Some extensions of the Riesz-Thorin theorem to generalized Orlicz spaces U

ROCZNIK1 PO LSK IFG O TOWARZYSTWA M ATEMATYCZNEGO Séria I: PR ACE MATEMATYCZNE XXII (1980)

H. Hu d z i k, J. Mu s ie l a k and R. Ur b a n sk i (Poznan)

Some extensions of the Riesz-Thorin theorem to generalized Orlicz spaces Um( T )

Abstract. In this paper we extend the Riesz-Thorin theorem for linear operators to generalized Orlicz spaces L ^(T) of functions with values in C" and X , generated by an N-function M ( t , x ) with parameter t on C" and X, respectively. The results obtained by other authors (e.g. Rutickil [8] and Kraynek [6]) concern only symmetric spaces. The results presented here go further and concern also some non-symmetric spaces.

0. Introduction. Rn denotes the «-dimensional real Euclidean space, R + = [0, oo), T is an arbitrary abstract set or T is a subset of R l . Z x and Z 2 denote cr-fields of subsets of T containing the cr-field of all Borel subsets of T(in the second case). fix and fi2 are non-negative, atomless and complete measures on Z x and Z 2, respectively. For every measurable set A, we denote by Xa the indicator function of the set A, X is a separable Hilbert space, C denotes the space of all complex numbers, i — the imaginary unit in C. R \ and C" denote the «-tuple Cartesian product of R + and C, respectively. By we denote the complex space of all functions defined and ^-measurable on Г with values in X (with equality almost everywhere).

denotes the subspace of //£ of all simple functions from .Æ*x with supports of finite measure ju. If x(-): T -> C”, then we write x ( ) = (x,-(•))”=!

instead of x () = (xt (•),..., x„( )) and we denote by |x(t)| the function with i-th component |x,(0l, i = 1, •••,«• <•, ■> denotes the inner product in C and in X .

Definition 1.1. A function M(-, •): T x R + -► R+ is said to be a (p-function with parameter t if:

(a) there exists a set T0 of measure zero such that M (f, u) = 0 is equivalent to и = 0 for every t e T\T0,

(b) there exists a set T0 с: T of measure zero such that

M ^ o q u j+ a a U j) ^ a x M (t, ux) + a2 M (t, u2) for every u1, u 2, a 1,<x2 e R + , oq-fa, = 1 and for every t e T \ T 0,

(c) M( t, u) is a /^-measurable function of t for every fixed u gR +.

Definition 1.2. A (^-function M ( t, u ) with parameter t, which satisfies the conditions:

(d) there exists a set T0 с: T of measure zero such that M( t, и)

t oo as и Î oo and M( t, и)

J. 0 as и I 0 for every t e T \ T 0, is called an N -function with parameter t.

The following conditions for «^-functions with parameter will be used:

(e) T = [a ,b ], M ( t , x ) is continuous on T for every fixed x e X and a non-decreasing or non-increasing function of t for every fixed x g X,

(0 T = [a , oo) , M ( t , x ) is a continuous and non-increasing function of t for every fixed x e X such that there exists the limit M(x) = lim M ( t, x)

t-* oo

for every fixed x e X and M () is a ^-function without param eteri1).

(g) T = ( — oo, oo), M (t, x) is an even function of t for every fixed x e X and satisfies condition (f) with T = [0, oo) instead of T — [a, oo).

A 2 there exists a set Г0 с Г of measure zero, a constant К > 0 and a non-negative function /zeL^(T) such that the inequality

M ( t , 2 x ) ^ K M( t , x) + h(t) holds for every t e T \ T 0 and for every ^{x e} X .

Obviously, we may assume that the sets T0 in the above conditions are the same.

Example. Let T = R + and let (p (u) be a ^-function without parameter.

