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We recall that a family 9C of functions has equiabsolutely continuous integrals if for arbitrary f > 0 an > 0 can be found such that for all functions in the family 9C we have
J 1
<
f
Iff
provided mes � < A general criterion for the equiabsolute continuity of the integrals of a family of functions is given by the following theorem. VALLEE POUSSIN'S THEOREM (d. , e.g. , NATANSON p. 1 59, American edition) . (0 � < 00)
h.
[I J, Let <1>(u) u be a monotonically increasing function which satisfies the condition . <1>(u) hm  = oo. u Suppose, for the funciions
[ I
[1 (x) IJ dx A < (
G
'.R
�
00
CHAPTER I I , §
11
95
This straightforward theorem is proved in NATANSON [ 1 ] under somewhat more restrictive hypotheses. Every Nfunction satisfies the conditions of Vallee Poussin 's theorem. Therefore, if the family of functions � is contained in LM and
M(u)
p(u; M) J M [u(x) ] dx A (u(x) E �) , (1 1. 2) then the functions u(x) have equiabsolutely continuous integrals. Now, let M(u) and M l (U) be two Nfunctions for which . M(u) = 00. hm Ml (U) Then it follows from (1 1. 2 ) that the family of functions {Ml [U(X)]} has equiabsolutely continuous integrals on G. To prove this, it suffices to note that the function cJ> (u) = M[Ml (u)] satisfies the conditions of Vallee Poussin's theorem inasmuch as (u) lim M [Ml1(u)] lim M(v) 00. lim cJ> Ml (V) U U 2. Steklov functions. Let u(x) be a function which is summable on G. The function ur(x) = _mr1_ f �t(t) dt (x E G) , ( 1 1. 3) where Tr(x) is the ndimensional sphere with radius r and center at the point x E G and mr is the volume of this sphere, is called a Steklov function. We set the function u(t) in integral (1 1. 3) equal to zero for t G. LEMMA 1 1.1. Suppose given a family of functions � C LM with uniformly bounded norms I lu l iM A , u (x) E �. Then the family �r of Steklov functions ur x), u(x) E �, is compact with respect to the �tniform norm (in the space( C of continuous functions on G) . PROOF. Since f M [ uA(x) ] dx f M [Iulu(lx)iM ] dx 1 (1 1. 4) =
�
G
u_oo
�1 =
l
=
=
=
u+oo
u+oo
u+ oo
T, (x)
E
�
G
�
G
�
96
CHAPTER I I , § 1 1
for u(x)
E
J U r (x) J
91, we have, in virtue of Young's inequality, that 1
:( __
mr
f J u(t) dt J
T, (x)
f mr
��
G
J u(1L dx :( A A :(  ( 1 + N ( I ) mes G) .
mr
Thus, the functions of the family 91 r are uniformly bounded. It follows from Vallee Poussin's theorem, in virtue of ( 1 1 .4) , that the functions of the family 91 have equiabsolutely continuous integrals, i.e. for given 8 > 0 an h > 0 can be found such that
f J u(xl J dx
( 1 1 .5)
8
for all functions in the family, provided mes Iff < h ( Iff C G) . We denote the set (Tr (x) U Tr (y) ) "'(Tr (x) n Tr (y) ) by Tx , y . Suppose that the volume of Tx , y is less than h for d(x, y) < 6. Then for all functions in the family 91r we have, in virtue of ( 1 1 .5) , that _ J ur (x)  U r (Y ) J � _ mr
f J u(t) J dt < _mB_r
1
Tx, lI
for d(x, y) < 6, from which it follows that the functions of the family 91r are equicontinuous. The assertion of the lemma now follows from Arzela's theorem. We shall use the following inequality in the sequel : ( 1 1 .6) To prove this inequality, we denote the sphere with radius r and center at the zero of the ndimensional space by To. Then 1
J J UrJ J M :( __ mr
sup p(v ; N ) ";;; l
J
f J u(t)v(x) dtdx J
G T,(x)
mr
sup p(v ; N ) ";;; l
Interchanging the order of integration
=
f f J u(x
G
III
+ s) v(xl J ds dx.
T.
the last integral and
CHAPTER I I , § I I
97
bringing the supremum sign under the integral sign, we obtain that
I l url lM
:(
_1m_r f [
sup p(v ; N ) ';; l
+
]
S)V (X) l dX ds
G
T.
and, since u (t)
f l u(x
=
=
I l u (x
+
s) i l M ,
0 for t E G, we have
I l u (x + s) I I M = sup
p(v ; N ) ';; l
:(
f l u (x
G
+
sup p(v ; N ) ';; l
s) v (x) I dx �
f I U(t)V(t  S) l dt :( lluI IM. (1 1.7)
G
This completes the proof of inequality
(1 1.6).
3. A . N. Kolmogorov ' s compactness criterion for the space EM. THEOREM A family in of functions of the space EM is compact
1 1.1 .
if, and only if, the following conditions are satisfied : a) I l u l I M :( A , u(x) E in ; b) for arbitrary e > 0, a t5 > 0 can be found such that the con dition r < t5 implies that I l u  Urll M < e for all functions of the family in.
PROOF. The sufficiency of conditions a) and b) follows directly from Lemma and Frechet 's theorem since a set of functions which is compact in C is also compact in every Orlicz space. [FRECHET'S THEOREM : A set is compact if it can be approximated
1 1.1
arbitrarily closely by a compact set.]
Suppose in is a compact family of functions in EM. Then one can construct, for this family, an (e/3) net consisting of continuous functions u(1) (x) , U( 2) (x) , . . . , u( n) (x) . The functions u(i ) (x) (i 1 , 2, . . . , n) may or may not belong to the family in. We denote by c the norm of the characteristic function K (X ; G) of the entire set G. Let r > 0 be a number such that for d(x, t) < r we have I U(i) (X)  U(i ) (t) I < e/ (3 c) (i = 1 , 2, . . . , n) . Then =
I U(ir ) (x)  u(i) (x) I
:(
_m1_r f l u(i) (t)  u(i) (x) I dt :( �e
3c
Co,wez JuncHons
(i = I 2 · · · , n) "
7
CHAPTER 11, § 1 1
98 and
( 1 1 .8) Suppose u (x) is an arbitrary function in in. A function u(io ) (x) can be found such that Il u  u(io ) I IM < 8/3. Then in virtue of ( 1 1 . 6) and ( 1 1 .8) , we have that
I l u  u rl lM � I l u  u(io ) I I M + I l u(io )  u�io ) I I M + I l u�io )  urll M � � 2 1 1 u  u( io) I I M + I l u( io )  u�io) I I M < 8. The necessity of condition b) is thus proved. The necessity of condition a) is obvious. * In the case when the Nfunction M (u) satisfies the L1 2condition, EM L M LM. Therefore, in the case when the L1 2condition is satisfied, Theorem 1 1 . 1 yields a necessary and sufficient criterion for the compactness of a family of functions in LM. Since, in this case, convergence in norm is equivalent to mean convergence, the following theorem is valid. THEOREM 1 1 .2. Suppose the Nfunction M(u) satisfies the .1 2=
=
condition. A necessary and sufficient condition that a family of functions in C LM LM be compact is that the followinR two con ditions be satisfied : =
a)
f M[u(x)] dx �
G
A,
u (x)
E
in ;
b) for arbitrary 8 > 0, a b > 0 can be found such that
f M[u(x)  ur (x)] dx < 8
G
for all functions in the family in, provided r < b. 4. A second criterion for compactness. We shall say that the family in of functions u(x) E LM have equiabsolutely continuous norms if for every 8 > 0 a b > 0 can be found such that I l u K (X ; 6") 11 M < 8 for all functions of the family in provided mes Iff < b. Clearly, in this case we have that in C EM . If in is a compact set in EM, then it can be shown, by means
of usual reasoning, that it has equiabsolutely continuous norms .
CHAPTER II, § 1 1 LEMMA 1 1 .2. A necessary and sufficient condition that a sequence of functions u n (x) E EM (n = 1 , 2, . . . ) , which converges in measure, converge in norm is that it have equiabsolutely continuous norms .
PROOF. The necessity of the condition follows from the fact that a convergent sequence is compact. We shall prove the sufficiency of the condition of the lemma . Suppose the 'sequence un(x) E EM ( n = 1 , 2, . . . ) converges in measure and has equiabsolutely continuous norms. Let I> > 0 be prescribed. We denote by tffm n the sets G{lu n (x)  um (X) I > 17}, where 17 = 1>/{3 mes GN l ( l /mes Gn. Let b > 0 be a number such that 
provided mes tff < b. Since the sequence u n (x) (n = 1 , 2, . . ) converges in measure, an n o can be found such that mes tffmn < t5 for n, m > no . Then, for n, m > n o , we have that .
lI u n  umllM :( :( I I (u n  Um) K(X ; tffm n) I IM
+
I I (u n  Um) K (X ; G/tffm n ) I I M ::(
:( I I UmK(X ; tffm n ) 11 M + Il umK (X ; tffm n) I I M + 17 I I K(X ; G) I I M <
8.
This means that the sequence u n (x) (n = 1 , 2, . . . ) converges in LM. * THEOREM 1 1 .3. If the family in C EM has equiabsolutely con
tinuous norms and is compact in the sense of convergence in measure, then the family in is compact in Lif.
PROOF. From each sequence of the family in we can select a subsequence which converges in measure. In virtue of Lemma 1 1 . 2, this subsequence converges with respect to the norm in Lif. * Verification of the second condition of the theorem on compact ness in the sense of convergence in measure usually reduces t o the proof o f the fact that the family in i s compact i n some Orlicz space which is distinct from Lif.
S. F. Riesz ' s criterion for compactness for the spaces EM. We shall give one more criterion for compactness of a family of functions in EM.
1 00
CHAPTER II, § 1 1
THEOREM 1 1 . 4. A family in of functions in the space EM is
compact if, and only if, the following two conditions are satisfied : a) I l u l i M � A , u (x) E in ; b) for arbitrary e > 0, a b > 0 can be found such that d(h, 0) < b implies that I l u(x + h)  u(x) I IM < e for all u(x) E in. PROOF. Suppose u (x) E in and that ur (x) is the corresponding
Steklov function. Then
1
l u(x)  ur (x) I � __ mr
J l u(x)  u(t) I dt,
from which it follows, for v (x) E LN , p(v ; N) � 1 , that
J l ur (x)  u(x) I v(x) dx
T,(x)
�r J [ J l u(t)  u(x) I dt] v(x) dx.
�
G
G
T,(x)
Interchanging the order of integration and making a change in variables, we obtain that
J l u (x)  ur (x) I v (x) dx
G
�
�
�r J [ J l u(x + s)  U(X) I V(X) dX] ds � To
1 � __
mr
G
J I lu(x + s)  u(x) I IM ds,
To
from which it follows that
Il u  u rl l M
=
sup p (v ; N ) � l
I J [u(x)  ur (x)] v (x) dx I J I lu(x + s)  u(x) I IM ds. � mr �
G
1 __
To
It follows from the last inequality that condition b) of Theorem 1 1 . 1 is satisfied provided condition b) of the present theorem is satisfied. Conditions a) of these theorems coincide. This completes the proof of the sufficiency of the conditions of the theorem.
CHAPTER II, § 1 2
101
We shall now prove the necessity of these conditions . Suppose 9'l is a compact family of functions in EM . Then, for this family, we can construct an (e/3) net consisting of continuous functions u( I) (X) , U( 2) (X) , " ' , u( n ) (x) . We denote by c the norm of the charac teristic function of the entire set G in the space L!f . Let 15 > ° be a number such that
lu(i) (x + h)  U (i ) (x) I
<
e

3c
(i = 1 2 " ' , n) "
provided d(h, 0) < 15. It is then clear that e l I u(i ) (x + h)  u(i) (x) I IM � 3"
(i
=
1 , 2, " ' , n) .
( 1 1 . 9)
Let u(x) be an arbitrary function in 9'l . A U( io)(X) can be found such that I l u  u (io)II M < e/3. Then, in virtue of ( 1 1 . 9) and ( 1 1 . 7) , we have that
I l u(x + h)  u(x) I I M � I l u(x + h)  u(io)(X + h) I IM
+
+ I l u( io)(X + h)  u(io)(x) I IM + I l u(io)(X)  u(x) I I M
<
e.
The necessity of condition b) is thus proved. The necessity of condition a) is manifest . * § 12. Existence of a basis
1 . Transition to the space of functions defined on a closed interval .
Below, we shall consider Orlicz spaces of functions defined on a finite segment . This will not disturb the generality of the results . This follows from the fact that the space L!f(G) is linearly isometric to the space L!f([O, mes G] ) , i. e . there exists a linear onetoone correspondence between the elements of the spaces L!f([O, mes G] ) and L!f (G) , under which norms are preserved. For the sake of simplicity of exposition, we shall show this here for the case when G is a bounded closed set situated in the plane with the cartesian coordinate system (GI, G2 ) . We enclose the set G in a square Bo : I GI I � b, I G 2 1 � b . We consider the sequence Tn of subdivisions of the square Bo into 4 n
CHAPTER I I , § 1 2
1 02 squares
B!
(k
=
I , 2, . . " 4 n ) by means of the straight lines
b . b . 1.0 1 t an d 0 2 = 81 = 1 n n 2 1 2 
(i, j
=
0, ± I , ± 2, " ' , ± 2 n  l ) .
I n the transition from the subdivision Tn t o the subdivision T n+ l , clearly, every square in the subdivision Tn subdivides into four equal squares in the subdivision T n+l (Fig. 9) . We shall construct a mapping of the set of squares of all sub divisions onto the totality of certain segments  subsegments of the segment 10 = [0, mes G] , in which the length of the segment I!  the image of the square B!  equals mes (G (\ B!) . Suppose the entire segment 10 corresponds to the square Bo. Let the segment [or;, P] C 10 correspond to some square B! in the subdivision Tn.
8, 2
822
82'
82
824
J
] 8I
2 8 ] 8 ]' 8 1] 8 ]4
Fig. 9.
We enumerate the four squares in the subdivision T n+ l which comprise B� as the quadrants are numbered in analytic geometry and we denote them by BI, BlI, BIll, BIV. The segment [or;, P] is sub divided by means of the points Y l. Y2 , ya , where or; � Y l � Y2 � ya � p, into four parts such that Y l  or; ya  Y2
BI ) , Y2  Y l = mes (G (\ BlI ) , = mes (G (\ BIll) , P  ya = mes (G (\ BIV) . =
mes (G
(\
We correspond the segments [or;, rr] , [Y l, Y2] , [Y2 , ya] and [Y2 , with the squares BI, BlI, BIll and BIV, respectively.
PJ
1 03
CHAPTER I I , § 1 2
peG
Q
Gl
G
To every set we assign the set C 10 of points defined by the systems of contracting segments which correspond to systems of squares (i .e. subdivisions Tn) , which contract to the points of the set P. Under this mapping, the measure of sets is preserved (this becomes clear if we note that the image of the totality of all sides of the squares of all subdivisions Tn has measure zero) . the set of points x E any neighborhood of If we denote by which contains a part of positive measure of the set and which does not lie on the sides of the squares in the subdivision Tn, then mes = mes and the mapping a, constructed above, of the set onto the segment 10 will be onetoone on The image of the set will be denoted by Since the functions belonging to an Orlicz space are defined to within a set of measure zero, we can assume that all of them are equal to zero on We correspond to each function the function u(t) , t E l 0, defined by u(x) E
G Gl
G
G,
Gl
Gl .
Ql.
G"'.Gl .
LM(G)
_
u(t)
=
(U(X) °
for t
=
for t E
a(x ) E
Ql.
Ql,
This correspondence is manifestly linear. Inasmuch as the measure of sets is preserved under the mapping a, the functions in are mapped, with preservation of norm, into functions in the space mes It is also clear that the space maps, under this mapping, into the space mes
LM(G)
LM([O, GJ). 2.
E (G) EM ([O, GJ). M
Haar functions. The functions defined on the segment [0, 1 J by means o f the following formulas are called H aar functions :
2, ,/2n for  ,/2n for
and, further, for n
X�) (x)
=
=
°
1,
for ° :( x < l, for x = t, for t < x :( 1
k 1 , 2, . . . , 2 n, 2k n 2 :( x < 2k n 1 , 2 +1 2 +1 2k n 1 < x 2kn ' 2 +1 "" 2 +l =
& 
for the remaining values of x.
CHAPTER I I , § 1 2
1 04
We write the Haar functions X�k) (X) as a sequence in the order of increasing n, and for given n in the order of increasing k . The sequence thus obtained is denoted by lPi (X) (i = 1 , 2, . . . ) . This sequence of functions, as is easily seen, is orthogonal :
f lPi (X)lPi (x) dx 1
o
=
(jtj
=
!
0 if i =F j, . , . 1 If Z = J .
Since all Haar functions are bounded, the Fourier coefficients
f U(X) lPi (X) dx
( 1 2. 1 )
1
Ci
=
(i
o
=
1 , 2,
•
.
.
)
are defined for every summable function u(x) . We denote the operators Sm by means of the equality
Smu(x)
=
m
�
i= 1
(m
Ci lPi (X)
=
1 , 2,
•
•
.
).
We denote the points of discontinuity of the functions IP I (X) , 1P 2 (X) , " ' , IPm (x) to which the points 0 and 1 are added by a I , a2, " ' , a p arranged in increasing order. LEMMA 1 2. 1 . For as < x < asH, the value of the piecewise
constant function Smu(x) is defined by the equality Smu(x)
=
f u(x) dx. a. + 1
1
as + l  as
a,
PROOF. It follows directly from the definition of Haar functions that the function m
i= l
assumes, for as < x < asH, the following values :
Fm (x, y)
=
I
1
asH  as o
. If a s if y
< <
y < asH, as or asH < y
( this is verified most easily by induction on
m) .
Therefore, for
1 05
CHAPTER II, § 1 2 as <
x
Smu(x)
< aBH , =
we have that
i�l CPi (X) f U(Y) CPi (Y) dy f u(y)Fm (x, y) dy 1
1
=
0
o
=
3.
1 aB+l  as
=
f u(y) dy.
*
a.
B a sis in
EM. Let M(u) and N(v) be mutually comple
mentary Nfunctions. Suppose u(x) belongs to the space L M([ O, I J ) . From the Jensen integral inequality and Lemma 1 2. 1 , it follows that
M[Smu(x)J
=
M
[
1 as+l  as
f U(X) dX] �
a.
� for every
m
and
as
x<
<
as H ,
f M[u(x)J dx a.+l

as+ l  as
a,
which implies that
f M[Smu(x)J dx � f M[u(x)J dx a. + !
a,+1
and
a,
a,
I M[Smu(x)J dx � I M[u (x)J dx. 1
1
( 1 2.2)
0
o
If lI u llM � I , then, in virtue of (9. 1 3) , we have that
I M[u(x)J dx 1
o
� 1.
It therefore follows from inequalities ( 1 2.2) and (9 . 1 2) that
I M[Smu(x)J dx � 1 + I M[u(x)] dx � 2 1
1
I I Smu l lM � 1 +
o
0
CHAPTER II, § 1 2
1 06 for I l u l iM � 1 . Consequently,
I I Sml 1
=
(m
sup I I Smu l l M � 2
l IullM ';;; l
=
1 , 2, . . . ) .
Thus, the norms of the linear operators Sm , acting in the space LM ( [O , I J ) , are uniformly bounded . Now suppose the function u(x) is continuous. In virtue of Lemma 1 2. 1 , the sequence Smu(x) converges to u (x ) uniformly on the segment [0, I J if one discards from it the denumerable number of points of discontinuity of all Haar functions. Therefore the functions Smu (x ) also converge in norm to u (x) in any Orlicz space. So, the sequence of operators Sm ( m 1 , 2, . . . ) has uniformly bounded norms and converges strongly to the identity operator on a set of continuous functions which is dense in EM ([O, I J ) . In virtue of the known BanachSteinhaus theorem, the operators Sm converge to the identity operator on all of E M ([O, I J ) . This means that the series =
00
( 1 2.3)
where the Ct are defined by equality ( 1 2. 1 ) , converges in LM to
u (x) for arbitrary function u(x) E EM ([O, I J ) . Thus the Haar functions 'Pi (x) (i 1 , 2 , . . . ) form a basis in EM ( [O, I J ) . I t is clear that the functions 'Pt (x /mes G ) (i = 1 , 2 , . . . ) , defined on the segment [0, mes GJ , form a basis in EM ([O, mes GJ ) . This fact =
and the line of reasoning followed in the first subsection of this section lead to the following assertion. THEOREM 1 2. 1 . A basis exists in the space EM (G) . It follows from this theorem that a basis exists for the entire space LM if the Nfunction M(u) satisfies the L1 2condition inasmuch as EM = LM in this case. The Fourier coefficients ( 1 2. 1 ) are defined for all functions u (x) E LM ( [O, I J ) and not only for functions in EM ([O, I J ) . For those functions u (x) E LM ( [ O , I J ) for which series ( 1 2.3) converges, one could have defined an operator P by setting Pu(x)
=
00
� Ci'Pi (X) .
i�l
The operator P would b e an operator o f proj ection o n EM. It turns out that the set LM "'E M contains no functions for which series ( 1 2.3) converges in LM. In fact , suppose series ( 1 2.3) converges
1 07
CHAPTER II, § 1 2 for some function u(x) . We then set
g(x)
=
u(x) 
00
�
i� l
C i (j?t(X) .
All Fourier coefficients ( 1 2 . 1 ) for the function g(x) are equal to zero. In virtue of Theorem 1 2 . 1 , every function u(x) E EN([O, I J ) can be represented in the form of a series, v(x)
=
00
�
i=l
di (j?i (X) , which
converges i n LN([O, I J ) . This enables us to compute the norm of the function g(x) : in virtue of ( 1 0 . 4) , we have that
I f g(x)v(x) dx I
I l g i lM
=
sup
p(v ; N ) ';; l 0 v EEN
This means that g(x)
=
I � dt f g(x) (j?i (x) dx I 1
1
=
° and
sup p(v ; N ) ';; l vEEN
0
i�l
=
0.
u(x) E EM ([O, I J ) .
4 . Further remarks on the conditions lor separability . It was shown in the preceding section that the space Lid is not separable if the Nfunction M(u) does not satisfy the L1 2 condition. This same
fact can be proved by effectively demonstrating a continuous totality of functions in the space Lid the mutual distances between which is greater than some fixed number. The reasoning of the first subsection permits us to limit ourselves to the construction of such a set in the space Lid( [O, I J ) . Suppose the Nfunction M(u) does not satisfy the L1 2condition ; then a sequence of positive numbers Ul < U2 < . . . < Un < . . + = can be found such that M(2u) > 2 nM(u n ) (n 1 , 2, . . . ) . We construct on [0, 1 ) a sequence of disj oint segments t5n arranged in order of increasing indices  from left to right  whose lengths are determined by the equalities .
=
(n
=
1 , 2, . . . ) . 00
Such a construction is possible inasmuch as � mes t5 n < 1 . Suppose, n� l
00
furthermore, that unity is a limit point of U t5 n . n�l
1 08
j
CHAPTER II, § 1 2
We define a function u (x) on [0, I J by setting u (x) =
(n = 1 , 2, . . . ) ,
2Un for X E l5n o
.
00
for x E U I5 n. n= l
The function u (x) belongs t o LAt since !u (x) E L M :
J M[lu(x) J dx = } J M[lu (x)J dx n� M(un) mes I5 n 1
l
o
Suppose, for all 0 by the equality /Pl¥(X)
<
=
{
=
�
l
<
00 .
IX � I , that the functions /Pl¥ (X) are defined
u (x + 1  IX ) for 0 � x � IX, u (x  IX ) for IX < X � 1 .
All these functions belong to LAt. We consider two functions /Pl¥(X) and /pfJ(X) , where IX < p. By construction, the function /PfJ(x) is bounded on the segment [0, IX] . Suppose l /pfJ(x) I = /pfJ(x) � A (0 � x � IX ) . Then a positive number 'YJ < IX can be found such that ( 1 2.4)
where Kl1(X) is the characteristic function of the segment [ IX  'YJ, IX] . In fact , in virtue of formula (9. 1 1 ) , we have I I /pfJKl1 I IM � A l l Kl1 l lM = A 'YJN1
(+)
<
!
for the norm of the characteristic function, for sufficiently small 'YJ. We shall estimate the norm of the function /Pl¥(X) Kl1 (X) from below. Clearly, this norm is equal to the norm of the function U(X) �l1 (X) , where �l1 (X) is the characteristic function of the segment [ 1  'YJ, I J . We consider the sets Fn = [ 1  'YJ, I J n ( U l5i ) n
i= l
(n
=
1 , 2, . . . ) .
The characteristic functions of these sets will be denoted by Kn(X) (n = I , 2, . . . ) . The function u (x) is bounded on each of the sets F n so that the functions vn (x) = P[u(X) Kn (X)] (n = 1 , 2, . . . )
1 09
CHAPTER I I , § 1 2
are also bounded and, a fortiori, belong to the class LN . In virtue of (2.7) , we have that
f
1
o
f M[u (x)] dx + f N[vn (x)] dx.
U(X) v n (x) dx =
1
F"
0
We shall show that
f
1
lim n>oo
N[vn(x)] dx > 1 .
( 1 2.5)
o
In fact , if the inequality
f N [vn(x) ] dx � 1 1
o
were true for all n (n = 1 , 2, ) then the function U (X) l
.
.
,
f M[U(X) l
.
o
+
f
F"
N[ V n (X)] dX =
}
F"
�
sup p(v ; N )';;;; 1
f
1
o
f u (X) Vn (X) dx � 1
n ��t . .
0
U (X) v (X) dx = I l u l i M < 00,
which is a contradiction because
f M[U(X) l
1
=
o
0
f
M[u (x) ] dx =
1  '1

0
1 10
CHAPTER I I , § 1 3
In virtue of ( 1 2.5) , an
no
can be found such that
J N[vn.(x)] dx 1
= p
o
> I.
Then, in virtue of ( 1 . 1 7) , we have that
J N [ Vn�X) ] dx � I . 1
o
Thus, the following estimate is valid for II U I<1/ IIM :
I I U I<1jI l M
=
sup p(v ; N ) ";; l
=
J U (X)I<1j (X)V(X) dx 1
o
� 1
J U(X)I<1j(X) Vno (Xl dx 1
P
0
+ { J M[u(x)] dx + f N[Vn.(X)] dX} > 0
F"o
1.
=
( 1 2.6)
From this inequality and ( 1 2. 4) , it follows that
> 1 Thus, the functions !P£¥ (x) (0 < than t from one another.
oc

