) 1 f(x) -+ 0 xa-+> 0, f E C2 (o,b), b > 0, x'1 j( )(x), 0, 1,
the results = = ) and { - { that another form of the first relation was indicated by Vekua [ p. 69). Lowndes [ ) also proved two statements similar to Lemma oo ,
Lemma 43.2. Let oo.
48
= O,
0,
a.nd
.X �
Then
Lemma 43.3.
oo ,
Let
=
2. Then
We mention
=
a.nd
48 X -
oo ,
.X �
All these results were applied to solving certain boundary value problems for the Laplace equation with mixed boundary conditions. It was noted that on the basis of Le we can and = of the generalized Helmholtz-type obtain the fundamental solutions = equations = and = from the fundamental solutions = - In = = = of the Laplace equations and = = and = u�� using the result = Uz z = Let be the operator given in § (note M
Lowndes at and
�
=
proved that if as
p
(
=
dx
m
=
2, is integrable
O, then
i..X, 0.
where = .X or p = .X > He obtained a similar statement in the case a < O, and applied these results to finding the complete set of solution of ( ) and the equation
40.18
p > -1/2, 0. �) 40.2, 40.3 43.1 L� I�,{J,e 18.6 a > o, e 1 f E 02 (0, b), b > 0, x11+ A: (lnx)6 j(A:) (x), k II 1,=
from the corresponding complete set of solutions of the above equations with .X = The results similar to Lemmas and and connecting the operator the operators 1;: •e
with
and defined in § 23.2, note were obtained by Kilbas, Saigo and Zhuk [ ] For example, they proved that if and {3 are real nwnbers, min(O,( - {3), 5 = if ( "¢ {3 and 5 = if ( = {3, = O, 2, is integrable at then
00, 1 .
40.2--40.3
40.2. The method used in §§ and based on Lemmas applied by Erdelyi [ ] to the generalized Stokes-Beltrami system
11
11) {2p 1
(2p
40.1--40.3, 43.1--43.3 { 43.1 ) {41.3" )
was
the solutions (u, of which called + 2)-dimensional conjugate symmetric potentials. Such an idea is conditioned by the fact, follows from ( ) that u is a solution of with r = called a + 2 )-dimen8iona.l 8ymmetric potential. Extending the investigations by Pahareva and Virchenko [ ] and also Polozhii [ ] Erdelyi [ p. 221] proved that if (u, 11)
y,
43.1 , 1-3 , 11,
860
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
is a (2p + 2)-dimensional potential, then
(f71>p- 112 u, ,cr
y 2cr L0(Ycr) v) is a (2p + 2a + 2)-dimensional ,
potential provided that p > -1/2, p + 01 > -1/2. Here ���l is the Erdelyi-Kober operator I, ,a given in (18.8) and applied with respect to y. Erdelyi [14] used the technique of fractional integro-differentiation to extend the results of Friedlander and Heins [1] who considered (40.19) with 0, i.e. (41.25) with q 0. Erdelyi applied his idea to derive representations for the solutions of the form (41.6) from the solution of the wave equation. Earlier this idea was used by Copson and Erdelyi [1] to investigating solutions of a certain boundary value problem for (40.19) with � = 0. 40.3. Representations more general than (41.22) of the solution via analytic functions of the form (41.22) for (40.18) with 0 and of the form (40.33) for (40.18) with � = 0 were obtained by Krivenkov [1] and Henrici [1], respectively. 40.4. Copson [5] considered the Dirichlet problem for the hyperbolic equation (40.19) with � 0 in the quadrant > 0, y > 0. By using the Riemann method he constructed the solution of this problem for < y and > y. He then showed that this solution and its derivatives are continuous for sufficiently large I" + p when they pass across the line y = provided that the given boundary values and ( , y ) satisfy the equation if p, and 0) if p > Here I, , a is the Erdelyi-Kober the equation = operator defined in (18.8).
I"= � =
=
I" = � =
=
xx
x u(x,O) u O u(O, x) H�!���� x-1 111- l ,p-llu(x,
I"·
u(x,O) = u(O,x)x, I" =
40.5. Weinstein [7) studied certain properties of the operator applied them to investigating the solutions of the equation
Pk = In particular, he proved that if
u(xl tx2 , ... ,xn)
is any
L�z) defined in (40.22) and
const .
(43.2)
solution of this equation, then
Pl = · · · = Pn =
is another solution. In the case 0 this property is sometimes called Kelvin's theorem. 40.6 . Radzhabov [1-4] and Radzhabov, Sattarov and Dzhabirov [J-2], and also the papers cited there, investigated in detail the properties of solutions, including fundamental solutions, for equations of elliptic type of the form (40.18) and (43.2) with singularities in coefficients, and for certain their analogues and iterative generalizations. They obtained integral representations for the solutions of these equations and the solutions of the certain boundary value problem such as Dirichlet, Neumann, "mixed" , etc. They used and developed the ideas connected with the relations given in (40.20) and Lemma 40.2, and the methods by Gilbert [2] and Weinstein [6]. They found the structure of solutions for the iterative equations of high order decomposed as compositions of equations of the form (43.2). Such equations of mainly hyperbolic type, and the Cauchy and Cauchy-Goursat boundary value problems for them were investigated by Kapilevich [1-4) , etc. He used the technique of hypergeometric functions of several variables. In particular, in the papers [1-3] he first constructed the Riemann and Green-Hadamard functions for (40.19), and found the solutions of the Cauchy and Cauchy-Goursat problems. In [4) Kapilevich obtained the solution of an analogue of a half-homogeneous problem given in ( 41.36) for the equation
(-asa22 asa 2 - -aax22 ) m u - c2m u = O a
+ -- + b 8
861 8 oFm -t(at. ··· •am -1 •z) [1]. = [1, 2] e = emU(x,uzz =UeeU0(x) U {x,O) = U (x), [1]. 2p = e 2)-1 1, r = (1p - 1/2, 2p)e1/(1-2P) U(x, e) = u(x,r) [1 . 2] [) U(x, '7) e= U0(x) Ut (x). [2] U(x, '7), Ut(x) ���-1 (U0(x) [1] -U0(a)) em uzzLq .uu = = e= § 43. ADDITIONAL INFORMATION TO CHAPTER
in terms of the integral operator involving the function in the kernel Erdelyi,. Magnys, Oberhettinger and Tricomi + 0 40.7. Chen investigated properties of the solutions of the equation near the singular line 0 with dependence on properties of the functions 0} and in particular, on their Holder nature - Chen Making the changes and he reduced this equation to the m (m + one given in (41 .311 ), 0 < < and applied the method of analytic continuation in a complex domain in two variables to the latter equation. This method is presented in the papers by Lewy and it is adjoined to the known method given in the book by Vekua 3 . According to such an approach each solution of the above equation corresponds to a certain analytic function, the real and imaginary parts of which are expressed for 0 via fractional integrals of the functions and In this paper Chen indicated such integer solutions fractional derivatives of which have the norms satisfying special estimates in terms of the norms of or in the space Usanetashvili proved the existence and uniqueness of the regular solution of the mixed boundary value problem for the equation + o, m const > 0, in which the values of the conormal derivative of the desired solution are specified in the elliptic part of thedmundary, and the conditions on the line of degeneracy o contain the ruemann-Liouville fractional integrals and derivatives defined in subsection 2.3. 40.8. The uniqueness and existence of the solution of certain boundary value problems with boundary conditions involving the Riemann-Liouville fractional inte�differentiation operatoiS were proved for equations of "mixed" type by Hasanov lsamukhamedov and Oramov Ivashkina and Nevostruev Ktunykova Nahushev Salahitdinov and Mengziyaev For the generate hyperbolic equations they were proved by Kumykova Orazov Salahitdinov and MiiSubarov Nahushev and Borisov and Zhemukhov investigated the first, second and mixed boundary value problems for the loaded parabolic equation, and the Darboux problem for the second-order degenerate loaded hyperbolic equations, respectively. The loaded parts contained the Riemann-Liouville fractional integrals and derivatives - § 43.2, note 42.5. 40.9. Kochubei considered the Cauchy problem
[1, 2].
[1],
[6],[1], [2. 3],
[1],
[1]
[1]
[1), [1]. [2],
(2]
(D(a)x)(t) = x{ ), A t
0<
t
<
T,
x(O) = xo ,
for the equation with the closed linear operator A in a Banach space, and the "regularized" fractional derivative
vg+
(2.22).
where is the fractional differentiation operator defined in He found conditions on the resolvent (A - ..\E)- 1 of the operator A which yield the unique solvability of the Cauchy problem. The results were applied to the ''mixed" problem for differential equations in partial derivatives with fractional differentiation in the "time" variable. The case when A = L is the second order elliptic differential operator in n variables was especially treated in Kochubei The corresponding Cauchy problem:
(3]. (D�a)x){y, t) = Lx(y, t), y E Rn , �p(y) y [1].Rn
t
0< <
T,
u(y,O) = �p(y),
known as the fractional diffusion problem, was proved to have a unique solution under the appropriate assumption on and the coefficients of the operator L. In the case when L is Laplacian, the fundamental solution was explicitly found in term of the Fox H-function, and its integrability in over was proved. In this connection we refer to the papeiS by Wyss and Schneider and Wyss
[1]
862
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
40.10. Chanillo and Wheeden (1] applied two-weighted estimates for the fractional integration operators - § 29.2, note 25.8 - to obtain infonnation about the number of negative eigenvalues of generate elliptic operators in divergence fonn. Kerman and Sawyer [1) used their characterization of the so-called trace inequality for potential operators to study the domain and essential spectrum of the SchrOdinger operator H = - 6 - V, where 6 is Laplacian and V is a nonnegative measurable function on Rn . 4:1.1 . Following Vekua [2] (Theorem 41 .5), Olevskii [1] solved the Dirichlet problem for (41.2) with q = 0 in the cases of a half-sphere and a half space in � , under certain conditions on p. In the case of a half-sphere this result was extended by Huber [1] to arbitrary p. The analogues statement for the equation of the form (40.18) with � = 0 in the case of a quarter sphere in Rn was obtained by Hall, Quinn and Weinacht [1) for any p and J.l. · Vollrodavov [1] and Evsin (1 , 2] constructed the fundamental solutions of (41.2) , and solved the Dirichlet problem for the semi-disk {:�:2 + 112 < 1 , 11 > 0} and the Neumann-Dirichlet problem for the domain for which the boundary contains the interval (- 1, 1] of the axis O:z:, respectively. 4:1.2. The Cauchy problem given in (41.35) and (41 .36) was studied by Diaz and Weinberger [1] , Blum (1] considered all values of p including the singular ones, 2p = -1, -3, -5, . . . . We also note that one may find reviews of investigations concerning the singular Cauchy problem given in (41 .36), and the regular Cauchy problem with the conditions of the fonn (41 .36) on the line 11 = � > 0, in the papers by E.C. Young [1] and Asral [1], respectively. 4:1.3. Saigo [2, 4, 5], see also [6-8], studied the three boundary value problem Jor the Euler-Poisson-Darboux equation (41 .1) in the domain < e < 11 < 1 with boundary conditions involving the integral operators I�f·" and I�!·" given in § 23.2, note 18.6. The first Goursat-type problem the following boundary conditions
o
has
IOa,b, {J * -a
+
-1
u (O, Tl ) = cp1 (Tl ) ,
Two other problems are the so-called problems with shift, and their investigation was begun by Nahushev [1]. In this context we refer also to Bzhikhatlov, Karasev, Leskovskii and Nahushev [1] . The boundary conditions have the form
in the second problem and
Aeb+ fJ+fJ* - 1 �: ,p• -a - 1 u (O , t) + B(1 - t)c+fJ+/3* - 1 I;�fJ -/3 * ,c ,p• -a - 1 u (e, 1) = cp2 (t ) Here A , B, a, b, c, d are given constants and cp 1 and cp2 are given functions in
in the third one. all three problems. The second problem coincides with that in the paper by Nahushev [1] in the case when A and B are given functions and {3* = {3, a = -b = -c = {3 - 1. All these problems were reduced to singular integral equations with a Cauchy kernel, and methods from the book by Gahov [1] were applied to their solution. Srivastava and Saigo [1] expressed solutions of the above problems in terms of Kampe de Feriet functions of two variables. They also considered some of the special circumstance in which the Kampe de Feriet function can be reduced to the relatively simpler hypergeometric functions. We also note that Orazov [l] generalized the results of Nahushev [1]. Volkodavov and Repin [1] solved one more problem of such a type with the operators I�/·" and I�!·" . Repin [1)
863
§ 43. ADDITIONAL INFORMATION TO CHAPTER 8
investigated the above second boundary value problem for the equation y 2 uzz - uyy + duz = 0 in the special domain. 41.4. By using a Fourier transfonn in the space of the generalized functions Bresters [3) constructed the solution of the Cauchy problem given in (41.36) for the equation
n 2 2 """" a u - a u 2 2 � ax i 8y i =1
au - ..\2 u = 0. - 2p !I 8y
A relation for the solution has the form
u(x, y) =
2 ISn l
f1 -r(x + yt)
-1
cos
( ..\y y'i":ti)
y'i":t2
dt
- (41 .37) - and generalizes the classical solution for such a problem from the paper by E.C. Young
[1].