Let Pi (t) and p2 (t) be continuous functions on T such that 1 ^ Pi (t) ^ p2 (t)

< go for any t e T . Moreover, let Pi(t) be a non-increasing and p2{t) be a non-decreasing function of t. We define a (^-function M (f, и) with parameter t by the formula

1 [<P(M)]P2 f°r </>(«)> 1.

Obviously, M ( t , u ) is a continuous and non-decreasing function of t for every fixed и ^ 0.

Definition 1.3. By SOlft (sDîg) we denote the class of all «^-functions (АГ-functions) M ( t , u ) with parameter t, which satisfy conditions (a)-(c) ((a)-(d)) with respect to the measure p.

Now, we introduce a partial order “ -3” in the class of all (^-functions with parameter. We say that a ^-function M 2( t , x ) is weaker than a q>-

(x) For definition of «^-function with parameter on X, see [5].

function M l (t,x) if there exist a constant К > 0 a set T0 a T of measure zero and a non-negative function heL'ï('T) such that

M2(f,x) ^ M x (t, Kx) + h(t) holds for every t e T \ T 0 and for every u e R +.

If for two ^-functions M ^ ^ x ) and M 2(t,x) with parameter t there holds M 2 -3 and -3 M 2, simultaneously, then we call the (^-functions M i(t,x ) and M 2( t , x ) to be equivalent.

The condition M 2 -3 М г is sufficient and necessary in order that (T) <= L!u2(T) (for the definition of Lfo(T), see below). This proved J. Ishii in [3].

For every N-function M ( t , u ) with parameter t the complementary N-function M( t, u) , if defined by

(0.1) M (f, u) = sup {uv — M(t, v)}, V и ^ 0, V fe T \T 0.

v^O

It is easily seen that the following Young’s inequality

(0.2) uv ^ M{t, u) + M( t, v)

holds for every u , v e R + and for every te T \T 0.

Definition 1.4. By (90T1) we denote the class of all (^-functions (iV-functions) M( t, u) with parameter t of the form

m

M ( t , u ) = £ Mi{u)xTl(t), V O 0, /= l

where M l are </>-functions (N-functions) without parameter, Tt are pairwise disjoint and //-measurable sets in T for l = 1 ,..., m.

Definition 1.5. By SJR^ ($W[) we denote the class of all tp-functions M { t, x ) on R n+ with parameter t, of the form

M (t,x ) = £ M k( t , x k), V x = {xkyk=le R n+,

k= 1

where М кеШц (9Л") for every к = 1 ,...,n .

Definition 1.6. By (ÜDl£) we denote the class of all ^-functions M(t, x) on X with parameter t, of the form

M ( t , x ) = M l (t, ||x||), V x e l , V te T , where ||x|| is the norm of x and М 1е Щ 1 (90T*).

Definition 1.7. By iOh* (99^) we denote the class of all ^-functions (N-functions) M( t, u) from the class sJRg (901ft) satisfying conditions (e) or (0 or (g).

Definition 1.8. (9tRD denotes the class of all (^-functions M ( t , x ) on R \ with parameter t, of the form

M ( t , x ) = £ M k( t , x k), x = (xkTk = l e R n+, V t e T , ic= i

where the ^-functions (N-functions) М кеЩ (SJJÎ^).

Definition 1.9. A function M(-, •): Tx X -> R + is called a (p-function on X with parameter t e T if it satisfies conditions (a), (b), (c) and the condition

(h) there exists a set T0 с T of ^-measure zero such that M (f,x )

= M ( t , e lxx) for every t G T yT0 and for every т e R 1 and x e X .

Definition 1.10. (^ 5 ) denotes the class of all ^-functions M ( t , x) on X with parameter t, of the form

M (f,x ) = M j(f, ||x||), V re Г, V x e N , where M l ( t , u) belongs to the class (ЩО-

D^efinition 1.11. For every (^-function M ( t , x ) of the classes

and we define the complementary ^-function M ( t , x ) with respect to the variable x to the ^-function M ( t , x ) by

M ( t , x ) = sup{|<x, y } \ - M ( t , y ) : y e l } . In this definition X may denote also the space C".

