Convex Functions and Orlicz Spaces

<

f

Iff

provided mes � < A general criterion for the equi-absolute 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 N-function 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 equi-absolutely continuous integrals. Now, let M(u) and M l (U) be two N-functions for which . M(u) = 00. hm Ml (U) Then it follows from (1 1. 2 ) that the family of functions {Ml [U(X)]} has equi-absolutely 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 n-dimensional 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 equi-absolutely 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 equi-continuous. 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 n-dimensional 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 N-function M (u) satisfies the L1 2-condition, EM L M LM. Therefore, in the case when the L1 2-condition 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 N-function 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 equi-absolutely 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 equi-absolutely 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 equi-absolutely 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 equi-absolutely 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 equi-absolutely 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 one-to-one 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 one-to-one 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 N-functions. 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 Banach-Steinhaus 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 N-function M(u) satisfies the L1 2-condition 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 N-function 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 N-function M(u) does not satisfy the L1 2-condition ; 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 'YJN-1

(+)

<

!

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 N-functions

1 . Comparison of spaces. Generally speaking, distinct N­ functions determine distinct Orlicz spaces. For example, the spaces L £¥ determined by the N-functions 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 N-functions. 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 f-t > 0, f-tu(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/f-t} by Go . Then, in virtue of

G�Go

G.

:::;; M 2 (UO) mes Go

+

f Ml[,uU(x)] dx <

G

00.

This means that (f-t/k)u (x) E LM. from which it follows that

u (x) E LM •.

*

Let us recall that the N-functions 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 N-functions Ml (U) and M 2 (u) are equivalent. This theorem is the basis for the introduction of the concept of equivalent N-functions . 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 N-functions 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 N-functions 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 N-fundion Ml(U) increases essentially more rapidly than the N-function 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 N-fundions M(u) and Q(u) increases essentially more rapidly than the N-fundion 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 N-function M1 (u) increases essentially

more rapidly than the N-function 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 equi-absolutely 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 equi-absolutely 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 equi-absolutely continuous norms in Li.t. Then the N-function 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 N-function 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 equi-absolutely 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 N-functions 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 N-functions 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 equi-absolutely 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 equi-absolutely 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 N-function 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 equi-absolutely 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

N-functions 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 N-functions 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 a-condition 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 N-function M I (U) satisfies the LJ 2-condition and the N-function M 2 (U) the LJ 2-con­ 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 N-function which satisfies the Ll 2-condition increases more slowly than some ex­ ponential function. If the N-function M2 (U) satisfies the LI '-condition, then another sufficient condition can be given, as follows. THEOREM 1 3. 8. Suppose the N-function 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 N-functions 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 N-function 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 N-functions. 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 N-functions 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 N-func­ 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)N-1 (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 N-functions 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[N-l (y)] l l (y)J. N- (y) P[MM-l(y)N-l(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 N-function M(u) does not satisfy is not the general form of a linear the iJ 2-condition. 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 Radon-Nikodym 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 N-function M (u) satisfies the Ll 2-condition, then EM coincides with Lit = L M . In this case, ( 1 4. 1 ) yields the general form of a linear functional on Lit. 3. EN-weak convergence. We shall say that the sequence u n (x) E Lit (n = 1 , 2, . . . ) EN-weakly 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 N-functions M(u) and N (v) satisfy the Ll 2-condition. If we consider weak convergence in the space EN , then the definition introduced above coincides with the usual definition if the N-function N (v) satisfies the Ll 2-condition. In the general case, EN-weak 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 one-to-one

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 EN-weakly 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 EN-weakly convergent, then the norms l !u nl l M 1 , 2, . . . ) are uniformly bounded . THEOREM 1 4.4. Every space LM is EN-weakly complete in the sense that for every EN-weakly 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 EN-weakly compact, i . e . every bounded sequence contains an EN-weakly convergent subsequence .

We note that the EN-weak 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 EN-weakly 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 EN-weakly 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 EN-weakly 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 EN-weakly convergent and compact in the sense

of convergence in norm in Lid, then it also converges in norm.

The following criterion for EN-weak 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 EN-weakly convergent to u (x) . PROOF. In virtue of the fact that a sphere in Lid is EN-weakly compact, every sequence of functions in Lid which are bounded in norm contains an EN-weakly convergent subsequence. It is therefore sufficient in our case to show that for any subsequence u n .(x) which is EN-weakly 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 ) , equi-absolutely 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 EN-weak 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. EN-weakly continuous linear functionals. We shall say that the linear functional l(u) is EN-weakly continuous on Lit if for every sequence u n (x) E Lit (n 1 , 2, . . . ) which is EN-weakly 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 EN-weakly continuous. The converse assertion also holds. THEOREM 1 4.7 . Every EN-weakly 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 one-to-one 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 EN-weak 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 one-to-one 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 EN-weakly 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 one-to-one correspondence established between the functionals on EN and the functions in the space LM. It is obvious that the EN-weak 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 non-zero 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 N-functions M l (U) and M 2 (U) . THEOREM Let !P(u) be an N-function 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 N-function to the N-function !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 N-function to the N-function ( 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 N-function to the N-function ( 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 N-functions which satisfy the LJ '­ condition. In the investigation of linear integral operators, an important role is played by N-functions 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 N-functions possess this property. Clearly, the N-functi 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 N-function M (u) satisfies the LJ 2-condition . PROOF. Let us assume that the N-function M (u) does not satisfy the LJ 2-condition. 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 2m-1 > 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 N-function M(u) satisfy the Lt ' -condition . Proof of sulliciency. Suppose the N-function 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 2-condition. 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 N-function M (u) satisfies the ,1 2-condition. Therefore Lif = L M . Let us assume that the N-function 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 N-functions (even those satisfying the Ll 2-condition) . LEMMA 1 5.4 . Suppose the N-function 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 N-functions. 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 N-function 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 N-functions. 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 N-function 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 N-function whose complementary

function satisfies the L h-condition. 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 N-functions 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 N-functions N 1 (u ) = e1ul - l u i - 1 and M 2 (U) = eU ' - 1 ; the N-functions 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 self-adj oint linear operator acting in the space L 2 of functions which are square-summable 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 non-zero characteristic numbers Ai . We denote the scalar product of the functions e(x) and