41.5. Bureau [1-3] investigated the Cauchy problem for the equation in partial derivatives of the hyperbolic type. In particular, the wave equation ( 40.19) and the Euler-Poisson-Darboux equations (41 .1) and (41.35) were considered. He used the concepts of a finite and logarithmic parts of divergent integrals denoted by pJ and pi respectively, and related to the Hadamard definition - § 5.5. In particular, by using these ideas some results from the papers by Weinstein [1-3) were extended. We give some of his definitions. Let A (t) e em be represented in the fonn
m-1
A(t) = L: A��x) (t - x) i + Bm(t), Ai (x) = A( i ) (x), i =O in the neighbourhood of a point x and let
J A(t)(x - t)•dt, Z'
I.(x) =
Q
Then we set
/ A(t)(x - t)-m-l' dt - P(e; m - 1, m + l')]
z-�
[ p fl-m -l' (x) = �lim -o
Q
J Bm(t)(x - 1)-m-l' dt, Z'
=P(t� - x; m - 1, m + I') +
Q
.
- 1 (x) p /I-m (x) =P(t� - x; m - 2, m) + (-1)m - 1 Am (m - 1 ) ! ln(x - t�) +
Z'
f Bm t(t)m dt, (x - )
Q
864
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
(x) . pll.(x)
A p II-m ( ) =(-1)m m- 1 (m - 1)!
x
'
= 0,
where 0 < I' < 1, m = 1, 2 , 3, . . . .
•
#; -m,
ll.
Bureau proved basic properties of p I I. and p such as various estimates, commutation with the operator of differentiation and connection with the Cauchy principal value of an integral. He extended all of these results to multidimensional integrals, and made a survey of the applications of these ideas to partial differential equations.
41.6. Cheng [1, 2] investigated some "mixed" boundary value problems with a condition on the singular line for (41.25) with q = 0 and its generalization. 41.7. Wood [1] investigated the problem of finding the fundamental solution of the equation = 0 provided that the fundamental solution for the simpler equation uxx = 0 is known. In the case = this problem was solved by using Uxx fractional integration.
2 -2 ++ uyyu.u+uu + w2+w c-2 (z)uc (z)u
c(z) z
41.8 . Clements and Love [1] considered two "mixed" Heumann-Dirichlet problems concerning finding a function = harmonic in the half-space > 0 provided that the values and are given in different parts of the plane = 0 with a jump for r = a and = b. According to Copson [3] the solution of these problems the form
2 2 Vz V V( r, z ), r Jx + y ,
r
hasz
z
!p fP) l /2 V(r,z) ..!.. f u(p)p8p f (z2 + r2 - d2rpcos 00
=
211'
0
1r
- 1r
and therefore can be reduced to solving in succession, the Abel-type equations
f Jxpu(2p)dp2 (x), f Vpu (2p)-xdp2 h(x), p b
X
a
and the equation
I(
X
)+
= It
_
P
X
2( -1) 6 11'
where 6 = 0 or 6 = 1.
=
f (xt)6xl(t) 2t2 dt - ( ) O < x < c= �, c
0
1-
-
1/
X '
41.9. Belonosov [1, 2] investigated certain boundary value problems for the biharmonic equation = 0 in the case of plane two-connected domains. To solve these problems he used fractional differentiation in a complex plane defined in (22.4) with £ = (-ioo , ioo) and the expressions connecting with the Laplace transform - § 7.2 and § 9.2, notes 7.2 and 7.4.
�2u
a) j(l(a)
41.10. Shinbrot [1] indicated certain sufficient conditions for the existence of a weak solution with the fractional derivative 1)�+ 0 < 01 < 1/2, in the "time" variable for the Navier-Stokes equation. He obtained the norm estimate in for such a derivative. This problem was first considered by Lions [1) who found such an estimate in the case 0 < 01 < 1/4 under certain assumptions on the dimension of the space.
u
u,
L2
u
41.1 1 . Senator [1] obtained Schauder and Lp estimates for the solution of elliptic boundary value problems with the boundary conditions containing pseudodifferential operators with non-smooth symbols and normal derivatives of fractional order.
8
§ 43. ADDITIONAL INFORMATION TO CHAPTER
[1] w(x, + V0+w(x, = 0, x 0, 0, 0 < < 1, �: o llw(x, - /o(t)I IL = 0, 1 � < oo,
41.12. Berens and Westphal d
dx
>
t)
t)
-
1>6+
considered the Cauchy problem
"Y
t)
lim
where
865
-y
t>
p
P
(5.�:6) = W� w ) [1] y(x)
is the Riemann-Liouville fractional differentiation operator defined in
applied
with respect to t. They constructed the solution of this problem in the form fo where in Lp(O, oo) for any > w.., is a semigroup of a class 42.1 . By using the Picard method and the Schauder fixed-point principle, Tazali proved two theorems which yield conditions for the existence of the solution of the Cauchy-type problem
Co
x 0.
('V:+ y)(x) = f(x, y(x)), a < x � a+ h, h > 0; (v:_;1 y)(x) l�:=a = b, 0 < a � 1.
a = 1. [a(10. , b) 19)
These statements generalize the known results by Caratheodory in the case Grin'ko proved the existence and uniqueness theorem for the solution of a nonlinear differential equation with a generalized fractional derivative inverse to in the Holder weighted space on a finite interval of the real line. He also constructed the approximation solution of this equation, and obtained the estimate for such a solution. Arora and Alshamani investigated the stability properties for the Cauchy-type problem given in and Hadid and Alshamani proved the estimate at infinity for the solution of this problem under the appropriate assumptions on 42.2. A series of papers is concerned with the investigation of systems of linear differential equations of fractional order mainly in the space V' of generalized functions - § notes to §§ and Veber found the solution of the Cauchy problem for the system of equations 0 with a constant matrix The asymptotic behavior as oo of different solutions of this system including the fundamental matrix was studied by lmanaliev and Veber and Veber The Cauchy problem for the system
[1] H�([a, b), p), p(x) = (x - a)�'(b - x)", (42.2) (42.3). [1] [1]
8.1 8.[32]. Ay(x), < a < 2, [1] [5, 6].
f(x, y).
A.
y(a) (x) = A(x)y(x) f(x), 1 < a � = 1,2, ... , A(x) x A(x) = A = +
n-
n, n
9.1 , x - y(a)(x) = [7), [1].
(43.3)
with the continuous matrix-function for � 0 was investigation by Veber and the fundamental solution of such a system with a constant matrix const was considered by Veber The case of a single equation was considered in the paper by Veber Veber proved the criterion of "passivity" for the systems
[8].[6]
Ay'(x) By(a) (x) + Cy(x) = g(x), < a < 1, (43.3) x - oo. (2; A, B,86,C,278) 0 < a[1;< 1 34, 58, 209]. [1], +
0
(43.4)
more general than with constant matrices and found quasi-asymptotic expansions of their fundamental solutions as The definition of the above concepts can be found, for example, in the books by Vladimirov p. and Vladimirov, Drozhzhinov and Zav'yalov p. We also note that the solution of the equation of the form with was constructed by Seitkazieva who used the method based on the Laplace transform.
(43.4)
866
CHAPTER
8. [1]
APPLICATIONS TO DIFFERENTIAL EQUATIONS
Bykov and Botashev equation of the form (43.4):
y'(x)
reduced one problem in studying (watering the furrows) to an
,\ I[1 + b(x-t)-a]y'(t)dt, 0 < a < 1. �
= q -
0
One may find other examples of applied problems reduced to differential equations and systems of differential equations with fractional derivatives in the paper by Veber 42.3. Srivastava, Owa and Nishimoto proved the following statement in terms of the fractional differentiation operator fv(z) := (V + lr f)(z) and - in the case , when = .C"' (z) or = .C o (z).
0
0
Theorem 43.1. Let
=
[8].
[1]
exp
0
(22.17), (22.18) (22.21)
be 4n41ytic in 4 dom4in
D in the complex pl4ne z.
{ I "'(+11· 1; } I ( { I +· 1; d })
z)
..,..
'
z)
dz
z)
exp
4nd f11 (z) exiat., then the equ4tion fv(z) = �.p(11 ; z)f(z), z
'"
If
- II
E D, h4a 4 aolution of the form
[1]
In the paper by Nishimoto, Owa and Srivastava such a result was obtained for more general equation fv(z) =
+[1 , 2]
was considered. Special solutions for such equations with power-type coefficients were found by Wiener Hadid and Grzaslewicz and Campos 42.5. Nahushev considered the problem of the proper solution for the equation
[3],
[1] [2] Ka = vg+ xP
j= l
(9].
a; ( x) V��
+ b(x)
where
o1 ([o, 1]), a; > 0 0 < a < 1, f3 > 0, a > a1 > · · · > 0 > · · · > am , { b(x), c(x) E 0([0, 1]). o([o, 1]), a; < 0. 0-ylx({O,"Y
Let max �e[o , t) then for
Op {(O ,
1)) be a Banach space of functions and let "'Y be a constant. Nahushev proved that if
Oo ({O,
if
if
with the norm l l
=
there exists a unique solution of the above equation in the space
§ 43. ADDITIONAL INFORMATION TO CHAPTER 8
867
[3] ym uz:z: uyy a(x,y)uz: b(x, y)uy c(x, y)u = 0 D E {y 0} 0 << xx << 1,1 Ox. K u = \li(x), 0 1 K1 Ka a = 1. y-o 42.6. [1, ] u"(x) a(x)'D�\u(x) = f(x), 0 x 1, 0 <= 0,< a(x)1. = A P 0 u(O) Pu'(O) = u(1) = 0 (x) f 0(0,1]= A c2 (0,1], au(O) Pu'(O) = , au(1) ,8u'(1) = "(, a(x) '"" I AA: , -2 1 2 a) A:= l A = AA: a(x) =A A,= A u(O) = 0, u(1) = 0 A:E , ( - A) AA: (1.91). 2-a 2 AA: = O(k2-a ) k 142]. y { n ) - Ay (1] [1]. 42.7. 151), (1) In another paper Nahushev
studied the ''mixed" boundary value problem for the equation
+
+
+
+
in a bounded domain > the boundary of which contains the interval axis The boundary condition on this interval has the form lim
of the where
is the value of the above operator when On the basis of the extremum principle obtained in the paper the stability and uniqueness of the solution of this problem were proved, and a question of the existence of such a solution was considered. Aleroev 2 investigated the spectrum of the Dirichlet problem for the differential equation �
+
when and
(43.5)
�
He proved that the problem with the conditions + for � does not have negative eigenvalues in the space of functions n and the problem with the conditions "( + + has eigenvalues but not more than a continuum set. The inequality at
L.J
� 16
00
r
(2 -
for the eigenvalues of the problem was also proved. We also note that earlier Nahushev (5) showed that is an eigenvalue of the latter problem if and only if is a zero of the Mittag-Leffier function defined in All these zeros, the moduli of which are sufficiently large, are simple zeros and they have the estimates as - oo - Dzherbashyan [2, p. The asymptotic behaviors of the eigenvalues, and problems of the uniform convergence for the differential operator given on a finite interval of the real axis with boundary conditions concentrated at the end points of this interval and containing the Riemann-Liouville fractional derivatives given in § 2.3, were considered by Bogatyrev and Amvrosova Using the relations of realization of operators, for example, see Gel'fand and Shilov [2 . p. Malakhovskaya and Shikmanter suggested a method for constructing a generalized solution of the Cauchy problem for the integro-difl'erential equation
y(t))a dt = f(x), x 0, Qn (_!!_dx_) y(x) - a I (x-t (O) = a , = 0,1, ... ,n 1, 0 < a < 1 Q42.8. n (x) Yj [1] j n. Otj , -1 Otj < O, (42.34} (1.91}. Otj = 1}/n ] ( } 30 z:
>
0
with the conditions j where and at is a rational number and is a polynomial of order Leskovslcii constructed the linear independent solutions of the homogeneous equation with specific distinct exponents in the form of a Mittag-Leffier � function defined in The special solution of the inhomogeneous equation (42.34) with (j was first found by Davis [2 - 4 .6 and § 4.2, note 2.5 and §§ and 34. An equation of the same type with constant coefficients
Aof(x) �n ;�J 1(y x)OtJ -l f(y)dy = o, Reaj J =l +
r
.)
c
_
�
>
o,
868
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
was studied by Alonso [1]. He proved that the solution of this equation has the form /(z) = e-'1% if and only of the parameter '1 satisfies the conditions Re, > 0,
�0 +
E �j'l-a; j=l n
= 0,
c = oo.