t
=
t·
� 1 ) are at a distance greater
§ 13. Spaces determined by distinct Nfunctions
1 . Comparison of spaces. Generally speaking, distinct N functions determine distinct Orlicz spaces. For example, the spaces L £¥ determined by the Nfunctions M (u) l u l £¥/oc are distinct for distinct oc > 1 . THEOREM 1 3. 1 . Let M l (U) and M 2 (U) be two Nfunctions. A necessary and sufficient condition that LM, be e LM, is that the relation M 2 (u) < M 1 ( u) be satisfied, i.e. that there exist constants uo, k > 0 such that =
( 1 3. 1 )
CHAPTER I I , § 1 3
111
PROOF. Let us assume that condition ( 1 3. 1 ) is not satisfied . Then an indefinitely increasing monotonic sequence of numbers Un (n = 1 , 2, . . . ) can be found such that M2 (un) > Ml (2 nnun) (n = 1 , 2, . . . ) . ( 1 3.2) In virtue of ( 1 . 1 7) , Ml (nUn) / (nun) � M l (2 nnun) / (2 nnun) , from which it follows that M l (2 nnun) � 2 nMl (nUn) . Combining the last inequality with ( 1 3.2) , we obtain that ( 1 3.3) Suppose Gl, G2,
•
•
•
are disj oint subsets of the set G for which (n = 1 , 2, . . . ) . 00
Such sets can be constructed inasmuch as � mes Gn < mes G. n= l Now we consider the function u (x) defined by the equality u (x) =
l
(n = 1 , 2, . . . )
nun for x E Gn o
,
00
for x E U Gn. n= l
This function belongs to the space LAt , inasmuch as
J Ml[u(x)J dx = n�l J Ml[U(X) J dX · =n�lMl (nUn) mes Gn =
G
Gn
00 1 = M(Ul) mes G � n < 00. 2 n= l 
This function does not, however, belong to the space LAt . since, for all A � 1 , the functions ( I /A) u (x) do not belong to L M• • In fact, suppose m is an integer greater than A. Then, in virtue of ( 1 3.3) , we have that
00
� � 2 nMl (nun) mes Gn n=m
=
00.
CHAPTER II, § 1 3
1 12
This proves the necessity of condition ( 1 3. 1 ) . We shall now prove the sufficiency of this condition. Suppose condition ( 1 3. 1 ) is satisfied. Then the function u(x) belongs to the space LM,. This means that for some ft > 0, ftu(x) E LM" i.e. that
f M [,uu (x) ] dx <
G
1
We denote the set G{ l u (x) I ( 1 3. 1 ) , we have that
G
<
00 .
kuo/ft} by Go . Then, in virtue of
G�Go
G.
:::;; M 2 (UO) mes Go
+
f Ml[,uU(x)] dx <
G
00.
This means that (ft/k)u (x) E LM. from which it follows that
u (x) E LM •.
*
Let us recall that the Nfunctions Ml (U) and M 2 (U) are said to be equivalent , Ml (U) '" M 2 (u) , if Ml(U) < M 2 (U) and M 2 (u) < < M 1 (u) , i.e. if there exist positive constants k l ' k 2 and Uo such that Theorem 1 3. 1 implies the next theorem. THEOREM 1 3 .2. The spaces LM, and LM• consist of the same
functions if, and only if, the Nfunctions Ml (U) and M 2 (u) are equivalent. This theorem is the basis for the introduction of the concept of equivalent Nfunctions . 2. A n inequality for norms. THEOREM 1 3.3. If LM, e LM., then there exists a constant q > 0
such that
( 1 3.5) PROOF. In virtue of Theorems 1 3. 1 and 3. 1 , LN. C LN, in which connection positive numbers k and vo can be found such that
1 13
CHAPTER I I , § 1 3 Nl (VJk) .:::;; N2 (V) , v � Vo. Then, for all v, we have that Nl Now suppose v (x) ( 1 3. 6) , we have that p
( � ) .:::;; Nl ( :0 ) + N2(V) . E
LN and p (v ; N 2) .:::;; 1 . Then, in virtue of
(� ) f [ r J f ; Nl =
G
Nl
( 1 3 . 6)
V )
+
G
dX ':::;; N l
(: ) O
mes G +
N2 [v (x)J dx .:::;; Nl
( � ) mes G + 1 = a.
We set q = ak. Then, in virtue of the last inequality and ( 1 . 1 7) , we have that p
(q ; ) f [ V
Nl =
G
]
1 v (x) Nl ;;k dx .:::;; ;;
f [
G
v (x) Nl k
Relation ( 1 3.5) follows from the inequalities
I l u lI M. = sup
p(v ; N. ) ';;; l
1f
u (x) v (x) dx
G
.:::;; q
1
=
sup
q
p(v ; N. ) ';;; l
sup p(v ; N ) ';;; l
If G
1f
u (x)
G
u (x) w (x) dx
] dx .:::;; 1 .
I .:::;; I = l l l lM, *
V (x) q
dx
q u
.
It is easily seen that inequality ( 1 3.5) is satisfied with the constant q = 1 : I l u I I M. ':::;; I l u l I M" u (x) E L!t" if the inequality M2 (U) .:::;; Ml (U) is satisfied for all u . From Theorem 1 3.3, it follows, in particular, that the norms generated by equivalent Nfunctions Ml (U) and M2 (U) are equivalent : ( 1 3.7)
In the solution of many problems, one can therefore choose from the class of equivalent Nfunctions that one which, in virtue of any considerations, is the most convenient . Convex
I unctions
8
1 14
CHAPTER II, § 1 3
Let M(¥(u)
=
M(ocu) , oc > 0. Clearly M(¥(u) ,...., M(u) . The equality ( 1 3.8)
holds, for the proof of which it suffices to note that N(¥ (v) and
I l u l IM� = sup
p(v ; N � ) <;; l
I f u(x)v (x) dx I G
=
oc
=
sup
N
(:)
I f u(x)v(x) dx I v(x) dx I oc l l u l lM . I f u(x) oc
sup
p(V/" ; N ) <;; l
p(V/" ; N ) <;; 1
=
G
=
=
G
3. Concerning a criterion for convergence in norm. We shall say that the Nfundion Ml(U) increases essentially more rapidly than the Nfunction M(u) if, for arbitrary A > 0, lim u.. oo
M( AU) M 1 (u)
=
0.
( 1 3.9)
For example, the function l u l (¥ increases essentially more rapidly than the functions l u l P if oc > (3. It is also easily seen that the composition Ml (U) = M[Q (u)] of two Nfundions M(u) and Q(u) increases essentially more rapidly than the Nfundion M (u) . It is not difficult to see that M l (U) increases essentially more rapidly than M (u) i f , and only if, for every s > 0, M(u) < Ml(Sl�) for large values of the argument . LEMMA 1 3. 1 . If M 1 (u) increases essentially more rapidly than M(u) , then N(v) increases essentially more rapidly than N 1 (v) , where N (v) and N 1 (v) are functions complementary to the functions M ( u ) and M 1 (u) , respectively. PROOF. Let an arbitrarily small s > ° and any ft be prescribed. In virtue of ( 1 3 . 9) , the inequality M(l�/S) � M 1 (u/ft) is satisfied for large values of u. In virtue of Theorem 2. 1 and (2.5) , the inequality N1 (ftv) � N (sv) is satisfied for the complementary functions for larg e values of v, from which it follows, III virtue of ( 1 . 1 7) , that N l (ft V ) � sN(v) . This means that N 1 (ft V ) lim O. * ( 1 3 . 1 0) N (v) v.. oo =
CHAPTER I I , § 1 3
1 15
If Ml (U) increases essentially more rapidly than M(u) , then the inclusion ( 1 3. 1 1 ) holds. Since EM is the maximal linear subset of the class L M and LJ.t, is the linear hull of the class LM" it will suffice to show that , for arbitrary Il, the function Ilu(x) belongs to LM, if u(x) E LM,. The last assertion follows from the fact that, in virtue of ( 1 3. 9) . M(llu) � Ml (U) for large values of u . LEMMA 1 3.2. Suppose the Nfunction M1 (u) increases essentially
more rapidly than the Nfunction M (u) . Suppose also that the family of functions in is uniformly bounded in the space LJ.t, : I lu l lM , � a, u(x) E in. Then the family has equiabsolutely continuous norms in LJ.t. PROOF. Let e > 0 be an arbitrarily small prescribed number . We set ft = 2aj e. In virtue of ( 1 3. 1 0) and the de la Vallee Poussin theorem (see § 1 1 , subsection 1 ) , the functions N1[ftv(x) ] , where p(v ; N) � 1 , have equiabsolutely continuous integrals, i. e. a 15 > 0 can be found such that for all functions v (x) E LN, which satisfy the condition p (v ; N) � 1 , we have that
f N1[ftv(x)] dx
8
<
;
,
provided mes C < 15 (C C G) . Let u(x) E in and v(x) E L N, p(v ; N) � 1 . It then follows from Young's inequality that
I f u(x) v(x) dx I 8
�
f M1 [ U�) ] dx f N [ftV(X)] dx � +
8
8
1
� for mes C < 15 , which implies that
I l u K (X ; 6") I I M =
sup p(V ; N ) < l
I f u(x) v (x) dx I &
I : 1 M, �
+
e
for all functions u(x) E in provided mes C < 15 (C C G) . * The converse assertion also holds.
;<
e
CHAPTER I I , § 1 3
1 16
LEMMA 1 3.3. Let M(u) < Ml (U) . Suppose every set of functions bounded in Lid, has equiabsolutely continuous norms in Li.t. Then the Nfunction Ml (U) increases essentially more rapidly than M(u) . PROOF. Let us assume that M l (U) does not increase essentially more rapidly than M(u) . Then, in virtue of Lemma 1 3. 1 , the N function N(v) , complementary to the function M(u) , does not
increase essentially more rapidly than the complementary function N 1 (v) to Ml (U) . This means that an eo > 0 and a sequence of num bers Vn � 00 can be found such that N1 {v n) > N(eov n ) (n = 1 , 2, . . . ) . It follows from these inequalities that the inequalities =
( 1 3. 1 2) are valid for the numbers Wn = N1 (vn) . We shall denote by G n (n = 1 , 2, . . . ) those subsets of G for which mes G n = 1 JWn . Let
un (x)
=
I
Wn . If x E G n, 1 Nl (w n) o if x E G n.
Clearly, I l u nl lM, = 1 . Since mes G n
�
0, the relation lim Il u nl lM
=
0
n+oo
must be satisfied in virtue of the condition of the lemma. This contradicts the following inequality, which is implied by ( 1 3. 1 2) :
N l ( Wn ) Wn Gn) = ; K(X I IM I l u n l lM = I I Nl l (wn) Nl l (wn)
> eo ·
*
THEOREM 1 3.4 . Suppose the Nfunction Ml (U) increases essentially more rapidly than the function M (u) . Suppose also that the sequence u n (x) E LM, (n = 1 , 2, . . . ) is mean convergent to zero : lim n+oo
f M1[un(x)J dx = O.
G
( 1 3. 1 3)
Then this sequence converges to zero with respect to the norm of the space Lid : lim I l u nl lM = O. n+oo
PROOF. It follows from condition ( 1 3. 1 3) that the norms Il u nllM, (n = 1 , 2, . . . ) are uniformly bounded. In virtue of Lemma ( 1 3. 2) ,
CHAPTER I I , § 1 3
1 17
the sequence u n (x) (n = 1 , 2, . . . ) has equiabsolutely continuous norms in L�. Since, in virtue of the same condition ( 1 3. 1 3) , the sequence u n (x) (n = 1 , 2, . . . ) converges in measure to zero, it converges, in virtue . of Lemma 1 1 .2, to zero with respect to the norm in L�. * 4. The product of functions in Orlicz spaces. Let u(x) E L�" w(x) E L�. Generally speaking, the product u(x) w (x) is not even a summable function. However, if the Nfunctions M l (U ) and l/J (u) are interrelated in a definite way, then the product u(x)w(x) may
turn out to be not only summable but it may also belong to a third Orlicz space L�., where M 2 (u) is defined by the functions M l (U) and l/J(u) . In the present subsection, we shall investigate a number of problems which arise in this connection. As the first example, we consider the case when L�, = Dx" L� = Dx A necessary condition that the product u(x)w (x) of the functions u(x) E La', and w (x) E LIX , be a summable function is, obviously, that the inequality I /IXl + I /IX2 :::;; 1 be satisfied. In this connection, u(x)w(x) belongs to all the Ly, where 1 :::;; Y :::;; :::;; IX I IX2/(IXl + IX2) . The verification of these evident facts is left to the reader. We proceed to the general case. LEMMA 1 3.4. Let u(x) be a function. Suppose u(x)w(x) E E for all functions w(x) E L�, where E is an Orlicz space L�. or the space L •.
of summable functions. Then there exists a constant k > 0 such that I l uw llE :::;; k l l w l l!ll · PROOF. The formula
Aw(x) = u(x)w (x) , w (x) E L�
defines an additive, homogeneous operator which acts from L� into E. We shall show that this operator is closed. [We recall that the operator A is said to be closed if I l wn  wo l l !ll + 0 and I I Awn  v i lE + 0 imply that v = Awo. ] In fact , let I l wn  wo ll!ll + 0 and I l uw n  v il E + O. Then the functions w n (x) converge in measure to wo(x) in virtue of which the functions u(x) w n (x) converge in measure to u(x)wo (x) . On the other hand, the functions u (x)wn(x) converge in measure to v (x) . It thus follows from this that v(x) = u(x)wo(x) almost everywhere. Since the closed operator A is defined on the entire Banach space L�, it is continuous. [Every
CHAPTER I I , § 1 3
1 18
closed, additive, homogeneous operator A defined on a complete metric space is continuous (see, e.g . , HILLE [ 1 ] ) .] * LEMMA 1 3.5. Let u (x) w(x) E E for all functions u(x) E Lk" w(x) E L';p, where E is an Orlicz space Lk. or the. space L. Then there exists a constant k > 0 such that
( 1 3. 1 4 )
PROOF. As in the proof of the preceding lemma, we consider the linear operators A u w (x) = u(x) w(x) , w (x) E L';p, defined by the functions u(x) E Lk" I l u l I M, � 1 . The values of these operators on every fixed element w (x) E L';p are bounded since, in virtue of Lemma 1 3.4, the linear operator Bu(x) = u(x) w(x) , acting from LM, into E, is bounded. According to a known theorem (see, e . g . , BANACH [ 1 ] ) , the norms of the operators A u are uniformly bounded. Let I I A ul 1 � k. Then for arbitrary functions u (x) E Lk" w(x) E L';p, we have that
I l uw l lE
= I l l Iu�M1 w i lE l I u l l M,
=
I A I IU�M1 w liE l IullM, �
�
k I l u I I M, l l w I 14l . *
This lemma implies, in particular, the following assertion :
If the product u (x) v(x)w (x) is summable for any triple of functions u(x) E Lk" v (x) E Lk., w (x) E LM., then the following inequality holds :
I u(x) v(x)w (x) dx
G
�
k I l u II M, ll v I I M. ll w I I M.,
This inequality is a generalization of the known HOlder equality :
I u (x)v (x) w(x) dx k ( I l u (x) I IXl dX) ( I I v(x) locl dx) ( I I w(x) I dx )
lll
�
G
�
G
1 /",
G
1 /"2
G
OCI
1 /(%',
where 1 j(¥.l + 1 I(¥. 2 + 1 j(¥.3 � 1 . This assertion is valid for an arbitrary finite number of factors.
1 19
CHAPTER I I , § 1 3
It is easily seen that the product u(x)w(x) will be summable for any functions u(x) E LM" w (x) E L� only in the case when lJI(u) < M l (U) or, equivalently, when N l (u) < C1>(u) . Here, as usual, lJI(u) , N l (U) denote the Nfunctions which are complementary to C1>(u) , M l (U) , respectively. We note that it is impossible to find an Orlicz space to which all the products u(x)w(x) belong if C1>(u) = Nl(U) . This follows from the following theorem. THEOREM 1 3.5 . Suppose u(x) w(x) belongs to some Orlicz space LM• for any pair of functions u(x) E LM" w (x) E L�. Then the N
function M l (U) increases essentially more rapidly than lJI(u) , or, equivalently, C1>(u) increases essentially more rapidly than N 1 (u) .
PROOF. In virtue of Lemma 1 3 . 3, it suffices to show that every bounded set of functions in LM, has equiabsolutely continuous norms in the space L 'Ip. Let T be a sphere of radius r in the space LM,. In virtue of Lemma 1 3.5, we have that I l uw l l M. � 2rk, u(x) E T, p(w ; C1» � 1 . It then follows from de la Vallee Poussin's theorem that the functions u(x)w (x) have equiabsolutely continuous integrals, i.e. to every e > 0 there corresponds a () > 0 such that
I I u(x)w(x) dx 1 < I I u(x)w(x) dx I
e for every
0,
inequality
0,
I I U (X) K(X ; G1) I I IJ1
<
=
e
G l C G, where mes G l
< I I u(x)w(x) dx I <
signifies that mes G l
sup p(w ;
0,
<
(). The
() implies that e. *
THEOREM 1 3. 6. If for every pair of functions u(x) E LM" w (x) E L�, the product u(x) w(x) belongs to some Orlicz space LM., then the N functions M l (U) and C1> (u) increase essentially more rapidly than the Nfunction M 2 (U) . PROOF. We shall prove the assertion of the theorem, for example, for the function M l (U) . Let G l C G and let K(X ; G 1 ) be the characteristic function of the set G l . Since K (X ; G l ) E L�, we have that U(X) K (X ; G l ) E LMz and, in virtue of Lemma 1 3.5 , that
I I U (X) K (X ; G l ) 1 1 M.
�
k l I u IlM, I 1 K I I
Suppose the family o f functions u(x) E LM, has bounded norms, lI u l lM, � a . Then, in virtue of the preceding inequality and formula
CHAPTER II, § 1 3
1 20 (9 . 1 1 ) , we have that
I l u(X) K(X ; G l ) I I M. � ak mes G 1 P l ( l /mes G 1) ,
from which it follows that
lim
mes Gl+ O
Il uK I I M. = 0, i . e. that the
functions u(x) have equiabsolutely continuous norms in LM •. *
5 . Sufficient conditions. We now point out some sufficient conditions for the product of two functions u(x) E LM" w(x) E L� to belong to an Orlicz space LM•. THEOREM 1 3.7. Suppose there exist two mutually complementary
Nfunctions R (u) and Q (u) such that the inequalities
( 1 3. 1 5)
and ( 1 3. 1 6)
are satisfied for u � Uo, where ex , f3 are certain constants. Then, for every pair of functions u(x) E LM" w (x) E L�, the product u(x)w(x) belongs to LM •. PROOF. In virtue of ( 1 3. 1 5) and ( 1 3. 1 6) , the inequalities
( 1 3. 1 7) and ( 1 3. 1 8) are satisfied for all values of the argument . Suppose v (x) E LN •. Then
f u(x)w (x)v (x) dx = exf3 l l u I IM, l l w ll!ll f ex u(x) I l u l I M,
G
G
w (x) v(X) dx . f3 ll w l l!ll
Applying Young's inequality twice to the last integral, we obtain that
I f u(x) w(x) v(x) dx I � exf3 llu I IM, l lw ll!ll { f M2 [R ( ex ;���, )] dx + f M2 [ Q ( f3���!Il )J dX + 2 f N2 [V(X)] dX} , G
G
+
G
G
CHAPTER I I , § 1 3
121
and, in virtue of ( 1 3 . 1 7) and ( 1 3. 1 8) ,
IJ G
u(x) w (x) v (x) dx
I<
{
ocfJ I IU I IMJWI 1 4l M2[R (U O )] mes G +
+ M2[Q (UO) ] mes G + +
J
G
f./J
=
sup p(v ; N.) .;;; 1
IJ G

G
J dx + 2 J N2[V (X)] dX} < =. [�� I l w l l 41
Thus, u(x) w (x) E LM., with I luw l lM.
J MI [ I lUu(X)l lM, ] dx +
u (x) w (x) v (x) dx
G
1<
k l l u l l M, l l wl 1 4l ,
( 1 3. 1 9)
where k = ocfJ(4 + {M2[R (uo)] + M2[Q (UO)]} mes G) . * In particular, the assertion introduced above that under the condition I /OC I + l /oc2 1 the product u (x) w (x) belongs to all Ly, where 1 < I' < y o = OC I OC2/(OC I + OC2) , follows from this theorem. In fact, if M I (U) = l u l <X', f./J (u) = l u l <X2, M 2 (U) = l u l Yo, then the Nfunctions R (u) = M2" 1 [MI (U)] = l u l o,, /)'o and Q (u) = M2" l [f./J(U)] = = l u l « ' /)'o will be mutually complementary since YO/OCI + YO/OC2 = 1 . In the sequel, we shall be interested in the case when the N function MI (U) satisfies the LJ acondition and M2 (U) = N I (U) . As is easily seen, in this case the assertion of Theorem 1 3.7 is satisfied if the inequality
<
( 1 3. 20) is valid for large values of u. In fact, in virtue of Theorem 6.3, we have that N I [M1 (U)] < < MI (ku) for large values of u, i.e. MI (u) < Nl 1 [MI (ku)] . Setting M I (U) = R (u) , N I (u) = Q (u) , we convince ourselves of the fact that the conditions of Theorem 1 3.7 are satisfied. It is also easily seen that in the case when the Nfunction M I (U) satisfies the LJ 2condition and the Nfunction M 2 (U) the LJ 2con dition, then the assertion of Theorem 1 3.7 is valid if the inequality ( 1 3.2 1 ) is satisfied for large values of the argument . To prove this, one must
CHAPTER I I , § 1 3
1 22
set R (u) = MI (U) , Q (u) = NI (U) and note that an Nfunction which satisfies the Ll 2condition increases more slowly than some ex ponential function. If the Nfunction M2 (U) satisfies the LI 'condition, then another sufficient condition can be given, as follows. THEOREM 1 3. 8. Suppose the Nfunction M2 (U) satisfies the LI '
condition . Let the inequalities
( 1 3.22)
and ( 1 3.23)
be satisfied for u � uo, where R (u) and Q(u) are mutually comple mentary Nfunctions and ex , fJ are constants . Then, for every pair of functions u (x) E LM" w (x) E L�, the product u (x)w(x) belongs to LM •. PROOF. In virtue of ( 1 3.22) and ( 1 3.23) , the inequalities
( 1 3.24) and
Q [fJM2 (u) ] < Q({Juo) + C/J (u)
( 1 3 . 25)
are satisfied for all values of u. Since the Nfunction M2 (U) satisfies the LI 'condition, there exist constants a, a I , b l and C such that
for all u, w . Let v (x) E L N •. Then
f u(x) w (x) v (x) dx = l I u I lM, ll w l l41 f u(x)
G
G
w(x) v (x) dx. I l u l I M, I l w l l 41 
Applying the Young inequality to the last integral, we obtain that
I f u (x) w(x) v(x) dx I G
�
�
I l u I I M, ll w l l 41
{ f M2 [ U (X) G
]
w(x)  dx + I l u l I M, I l w l l 41

f N2 [v(x) ] dx} .
G
1 23
CHAPTER II, § 1 3 In virtue of ( 1 3.26) , we have that
I M2 [ I luu (x)llM,

G
w (x) I I w lI 4l

] dx � a mes G + al I M2 [ U (X) ] dx + 
l I u llM,
G
Inequalities ( 1 3.22) and ( 1 3.23) imply that
M 2 (u) and
<
1
a l +  M I (U) a
for all u, where a I , fh are certain constants. Therefore,
I M2 [ l Iu(x) u l lM,

G
al
+a
w(x) I I w lI 4l

] dx � a mes G + a la I mes G +
U(X) J dx +' fhbl mes G + bl I (/J [ w (X) ] dx + I MI [ l I u l lM, f3 I I W I I 4l 
G

G
In virtue of Young's inequality and inequalities ( 1 3.24) and ( 1 3. 25) ,
u (x) ] [ w(x) ] 1 [ u (x) ) I M2 [ M2  dx �  { I R ( aM2 J dx + af3 lI u l lM, l I u l lM, I I w l I 4l
G
G
+
1 I Q (f3M2 [ wI I w(x)lI 4l J) dX} � _ {CR(auo) + Q(f3uo)] mes G + af3
G
CHAPTER I I , § 1 4
1 24
�
1

cx/J
{2 + [R (cxuo) + Q( /Juo}] mes G} .
Combining all the estimates obtained, we see that
J u(x) w(x)v(x) dx <
G
for all v (x) E LN. with
l I uw l lM. = sup
p(v ; NI) ';;; l
where
{
k= 1 +
00
I J u(x)w(x) v (x) dx I G
�
k ll u I I M, ll w l lQ) ,
2c + [a + CX l a l + /J I bl + CR(cxuo) + cx/J

( 1 3.27)
}
aI bI + CQ(/Juo) ] me as G + ;; + 73 . *
§ 14. Linear functionals
1 . Linear functionals in Lid. Suppose M(u) and N(v) are mutually complementary Nfunctions. Let v(x) be a fixed function in Lid. It follows from Holder's inequality (9. 1 6) that the linear functional I (u)
=
(u, v)
=
J u(x) v (x) dx, u(x) E LM,
G
( 1 4. 1 )
is defined on the entire space Lid . The inequality 1 1I1 1 � Il v i i N � 2 1 1 1 1 1 holds, where 1 1 I1 1 denotes the norm of the functional I (u) : 1 1 I1 1
=
sup l !ullM.;;; l
I I (u) l .
( 1 4.2)
1 25
CHAPTER I I , § 1 4
2
The left inequality in ( 1 4. ) follows from the Holder inequality :
The right inequality in ( 1 4.2) follows from (9. 1 2) : =
1 2 1 11 .
I (u, v) l ::( sup I (u, v) = l Iul lM < 2 l We introduce the notation : k(v) = I l v I I N/ I III I . Then inequality ( 1 4.2) can be rewritten in the form 1 ::( k(v) The function k(v) was studied in detail by D. V. Salehov. The simplest properties of this function are given below. As an example, we shall compute the value of the function k(v) for the Orlicz spaces defined by the Nfunctions M (u) = l u l lX / ex (ex > 1 ) , i.e. for the case of the spaces LIX. We recall that for every function u(x) E LM we have I l v i iN
sup
p (u ; M ) <
::( 2.
I l u liM where l /ex + I /P
1 I1
=
sup
l I u l lM < 1
If G
=
=
ex 1 /<X PI /Il
{ f M[u (x) ] dxf /<x , G
1 . Therefore,
u(x) v (x) dx
I
=
sup
",l/ �lll / P{
I M[u(x)]dx}l/� < I
G
sup
If If
I M[u(x)] dx < 1
G
u(x) v (x) dx
G
G
I
u (x) v (x) dx
=
I
=
from which it follows that k(v) = ex l / <xp l /ll, v (x) E LN . The constant exi / "'p l /ll ( l / ex + l iP = 1 ) can assume any value in the interval ( 1 , Let M (u) and N (v) be arbitrary mutually complementary Nfunc tions. Let G 1 be a subset of G. We set v (x) = N l ( l jmes G I ) K (X ; G 1 ) , where K(X ; GI) is the characteristic function of the set G1 . We shall show that
2J.
k (v)
=
mes GIM I ( I /mes GI) N I ( l jmes GI) .
( 1 4.3)
It follows from the formula for the norm of the characteristic
1 26
CHAPTER II, § 1 4
function that IlviiN equals the number appearing in the right member of formula ( 1 4. 3) . It therefore suffices to show that the norm of the linear functional ( 1 4. 1 ) , defined by the function v ) , equals unity . Since p (v ; N) = 1 , we have that
I I/I I = sup I (u , v) 1 � sup
{
sup
Ilul lM "';; l p(w ; N ) "';; l
I lullM ",;; l
On the other hand,
II/I I = sup I (u, v) I �
I (u,
w) l } =
(x
1.
( 1 1 K�x(x; ;:1)11)1M , v) = 1 .
This means that II/II = 1 , and formula ( 1 4.3) i s proved. Thus, the set of values of the function k (v) comprises at least the set of values of the function I lullM "';; l
fey)
M 1 (y)N1 (y) =
y
,
<
mes G
y <
=
.
D . V. Salehov showed that fey) = constant only in the case when M (u) k l u l 'X, IX > 1 . This result signifies that the function k (v) is constant only in the case of the spaces L iX . We shall prove this assertion for the simplest case  when fey) = 2, y � l imes G. We note first of all that the equality fey) = c implies the con tinuity of the right derivatives P (u) and q(u) of the Nfunctions M (u) and N (u) . In fact, it follows from the equality
=
M 1 (y)N 1 (y)
that
=
M 1 (y) q[N 1 (Y)]
cy,
1 � y < =, mes G
N 1 (y) c. + P [M1 (y)] =
Since the derivatives q (u) and P (u) have positive salt uses at their points of discontinuity, it follows from the last equality that q(Zt) and P (zt) are continuous. 2y In virtue of Young's Now let fey) = 2, i . e . , M 1 (y) N 1 (y) inequality, we always have that M 1 (y)N 1 (y) � 2y. We t h er e fo re have the case when the equality sign is attained in Young's in equality. In virtue of the continuity of the derivatives P (lt) and q(u) ,
= .
1 27
CHAPTER II, § 1 4
M l (y) q[Nl (y)] l l (y)J. N (y) P[MMl(y)Nl(y) y 2y l 1 l M (y)P[M (y)] I jmes G. We set M l (y) for all y u. Then P(u)jM(u) 2ju for u Uo M l ( l /mes G) . Integrating the equality P(u)/M(u) 2j u between the limits from Uo to l u i , we obtain that M(u) M(uo) Uo l u i , u Uo·
= the equality sign is attained if or, equivalently, = if Making use of the last equality and the relation = = 2 , we obtain that =
=
�
�
=
=
=

=
2
2
=
�
It follows from ( 1 4.2) that the space Llv can be considered as a linear subset of the space of functionals conj ugate to LM . In this connection, the norm of the space Llv is equivalent to the norm induced on LN as a subset of the space of functionals. Since the space LN is dense with respect to the Orlicz norm, it forms a closed subspace, LN, in the space of functionals, which, generally speaking, does not coincide with the space of functionals on LM, as is indicated by the following theorem. THEOREM 1 4. 1 . ( 1 4. 1 ) LM. PROOF. Let EM be a linear subspace of LM which is the closure in LM of a set of bounded functions. In virtue of Theorem 9.4 and ( 1 0. 1 ) , EM is a proper subset of LM. Let E LM�EM. We define a linear functional on LM by setting = 1 and = 0 for (x E EM, and then extending it by the Hahn Banach theorem, with preservation of norm, to all of LM (see BANACH [ I J ) . We assume that this functional l(u) is representable in theJform
Suppose the Nfunction M(u) does not satisfy is not the general form of a linear the iJ 2condition. Then functional on u)
I(u)
I(u)
uo(x)
I(u)
=
f u (x)v (x) dx, u (x)
l(uo)
E LM,
G
where v(x) is some function. We form the following sequence of bounded functions :
v n (x)
=
{V(X) o
for I v(x) I :(; for I v(x) I >
n (n n.
=
1 , 2, .
. ), .
128
CHAPTER
II, § 14
I(u), we have that I Vn(X) V (X) dx = 0 (n = 1 2 . . . ) , from which it follows that the function vex) equals zero almost everywhere. But then we also have that I(uo) = 0 which contradicts the fact that I(uo) 1. 2. General form of a linear functional on EM. THEOREM 14. 2 . Formula (14.1), where vex) E LM, yields the general form of a linear functional on EM. By the construction of the functional
"
G
=
*
PROOF. We shall carry out the proof of this theorem by following the line of reasoning customary for such theorems. Let be a linear functional defined on EM. We define, on the totality of all measurable subsets eS' of the set G, a set function eS') ] , where (X eS') by means of the equality is the characteristic function of the set eS'. The additive function eS' is absolutely continuous inasmuch as, in virtue of we have that
I(u)
F(eS')
F(eS') I[K(x ;
K ;
=
F( ) (9.1 1), 1 ), [ F( eS') [ = [ I[K(X ; eS')] [ � [ [1[ [ mes eS'N l ( mes eS'
from which it follows that
lim [ ( cf� O
F t&") [
=
O.
In virtue of the RadonNikodym theorem (d. HILLE function is representable in the form mes
F(t&")
F(t&") I v (x)dx, =
[1]), the ( 14. 4)
cf
vex)
where is a summable function on G. It follows from that the equality
( 14. 4)
I(u) I u(x)v(x) dx =
(14.5)
G
is also valid for every measurable function finite number of values.
u(x) which assumes a
CHAPTER I I , §
14
1 29
Let u(x) be an arbitrary function in LM . A sequence of bounded functions u n (x) (n = 2, . . . which converges almost everywhere to u(x) can be found such that I � l u(x) I almost everywhere so that I l un l l M � I l u lIM . The sequence of positive functions I also converges almost everywhere to the function In virtue of Fatou's theorem and we have that
)
1,
I U n(x)
I U n(x)v(X) (16. 5 ), l u (x)v(x)I . I f u(x)v(x) dx I � s�p { f l un(x)v (x) I dX} sup 1 / ( l un (x) I sgn v(x)) I � IIIII sup Il un l lM � 1 I/I I I I u liM < 00 This means that v(x) E LN. We denote the functional f u(x)v(x) dx, defined on LM, by 11 (u) : G
=
G
=
.
n
n
G
i I (u) =
(14. 5 ),
f u(x)v(x) dx.
G
In virtue of the continuous linear functionals I (u) and 1 1 (u) assume the same values on a set of bounded functions which is everywhere dense in EM . This signifies that they take on the same values on all of EM , i.e. formula is valid for all functions u(x) E EM . Distinct functions E LN, obviously, generate distinct functionals on EM . * We shall momentarily denote the norm of the functional considered only on EM , by 1 1/ 1 i I. For every s > 0 , a function u(x) E LM, I l u l iM = can be found such that
(14. 5 )
v(x)
(14.1),
1, f u(x)v (x) dx G
We set
for
I IIII
:(
n,
l u (x) I > l Iunl lM J ul M and un(x)
un (x) Clearly,
{ U(X) l u(x) J
�
Convex functions
=
�
0
for
n
E

s.
(n = I , 2, · · · ) . EM. It follows from the 9
CHAPTER I I , § 1 4
1 30 absolute continuity of the integral that
I un (x)v (x) dx ;:?:: I u(x)v(x) dx
 E
G
G
for sufficiently large n, from which it follows that
1 111 1 1 ;:?:: I l u n l lM 1 11 1 1 1 ;:?::
I Un (X) V(X) dx ;:?:: I I II I
G

