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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
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
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 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 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
(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
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 ;
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) 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
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
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 ()
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.
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
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
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
(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
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 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
[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
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
�
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
�
�
�
•
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
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 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
[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
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