Alonso also investigated properties of solutions of the corresponding inhomogeneous equation in the case n = 1 . By analogy with linear ordinary differential equations with constant coefficients, Campos [9] presented a method for solving simple equations of the form (42.34) with fractional derivatives in a complex plane given in § 22.1, in terms of the roots of the characteristic peeudopolynomial
L: a; r a; j=l n
= 0.
42.9. Equations of the form (42.34) with piecewise constant coefficients were investigated by Kochura and Didenko [1]. 42.10. The abstract Cauchy problem for the equation of bypergeometric type
II (
II (
)
)
p d 9 d t d + fJ; - 1 u(t) - At t dt t dd + a; u(t) = 0, t > 0, t t j=l j=l with the closed linear operator A and constant coefficients aj and fJ; was considered by Bragg [1 , 2]. By using fractional integrals he obtained the expressions connecting the solutions of this equation for different values of parameters and applied these expressions to investigating the Cauchy problem for the degenerate equation of hyperbolic type
uu - tm u%% + vtml 2- 1 u% = 0, t > 0, m � 2, u(z,O) = cp(x) , u , (z , O) = cp( z) . 42.1 1. Yu. Rabinovich and Nesterov [1] considered the differential operators Kn u: (43.6) where Pn (z) are polynomials of certain degrees. They found the conditions under which the order n of the operator in (43.6) can be lowered by using fractional derivatives defined in (22.30) and (22.331 ), and in the relation similar to the one in (22.331 ) in the case zo = oo. In the paper by Nesterov [1] such derivatives were applied to constructing solutions of differential equations of the Fuchs type with • singular points
•
where Qm (z)
•
n u + E Q <• - l )j IT n -j u
o
is a polynomial of degree m .
42.12. A series of papers is concerned with the application of methods of fractional integro.differentiation to solving linear differential equations of second order of the form ( 43.6) with polynomial coefficients. Al-Bassam [2, 3, 5, 7, 9-12, 14-15], Al-Bassam and Kalla [1] considered differential and integro.difl'erential equations reduced by the Leibniz rule given in (17.11), and other properties of
§ 43. ADDITIONAL INFORMATION TO CHAPTER 8
869
fractional integrals and derivatives to operator equations of compositional type (43.7)
where
p(x) q(x) and
m
n (ak
are products of the form
+
b x)<XJc ei-'Z . Necessary and sufficient
k = l conditions under which certain classes of differential equations of second order are equivalent to
{ (
k
43. 7) were proved, and their solutions were constructed. Examples of the equations for Gauss, Hermite, Kummer, Laguerre, Legendre and Jacobi functions, for the generalized hypergeometric functions and aF , and for orthogonal polynomials were considered. On the basis of 43. 7) solutions of these equations are represented via the corresponding fractional integrals and their compositions - § 9.3 and § 10. In this connection see also the paper by Al-Bassam 1 3 where equations with solutions including generalized power series and, in particular, analogues of exponential, trigonometric and hyperbolic functions were considered. Higgins suggested an original method for obtaining general solutions of the inhomoge neous hypergeometric equations. This method is based on the application of direct and inverse Laplace transforms given in {1.119) and {1.120) , and Erdelyi-Kober transforms defined in 18.1 ), {18.2) with respect to parametel'S. Nishimoto 7)- 1 3 , and also his papel'S published in J. Coli. Engng. Nihon Univ. 1988. given in § 43.2, note B-29, 1989. B-30, applied the fractional integro-differentiation operators 42.3 to studying special solutions of ordinary and partial differential equations of the Fuchs type connected with Gauss, Kummer; Laguerre , Legendre, etc. special functions. In this connection we refer also to Nishimoto and Kalla [1, 2 and Nishimoto and Tu Shih-Tong [1, 2]. 42.13. Fedosov and Yanenko proved that the equation in partial derivatives of half-integer order
2F2
2
[]
[6]
{
[ []
f11
[1]
]
a k - const , - § 24.2 - can be investigated by means of representating their operators as compositions of invertible operators of the form v where + a;v
¥,!
¥,�
n
z
1)+1/'!u(x, y) = y�1f .!!._dx j {x -t)-1.2u(t, y)dt n= = a1 = u(x, y) =1f-l/2 J J (6-e) 112 :e [J(e, y + at2 {x - 6)} - atf(s, y + at2 {x - 6) + e- s)]ded6 + { o,q(y + at2x), at � o, q(t) n= 42.14. [ 1 ] F; , j = D2 , F Fv = FD2 v
and ao
-co
Otj
0,
are the roots of a certain characteristic polynomial. In particular, in the case 1 the relation for the general solution of this equation z
-
oo
'
-
oo
Ot
where
1,
0,
>
is an arbitrary function was indicated. The case
Yaroslavtseva constructed the operators operator P into i.e. the operators with the property P
2 was also considered in detail. 1 , 2, for transformation of the on the elements 11 of a
870
CHAPTER 8. APPLICATIONS TO DIFFERENTIAL EQUATIONS
certain function space where
P
=
-
(
d2 d -- + 211ctg
In particular, the operator F1 has the fonn
F1 v =
!p
2''r(11 + 1 / 2 ) . (sm
I0
)
,
v(t)(cos t - cos
for real 11 > 0, and the fonn of a convolution of a test function for other values 11.
v
with the functional sin2 11 (
These results were applied to reducing the Cauchy problem for the differential equation
n
E ak pn -k u
=
0,
k=O
a k - const ,
0 <
1r,
to the Cauchy problem for the ordinary differential equation
n
E (- 1)n- k ak w2n-2k
=
0.
k=O
42.15. Biacino and Miserendino [2] considered properties of the operator Lu =
E aa (x)D a u,
la l�4
lal = a1 + a 2 ,
with fractional derivatives D a u defined in the papers by Biacino and Miserendino [1 , 3] - also § 29.2, note 24.10. On the basis of the representation
Lu =
E
l a le{2 ,3,4}
aa (x)D a u +
E aa (x)D a u
l a 1< 2
the conditions for the index Lu to be equal to zero were indicated, and the mapping properties of the operator L is Sobolev spaces were studied.
42.16. Sprinkhuizen-Kuyper [1] proved a series of theorems about the solution of the Cauchy problem for the equation
( -) I ( - -) 1 d -x dx
d2 II d +dx2 x dx
k /(x)
.
=
g(x),
with conditions of the form J(j ) (1) = 0, j = 0, 1 , . . , l + 2k - 1 . For g(x) E C{[O, 1 )] the above problem has the solution f( x) = I�k ,I g (x) where
§
43.
ADDITIONAL INFORMATION TO CHAPTER
8
871
Iff• >. f(x)
E
1]}.
in the space f (x) c 2k+ '{[O, Some properties of this operator with a Gauss hypergeometric function in the kernel were also studied in this paper. These results were developed by McBride 9 and Dimovski and Kiryakova who considered operators more general than (9.5). For such operators their integral representations in the fonn of compositions of the Erdelyi-Kober type fractional integro-differentiation operators . given in and and in terms of the Meyer G - function of the fonn were found. Special cases of these representations which lead to operators with a Gauss hypergeometric function of the fonn were earlier considered by Sprinkhuizen-Kuyper 42.1 T. Tremblay and Tremblay and Fugere investigated properties of the operators
[7, ]
{18.1) (18.2) (10.[21]8)
where D = S = 0 or S relations
1; r, n, rjE (j 1,2,
{3, aj C(j = . . . , m) , nonnegative integers. In particular, the operator
fz, nr, a C, is the fractional derivative given in =
[1].
[1]
=
. . . , m)
{22.4); E
1,2,
{zD + aj - {38n + f3i )r; z68,8n D�l -6 )( 1 -8),8n , II II i
n-1 X
are
[2] {10.48),
m
=l i=l
where 8 = 0 or 8 = were obtained together with the representation for D�;�;�.� r�'01 "' in terms of the operators z ( 1 - 2"l' )w +( l -"Y) k n k z(2"Y- l )w +"l' k where 'Y = or 'Y = w C, k= . . . , n( rt + · · · + rm). A s examples the new operator relations for the usual differential operators D and integral operators n; 1 were given.
1
0, 1,
0
1, E
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Abel, N.H. 1) {1881) Solution de quelques problemes a l'aide d'integrales definies. In Ge•a.mmelte ma.thema.ti•che Werie. Leipzig: Teubner, 1, 11-27. (First publ. in Mag. Naturvidenbbeme, Aurgang, 1, no 2, Christiania, 1823). 2) {1826) AuftOsung einer mechanischen Aufgabe. J. fiir reine und angew. Math., 1 , 153-157.
Adamchik, V.S. and Marichev, 0.1. 1) {1990) The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system. In Proc. Intern. SJmp. SJmbolic a.nd A lgebraic Comptda.tion (ISSAC-90) , (Tokyo, 1990}, ACM Press, 212-224.
D.R. 1) {1975) A note on Riesz potentials. Duke Ma.tla.. J., 42, no 4, 765-778. Adams, D.R. and Bagby, R.J . 1) {1974) Translation-dilation invariant estimates for Riesz
Adams,
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Adams,
R., Aronszajn,
Agarwal,
N. and Smith, K.T. 1) (1967) Theory of Bessel potentials, II. Ann. ln•t. Fourier, Grenoble, 17, no 2, 1-135.
R.P. 1) {1953} A propos d'une note de M. Pierre Humbert.
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2) {1969) Certain fractional q-integrals and · q-derivatives. Proc. Cambridge Phil. Soc. , 66, no 2, 385-370. 3} {1976) Fractional q-derivatives and q-integrals and certain hypergeometric transforma tions. Ga.nita., 27, no 1-2, 25-32. Ahern, P. and Jevtic, M. 1) {1984) Mean modulus and the fractional derivative of an inner function. Comple� Variable•, Theor, Appl., 3, no 4, 431-445. Ahiezer, N.I. 1) (1954) On some dual integral equations (Russian). Dokl. Aka.d. Na.uk SSSR, 98, no 3, 333-336. 2) {1957) To the theory of dual integral equations (Russian). Ucla. ebn. zap. Khar'kov. Univ., Ser. 4, 80{25), 5-31 . Ahiezer, N.I. and Shcherbina, V.A. 1) {1957) About inversion of some singular integrals (Russian). Ibid., 191-198.