D^efinition 1.12. For every pair of N-functions M , N eURft, we define т-intermediate N-functions M x{t,u) and M x(t, u) with parameter t between M and N by

(0.3) = {[M - 1 (t, u)]1~x\_N~1 {t, u)]r}-1 , V te T \T 0, Vm ^ 0 and MT(f,u) as the complementary <p-function to

(0.4) M x(t,u) = Vte Г \Т 0, Vu ^ 0,

where the inverse functions are taken with respect to the variable u for every fixed t e T \ T 0. Moreover, the first definition of т-intermediate function between M and N is meaningful also for M , N e . The fact that Mr and M x also belong to follows from the results of [8], because M ( t , u ) and N (r,u ) are N-functions without parameter for every fixed t.

Analogously, for M , N e sJQ^ (9JΣ) we define i-intermediate N-functions between M and N by the formula

(0.5) M x(t,x) = £ M kx( t , x k), M x( t , x ) = £ M kx( t , x k),

k = l fc=1

where x k is the k-th component of x, M kx and M kx are т-intermediate N-functions with parameter between M k and N k appearing in the definition of the N-functions M and N.

If M , N belong to the class 99?^ and the sets Tt in the definition of these ЛГ-functions with parameter are the same, then we have

m m

(0.6) M z(t,u) = £ Л?1т(м)Хт,(0» M x( t , u ) = £ M,t (u)xr (f),

/=1 j=i

where M lz and M lt are t-intermediate ЛГ-functions without parameter be­

tween the ЛГ-functions M, and N t appearing in the definition of the ЛГ-functions M and N, l = 1 , m.

If М,ЛГе9И^ (90^) and the sets Tt appearing in the definition of the

<p-functions M and ЛГ coincide (in the case 5CR^), then we define т-interme­

diate (p-functions between M and N by

(0.7) M z(t, x ) = M u (t, ||x||), M z(t, x) = M u (t, ||x||) in the case of the class 99IÇ, and

m m

(0.8) M x(t,x) = ^ M lt{\\x\\)xTl(t), M z(t,x) = X MlT(l|x||)*r (0

1 = 1 /=i

for every M , N belonging to the class 99î^, where M lz and M lz are т-inter­

mediate ЛГ-functions between M, and N t appearing in the definition of the ЛГ-functions M and N.

For every pair of N -functions M, ЛГ with parameter the N -functions Mz and M z are equivalent, and the inequality

(0.9) J®t M - ) ^ M z(t, •) ^ Ü z(t, 2-), V t Gr \T 0

holds. This follows applying the case of <p-functions without parameter (see [8]).

We define on Jt^n and ,<#£ a convex modular by Q^ix) = ( M ( t, \x{t)\)dg

T

if x eJ ^ c n and M ( t , x ) is a (p-function on Rn+ with parameter t, and eM *) = GfojflMI) = J M i ( r ’ ll*(0ll)^

T

if x e J J and M ( t , x ) is a <p-function from the class <W2 or

Now, we define the generalized complex Orlicz space generated by the Ф-function M ( t , x ) with parameter t, by

Lm (T) = {x eM ^ . ЗА > 0 ,е&(Ях) < oo}, where X may denote also C". The functional || • defined by

||x||M,„ = inf {e > 0: gfaie'1 x) ^ 1}, VxeLMM(T)

is a finite norm on Lfo(T), if M ( t , x ) is an arbitrary <p-function with parameter t. This functional is called the Luxemburg-norm.

If M ( t , x ) and M ( t , x ) are complementary N -functions on X with parameter t (as X we understand here also C"), then we may define the Orlicz-norm

IMIm = SUP {| f <x(t),y(t)}dp\: Qü(ÿ) ^ 1}.