2e.
It follows from the inequality j ust obtained and the obvious relation 1 111 1 1 � I I II I that 1 111 i I = l ill i , i.e. that
I III I
=
sup I I (u) l .
( 1 4.6)
I luIlM";; l , u eEM
Equality ( 1 4.6) allows us to also use the same notation I I I I I when the functional ( 1 4. 1 ) is considered on all of Lit as when it is considered only on E M . If the Nfunction M (u) satisfies the Ll 2condition, then EM coincides with Lit = L M . In this case, ( 1 4. 1 ) yields the general form of a linear functional on Lit. 3. ENweak convergence. We shall say that the sequence u n (x) E Lit (n = 1 , 2, . . . ) ENweakly convergent if the sequence
is
of numbers
I (un)
=
I un (x)v (x) dx
(n
=
1 , 2, . . . )
G
converges for every function v(x) E EN. The definition j ust introduced differs from the usual definition inasmuch as the functions v(x) E EN do not , generally speaking, define all linear functionals on Lit. This definition coincides with the usual definition if both Nfunctions M(u) and N (v) satisfy the Ll 2condition. If we consider weak convergence in the space EN , then the definition introduced above coincides with the usual definition if the Nfunction N (v) satisfies the Ll 2condition. In the general case, ENweak convergence is weak convergence in the space Lit considered as the space of functionals on EN. I n fact, as was already shown, there exists a linear onetoone
CHAPTER I I , § 1 4
131
correspondence between the elements of this space of functionals and the elements of LM in which connection the norms of the corresponding elements are equivalent . Thus, to every ENweakly convergent sequence of elements in LM there corresponds a weakly convergent sequence of linear functionals on EN . The following assertions follow from general theorems (see, e.g. , LYUSTERNIK and SOBOLEV [ I J ) . THEOREM 1 4.3. If the sequence of functions u n (x) E LM (n = =
1 , 2, . . . ) is ENweakly convergent, then the norms l !u nl l M 1 , 2, . . . ) are uniformly bounded . THEOREM 1 4.4. Every space LM is ENweakly complete in the sense that for every ENweakly convergent sequence of functions u n (x) E LM (n = 1 , 2, . . . ) a unique function u(x) E LM can be found such that (n
=
!��
J un(x)v(x) dx J u(x)v(x) dx, v(x) =
G
G
E
EN .
Every space LM is ENweakly compact, i . e . every bounded sequence contains an ENweakly convergent subsequence .
We note that the ENweak closure of the space EM in the space LM is the entire space LM. This follows from the fact that for every function u(x) E LM the sequence of bounded functons
u n (x)
=
( u(x) o
for l u(x) I � n, for l u(x) I > n
(n = 1 , 2 , . . . )
converges ENweakly to u(x) inasmuch as
!��
J [u(x)  un(x)Jv (x) dx
G
=
0
for all v (x) E EN (and also for all v (x) E LN) . As was already shown (see p. 9 1 ) , the norms of the functions un(x) also converge to Ilul IM . Thus, the sequence u n(x) converges almost everywhere to u(x) , is ENweakly convergent to u(x) , and I l u nl lM + I l u l IM . However, generally speaking, u n(x) does not con verge in norm to u (x) . Every sequence of functions which converges with respect to the norm in LM, obviously, is ENweakly convergent . The converse,
CHAPTER I I , § 1 4
1 32
of course, does not hold. We shall use the following obvious assertion in the sequel. THEOREM 1 4.5. If the sequence of functions un(x) E Lid ( n = = 1 , 2, . . . ) is ENweakly convergent and compact in the sense
of convergence in norm in Lid, then it also converges in norm.
The following criterion for ENweak convergence is sometimes convenient to use. THEOREM 1 4.6. Suppose the sequence u n (x) E Lid (n = 1 , 2, . . . )
converges in measure to the function u(x) with the norms I l un llM (n = 1 , 2, . . . ) uniformly bounded. Then u(x) E Lid and the sequence u n(x) (n = 1 , 2, . . . ) is ENweakly convergent to u (x) . PROOF. In virtue of the fact that a sphere in Lid is ENweakly compact, every sequence of functions in Lid which are bounded in norm contains an ENweakly convergent subsequence. It is therefore sufficient in our case to show that for any subsequence u n .(x) which is ENweakly convergent in Lid to uo(x) E Lid, we have uo(x) = u(x) . We denote by Km (X) the characteristic function of some fixed set of points on which l u(x)  uo(x) I � m, and the function sgn [u(x)  uo(x)] by vo(x) . Suppose 8 > 0 is prescribed. Since the functions uO(X) , u n.(x) (k = 1 , 2, . . . ) have, in virtue of the de la Vallee Poussin theorem (see § 1 1 , subsection 1 ) , equiabsolutely continuous integrals, a (j > 0 can be found such that
I l uo(x) l dx ; , I l un.(X) l dX ; <
<
(k  1 , 2 . . . ) 
,
provided mes tff < (j (tff C G) . We shall assume that (j < 8/(5m) . It follows from the convergence in measure of the subsequence u n. (x) to the function u(x) and the ENweak convergence of this sequence to the function uo(x) that there exists a ko such that , for k > ko,
I [Un.(X)  uo(x)]vo(X) Km(x) dx ; <
G
and mes G k < (j, where
Gk
=
G{ l un.(x)  u (X) I
�
8/(5 mes G)}.
CHAPTER II, § 1 4
1 33
Then, for k > ko, we have that
f l u(x)  uo(x) I Km(X) dx I f [un.(x)  uo(x)]Vo(X) Km(X) dx I + + f l u(x)  u n.(x) I dx + f I Un.(x) I dx + + f l uo(x) I dx + f l u(x)  uo(x) I Km(X) dx ::;:;;
0
o
<
<
Since
e
e
5
+
e
S mes G
mes (G"",G k )
e
e
+  +  + m mes Gk 5
5
<
e.
is arbitrary, we have that
f l u(x)  uo(x) I Km(X) dx
=
0,
o
i.e. uo(x) u (x) almost everywhere. * 4. ENweakly continuous linear functionals. We shall say that the linear functional l(u) is ENweakly continuous on Lit if for every sequence u n (x) E Lit (n 1 , 2, . . . ) which is ENweakly convergent to the element uo(x) E Lit the equality lim l(u n) l (uo) n+ oo holds. It follows from this definition that functionals of the form ( 1 4. 1 ) , where v (x) E EN, are ENweakly continuous. The converse assertion also holds. THEOREM 1 4.7 . Every ENweakly continuous linear functional on Lit is representable in the form ( 1 4. 1 ) , where v (x) E EN. PROOF. We consider the space EN of functionals defined on the space EN. In virtue of Theorem 1 4.2 on the general form of a linear functional in EN, there exists a onetoone correspondence, defined by the formula =
=
=
f(v)
=
f v(x)u(x) dx, u (x) E Lit, v(x) E EN,
o
( 1 4.7 )
between the functionals f E EN and the functions u (x) E Lit. In this connection, in virtue of the definition of ENweak con vergence, to the sequence of functions u n(x) E Lit which is EN
CHAPTER II, § 1 4
1 34
weakly convergent t o the function u(x) E LM there corresponds in onetoone fashion a sequence of functionals
fn(v) =
f v(x)un(x) dx
G
which converges weakly to the functional
f(v) =
f v (x)u(x) dx.
G
Let lo(u) be an ENweakly continuous functional defined on the space LM. We define a functional cP (f) , f E EN, on the space EN by means of the equality
cP (f) = lo (u) ,
( 1 4.8)
where u (x) E LM is the function corresponding to the functional f in virtue of the onetoone correspondence established between the functionals on EN and the functions in the space LM. It is obvious that the ENweak continuity of the functional lo(u) on LM implies the weak continuity of the functional cP (f) on EN. Since the space EN is separable, we have, in virtue of Banach's theorem on the general form of a weakly continuous functional in the conj ugate space, that there exists a function vo(x) E EN such that cP (f) = f(vo) for all f E EN, i.e.
lo(u) =
f vo(x)u(x) dx
G
for all u (x) E LM. [Banach's theorem (see BANACH [ I J ) asserts that if E is a separable space and tf.J (f) is a weakly continuous functional on the space E conjugate to E, then there exists an element Vo E E such that cP (f) = f(vo) for all f E E . * 5 . Norm of a functional and Il vl l (N) . If we consider the Luxem burg norm I l u l l (M) (see § 9, subsection 7) in the space LM, then the norm of every linear functional l(u) , which admits of an integral representation ( 1 4. 1 ) , is defined by the equality
J
1 11 1 1 (M) = sup
IlulkM, ';;;; l
I f u (x)v(x) dx I , G
and, in virtue of (9.25) , I l l l l(M) = I l v l l(N) .
CHAPTER I I , § 1 4
1 35
The equality
I I III = Il v l l (N)
( 1 4. 9)
also holds, where I III I denotes the usual norm of the functional ( 1 4. 1 ) . This equality coincides with the equality
I l v l l (N) = sup
I lullM ';; l
1 f u(x) v(x) dx I ·
( 1 4. 1 0)
G
To prove ( 1 4. 1 0) , we note that (9.26) implies the inequality sup l Iul lM ';; l
I f u(x)v (x) dx I G
:::;;;
I l v ll (N) .
�
I l v ll (N) .
Thus, it will suffice to show that sup I lullM ';; l
1 f u(x) v (x) dx 1 G
( 1 4. 1 1 )
We denote by T the unit sphere I l v l l(N) :::;;; 1 considered as a subset of the space Ll of summable functions on G. The convex set T i s closed with respect to the norm i n Ll inas much as lim n> oo
f I vn (x)  w(x) I dx =
G
0, v n (x) E T
implies that v n (x) converges in measure to w (x) , in which connection, in virtue of Fatou's theorem and (9 . 2 1 ) , we have that
f N[w(x)J dx
G
:::;;; s �p
f N[vn(x)J dx
G
:::;;; 1 .
Let vo(x) be a fixed nonzero function in the space LN. Clearly, the function ( 1 + e)vo(x) / l l vo ll (N) , where e > 0, is not in T. As we know (see BANACH [ 1 ] ) , a linear functional 1(v) , defined on Ll, can be constructed such that
[
1 ( 1 + e)
vo(x) Il vo l l (N)
]
> 1(v) ,
v (x) E T.
( 1 4. 1 2)
CHAPTER I I , § 1 4
1 36
The functional f(v) admits of the integral representation
f(v)
=
f v(x)h(x) dx, v (x) E L I,
G
where h(x) is an essentially bounded function (in this connection, see AHIEZER and GLAZMAN [ 1 J ) . It therefore follows from ( 1 4. 1 2) that 1 + s
I lvOII(N)
f vo(x)h(x) dx
G
and, in virtue of (9.25) ,
1 + s
( 1 + s) and, since
s
veT
f v(x)h(x) dx =
G
=
SUp I Ivl l(N) ';;; l
f I v(x) I Ih(x) dx
G
f vo(x)h(x) dx
�
I I h I lM
f vo(x) IIh(x)h l lM dx
�
I I vo l l (N) ,
I I vo I I (N) It follows that
� sup
G
G
is arbitrary, we have that
f vo(x) Ilh(x)hllM dx
G
Now, this inequality implies ( 1 4. 1 1 ) .
�
I I vo l l (N) .
.
I
C H A P T E R III
O P E R A T O R S IN O R L I C Z S P A C E S § 15. Conditions for the continuity of linear integral operators
I . Formulation of the problem. This entire chapter will be devoted to the study of linear operators A operating from one Orlicz space Lid, into another Orlicz space Lid• . We shall denote the class of linear operators operating from the space B l into the space B 2 by {B l + B 2}. The class of continuous operators will be denoted by {B l + B 2 ; c.} and the class of completely continuous operators will be denoted by {B l + B 2 ; compo c.}. Basically, we shall be interested in integral operators of the form Au(x)
=
f k (x, y)u(y) dy.
G
( 1 5. 1 )
The fundamental problem of the present section consists in elucidating the conditions under which the operator ( 1 5. 1 ) is continuous considered as an operator operating from Lid, into Lid., i .e. that it satisfies the condition
I I Au I IM. � I I A l l l l u l lM" where I I A I I is some number. We shall naturally search for in the various characteristics suitable such characteristic is Orlicz space, i.e. the finiteness
conditions for the continuity of A of the kernel k(x, y) . The most that the kernel belongs to some of the integral
f f P[exk(x, y)] dx dy
G G
for some ex. Below, we denote by G the topological product G X G equipped with the natural measure . By L M , LM, E M we will denote the corresponding class and spaces LM(G) , Lid (G) and E M (G) .
138
CHAPTER I I I , §
15
2 . General theorem. As usual, we shall denote by N l (V) and N 2 (V) the complementary functions to the given Nfunctions M l (U) and M 2 (U) . THEOREM Let !P(u) be an Nfunction such that for u(x) E L111,
15.1.
v(x) E LN. we have
w(x, y) = u (y)v(x) E L�,
with
15.2) ( 15. 3 ) (
where 1 is a constant. Suppose the kernel k(x, y) of the linear integral operator belongs to the space L rp , where P(v) is the comple mentary Nfunction to the Nfunction !P(u) . Then the operator belongs to {L111 + LM. ; c.}.
(15.1)
(15.1)
PROOF. In virtue of HOlder's inequality and for u(x) E L111, v(x) E LN., that
I Au(x)v (x) dx I I k(x, y) u(y)v(x) dx dy
o
=
0 0
(15. 3) we have,
�
from which it follows that the operator A acts from L111 into LM• . Since I l v i I N. � 2 for p(v ; N 2 ) :::;;; it follows from that
I I Au I IM. = sup
p (v ; N.) .;;; 1
1,
I I Au(x)v (x) dx I
(15. 4)
� 21 1 I k(x, Y ) I I � l l u I l Ml ·
(15. 5)
o
Thus, the operator A is bounded and, consequently, it is con tinuous. * From inequality we obtain an estimate for the norm of the operator A :
(15. 5 ),
I I A I I = sup I I Au I IM. � 21 1 I k(x, y) ll op .
( 1 5.6)
This estimate is, of course, too high. It can be sharpened in many cases. One way of sharpening the estimate of the norm of the operator A can be based on the application of the strengthened Holder inequality (9.26) .
3. Existence of the function !P(u) . The application of Theorem 15.1 requires knowledge of a function !P(u) for which conditions ( 1 5.2) and (15. 3 ) are satisfied.
CHAPTER III,
§ 15
1 39
LEMMA 1 5. 1 . Let tP (u) be defined as the complementary Nfunction to the Nfunction ( 1 5.7) Then conditions ( 1 5.2) and ( 1 5.3) are satisfied . PROOF. Let u(x) E Lid" v (x) E LN. . We shall first show that condition ( 1 5.2) is satisfied, i.e. that the function w(x, y) = u(y)v(x)
belongs to the space L�. Let g(x, y) E L IJI • Since
I I I w(x, y)g(x, y) dx dy I :;:;; (j
:::::: Il u llM, l l v I I N . "'"
1 dX dy, II I g(X, y) I lI ulu(IYI ) l llv(x) l v I I N. M,
(j
we have, in virtue of Young's inequality (2.6 ) , applied to the first of the two factors appearing under the integral sign in the right member, that
I I I w(x, y)g(x, y) dx dy I:;:;; (j
:;:;; I l u I IM, ll v I l N.
+
I v (x) I dx dy + { I I Nl[g(X, y)] I l v i I N. (j
I v (x) I dX dY} . J II M1 [ � Il u l IM, I l v i I N. ()
Applying, once more, Young's inequality to the first term in the curly brackets, we obtain that
I I I w(x, y)g(x, y) dx dy I :;:;; :;:;; I I U I I M, I I V II N. { I I M 2 [Nl[g(X, y)]] dx dy + () + I I N 2 [ ��� ] dx dy + I M l [ I:�� ] dy I :���� dX} . , I . I (j
()
G
G
( 1 5.8)
CHAPTER III, § I S
1 40 Since
V(X) ] dx ::::;; I , J M J N2 [I l v i I N. G
G
1
J dy ..::: [� I l u l I M,
� I
and the function v(x) is summable, it follows from ( 1 3.8) that
I J J w(x, y)g(x, y) dx dy I G
::::;;
{ J J P[g(x, y)] dx dy + I v(x) 1 } , ( 1 5.9) dX mes G + J
I l u I l M, l l v IIN. +
G
G
Il v i I N.
which implies ( 1 5.2) . In virtue of Young's inequality, we have that
+ M 2 ( 1 ) mes G
::::;;
I + M 2 ( 1 ) mes G .
Therefore, if p (g(x, y) ; P) ::::;; 1 , then it follows from ( 1 5.9) that
I l w(x, y) l l $ where
=
sup
p((J ; 'P)<;; 1 1 =
I J J w(x, y)g(x, y) dx dy I ::::;;lllu I IM. llv I lN., G
2 + mes G + M 2 ( 1 ) mes G.
*
( 1 5. 1 0)
The next lemma is proved in an analogous manner.
LEMMA 1 5.2. Let tP(u) be defined as the complementary Nfunction to the Nfunction ( 1 5. 1 1 ) Then conditions ( 1 5.2) and ( 1 5.3) are satisfied. Under the conditions of Lemma 1 5.2, the constant I is defined
by means of the equality 1 =
2 + mes G + N1 ( 1 ) mes G.
( 1 5. 1 2)
4. Concerning a property of Nfunctions which satisfy the LJ ' condition. In the investigation of linear integral operators, an important role is played by Nfunctions M(u) which are such that u(x) , v(x) E LM implies that u(y)v(x) E LM. In this subsection,
we elucidate what Nfunctions possess this property. Clearly, the Nfuncti ons M (u) k l u l lX ( ex > I ) possess the indicated property. =
CHAPTER I I I , § 1 5
141
LEMMA 1 5.3. Suppose u (x) , v (x) E Lll implies that the function w(x, y) = u(y) v(x) belongs to Lll. Then the Nfunction M (u) satisfies the LJ 2condition . PROOF. Let us assume that the Nfunction M (u) does not satisfy the LJ 2condition. Then a monotonic indefinitely increasing sequence of numbers U n, U n + 00 , can be found such that
(n = 1 , 2, . . . ) . ( 1 5. 1 3) We construct disj oint sets G n C G such that mes G n = M(2) · · mes Gj{2 nM(2 n )} (n = 1 , 2, . . . ) and disj oint sets rff n C G such that mes rff n = M(U l ) mes Gj{2 n M(u n)} (n = 1 , 2, . . . ) . M(2un)
We set
v(x) and
>
1
u(x) = Then
I
2 2 nM(2 n )M(u n)
(n = 1 , 2, . . . )
2 n if x E G n o
00
if x E U G n
n= 1
Un if X E rff n o
,
(n
=
1 , 2, . . . )
00
if x E U rff n .
,
n= l
I M[v (x)] dx = n�l I M[v (x)] dx n�1M (2n ) mes G n
a
=
= M (2) mes G
Oft
I M[u(x)] dx n�l I M[u(x)] dx n�lM (Un) mes rffn =
=
G
=
On the other hand, for arbitrarily large k, we have that
II ()
M
[ U(y� (X) ] dx dy =
= � � M 00
00
i = 1 i= 1
� � II
i 1i 1
( 2i Uj ) mes G i mes rffj k
G.
�
Ifl
M
00 ,
=
= M(U l ) mes G
If.
<
<
[ U (Y� (X) ] dx dy =
00 .
CHAPTER III, § I S
1 42
where 2m1 > k . I t follows from this, in virtue of ( 1 5 . 1 3) , that
f f M [ u(y�V(X) ] dx dy o
=
00 .
Thus, u (x) , v ex) E LM, whereas u(y)v(x) E LM. * THEOREM 1 5.2. A necessary and sufficient condition that for an arbitrary pair of functions u(x) , v ex) E LM the product w(x, y) = = u (y)v (x) belong to the space LM is that the Nfunction M(u) satisfy the Lt ' condition . Proof of sulliciency. Suppose the Nfunction M(u) satisfies the Lt ' condition, i.e. there exist positive constants Uo and C such that
M (uv)
�
CM(u)M(v)
(u, v
�
uo) .
( 1 5 . 1 4)
In virtue of Lemma 5 . 1 , the function M(u) satisfies the Lt 2condition. Therefore LM = L M . Let u (x) , vex) E L M . We denote by G u (respectively G v ) the set G{ l u(x) I � uo} (respectively G{ l v (x) I � uo}) , where Uo is the number appearing in condition ( 1 5 . 1 4) . Then, in virtue of this condition, for x E G v , y E G u , we have that M[u(y)v(x)] � CM[u(y)]M[v(x)] . From this inequality and the obvious equality
f f M[u (y)v(x)] dx dy f f M[u(y) v (x)] dx dy + f f M[u(y)v(x)] dx dy + + f f M[u(y)v (x)] dx dy + f f M[u(y)v(x)] dx dy =
o
=
au a o
it follows that
f f M[u(y) v(x)] dx dy C f M[u(y)] dy f M[v(x)] dx + f M[uov (x)] dx + + f M[uou(y)] dy + M(u�) ( e G) , ( 1 5 . 1 5) �
(j
�
mes G
a
a
a
mes G
m s
a
2
1 43
CHAPTER III, § 1 5
and since uov (x) , uou(x) E LM , we have u(y)v(x) E LM = Lk Prool 01 necessity . Let the function w (x, y) = u(y)v (x) E Lif if u(x) , v ex) E Lif. In virtue of Lemma 1 5.3, the Nfunction M (u) satisfies the ,1 2condition. Therefore Lif = L M . Let us assume that the Nfunction M (u) does not satisfy the ,1 ' condition. Then monotonically indefinitely increasing sequences of positive numbers U n , Vn (n = 1 , 2, . . . ) can be found such that
M(u n v n ) > 2 2 n M(u n )M(v n )
(n = 1 , 2 , . . . )
We construct sets G n C G and tt n e G for which mes G n = with Gt
n
M( Vl ) mes G M(Ul) mes G , mes tt n = 2 nM( vn) 2 nM(u n )
j j
G, = 0, tti u(x)