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AUTHOR INDEX
A
Abdullaev, S.K. 590 Abel, M.H. xxi, xxiii, xxiv, xxv, 82 Adamchik, V.S. xxii, 174 Adams, D.R. 589 Adams, R. 583 Agarwal, R.P. 87, 317, 442, 443 Aggarwala, B .D. 793 Ahern, P. 451 Ahiezer, N .l. 302, 685, 693, 774, 779 Ahmad, M .I. 794 Ahner, J .F . 780 Ahuja, G. 172 Ahuja, O .P. 454 Akopyan, S.A. 440, 772 Al-Abedeen, A.Z. 857 Al-Amiri, H.S. 454 Al-Bass am, M .A. xxii, 316, 454, 857, 868, 869 Alekseevskii, V .P. 858 Aleroev, T .S. 857, 867 Alexitz, G. 445 Aliniov, S.A. 583 Alonso, J . 868 Al-Salam, W.A. 317, 442, 443 Alshamani, J .G. 865 Amvrosova, 0 .1. 867 Andersen, K.F. 166, 168, 590 Anderson, T .P. 85 Anderssen, R.S. 85 Andreev, A.A. 798
Ang, D.D. 659 Arestov, V.V. 313 Aronszajn, N. 432, 583 Arora, A.K. 317 Arora, H.L. 857, 865 Arutyunyan, M.H. 689 Askey, R. 314 Asral, B. 862 Atiyah, M.F. 584 Atkinson, F .V. 685 Atkinson, K.E. 85 B
Babenko, K.l. 821 Babenko V .F. 444 Babenko, Yu.l. xxii Babich, V .M. 798 Babloyan, A.A. 793 Badalyan, A.A. 89 Bagby R.J. 584, 589, 594, 601 Bagley, R.L. xvii, xxii Bakaev, N.Yu. 161 Baker, B .B 583 Bakievich, M.l. 774 Balakrishnan, A.V. 120, 161 Balasubramanian R. 85 Banerji, P.K. 793 Bang, T. 432, 433 Bari, N .K. 249, 443 Barrett, J .H. 857 Bavinck, H . 445 Beatroux, F . 603
954
AUTHOR INDEX
Beekmann, W. 315 Belinskii, E.S. 436 Belonosov, S.M. 864 Beltrami, E. 774, 856 Bel ward, J .A. 302 Belyi, V .1. 436, 449 Benedek, A. 580, 588 Berens, H . 161 , 170, 865 Berger, N . 305, 318, 319, 857 Bernstein, I.N. 584 Besov, O .V. 580, 589, 591 Bessonov, Yu.L. 580 Betilgiriev, M .A. 320 Bharatiya, P.L. 773 Bhatt, S.M. 315 Bhise, V .M . 170, 438, 782 Bhonsle, B .R. 776 Biacino, L. 432, 587, 870 Bitsadze, A.V. 820, 843, 844, 856, 857 Bleistein, N. 294, 305 Blum, E.K. 862 Blumenthal, L.M. 435 Bocher, M. 82 Bochner, S. 581 Bogatyrev, S .V. 867 Boman, J . 448 Bora, S.L. 170 Borisov, V .N. 861 Borodachev, N .M. 794 Bosanquet, L.S. xxxii, 83, 84, 160, 163, 164, 277, 312, 313, 315, 434 Botashev, A.l. 866 Bouwkamp, C.J . 790 Box, M .A. 783 Braaksma, B.L.J . 438, 777 Bragg, L.R. 868 Brakhage, H. 86 Bredimas, A. 162, 172, 371 , 434, 591 Brenke, W.C. xxii Bresters, D.W. 584, 863 Brodskii, A.L. 587 Brunner, H. 85 Brychkov, Yu.A. 15, 18, 20, 22, 27, 172, 174, 198, 201, 206, 282, 284, 325, 355, 372, 551 , 584, 586, 711 , 733, 739, 740, 742, 743, 755, 773, 774, 781
Bugrov, Ya.S. 371, 434 Bukhgeim, A.L. 90 Burbea, J. 603 Bureau, F .J. 863 Burenkov, V .1. 371 , 434 Burkill, J .C. 312 Burlak, J. 773, 774 Busbridge, I,W. 774 Buschman, R.G. xxii, 23, 431 , 695, 772, 776, 777, 784, 786, 787, 793 Butzer, P.L. xix, xxxii, 161 , 303, 371 , 433, 434, 444, 446-448 Bykov, Ya.V. xxii, 866 Bzhikhatlov, H.G. 689, 862 c
Calderon, A.P. 583 Campos, L.M.B.C. 452, 866, 868 Carbery, A. 163 Carleman, T. 449, 685, 687 Cartwright, D .1. 90 Cayley, A. 83 Center, R. W. xxv Chakrabarty, M. 789 Chan, C.K. 85 Chandrasekharan, K. 277 Chanillo, S. 584, 588, 590, 601 , 862 Chen Yung-ho 593 Chen, Y.W. 82, 84, 301 , 302 Cheng, H. 864 Cheng Min-teh 593 Cherskii, Yu.l. 695 Chiang, D. 689 Cho Nak Eun 454 Choudhary, B . 163 Chrysovergis, A. 432 Chumakov, F.V. , 688, 689 Chuvenkov, A.F. 168, 591 Civin, P. xxxi, 433 Clements, D.L., 864 Cohn, W.S. 451 Colzani, L. 602 Conlan, J ., 171 Cooke, J .C. 793 Copson, E.T. 583, 772, 774, 775, 779, 860, 864 Cossar, J . xxxii, 163, 315 Cotlar, M., 589
AUTHOR INDEX D
Darboux, G. 856 Davis, H.T. xix, xx, xxx, 87, 302, 434, 867 Davtyan, A.A. 595, 596 Delerue, P. 580 Deng Dong-gao 593 Di Giorgio, M. 587 Diaz, J .B. 453, 862 Didenko, A.V. xxii, 857, 868 Dimovski, I.H. xxii, 302, 436, 454, 871 Din Khoang An 777 Dinghas, A. 87 Ditkin, V .A. 24, 28 Dixit, L.A. 776 Doetsch, G. 162 Doktorskii, R.Ya. 85 Domingues, A.G. 600 Drianov, D.P. 447 Drozhzhinov, Yu.N. 602, 865 Duduchava, R. V. 302 Dunford, M. 642 Dveirin, M .Z. 449 Dwivedi, A.P., 794 Dyckhoff, H. 434, 444, 447, 448 Dyn'kin, E.M . 590 Dzhabirov, D.K. 860 Dzherbashyan, A.M. 453 Dzherbashyan, M.M. xix , xxvii, 22, 24, 83, 84, 87, 88, 89, 303, 345, 432, 435, 453, 661 , 686, 709, 713, 772, 841, 857, 867 Dzyadyk, V .K . 432, 444 .
E
Edels, H . 85 Efimov, A.V. 433, 444 Eggermont, P.P.B. 85 Elliott, J . 602 Elrod, H .G., Jr. 773 Emgusheva, G.P. 593, 596, 597 Erdelyi, A. xxxi , xxxii, 14, 18, 22, 88, 91, 160, 162, 163, 165, 172, 180, 193, 196, 199, 201 , 302, 319, 431 , 432, 436, 437, 441 , 455, 539, 551 , 661 , 677, 724, 733, 735, 751 , 772, 774-777, 783, 851, 856, 858, 859, 860
955
Eskin, G.l. 582, 599 Esmaganbetov, M .G. 444, 447, 568 Estrada, R., 162 Euler, L. 856 Evsin, V .1. 862 Exton, H. 443 F
Faber, V. 86 Fabian, W. 89, 448, 449 Far aut, J . 603 Fattorini, H.O. 161 Favard, J . 444 Fedoryuk, M.V. 285, 584 Fedosov1 V .P. xxii, 584, 869 Feller, W. 216, 303 Fenyo, S. xix Ferrar, W .L. 434 Fettis, H.E. 85 Fihtengol'tz, G.M. 485 Fisher, B. xxii Fisher, M .J . 83, 582, 599 Flett, T.M. xxxii, 84, 91, 160, 315, 450, 452, 583, 591 Fofana, I. 590 Foht, A.S., 587 Fomin, S.V. 3, 24, 328 Fourier, J. xxiii, 32 Fox, Ch. xxxii, 437, 442, 772, 773, 781, 783, 794 Fremberg, M.E. 583 Frie, W. 85 Friedlander, F .G. 860 Frostman, 0. 581 Fugere, B .J . 871 Fujiwara, M. 857 Fukui, S. 454 G
Gabidzashvili, M.A. 581 , 590 Gaer, M.C. 317, 453 Gahov, F.D. 199, 201 , 202, 605, 617, 622, 651, 684, 685, 695, 848 Gaimnazarov, G. 447 Gamaleya, R.V. 793 Ganeev, R.M. 685, 688 Garabedian, H.L. 315 Garding, L. 581 , 603
956
AUTHOR INDEX
Garnett, J .B. 92 Gasper, G. 163, 164, 783 Gatto, A.E. 590 Gearhart, L. 168 Geisberg, S ,P. 167, 433 Gel'fand, I.M. 162, 487, 548, 556, 576, 583, 584, 867 Gel'fand, S.l. 584 Gel'fond, A.O. 436 Gel'man, I.V. 167 Genebashvili, I.Z. 590 Gerlach, W. 85, 659 Gilbert, R.P. 857, 860 Gindikin, S .O. xxiv, 602, 603 Ginzburg, A .S. 600 Glaeske, H.-J. xxii, 440, 773, 792 Gliner, E.B. 798 Godunova, E .K. 314 Gohberg, I.Ts. 199, 302, 630, 646, 656, 675, 684, 685 Gomes, M .l. xxii Gopala Rao, V.R. 584, 601 Gordeev, A.M. 746, 857 Gordon, A.N. 774, 775 Gorenflo, R. xvii, xviii, xxii, 85, 86, 87, 659, 691 Gorlich, E. 434, 444, 447, 448 Goyal, S.P. 439, 777 Graev, M .I . 576, 584 Greatheed , S .S . xxv Grinchenko, V.T. 793 Grin'ko, A.P. 305, 440, 865 Grunwald, A.K. xxiv, xxv, xxx , 304, 371, 434, 435, 479 Grzaslewicz, R. 866 Gupta, H.L., 786 Gupta, K .C. 171, 782 Gupta, S .B . 786 Gupta Sulaxana K. 315 Guseinov, A.l. 251, 303, 304 Gutierrez, C.E. 590 Gwilliam, A.E. 451 H
Habibullah, G .M . 777, 784, 785 Hadamard, J . xxvii, 112, 345, 435 Hadid, S.B. 865, 866 Hai, D.D. 659, 691
Hall, M.S. 862 Handelsman, R.A. 288, 305, 318320, 857 Harboure, E. 590 Hardy, G.H. xxvii, xxviii, xxix, 84, 86, 88, 91 , 160, 161, 277, 304, 314, 315, 432, 433, 435, 450, 451 Hasanov, A. 861 Hatcher, J.B. 689 Havin, V.P. 582 Hearne, K. 85 Hedberg, 1.1. 581 Heinig, H.P. 166, 590 Heins, A.E. 860 Helgason, S. 581 , 591 Henrici, P. 774, 860 Herson, D,L. 161 Herz , C.S. 591 Hess, A. 792 Heywood, P. 161 , 685, 789 Higgins, T.P. 172, 302, 439, 772, 773, 775-777, 869 Hilbert, R.P. 805 Hille, E. 83, 85, 120, 161, 301, 315, 445 Hirsch, F. 161 Hirshman, 1.1. 452, 695, 772, 792 Holmgren, Hj . xxiv, xxv, xxvi, 82, 83, 316, 431 , 435, 580, 851 , 857 Botta, G.C. 315 Horvath, J. 582, 594 Hovel, H.W. 161 Hromov, A.P. 86 Huber, A. 862 Hughes, R.J . 161, 313 Humbert, P. 87 Hvedelidze, B .V. 302 I