T

If M belongs to the classes ЯЙ*1, or if M belongs to the classes

$№(, StJÎÇ and satisfies conditions (e) or (f) or (g), then the Luxemburg- norm and Orlicz-norm are equivalent and the following inequalities

(0.10) 1М1м,д ^ 1М1м,д ^ 2 ||x||M<|i

hold for every x e L ^ ( T ) (see [1] and [5]). Moreover, for all pairs of complementary N -functions considered in this paper there holds the following generalized Holder’s inequality

I J ( x ( t ) , y { t ) } d n I ^ J \(x(t),y(t))\dp ^ ||у||м,д

T T

for every x e Lm(T), у e Lm(T).

Definition 1.13. For every (^-function M with parameter such that cz Lfo(T) we denote by E ^ ( T ) the closure of  £ in L ^(T ) with respect to the norm || • \\Мф. Since Lft,(T) is a Banach space (see [4]), so E*M { T )

is a closed subspace of Lft,(T). Moreover, if a (^-function M with parameter satisfies condition A2, then E % f ( T ) = ^{L h ( T )} (see also [4]).

For every pair of complex Banach spaces X , Y and for any linear operation T the symbol “T: X -> У” denotes henceforth that T is a continuous map from X into Y.

1. Results. First, we shall prove some lemmas.

Lemma 1.1. I f Meffl*1 and ц{Т) < со or if M e l ' 1, n(T) ^— oo and for all sets Tx appearing in the definition of M, p(Tt) = oo, then there exists a (p-function Ф without parameter equivalent to M if and only if all (p-functions Мг appearing in the definition of M are pairwise equivalent, i.e. the class of all (p-functions without parameter is a proper subclass of

P ro o f. Let us suppose that there exists a ^-function Ф without parameter such that M ~ Ф, i.e.

Ф(к2 и) — h2 (t) ^ M ( t, и) ^ Ф(к1 u) + h1 (f), V и ^ 0, V t e T \ r 0 for positive constants k x, k 2 and for non-negative functions hx, h2 e E X(T).

Let t e T t. Then

Ф(/с2 w) —h2 (0 < М,(м) ^ Ф(к1и) + к 1^), Vu ^ 0, V t e T \ T 0.

Hence, we have

Ф(к2 и)— inf h2(t) ^ M. (и) ^ Ф(кх u)+ inf hx (t), Vu ^ 0.

teT i teT i

Let us denote

ot2 i = inf h2 (£), dq, = inf hi (t).

t e T i ’ t e T i

If /i(7]) = oo, then ot2J = a u = 0 (since hr, h2 € Ei(T)). Hence Ф(к2и) ^ М](и) ^ Ф(к1и)

for every u e R + if /i(7]) = oo.

Let /i(7]) < oo. It is obvious that а и < oo and a2>i < oo. Thus, there exist positive numbers tq and u2 such that а ы ^ Ф(к1 Ui) and а2>/ ^ Ф(к2и).

So for every и ^ max (ux, u2), we have

M {(u) ^ Ф(/с1и) + а1>/ ^ 2Ф(к1и) and

Ф(к2 и)-ос2Л ^ Ф(к2и)-%Ф(к2и) = \Ф{к2и) ^ Ф(^/с2и).

Hence, we obtain

Ф(?к2и) ^ МДа) ^ 2Ф(к1 и), Vu ^ max (tq, iq).

These inequalities finish the proof.

Lemma 1.2. / / M l5 ..., M„ are N -functions on R+ with parameter and if M ( t , x ) is a cp-function on R"+ with parameter t of the form

П

M (t, x) = £ M k( t , x k), Vx gRb+, V f e 7 \ k= 1

f/ien £/ie complementary N -function to M is of the form

M (t, x) = X M k( t , x k), V x e R + , V ie T\T0, k= i

where M k are complementary N-functions to M k, к = l , . . . , n .