and
vex) = Then
n
.
(n = 1 , 2 , . . . ) ,
tt,
= 0 (i =1= i) . We set Un if X E Gn ( n = 1 , 2, . . . ) , o
if X E U Gn 00
n= l
Vn if x E tt n o
if X E U tt n.
(n = I , 2, · · · ) ,
00
n= l
J M[u(x)] dx = n� l J M[u(x)] dx = n� lM (Un) mes Gn = G.
G
= M(U l ) mes G
and
J M[v(x) ] dx = n�l J M[v(x)] dx = n�lM(vn) mes
a
tt n
J J M[u(y)v(x)] dx dy = i�l i�l J J M[u(y)v (x)] dx dy = aj
<1,
< 00
=
= M(Vl) mes G
8.
On the other hand,
(j
( 1 5 . 1 6)
< 00 .
CHAPTER III, § 1 5
1 44 and, in virtue of ( 1 5 . 1 6) ,
f fM[U(y)V (X)] dX dY o
=
=.
Thus, u (x) E L M , V (X) E L M , and w (x, y) = u(y)v(x) E LM = LM. We have therefore arrived at a contradiction. * It follows from the theorem j ust proved that "u(x) E LM, v (x) E LM implies u(y)v (x) E LM" is not true for all Nfunctions (even those satisfying the Ll 2condition) . LEMMA 1 5.4 . Suppose the Nfunction M(u) satisfies the Ll '
condition. Then a constant a > 0 can be found such that I l u(y)v(x) I IM � a ll u l l M l l v l lM
( 1 5 . 1 7)
for u(x) , v(x) E LM .
PROOF. Suppose M(uv) � CM(u)M(v) for u, v � Uo > 1 . In virtue of ( 1 5. 1 5) ,
(x) ] dX dY '"'" f f M [ uo li uu(y)v ll M . Uo ll v ll M &
o
�C
J dy f M [ � J dx + f M [� I l u l iM Il v i lM
G
+ mes G
G
v (X) ] dX} + M(u�) (mes G) 2, dy + f M [ ] { f [ Iu(y) u liM I l v i lM l G
M
G
and, since p (u/ l l u I IM ; M) � 1 , p(v/ l l vI IM ; M) � 1 , we have that
v(x) ] dx dy � C f f M [ Uo Iu(y) l u l i M · l l v l lM o
2
+
2 mes G + M (u�) (mes G) 2 .
Consequently,
u(y)v(x) I ] dx dy � � 1 + ffM [ I I u�u(y)v(x) u � lI u l l M l l v l l M l l u l lM l l v l l M Iu o
� 1 + C
+ 2 mes G + M (u�) (mes G) 2 .
CHAPTER III, § 1 5 Inequality ( 1 5. 1 7) ,
1 45 which
a = u � [ 1 + C + 2 mes G + M(u5) (mes G) 2] , III
( 1 5. 1 8)
follows from this. * Let Q(u) and R(u) be two Nfunctions. We recall that the relation Q(u) < R(u) signifies the existence of positive constants k and Uo such that Q(u) ::( R(ku) (u � uo) . In virtue of Theorem 1 3. 1 , the relation Q(u) < R (u) is equivalent to the inclusion L'R C L'Q. We also recall (d. Theorem 1 3.3) that the inclusion L'R C L'Q implies the existence of a constant q > 0 such that I l u l l Q ::( q l l u l l R (u ( x) E L 'H) . THEOREM 1 5.3. Suppose the Nfunction lP(u) satisfies the ,1 '
condition with
( 1 5 . 1 9)
Then conditions ( 1 5 . 2) and ( 1 5.3) are satisfied. PROOF. Let u(x) E Lu., v(x) E LN,. In virtue of ( 1 5. 1 9) , we have that u(x) E Lq" v (x) E L q, . Then , in virtue of Lemma 1 5 .4, w(x, y) = u(z) v(x) E Lq, and I l u (y) v(x) 1 1q, ::( a I l u l l q, I l v l l q,. It follows from ( 1 5. 1 9) that there exist constants ql and q 2 such that I l u l l q, ::( ql l l ttl lM. (u(x) E L uJ , I l v ll q, ::( q 2 11 v 11 N . (v (x) E LN,) . Therefore, I l u(y) v(x) 1 1 q, ::( I l l u IIM. l l v II N" where I = aqlq 2 . * 5. Sufficient conditions for continuity . THEOREM 1 5 . 4 ( FUNDAMENTAL THEOREM ON CONTINUITY ) . Let (/)(u) and lJI(v) be mutually complementary Nfunctions. Suppose the kernel k(x, y) of the linear integral operator ( 1 5 . 1 ) belongs to the space L'ip. Then the operator ( 1 5 . 1 ) belongs to {Lu. _ Lu, ; c .} if any one of the following conditions is satisfied : M 2 [N1 (v)] < lJI(v) , a) ( 1 5 . 20) N1[M 2 (v)] < lJI(v) ,
b) c)
the function lP(u) satisfies the ,1 'condition and N l (V) < lJI(v) , M 2 (v) < lJI(v) .
( 1 5 .2 1 ) ( 1 5. 22)
PROOF. In virtue of Theorem 3. 1 , condition a) implies the condition of Lemma 1 5. 1 , condition b) implies the condition of Lemma 1 5.2, and condition c) implies the condition of Theorem 1 5.3. It is asserted in these lemmas and Theorem 1 5 .3 that the conditions of Theorem 1 5. 1 are satisfied and the proposition to be proved follows from the latter theorem. * Convex functions
IO
CHAPTER III, § 1 5
1 46
We note that the Nfunction P(v) appearing must satisfy the condition I v l lX < P(v) ,
where
IX
m
condition c)
( 1 5.23)
> 1 . This follows from the obvious fact that condition
( 1 5.23) is satisfied by every Nfunction whose complementary
function satisfies the L hcondition. Conditions a) , b) , c) of Theorem 1 5.4 are not equivalent. There fore, e . g . , in the choice of the function P(v) satisfying either con dition ( 1 5.20 ) or ( 1 5.2 1 ) , it is natural to make clear at the start which of the compositions M 2 [N l (V)] and Nl[M 2 (V)] increases "the slower. " The conj ecture arises that to answer this question it is sufficient to know which of the two relations N l (U) < M 2 (U) and M 2 (U) < Nl (U) holds. But it turns out that this is not so. For example, for N l (U) = u 2 , M 2 (U) = e1ul  l u i  1 we have that the relation N l (U) < M 2 (U) holds. In this connection, the Nfunctions N1 [M 2 (V) ] and M 2 [Nl (V)] are not equivalent and inasmuch as N 1 [M 2 (v) ] "" M 2 (V) , Nl[M 2 (V)] < M 2 [Nl (V)] M 2 [N l (V) ] "" e v'  1 and, for arbitrary k > 0, we have that lim 1)+ 00
Nl[M 2 (kv)] = o. M 2 [N l (V)]
The relation Nl (U) < M 2 (U) is also true for the Nfunctions N 1 (u ) = e1ul  l u i  1 and M 2 (U) = eU '  1 ; the Nfunctions N 1 [M 2 (V)] and M 2 [N l {V)] are again not equivalent  however, M 2 [Nl (V)] < N 1 [M 2 (v)] since, for arbitrary k > 0, we have that . 11m 1)+ 00
N l [M 2 (v)] = 00. M 2 [Nl (kv)]
A detailed comparison of conditions a) , b) and c) of Theorem
1 5.4 will be executed below. 6. On splitting a continuous operator. Let A be a positive
definite selfadj oint linear operator acting in the space L 2 of functions which are squaresummable on G. As is known (see AHIEZER and GLAZMAN [ 1 ] ) , the operator A admits of the spectral decomposition
A=
f )' dEA 00
o
CHAPTER III , § 1 5
1 47
where EA is the spectral function of the operator A . The operator A 2 then admits of the spectral decomposition 00
In the case when the operator A is completely continuous, the spectral decomposition is replaced by the infinite series 00
( 1 5. 24)
i=1
where the e,,(x) are the characteristic functions o f the operator A corresponding to the nonzero characteristic numbers Ai . We denote the scalar product of the functions e(x) and
(e,
=
A 2
=
In this case,
J e (x)
G
00
�
i=1
A; (e, ,,
W e say that the operator B, acting from the Banach space E l into the Banach space E 2 with domain of definition D (B) , admits of a continuous extension ii, if ii E {E l + E 2 ; c.} and B
Nfunctions with N(u) < u 2 < M (u) . Suppose the operator A 2 (where A is a positive definite selfadjoint linear operator from {L 2 + L 2 ; c.}) admits of a continuous extension to the operator .1 2 E {EN + LM ; c.}. Then ..4 E {L 2 + LM ; c.}. PROOF. Let
� I I A2
l.e. ( 1 5.25)
L 2 is dense in EN since it contains all bounded functions. It therefore follows from ( 1 5 . 25) that the operator A admits of a continuous extension to all of EN. The extended operator will be
CHAPTER III,
1 48
§ 15
denoted by A l . The scalar product l(rp) = ( A 1rp, "P ) , where "P(x ) is a fixed element in L 2 , defines a continuous linear functional on EN. According to Theorem 1 4. 2 on the general form of a linear functional on EN, a function u(x) E LM can be found such that
l (rp) = (rp, u) =
I rp(x)u (x) dx.
G
We define the operator A ; by means of the equality
A ; "P(x) = u(x) .
In virtue of ( 1 0. 4) ,
I I A ; "P I I M =
sup p ( tp ; N ) ';; l tp(x)EEN
I ( A ; "P, rp ) l :S;;
sup I l tp 1 1 N ';; 2 tp(x)EEN
I ("P, A 1rp ) I :s;; 2 k l l"PI I L"
Consequently, the operator A ; belongs to {L 2 + LM ; c.}. We shall show that A ; "P (x) = A "P(x) if "P (x) E L 2 . This follows from the fact that ( A ;"P' rp) = ("P, A 1rp ) = ("P, A rp ) = ( A "P' rp ) for arbitrary function rp(x) E L 2 . * The square root Ai of the operator A is defined as the operator whose square equals A . The operator A l has the spectral de composition 00
and, in the case of the completely continuous operator ( 1 5 . 24) , we have that
A lrp (X ) = � A.� (e i ' rp) e i (x) . 00
i=l
THEOREM 1 5. 6 . Suppose the positive definite selfadjoint operator A , which is continuous in L 2 , admits of a continuous extension to the operator .Ii E {EN + LM ; c.}, where N(u) < u 2 < M(u) . Then
the operator A admits of the representation A = HH* ,
( 1 5 . 26)
where H E {L 2 + LM ; c.} and H* is the operator in {EN + L 2 ; c.} which is adjoint to H. [The operator H* is defined with the aid of the equality ( H* rp, "P ) = (rp, H"P) ("P( x) E L 2 , rp(x) E EN) .]
CHAPTER I I I, § 1 6
1 49
PROOF . We set H = Ai. In virtue of Theorem 1 5.5, H belongs to {L 2 � LM ; c.}. The operator HH* assumes the same values on L 2 as does the operator A inasmuch as (HH*cp, 'IjJ) = (H*cp, H*'IjJ) = = (Aicp, A!'IjJ) = (Acp, 'IjJ) for an arbitrary pair of functions cp(x) , 'IjJ (x) E L 2 . Therefore, the operator HH* is a continuous extension of the operator A to an operator in {EN � LM ; c.}.
Equality ( 1 5 . 26) follows from the fact that a continuous extension of the operator A is unique since L 2 is dense in EN with respect to the norm of the space EN. * The representation A = HH* will be called the splitting ot the operator A . § 1 6 . Conditions for the complete continuity o f linear integral operators 1 . The case ot continuous kernels . We now continue with the study of the linear integral operator A u(x)
=
f k(x, y)u (y) dy .
( 1 6. 1 )
G
In the present section, we shall study the problem of the con ditions for the complete continuity of the operator ( 1 6. 1 ) , i.e. the conditions under which the operator ( 1 6. 1 ) maps the unit sphere of the space LM , into a compact set in the space LM•. LEMMA 1 6. 1 . Suppose the kernel k(x, y) is continuous on C. Let
LM, and LM• be two arbitrary Orlicz spaces . Then the operator ( 1 6. 1 ) belongs to {LM, � EM. ; compo c.}. PROOF. In virtue of Theorem 1 5.4, the operator A belongs to {LM, � LM. ; c.}. Let T be the unit sphere of the space LM,. Since
f l u(x) I dx � I lu IIM, I[K (x ; G) l iN �
G
for u(x)
,
E
T, we have, for x E G, u(x)
I Au(x) I where K
=
=
I f k (x, y)u(y) dy I G
max Ik(x, y) 1 (x, y
E
mes G Mi l E
T, that
� K mes
G) .
G Mi l
( me� G )
( me� G ) ,
C HAPTER III,
1 50
§ 16
This means that the functions I Au(x) I (u(x) E T) are uniformly bounded. Let E > 0 be prescribed. We choose a lJ > 0 such that ]k(x l . y)  k(x 2 , y) 1 < E/[mes GM1 1 ( 1 /mes G)] for d(X I , X 2 ) < lJ, where d(X I , X 2 ) denotes the distance between the points Xl, X 2 E G. Then, with d(x l . X 2 ) < lJ , for an arbitrary function u(x) E T, we have, in virtue of the formula for the norm of the characteristic function of the set G in the space LN" that
I Au(Xl)  AU(X 2 ) I �
f I k(xl . y)  k(X2 , y) l I u(y) dy
G
I
�
The functions Au(x) (u(x) E T) are thus equicontinuous. In virtue of Arzela's theorem, the set AT is compact in the space C of functions which are continuous on G and, a fortiori, compact on an arbitrary Orlicz space. Since the functions Au(x) are continuous, they belong to EM,. *
2. Fundamental theorem . Conditions for the complete continuity
of operators of type ( 1 6. 1 ) can be obtained by making use of criteria for the compactness of a family of functions in Orlicz spaces. A simpler way consists in establishing the possibility of an arbitrarily close approximation of the operator ( 1 6. 1 ) by a known completely continuous operator. It is convenient to consider, as such approxi mating operators, also integral operators but with continuous kernels. In some cases, the conditions of Theorems 1 5. 1 and 1 5 . 4 are sufficient for the complete continuity of the operator ( 1 6. 1 ) . But it is unknown if they are sufficient in the general case. It turns out that the complete continuity of operator ( 1 6. 1 ) will be guaran teed if the condition k(x, y) E VI! in Theorems 1 5 . 1 and 1 5.4 is replaced by the more severe condition k(x, y) E E'l'. We shall prove this fact, which will be utilized in the sequel, from the corre sponding two assertions. THEOREM 1 6. 1 ( C ONCERNING SUFFICIENT CONDITIONS FOR COM PLETE CONTINUITY ) . Let <J>(u) and P(v) be mutually complementary
Nfunctions. Suppose the kernel k(x, y) of the linear integral operator
CHAPTER III, § 1 6
151
( 1 6. 1 ) belongs to the space E 'l' . Then each of the following conditions a) , b) , c) of Theorem 1 5.4 is sulficient for the operator ( 1 6. 1 ) to belong to {LM. � EM, ; compo c.} : a) M 2 [N1(v)] < lJI(v) ; b) N l[M 2 (V)] < lJI(v) ; c) the function f/>(u) satisfies the t1 ' condition and N l (V) < lJI(v) ,
M2 (V) < lJI(v) .
PROOF . Since k(x, y) E E'l', a sequence kn(x, y) (n = 1 , 2, . . . )
of continuous kernels can be constructed such that
I l k(x, y)  k n (x , y) I I 'l' < l in .
We shall denote the linear integral operators
Anu(x) =
I kn(x, y)u(y) dy
G
by A n . In virtue of Lemma 1 6. 1 , these operators act from LM. into EM. and they are completely continuous. Under the conditions of the theorem j ust proved, the conditions of Theorem 1 5. 1 are also satisfied. Therefore, in virtue of ( 1 5 . 6) , we have that
I I A  A n l l � 2l l l k(x, y)  kn(x, y) I I 'l' < 2l/n
(n = 1 , 2, . . . ) .
This means that the operator A can be approximated arbitrarily closely in norm by a completely continuous operator with values in EM• . And this implies the assertion of the theorem. Theorem 1 6. 1 can be applied in two different variants . First , the problem can be posed on the properties of the function lJI(v) under which the operator ( 1 6. 1 ) acts from the given space LM, into the given space EM. and is completely continuous. In this case, the application of Theorem 1 6. 1 depends on the verification of the fact whether the kernel k(x, y) belongs to the space E'l', i .e. is the condition
I I lJI[Ak(x, y)] dx dy < G
satisfied for all A ?
<Xl
( 1 6.2)
CHAPTER III , § 1 6
1 52
If the Nfunction P(v) satisfies the Ll 2 condition, then ( 1 6.2) is equivalent to the condition
I I P[k(x, y)] dx dy < (j
00.
( 1 6.3)
But if the Nfunction P(v) does not satisfy the Ll 2 condition, then the verification of condition ( 1 6.2) becomes rather difficult. It is easily seen that ( 1 6.2) is satisfied if
I I P{Q[k(x, y)] } dx dy < (j
00 ,
( 1 6.4)
where Q (u) is an Nfunction. We also note that, under the conditions ( 1 6.2) , ( 1 6.3) and ( 1 6.4) , one can use, instead of the function P(v) , any other function R (v) which is the principal part of an Nfunction which is equivalent to P(v) . Second, the problem can be posed concerning finding Nfunctions Ml (U) and M 2 (U) such that the operator ( 1 6. 1 ) with given kernel k(x, y) acts from LM, into EM. and is completely continuous. The Nfunctions Ml (U) and M 2 (U) are then chosen so that one of the conditions ( 1 5.20) , ( 1 5.2 1 ) and ( 1 5.22) is satisfied.
3. Complete continuity and ENweak convergence. As is known, completely continuous linear operators acting from one Banach space into another transform a weakly convergent sequence of elements into a sequence which converges in norm. Since the class of ENweakly convergent sequences in an Orlicz space is, generally speaking, more extensive than the class of sequences which converge weakly in the usual sense, supplementary conditions are required in order that a completely continuous linear operator acting from one Orlicz space LM, into another Orlicz space LM2 possess the analogous property with respect to ENweak convergence. Here, we shall elucidate this problem for the linear integral operator ( 1 6. 1 ) . In this connection, we shall assume that the kernel k(x, y) of the operator A satisfies the following condition :
I I k(x, y)u(y)v (x) dx dy ()
<
00
( 1 6.5)
CHAPTER III, § 1 6
1 53
for an arbitrary pair of functions u(x) E LM., v (x) E LN •. In virtue of this condition, the operator A acts fro m Llt, into Llt. and the operator A*, defined by the equality
A*v (x) =
f k(y, x) v(y) dy,
( 1 6.6)
G
acts from LN, into LN" in which connection, the inequality
f Au(x)v (x) dx f u(y) A *v (y) dy =
G
( 1 6.7)
G
holds for an arbitrary pair of functions u(x) E LM" v(x) E LN•. LEMMA 1 6.2. A necessary and sufficient condition that the con
tinuous linear integral operator A with kernel k(x, y) , satisfying condition ( 1 6.5) , transform every EN, weakly convergent sequence of functions in LM, into an EN. weakly convergent sequence of functions in LM• is that the operator A* act from EN. into EN,. PROOF. The sufficiency of the condition of the lemma is evident inasmuch as, in virtue of equality ( 1 6.7) , the sequence of numbers I (Aun) =
f A nu(x) v (x) dx
G
converges for arbitrary function v (x) E E N. in virtue of the fact that A*v(x) E EN, . We shall now prove the necessity of the condition of the lemma. Suppose the continuous linear integral operator A transforms every EN,weakly convergent sequence of functions in LM, into an EN.weakly convergent sequence of functions in LM •. Let v(x) E EN, . Then the functional l(u) , defined on LMl by means of the equality
l(u) =
f Au (x)v (x) dx,
G
will be an EN.weakly continuous functional on LMc In virtue of Theorem 1 4 . 7, this functional is representable in the form
l(u) =
where v* (x) E EN, .
f u(x)v*(x) dx,
G
CHAPTER I I I , § 1 6
1 54 Since, on the other hand, in virtue of ( 1 6.7) ,
f u(x)v*(x) dx f u(x) A*v (x) dx =
G
G
for arbitrary function u(x) E LM" we have that v* (x) A*v(x) for almost all x E G, i.e. that A*v(x) E EN,. * The following theorem can be easily obtained from this lemma. =
THEOREM 1 6.2. A necessary and sufficient condition that the completely continuous linear integral operator A with kernel k(x, y) , satisfying condition ( 1 6.5) , transform every EN, weakly convergent sequence of functions in Llf, into a sequence of functions which converges with respect to the norm in LM. is that the operator A * act from EN. into EN,. PROOF. The necessity o f the condition o f t h e theorem follows,
in an obvious manner, from Lemma 1 6.2. The sufficiency follows from Lemma 1 6.2, Theorem 1 4.3 and Theorem 1 4.5. * We note that , under the conditions of Theorem 1 6. 1 on the complete con tinuity of the operator ( 1 6. 1 ) , condition ( 1 6.5) is satisfied and the operator A *, defined by equality ( 1 6 . 6) , acts from LN, into EN, (and, a fortiori, from EN. into EN,) . We shall prove the last assertion. Under the conditions of Theorem 1 6. 1 , conditions ( 1 5. 2) and ( 1 5 .3 ) are satisfied. Suppose v(x) E LN, and let e be an arbitrary positive number. Since the kernel k(x, y) , belonging to E'I', has an absolutely continuous norm in L t" a 15 > 0 can be found such that
mes S < 15 implies that I I k(x, Y ) K (X, y ; 2' ) 1 1 '1' < e /{2 1 I v I I N.t } (i C G) , where l is the constant in condition ( 1 5.3) . Let S C G, u(x) E LM1 . Then
f A*v(x)u(x) dx f f k(y, x)v(y)u(x) dx dy f f k(y, x)v(y)u(x) dx dy, =
8
8
=
G
=
where S S X G. Now let mes S < t5/mes G and p(u ; M) � I . Then mes S mes S mes G < 15, Il u l i M � 1 + p(u, M) � 2. Applying the Hol=
=
=
CHAPTER I I I , § 1 6
1 55
der inequality and ( 1 5.3) to the last integral, we obtain that
I f A *v (x)u(x) dx I
::s;
l l l k (x , Y) K(X, y ; ti) 1 I 'l' · 2 1 1 v 1 l N. <
e.
of
It follows from this that I I A *v (x) K(X ; t9') l iN, < e provided mes t9' < �/mes G. Thus, the function A *v (x) has in LN, an absolutely continuous norm and, consequently, it belongs to EN,. Thus, under the conditions of Theorem 1 6. 1  without any other, supplementary, assumptions  the operator ( 1 6. 1 ) trans forms every EN,weakly convergent sequence of functions into a sequence of functions which converges with respect to the norm in LM• . We note further that the assertion of Theorem 1 6.2 remains valid for abstract linear completely continuous operators A if the Nfunction Ml (U) satisfies the ,1 2 condition and the operator A* is defined in the following way. Let v (x) E LN• . We consider the linear functional, defined, on LM, = EM" by means of the equality
l(u)
=
f Au(x)v (x) dx.
G
In virtue of Theorem 1 4.2, this functional can be represented in the form
l(u) =
f u(x)v*(x)dx,
G
where v* (x) E LN,. We set A *v (x) = v*(x) . Thus, the operator A* acts from LN. into LNt and ( 1 6.7) holds.
4. Zaanen ' s theorem.
LEMMA 1 6.3. Suppose the kernel k(x, y) satisfies either one of the following two conditions : a) for almost all x E G, the kernel k ( x, y) , as a function of y, belongs to the space LNt, where the function q;(x) = I l k (x, y) I IN , belongs to the space LM. ; b) for almost all Y E G, the kernel, as a function of x, belongs to the space LM., where the function P(y) = I Ik(x, y) I I M. belongs to the space LN,. Then the operator ( 1 6. 1 ) belongs to {LM, + LM. ; c.}. PROOF. Suppose the condition a) is satisfied. Then, m virtue
CHAPTER I I I , § 1 6
1 56 of Holder's inequality,
I f Au(x) v (x) dx I G
�
�
I I I k(x, y)u(y) dy I I V (X) I dx � l Iu llM1 I cp (x) I v(x) I dx
G
G
G
for arbitrary functions u(x) E LM" v (x) E LN. , from which it follows that
I I Au I iM. = sup
p(v ; N.) ';;; l
I I Au(x) v (x) dx I
� I lcpI l N. ll u I l M"
G
If condition b) is satisfied, then, interchanging the order of integration (which is possible on the basis of Fubini 's theorem) , we obtain that
I f Au(x)v (x) dx I I I I k(x, y)v (x) dx I I U(y) I dy �
G
G G
� I l v i I N.
I tp (y) l u(y) I dy
� �
I l v i l N. !ltpI I N, ll u I IM,
G
for arbitrary functions u(x) E Lk, v(x) E LN. , from which it follows that
: I Au I iM. = sup
p(v ; N.) .;;; l
I I Au(x) v(x) dx I I I Au(x) v(x) dx I G
�
�
.;;; 2
sup
' I v I IN •
G
� 2 1 I tp I I N, l l u I l M, . *
THEOREM 1 6.3. Suppose the kernel k(x, y) , as a function of y, belongs to EN, for almost all x E G, where the function cp (x) Then the operator ( 1 6. 1 ) belongs to = I I k(x, y) I I N , belongs to EM •. {LM, � EM. ; compo c.}. PROOF. In virtue of the preceding lemma, the operator ( 1 6. 1 ) =
acts from LM, into LM• and is continuous. It remains to show that this operator transforms the unit sphere T of the space Lk into a set of functions which is compact in EM•. In virtue of the EN,weak compactness of the sphere T, it suffices to show that the operator A transforms an EN,weakly convergent sequence of
1 57
CHAPTER III, § 1 6
functions on the sphere T into a sequence of functions in EM, which converges in norm. Suppose the sequence u n (x ) E T (n = 1 , 2, . . . ) is EN,weakly convergent to the function uo (x) . In virtue of the condition of the theorem, the sequence of function'> A U n (x) (n = 1 , 2, . . . ) converges to the function Auo(x) almost everywhere and, conse quently, it converges to this function in measure. We shall show that the functions A u n (x) have equiabsolutely continuous norms in LM, . Let K (X ; rff ) be the characteristic function of the set rff C G, v (x) E L N, . Then
If
I
A U n (x) v �x) dx �
,ff
�
f I f ll(x,
8
G
y) 'u n(Y) dy
I I V (X) I dx f !p (x) I v (x) I dx, �
8
from which it follows that
x
Since the function !p (x) belongs to EM" it follows that the functions A u n ( ) have equiabsolutely continuous norms. In virtue of Lemma 1 1 .2, the sequence A u n ( x ) , which belongs to EM" converges in norm. * Theorem 1 6.3 can be augmented by the following assertion. THEOREM 1 6.4. Suppose the kernel k(x, y) , as a function of x, belongs to EM, for almost all y E G, where the function "P (y) y) I IM, belongs to EN, . Then the operator ( 1 6. 1 ) belongs to = {EM, + EM, ; comp o c.}. PROOF. In virtue of Lemma 1 6.3, A E {LM, + LM , ; c.}. We consider the operator A* defined by means of equality ( 1 6.6) . In virtue of Theorem 1 6.3, the operator A * belongs to {LN . + EN, ; compo c.} and transforms every EM.weakly convergent sequence of functions in LN, into a sequence of functions in EN, which converges in norm. In virtue of Theorem 1 6.2, the operator A = (A*) * acts from EM, into EM• . In order to complete the proof, it remains to prove that the operator A transforms every EN,weakly convergent sequence of functions from the unit sphere
I l k (x,
=
1 58
CHAPTER I I I , § 1 6
T of the space LM. into a sequence which converges with respect to the norm in LM • . Suppose the sequence of functions u n (x) E T (n 1 , 2, . . . ) is EN.weakly convergent to the function uo(x) . It is easily seen that I l uo l lM . ::;;; 2. The sequence AUn (x) is EN.weakly convergent to the function Auo(x) inasmuch as for every function u(x) E EN. we have, in virtue of equality ( 1 6 . 7) , that =
lim It+OO
J A [u n (x)  uo(x)]v (x) dx =
G
=
lim n,>oo
I [u n (x)  uo(x)] A*v(x) dx = O.
G
Let e > 0 be given. We denote by A *Vl (x) , A *V 2 (X) , . . . , A *vs(x) a finite ( e /6) net of the set {A *v} (p(v ; N 2 ) ::::;; 1 ) , which is compact in virtue of the complete continuity of the operator A*. Then, for every function v(x) E LN. (p(v ; N 2 ) ::::;; 1 ) we can find a function v i (V ) (x) such that I I A *v  A *v i (v ) II N 1 < e /6. Suppose the inequalities
I [un(x)  uo(x)] A *v,/ (x) dx < ;
G
(i = 1 , 2,
· · · , s
)
are satisfied for n � no . Then, for n � no, we have that
I I A [un(x)  uo(x)]v (x) dx I I [un(x)  uo(x)]A*v(x) dx I ::::;; p (v ; N. ) ';;; 1 I I I [un(x)  uo(X)]A*Vi(v) (x) dx I + p ( v; N. ) ';;; 1
I I Aun  Auo llMa = ::::;;
+
=
sup
p ( v ; N . ) ';;; 1
sup
sup
sup
p ( v ; N. ) .;;; 1
G
=
G
G
I l u n (x)  uo(x) IIA *v (x)  A *vi (V) (x) I dx <
G
i.e. the sequence Au n (x) converges in norm to Auo(x) . *
CHAPTER III,
§ 16
1 59
We note that the sufficiency of condition a) of Theorem 1 6. 1 can also be obtained as a consequence of Theorem 1 6.3 . In fact , suppose the kernel k(x, y) belongs to the space EMalN,] . Then, for arbitrary A > 0, we have that
f f M2{Nl[Ak(x, y)]} dx dy < (j
00.
( 1 6 .8 )
It follows from this that the conditions of Theorem 1 6.3 are satisfied. In fact, it follows from ( 1 6 . 8 ) that
f N1[Ak(x, y)] dx <
G
00
for arbitrary A > 0 and for almost all x E G. This means that for almost all x E G the kernel k(x, y) , as a function of y, belongs to the space EN,. I n virtue of (9. 1 2) , the estimate
q;(x) � 1 +
f Nl [k(x, y)] dy
G
is valid for the function q;(x) = Il k(x, y) I I N ,. In order to complete the proof of the theorem, it suffices to show that
f M2 [,uq; (x)] dx <
G
for arbitrary ,u > (2 mes G)  I . inasmuch as
00
This fact follows from t I 6.8)
f M2 [,uq;(x)] dx f M2 {,u + ,u f Nl [k(x, Y)] dY} dx { f N1[2,u Gk(x, y)] dX ) dx � � !M2 (2,u) mes G + ! f M 2 �
G
G
� !M 2 (2,u) mes G + +
1 2 mes G
G
G
�
G
mes
mes G
ff M2 {Nl [2,u mes Gk(x, y)]} dx dy < (j
00.
CHAPTER I I I , § 1 6
1 60
5 . Comparison 0/ conditions. As a first example, we shall consider the simplest case  when LlI , LlI. L 2 . In this case, conditions a) and b) of Theorem 1 6. 1 signify that the kernel k(x, y) must satisfy the relation =
=
f f k4 (x, y) dx dy <
00
f f k2 (X, y) dx dy <
00
G
.
But if we make use of condition c) , taking the function v 2 for the function P(v) , it turns out that the satisfaction of the relation ()
( 1 6.9)
L2.
is sufficient for the complete continuity of the operator ( 1 6. 1 ) in Condition ( 1 6.9) is less restrictive. We note that the known condition ( 1 6.9) is not necessary for the complete continuity of the linear integral operator in L 2 . It is clear that all the conditions, indicated above, for complete conti nuity in the case of arbitrary Orlicz spaces are only sufficient . As a second example, we note that it follows from this same condition c) of Theorem 1 6. 1 that the linear integral operator ( 1 6. 1 ) acts from L "' , into L "' . (CX1 > 1 , CX 2 > 1 ) and is completely continuous if the kernel k(x, y ) is pth power summable on G with p equal to max {jh , CX 2}, where I /CX1 + l /fh = 1 . In order to obtain this known result (which, of course, is proved directly quite simply) , it suffices to set P(v) = I v l max{pl, "" } . e1 u l ( 1 + l u i ) In ( 1 + l u i )  l u i , Now, let M 1 ( )  l u i  1 . In this case, N1 (u) M 2 (u) . From Theorem 1 6. 1 there follow three conditions which are sufficient that the operator ( 1 6. 1 ) act from Lk L M into E M. and be completely continuous. These conditions can be written in the form
U
M2 (U)
=
=
=
=
ff G
exp {exp l A.k (x, y) l } dx dy < 00
if we make use of conditions a) or b) , and in the form
ff ()
exp 1 A.k(x, y) l dx dy < 00
if we make use of condition c) , setting P(v)
=
( 1 6. 1 0)
M 2 (V) .
CHAPTER I I I , § 1 6
161
Thus, in this case also, condition c) turns out t o be less restrictive . The last example is an illustration of the following theorem (which follows directly from Theorem 1 6. 1 )  this theorem will play an important role in the study of integral equations with essentially nonpower nonlinearities. THEOREM 1 6.5. Suppose M (u) and N (v) are mutually comple
mentary Nfunctions where N(v) satisfies the LJ 'condition. Let
f f M[Ak(x, y)] dxdy < G
00
for arbitrary A > o. Then the linear integral operator ( 1 6. 1 ) belongs to {LN + EM ; compo c. } . In the next example, we set M1 (u) = l u l IX/ae (ae > 1 ) , M 2 (U) = = e1ul  l u i  1 . For the complete continuity of operator ( 1 6. 1 ) , it is sufficient : in virtue of condition a) that
ff G
exp I Ak(x, y) I P dx dy < 00
(� + �
= 1
)
for arbitrary A > 0, and in virtue of conditions b) or c) that
ff G
exp I Ak(x, y) I dxdy < 00
.
The second condition is, of course, less restrictive . We note, further, that it coincides with condition ( 1 6. 1 0) , which was obtained in the preceding example. This is not accidental since the following more general assertion holds. THEOREM 1 6.6. Let
( 1 6. 1 1 )
Then every linear operator in {Llr , + EM. ; compo c.} belongs to {L4> , + E� . ; compo c.}. PROOF. The fact that A E {L4> + E�.} follows from the inclusions L4> , C Llr" EM. C E�• . In virtue of Theorem 1 3.3 and ( 1 6. 1 1 ) , the inequalities lI u l lM, � q l l 1u l l�, (u(x) E L4> , ) and lI u l l �. � Q 2 I1 u I lM. (u(x) E Llr.) hold. Let T be a bounded set in L4> , . In virtue of the first of the above inequalities, T is also bounded in Llr, . Therefore, the set AT is compact in Llr •. In virtue of the second inequality, it is also compact in L4> • . * Convex junctions
II
CHAPTER I I I , § 1 6
1 62
I t follows from the preceding examples that condition c) is less restrictive, in a number of cases, than conditions a) and b) . We shall now give an example which shows that this is not always so. Let M l (U) = e1ul  l u i  1 , M 2 (U ) = ( 1 + l u i ) In (1 + l u i )  l u i . In this case, EM. = Lid. = LM •. It follows from conditions a) and b) of Theorem 1 6 . 1 that the operator ( 1 6 . 1 ) belongs to {Lid, + EM . ; compo c.} if the inequality
f f I k(x, (j
Y) I In 2 ( 1 + I k(x, y) I ) dx dy < =
( 1 6. 1 2)
is satisfied. Application of condition a) leads to the assumption that the kernel k (x, y) is summable with some power 01: > 1 , i.e . it leads to a more restrictive condition (see Remark, p. 1 46) . Summarizing everything stated above, we can formulate a rule for finding a function lJI(v) satisfying the conditions of Theorem 1 6. 1 . In this connection , we shall assume that any two Nfunctions q} l (U) and q} 2 (U) , which may be under consideration, are "compa rable, " i. e. that one of the relations : q} l (U) < q} 2 (U) and q} 2 (U) < q} l (U) is satisfied. In the first case, we shall say that q}l (U) is "smaller" than q} 2 (U) and in the second that q}l (U) is " larger" than q} 2 (U) . Let the spaces Lid, and Lid. be given. We consider the Nfunctions Ml (U) and N 2 (U) . If the "smaller" of them satisfies the LJ 'condition, then we shall consider the complementary Nfunction to it to be equal to lJI(v) . Obviously, in this case condition c) is satisfied which leads to the least (in comparison with conditions a) and b)) restrictions on the kernel k(x, y) . But if the "smaller" of the functions Ml (U) and N 2 (u) does not satisfy the LJ 'condition then two cases are possible : 1 . Both of the functions M1 (u) and N 2 (U) increase more rapidly than an arbitrary power l u l <% ( 01: > 1 ) . We shall assume that they satisfy the LJ 3condition. Then the functions N l (V) and M 2 (v) increase less rapidly than any power I v l P ({J > 1 ) . Making use of condition a) or b) of Theorem 1 6. 1 and Theorem 6.9, we can set lJI(v) = N l (v) M 2 (v) / l v l which is equivalent , in this case, t.o one of the functions M 2 [N1 (v)] or N1[M 2 (v)] and is "smaller" than the other. It is easily seen that the function lJI(v) also increases less rapidly than any power of the form Iv l P ({J > 1 ) . But any function lJI(v) , which satisfies condition c) in virtue of the
CHAPTER III, § 1 6
1 63
remark on p. 1 46, increases more rapidly than some power function I v l fio ({Jo > 1 ) , which leads to a greater restriction on the kernel k(x, y) . 2. The " smaller" of the functions M 1 (u) and N 2 (u) does not satisfy the LI /condition but it does not satisfy the LI scondition either. In this case, we can make use of condition c) , taking as the function P(v) some Nfunction which is larger than N1 (v) and M 2 (v) but whose complementary function satisfies the LI condition. One can also make use of conditions a) or b) , taking as the function P(v) the function M 2 [N l (V)] or N l [M 2 (V)] . In some cases, the first method leads to a lesser restriction with respect to the kernel k(x, y) and, in other cases, the second. We shall elucidate this by means of examples. Let M1 (u) = u 2 /2, M 2 (U) = u 2 ( l ln l u l l + 1 ) . In this case, the function N 2 (u) is equivalent to the Nfunction zf 2 /ln ( l u l + e) < < M l (U) and does not satisfy the LI /condition. Making use of condition c) , setting P(v) I v l 2 + e, where e is a positive number, we arrive at the following restriction on the kernel k(x, y ) : I
=
I I I k(x, y) 1 2 + e dx dy < 00 (j
.
Application of condition a) or b) leads to the worse conditions :
I I I k(x, Y) 1 4 ln2 ( l k(x, y) 1 + l ) dx dy < 00 (]
or
I I I k(x, y) 1 4 ln ( l k(x, y) 1 + l ) dx dy < 00, (j
respectively. Now let M l (U) = elul  l u i  1 , M 2 (u) = u 2 ( l ln lu l l + 1 ) . In this case, the Nfunction N 2 (Zf) is also "smaller" than M1 (u) and does not satisfy the LI /condition. Application of condition a) leads to the following restriction on the kernel k(x, y) :
I I I k(x, Y) 1 2 1n3 ( l k(x, y) 1 + l ) dx dy < 00 ()
CHAPTER I I I, § 1 6
1 64
and application of condition b) leads t o the restriction
I I I k(x Y) 1 2 G
,
ln 2 ( l k(x , Y ) 1 + l ) dxdy <
=.
But to apply condition c) one must take as the function tP(u) , complementary to P(v) , a function which is "smaller" than u 2 /ln ( l u l + e) and which satisfies the LI 'condition. Such a function can be, for example, the function l u I 2 ., where 0 < e < 1 . But then the function P(v) will increase as I v 1 2+B1 (el > 0) which leads to a worse condition. 6 . On splitting a completely continuous operator. THEOREM 1 6.7. Let M(u) and N(u) be mutually complementary Nfunctions with N(u) < u 2 < M(u) . Suppose the operator A 2 (where A is a positive definite selfadjoint linear operator in {L 2 _ L 2 ; compo c. } ) admits of a continuous extension to the operator ..42 E {EN  Lif ; compo c. } . Then A E {L 2  Lif ; compo c. } . PROOF. Since the conditions of Theorem 1 5.5 are satisfied, we have that A E {L 2  Lif ; c. } . Let A l be the operator acting from EN into L 2 , introduced in the proof of Theorem 1 5.5. As was proved, A is the adj oint operator to A I. Therefore, to prove the complete continuity of the operator A, i t suffices to prove that the operator A l i s completely continuous. Let !p l (X} , !P 2 (X) , . . . be an ENweakly convergent sequence of functions in L 2 . Then A 1 !Pt (x) = A!pt(x) and
I I A 1 ( !P n  !Pm) I J l.
=
(Ai {!P n  !Pm) , !pn  !Pm ) < < I I A 2 (!p I�  !Pm ) I I M I I !P n  !Pm I I N .
The first factor in the right member tends to zero since the operator A 2 is by assumption completely continuous and trans forms every ENweakly convergent sequence of functions into a sequence whi ch converges with respect to the norm in Lif. The second factor in the right member is bounded. Consequently, the sequence A1 !P n (x) (n = 1 , 2, . . . ) converges in norm in L 2 . N ow let "1'1 (x) , "1' 2 (x) , . . . be an arbitrary sequence which con verges weakly in EN. Since L 2 is dense in EN, a sequence !p I (x) , !P 2 (X) , . . . can be found in L 2 such that I I V'n  !P n i l N  O. The sequence !P l (X) , !P 2 (X) , . . . is also weakly convergent . As was already
CHAPTER III, § 1 6
1 65
proved, the sequence AI !P n (x) converges in norm in L 2 . Therefore, the sequence AI1J' n (X) also converges in norm in L 2 inasmuch as
I I A1 (1J' n  1J'm) l l L ' � I I A1 ( !P n  !Pm) ll L ' + + I I A1 ( !P n  1J' n ) l l L ' + I I A1 (!Pm  1J'm ) II L2 '
*
Following a line of reasoning analogous to that used in the proof of Theorem 1 5.6, we arrive at the following proposition. THEOREM 1 6.8. Suppose the positive definite selfadjoint completely
continuous linear operator A in L 2 admits of a continuous extension to an operator A E {EN _ LM ; compo c.}, where N(u) < u 2 < M(u) . Then the operator A splits, i.e. A = HH*, where H E {L 2 _ LM ; compo c.} and H* is the operator from {EN _ L 2 ; compo c.} which is adjoint to H. 7 . On operators of potential type. We shall assume that G is a closed bounded region (with a sufficiently smooth boundary) in ndimensional space. An operator of potential type is a linear integral operator whose kernel is symmetric and satisfies the condition
a I k(x, y) 1 � ,r). ,
(x, Y E G) ,
( 1 6. 1 3)
where r is the distance between the points x and y. A detailed theory of operators of potential type is developed in S . SOBOLEV [ 1 ] in connection with embedding theorems. S . Sobolev and other authors showed that under definite relations among the numbers A, oc and the dimension n of the space, an operator of potential type acts from the space LOI into a definite space LOll or into the space C of continuous functions. In particular, for A = nj2, an operator of potential type acts, according to S. Sobolev's theorems, from the space L 2 into an arbitrary space LOll (OCI > 1 ) . Theorems on the splitting of linear operators enable one to strengthen this assertion. Let A be a linear integral operator acting in L 2 whose kernel satisfies condition ( 1 6. 1 3) in which A = nj2. Then the operator A 2 will also be a linear integral operator with kernel
k 2 (x, y)
=
I k(x, z) k(z, y) dz
G
.
CHAPTER III, § 1 6
1 66
A direct calculation, using condition ( 1 6. 1 3) , shows that ( 1 6. 1 4)
Consequently, k 2 (x y) E Lt" where P(u) = e1uI  l ui  1 . In virtue of Theorem 1 5 .4, the operator A 2 , defined by the kernel k 2 (x, y) , acts from the space E� = L�, where
Theorem 1 5.5, we arrive at the conclusion that A E {L 2 � Lt, ; c.}. Let Pl (U) = l u l (e'U,H  I ) , where 0 < e < 1 . It is easily seen that P1 (u) satisfies the LJ 2 condition. Therefore, the comple mentary function
f f Pl{l k2 (x, y) l l /(l£) } dx dy o
<
00.
In virtue of Theorem 1 6. 1 , the operator .,4 2 acts from E�, = L� , into Lt" . Applying Theorem 1 6.7, we arrive at the conclusion that A E {V � Lt, , ; compo c.}. Thus, the following theorem is valid. THEOREM 1 6.9. Suppose the symmetric kernel k (x, y) satisfies the condition ( 1 6. 1 3) , where A = nj2. Then the linear integral operator A,
defined by the kernel k(x, y) . acts from V into Lt" where P(u) = = e1uI  l u i  I , and is continuous  and, as an operator acting from V into Lt" , where Pl (U) = l u i (e1ul' '  1 ) (0 < e < I ) , it is completely continuous. As is known, the Green function k(x, y) of the Laplace operator (in the case of homogeneous boundary conditions) for a four dimensional region satisfies the inequality I k (x, y) 1 � ajr2 . We find ourselves in a situation where the conditions of Theorem 1 6. 9 are satisfied. Consequently, the Green function in the case of a fourdimensional region gen erates a linear integral operator belonging to {L 2 � Lt, ; c.} and {L 2 � Lt, , ; compo c.}, where P(u) and P1 (u) are the Nfunctions defined above. We note one more fact. The Green function of the Laplace operator for a twodimensional region satisfies the condition I k (x, y) I < b + c l In I r l l . It follows from Theorem 1 5. 6 that the linear operator A, defined by this Green's function, is splittable
CHAPTER III, § 1 7
1 67
in the sense that it is representable in the form A = HH*, where H E {V + Ly., ; c.} and H* is the operator in {Edl + V ; c.} which is adj oint to H. An analogous representation A = HH*, where H E {L 2 + Lir , ; compo c.}, H* E {EdI , + V ; compo c.}, follows from Theorem 1 6.8. In the last two assertions, the Nfunctions P(u) and Pl (U) are the functions indicated in the formula of Theorem 1 6.9. § 17. Simplest nonlinear operator 1 . The CaratModory condition. We say that the realvalued function f(x, u) of the two variables x E G,  = < u < =, satisfies the CaratModory condition if it is continuous in u for almost all x E G and measurable in x for every fixed u. The following theorem holds. THEOREM 1 7. 1 . A function f(x, u) satisfies the CaratModory condition if, and only if, to a prescribed e > 0 there corresponds
a closed set Go C G such that mes (G"'Go) < e and the function f(x, u) is continuous in aU variables on the set Go X (  =, =) .
The sufficiency of the condition in the theorem is obvious. For a proof of the necessity, see KRASNOSEL'SKII and LADYZENSKII [ I J and KRASNOSEL'SKII [5J . We shall denote by f the operator defined on the set of real valued functions u(x) (x E G) by means of the equality
fu(x)
=
f[x, u(x) J .
( 1 7. 1 )
If the Caratheodory conditio ri is satisfied, then, in virtue of Theorem 1 7. 1 , the operator f transforms measurable functions into measurable functions  and it transforms sequences of functions which are convergent in measure into sequences of functions which are also convergent in measure. In the sequel, when we consider operators defined by formula ( 1 7. 1 ) , we shall always assume that the corresponding functions satisfy the Caratheodory condition . 2. Domain of definition of the operator J. We recall that JI(EM, r) denotes the totality of functions u(x) E LM for which d(u, EM) < r (see p. 82) . We shall denote the sphere of radius r and center at the point u of the Banach space E by T(u, r ; E) .
CHAPTER Ill, § 1 7
1 68
LEMMA 1 7 . 1 . Let h (x, 0)
=
(x E G) .
0
( 1 7 .2)
Suppose the operator h acts trom a sphere T(O, r ; L"M , ) into the space L"M. , the class LM. or EM • . Then the operator h acts trom II(EM" r) into the space L"M. , the class L M. or EM., respectively . It the operator h acts trom the sphere T(O, r ; EM,) into L"M. , L M. or EM. , then it acts trom aU ot EM, into L"M., L M. or EM. , respectively. PROOF . Let u(x) E II(EM " r) . In virtue of Lemma 1 0. 1 , a set Go C G can be found such that U(X) K(X ; Go) E EM, and I l uo l I M, < r, where uo(x) = U(X) K(X ; G""Go) . The function U(X) K(X ; Go) has an absolutely continuous norm . Therefore, the function u(x) can be written in the form U(X)
=
where I l u�II M, < r (i
uo(x) + Ul (X) + . . . + U k (X) ,
=
0, I , . . . , k) , and, for i ¥= /,
u,! (x)Uj(x)
=
0
(x E G) .
In virtue of ( 1 7.2) ,
hUi (X)huj(x) = 0
(x E G, i ¥= i) ,
( 1 7.3)
and
If hT(O, r ; L"M, ) C L"M . , then each of the terms in the right member of formula ( 1 7 . 4) is a function in L"M • . Therefore, hu(x) also belongs to L"M •. If hT(O, r ; L"M , ) e L M. , then, in virtue of ( 1 7.3) ,
f M[hu(x)] dx i�O f M[hui (X)] dx <
G
=
G
00 ,
i.e. hu(x) also belongs to LM• . If hT(O, r ; L"M, ) C EM. , then all the terms in the right member of ( 1 7 . 4) are functions in EM• . Therefore, hu(x) is also an element in EM • . Finally, if h is considered only on the functions in EM" then 149(X) = o. Therefore, hT(O, r ; EM,) C L"M. implies that all the
1 69
CHAPTER I I I , § 1 7
terms in the right member of ( 1 7 . 4) belong to Llr . and, conse quently, JIu(x) is also an element in Llr • . The remaining cases are considered analogously. * Let us now assume that the Nfunction M1 (U) satisfies the Ll 2 condition. It then follows from Lemma 1 7 . 1 that the operator JI is defined on the entire space Llr, = LM, = EM, provided it is defined on any sphere of this space. The question naturally arises whether or not this fact holds in the case when M 1 (U) does not satisfy the Ll 2 condition. It turns out that this is not so. In fact , if M l (U) does not satisfy the Ll 2 condition, then the operator ( 1 7 .5) JIu(x) = M2" 1 {M1[U(X)]} acts from the unit sphere in the space Llr , into L M. since
f M2 [JIu(x)] dx f M1[u(x)] dx ::::;; Ilu l lM,
G
=
G
::::;; 1
for lI u l l M, ::::;; 1 . However, in an arbitrary sphere of radius r :::;;; 1 , a function u(x) can be found such that JIu(x) E L M •. To this end, it suffices to choose a function u(x) , with norm Il u ll M, < r, which does not belong to LM, . If the Nfunction M2 (U) satisfies the Ll 2 condition, then the operator ( 1 7 .5) from the unit sphere of the space Llr, acts simul taneously into Llr. , LM. and EM. since Llr. = LM. = EM• . How ever, not all its values on a sphere of larger radius belong to Llr• . THEOREM 1 7.2. Suppose the operator f acts from a sphere
T((), r ; Llr,) into the space L M• or into the space EM• . Then the operator f acts from II(EM" r) into the space Llr. or into the space EM. , respectively . If the operator f acts from the sphere T((), r ; EM) into Llr. or into EM. , then it acts from all of EM , into Llr. or into EM. , respectively . PROOF. The function h (x, u) = f(x, u)  f(x, 0) satisfies con
JIu(x) = /I [x, u(x)] . This implies the assertion of the theorem. * 3. Theorems on continuity . A nonlinear operator B, acting from the Banach space E1 into the Banach space E 2 , is said to be continuous at the point Uo E E1 if lim I I Bu  Buo iI E . = o .
dition ( 1 7.2) . Therefore, Lemma 1 7. 1 can be applied to the operator
u eE, Ilu  uoI I E ,""' O
CHAPTER I I I , § 1 7
1 70
For nonlinear operators, in distinction from the case of linear operators, continuity at one point does not imply continuity at other points. In general, a nonlinear operator can be defined, as we saw in the example of the operator I, only on a part of the space E l. A nonlinear operator B is said to be bounded on the sphere T of the space E l if sup I I Bu I I E. < 00 . xeT
In distinction from linear operators, the concepts of boundedness and continuity are not connected with one another in the case of nonlinear operators : there exist discontinuous operators which are bounded and, conversely, there exist operators which are continuous on the entire space but which are not bounded on any spheres. In the sequel, we shall find the following assertion from KRASNO SEL'SKII [5] , pp. 3 1 , 35, 36, useful. LEMMA 1 7 .2. Suppose the operator g acts from the space L of
summable functions into the space L . Then the operator g is continuous and bounded with respect to the norm of the space L and the function g(x, u) satisfies the condition (x E G,  00 < u < 00) ,
Ig(x, u) 1 � a(x) + b l u i
where a(x) E L and b is a positive number . THEOREM 1 7.3. If the operator I acts from II(EM" r) into EM" then I is continuous at every point of II(EM" r) . PROOF. We shall first assume that If) = f) and show that the operator I is continuous at the zero f) of the space LM, . If we assume the contrary, there exists a sequence ot funct i ons u (x) E LM, (n = 1 , 2, . . . ) such that l i m I l u l l M, = 0
whereas
n�oo
n
I I Iu I I M. > oc
n
(n = 1 , 2, . . . ) ,
n
( 1 7 . 6)
( 1 7.7)
where oc is some number . It follows from Lemma 9.2 and ( 1 7 . 6) that lim n>oo
f M [�ttn(x) ] dx = O. r
G
1
( 1 7.8)
171
CHAPTER I I I , § 1 7 I t follows from ( 1 7.7) that
I M2 [ � Jun(X) ] dx � " � Jun 1 M2  1
G
> I.
( 1 7 . 9)
Thus, if the operator J is discontinuous at zero, then there exists a sequence of functions u n (x) E LM, such that relations ( 1 7.8) and ( 1 7. 9) hold. We now consider the operator g defined by means of the formula
If u(x) is a summable function, then Ml l [u(x)] E LM,. In virtue of Theorem 1 0. 1 , Ml l [u(x)] E JI(EM" 3/2) . Therefore,
By the condition of the theorem, we have that
J
( ; M 1 1 [u(x)] ) E EM
•.
This means that gu(x) E L . In virtue of Lemma 1 7.2, the operator g, acting from L into L, is continuous at zero. It therefore must follow from ( 1 7.8) that
l��
I g {Ml [ � ]}
G
U n (X )
dX =
l��
I M2 [ � Jun(X) ] x
d = o.
G
We have thus arrived at a contradiction with ( 1 7.9) . This means that the operator J is continuous at the zero of the space LAt, if JO = O. We now proceed t o the consideration o f the general case : we shall show, without any additional assumptions, that the operator J is continuous at an arbitrary point uo(x) of the set JI(EM" r) . Let d = d (uo, EM , ) . Clearly, d < r. The continuity of the operator J at the point uo(x) is equivalent to the continuity of the operator /Iu(x) = J[uo(x) + u(x)]  Juo(x) at the zero of the space LM, . The operator /I acts from the sphere T(O, r  d ; LAt, ) into EM• . d ) into EM• . In virtue of Theorem 1 7.2, it acts from JI(EM" r