Ibragimov, 1.1. 315 Il'in, V .P. 580, 589 lmanaliev, M .1. 865 Isaacs, G.L. 83, 312 Isamukhamedov, S.S. 861 Ivanov, L.A. 600 lvashkina, G.A. 861 Izumi, S. 444
AUTHOR INDEX J
Jackson, F.H. 442 Jain, N .C. 580 Jevtic, M. 451 , 604 Joachimsthal, F. xxv Johnson, R. xxii, 583 Jones, B.F., Jr. 584 Jones, D.S. 685 Joshi, B.K. 786 Joshi, C.M. 88 Joshi, J . M . C . 437 J uberg, R.K. 303 K
Kabanov, S.N. 86 Kae, M . 602 Kalia, R.M. 774 Kalisch, G .K. 162 Kalla, S.L. xxii, xxxii, 170, 439, 778, 782, 786, 792, 868, 869 Kant, S. 786 Kanwal, R.P. 162 Kapilevich, M.B. 746, 747, 798, 860 Karapetyants, N .K. 84, 91, 92, 169, 308, 311, 432, 685 Karasev, I.M. 858, 862 Kashin, B.S. 92, 310 Kashyap, N .K. 778 Kato, T. 161 , · 685 Katsaras, A. 443 Kaul, C.L. (see also Koul, C.L.) 317, 318, 584, 782. 786 Kelland, P. xxv Kerman, R. 862 Kesarwani, R.N. (Narain Roop) 793 Khan, A.B. 443 Khan, M .A. 443 Khandekar, P.R. 786 Kilbas, A.A. xvii, xxii , 84, 303, 305, 319, 434, 435, 440, 448, 589, 603, 859 Kim Hong Oh 451 Kim Yong Chan 451 King, L.V . 774 Kipriyanov, I.A. xxxii, 580, 585, 600 Kiryakova, V.S. 302, 439, 454, 580, 778, 871
Kishore, N. 315 Klyuchantsev, M.l. 441 Kober, H. xxxi i, 83, 160, 162, 302, 303, 431 , 436, 453, 772 Kochubei, A.M. 597, 861 Kochura, A .I. 857, 868 Koeller, R.C. xxii Kofanov, V.A. 315 Kogan, H.M. 690 Koh, E.L. 171 Koizumi, S. 452 Kokilashvili, V.M. 308, 581, 590, 591 Kolmogorov, A.N. 3, 24, 328 Komatsu, H. 161 Komatu, Y. 454 Komori, Y. 588 Korobeinik, Yu.F. 436 Kosarev, E.L. 85 Koschmieder, L. 584 Koshljakov, N .S. 796 Kostitzin, V. 86 Kostometov, G.P. 312 Koul, C.L. (see also Kaul, C.L.) 318, 584, 782, 786 Kovetz, Y. 85 Kolbig K.S. xxii Kralik, D. 444, 445 Krantz, S.G. 590 Krasnosel'skii, M.A. 120, 161 , 168, 169, 655 Krasnov, V.A. 432, 587 Krbec, M. 591 Krein, M.G. 685 Krein, S.G. 5, 630 Krepkogorskii, V .L. 466 Krishna, S. 585 Krivenkov, Yu.P. 860 Krug, A. 435 Krupnik, N .Ya. 199, 302, 630, 656, 675, 684, 685 Kudryavtsev, L.D. 444 Kufner, A. 590 Kumbhat, R.K. 438, 439, 776, 786 Kumykova, S.K. 861 Kurokawa, T. 592, 595 Kuttner, B. 302, 315 Kuvshinnikova, I.L. 597
957
301 ,
589,
317,
167,
646,
782,
958
AUTHOR INDEX
L
Lacroix, S.F.171 Lamb, W. Lambe, C.G. 162, 171 Landau, E. 313 Landkof, O.M.ES.. 161 581 , 593 Lanford, Laplace, P.H.S . 415,xiii 435 Laurent, Lavoie, J .L.N.A.172,429,452436 Lebedev, Lebedev, N.N. 23, 24, 7'74, 778, 793, 794 Lee SangG.Keun 454 Leibniz, W . xxiii Leont'eJ.v, A.581F ., 600426, 436 Leray, Leskovskii, I.P. 862 Letnikov, A.V. xxi, xxiv, xxv, xxvi, 83, 160, 174, 302, 304, 371 , 433, 435, 452, 479 Levin, V.B.I . M314 Levitan, . 431 Levy, M. E . 85 288, 318 Lew, JH..S. 861 Lewy, Lieb, B.H.N .E588. 454 Linchuk, Linfoot, E.H. 315 Linker, A. l . 312, 692 Lions, J .L. J. 161,xxi,583xxiii, xxiv, xxv, Liouville, xxvi, xxviii,J.E82,. 83 84, 91, 160, Littlewood, 304, 433, 435, 450, 451 Liu, D. 443 597 Liu Gui-Zhong Liverman, T.P. Gxxx. i432 Lizorkin, P. l . i, 147, 162, 171, 433, 467, 475, 477, 489, 580, 58� 583, 586, 588, 591 , 594, 598, 601 Loo, -T. xxii, 315 xxx, xxxi, xxxii, 23, Love,C.E.R. XXIU
.xix,
xxix ,
83, 89, 92, 301, 302, 432, 593, 772, 777, 783 689, 774, 794 431, 774, 775, 780, 794
Lowengrub, Lowndes, J .M.S.
Lu, P. Ch. 85 Lubich, 85, 371 , 448 Luke, Y. L . 17, 22 Lundgren,J. T.xxiv689 Liitzen, M
Macias, R.A.AG.. 588, 590 Mackie, 788, 791 Madhavi Dinge 782 Magaril-Il' y aev, G. G . 580, 586 Magnus, W. 14, 18, 22, 88, 91, 160, 172, 180, 193, 196, 199, 201, 441 , 455, 539, 551 , 861 Mainra, V.PL.G. . 782 775, 793, 794 Makarenko, Malakhovskaya, R.691 M . 867 Malamud, M.M. Malinovski, H.V.A.85 439 Malovichko, Malozemov, V.GM. . 442443 Mamedov, R. Manandhar, R.S.P. 857783 Mandelbrojt, Manocha, H.M.M. L . 88,689318 Manukyan, Marchaud, A. 109, 1 10, 1 1 1 , 1 16, 168, 303, 469, 480, 580 Marichev, xxii, 27, 28, 172, 174, 190, 191 , 198, 201 , 206, 282, 284, 0.1.
301 , 700, 739, 781 ,
xxix ,
325, 355, 372, 437, 551, 584, 701 , 707, 711, 725, 726, 733, 740, 742, 743, 755, 772-774, 784, 785, 792, 856, 857 .xix
Marke, P.B.W.170 Martie, Martin-Reyes,V.M. F.J . 453 167 Martirosyan, Mathai A.B.L. M . 781 Mathur, 170 Mathur, S. L . 170, 171, 585 Matsnev, L.B.N. 86445 Matsuyama, Maz' ya, V.A.GC. . 582xix, xxii, xxxii, 157, McBride, 162, 172, 301 , 431, 439, 777, 871 McClure, P. 288,783305, 319 McKellar,J.B.H.J.
AUTHOR INDEX
McMullen, J .R. 90 Meda, S. 581 Medvedev, N. V, 85 Mengziyaev, B. 861 Mhitaryan, S.M. 685 Mihailov, L.G. 87 Mihlin, S.G. 518, 521 , 570, 584, 589, 661 , 724, 733, 735, 798 Mikolas, M. xx, xxii, xxxii, 162, 315, 432-434, 452, 581 Miller, J .B. 160 Minakshisundaram, S. 277, 315 Minerbo, G .M. 85 Miserendino, D. 587, 870 Misra, A.P. 9 1 Mittal, P.K. 171 Mizuta, Y. 588, 592 Modi, G .C. 585 Mohapatra, R.M. 315 Montel, P. xxviii, 315, 433, 580, 586 Moppert, K.F . 434 Most, R. 433 Mourya, D.P. 584 Muckenhoupt, B. xxii, 199, 202, 3 1 1 , 495, 581 , 588, 590, 596, 605, 617, 622, 684, 685 Muhtarov, B .S. 251 , 303, 304 Mulla, F. 583 Miiller, C. 776 Murdaev, H.M. 303, 304, 433 Murray, M.A. 588 Muskhelishvili, N .1. 5, 6, 302 N
N agnibida, N .1. 436 Nagy, B.S. xxxi, 432, 433, 444 Nahushev, A.M. 442, 857, 861 , 862 , 866, 867 Nair, V .C. 787 Narain Roop (see Kesarwani Roop N arain) 23, 772, 781 Nasibov, F.G. 315 N asim, C. 782, 793 Natanson, I.P. 798 Nauryzbaev, K .Zh. 444 Nekrasov, P.A. 302 , 415, 435, 858 Nersesyan, A.B. 83, 88, 89, 303, 440, 453, 841 , 857
959
Nessel, R.J. xix, 444 Nesterov, S.V. 868 Neugebauer, C.J . 371 Neunzert, H. 689 Nevostruev, L.M. 861 Nickel, K. 86 Nieva del Pino, M .E. 438 Nikolaev, V.P. 581 Nikol 'skaya, N .S. 580, 586 Nikol'skii, S.M. xxxii , 24, 368, 433, _ 444, 477, 528, 580, 583, 586, 589, 593, 595, 596, 597, 600, 601 , 685 Nishimoto, K. xvii, xxii, xxxii, 172, 435, 454, 866, 869 Noble, B. 774 Nogin, V.A. xxii, 552, 582-584 Norrie, D.H. 85 Nozaki, Y. 583 Nunokawa, M. 454 0
Oberhettinger, F. 14, 18, 22, 88, 9 1 , 160, 172, 180, 193, 196, 199, 201 , 441 , 455, 661 , 677, 724, 733, 735, 751 , 772, 783, 851 , 861 Obradovic, M. 454 Ogievetskii, 1.1. xxxii , 433, 444 Okikiolu, G .O. xix, 169, 303, 307, 581, 783 Oldham, K.B. xix, xxii, 172 Olevskii, M.M. 862 Olmstead, W.E. 320 Olver, F. 285 O 'Neil, R.O. xxxii, 167, 168, 591 Onneweer, C.W. 443 Oramov, Zh. 861 Orazov, I. 861, 862 Ortner, M. 582, 594 Orton, M. 689 Orudzhaev, G .N. 442 O 'Shaughnessy, L. 857 Osilenker, B .P. 590 Osipov, A.V. 85 Osler, T.J. xxxi i, 89, 90, 304, 305 , 316, 317, 432, 452, 453 Ovesyan, K.R. 453 Owa, S. xxii , xxxii, 435, 436, 454, 866
960
AUTHOR INDEX
p