CHAPTER I I I § 1 7
1 72
Since hO 0 , we have, by what has already been proved, that the operator h is continuous at the zero of LM,. * We note that Theorem 1 7.3 yields a rough criterion for the continuity of the operator f. Even the continuity, for example, of the operator of the identity transformation (i.e. of the operator fu(x) u(x) ) in the spaces LM does not follow from it if M (u) does not satisfy the L1 2 condition. If, under the conditions of Theorem 1 7 .3, the Nfunction M 2 (U) satisfies the L1 2 condition, then this theorem signifies that the operator f is continuous if it acts into the space LM. = LM• = EM•. The question arises whether the operator f is always continuous if it acts from II(EM" r) into the space LM. when M 2 (U) does not satisfy the L1 2 condition. It turns out that the operator f is not always continuous. Here is an example. Let =
=
( 1 7. 1 0) where M l eU) satisfies but M 2 (U) does not satisfy the L1 2 condition. Clearly, the operator f acts from II(EM" I ) = L M , into LM•. Suppose v(x) belongs to LM• but that v ex) E EM• . Then, in virtue of Lemma 1 0. 1 , we have that lim I l v  v nl l M. = d(v, EM.) = d > 0,
where
vn(x)
=
{
vex) if I v (x) I o if I v (x) I
�
>
( 1 7. 1 1 )
n, n.
1 1 7 . 1 2)
The functions u n (x) = M l l {M 2 [V (X )  Vll (x)]} belong to LM" with lim ft,+OO
f Ml[Un (x)] dx
o
=
lim n�oo
f M [V
0
2 (X )  vn(x)] dx
=
O.
Since, in L M" mean convergence implies convergence in norm, we have that lim I l u nllM, O. From this equality and ( 1 7 . 1 1 ) , =
n.. oo
it follows that the operator f is discontinuous at the zero of the space LM, since fU n (x) = v ex)  v n (x) .
4. Boundedness 0/ the operator f. The boundedness of the set of values of the operator f on a sphere of the space LM, can be
1 73
CHAPTER III, § 1 7
proved under fewer restrictions that those under which the continui ty of the operator 1 was proved in Theorem 1 7.3. THEOREM 1 7.4. Suppose the operator 1 acts trom the sphere
T(O, r ; LM, ) into the class L M• . Then 1 is bounded on any sphere T(O, r l ; LM,) (rl < r) : sup 11 1u 1 1 M. < 00 .
I luIlM,
PROOF. In virtue of Lemma 1 7. 1 , the operator [fu(x)  10]/2 acts from II(EM" r) into the class L M• . Therefore the operator
gv(x) = M2{t1(r l M1 1 [v (X)] )  110}
acts from L into L . For functions u (x) E T(O, r l ; LM,) we have that
f M [ ;U(X);] dx � l .
G
1
In virtue of Lemma 1 7.2,
i.e. sup Ilul l M, < T1
f M2[l1u(x)  110] dx < 00
.
G
Consequently, sup IluIIM,
1 1 1u 1 1 M. � 1 1 10 11 M. + sup
IluIlM,
1 1 1u  10 1 1 M. �
The question arises whether the operator 1 is bounded on spheres with larger radii. It turns out that this is not necessarily so. As an example, we consider again the operator ( 1 7 . 1 0) in which connection we shall assume that M1 (u) does not satisfy but M 2 (U) satisfies the .1 2condition. The operator ( 1 7. 1 0) acts from the class LM, into L M• = EM, and, in virtue of Theorem 1 7.3, it is continuous on EM, . We shall show that the values of this operator 1 are not bounded on every sphere T(O, 1 + e ; EM,) ' where e is an arbitrary positive number.
1 74
CHAPTER I I I , § 1 7
Since M 2 (U) satisfies the L1 2 condition, boundedness in norm is equivalent to boundedness in the mean. It therefore suffices to show that there exists a sequence of functions u n (x) E T((), 1 + e ; EM,) such that
( 1 7 . 1 3) The functions u n (x) are defined by equalities similar to ( 1 7. 1 2) where u (x) is a function not belonging to LM, such that I l u l I M, < 1 + e. The existence of the function u(x) follows from Theorem 1 0. 1 . If relation ( 1 7 . 1 3) were not satisfied for the functions u n (x) , then, in virtue of Fatou's theorem (see p. 7 1 ) , the function u(x) would belong to L M,. 5. General form of the operator J. THEOREM 1 7 .5. The operator f acts from the class LM, into the
class L M• if, and only if, M2[f(x, u)]
:s;;
a(x)
+
bM l (U)
(x E G,  00 < u < 00) , ( 1 7. 1 4)
where a(x) E L, b � O . PROOF. If f acts from L M , into L M" then the operator gv(x) = = M2{j(M l 1 [v (x)J)} acts from L into L and, in virtue of Lemma 1 7.2, we have that . M2{t[x, Ml 1 (v)]} :s;; a(x) + b l v l . Setting v = M(u) in the last inequality, we obtain ( 1 7. 1 4) . The sufficiency
of condition ( 1 7 . 1 4) is obvious. * We now assume that condition ( 1 7. 1 4) is satisfied. Then, in virtue of ( 1 . 20) , we have that
I f(x, u) 1
:s;;
c(x)
+
b 1M2" 1 [M l (U)]
(x E G,  00 < u < 00 ) , ( 1 7 . 1 5)
where c(x) = M2" l [a(x)] E LM, . Thus, condition ( 1 7 . 1 5) is necessary in order that the operator f act from LM, into LM,. In the case when the Nfunction M 2 (U) satisfies the L1 2 condition, condition ( 1 7. 1 5) is at the same time a sufficient condition inasmuch as in this case ( 1 7 . 1 5) implies ( 1 7 . 1 4) . 6 . Sufficient conditions for the continuity and boundedness of the operator J. The assertions proved above imply the next theorem.
CHAPTER I I I , § 1 7
1 75
THEOREM 1 7 .6. Suppose the function f(x, u) satisfies the zn
equality
[
I f(x, u) 1 � c(x) + b 1M;; 1 M I (X E G,  =
<
u
( : )]
( 1 7 . 1 6)
< =) ,
where c(x) E L M2 , bi � 0, in which connection the Nfunction M2(U) satisfies the iJ 2condition. Then the operator f acts from II(EM" r) into the space L'ivl , = EM2 , is continuous at all points of II(EM " r) , and is bounded on every sphere T((), r l ; LM.) (ri < r) . If the Nfunction M2(U) does not satisfy the iJ 2 condition, then one must require, in order to have continuity of the operator f, that the set of its values belong to E M2 . Therefore , we must place more severe restrictions on the function f(x, u) . THEOREM 1 7 .7. Suppose the function f(x, u) satisfies the in equality ( 1 7. 1 7) I f ( x, u) I � b (x) + aQ I M;; 1 M I
{
(x E G,  = < u
[ ( : )]}
<
=) ,
where b(x) E EM" Q (u) is an Nfunction, and a, r are positive numbers. Then the operator f acts from II(EM1, r) into EM2 , is continuous at all points of II(EM1 , r) , and is bounded on every sphere T((), r I ; LM.) (rr < r) . 7. The operator f and EN weak convergence . We proved the continuity of the operator f under the assumption that it acts from the sphere T((), r ; LM.) into EM,. The boundedness of the operator f was established under weaker hypotheses. We shall show that the continuity of the operator f in some weakened sense can be proved under these weaker assumptions. THEOREM 1 7 .8. Suppose the operator f acts from the sphere T((), r ; LMJ into the class LM2 . Then this operator transforms an arbitrary sequence u n (x) E T((), r ; L'ivI.) , which converges in norm in the space LM I to the interior point uo(x) of this sphere, into the
sequence of functions fUn (x) which is EN 2 weakly convergent in the space LM2 to the function fuo(x) . PROOF. Since the functions u n (x) converge in norm to the function uo(x) , they also converge to this function in measure .
CHAPTER I I I , § 1 8
1 76
Therefore, the sequence of functions JU n (x) also converges in norm to the function Juo(x) . Since the function u o (x) is an interior point of the sphere T((), r ; LM,) , we may assume, without loss of gener ality, that all u n ( x) E T((), r l ; LM,) , where r l < r. It follows from Theorem 1 7.4 that the norms of the functions JU n (x) are uniformly bounded. In virtue of Theorem 1 4.6, this sequence is ENweakly convergent to the function Juo(x) . * § 18. Differentiability. Gradient of the norm
1 . Differentiable functionals. We shall consider realvalued functionals defined on Orlicz spaces LM (or on parts of these spaces) . We say that the functional F(u) is differentiable at the point uo (x) , which is an interior point of the domain of definition of this functional, if the increment of the functional can be written in the form F (uo + h)  F(uo) = l u o (h) + w (uo, h) , ( 1 8. 1 ) where l u o(h) is a linear functional on LM and the remainder w (uo, h) satisfies the condition lim IIhll� O
w (uo, h) ''  0
I l h i lM
.
( 1 8.2)
The functional F(u) is differentiable on the set m if it is differ entiable at every point of m. The operator r which assigns the functional l u o to the element uo (x) E m is called the gradient of the functional F(u) . The gradient acts from the space LM into the conj ugate space. The space LN, where N (v) is the complementary function to M(u) , can be con sidered as a part of the conj ugate space. In the case when the gradient acts into LN, equality ( 1 8. 1 ) can be rewritten in the form
F(uo + h)  F(uo)
=
(ruo, h) + w (uo, h) ,
( 1 8.3)
where, as usual, the symbol (w, h) denotes the inner product :
(w, h)
=
f w(x)h(x) dx
G
(w(x) E LN, h(x) E LM) .
We shall retain the terminology gradient for the operator which assigns the function ruo (x) to the function uo (x) .
CHAPTER I I I , § 1 8
1 77
2. Measurability of the function O (x} . Suppose the function F(x, u} (x E G, 00 < u < oo ) satisfies the Caratheodory con ditions and has a continuous derivative with respect to u : F�(x, u) . Clearly, the function F�(x, u } also satisfies the Caratheodory 
conditions. Suppose u(x} is a measurable function on G. On the basis of th e meanvalue theorem, there exists a function O (x} such that
F (x, u(x} }  F(x, O}
=
u(x}F�(x, O (x}u(x} } .
( 1 8.4)
Generally speaking, the function O(x} is not uniquely defined. We shall use the following assertion in the sequel. LEMMA 1 8. 1 . There exists a measurable function O(x) , satisfying
the inequalities
o
� O (x} � 1 ,
( 1 8.5)
for which equality ( 1 7. 2 1 ) is satisfied . PROOF. We define the function O (x} for all x E G as the minimum of those 0 E [0, I J for which the equality =
F(x, u(x}}  F(x, O}
u(x}F� (x, Ou(x} }
is satisfied. This minimum exists inasmuch as F�(x, u} is continuous with respect to u. Let [; > 0 be given. In virtue of Luzin's Cproperty and Theorem 1 7. 1 , a set GI C G can be found such that mes (G",GI) < [; and the functions F�(x, u} , F(x, O} , F(x, u} and u(x} are continuous in all variables simultaneously for x E GI, 00 < u < 00 Since [; is arbitrary, it will suffice to prove the measurability of O (x} on GI. Let Xo E G l , Xn E GI (n = 1 , 2, . . . ) and 
lim Xn
=
lim O(x n}
x,
�oo
n+oo
=
.
00.
Passing to the limit in the equality
F(xn, u(xn}}  F(x n, O}
=
U(Xn) F� (xn, O(x n }u(xn} } ,
we obtain that
F(xo, u(xo} }  F(xo, O}
=
u(xo}F� (xo, Oou(xo} } .
The last equality implies that O (xo} � 00. This means that the function O (x} is lower semicontinuous on GI and, consequently, that it is measurable. * Co...es
'....clUJ"s
CHAPTER I I I , § 1 8
1 78
3 . Functional for the operator J. F 1 (u)
=
We now consider the functional
I dx I f(x, s) ds . u (x)
G
0
( 1 8. 6)
We shall be interested in the case when the functional Fl (U) is defined on a part of the space LM which is a part of the space L 2 . In this case, the Nfunction M (u) and the function N (v) comple mentary to it are connected by the relations N (u) < u 2 < M(u) . Since N (u) does not increase more rapidly than u 2 , it will satisfy the ,1 2 condition in the fundamental cases. It is under this as sumption that the following theorem is proved. THEOREM 1 8. 1 . Let the operator J act from IJ(EM, r) into the space Liv = LN . Then the functional ( 1 8. 6) is defined and differ
entiable on IJ(E M , r) , in which connection its gradient is the operator J. PROOF. Let u(x) E IJ(EM" r) . In virtue of Lemma 1 8. 1 , a measurable function O (x) can be found, satisfying condition ( 1 8.S) , such that
Fl (U) I dx I f(x, s) ds I f[x, O (x)u (x)]u(x) dx. u(x)
=
=
G
G
O
Clearly, O(x) u(x) E IJ(E M, r) . Therefore, f[x, O (x)u(x)] E L N . Conse quently, J l (U) I � I I JOu l l N I l u l i M < 00, from which it follows that the functional l (U) is defined on IJ(EM, r) . In virtue of the same Lemma 1 8. 1 , to each function h (x) E LM such that I l h i l M < r  d(u, EM) there corresponds a measurable function O h (X) , satisfying condition ( 1 8. S ) , such that
F
F
J Fl (U + h)  Fl (U)  (Ju, h) I =
=
I I [I(x, u(x) + Oh(x)h(x) ) G
I
 f(x, u(x) )]h(x) dx �
,;;;; I I J(u + O hh)  Ju l i N I l h i I M . It follows from this inequality and the continuity of J (see Theorem 1 7 .3) that J is the gradient of the functional Fl (U) . * 4.
The linear operator J. The linear operator J: Ju(x)
=
a(x)u(x) ,
( 1 8.7)
CHAPTER I I I , § I B
179
where a(x) i s a fixed function, i s one o f the simplest forms o f the operators J. This operator was, in essence, already studied in §. 1 3 (subsections 4 and 5) . If the function a (x) is bounded, then, obviously, the operator f transforms every Orlicz space into itself and is a continuous operator. The next assertion follows from Theorem 1 3.7, Theorem 1 3.B and inequalities ( 1 3. 1 9) and ( 1 3.27) . THEOREM 1 B.2. Let a(x) E L�. A sufficient condition for the
operator ( l B.7) to act from LA!, into L1.t. and be continuous is that there exist mutually complementary Nfunctions R (u) and Q (u) such that or, if the Nfunction M2(u) satisfies the iJ 'condition, that R (u) < M I [M2" l (ocU)] , Q (u) < q>[M2" l (ocU)]
( l B.9)
for large values of the argument. Under these conditions, we have that I l au l lM. � k Il a l l � I l u I I M" where the constant k does not depend on the function u(x) . Under the conditions of this theorem, oc is a positive number. 5. The Frechet derivative. Suppose the nonlinear operator A acts from the Banach space E into the Banach space E1. We say that the linear operator B is the Frechet derivative of the operator A at the point Uo E E if A (uo + h)  Auo = Bh + w(uo, h) , where lim I lw (uo, h) I IEjl l h l lE = O. Ilhll.sO
In this connection, the linear operator B also acts from the space E into the space E1. The expression Bh is called the Frechet differential. Operators which have a Frechet derivative are said to be differ entiable. If an operator is differentiable on some set then it obviously is continuous on this set . THEOREM 1 B.3. Suppose the function f(x, u) together with its
derivative f�(x, u) , which is continuous with respect to u, satisfy the CaratModory conditions. Suppose, further, that the operator f acts from some sphere T(uo, r ; L1.t, ) into the space L1.t. and that the operator JIu(x) = f� (x, u(x) )
( l B. 1 O)
CHAPTER I I I , § 1 8
1 80
acts from the sphere T(uo, r ; LM.) into the space L 'fr, and is continuous. Finally, let the functions M1 ( u) , M2(U) and $(u) satisfy one of the conditions ( 1 8.8) or ( 1 8. 9) . Then the operator f is Frichetdifferentiable at every interior point of the sphere T ( uo, r ; LM.) , in which connection the Frechet differential Bh at the point u(x) E T is defined by means of the equality Bh(x) = iI u(x)h(x) , (h(x) E LMJ PROOF. In virtue of the meanvalue theorem, we have that
f[x, u(x) + h(x)]  f [x, u(x)]  f� [x, u(x)]h(x) = = {t�[x, u(x) + O h(x)h(x)]  fJx, u ( x)] } h(x) ,
( 1 8. 1 1 )
where 0 � O h (X) � 1 , in which connection the function O h(X) can, in virtue of Lemma 1 8. 1 , be considered measurable. Let h(x) E LM , and suppose I l h iI M. is sufficiently small. Then the functions u(x) + h(x) and u(x) + O h (x)h(x) belong to the sphere T(uo, r ; LMJ, In virtue of Theorem 1 8.2, f�[x, u(x)]h(x) and the entire right member of equality ( 1 8. 1 1 ) belong to the space LM •. It follows from this same theorem that
From this, in virtue of the continuity of the operator iI , it follows that
I l f(u + h)  fu  iItt · h I I M, I l h iI M•
lim IlhllM.> o
=
O.
*
As an example, we consider the operator fu(x) = eU(x ) . It follows from Theorem 1 7 . 6 that the operator f acts from the sphere T(O, t ; LM.) , where M1(u) = e1uI  I tt l 1 , into LM, = V. In virtue of Theorems 1 7. 2 and 1 7 .3, it acts from [[ ( EM" t ) into L 2 and is continuous. We shall show that this operator is differentiable and that its Frechet differential Bh(x) = iI u(x)h(x) at the point u(x) E [[ (EM. , t ) has the form Bh(x) = eU(X )h(x) . We apply Theorem 1 8.3, taking into consideration that $(u) = l u l 2 +/l, where 0 < P < { l jd (u , EM.)} 2. As T, we consider the sphere with center at the point u(x) and radius 

r
=
{ l j(2 + P)}  d(u, EMJ
The operator f acts
fro m the
sphere
T into L 2 smce
CHAPTER I I I , § 1 8
181
T C JI(EM" i) . The operator /Iu (x) eU(X ) acts from the sphere T((), 1 / (2 + (3) ; Li..c, ) into L� L 2 + fJ. It follows from Theorems 1 7 .2 and 1 7 .3 that it acts from JI(EM" 1 / (2 + (3) ) and, in pa rticular, from the sphere T into L� = L 2 + fJ and is continuous. =
=
In order to complete the proof, it suffices to verify that condition ( 1 8.8) is satisfied if we set R (u) M 1 (u) , Q (u) N1 (u) . As a second example, we consider the operators fu (x) sin u (x) , /Iu(x) cos u(x) . These operators act from an arbitrary Orlicz space Li..c , into every set E M, and, consequently, they belong to {LM, + Li..c , ; c.} . Let Li..c, C Li..c . . Then the operator Bh (x) = /Iu (x) . h (x) is the Frechet differential of the operator J. In the examples introduced above, the operator f was differ entiable at every point of some sphere T. This explains the simplicity of the proof of differentiability. It is quite easy to give examples of operators which are differentiable at one point of an Orlicz space but which are not differentiable on a set for which the given point is a limit point . For example, let =
=
=
=
fu (x)
=
sin (eU ' (X )  1 ) .
( 1 8. 1 2)
We shall consider f to be an operator in {L4 + L2} . At those points in which this operator is differentiable, its Frechet differential obviously has the form Bh(x) = 2u(x) eU ' (X ) cos (eu ' (X )  1 ) h (x) .
It is clear that the right member belongs to the space L2 only for certain functions u. (x) : for h(x)  1 the right member does not belong to L2 for a set of functions u (x) which is everywhere dense in L4. It will be shown below that the operator ( 1 8. 1 2) is differentiable at the zero of the space L 4. 6. Special condition for differentiability . In this subsection, special condition for the differentiability of the operator f at one point of an Orlicz space will be given. We shall restrict our selves to a condition for the differentiability at the zero () of the space since the differentiability of the operator gu (x) g (x, u (x) ) at the point uo (x) is equivalent to differentiability at zero of the operator fu (x) g[x, uo (x) + u (x) ] . a
=
=
1 82
CHAPTER I I I , § 1 8
Below, we shall consider functions t(x, u) which satisfy con dition A) : A) The inequality
I t(x, u)  t(x, 0)  t� (x, O) u l � R ( l u l ) (X E G,  00 < u < 00) ,
( 1 8. 1 3)
where R (u) is a continuous nondecreasing function such that R' (O) 0, holds, in which connection there exists an Nfunc tion P(u) which satisfies the J 'condition
R (O)
=
=
P(uv) � CP(u)P(v)
( 1 8. 1 4)
for all values of the argument , such that for sufficiently large values U l � UO, U 2 � Uo we have that U l < U2 implies ( 1 8. 1 5) where fl, v are positive constants. As the functions P(u) , it is most convenient to consider the functions P(u) Ite lT (r > 1 ) . Condition ( 1 8. 1 5) i s satisfied for such functions P(u) if R (u) is the principal part of an Nfunction satisfying the J 2 condition. In fact , in this case, for large values of the argument , we have that urR(u) < R (vu) , where v is some constant . Therefore , for large values of U l and U 2 , it follows from Ul < U 2 that =
In the theorem to be proved below, the fundamental condition of the differentiability of the operator j, considered as an operator acting from Llt, into LM" will be based on inequality ( 1 8. 1 3) . A necessary condition for this inequality to guarantee the differ entiability of the operator j is , obviously, that lim IlhI I JI,+ O
I IR ( l h(x) I ) I IM, I l h l lM,
=
O.
Considering the characteristic functions of sets whose measures
1 83
CHAPTER III, § 1 8
tend to zero as the functions h(x) , we arrive at the condition 1 1m ·
HOO
N 2 l ( V )  0, N i 1 (v)
( 1 8. 1 6)
where N l (V) and N 2 (V) are the complementary functions to M l (U) and M 2 (U) , since for the characteristic function K (X ) of the set G1 (mes G 1 = l /v) , we have that N:; l (V) I I R(K) I I M. = Rl I ) . Ni l (V) I I K I I M,
Let 13 > 0 be given. It follows from ( 1 8. 1 6) that N:; l (V) < I3Ni l (V) for large values of the argument . From this inequality, it follows, in turn, that the Nfunction N 2 (v) increases essentially more rapidly than the Nfunction N1 (v) . In virtue of Lemma 1 3. 1 , the Nfunction M 1 ( u ) increases essentially more rapidly than M 2 (u) , i.e. for arbitrary 13 > 0, the inequality M2 (U) < M1 (l3u) is satisfied for large values of the argument . A sufficient condition that this inequality be satisfied is that the functions M l (U) and M 2 (U) be connected by the relation ( 1 8. 1 7) where Q (u) is an Nfunction. In the sequel, we shall assume that condition ( 1 8. 1 7) is satisfied. In this connection, in virtue of Theorem 1 3.3, a constant q > 0 can be found such that ( 1 � . 1 8) We consider, further, the condition B) : B) The operator hh(x) f�(x, O)h(x) acts from LM , into LM• and is continuous. We note that , in virtue of ( 1 8. 1 7) , condition B) is always satisfied if the function a(x) = f�(x, 0) is bounded. If the function a (x) is not bounded, then a sufficient condition for condition B) to be satisfied is that the fUIlction a(x) belong to the space L�, where (/J(u) satisfies one of the conditions in Theorem 1 8.2. THEOREM 1 8.4. Suppose conditions A) , ( 1 8. 1 7) and B) are =
satisfied. Let
R (u ) :::::;; b + aM:; l [Ml (ku)]
(0 :::::;;
u
<
00) ,
( 1 8. 1 9)
CHAPTER I I I , § 1 8
1 84
where a , b , k > O . Let t(x , 0) E LM, . Then the operator f acts trom some neighborhood ot the zero () ot the space L'M, into the space LM. and has at the point () the Frechet ditferential Bh(x) hh(x) t� (x, O)h(x) . =
=
PROOF . In virtue of ( 1 8. 1 3) and ( 1 8. 1 9) , we have that
I t(x, u) 1 � I t(x , 0) 1 + l a(x)u l + b + aM2 1 [Ml (ku)] (  00 < U < 00) .
By hypothesis, t(x , 0) E LM •. In virtue of B) , a(x)u(x) E LM, for every u(x) E LM,. If I lullM, � 1 1k , then R ( l u(x) 1 ) E LM" in which connection we have that
I I R ( l u(x) I ) I IM.
=
2a
I �a R ( l u(x) I ) 1 M, { + f M2 [ �a R ( I U(X) I ) ] dX} �
� 2a 1
�
G
� 3a + aM 2
( : ) mes G
G
=
(3.
( 1 8.20)
Consequently, the operator f acts from the sphere T((), 1 1 k ; LM,) into the space LM, . To prove the differentiability of the operator f at the point () we must show that
I I R( l u(x) I ) I IM, I l u l lM ,
=
o.
( 1 8.2 1 )
Let e > 0 be given . Since R ' (0) = 0 , a C l > 0 can be found such that ( 1 8.22) R(lul) � e lui for l u i < Cl. To each function u(x) E Lid we assign the function u (x) defined by means of the equality
u(x) = _
{
U(X) if l u(x) I � Cl , o if l u(x) I > Cl.
CHAPTER I I I , § 1 8
1 85
In virtue of ( 1 8.22) and ( 1 8. 1 8) , we have that
I IR ( l u (x) I ) I I M. � e ll u l lM. � e l i u l lM. � eq l l u I I M"
( 1 8.23)
We can assume, without loss of generality, that the constant Uo appearing in condition A) is greater than C l . It follows from con dition A) that, for u ;;:,: Cl and y � Cl /UOV,
R (u) � R
( :: u) � #
and, in virtue of ( 1 8. 1 4) , that
R (u) � C#R Let lI u l lM1
R( l u(x)
< _
( ; ) P ( Uco; ) .
cl/ (uovk) . We set u(x) I ) � C#R
y
=
( 1 8. 24)
k il u l lM1 in ( 1 8.24) . Then
kuo I I u l l M1 I ( l u(x)k i lullMu(x) ) ), P( Cl 1
from which, in virtue of ( 1 8.20) , it follows that
I I R (u  u) I I M. � C#fJP
( kuo �� "M ) , l
and, since lim P(u) u = 0, we have that
/
M I I R"" u) I I� " (u "'  = . = O . I I u l l M1
From the last relation and ( 1 8.23) it follows that
I I R(u ) 1 1 M. II u l lM1 +
+
11' m l IuIIM,+O
) M.  'I I R (u , , u 11=I l u lI M ,

""" ./
eq.
Since e is arbitrary, equality ( 1 8. 2 1 ) is valid. * As an example, we consider operator ( 1 8. 1 2) as an operator in
{L 4 _ L 2} .
1 86
CHAPTER I I I , § 1 8
The function = sin  I ) satisfies inequality ( 1 8. 1 3) with the function R (u) = Therefore condition A) is satisfied in which we can set = and Since = Finally, condition ( 1 8. 1 7) is satisfied with the function = condition B) is satisfied inasmuch as 0) == ° in the case under consideration. The validity of condition ( 1 8. 1 9) is obvious. It thus follows from Theorem 1 8.4 that the operator ( 1 8. 1 2) in � is differentiable at the zero of the space
f(x, u) P(u)
(eU2 2 eu 2. u . M1(u) u4 M22 (u) u2 , Q(u) u . f�(x, =
{L4 L 2} fJ L4. 7. Auxiliary lemma. We shall need an operator defined by the function P(u), which is the derivative of an Nfunction M(u). LEMMA 1 8.2. Let M(u) and N(v) be mutually complementary N functions the second of which satisfies the J 2condition. Suppose that the derivative P(u) of the function M(u) is continuous. Then the operator p, defined by means of the equality pu(x) P( l u (x) I ) , acts from JI(EM, 1 ) into L'N LN and is continuous. PROOF. In virtue of Lemma 9. 1 , the operator acts from T(fJ, 1 ; LM) into LN. It then follows from Theorem 1 7 . 2 that the operator p acts from JI(E M , 1 ) (and, by the same token, from EM) into LN. The continuity of the operator follows from Theorem 1 7.3. 8. The Gateaux gradient. We shall say that the functional F(u), defined on the Banach space E, is differentiable in the Gateaux sense, or Gateauxdifferentiable, at the point u E E if, for arbitrary h E E, the function F(u + th) is differentiable with respect to t and the derivative of this function, for t 0, has the form d dt F(u + th) l t�o (v, h), where the element v from the conjugate space to the space E does not depend on h. The element v will be called the Gateaux gradient of the functional F(u) at the point u. The operator r, defined by the formula ru v on all elements at which F(u) is Gateauxdiffer =
=
p
p
*
=
=
E
=
entiable, will also be called the Gateaux gradient . Clearly, the Gateaux gradient acts from E into the conj ugate space E. The following assertion holds (see, e.g. , VAIN BERG [ I J ) . LEMMA 1 8.3.
Suppose the functional F(u) is Gateauxdifferentiable on some sphere T of the space E and that its Gateaux gradient is a continuous operator. Then the functional F(u) is differentiable in the
1 87
CHAPTER I I I , § 1 8
usual sense and its gradient (see subsection I ) coincides with the Gateaux gradient. PROOF. Let u E T, u + h E T. By definition, dtd F(u + th) (r(u + th), h) (0 :( t I ) . =
:(
Integrating this equality, we obtain that
f (r(u + th), h)dt. 1
F(u + h)  F(u)
=
o
Therefore,
i F(u + h)  F(u)  (ru, h) I
I f (r(u + th)  ru, h) dt I :( :( I lh i l E f I r(u + th)  ru l X' dt. 1
=
o
1
o
r that i F(u + h) F (u) (ru , h) I lim ______________  0 . I l h i lE Gradient of the Luxemburg norm. Suppose M(u) and N(v)
It follows from the continuity of the operator
_
Ilhlls+ O
*
9. are mutually complementary Nfunctions the second of which satisfies the L hcondition. It is assumed everywhere below that the function = is continuous. It will be convenient for us to consider the function also for negative values of the argument. Clearly, = Let E EM. Then the function
P(u) M' (u) P(u) P(  u)  P(u). u(x), h(x) rp (t, k) f M [ u(x) : th(x) ] dx =
t
G
( 1 8.25)
is defined for all and k oF o. It is easily seen that
orp (t,ot k) = kI f P [1f (X) +k th (x) ] h(x)dx G
___
( 1 8.26)
CHAPTER III, § 1 8
1 88 and that
ckp(t, k)
k2
ok
J p [ u(x) : th(x) ] [u(x) + th(x)Jdx.
( 1 8.27)
G
(The legitimacy of differentiating under the integral sign is proved here and in the sequel by means of standard lines of reasoning. ) I n virtue o f Lemma 1 8.2, each o f the derivatives found, above, is continuous in all arguments simultaneously. Let be a function for which
u(x) E LM
( 1 8.28)
G
for some k > O . We recall (see p. 78) that in this case the number k coin cides with the Luxemburg norm : k = Clearly, the Luxemburg norm can be defined with the aid of equality ( 1 8.28) for all functions =F 0) . This E follows from the fact that the integral appearing in the right member of equality ( 1 8.28) is finite for all k =F 0, depends continuously on k, and is such that
I lu l (M) . u (x) EM ( 1 Iu l (M)
lim k+ O
J M [ U(X)k ] dx 
=
=,
G
lim k+ oo
J M [ U(kXl J dx
G
=
O.
The Luxemburg norm is a differentiable functional in EM . The gradient r of the Luxemburg norm is defined by means of the formula P ( I luu(x)l (M) ) ( 1 8.29) ru(x) ,(U(X))U(X)J p I l u l (M) I l u l (M) dX THEOREM 1 8. 5 .
=
G
PROOF. We first find the Gateaux gradient of the Luxemburg norm. To this end, we consider the equality
J M [ u(x) k th(x) ] dx  1 (u(x), h(x) E EM)' ( 1 8. 30) +
G

CHAPTER III, § 1 8
1 89
t.
This equality defines k as an implicit function of Since the partial derivatives ( 1 8.26) and ( 1 8.27) in the left member of equality ( 1 8.30) are continuous and
orp(O, k) ok
=
_
I
1
_
k2
( l u(x) I ) l u(x) I dx < 0
p
k
G
( l I u l l (M)
=1= 0) ,
we have, on the basis of the implicit function theorem (see, e.g. FIHTENGOL'C [ I J ) , that k (O ) ;]t
d
h), ( u(x) ) P i l u ll (M) u(x) u( x ) ( I P I lu l l (M) ) I lu l l (M) dx
where
V = �
=
(v,
������
G
We have shown that formula ( 1 8.29) defines the Gateaux gradient . Since it follows from Lemma 1 8. 2 that this Gateaux gradient is a continuous operator, it is the ordinary gradient  in virtue of Lemma 1 8.3. *
Gradient ot the Orlicz norm.
1 0. The Nfunctions M(u) and N (v) considered in this subsection satisfy the same restrictions as in the preceding subsection. In virtue of Lemma 1 8.2, for every function E EM, the function
u(x)
J (k)
=
=
I
G
N[P (k l u (x) I )J dx
=
is defined for all values of k and is continuous. Since J(O) = 0, .1(00) 00 , a k* can be found such that J (k*) 1 . This signifies, in virtue of Lemma 1 0.4, that the Orlicz norm can be defined with the aid of the equality
I l u l iM where
I
G
=
IG P (k * l u (x) I ) l u(x) I dx,
x dx
N[P (k* l u( ) I )J
=
1.
( 1 8.3 1 )
( 1 8.32)
1 90
CHAPTER I I I , § 1 8
I t will b e convenient for us t o use, in place of formula ( 1 8.32) , the equivalent (see ( 1 0.7) ) equality
k* J l u (x) 1 P(k* l u (x) I ) dx  J M[k*u(x)] dx I . ( 1 8.33) As was already remarked, the constant k* is not , generally speaking, uniquely defined. In this subsection, we assume that P(u ) does not have intervals of constancy. Then, obviously, k* is uniquely defined. It is easily verified that k* is a continuous functional on EM. Let u(x), h(x ) E EM. We denote by k(t) the solution of equation ( 1 8.33) corresponding to the function Ut(u) u(x) + th(x). LEMMA 1 8.4. Suppose the function k(t) has the derivative k'(t). Then the Orlicz norm is a differentiable functional on EM . The gradient r of the Orlicz norm is defined by the formula ( 1 8.34) ru(x) P(k*u(x)) (u(x) E EM) , where k* satisfies equality ( 1 8.33) . PROOF. Since, in virtue of Lemma 1 8.2 and the continuity of the functional k*, the operator r defined by formula ( 1 8.34) is a continuous operator acting from EM into LN , it suffices to prove that r is the Gateaux gradient of the Orlicz norm. In virtue of ( 1 8.3 1 ) and ( 1 8.32) , the Orlicz norm for the function u(x) E EM can be defined by the equality I l u l iM � ( 1 + J M[k*U(X )]dX) , where k* satisfies equality ( 1 8.33) . Let h(x) E EM. Consider the function F(u + th) _k(t)1_ ( 1 + J M[k(t)Ut(X)]dX) , G where Ut(x) u(x) + th(x). In virtue of the differentiability of the function k(t) (see remark on p. 1 88) , we have that � dt F(u + th) _ k21(t)_ {k(t) GJ P[k(t)ut(x) J [k'(t)ut(x)  k (t)h(x)] dx =
G
G
=
=
=
G
=
=
=
191 18  k'(t) ( 1 + I M[k(t)Ut (X)] dX)} 1 = k 2 (t) { k 2 (t) I P[k(t)U t (x )]h(x)dx + k'(t) [k(t) I P[k(t)Ut(X)]Ut(X) dx  1  I M[k(t)Ut(X)] dxJ } , from which it follows, in virtue of (18. 3 3), that � F(u + th) I P[k(t)U t (x)]h(x)dx. dt Since k(O) k* and ut(x) l t =o u(x), we have that d F(u + th) l t =o (v, h), dt where v P(k*u(x)). To apply this lemma, one must know under what conditions the function k (t) is differentiable. LEMMA 18. 5 . Suppose the Nfunction M(u) has a continuous second derivative P'(u) which is positive for u i= 0 and which satisfies the inequality l uP'(u) 1 � a + bP(c l u l ) (  00 < u < 00) (18. 3 5) Then the solution k(t) of equation (18. 33) corresponding to the function ltt(X) u(x) + th(x) is a dilferentiable function. PROOF. We note first of all that , in virtue of (18. 3 5), the operator u(x)P'(u(x)), as also the operator P(u(x)), acts from EM (even from II(EM. 1)) into LN and it is a continuous operator. Therefore, for an arbitrary pair of functions u(x), h(x) E EM, with I l u l i M i= 0, the integrals CHAPTER I I I , §
=
G
G
+
G
G
=
G
=
=
=
*
=
.
=
I ul(x)p'[kUt(X)] dx and I Ut(x)h(x)P'[kUt(X) ] dx, where Ut(x) u(x) + th(x), are finite for arbitrary t and k > 0 G
G
=
and they are continuous functions of these variables. From this
1
92
CHAPTER III, § 1 8
and Lemma 1 8.2, it follows that the function
x (t, k) k f Ut(x)P[kut(x)]dx f M[kut(x)]dx =

G
G
has the continuous partial derivatives
_ox_(att_,k_) k2 f ut(x)h(x)P'[kut(x)]dx ox ( t, k) f u;(x)P'[kut(x)] dx ok =
and
G
=
k
G
(in this connection, see remark on p. 1 88) . Since
ox (O,'c'k) k f u2 (x)P'[ku(x)]dx > 0, ''ok =
G
the equation
k f Ut(x)P[kut(x)]dx f M[kut(x)]dx I defines k as an implicit function of t, in which connection the function k(t) has a continuous derivative k'(t). Condition ( 1 8.35) is satisfied if the function P'(u) is monotonic. In fact , if P'(u) decreases, then, in virtue of the evenness of P'(u), we have that P( [ u l ) f P'(t)dt > [ u [ P '(u). 
G
=
G
*
l ui
=
o
But if
P'(u) increases, then P(2 [ u l ) f P'(t) dt > f P'(t) dt > [ u [ P ' (u). 2 1ul
=
o
2 1ul
lui
Lemmas 1 8. 4 and 1 8. 5 imply the next theorem.
III, § 18 193 THEOREM 18.6. Suppose the Nfunction M(u) has a continuous second derivative P'(u) which is positive for u 0 and which satisfies inequality (18.35). Suppose further that the complementary function N(v) satisfies the fhcondition. Then the Orlicz norm is a differentiable functional on EM. The gradient r of the Orlicz norm is defined by the equality ru(x) P(k*u(x)) (u(x) E EM) , where I N[P(k* l u (x) I ) ] dx 1. CHAPTER
;::j=
=
G
Conve" IUKelions
=
I3
C H A P T ER IV NONLINEAR INTEGRAL EQUATIONS § 19. The P. S . Uryson operator
The P. S . Uryson operator. The operator defined by the (19.1) Ku(x) = I k[x, y, u(y)]dy G will be called the P. S . Uryson operat . Concerning the function k(x , y, u) we shall assume that it satisfies the Caratheodory con ditions, i.e. that it is continuous in u for almost all x, y E G and it is measurable in both variables x, y, simultaneously, for every u. 1.
formula
or
(We shall not stop to prove the measurability of the sets and functions which are encountered in the subsequent constructions.) We shall assume that the function satisfies the inequality
k(x, y, u) (19. 2) I k (x, y, u) 1 � k(x, y)[a(x) + R( l u l ) ] (x, y E G,  00 u 00) , where (19. 3 ) I I M[k(x , y)]dxdy b 00, M(u) is some Nfunction, a(x) is a nonnegative function, R(u) is a nonnegative monotonically increasing function for u > 0 <
<
�
(j
<
which is continuous. We shall be interested in the problem in what cases are the conditions and sufficient that the Uryson operator operate in some Orlicz space L� and be continuous, bounded, compact , or completely continuous in this space . (An operator is said to be on the set if maps every bounded subset TI C T into a compact set . An operator is said to be if it is compact and continuous. We note that for nonlinear operators compactness does not imply continuity.)
(19. 2)
compact
continuous
(19. 3 )
T
completely
CHAPTER IV, § 1 9
1 95
We shall assume that the conditions ( 1 9 . 2) and ( 1 9 . 3) imply the boundedness of the values of the operator ( 1 9 . 1 ) on the sphere
T((), r ; L4I) :
( 1 9 . 4) It is natural to assume that the constant c depends only on the number and on the functions and Let be a nonnegative function in Lli for which
b
a(x), R(u) I M[k(x)] dx b �
G
mes G
M(u) .
k(x)
•
Then conditions ( 1 9 . 2) and ( 1 9 . 3) are satisfied for the operator
Ku(x) I k(x) : k(y) R(l u (y) I ) dy. =
G
Therefore, in virtue of condition ( 1 9 . 4) , we have that
I I k(y)R( l u (Y) I ) dy k(x) I R( l u (y) I ) dy l ., � 2c +
G
G
( l iu l ., � r) .
( 1 9 . 5)
k(x) E L4I. This means that Lli C L4I and, ( 1 9 . 6) tP(exu) < M(u) for large values of the argument and some ex > O . It also follows from ( 1 9 . 5) that the integrals f k(y )R( l u (y ) I ) dy are finite for arbitrary function k(x) E Lli and, furthermore, that the operator Ru(x) R(r l u (x) 1 /2) maps the sphere T((), r ; L4I} into some bounded set in the space LN, where, as usual, N (v) denotes the Nfunction complementary to M(u). In virtue of It follows from this that in virtue of Theorem 1 3 . 1 ,
G
=
Theorem 1 7.5, positive constants Cl, N
[l; R ( r; ) ] �
C2
C2
and C3 can be found such that
+ C3tP (U)
(
00
<
u
<
00
)
.
196
CHAPTER IV,
I t follows from this inequality that
§ 19
(19. 7 )
N[fJR(yu)] < k(/J(a.u)
for large values of u . In combination with the inequalities established above, further investigation is made under the assumption that
(19. 8) N[fJR(yu)] < kM (u) for large values of u and that the Nfunction (/J(u) satisfies con ditions (19. 6 ) and (19. 7 ). 2. Boundedness of the Uryson operator . Our main attention will be centered on the case when the function N(v), complementary to M(u), satisfies the .1 'condition. LEMMA 19.1. Suppose the Nfunction N(v) satisfies the .1 ' condition . Assume that conditions (19. 2), (19. 3 ), (19.6) and (19. 7 ) are satisfied. Finally, let a(x) E LN LN. The operator (19.1) is defined on the sphere T((), y/a. ; L�) and the set of its values have uniformly bounded norms in L� : (19. 9) ( 11 U I I!ll � : ) , I I Ku l l!ll � C l k (x, y ) I I where the constant C does not depend on the kernel k(x, y) . PROOF. Suppose inequality (19. 7 ) is satisfied for u Uo . Since l I a (x) + R ( l u(x) I ) l i N � I l a i N + 7i1 I lfJR ( l u(x) I ) l iN � =
M
�
� we have, in virtue of
I l a i N + i {1 + f N[fJR ( l u(x) I ) ] dX} , G
(19. 7 ), that for I l u ll!ll � y/a.
l I a (x) + R ( l u(x) I ) I IN � � I l a i i N + i { 1 + N[fJR(yuo)] mes G + k f (/J [ ; u(x) ] dX} � G
�
I l a i N + � {I + k + N[fJR(yuo)] mes G} ,
CHAPTER IV,
§ 19
197
i.e.
I la (x) + R(l u (x) I ) I N C1 ( 1 U I !ll � : ) . (19 . 10) In virtue of Theorem 1 S.4, the linear integral operator Av(x) f k(x, y)v(y) dy �
=
G
operates from LN into Ljp where
(19. 1 1)
(19 . 2), I Ku(x) I � A [a(x) + R ( l u (x) I ) ], from which it follows, in virtue of (19 . 1 1) and (19.10), that I Ku l !ll � 2l l k (x, y) 1 M l I a (x) + R( l u (x) I ) l iN � 2lC 1 1 I k (x , y) 1 M. 3. Transition to a simpler operator. We shall assume that the conditions of Lemma 19.1 are satisfied. In virtue of Theorem 17 . 1, one can find a sequence of closed sets &n & such that mes (&'" en) < 1 In, the function k(x, y, u) is continuous in all variables on the sets & n () and the functions k(x , y) and k x, y)a(y) are continuous on the sets en. We set kn(x, y, u) { k (x, y, u) ifif {x{x,, y}y} E een,n . Each of the functions kn(x, y, u) satisfies condition (19. 2) and , i n virtue o f Lemma 19.1, i t determines the Uryson operator Knu(x) f kn [x, y, u(y) ]dy, acting from the sphere T((), Y la. ; Ljp) into Ljp . Clearly, Ku (x)  K7Iu(x) f k[x, y, U(Y)] K(X, y ; &"'& n) dy. In virtue of
*
C
X
(
=, = ,
=
o
=
=
G
G
E
1 98
CHAPTER IV, § 1 9
I n virtue of ( 1 9.2) and Lemma 1 9. 1 , we have that
I Ku  Knu l ql I I k(x,y) K(X, y; G ""G n) [a(x) + R( l u (y) I ) ] dy I . I lu l ql :(; We shall assume, in addition, that k(x, y) E EM. In virtue of Theorem 1 0.3, lim I l k (x, Y ) K(X, y ; C"" Cn) 1 M O . Therefore, O. lim sup I Ku  Knu l . We showed that the operator K can be uniformly approximated by the operators Kn if k(x, y) E EM under the conditions of :(;
:(;
G
y
 .
oc
=
n�oo
=
n� oo lIull .. ';;; ( l'/"' )
Lemma 1 9. 1 . Thus, to prove the continuity or the compactness of the operator it suffices to prove the continuity or the compactness of the operators The operators are defined by the functions for which the inequalities
K, Kn. kn(x, y, u) I k n(x , y, u) 1
:(;
Kn a n + bnR( l u l ) ,
 00
<
u<
00,
are valid, where
an
=
max
{x, y}eGn
I k (x, y)a(x) I , bn
=
max
(x, y}eGn
I k (x, y) l .
Thus, the continuity and compactness of an arbitrary operator
K under the assumptions made above will be proved if the corre
sponding properties are proved for the Uryson operator
Ku(x)
=
I k[x, y, u(y)]dy
G
under the assumption that
( 1 9. 1 2) I k (x, y, u) 1 :(; a + R( l u l ) ; x, y E G,  u 4. A second transition to a simpler operator. Suppose condition ( 1 9. 1 2) is satisfied. We define a new sequence of functions kn(x, y, u) 00 <
< 00.
CHAPTER IV, §
19
1 99
by means of the equality if l u i � n, k(x, y, u) k(x, y, n)(n + 1  u) if n < u < n + 1 , kn(x, y, u) k(x, y,  n)(u + n  1) if  n  1 < u < if 1 1t l ;;;::: n + 1. Suppose the Uryson operators Kn are defined by the functions kn(x, y, u) . We shall show that sup I Ku  Knu l 4l (19.13) lim .. Let u(x) E T({), y ia ; Lq,) . We denote the set G{ l u (x) 1 n} by G'. Clearly, 1 ) f cP [au(x)] dx � mes G' an y cp ( y G =
n,
o
=
n+oo
Ilull ";;; (y/a)
o.
>
�
I Kthatu  Knu l 4l . It follows from the definition Kn I Ku(x)  Knu(x) I � f I k [x, y, u(y) J  kn[x, y, u(y)J l dy � f f � I k [x, y, u(y) Jl dy + I k [x, y, V' (y) J l dy,
We shall estimate of the operators
G
G'
G'
u(x), 1 V' 1 4l � I lu l 4l � yia . K (x ; G') wesgn have V' (x) n(19.12), that I Ku(x)  Knu(x) I � f K(X, y; G ')[a + R( l u (y) I ) J dy + + f K(X, y ; G ')[a + R( I V' (y)I ) Jdy .
where In virtue of =
G
G
200
CHAPTER IV, §
19
C' G G' . Lemma 19.1 implies the inequality I Ku  Knu l !ll � 2C I K {X, y; CI) I .M = 2C mes C'N1 ( mes1 C ) . Making use of the estimate for mes G ' , we obtain the inequality ) ( G N1 [ � ] I lu l !ll yjlX, I Ku  K.u l _ 2C mes ) � mes G ( from which 19 . 13 does indeed follow. Thus, the continuity and compactness of the Uryson operator under the conditions of Lemma 18.1 with the additional condition that k(x, y) E EM will be proved if we prove the corresponding properties of Uryson operators with bounded function k(x, y, ) I k (x, y, u) 1 � d (x, Y E G, 00 u 00) (19.14) where
=
X
I
�
� <1>
�
•
u :
<

satisfying the condition
k(x, y,
u)
<
,
0, l u i uo, (19.15) where Uo is a positive number. 5. third transition to a simpler operator. Suppose the condi tions ( 19.14) and (19.15) are satisfied. In virtue of Theorem 17.1, there exists a sequence of closed sets Cn C such that mes (C"",Cn) = 0 and the function k(x, y, ) is continuous in all variables {x, y} E Cn, 00 u 00 By Uryson's theorem, there exist functions kn(x , y, u), which are continuous in all varia bles, coincide with k(x , y, u) for {x, y} E Cn, and satisfy the con ditions I k n(x, y, u) I � d for x, y E G, 0 00 00 kn{x, y, ) for l u i Uo. We now consider the Uryson operators Kn defined by the func tions kn{s, y, u) . Each of the operators Kn maps an arbitrary Orlicz space into a uniformly bounded, equicontinuous family of functions. Therefore, the set of values of every operator Kn is compact in C and, a fortiori, in an arbitrary Orlicz space. Suppose the sequence of functions Ut(x) (i 1, 2, . . . ) converges to the function uo(x) with respect to the Orlicz space norm. Then this sequenc� q.lso converges to the function uo(x) in measure . �
=
A

<
<

�
u
C
.
< u <
=
,
u
==
§ 19 201 The operator Kn maps every sequence of functions which converges in measure to uo(x) into a sequence of functions which converges for every x, which fact follows from the possibility of passing to CHAPTER IV,
the limit under the integral sign :
Knuo(x) J kn [x, y, uo(y)] dy �im J kn[x, y, ui (y)]dy �im KnUi (X). The sequence KnU (X) converges uniformly to the functions Knuo(x) inasmuch as iti is compact . Thus, Kn E {L� C ; compo c.} and, a fortiori, Kn E {L� L�; compo c.} . For any function u(x) E L�, the obvious inequality I Ku (x)  Knu(x) I � 2d J K(X, y ; G"/;n)dy =
=
G
=
1.+00
=
G
1.+00
+
+
G
is valid, from which it follows that
The inequality obtained signifies that the continuous and compact operators on converge uniformly to the operator the entire space This means that the operator is continuous and compact .
Kn
K
L�. K 6 . Fundamental theorem on the complete continuity of Uryson 's operator. We now formulate the result obtained in the preceding subsections. THEOREM 19.1. Suppose M(u) and N(v) are mutually comple mentary Nfunctions the second of which satisfies the ,tj'condition. Let (19.16) I k (x, y, u) I k(x, YHa(x) + R( l u l ) ] ; x, Y E G,  < u < where k(x, y) E EM, a(x) E LN, and R(u) is a nonnegative non decreasing function. Finally, suppose positive numbers fJ, y and K can be found such that N[fJR(yu)] KM(u) ( 19.17) �
=
�
=,
CHAPTER IV
202
§ 19
for large values of the argument. Then the operator (19.1 8) Ku(x) J k[x, y, u(y)] dy belongs to {T(O, y ; L�) � L�; compo c.}, where cP(u) zs any N function which satisfies the inequalities N[(JR (y u)] � KcP(u) � KM (u ) ( 19.19) for large values of the argument. The natural question arises under what additional assumptions is the operator K defined not only on the sphere T(O, y ; L�) but also on the entire space L�? This will occur if arbitrarily large y's can be taken in conditions (19.17) and (19.19). I n particular, the operator K acts in all LfP if the Nfunction cP(u) satisfies the Ll 2condition since, in this case, =
G
N[(JR (2ByU)] � KcP (2Bu) � K1cP (u) � KIM(u)
for large values of the argument. I t is verified analogously that the U ryson operator is completely continuous in the entire space if the function R( ) satisfies the Ll 2condition : R 2 ) � KIR(u) for large values of the argument . Conditions and can, in some cases, be rewritten in a simpler form. Let us assume that the function cP (u) is such that  cP(u) , i.e. that � cP (cxu) for large values of the argument . Then conditions and are satisfied if
(u (19.17)
N[cP(u)] (19.17)
L� (19.19)
u
N[cP(u)]
(19.19)
(u
(19. 20)
u.
R (cxyu) � KcP ) � KM( ) This follows from the obvious inequalities
Various conditions for the complete continuity of nonlinear integral operators in the spaces are given in KRASNOSEL'SKII All these theorems are concerned with the case when nonlinearities of polynomial type are investigated. The fundamental theorems, given in KRASNOSEL'SKII are comprised in Theorem proved above. However, Theorem also gives conditions for complete
La
[5], 19.1
[5].
19.1,
CHAPTER IV,
§ 19
203
continuity in certain Orlicz spaces for integral operators which contain essentially nonpower, e .g. exponential, nonlinearities. For example, let
I k (x, y, u)1 ::::;;; k(x, y)e� I UI ; x, y E G,  00 < u < 00, ( 1 9.2 1 ) with ( 1 9.22) I k (x, y) 1 ::::;;; a l ln r l 1  o + b, where r is the distance between the points x, y E G . Let p.p. cP(u) el u l l U, where 0 ::::;;; fJ ::::;;; fJo. Let M(u) cP(u). Clearly, k(x, y) E EM since I I exp I Ak(x, y)I1 + lIdxdy < 00 for arbitrary A O. Since M(u) satisfies the ,d 2condition, the function N (v) comple mentary to it satisfies the ,d 'condition. In virtue of Theorem 6.3, we have that N[cP(u)] I'".J cP(u). Therefore, we can use Theorem 1 9 . 1 , in which conditions ( 1 9. 7) and ( 1 9.9) are replaced by condition lI
=
=
(j
>
( 1 9.20) . Condition ( 1 9.20) is satisfied and, furthermore, it is satisfied for arbitrary y > 0 if > o. Thus, if conditions ( 1 9.2 1 ) and ( 1 9.22) are satisfied, then the Uryson operator is completely continuous in some sphere of the space where  1 and it is completely continuous in the entire space where is an arbitrary function of the form  1 , with 0 < < . An analogous line of reasoning shows that the U ryson operator is completely continuous in the entire space where  1) if
fJ
L�,
cPo(u) e1 u l l u i L� cP(u) el u l l U o ' =
=
cPo(u)
fJ fJo
L�o ' cPo(u) lui ( 1 9.23) I k (x, y, u) 1 ::::;;; (a l In r l + b)e IU 1 H, with fJo fJ < 1 . 7. The case of weak nonlinearities. In the case considered above, the function M(u) increased more rapidly than some =
=
(e IUl '�o
<
power function inasmuch as the Nfunction N (v) comple mentary to it satisfied the L1 ' condition. This signifies that for the cases studied the function in condition ( 1 9.2) belongs to some 1 ) . In the present subsection, we shall asume
LIX (ex >
k(x, y)
CHAPTER IV, § 1 9
204 that
M(u)
IX
J J M[k(x, y)]dxdy < ()
( 1 9.24)
00,
where < l u i for all IX > 1 . Then the Nfunction N(v) comple mentary to it increases more rapidly than any power function. We shall assume that N (v) satisfies the Ll acondition. In this con nection, the function M(u) itself satisfies the Ll 2 condition. As was elucidated above, in the study of Uryson operators, it is natural to assume that condition ( 1 9.8) is satisfied for large values of the argument : N[{1R (yu)] < KM(u) < M(Ku) .
( 1 9.25)
In virtue of Theorem 6.3, the inequality N l [M(u)] < K1N l (u) is valid for large values of the argument . It therefore follows from ( 1 9.25) that {1R (yu) < N l [M(Ku)] < K1N l (Ku) for large values of u. In virtue of Theorem 6. 1 , {1R (yu) <
Kl KuN l (Ku) M(K 2U) < Ku K u Kl
for large values of u. Since the Nfunction M(u) satisfies the Ll 2 condition, it follows from ( 1 9.25) finally that R (u) <
C M(u) u
( 1 9.26)
for large values of u. We note that condition ( 1 9.25) follows from ( 1 9.26) and , further more, ( 1 9.26) implies the validity, for large values of u and certain {1 and y, of the inequality N[{1R (u)] <
1 u.