Pacchiarotti, N . 88 Padmanabhan, K.S. 454 Pahareva, N .A. 859 Palamodov, V .P. 584 Paley, R.E.A.C. 315 Panzone, R. 580, 588, 589 Parashar, B .P. 439, 778 Pathak, R.S. 783, 794 Pavlov, P.M . 602 Peacock, G. xxxii Peetre, J . 92, 161 Pekarskii, A.A. 451 Pennell, W.O. 90 Penzel, F. 303, 690 Peres, J . 686, 693 Peschanskii, A.l. 449, 689 Pestana, D.D. xxii Peters, A .S. 688, 689, 774, 775 Petunin, Yu.l. 5 Phillips, R.S. 83, 84, 86, 120, 161 Pichorides, S .K. 200 Pilidi, V.S. 580 Pinkevich, V.T. 444 Pinney, E. 789 Pitcher, E. 857 Plessis, N.du 581 , 588, 589 Poisson, S.D . 856, 85 7 Pokalo, A.K. 444 Polking, J .C. 317 Pollard, H . 789 Polozhii, G .N. 82, 859 P6lya, G. 160, 308 Ponce, G. 597 Ponomarenko, S.P. 794 Ponomanenko, V .G . 447, 775, 794 Popov, G .Ya. 762 Popoviciu, T. 168 Post, E.L. 316, 375, 434, 447, 857 Prabhakar, T.R. 302, 773, 776, 786, 789, 791 Preobrazhenskii, N.G. xxii, 584 Privalov, 1.1 302 Prudnikov, A .P. 15, 18, 20, 22, 24, 27, 28, 172, 174, 198, 201, 206, 282, 284, 325, 355, 372, 551, 584, 711, 733, 739, 740, 742, 743, 755, 773, 781
Pryde, A.J . 595 Pustyl'nik, E.I. 161 , 655 Q
Quinn, D.W. 862 R
Rabinovich, V.S. 603 Rabinovich, Yu.L. 434, 868 Rabotnov, Yu.M. xxii Radzhabov, E.L. 582 Radzhabov, N. 860 Rafal'son, S.Z . 442 Raina, R.K. 584, 580, 782, 786 Rakesh, S.L. 171 Reddy, G.L. 454 Reimann, H.M. 92 Ren, F. 454 Repin, O.A. 862 Ricci, F. 604 Richberg, R. 776 Rieder, P. 86 Riekstyn'sh, E.Ya. 288, 305, 318, 319, 448 Riemann, B . xxiv, xxv, xxvi, 82, 83, 88, 92, 856 Riesz, M. xxvii, xxx , 84, 277, 301 , 302, 314, 481, 483, 498, 502, 58 1, 583, 600 Roach, G.F. xxxi i Roberts, K.L. 168 Robinson, D.M. 161 Rogosinski, W.W. 315 Romashchenko, V.A. 794 Rooney, P.G, 441, 773, 782, 789 Ross, B. xix, xx, xxii, xxiii, xxxii Rothe, R. xxii Rozanova, G .1. 314 Rozet, T.A . 773, 788 Rubel, L.A . 317, 453 Rubin, B .S. xxii, 83, 84, 89, 91, 92, 160, 161 , 303, 308, 311, 432, 554, 582-584, 589, 592, 595, 597, 599, 600, 601 , 686, 687, 690, 692-694 Ruhovets, A.M. 793 Ruiz, F.J. 590 Rusak, V.N. 444 Rusev, P.K . 455
AUTHOR INDEX
Rusia, K.C. 773, 776, 786 Rutitskii, Ya.B. 167, 168, 169 Rychener, T. 92 s
Saakyan, A.A. 92, 310 Saakyan, B .A. 89 Sadikova, R.H. 314 Sadowska, D. 690 Saigo, M . xxii, xxxii, 440, 777, 786, 859, 862 Saitoh H. 454 Sakalyuk, K .D. 84, 685, 688, 689 Saksena, K.M. 170, 438, 439 Salahitdinov, M .S. 442, 861 Samarskii, A.A. 318 Samko, S .G. xvii, xix, xxi, xxii, 83, 161, 164, 168, 171, 302, 303, 304, 306, 311 , 432, 433, 434, 447, 448, 485, 507, 528, 529, 530, 537, 544, 546-548, 582, 583, 587, 588, 591, 593-597, 602, 685, 686, 691 Sampson, C.H. 584 Sargent, W .L.G. 3 12 Sato, M . 444 Sato, T. 82 Sattarov, A.S. 860 Savelova, T .1. 659 Sawyer, E. 167, 590, 862 Saxena, R.K. 78�, 786, 794 Saxena, V .P. 782, 783 Schneider, W .R. 861 Schuitman, A. 438, 777 Schwartz, J .T. 467, 642 Schwartz, L. 154, 581 , 583 Segovia, C. 588, 590 Seitkazieva, A. 865 Sekine, T. 454 Semenov, E.M. 5 Semyanistyi, V .1. 162, 581 Senator, K. 864 Sethi, P.L. 793, 794 Sewell, W .E. xxviii, 432, 435, 448, 857 Shabunin, M .l. 285 Shah, M. 774 Shanmugam, T.N. 454 Shapiro, B.S. 448 Sharma, B .L. 318
961
Sharma, S. 442 Sharpley, R. 167 Shcherbina, V .A. 302, 685, 693, 779 Shelkovnikov, F .A. 432 Shen, C.Y. 454 Shermergor T.D. xxii Shevchenko, G .N. 798 Shikhmanter, E.D. 867 Shilov, G.E. 162, 487, 548, 556, 583, 867 Shinbrot, M. 864 Sidorov, Yu.V. 285 Simak, L.A. xvii Singh, B. 782 Singh, C. 776, 786 Singh, Rattan 782 Singh, R.P. 776 Singh, Y.P. 786 Sintsov, D .M . 435 Sirola, R.O. 85 Skal'skaya, I.P. 793 Skorikov, A.V. 442, 580, 587, 691 Sk6rnik, K. 171 Slater, L.J . 789 Sludskii, F. xxv, 433 Smailov, E.S. 444 Smarzewski, R. 85 Smirnov, M.M. 777, 798, 820, 856 Smirnov, V.I. 429, 436 Smith, C.V.L. 171 Smith, K.T. 583 Sneddon, I.N. xix, xxxii, 431, 762, 773-775, 777, 793 Sobnak, Sh.D. 371 , 434 Sobolev, S.L. 494, 495, 528, 581 , 589 Sobolevskii, P.E. 161, 655 Sohi, N .S. 454 Sohncke, L. 857 Solonnikov, V.A. 160 Soni, K. 773, 788, 790 Sonine, N.Ya. xxvi, xxvii, 83, 85, 415, 431 , 435, 696, 773, 787, 789 Spain, B. 90 Spanier, J. xix, xxii, 172 Sprinkhuizen-Kuyper, I.G. 602, 870, 871 Srivastav, R.P. 773 Srivastava, H.M. xxxii, 440, 454, 584, 695, 783, 785-787
AUTHOR INDEX
962
Srivastava, K.N. K .J . 784 776, 777, 786, 794 Srivastava, Srivastava, R. 584 Srivastava, TA.P. .N. 786316 Starovoitov, Stechkin,E.MS.B. 249, 444 Stein, . 92, 160, 593, 498, 604, 528,685546, 550, 566, 581, 590, Steinig,R.L. J. 312434, 444, 44�448 Stens, Stepanets, V.A.l.D . 444, 445 Stepanov, 166 Stolle, H.W. Strichartz, R.S. 588,167,594,590597 Stromberg, Stuloff,Yung-Sheng M. 434 444 Sun' Sunouchi, R.G. 785452 Swaroop, Szego, G . 308 Szeptycki, P. 432, 583 xix
J .-0 .
T
Ta Li 772,R. 776444, 44�448 Taherski, Taibleson, M.S. 432 590 Takahashi, Takano, K. 602 Tal e nti, G. 432 Tamarkin, 445 J.D . 82, 83, 85, 303, Tanno, Y.R.P792. 161 Tarasov, Tardy, P.A.Z.xxv Tazali, -A.M. 865 Tedone, 773, 792 Telyakovskii,H .PS.A. 433, 444 Thielman, . 90 Thorin, G 581 Tihonov, A. N . 318 Tikhomirov, V . M . 586315, 444 Timan, A.F. xxxii, Titchmarsh, E.C.774,24,775,86, 780, 169, 857695, 708, 709, 713, Tonelli, TL..L. xxix590 Torrea, Torvik, P.C.J. J . xxii774, 793 Tranter, xxx ,
0.
.0 .
,
xix ,
163, 164, 303, 313, Trebels, W. 444, 783 Tremblay, R. xx,14,172,18,301,22, 452,88, 87191, Tricomi, F.G. 160, 172, 180, 193, 196, 199, 201,
307, 441, 455, 661, 677, 724, 733, 735, 751, 772, 783, 791, 820, 851, 856, 861 Triebel, H. 599 Trimeche, K.. 583, 441 Trione, E. S 600 Tripathy, N. 315 Trivedi, TA.l..N. xxii 794 Tseitlin, Shih-Tong 869 Tiirke, H. 312 Tu u
Uflyand, Ya.S.S. 85762, 793 Ugniewski, Ulitko, A.F . 793S.M. xxi, 595, 597 Umarkhadzhiev, Upadhyay, M.A.B.442162 Urdoletova, Usanetashvili, M.A. 861 v
Vainberg, B.R. 602 Vakulov, B .G. xxii, 497, 581, 602 Varma, V.K. R.S. 170 Varma, 791 Vashchenko-Za:kharchenko, M. xxv Vasilache, S. 580 Vasilets, S.I. 693 Vasil'ev, I.L. 688, 689-691 162, 693, 865, 866 Veber, V. K . 774, 796, 798, 856, 857, Vekua, 859, I.N. 861, 862 Verblunsky, S.317,312, 314 Verma, A. 442 Verma, R.U. 584, 781, 782 Vessella, S. xvii, xxii, 87, 691 Vilenkin, N. Y a. 576 Virchenko, 859 N .A . xxii, 762, 775, 793, Vladimirov, 847, 865 V.S. 172, 583, 602, 846, Volkodavov, V.F. 798, 862
AUTHOR INDEX
Volkov, Yu.l. G.F. 449 Volodarskaya, 692, 694 Volterra, V. 83, 85, 89, 686, 693 Voskoboinikov, Yu.E. 85 Vries, G.de 85 Vu 711Kim, 712, Than757,xxii, 162, 584, 709, 773, 774, 784, 794 w
Wagner, P. 594 Wainger, S. 593 Wall, B.S. 315 Walton, 689, 793, 794 Wang, F .T. 315 Watanabe, 161 Watanabe, S. 602 Watanabe, T.Y. 603 Watanabe, 88, 304, 316 Watson, G .N. 2 1 Weber, H. 774 862 Weinacht, Weinberger, H. F856, . 862857, 860, 863 Weinstein, A. Weiss, G. 581 , 590 Weiss, 85 Weiland, G V 581 Westphal, U. xxxii, 161, 170, 312, 371, 434, 446, 865 Weyl,161,H.432,xxviii, 84, 91, 433 167, 495, 581 , Wheeden, 582, 588, 590, 591 , 595, 862 Whittaker, E.T. 21, 84 Wick, 689 Widder, D.V. 162, 169, 170, 302, 695, 772, 786, 789, 792 Widom, K.H. 866 685 Wiener, Williams, W.E. 685, 687, 779, 780, 790 Wilmes, 433, 582, 587 Wimp, G. 24,371773,, 447,774-776 J .R.
J.
R.J .
R.
. .
R.L.
J.
J.
xxix,
xxii ,
xxx ,
963
.M
Wing, G . L.von 86 Wolfersdorf, 85, 302, 659, 685, 688, 689 Wong, 288, 294, 305, 319 Wood, D.H. Wyss, W. 861864 R.
303,
X
Xie Ting-fan
444
y
Yakubov, A.Ya. 303, 304, 447, 448, 594 Yakubovich, S.B. xxii, 757, 772-774, 792 Yanenko, N.N. 869 Yaroslavtseva, V167.Ya. 869 Yasakov, A. l . Yoshikawa, A. 161 161, 581 , 162, 580 Yoshinaga, K. Yosida, K. 120, 161 Young, A.E.C. 85862, 863 Young, Young, L.C. xxxi , 83, 301 z
Zabreiko, P.P. 120, 161, 328, 655, 671 Zag&nescu, Zanelli, V. .l.90 602, 865 Zav' y alov, B Zeilon, N . xxvii, 302, 684, 689 Zeller, K. V312.A. 85, 371 Zheludev, Zhemukhov, Kh. Kh. 861 V . A . 859 Zhuk, V.V. 444 Zhuk, Zhong Zhu Zo 454 Zygmund, A. xxxii, 51, 91, 92, 350, 354, 367, 432, 433, 445, M.
xxii
xix,
581 , 685
xxx ,
SUBJECT IND EX
A
Abel integral equation 29, 30 asymptotic solution 299 generalized 85, 610 multidimensional 458 pyramidal analogue 571 Abel-type integral operator 731 Absolutely continuous function 2 space (see space of absolutely continuous functions) Associated Legendre function 19 Asymptotic expansion (series) 285 of fractional integral 288, 289, 292, 294, 297, 318, 319 of integral with power-logarithmic kernel 411 power 286 Asymptotic sequence 285 Asymptotic solution of Abel equation 29 Axially-symmetric potential (see potential)
Pf(z)
B
Banach theorem 13 Bernstein inequality 368, 370 analogue 368, 370
Bernstein type inequality 597 Bessel-Clifford function J, (z) 731 Bessel fractional derivative (E ± V)a J , ( E ± D)af 335 Bessel fractional integral aar.p 333, 482, 540, 598 Bessel fractional integration operator (see operator of Bessel fractional integration) Bessel function of the first kind 19, 20 modified 20 Bessel kernel 539 Bessel-Maitland function 437 Bessel potential Gar.p 333, 540 anisotropic 598 modification G%r.p, esa r.p 334, 541 space Lp (R1 ) = Ha•P (R l ) aa (Lp) 336, 541 unilateral G%r.p 334, 482, 598 Bessel-type potential 482 Beta-function B(z, w) 17 Binomial coefficient (6) 14 Bisingular integral operator 467 Bochner formula 484 Boundary value problem (see Cauchy, Dirichlet, Hilbert and Neumann boundary value problem)
Jv(z)
lv(z)
Je (z) =
966
SUBJECT INDEX
c
Carleman equation 626, 627 Cauchy boundary value problem 812 , 819, 823, 830 Cauchy-type boundary value problem 829, 832, 837 Cauchy formula xxvi, 421 Cauchy integral formula 415 Chen fractional derivative c , c 339, 340 Chen fractional integral 338 Cone characteristic 556 Cone light 556 Confluent hypergeometric ( Kummer) function 1F1 (a ; c; z) 19 Continuity modulus (integral) wp (f' t) 131 , 233 of fractional order 447 Convolution 12, 25, 28, 154 , 465 , 484, 722 Cossar fractional derivative 163 ( C, a)-met hod ( see method of summation)
'D01J DOIJ l�cp
D D 'Alambert wave equation 800 Difference finite 1 16 of a fractional order 371, 385, 453 of the vector order 469, 479 weighted 553, 567 with a vector step centered h 499 with a vector step non-centered 499 Differential equation of an integer order, ordinary 849 Differential equation of fractional order 829 Cauchy-type problem 829 Dirichlet-type problem 829 linear 837
(��/}(x) c�:/)(x) (�� J)(x) �LJ, �L.'7 ! (�a J)(x)
c�:J)(x)
linear w i t h constan t coefficients 846 ordinary 850 Dirichlet formula 9 pyramidal analogue 571 Dirichlet series 421 Dirichlet-type boundary value problem 829 Dirichlet weighted boundary value problem 826, 843, 845 Distribution function 496 Dual equations 762 Dzherbashyan's generalized fractional integral £(w ) 345
cp
E
1'7,01 , K11 ,01
Erdelyi-Kober operator 332 analogue 766 generalized J� (77 , a) , R� (71 , a) 738 Erdelyi-Kober-type operator JOta+·q >"In ! ' JOtb - ;u ,'l 322 Erdelyi operator , u,'l b-; u,f1 322 Euler-Poisson-Darboux generalized operator 813 elliptic 813 hyperbolic 819 Euler t/l(z)-function 17 I
!001+. , 101
F
Factorization of a-transform 713 of W-transform 754 Favard inequality 370 Feller potential 216, ,v , 623 356 analogue I� Finite part of integral by Hadamard p.j., pf 112 Formula (see also under Bochner ' Cauchy and Funk-Hecke )
M: cp M01cp 01) cp
SUBJECT INDEX
Formula of conformity 800 of fractional integraltion by parts 34 Fourier convolution theorem 25 Fourier kernels 781 Fourier-Laplace series 528 Fourier-multiplier theorem 13 Fourier series 347, 476 Fourier transform :F!fJ = :F{!fJ(t); x} = tP(x) 24 convolution 25 multidimensional 473 of fractional integral and derivative 137, 473, 474 of generalized function 487 of singular integral 200 sine- and cosine- (see sine- and cosine- Fourier transforms) 25 Fox H-function 781 , 787 Fractional derivative (see also Bessel, Chen, Cossar, Griinwald.:.Letnikov, Griinwald-Letnikov-Riesz, Hadamard, Liouville, Marchand, Riemann-Liouville, Ruscheweyh, Weyl, Weyl-Liouville, Weyl Marchaud fractional derivatives) in a given direction J�01), 'IJ� J 585 in the complex plane 1J�0 / , V% ,8!, 1J�f 35, 415, 419 of a function by another function 1J�+ ;g f 326 of absolutely continuous functions 267 of analytic functions 414-416 of complex order 38 of periodic functions (see Weyl fractional derivative) 8 of purely imaginary order pai + J 38 Fractional q-derivative 443 Fractional integral (see also under Bessel, Chen, Griinwald--Letnikov, Hadamard, Liouville, Riemann Liouville, Weyl) 33 in a given direction I� J 216, 470
967
in the complex plane I.?0 !fJ, I± , 8!fJ, I�!(J 94, 415, 416, 418, 419, 502 of a function by another function I�+ ;g !fJ 35, 326 of complex order 38 of generalized functions 146, 151, 154, 157, 158 of purely imaginary order �'+ J, ·s I+ip 38, 89, 97 Jb_!, Fractional q-integral ,I: " 443 Fractional integra-differentiation of analytic functions 419, 421 generalization 422 Fractional power of operator 120, 555 Fubini theorem 9 Function (see also Bessel, Bessel Clifford, Bessel-Maitland, Beta, Fox, Gauss, Gorn, Humbert, Kummer, M cDonald, Meijer, Mittag-Leffler, Riemann, Riemann Gurwitz functions) of bounded mean oscillation 92 Funk-Hecke formula 551 G
Gamma-function r(z) 15 Gauss hypergeometric function 2F1 (a, b; c; z) 18 Gauss-Weierstrass integral Wt!fJ 497 kernel W(x, t) 497 G el'fo n d - L e o n t'e v g e n e r al i z e d integration and differentiation P- (a; !), vn (a; /) 426, 428 operator of generalized integration and differentiation F (a; f), p n (a; f) 426, 428 Generalized function (some notions) Generator of semigroup (see infinitesimal operator) Gorn hypergeometric function Fa( a, a ' , b, b' ; c; x,y) 193 Griinwald-Letnikov differentiation in a region 587
968
SUBJECT INDEX
G rii n w a i d - L e t n i k o v fr a c t i o n al derivative 1:> 373
1;? ± 479 on a finite interval 1:�, 1:�
multidimensional
..
386 Griinwald-Letnikov fractional integral J:+ 1{), Jf_ I{) 387 Griinwald-Letnikov-Riesz fractional derivative j01 ) 374 H
Hadamard fractional derivative D%!, D:+J, D�_/ 332 Hadamard fractional integral J%'P, J:+ 1{), J�_'P 330, 331 Hadamard property 1 12 Hankel contour £6 424 Hankel transform , b ,y ) J 738 generalized s(,a,a,u modified s,, a;u J 325 of modified form {J, (2./i)} J 722 truncated 437 Hardy inequality 104 Hardy space Hp 424 Hardy-Littlewood theorem 66, 102, 103 analogue 91, 466 Heat operator 554, 564 Helmholtz generalized two-axially symmetric equation 800 Hilbert boundary value problem 807 Holder condition 2, 3 generalized 7 Holder inequality 2, 8 analogue 11 generalized 8 Holder space H>.. = H>.. ( o) 2, 3 generalized Hw, H't ([o, 21r]) 249, 364 weighted H>.. (P)= H>.. ( o; p) , � (p) = H6(0 ; p) 1 , 4, 5 Homogeneous function 12
Hyperbolic Riesz potential IP i o f, ± 101 558, 559 Hypergeometric function (see also under Gauss, Gorn, Humbert, Kummer) generalized pF9((ap ); (,B9); z) 91 Hypersingular integral D 01J, T01 J 498, 499 annihilation 510 characteristic 518 parabolic T01 J, '!01 J 500 symbol V0(x) 521 truncated n: J 500 with homogeneous characteristic 518 with weighted differences T,a J 553 Humbert hypergeometric function � 1 ( · . . ) , 81 ( . . . ) , 82(· . . ) 199, 750, 802, 809 I
Identity approximation theorem 1 1 Index laws 177, 301 , 307, 727 Index of complexity of G-transform p + q 706 of a-transform Tl 106 of operator x 601, 631 of singular integral equation x 601 Inequality (see under Berstein, Favard, Holder, Kolmogorov, Minkowsky, Opial) Infinitesimal operator of semigroup 120 Integral equation (see also under Abel, Carleman, Volterra) of compositional type 746 of the first kind 606, 696 singular 606 characteristic 608 Noether theory 631 , 632 with Bessel function 722, 723
SUBJECT INDEX
with Gauss hypergeometric function 696 with Legendre function 699 with logarithmic kernel 694 with power kernel 605, 610, 629 Noether nature 630 with power-logarithmic kernel 550, 672 Noether nature 672 Integral transform (see also Fourier, Hankel, Kon torovich-Le bedev, Laplace, Mehler-Fock, Mellin, Meijer, Stieltjes, Varma transform) 23 convolution type 23, 704 with respect to index (parameter) 23, 752 F3-transform 758 G-transform 705, 711, 713 characteristic ( c• , 'Y* ) 706 index of complexity (see index of complexity) H- and -transform 723, 724 W-transform 752
Y
K
Kernel of operator Zx (A) 630 Kober (Kober-Erdelyi) operator It, a !A K;, a (/) 322 Kolmogorov inequality 275, 317 Kontorovich-Lebedev transform K�1 {J(t)} 753 Kummer function (see confluent hypergeometric function) L .