( 1 9.27)
y
C
It follows from the last inequality that R (u) < 1 N l (u) for large values of the argument . This means that in the case under consideration the nonlinearity of R (u) must be "very" weak. In fact , it follows from the fact that N (v) satisfies the Ll acondition that N (v) increases more rapidly than any power function. There
CHAPTER IV, § 1 9
205
R(u) e> v e'V� ( >
fore the inequalities obtained signify that increases more slowly than any power function where 0, If N ( ) satisfies the Ll 2 condition, then, as was noted above (on p. 43) , N (v) increases more rapidly than some function 01: 0) . From this it follows that , in the case under consideration, increases more slowly than (In U ) l / �. THEOREM 1 9.2.
l u i ",
R(u) Suppose M (u) and N(v) are mutually comple mentary Nfunctions the second of which satisfies the Ll3condition . Let ( 1 9.28) I k (x, y, u) 1 � k(x, y)[a(x) + R ( l u l ) ] ; x, E G,  00 < U < 00, where k(x, y) E LM LM, a(x) E LN, and R(u) is a nonnegative function which does not decrease for u > Finally, suppose a C > 0 can be found such that inequality ( l 9.26) is satisfied for large values of the argument. Then there exists an Orlicz space L� in which the completely continuous operator ( 1 9.29) Ku(x) I k[x, y, u(y)]dy acts. PROOF. Since the function M[k(x, y)J is summable on G, an Nfunction tP(u) , satisfying the Ll 2 condition (and even the LI ' Y
=
o.
=
G
condition) , can be found (see p. 6 1 ) such that
I I tP{M[k(x, y) J }dxdy (j
< 00.
( 1 9.30)
L� .
We shall show that the operator ( 1 9.29) acts in the space The validity of inequality ( 1 9.27) for large values of follows from inequality ( 1 9.26) . Suppose it is satisfied for � Uo. Let E = Since
u
u u(x) L� L�. I l a(x) + R( l u (x) I ) l iN � I l a i iN + 7i1 I IPR ( l u (x) I ) l IN � � I la i iN + � { I + I N[ PR(l u (x) I ) ] dX} , G
206
CHAPTER IV, § 1 9
we have, in virtue of
( 1 9.27) ,
that
I la (x) + R ( l u (x) I l I N � � l I a l N +{I + N[tJR(uo)J mes G + + J I U (X ) l dX} � 1 � l I a l N + 7i {I + N[tJR(uo)J mes G + fl i l u l ll>} , where fl is some constant . Thus, for I l u l ll> � r, we have that (19.3 1) I l a (x) + R ( l u (x) I l I N � C(r). +
G
Applying to the linear integral operator
Av(x) J k(x , y)v(y) dy condition a) of Theorem 15. 4 (setting M 1 (u) N(u), M (u) C/>(u), P(u) C/>[M(u)J), we convince ourselves, in virtue 2 of (19. 30), that the operator A acts from the space L'N into the space LII> and is continuous, with (19. 32) I A v l 1l> � 2l l k (x, y) I q, l v I N. In virtue of (19.28), we have that I Ku(x) I � A[a(x) + R( l u (x) I ) J , from which it follows, in virtue of (19. 3 1) and (19. 3 2), that I Ku l 1l> � 2l 1 I k (x, Y) I q, l a (x) + R( l u (x) I ) I N � � 2lC(r) I l k (x, y) I q, . The continuity and compactness of the operator (19. 29) are proved the same way that Theorem 19.1 was proved. We now consider a simple example. Suppose (19. 33) I k (x, y, u) 1 � k(x, y) [a + In ( l u i + I )J and (19. 34) J J I k (x, y) l In ( I k (x, y) I + 1) dxdy =
G
=
=
=
*
(1
< 00 .
CHAPTER IV, § 1 9
207
Then the operator ( 1 9.29) acts in some Orlicz space and is completely continuous there. To prove this, we must apply Theorem 1 9.2 in which the function
M(u) ( 1 + l u i ) In ( 1 + l u i )  l u i . Hammerstein operators . We now consider Uryson operators =
8. of the special form
Ku(x) J k(x, y)t[y, u(y)] dy. Such operators are called Hammerstein operators .
( 1 9.35)
=
G
The conditions found above under which the Uryson operators act in some Orlicz space and are completely continuous there are, naturally, applicable also in the investigation of the operator ( 1 9.33) . However, another method can be utilized in the study of this operator in certain cases. Suppose E l and E 2 are two Banach spaces. We shall assume that the operator acts from some sphere C E l into the space E 2 , is continuous and is bounded on this sphere. Suppose the linear integral operator
T
j: ju (x) t[x, u(x)], =
Av(x) J k(x, y)v(y) dy =
G
acts from E 2 into E l and is continuous. Since the operator ( 1 9.35) can be represented in the form of the composition under the indicated conditions, it obviously acts from the sphere into E1, is continuous and bounded. If the operator is comple tely continuous, then the operator ( 1 9.35) is also completely continuous. E l and E 2 can be considered to be two Orlicz spaces. In § 1 7 , we found conditions for the continuity and boundedness of the operator j. Combination of these conditions with the conditions for continuity (§ 1 5) and complete continuity (§ 1 6) of the operator yields sufficient conditions for the continuity and complete con tinuity of the Hammerstein operator.
K Aj, =
T
A
A
208
CHAPTER IV, § 20 § 20. Some existence theorems
1. Problems under consideration.
Suppose A is an operator, which, generally speaking, is nonlinear and acts in some Banach space E. We shall point out some problems which arise in the consideration of the equation
A rp
=
Arp.
(20. 1 )
The first problem is t o find conditions under which equation (20. 1 ) has solutions for fixed values of A. As a rule, it is desirable to supplement the conditions for the existence of solutions by the conditions for uniqueness of the solution . In many cases the operator A possesses the property that AO = 0, where 0 is the zero of the space E. Then equation (20. 1 ) has the trivial, zero solution for all values of the numerical para meter A. In these cases, solutions different from the trivial solution are of interest . Such solutions exist only for isolated values of the parameter A. It is customary to call the nonzero solutions of equation (20. 1 ) (or of the operator A . The numbers A for which equation (20. 1 ) has nonzero solutions are called of the operator A . The second problem is t o find conditions under which the operator A has characteristic vectors. The totality of characteristic values of the nonlinear operator A is called its (in analogy with the linear operator case) . If the spectrum of an operatol fills an interval, it follows from this that the operator has a continuum of characteristic vectors. There can be cases when an infinite (countable or continuous) set of characteristic functions corresponds to one characteristic value . The third problem is the investigation of the spectrum of a nonlinear operator and the study of the topological structure of the set of characteristic vectors. It turns out that under rather general assumptions the sets of characteristic vectors are entities of the same type as continuous curves ; these entities are called continuous branches. We now introduce the corresponding definition . A set m e E is called a in the spherical layer < < if the intersection of the set m with the boundary 5 of any region m which contains the sphere and is contained with its boundary in the sphere < is nonvoid.
characteristic vectors characteristic functions) characteristic values
spectrum
continuous branch
a I lu  uo l b I lu  uo l ::::;;; a I lu  uo l b
CHAPTER IV, § 20
209
Of basic interest are the conditions under which a nonlinear operator has characteristic vectors with arbitrarily small norms. Let AO be some number and suppose to every B > 0 there corre sponds a A such that IA  Ao l < B and that for this value of A equation (20. 1 ) has at least one nonzero solution rp satisfying the condition Ilrpll < B . Then the number A O is called a of the nonlinear operator The fourth problem i s the investigation o f branch points . T o solve the problems enumerated above (and a number of others which we did not mention) qualitative methods of non linear functional analysis are being worked out at the present time. The application of general propositions to the investigation of concrete equations (20. 1 ) requires that the operator possess definite "good" properties : that it be continuous and bounded, in other cases that it be completely continuous, that it be differenti able, that it be the gradient of some functional, and so on. In connection with this, the application of general theorems of nonlinear functional analysis to the study of concrete non linear integral equations requires the construction of a functional space in which the integral operator acts and possesses certain "good" properties. In the maj ority of known investigations, the space of continuous functions and various La spaces are taken as the functional space E. This circumstance leads to the fact that various restrictions are placed upon the functions which appear in the equation . The application of Orlicz spaces leads to other (sometimes weaker) restrictions and permits us to consider new classes of equations. Combination of the results of the preceding sections with the general propositions of nonlinear functional analysis leads to new existence theorems, theorems on characteristic functions and branch points, and so on. Below, we shall introduce some examples of such a combination. The reader who is familiar with nonlinear functional analysis can easily extend the list of such examples. 2. One of the most common methods to prove existence theorems consists in utilizing Schauder's fixed point principle. SCHAUDER'S PRINCIPLE.
point
branch
A.
A
C
The existence ot solutions.
Suppose the completely continuous operator A maps the sphere T ot some Banach space B into a subset Convex functions
I4
CHAPTER IV, § 20
210
T. Then the sphere T contains at least one element Uo such that Uo Auo. We consider the equation (20.2) u(x) A f k[x, y, u(y )]dy + lo(x).
01
=
=
G
Suppose the conditions (see § 1 9) are satisfied under which the operator
(20 .3) Ku(x) f k[x, y, u(y)] dy is defined on the sphere T((), y ; L�) and is a completely continuous operator with its set of values in L�. Let a. sup I Ku l 1ll We shall assume that lo(x) E L� and that 1 1/01 1111 t5 < y. Then , for (20.4) I A I y a t5 , the operator defined by the right member of equation (20.2) maps the sphere T((), y ; L�) into a subset of itself : I AKu + 101 1111 I A I a + 1 1/0 1 1 111 Y (1 I ul lll ::( y) . The operator AKu(x) + lo(x) is completely continuous since the operator K is completely continuous. It thus follows from the Schauder principle that equation (20.2) has at least one solution in the space L� lor sufficiently small A. If the operator K is defined on the entire space L�, then, in =
G
=
I lull .. ";; "
=
::(

::(
::(
making use of the Schauder principle, one can consider spheres of various radii y. It is natural in this case to consider spheres with radii for which the right member of (20.4) takes on the largest possible value. In particular, equation (20.2) has a solution for all if
A
lim y. oo
y
 = =,
sup
I Ku l 1ll
since if this condition is satisfied for every can be found for which (20.4) is satisfied.
(20.5)
A a sufficiently large y
CHAPTER IV, §
20
21 1
The line of reasoning introduced above enables us to prove , for example, the following assertion :
equation (20.2) has a solution for arbitrary if the conditions of Theorem 19. 2 are satisfied in which N (v) satisfies the fl2condition and if fo(x) is a summable function. In fact , we choose the Nfunction (u), which satisfies the fl' condition, so that , on the one hand, inequality (19. 30) is satisfied and, on the other hand, fo(x) L�. Equation (20. 2 ) can then be A.
E
considered as an operator equation in the space L�. It follows from the line of reasoning followed in the proof of Theorem that
19.2 I Ku l IP � 21 I lk (x, Y) I q, l Ia (x) + R ( l u (x I ) l iN � � + PI I R ( l u (x ) I ) l i N � OC l + P I I Nl ( l u (x ) I ) l iN . Therefore, to prove equality (20. 5 ), it suffices to prove that I N  l ( l u (x ) I ) I N lim (20.6) l Iu l IP Since the Nfunction N(v) satisfies the fl 2 condition, a Uo can be found ( see (6 . 12)) such that N(u)Ntv) � N (uv) for v Uo. Setting N(u) t in this inequality and applying the p, N ( v) )
oc
t
=
Ilul l . O
=
O
.
1'(' ,
=
�
function N l (u) to both members of the inequality we arrive at the inequality N l (pt) � N l (p)Nl (t) , which is valid for p , � p If p � p and < p , then
t o N(uo).
o
=
t o
N l (pt) � N l (pPO) � N l (p)N l (p O ) . Thus, the inequality N l (pt) � Nl (p)N l (t ) + N l (p)N l (PO )
po and for all t > It follows from this inequality N l ( l u (x) I ) N l (p U�X) ) � IU ) I (20. 7 ) N l (p) [N l ( � ) + N l (PO ) ] . O.
is valid for p � that , for p � po , =
�
212
CHAPTER IV,
Let
1).
I I N l ( l u(x) I ) li N :::;;; a
( l l u I1 4l :::;;; lt then follows from that , for I l u ll 41 = p � po, I I N l ( l u(x) I ) I I N :::;;; N  l (p) [a + N l (P O) I I K(X ; G) I I NJ
(20. 7 )
Now
=
(20. 6) follows from this inequality inasmuch as lim 1>> 0
N l (p ) P
§ 20
=
bN l (p) .
o.
(20.2),
The conditions for the existence of solutions for equation obtained with the application of Orlicz spaces, differ from those conditions for the existence of solutions which appear when one uses the space or (see, e.g. , NEMYCKII I , KRASNOSEL'SKII and SCORZA DRAGON! [ I J ) . As we already saw above, this is explained by the fact that completely continuous Uryson operators, which do not act in the spaces or act in Orlicz spaces. Thus, it follows from Theorem that equation has solutions for sufficiently small A if
C Lrx
[5J,
[ J [2J,
C Lrx, 19.1
f
(20.2)
exp I /o(x) l l +ll dx <
00
G
provided (see
(19.21) and (19.22))
I k(x, y, u) 1 :::;;; k(x, y ) e � I U I
(x, y E G,  00 < u < 00) ,
(20. 8 ) and (20.9) I k(x, y) 1 :::;;; a lln r l 1  Po b. I n this connection, the solution u (x) belongs to all spaces L� where p.p. tP(u) e 1u11+P and 0 :::;;; {J (Jo. lt follows from Theorem 19. 2 , e.g. , that equation (20. 2 ) has solutions for all A if lo(x) is summable and (see (19. 3 3) and (19. 3 4)) (20.10) I k(x, y, u) 1 :::;;; k(x, y) [a + In ( l u i + I )J +
=
and
�
(x, y E G,  00 < u < 00)
f f I k (x, y) l ln ( I k(x, y) 1 + l ) dxdy < o
00 .
(20.1 1)
We have shown two examples. In each of them, the operator
CHAPTER IV, § 20
213
Lot
without additional (20.3) cannot be considered in the spaces assumptions : in the first example  because of the " <;trong" nonlinearity, and in the second  because of the fact that the kernel y ) may have "strong" singularities. Other fixedpoint principles can also be used in the proof of theorems on the existence of solutions.
k(x,
Positive characteristic functions. cone t u,  u
A closed convex set Sf in 3. the Banach space E is called a if u E Sf implies that E Sf for all > 0 and if E Sf imply that = e. It is easily seen that the totality of nonnegative functions in any Orlicz space forms a cone. We write « if E Sf. The operator acting in the space E with cone Sf, is said to be if Sf C Sf. The operator implies that is said to be if
u
tu
A, U l Uz Uz  U l positive A A monotonic U l <. U z AUl « Auz.
Numerous conditions for the existence of characteristic vectors for positive completely continuous operators are known (see KRASNOSEL'SKII [5J , Chapter V) . All of them are applicable to operators acting in Orlicz spaces. We mention one of these general theorems. Let the positive completely continuous operator Sf satisfy the inequality ( 0. ) « E Sf) ,
A
2 12
Au Ku (u
where is a linear completely continuous positive operator possessing the property that for arbitrary E Sf =1= e) positive numbers IX, {J can be found such that
U (u
IXUo «
Au « (Juo. (20.13) Then the characteristic vectors U of the operator K: (20. 1 4) Ku AU , =
corresponding to po"itive characteristic values A, form in the cone Sf a continuous branch of infinite length, i.e. on the boundary of every bounded region, containing e, there is at least one charac teristic vector of the operator lying in the cone Sf. Suppose the conditions of Theorem are satisfied and let k (x, y, satisfy the additional condition
K
u)
19.1 k(x, y , u) ;;:::: aZu (x, y E G, u ;;:::: 0),
(20.15)
214
CHAPTER IV, § 20
where which
a
i= o. Then the operator K satisfies condition (20. 1 2) in
Au(x) a2 f u(x) dx. =
G
The operator A , obviously, satisfies condition (20. 1 3) , where
uo(x) 1 . It follows from the theorem j ust formulated that under these conditions, the equation (20. 1 6) f k[x, y, u(y)]dy AU(X) has a continuum of positive solutions, corresponding, possibly, to distinct values of the positive parameter A. These solutions form in the corresponding Orlicz space a continuous branch of infinite length. In this continuous branch, there are, in particular, solutions with arbitrarily small and with arbitrarily large norms. ==
=
G
Thus, for example, it is sufficient for the existence of a continuous branch of positive solutions of equation (20. 1 6) that conditions (20.8) , (20.9) and (20. 1 5) are satisfied.
Characteristic functions of potential operators. potential operator
4. An operator is called a if it is the gradient of a functional. A number of theorems on the existence of characteristic functions for a potential operator are known. The utilization of Orlicz spaces in the application of these theorems allows us to extend the class of operators that can be studied. This utilization is carried out according to the following scheme. Let H be a linear operator acting from L 2 into the Orlicz space L�. Suppose Fl ( U) is a real functional on L� (or on some sphere in this space) . Then F(u) F l ( H ) will be a functional defined on L 2 (or on some sphere in the space L 2 ) . If the operator H is completely continuous, then it is easily seen that the functional F(u) is weakly continuous, i.e. its values converge on every weakly convergent sequence of functions in L 2 . If the functional Fl (U) is differentiable in an Orlicz space and has the operator r1 for its gradient, then the functional F ( ) is differentiable in L 2 and its gradient is the operator =
u
u
(20. 1 7)
CHAPTER IV, § 20
215
where H* is the operator adj oint t o H which acts from the conj ugate space L� into L 2 . In fact, if where lim
1101 1 _ 0
then
F(u
)1 _lw,(v_,_g_ I l g l l lfI
=
0,
+ h)  F(u)  (ru, h) F l(Hu + Hh)  Fl (Hu)  (rl Hu, Hh) =
=
=
h
w(Hu, H ) ,
where
Iw (Hu, Hh) I I w (Hu, Hh) I :::;; I I HI I lim h I l l IL" I I Hhl 1 ,,>0
I Hh l 1fl
If u(x) is a solution of the equation
H*rlHu(x)
=
AU(X) ,
=
o.
(20. 1 8)
then Hu(x) will be a solution of the equation
HH*rl V(X)
=
AV(X) .
(20. 1 9)
Thus, the proof of theorems on the existence of distinct solutions of equation (20. 1 9) can be reduced to the proof of theorems on the existence of solutions in of equation (20. 1 8) . But , in the proof of the existence of solutions in of equations with potential operator one can apply, as we have already pointed out , various general propositions (see KRASNOSEL'SKII [5J , Chapter 6) . Theorems 1 6.8 and 1 8. 1 contain the conditions under which the operator
L2
Ku(x)
=
L2
f k(x, y)t[y, u(y)J dy
G
(20.20)
with symmetric positive definite kernel k(x, y) is representable in the form K HH*rl , where r1 f is the gradient of a functional. It is known that the gradient of every weakly continuous functional in has a continuum of characteristic functions. Therefore, it follows from the above discussion that t =
L2
=
i the condition!j
CHAPTER IV, § 20
216
of Theorems 16. 8 and 18.1 are satisfied, then the equation I k(x, y)f[y, u(y) ] dy AU (y) has a continuum of distinct characteristic functions, each of which corresponds to a value of the parameter A. =
G
Variational considerations can also be applied in the proof of theorems on the existence of solutions. Suppose the completely con tinuous operator K (KO = 0) has the Frechet derivative B at the point O. In this connection, the operator B is also completely continuous. It is known that every characteristic value of odd multiplicity of the linear operator B is a branch point of the operator K. In order to apply this assertion to the investigation of nonlinear integral operators K, which are completely continuous in some Orlicz space, one must know the conditions under which this operator K is differentiable at the zero of this space. Let us consider, for example, the operator (20.20) . It can be represented in the form of the composition K = where is a linear integral operator defined by the kernel If the operator acts from the space into the space and has the Frechet derivative /I at the point 0 : /I u ( x ) = and the operator acts from into then, as is easily seen, the operator K is differentiable at the zero 0 of the space and its Frechet derivative B has the form
5. Theorem on branch points.
j
L'M,
A
L'M.
L'M"
Aj, A k(x, y). L'M. f�(x, O)u(x), L'M,
Bu(x) A/I (x) I k(x, Y)f� (y, O)u (y) dy. =
u
=
G
(20.2 1 )
Combining the conditions for the complete continuity of a linear integral operator (see § 1 with the conditions under which the operator j acts from into and is differentiable at the zero 0 of the space (see § we arrive at the conditions under which every characteristic value of odd multiplicity of the kernel 0) is a branch point of the operator (20.20) .
6) L'M, L'M. L'M, 18),
k(x, Y)f�(Y ,
SUMMARY OF FUNDAMENTAL RESULTS Convex functions
1 . Let and be two rightcontinuous > 0) non decreasing functions which are the "inverses" of one another in the sense that
P(t)
q(s)
q(s)
(s, t
t, P(t) sup and which satisfy the conditions P(O) q(O) q(+ ) + P(+ ) sup
=
=
p(t ) .;;; s
q(s) .;;; t
=
00
00
=
(1)
s
=
0,
00 .
=
The convex functions M(u) and N (v) , defined by the equalities
f P(t) dt,
Ivl
lui
M (u)
=
N (v)
=
o
f q(s) ds,
o
are called mutually complementary Nfunctions. The following pairs of functions : M1 (u)
=
 ( 0(
l u l iX 0(
>
1),
N l (V)
(2)
IvlP
= 
fJ
(1

0(
+
) 1 , = fJ 1

can serve as examples of mutually complementary Nfunctions. If M1 (u) � M 2 (U) for large values of the argument , then the reverse inequality N 2 (V) � Nl (V) is satisfied by the complementary Nfunctions for large values of v. The Young inequality is valid for mutually complementary N functions : uv � M(u) + N (v) .
(3)
218
SUMMARY OF FUNDAMENTAL RESULTS
This inequality transforms into an equality only for values of U and v which are related in a definite way : lu IP ( l u l )
=
M (u) + N[P ( lu l ) ] ,
(4)
q( l v i ) I v l
=
M[q ( l v l ) ] + N (v) .
(5)
2. If M l (U) � M 2 (ku) for large values of the argument then we write M l (U) < M 2 (U) . If M1 (u) < M 2 (U) or M 2 (U) < M1 (u) , then the Nfunctions M1 (u) and M 2 (U) are said to be The relation M l (U) < M 2 (U) implies the relation N 2 (V) < N1 (v) for the complementary Nfunctions. The Nfunctions M l (U) and M 2 (U) are said to be (and we write M1 (u) ,.." M 2 (u) ) if M l (U) < M 2 (u) and M 2 (U) < < M l (U) ; Nfunctions which are complementary to equivalent functions are also equivalent . There are various criteria for the equivalence of Nfunctions. . . . ) there For every sequence of Nfunctions M 1, can be found Nfunctions cI>(u) and 'JI(u) such that the relations cI>(u) < M n (u) < 'JI(u) are valid for all
comparable.
equivalent
n(u) (n 2, n. We say that M(u) satisfies the fhcondition if =
3.
M(2u) � kM(u)
(6)
for large values of the argument . Nfunctions which satisfy the L1 2 condition can be maj orized by a power function for large values of the argument . A necessary and sufficient condition that M(u) satisfy the L1 2 condition is that .
u
uP ( ) hm  < 00. M (u) u>oo
(7)
A necessary and sufficient condition that M(u) satisfy the L1 2 condition is that the complementary function N (v) satisfy the inequality 1 N (v) �  N ( v ) ,
2l l
where
l > 1 , for large values of the argument .
(8)
219
S U M M A R Y O F F U N D A M E N T A L R E S U LT S The functions M1 (u)
=
Ma(u)
=
 ( 0( > 1 ) , M 2 (U) lula 0(
l u l a ( l ln l u l l +
1) (
0(
(1 + l u I ) In (1 + l u I )  l u i , 1), M4 (U) In ( l uui2+ e)
=
>
(9)
=
can serve as examples of Nfunctions which satisfy the Ll 2 condition . The Nfunction M(u) complementary to N (v) eV"  also satisfies the Ll 2 condition ; the explicit fo rm of the function M (u) is unknown. There also exist mutually complementary Nfunctions M (u) and N(v) neither of which satisfies the Ll 2 condition. =
1
4 . We say that the Nfunction M (u) satisfies the LI 'condition if M(uv) � CM(u)M (v)
(10)
for large values of u and v. If the Nfunction M(u) satisfies the LI 'condition, then it also satisfies the Ll 2 condition. For example, the Nfunctions M1 (u) , M 2 (U) and Ma (u) in satisfy the LI 'condition. The function M4(U) does not satisfy the LI 'condition. If the function
(9)
h(t)
=
(ut P L P (t)
(1 1 )
does not increase for large t for every fixed, sufficiently large , lui
u, then the Nfunction M(u)
=
f P(t) dt satisfies the LI 'condition. o
If the function P ( t) is differentiable for large values of t, then a sufficient condition for the LI ' condition to be fulfilled is that the function
g (t)
=
tp'
(t)
p(i)
(12)
does not increase for large values of t. If g (t) does not decrease , then the Nfunction N(v) complementary to M (u) satisfies the LI ' condition.
220
S U M MARY OF F U N DAMENTAL RESULTS
l u I M (u),
M(u) M(u) satisfies the acondition.
then we 5 . I f the Nfunction is equivalent t o say that ,1 Functions which satisfy the j acondition increase more rapidly than any power function for large values of the argument . However, not all Nfunctions which increase more rapidly than an arbitrary power function 'iatisfy the ,1 acondition. If satisfies the ,1 acondition, then the Nfunction N (v) complementary to it satisfies the j 2 condition and it satisfies the inequalities
M(u)
t J 3)
for large values of the argument , where inverse to If satisfies the j acondition and
M(u)
M(u).
Ml (v) is the function
2P2 (u) M(u)P'(u)
(14)
�
for large values of the argument , then the Nfunction N (v) comple mentary to is equivalent to the Nfunction which equals for large values of the argument . For example, if then N (v) is equivalent to the Nfunction which equals v v ln for large values of If satisfies the ,1 acondition, then From the class of Nfunctions satisfying the ,1 acondition, we select a narrower class of functions, ,1 M 2 (U) M 2 (U) . The Nfunctions can serve as examples of such functions. The Nfunction which equals for large values of satisfies the j a condition, but it does not satisfy the j 2 condition. A sufficient condition for to satisfy the j 2 condition is that the inequality < be satisfied for large values of the argument. A necessary and sufficient condition that the N function satisfy the j 2 condition is that the Nfunction complementary to satisfy the inequality
vMl(v) eU' 1, v M(u) =
M(u)
M(u) ,...." eU' 1 2If(u) =
M(u)


M(u)
=
v.
M(u) ,...." N [M(u)J. satisfying the 2condition: M1(u) e1 u I l u i 1, u1nu u M(u) P2 (u) P(ku) =
M(u)
N (v)