Laplace convolution theorem 28 Laplace equation 813 Laplace transform L!{) = L{({)(t); p} 27, 140 convolution 28 generalized 725 modified A±f, A ± 1 J 714 multidimensional 474
969
of fractional integral and derivative 140, 141, 474 Lebesgue dominated convergence theorem 10 point 51 Leibniz rule 277 analogue 317 generalized 280 integral analogue 283 Liouville fractional derivative on axis 'D�f 332 partial and mixed D±'" J 332 Marchand -form 468 truncated D+ ... +,e J 468 Liouville fractional integral 1�!{), ��-·· ± (/) 94, 419, 462, 502 Liouville space of fractional smoothness aa(Lp) 543 Lipschitz space nt(o), n; , h; , iit 2, 254, 255 Lizorkin 's space � 148 of generalized functions �, , '11 ' 150 of test functions �, \11 147, 475 Lorentz distance 556 M
McDonald function Kv(z) 20 Marchand fractional derivative 110, 1 19, 468 analogue 225 generalized 168 on half-axis 111, 1 12 on interval D�+ J, D6_! 225 truncated D�+, ef 226, 469 on real line D�J 109, 1 10, 119 truncated D% , ef 111, 118 Mehler-Fock integral transform 23, 753 Meijer transform 23 generalized 23 Meijer a-function �n
(z l ��:�)
22
970
SUBJECT INDEX
Mellin convolution theorem 26 Mellin transform IP*(s) = rot{rp(t); s} 25, 142 convolution 26 multidimensional 474 of fractional integral and derivative 143, 144, 165, 475 Method of summation of integrals or ( c, a )-method 277 Minkowsky inequality 8 generalized 9 Mittag-Leffler function Ea (z), Ea p(z) 21 Mixed norm space Lp(Rn ) 465 Modulus (see continuity modulus) Multiplier 598 Muckenhoupt condition 167 Muckenhoupt-Wheeden condition 495 M uskhelishvili space H* 246 I
N
Neumann weighted boundary value problem 510, 826-828 Neutral periodic function 477 Noether operator 631 0
Operator (see also Abel, Bessel, Erdelyi-Kober, Erdelyi, Gel'fond L e o n t 'ev, Kober, N oether, Riemann-Liouville operator) d-characteristic (n , m ) 631 cokernel Zy. (A * ) 630 dificiency numbers 631 heat (see heat operator) non-convolution with B essel functions in the kernel J± "'� , J± >. a, a, 741 normally solvable 630 polysingular 482 of Bessel fractional integration aa 123, 333 of truncation PalJJ P+ , P- 211
singular (see singular operator) tensor product (see tensor product of operators) transposed 647 with homogeneous kernel 12 with power-logarithmic kernel 1:/ , f!• 6-P 388 0pial's inequality 313 0-symbolism f = O(g ) , f = o (g ) , J "' g 16 p
Parabolic potentials Ha�P, 'H.a iP 563, 565 space (see space of . parabolic potentials) Parseval relation 26 Pochhammer loop 424 symbol (a) n 14 Poincare-Bertrand formula for interchanging 201 Poisson integral PtiP 497 Poisson kernel P(:c, t) 497 Post generalized differentiation a( 'D)/, a [V)f 375, 376 Potential (see also Bessel, Feller, Riesz potential) axially symmetric p-dimensional 813 R
Radial function 484 Regularization of divergent integral 529 Riemann function R(e , '7i eo, '7o) 797, 816 Riemann-Gurwitz function ((s, a) 20 Riemann-L io u v i ll e fr a c t ion al derivative left-sided and right-sided v:+ J, V6_f, 35, 37, 225, 460 mixed and partial 459, 460 pyramidal analogue VAcf, V£ 1 f 576, 577
SUBJECT INDEX
of a function by another function 'D:+ ;g f 326 M archand form 327 Riemann-Liouville fractional integral left-sided and right-sided 1:+ ({), If_({) 28, 33, 232 mixed and partial 1::+ ({)' I�+ ({) 459, 462 pyramidal analogue I_A c ({), I_E1 ({) 576, 577 Riemann-Liouville operators 1:+ ' If_ , 'D:+ ' 'Df_ 28, 33, 35, 37, 225, 232 Riemann zeta-function generalized {see Riemann-Gurwitz function) 20 Riesz differentiation 498 Riesz fractional derivative n a 1 499, 500 Riesz fractional integra-differentiation 483 Riesz kernel ka (x) 492 Riesz means (see Riesz normal means) Riesz mean value theorem 270 Riesz normal means ca(x) 276 Riesz potential 1a ({) 214 analogue J( a ) ({) 355 anisotropic 588 generalized 10({) 589 hyperbolic (see hyperbolic Riesz potential) modified Ha({) 481 , 565 on half-axis 221 on interval 223 space {see space of Riesz·potential) unilateral 502 modified I�+ ({)' B�({) 592 with Lorentz distance {see hyperbolic Riesz potential) with radial density 589 Riesz-type polypotential operator K:,a 480 Riesz-type potential operator J( a ) ({J, �r> ({) 94, 355, 419, 552
971
Ruscheweyh fractional derivative 430 s
Semigroup property 35, 48 Sine- and cosine- Fourier transforms :F, ({), :Fe ({) 25 Singular integral with Hilbert kernel H({) 354 with power-logarithmic kernel Sa , a,m({), Sb, a ,m ({) 675 with weight 618 Singular operator S 199 Sobolev fractional space aa(Lp) 544 Sobolev limiting exponent 495 Sobolev theorem 494 weighted case 495 Spherical harmonic Ym(u) 529 Spherical mean Mn (x, y; -r) 823 Space ( see also Hardy, Holder, Liouville, Lipschit z, Liz or kin, Sobolev, Zygmund an d mixed norm space) of absolutely continuous functions A C(O) 2 of bounded mean oscillation BMO(a,b) 92 of parabolic potentials Ha(Lp) 564 generalized 600, 601 of Riesz potentials 1a (Lp) 43, 122 of summable functions Lp = Lp(O) 7 with exponential weight Lp.., 108 Stieltjes transform 23 T
Table of fractional integrals and derivatives 172 Taylor formula analogue 46 generalization 88 Tensor product of operators 463 Triple equations 768 Truncated power function Y± 22, 94
972
SUBJECT INDEX
v
Volterra functions #J.(X, O', a ) , #J.t,{J(x), v(x) 661, 662 Volterra integral equation 656 w
Weyl fractional derivative v<; )J, D �a )f 109, 332, 360, 505 multiple and mixed 477 truncated 357 Weyl fractional integral 4_a ) cp 348, 477 multiple and mixed 477
Weyl-Liouville fractional derivative 'D �Ot) f 348, 477 Weyl-Marchaud fractional derivative n c; ) J 353, 357 truncated D���f 357 y
Young's theorem 12 z
Zygmund generalized space A0 ([o, 21r]) 364 Zygmund type estimate 249
INDEX OF SYMBOLS
Latin and Gothic
A a
A,( A �+>./, A�:_>. I 731 AC(O) , ACn (n) 2 (an ) 14 (a)n , (a;+ t) 705 a (V ) f , a [V]f 375, 376 (a - dhf)(x) 375
B a
n:+>. B�:_>. I 731
/, BMO{a , b) 92 (bm ) , {b;n + t) 705 Cw = Cw(R1) 108 C(n) , cm (n) 2 cgo(n) 10 ca(x) 276 c:l J, c�:..>. 1 731 C� ( [a , b] ) , Cb' ( [a , b] ) 159, 160 c* 706 n a /, D � I 499, 500 D ±f , D ±;e f 109- 1 1 1 , 360, 505
D�) /, D�� / 352, 357 D± .. .±f, D± ·· · ± , e f 468, 469 D� /, D� / 340, 470 Dfi/, Dfi,e / 5 18, 524, 552 D�+ /, D6_/, D�+,e f 225, 226, 470 V = (a!� ' . . . ' a�.. ) 458 Vi = az�1al. i.azi !." 458 vn (a ; /) , V{b; /} 426, 428 V(a) f 411 Va f 428 V± f , V±,e f, V±_�. ±f 332 v1a) /, vC:. �- + 1 348, 477 V± . .± f, V+· · + ,e f 462, 468 V±,9 ! 419 v�/, V�f 339 v�of 415 vg(x) 521 VAcJ, VE1/ 576, 577 v�+ J, v:_, v�:+ J 35, 37, 225, 460 vc;+ I, v:_ ;z" I, V± ;z" I 326 n�:; /, D�:_>. I 73 1 a �,o a 2 ) I ' v ( a,O) v((a1 e 461 47 1 dn ,l (o:)a2 )+ 520 + , ' ;u
462
974
INDEX OF SYMBOLS
�±/, ��+ /, �1:- 1 330, 331 E 464 (E ± D ) a /, (E ± V) a f 335 Ea (z), Ea ,p (z) 21 E( /3, f3* )u 813 a , >.., v / , E6a , >.. , v / 731 Ea+ 1 F1 (a; c; z) 19 2 F1 (a; b; c; z) 18 F3(a, a', b, b', c; x, y) 193 Fp, F;:. ,JJ = FpJJ 158 p F.q [ a/3l !! . . .. ,, n/3p;q x ] 91 1 :F, :F- 1 1 = j 24, 473, . .
486
484,
:Fe
t:� , t:�
373, 374, 474 386, 587
/�a) 585 (a) Jl'b-( a) 386 /.a+ ' Ga
(a�n���=�lt(t))(x) �a
t
705
246, 608
H.. = H >.. ( O), H� (O) 2, 3, 6 HW , H� , H�(p) 249 Hp 424 Ha .. (p) = H>.. ( O; p), H� (p) = H� (O; p) 4, 5
H >.. Cfln ), H >.. ( Rn , p)
496
Ha , if a 246, 247 H� ([0, 21r]) 364 HP>.. , H- P>.. 254, 255 n;(tt�) 634 n; ([o, 21r]) 367 H a ·P(R1 ), H"•P ((a, b]), H�·P ([a, b]) 232, 336 H>.. ,lc(O), n; ·�c (p) 7 H;,JJ u 800 1l a .. = h>.. ( O), h�(p) 2 hp>.. , h- p>.. , hp>.. ([0, 211"]) 254, 367 h>.. , k (O), h�·lc (p) 7 h;,JJ u 800 { h( X)}
±
232
1± ,8 / 418 I:+;9 cp, lf- ;:c"
325, 329
INDEX OF SYMBOLS
I:f .. ' I-e >.. 731 I�a)• {0), I:'(a)• {0) 176 I:+ ,b- .. f 741 q T:. " I 443 J:+
.. (TJ, o:)f 738 J-±a , >.. f 741
20
Jp, :�:, Jp,, , :�; , 157, 158 :fa / 427 a-±
L�(Rn ) 457 Lp(Rn ) 464 Lpx, . (a ,b) 176 L� ) 801 L;(R1 ) 336 Lp ,w, 108 L�e -y ) 706 L01p ,r (Rn ) ' L01(R p n ) 505 £;, r 600
M01
,
11, 495
25
Pf, (P ± iO)>.. 555 P{x,t), Pt
.. (TJ, o:) 738 ra (x), rb (x) 206 s, s', s+ 46, 155 S
ll(y)) (x) 752 wp,, (x) 437 Wp(l, t) 131, 136, 233, 447 X(R1 ), X2,.. 371 Ym (l, x), Ym {u) 528, 529 Y± 22, 94 (z+, zo+, z-, zo-) 424 ...
K,, 01 , K;, 01 322 K, (z) 20
Kix{f(t)}, Ki-;1 {g(t)} 753 /C01
975
;
976
INDEX OF SYMBOLS
Greek
[o:J , { a: }, <;n B (z, ) 17 f(z) 15
�, (t) 496 Jt (x, u, o:), Jtt ,p (x) v(x), vh (x) 662
14, 36
w
Dp , ( -Dp ) �
,.. 706 'Yn ( o:) 490
-�x + :t 562 (� h f)(x), ( ��f)(x, p)
469, 479, 499, 553 � z.a (x, h) 513 6(x - zo ) 145 ((s, a) 20 X 601, 631 x( o:, l) 1 19, 469 A± , A± 1 7 14 A� ([O, 21r] ) 364 �� 367
661
1 16,
371 ,
� ' �' , �� 147, 150, 151, 475, 487 �+ ' �+, �� ' (�+) ' 156, 157, 162 �� 253 �1 (,8, 6 ; /; x, y) 199
1/; (z) O a ( Y)
17 523, 530 w(
S l (o:, o:' , ,B; 'Y; x, y) 750 S2 (o:, ,B; 'Y; x , y) 802, 809
Fractional Integrals and Derivatives Theory and Applications S.G. Samko, Rostov State University, Russ/a A.A. Kilbas and 0.1. Marichev, Belorusslan State University, Minsk, Belarus "This is the world's first comprehensive (indeed, encyclopedic) exposition of fractional calculus and its applications. It provides a full historical survey, from the earliest development of the subject to the frontier of current research.... No other . book approaches this one in its authoritative and encyclopedic coverage of the subject."- Dr N.M. Queen, University of Birmingham, UK
This comprehensive monograph is devoted to the systematic and comprehensive exposition of classical and modern results in the theory of fractional integrals and their applications. Various aspects of this theory, such as functions of one and several variables, periodical and non-periodical cases, and the technique of hypersingular integrals are studied. All existing types of fractional integra-differentiation are examined and compared. The applications of fractional calculus to first-order integral equations with power and power-logarithmic kernels and with special functions in kernels and to Euler-Poisson-Darboux-type equations and differential equations of fractional order are discussed. The clear presentation of historical background, the extensive analysis of the great number of cited papers (more than 3000), and the authors' own significant research gives this work the compactness of a handbook and the depth of an encyclopedia. It is an invaluable tool not only for undergraduates and postgraduates of mathematical physics and engineering but also specialists in different fields of calculus, mathematical physics, and differential equations. Titles of related Interest Tables of Indefinite Integrals Yu.A. Brychkov, 0.1. Marichev and A.P. Prudnikov
More Special Functions, in the series Integrals and Series, Volume 3 A.P. Prudnikov, Yu.A. Brychkov and 0.1. Marichev
Applicable Analysis, an international journal edited by R.P. Gilbert
ISBN: 2-88124-864-0
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London
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