v
<
k
N ( V0
Vv


=
(15)
for large values of v. If satisfies the j 2 condition, then the complementary function satisfies the j 'condition.
M(u) N(v)
S U MMARY OF F U N DAME NTAL R E S U LTS
221
If the Nfunctions Ml (U) and M 2 (U) increase more rapidly than an arbitrary power function, then, under certain additional assumptions, the composition Nl[N 2 (V) ] is equivalent to the Nfunction Nl (V)N2 (V) / l v l . This occurs, for example, if both the functions M1 (u) and M 2 (U) satisfy the LJ 2 condition.
6.
Suppose the Nfunction M(u) coincides with a function of the form u!¥ (ln u)1'l (ln In u) 1" . . . (In In . . . I n u) 1'n (oc > 1 ) for large values of the argument . Then the function N (v) which is complementary to M(u) is equivalent to the Nfunction vP[ (ln v) 1'l (ln In v) 1" . . . (In In . . . In v) 1'n] 1 p, where l /oc + I /P
=
1 , for large values of the argument . Orlicz spaces
1.
Suppose G is a bounded closed set in a finitedimensional Euclidean space and that M (u) is an Nfunction. The totality of functions u(x) for which p (u ; M)
Orlicz class
=
G
is called the LM LM(G) . Every function, which is summable on G, Orlicz class. Orlicz classes are convex sets. The class LM and only if, the Nfunction M(u) satisfies the A sequence of functions un (x) E LM is said vergent to zero if lim p (un ; M) = o.
2.
(16)
f M[u (x) ] dx < 00
=
belongs to some is a linear set if, LJ 2condition. to be mean con
n> oo
The linear hull of the Orlicz class LM transforms into a complete normed space Lit if a norm is introduced by means of the equality
f u(x) v (x) dx. The space Lit is called an Orlicz space. This space l
I ul iM
=
sup
p (v ; M) ';;; l
G
( 1 7) is separable if,
and only if, the Nfunction M(u) satisfies the LJ 2 condition.
222
S U MMARY OF FU ND AMENTAL RES ULTS
The norm of the characteristic function is calculated by means of the formula
I K (X; 8) I M
mes 8Nl
=
K (X; 8) o f the set 8
( me� 8 ) '
CG
( 1 8)
where Nl(U) is the function inverse to the Nfunction N (v) which is complementary to To calculate the norm, one can make use of the formula
M(u)
I l u l iM
=
inf k> O
.
k ( + f M[kU(X)]dX) .
� I
( 1 9)
G
A norm equivalent to the norm ( 1 7) can be introduced into the space this is the socalled
LA!;
Luxemburg norm: I lu l I (M) inf k. where the infimum ranges over those positive k for which ( � ; M) f M [ U�X) ] dx � I . =
P
=
G
The Luxemburg norm is connected with the Orlicz norm by means of the inequalities
These norms differ by a constant factor only in the case when is the space of functions which are ot:summable for some power ot: > I .
LA!
L�
3. The inequalities
I UI M and
hold.
�
I
+
f M[u(x)]dx
G
f M [±LJ dx I lul M
G
i
::::;;; 1
(20)
(2 1 )
S U MMARY O F F U N DA M E N TAL RESULTS If
223
l Iu lM � I , then
I M[u(x)] dx � l I u l M. For any pair of functions u(x) E LM, v(x) E LN, u(x) v (x) is summable and the old r inequality I u (x) v (x) dx � I l u l M l v l N
(22)
G
H
the function
e
G
(23)
is valid. A sufficient condition for the product (x w ( of the functions (x) w(x) to belong to the space is that there exist mutually complementary functions and such that the inequalities
u E LM,
u ) x) LM• R(u) Q(u)
E L�
R(cxu) < Mi l [Ml (U) ] , Q(cxu) < Mi l [4>(U)]
(24)
are satisfied for large values of the argument . If the Nfunction M2(U) satisfies the ,1 ' condition, then it is sufficient that the ineq uali ties < M l [Mi l (U)] , hold. The inequality
R(cxu) Q(cxu) < 4>[Mi l (U)]
(25)
holds in the cases described above. where is the topological product G X G, is The space denoted by A necessary and sufficient condition that , for an arbitrary pair of functions (x) , the product (x (x) belong to is that the Nfunction M(u) satisfy the ,1 'condition . In this connection,
LM(G), LM. LM
G
u v(x) E LM,
u )v
I lu (x)v(y) 1 1M � c l Iu l M I lv i IM. 4. The convergence of the sequence u n (x) t o the function uo (x) with respect to the norm in the space LM implies the mean con vergence to zero of the sequence u n (x )  u o (x) . Convergence in
224
S U MMARY OF F U N DAME NTAL RESULTS
norm is equivalent t o mean convergence t o zero of the sequence u n (x)  uo (x) if, and only if, M(u) satisfies the Ll 2condition.
5. If M(u) does not satisfy the Ll 2condition, then the set of bounded functions is nowhere dense in the space Livr. The closure in Livr of the set of bounded functions is denoted by EM. This space plays an important role. It coincides with Livr L M if M(u) satisfies the Ll 2 condition. EM is separable and has a basis. A necessary and sufficient condition that the function u(x) E Livr belong to E M is that its norm be absolutely continuous. The absolute continuity of the norm signifies that to every B > there corresponds a 0 > such that I l u(X) K(X ; c&") 11 M < B provided mes c&" < 0 (c&" C G) . If p (u) M ' (u) is continuous then to calculate the norm of a function in EM one can use the formula =
0
0
=
I lu l iM where k*
IS
=
f P (k* l u (x) I ) l u(x) I dx,
G
(26)
determined from the equation
f N[P (k* lu(x) I )J dx
G
=
1.
(27)
The space E M enables us to describe the disposition of the class LM in the space Livr : the class LM contains the totality II of all functions u for which inf I l u  w i lM < 1 and is contained
in the closure ii. If E M is a proper subset of the space Livr (i.e. lYJ(u) does not satisfy the Ll 2condition) , then II is a proper subset of LM and LM is a proper subset of II. 6. Under natural assumptions, the Orlicz norm and the Luxem burg norm are differentiable in the space EM (or in the space Livr LM if M(u) satisfies the Ll 2condition) . Suppose the Nfunction N(v) , which is complementary to M(u) , satisfies the Ll 2condition. Suppose the Nfunction M(u) has a continuous monotonic second derivative which is positive for u =1= O. Then the Orlicz norm is a differentiable functional on EM. The gradient r of the Orlicz norm is defined by the equality =
ru(x)
=
P (k*u(x) ) ,
(28)
S U M MARY O F F U N D A M E N TAL R E S U LTS
225
where
f N[P (k* l u(x) I )] dx
G
=
1.
The gradient r1 of the Luxemburg norm I l u l l (M) is defined by the equality
rl U(X)
=
( l Iuul(lx)(M) ) f ( Pu(x) ) u(x) P l Iu l l (M) lI u ll(M)
(29)
d
x
G
7.
We say that a family we C LM has equiabsolutely continuous norms if for every 8 > a 15 > can be found such that for all functions u(x) of the family we have that
0
Ilu(X)K(X; tf) IIM
0
<
8
provided mes tf
<
15.
A necessary and sufficient condition that a sequence of functions
un(x) E EM which converges in measure also be convergent in
norm is that it have equiabsolutely continuous norms. It follows from this fact that the family we C EM is compact in LM if it has equiabsolutely continuous norms and is compact in the sense of convergence in measure. The known criteria for compactness due to A. N . Kolmogorov and F. Riesz also generalize to families of functions situated in the space EM.
8. Generally speaking, distinct Nfunctions determine distinct
Orlicz spaces. A necessary and sufficient condition that the settheoretic inclusion LMl C LMa hold is that the relation M2(U) < Ml(U) be satisfied. In this connection, the norms turn out t o be comparable :
(30) The spaces LMl and LMa consist of the same functions if, and only if, M l (U) '"" M2(U) . Norms generated by equivalent N functions are equivalent .
Convex functions
IS
226
S U M M A R Y O F F U N D A M E N TA L R E S U L T S Functionals and operators
1 . The general form of a functional on the space EM is given by the formula
l ( u)
=
f u (x)v(x) dx,
(3 1 )
G
where v(x) E LN. The norm of the functional I coincides with the Luxemburg norm I l v l l (N) of the function v (x) . If M ( u) does not satisfy the L1 2 condition, then there exist linear functionals on which do not admit of an integral re presentation (3 1 ) . The space LAt is reflexive if, and only if, M(u) and the comple mentary function N ( v) satisfy the L1 2 condition.
LAt
2. ENweak convergence is introduced in the space LAt as follows : the sequence of functions un (x) E LAt is ENweakly con
vergent if the sequence of numbers
f un (x)v (x) dx converges for
G
every function v ( x) E EN. The ENweak convergence of a sequence of functions implies the boundedness of the norms of the elements of the sequence. Every Orlicz space is ENweakly complete and ENweakly compact . The space EM does not possess these properties : the ENweak closure of the space EM is the entire space Convergence in measure and boundedness of the norm imply ENweak convergence.
LM.
3. Various criteria have been found for the continuity and complete continuity of linear integral operators
AqJ (x)
=
f k (x, y)qJ(Y) dy
G
(32)
acting from one Orlicz space into another. We formulate the fundamental result . Let
SUMMARY OF FUNDAMENTAL RESULTS
227
continuous if any one of the following three conditions is satisfied :
M
a) 2 [N l (V)] < P(v) ; b) Nl[M2 (V)] < P(v) ; c) the function q; (u) satisfies the Ll 'condition and M 2 (v) < P(v) ,
N 1 (v) < P(v) .
If, under the conditions of this theorem, the kernel k (x, y ) belongs to the space £'1' ( i.e. to the closure in L r, of the set of functions which are bounded on G) , then the operator (28) is completely continuous. In this case, the operator (32) transforms every ENweakly convergent sequence of functions in LM, into a sequence which converges with respect to the norm of the space LM• .
4. Let M(u) and N (v) be mutually complementary Nfunctions where N( u ) < u 2 < M (u ) . Suppose the square A 2 of a positive definite linear continuous operator A, which is selfadj oint in L 2 , is continuously extendible to an operator which acts from EN into LM. Then the operator A acts from L 2 into LM and is continuous. If, under the conditions of the theorem formulated above, the extension of the operator A 2 is a completely continuous operator, acting from EN into LM. then the operator A is also a completely continuous operator, acting from L 2 into LM. These assertions imply various conditions for the decomposition of a linear operator A, acting from EN into LM, into the product A = HH·, where H acts from L 2 into LM and H* is the operator adj oint to H and acting from EN into L 2 .
5.
We investigate the operator j : ju(x) = I[x , u (x) ] , where I(x, u) (x E G,  = < u < 00) is continuous in u and measurable in x for every u . Conditions are found under which the operator j, acting from some region of the space LM, into the space LM., is continuous and bounded. In distinction from the case of LIX spaces, the operator j can be defined on a sphere but not be defined in the entire space. It can be continuous at every point of some bounded closed region but not be bounded in this region itself. We shall formulate several propositions on the properties of the operator f. We denote the totality of functions u (x) E LM, for which the distance to EM. is less than r by lIr. Suppose the operator j acts from lIr into EM• . Then the operator j is continuous at every point
228
S UMMARY OF F U N DAME NTAL RESULTS
o f IIr. The set o f its values on the sphere lI ul l M, � rl < r is bounded with respect to the norm in LMs. It is convenient to express concrete conditions for the continuity and boundedness of the operator f in terms of estimates of the growth of the function f(x, u ) . If
I f(x, u) 1 � b(x)
+ aQ1 {M2 1 [Ml ( : ) J }
(X E G , 
00
<
u
<
)
( 33)
00 ,
where b (x) E EMs' Q (u) is an Nfunction and a, r > o. then the operator f which acts from IIr into EM. is continuous at all points of IIr and is bounded on every sphere I l u ll M, � r l < r. In the case when M2(U) satisfies the L1 2 condition, we can set Q(u) u in (33) . =
6.
Suppose there exist mutually complementary Nfunction s
R (u) and Q (u) such that for large values of the argument inequalities (24) are satisfied or, if M2 (U) satisfies the L1 'condition, the in equalities (25) are satisfied. Suppose there exists a derivative f�(x, u) which defines a continuous operator h : h u(x) f� [x, u(x) ] , which acts from the sphere Ilu l l M , � r into the space L�. Under these assumptions, the operator f is differentiable in the Frechet =
sense at every interior point of the indicated sphere, where its Frechet differential Bh at the point u(x) has the form
Bh(x)
=
f�[x, u(x)]h(x) .
The conditions for the differentiability of the operator f not in a region but at an isolated point are more delicate.
7.
The nonlinear integral operator
Kcp(x)
=
I k(x, y)f[y, cp (y)] dy
G
(34)
can be represented as the composition of a nonlinear operator f and a linear integral operator (32) . Combining the conditions under which the operator f acts from the space LM, into the space LM• and is continuous and bounded on a sphere with the con ditions under which operator (32) acts from LMs into L M, and is completely continuous, we arrive at the conditions for complete continuity of the operator (34) in the space LM, .
S U M MARY OF F U N D A M E N TAL R E S U L T S
229
Conditions for complete continuity in Orlicz spaces are also established for nonlinear integral operators of a more general form :
Krp(x)
=
f k[x, y, rp (y) ] dy .
G
(35)
These conditions enable us, for certain operators with essentially nonpower nonlinearities, to choose Orlicz spaces in which they are completely continuous. Suppose, for example, that
u) I � k(x, y) [a (x) R ( l u l ) (x, y G, u ) where k(x, y) EM, a(x) E LN, R(u) is a nonnegative continuous function, M(u) satisfies the LJ 2 condition (e.g. , it increases as e fl) . Suppose that R ( u ) � KM(u) for large values o f t h e argument . Ik(x, y,
+
E
00
E
<
<
00 ,
y
Under these assumptions, the operator (35) acts from some sphere in the space LM into LM and is completely continuous. A number of other properties of nonlinear integral operators are proved. 8. Knowledge of the spaces in which integral operators (35) possess "good" properties (i.e. they are continuous, completely continuous, differentiable, potentials, and so on) enables us to apply the general methods of nonlinear functional analysis t o the investigation of the equation
Arp(X)
=
f k[x, y, rp(y)] dy.
G
(36)
The application of these methods leads to various theorems on the existence of solutions and characteristic functions, to theorems on branch points and structure of the spectrum, and so on. A peculiarity of the theorems proved consists in this that the nonlinearities in the equations considered can be of an essentially nonpower character.
B I B L I OGRA P H I CAL NOTES § § 1 , 2 . The fundamental concepts of the theory of convex functions were established by Jensen (see J ENSEN [ I J ) . The reader can find detailed discussions of the elements of the theory of convex functions (in particular of Nfunctions) in HARDY, LITTLEWOOD and POLYA [ I J , POLYA and SZEGO [ I J , and ZYGMUND [ I J . The definition of an Nfunction which we have accepted coincides with the definition of an N'function in BIRNBAUM and ORLICZ [ 1 ] . § 3. BIRNBAUM and ORLICZ [ I J call the Nfunctions M l (U) and M2 (u) equivalent if aM1 (u) � M2(U) � bM1 (u) for large values of
the argument . Equivalence in this sense signifies that the N functions M l (U) and M2 (U) determine the same Orlicz class (see § 8) . Equivalence in our definition (see KRASNOSEL'SKII and RUTICKII [ I J ) means that the Nfunctions M1 (u) and M2 (U) determine Orlicz spaces consisting of the same functions.
§§ 4, 5. It appears that classes of Nfunctions satisfying the L1 2 condition and L1 'condition were first isolated in the paper by BIRNBAUM and ORLICZ [ 1 ] . Criteria for the fulfillment of these conditions did not interest them. The propositions pointed out in §§ 4, 5 were published earlier in KRASNOSEL'SKII and RUTICKII [6J , [7] . An assertion, close to Theorem 4.2, was formulated by S. M. LOZINSKII [ I J (under the additional assumption that N(v) satisfies the L1 2 condition) . (When S. M. Lozinskii became acquainted with the manuscript of the present book, he graciously notified the authors of a number of tests found by him for the fulfillment of the L1 2 condition ; these investigations, executed about ten years ago, have still not been published. ) We note that up to the present time, sufficiently useful necessary and sufficient criteria for the fulfillment of the L1 'condition have not been found. It would be interesting to find such criteria in terms of the complementary function.
23 1
B I B L IOGRAPHICAL NOTES
§ 6. In this section, with essential additions, are discussed propositions which were published earlier (see KRASNOSEL'SKII and
RUTICKII [3] , [6] , [9] ) .
We know suitable criteria that the Nfunction M (u) satisfy the L1 scondition, expressed in terms of the complementary function N(v) . It would be desirable to obtain such criteria in a form analogous to Theorem 6.8. Narrower classes of rapidly increasing Nfunctions can be isolated with the aid of a condition analogous to the L1 2 condition. We shall say, for example, that the Nfunction M(u) satisfies the L1 11lcondition, where
=
§ 7. The results of this section were discussed earlier in KRASNO SEL'SKI1 and RUTICKI1 [4J for a somewhat more general class of Nfunctions. § 8. The classes of functions under consideration were studied in detail in the paper by BIRNBAUM and ORLICZ [ I J (also see YOUNG [ I J ) . Orlicz classes and spaces were used in a number of works by various authors (see, for example, ZYGMUND [ I J , KALUGINA [ 1 3J , KIPRIYANOV [ 1 2J , KORENBLYUM [ I J , LOZINSKII [ I J , MEDVEDEV [ 1 ] ) basically in connection with problems of the theory o f functions (theory of series, singular integrals, approximation theory, and so on) . Orlicz classes were utilized in connection with the theory of partially ordered spaces by KANTOROVIC, VULIH and PINSKER [ 1 ] . We also note the connection of the works of the Japanese mathematicians NAKANO [ 1 ] , YAMAMURO [ 1 ] , AMEMIYA [ 1 ] , and others, on the theory of modulared spaces, with the theory of Orlicz classes and spaces.
L'M
§ 9. The spaces were considered by ORLICZ [ 1 ] . In this article, spaces were studied under the assumption that the Nfunction M(u) satisfies the L1 2 condition. A number of properties of spaces are shown, without this assumption, in ORLICZ [2] and in ZYGMUND [ I J . The norm, defined b y formula (9. 1 8) , was introduced by LUXEM:
LM
L'M
232
BIBLIOGRAPHI CAL NOTES
BURG [ 1 ] . Norms can b e introduced in modulared spaces by an analogous formula ( see NAKANO [ I J and YAMAMURO [ I J ) . We note that a number o f authors studied the Orlicz space of functions defined on sets of infinite measure and on sets whose measure is not continuous ( see, for example, ZYGMUND [ I J , LUXEM BURG [ I J , SOBOLEV [2J ) . In the particular case when the set consists of sequences of points each of which has a measure equal to unity the Orlicz space is transformed into the space of numerical sequen ces. This space is denoted by 1M ; it is a generalization of the spaces 1p . These spaces were studied by ORLICZ [2J and recently by the young Kazan mathematician Yu. I . Gribanov. DINCULEANU [ 1 2J consider ed Orlicz spaces of abstract functions. § 1 0. The space E M was introduced and studied by KRASNO SEL'SKII and RUTICKII [2, 5, 6] . Necessary and sufficient conditions for the separability of Orlicz spaces were shown in ORLICZ [2J . Theorems 1 0. 1 and 1 0.3 were proved in KRASNOSEL'SKII and RUTICKII [2, 6] . Subsection 7 was written j ointly by the authors
and N. G. S imko.
§ 1 1 . The generalization of A. M. Kolmogorov's theorem on the conditions for compactness of families of functions in L p spaces to the case of Orlicz spaces was carried out by TAKAHASHI [ I J under the assumption that the Nfunction M (u) satisfies the Ll 2condition. More accurately : Theorem 1 1 .2 is due to him. Theorem 1 1 .3 for the case of L p spaces is shown in KRASNOSEL'SKII 3] . Theorem 1 1 .4, which is a generalization of the analogous theorem of F. Riesz for L p spaces ( see RIESZ [ I J ) , was proved by GRIBANOV [ 1 ] . We note that an assertion close to Theorem 1 1 .4 can be obtained as the immediate consequence of a general compactness criterion proved by SILOV [2J . We consider the linear normed space consisting of complex valued functions defined on a commutative bicompact group H (with addition as the operation) . According to SILOV [ I J , the space is called a homogeneous function space if, together with each function f (t) E all its translates f (t h) also belong to the space where
[
R
R
R,
anq
R,
+ Il f (t h) 11 I If (t) II (f (t) E R, h E H) lim 1 1f( t h)  f(t) 1 I 0 (f (t) E R) . +
,h>9
=
+
=
233
BIBLIOGRAPHICAL NOTES
S ILOV'S THEOREM. The set m is compact in the homogeneous tunction space it, and only it, m is bounded in and the elements ot the set m are equicontinuous with respect to translation. By equicontinuity we understand the following property : for arbitrary e > 0 there exists a neighborhood of the zero of the group H such that implies that I I t ct h)  t(t) I I < e for arbitrary function tCt) m.
R
R
U
hEU E
+
The proof o f S ilov's theorem utilizes harmonic analysis on groups. I . P. Natanson noted that the proof in TAMARKI N [ I J of the independence of the first condition in A. N. Kolmogorov's theorem on the compactness of families of functions in Lp from the remaining conditions of this theorem was erroneous. As V. N. Sudakov recently showed, the condition of the boundedness of the norm of the set of functions considered in A. N. Kolmogorov's theorem and in its subsequent generalizations to L spaces (cf. TULAIKov [ I J ) and Orlicz spaces i s a consequence of the other conditions and can be omitted. § 12. The fact that the Haar functions form a basis in the space
Lid if M(u ) satisfies the L1 2 condition was shown by ORLICZ [ 1 ] .
I t is shown in the paper by SOLOMYAK [ I J that a basis of Haar functions in separable Orlicz spaces possesses the orthogonality property. This signifies that the inequality n+m
n
I I � C 'l!P'l (X) 11 M � I I � C 'l !Pi (X) 11 M is satisfied for the partial sums of the series ( 1 2.3 ) . The second proof of the necessity condition for the separability of Lid, presented in § 1 2, is adapted from KRASNOSEL'SKII and SOBOLEV [ 1 J . The passage to the space of functions defined on a closed interval, utilized in this proof, can also be used in the proof of many other propositions. Thus, in a number of sections, we could have limited ourselves to the consideration of Orlicz spaces of functions defined on a closed interval. The authors did not do this for the reason that the consideration of arbitrary sets with continuous measure does not give rise to any additional difficulties . For a proof of the fact that an arbitrary set of finite continuous measure can be mapped in a onetoone fashion onto a closed
G
BIBLIOGRAPHICAL NOTES
234
interval in such a way that under this mapping the measure of every subset remains invariant see ROHLIN [ 1 ] . § 13. Some of the results of this section were discussed earlier in KRASNOSEL'SKI! and RUTICKI! [ 1 , 4, 6J . § 14. The theorem on the general form of a linear functional on the space in the case when the Nfunction M(u) satisfies the L1 2 condition was proved in ORLlCZ [ 1 ] . Theorem 4. 1 is proved in ORLlCZ [2] . For the case when the condition is not satisfied, the theorem on the general form of a linear functional on was proved in KRASNOSEL'SKII and RUTICKII [5J (also see LUXEMBURG [ I J ) . The problem of the general form of a linear functional on for the case when M(u) does not satisfy the L1 2 condition remains open. The theorem on the connection between the norm of a linear functional and the Luxemburg norm, generated by the Luxemburg function, is proved in LUXEMBURG [ I J . The function k(v) is studied in SALEHOV [ 1 , 2] . In the paper by AMEMIYA [ 1 ] , in connection with the theory of modulared spaces, the relation between the Luxemburg norm and the norm defined by formula ( 1 0. 1 1 ) is studied. It is assumed that these norms differ by a constant factor and it is proved that in this case the spaces considered are Lp. In virtue of Theorem 1 0.5, formula ( 1 0. 1 1 ) defines the ordinary Orlicz norm, and the Luxem burg norm coincides with the norm of the linear functional generated by it (see subsection 5) . Therefore, D. V. Salehov's theorem, introduced on p. 1 26, also follows from Amemiya's results. The consideration of ordinary weak convergence is inconvenient since the general form of a linear functional on is unknown. In this connection, it turned out to be convenient to consider that weak convergence which arises if is assumed to be the space of linear functionals on
LM
EM
LM
LM
EN.
LM
§§ 15, 16. We shall point out some of the numerous papers in which linear integral operators are studied. The most detailed analysis of linear integral operators for the case of space of continuous functions was carried out by RADON [ 1 ] . The simplest theorems in the case of the spaces Lp are given in BANACH [ I J and RIEszSz . NAGY [ I J . Linear integral operators of a special form (the socalled potential
C
235
BIBLIOGRAPHICAL NOTES
type operators) are studied in the works by S. V . SOBOLEV and V . I . KONDRASOV (see S. SOBOLEV [ 1 ] ) . Strong results in the study of integral operators, acting in LP, were obtained by KANTOROVI<� [ 1 ] who generalized the results of S . L. Sobolev and V . I . Kondrasov. A number of theorems on linear integral operators, acting on Orlicz spaces, are proved in ZAANEN [ 1 ] . The papers by KRASNO sEL'sKII and RUTICKrI [2, 7] , the expansion of which constitutes the basic content of §§ 1 5 and 1 6, are devoted to the investigation of linear integral operators acting in Orlicz spaces. Theorem 1 6.3 was proved earlier by Zaanen under the assumption that all the Nfunctions under consideration satisfy the L1 2 con dition. It is natural to expect that new conditions for the continuity and complete continuity of linear integral operators acting from one Orlicz space into another can be obtained in the way proposed by KANTOROVIC [ 1 ] in the study of operators acting in Lp spaces. Under these new conditions, the restrictions on the kernel must , apparently, be formulated in the following way : there exist func tions
qy (x )
=
f Rl[k(x, y)] dy E
L �l
G
and
'q) (y)
=
f R2[k(x, y)] dx E
L!II • .
G
y),
Then the linear integral operator, defined by the kernel k(x, acts from LM, into LM • . In this connection, the functions Ml (U) and M2(U) must be connected with the functions
]
236
B I BLIOGRAPHI CAL NOTES
in this book, was adapted from KRASNOSEL'SKII and KREIN [ 1 ] . We note that i n Theorems 1 5.5 and 1 6.7 it is possible not t o assume that the operator is selfadj oint ; then the operator rather than the operator must satisfy the condition of the theorems (see KRAsNosEL'sKII and KREIN [ l J ) . We note additions t o embedding theorems follow from Theorem 1 6. 9 on potentialtype operators, according to a known scheme (see SOBOLEV [ l J and KANToRovl(� [ l J ) . Other additions to the embedding theorems of S. L. Sobolev, V. I. Kondrasev, and S. M . Nikol'skir with the utilization of the Orlicz metric are pointed out in KALUGINA [ 1 , 2J ) . Subsection 3 of section 1 6 is taken from RUTICKII [3] .
A2 A
AA *
§ 17. The Caratheodory conditions were first used in the con sideration of nonlinear operators in CARATHEODORY [ l J . Theorem 1 7 . 1 , which generalizes the theorem of N . N . Luzin on the Cproperty of a measurable function, was apparently first proved in KRAsNosEL'sKII and LADYZENSKII [ l J . The operator i for the case of various functional spaces (in particular, for the LP spaces) was studied by a number of authors (see, for example, NEMYCKII [5J , KRASNOSEL'SKII [5J , VAINBERG [2, 4J ) . For the case of Orlicz spaces fun damental propositions on the continuity and boundedness of the operator i were pointed out in R UTICKII [ 1 , 2J , KRAsNosEL'sKII and R UTICKII [BJ . These propositions are discussed in subsections 26 in a somewhat more general form. Theorem 7 . B is taken from RUTICKII [3] . § 18. The functional ( l B.6) was used by a number of authors (see HAMMERSTEIN [ l J , GOLOMB [ I J , KRASNOSEL'SKII [5J , VAIN BERG [ 1 , 4J , SOBOLEV [ l J , RUTICKI! [2J , and others) . Theorem I B. l , for the case of Lp spaces, was proved by A. I. Povolockir (see KRASNO SEL'SKII [2J ) . The functional ( l B.6) , defined on Orlicz spaces, was considered in RUTICKII [2J and later in KRAsNosEL'sKII and RUTICKII [BJ . General facts of the differential calculus in functional spaces are discussed in LYUSTERNIK and SOBOLEV [ l J , HILLE [ l J , VAIN BERG [ 1 ] . Some of the theorems on the differentiability of the operator i, acting in Orlicz spaces, were published earlier in KRASNOSEL'SKII and RUTICKII [3, B] . Of the papers, in which the differentiability of the operator i in Lp spaces is studied, we
B I BLIOGRAPHI CAL N OTES
237
mention the article by VAIN BERG [2] . Formulas for the gradients of the Orlicz norm and the Luxemburg norm (see KRASNOSEL'SKII and RUTICKII [ 1 0] ) generalize the Mazur formula for the gradient of the norm in Lp. § 19. The theorems proved in this section are a generalization of some propositions obtained in KRASNOSEL'SKII and L ADY zENsKII [ 1 ] (also see KRAsNosEL'sKII [5J ) . The analysis of Hammer stein operators acting in Orlicz spaces is based on the representation of these operators in the form K = Ai carried out in KRASNO SEL'SKII and RUTICKII [8] . The investigation of various nonlinear integral operators acting in other functional spaces was accomplished in many works (see, for instance, GOLOMB [ I J , SCORZA DRAGONI [ I J , DUBROVSKII [ 1 ] , VAINBERG [4J , KRAsNosEL'sKII [ 5J , NEMYCKII [ 1 , 2J , LADYZENSKII [ I J , and others) . § 20. The papers by NEMYCKII [2] , KRAsNosEL'sKII [5J , and LERAY and SCHAUDER [ I J are devoted to the general methods of nonlinear functional analysis. All propositions to which references are made in this section are proved in KRASNOSEL'SKII [5] . The theorem, discussed in subsection 2, on the existence of solutions for an equation with weak nonlinearities is based on reasonings due to V. M . Dubrovskii, which were applied by him in DUBROVSKII [ 1 ] in the study of other classes of integral equations. The theory of Banach spaces with a cone was developed basically by M. G. Krein (the fundamental propositions of this theory are discussed in KREIN and RUTMAN [ I J ) . The theory of cones is interrelated to a significant degree with the theory of partially ordered spaces (see KANTOROVH';, VULIH and PINSKER [ 1 ] ) . The general theorem, utilized i n subsection 3 , o n the continuous branch of positive characteristic functions is a special case of more general theorems on operators with monotonic minorants (see KRAsNosEL'sKII [5] ) . The application o f variational methods in the study of equations with Hammerstein operators has its beginning in the works of HAMMERSTEIN [ 1 ] and GOLOMB [ 1 ] . The scheme with the de composition of the operator A into the product HH* , discussed in subsection 4, was proposed by one of the authors. The application
238
BIBLIOGRAPHICAL NOTES
of this scheme with the use of L p spaces is discussed in detail in KRASNOSEL'SKII [5] . The indicated scheme is applied in the case of Orlicz spaces in RUTICKI! [2J and later in KRASNOSEL'SKII and RUTICKI! [8] . Remark. A number of new results, related to Orlicz spaces, were obtained recently (see, e.g. , ALBRYCHT [ I J , WEISS [ I J , VIDENSKII [ I J , GRIBANOV [ I J , DINCULEANU [ I J , KRASNOSEL'SKII [6J , MILNES [ I J , and S RAGIN [ I J ) . VIDENSKII [ I J discovered an interesting connection between the theory of Nfunctions with the theory of entire functions. In particular, he showed that N (v) ,...., In � exp [nv  M(n) J , where M(u) and N (v) are arbitrary mutually complementary Nfunctions. It is shown in KRAsNosEL'sKII [6J that the set of functions u (x) for which M l [U(X)J is summable forms a ring if, and only if, M(u) satisfies the J 2 condition.
L I TE RATURE (An asterisk, * , indicates that the paper cited came to the attention of the authors after the proofs of the book had already been read ; two aster isks, * * , indicate that the book or paper was added by the translator. ) AHIEZER, N . and GLAZ M A N , 1 . [ I J The theory of linear operators i n Hilbert space, GITTL, Moscow Leningrad, 1 950 (Russian) ; Ungar, New York, 1 96 1 ; Theorie der linearen Operatoren im Hilbertraum, AkademieVerlag, Berlin, 1 954 ; MR, Vol. 1 3, pp. 358359 and Zentralblatt, Vol. 4 1 , pp. 229230. J. * [ I J Some remarks on the MarcinkiewiczOrlicz space, Bull. Acad . Polon. Sci . , I, Cl. I I I , 4 ( 1 956) , 1 3 ; MR, Vol . 1 7, p . 953 and Zentralblatt, Vol . 7 1 , p. 1 09.
ALBRYCHT,
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[ 1 ] Vorlesungen uber reelle Funktionen, LeipzigBerlin, de Gruyter, 1 9 1 8 . N. [ 1 ] Espaces d'Orlicz de champs de vecteurs . I, Atti Accad . naz. Lincei, Rend . , Cl. sci. fis . mat. e natur., 22 ( 1 957) , 1 35 1 39 ; Zentralblatt, Vol. 77, pp. 1 02 1 03 and MR, Vol . 1 9, p. 566. [2] Espaces d'Orlicz de champs de vecteurs . Fonctionnelles lineaires continues . I I, Atti Accad . naz . Lincei, Rend . , Cl. sci. fis . mat. e natur., 2 2 ( 1 957) , 269275 ; Zentralblatt, Vol. 78, p . 1 02 and MR, Vol. 1 9, p . l 066. * * [3] Espaces d' Orlicz de champs de vecteurs . I II. OPerations lineaires, Studia Math . , 1'1 ( 1 958) . 285293 ; MR, Vol. 2 1 , p. 420.
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* * [4J Espaces d'Orlicz de champs de vecteurs . I V, Studia Math . , 1 9 ( 1 960) , 32 1 33 1 . * [5J SPajii Orlicz de cimpuri de vectori, Studii �i cercetari mat. Acad. RPR, 8, 34 ( 1 957) , 3434 1 2 ; Zentralblatt, Vol. 85, pp. 9596. D UBROVSKII, V. [ I J On some methods of nonlinear integral equations, Uc. zap. MG U, 30 ( 1 939) , 4960 (Russian) . FIHTENGOL'C, G. [IJ Course in differential and integral calculus, Vols. I, I I , I I I , Gostehiz dat, MoscowLeningrad, 1 949, 1 95 1 (Russian) ; Zentralblatt, Vol. 4 1 , pp. 378379 and Zentralblatt, Vol. 33, p . 1 07 and Zentralblatt, Vol. 34, p. 3 1 9. GOLOMB, M. [ I J Zur Theorie der nichtlinearen Integralgleichungen, Integralgleichungs systeme und allgemeinen Funktionalgleichungen, Math. Zeitschrift, 39 ( 1 934) , 4575 ; Zentralblatt, Vol. 9, pp. 3 1 23 1 3 . [2J Ober Systeme von nichtlinearen Integralgleichungen, Publ. Mathern. de l' Universite de Belgrade, 5 ( 1 936) , 5283 ; Zentralblatt, Vol. 1 7, p. 404. GRIBANOV, Yu. [ I J Nonlinear operators in Orlicz spaces, Uc. zap . Kazansk. unta, 1 15, 7 ( 1 955) (Russian) . * [2J On the theory of 1M spaces, Uc. zap. Kazansk. unta, 1 1 '1, 2 ( 1 957) (Russian) . HAAR, A. [IJ Zur Theorie der orthogonalen Funktionensysteme, Math . Annalen, 69 ( 1 9 1 0) , 33 1 37 1 . HAMMERSTEIN, A . [ I J Nichtlineare Integralgleichungen nebst A nwendungen, Acta Math. , 5 4 ( 1 930) , 1 1 7 1 76. HARDY, G . , LITTLEWOOD, J. and P6LYA, G. [1 J Inequalities, IlL, Moscow, 1 948 (Russian) ; Cambridge, Cambridge Univ. Press, 1 934 ; Zentralblatt, Vol. 1 0, pp. 1 07 1 08. H I LLE, E.
[ I J Functional analysis and semigroups, Colloquium Publ. AMS, New York, Vol . XXXI, 1 948 ; Moscow, IlL, 1 95 1 (Russian) ; MR, Vol. 9, pp. 594595. and PHILLIPS, S. * * [ I J Functional analysis and semigroups, Colloquium Publ. AMS, Providence, Vol. XXXI, 1 95 7 ; MR, Vol. 1 9, pp. 664665.
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LITERAT URE
247
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I N D EX 1 36, 1 46, 239 Albrycht 238, 239 Amemiya 23 1 , 234, 239
Absolute continuity of the norm Ahiezer
87
1 86 1 86 Glazman 1 36, 1 46, 239 Golomb 236, 237, 240
1 27, 1 34, 1 35, 234, 239 1 34 Basis 1 0 1 Birnbaum 230, 23 1 , 239 Branch 208  point 209, 2 1 6 239  conditions 1 67
Gradient of a functional 1 76  of the Luxemburg norm 1 87,
Gribanov Haar
208
1 37
Compact operator 1 94 Compactness criteria 94 Comparison of Nfunctions
 of Orlicz classes  of spaces
1 10
63
Complementary Nfunction
1 49
1 37
II
70 Composition of Nfunctions 1 0 Cone 2 1 3 Convex functions 1 , 2 1 7 Ll 2condition 23, 2 1 8 LI 'condition 29, 2 1 9 LI scondition 35, 220 LI Bcondition 40, 220 Completeness of Orlicz spaces
1 94
1 76, 1 8 1 Differentiable operator 1 79 Dinculeanu 232, 238, 239 Dubrovskii 237, 240
ENweakly continu ous linear function
1 33
226 Equivalence criterion 1 7 Equ ivalent Nfu nctions I S
71 1 89, 240 Frechet derivative 1 79 Fatou ' s theorem Fihtengol 'c
 , J ensen' s
I , Young's 1 3
Inequality, Holder ' s
Integral
Differentiability
als
1 03 236, 237, 240  operator 207 Hardy 230, 240 H ille 1 28, 236, 240 Holder's inequality 72, 223 Hammerstein
Homogeneous fu nction space
1 4, 2 1 8
Completely continuous operator
 convergent sequence
240
 functions
Complete continuity of linear integral operators
225
 of the Orlicz norm
88
 of continuous operators
1 76
1 89 232, 238, 240
 of the norm
 vectors 208 Class of completely continuous operators Characteristic values
97
Gateaux gradient
Banach ' s theorem
CaratModory
1 79
Frechet's theorem
 differentiable
Banach
Calculation of the norm
 differential
3
re p resentation
function J ensen
72
230, 240
J ensen ' s inequality
a
convex
1
Kalugina 23 1 , 236 , 24 1 Kantorovic 23 1 , 235 , 236,
23 1 , 24 1 Kolmogorov 24 1
Kipriyanov
of
232
23 7, 24 1
225
Kolmogorov's compactness criterion
97,
235 23 1 , 24 1 Krasnosel'skii 1 67, 1 70, 202, 2 1 2, 2 1 3 , 2 1 5, 230, 23 1 , 232, 233 , 234, 235 , 236 , 237, 238 , 24 1 , 242 Krein 235, 236, 237, 242, 243 Kondrasov
Korenblyum
1 67, 236, 237, 242, 243 237, 243 Levi 's theorem 69 Linear functionals 1 24 Littlewood 230 , 240 Lozinskii 230, 23 1 , 243 Luxemburg 78, 232, 234, 243  norm 78, 222 Lady�enskii
Leray
INDEX 187 1 77 Lyusternik 1 3 1 , 236, 243
1 26, 234, 245 245 Schauder 237, 243 Schauder's principle 209 Scorza Dragoni 2 1 2, 237, 245 Sep arability of EM 8 1 Set of complete measure 1 7 S ilov 245 Silov 's theorem 233 S imko 232 S kvorcov 245 Sobolev 1 3 1 , 232, 233, 235, 236, 242, 243, 245 Solomyak 233, 245 Sp ace EM 80  L iI 67 , Orlicz 69 S p ectrum 208
 , gradient of
Salehov Schaffer
Luzin's Cproperty
23 1 , 243 Milnes 238, 243 Mean convergence Medvedev
75
Monotonic operator Naimark 244 Nakano 23 1 ,
213
232, 244 6, 69, 7 1 , 94, 95, 244 Nemyckii 2 1 2, 236, 237, 243 Nfunctions 1 Nonlinear integral equations 1 94  operators 1 67 Norm, Luxemburg 78  of a functional 1 3 4 Natanson
 o f the characteristic function
O perator, compact 1 94 , completely continuous
207
72
operator 1 64  of a continuous op erator S p litting
1 94
, potential 2 1 4 , Uryson 1 94 O p erators, completely continuous
, Hammerstein
137  , differentiable 1 79  in Orlicz spaces 1 3 7 , nonlinear 1 67 Orlicz 230, 23 1 , 232, 234, 239, 244  classes 60  norm 67, 22 1  , gradient of 1 8 9   space 69, 22 1  , operators in 1 37 , continuous
15 240 237, 24 1 P6lya 230, 240, 244 Positive operator 2 1 3 Potential operator 2 1 4 Povolockii 236, 242
S ragin
a
completely
238, 246
Steklov functions
1 37
233 S zeg6 230, 244 S z .  N agy 234, 244
95
246 233, 246 Theorem, Fatou ' s 7 1  , Fnichet ' s 97 , Levi 's 69 Takahashi Tamarkin
 on branch points  , S ilov's 233 , Vallee Poussin's
1 55 Tulaikov 233, 246 , Zaanen ' s
Uryson 246  o p erator
Partially ordered set
216
94
1 94
1 8 6, 236, 237, 246 94 Videnskii 238, 246 Vulih 237, 24 1 Vainberg
Vallee Poussin ' s theorem
Principal p art of an Nfu nction 1 6 Product of functions i n Orlicz spaces
234, 244
of
Sudakov
Phillips Pinsker
Properties of Nfunctions
249
7
1 17
Weiss
238 , 246
23 1 , 232, 247
Young 247 You ng ' s inequality
Riesz 234, 244 Ries z ' s criterion for compactness
Yamamuro
Rohlin
235, 243, 247 1 55 Zygmund 230, 23 1 , 232, 247
Radon
99, 225 234, 244 Rutickii 230, 23 1 , 232, 234, 235, 236, 237, 238, 242, 244 Rutman 237, 243
Zaanen
Zaanen ' s theorem
13
continu ous
1 46