P ro o f. We may extend the N -function M ( t , x ) from JR+ to C" by M(t, x) = £ Mk(f,|x k|), V x g C", V£gT.

k= 1 From Definition 1.11, we have M(t, x)

= sup { |< x ,y > |-M (£ ,y ): y e C n}

n

= sup (|<x, y > |- M ( t, y): у е Я + l = sup { £ [xkyk- M k(£, yk)]: у е Г +j fc= 1

£1 П

^ Z SUP (xfcyk- M k(£,yk): y e R \ } = £ M k( t , x k)

k= 1 fc=l

4 - Prace Matematyczne 22.1

for every xeR"+. Conversely, taking arbitrary s > 0, we conclude from the definition of M k( t , x k) that there exists y = (yk)2 = ie£ + such that

M k{t, x k) ^ xkyk- M k(t, yk)+ — , V к = 1 , и, V t e T\T0.

n Hence, we obtain

и n

£ M k( t , x k) ^ £ {*kyk- M k( t,y k)} + £ = \ < X , y } \ - M ( t , y ) + £

fc= 1 fc=l

^ M (t, x) + e, and the proof is finished.

Le m m a 1.3. I f X is a complex Hilbert space, M 1( t, u) belongs to ffîft and M ( t , x ) is a N-function on X with parameter t of the form

M ( t , x ) = M x(t, ||x||), V r e Г, V x e X (2),

then the complementary N-function M ( t , x ) to the (p-function M ( t , x ) with respect to x is of the form

M ( t , x ) = M 1(t, ||x||), V t e T \T0, V x e X . P ro o f. From Definitions 1.11 and (0.1), we have

. M ( t , x ) = sup (|<x, y>| —M (f, y): y e X } - and

M i(t, ||x||) = sup {||x|| v - M ^ t , v): v eR + } .

From the inequality |<x,y>| ^ ||x|| ||y|| it follows immediately that M(t, x) ^ M t (t, (Ixj|), V t6 T \T 0, V x e X .

In order to prove the opposite inequality it suffices to show that for every pair of elements x and y belonging to X there exists an element y x e X such that ||y|| = HyJ and ||x|| ||y|| = K x,yi> |. But for this purpose it suffices to put y 1 = x, and the proof is finished.

R e m a rk 1.4. If M (t , u) belongs to 9Л", i.e. M( t, u) = £ M ,(u )/Tt(f),

m 1 = 1

where T{ n Tj = 0 for every i Ф j and (J Tx — T, then the complementary z= 1

N-function M ( t, u ) with parameter t also belongs to 90^ and M ( t , u )

m

= £ MiiujXrSt).

1= 1

(2) This lemma holds also for a Banach space X.

R e m a rk s 1.5. Replacing R + by C", there holds lemma analogous to Lemma 1.2. This follows from Lemmas 1.2 and 1.3.

Le m m a 1.6. Let the (p-functions M ( t , u) and N ( t , u) satisfy condition A2.

Then for every 0 ^ т ^ 1 the x-intermediate (p-functions M x(t, и) and M x(t, и) between (p-functions M and N also satisfy condition A2.

P ro o f. We shall prove only that A?t satisfies condition A2. Let us suppose that the ^-functions M ( t , u ) and N ( t , u ) with parameter t satisfy condition A2 with constants K X, K 2 and non-negative functions ht , h 2, respectively.

Then putting К = max ( K lt K 2) and h(t) = max (hl (t), h2(t)) for every t e T we get that both these (^-functions with parameter satisfy condition A2 with the same constant К and function h, i.e.

M(t,2u) ^ K M ( t , u)+h(t), N(t, 2u) ^ K N ( t , u) + h(t)

for almost every t e T and for every и ^ h (t). These conditions are equivalent to

h M ~ 1{t,u) ^ M ~ 1 ^t, (u — h{t))j,

( u - h ( t) ) j

for almost every t e T and for every u ^ h ( t ) . Then, in order to prove that the т-intermediate (^-function M x(t,u) satisfies condition A2, it suffices to show that

%N~l (t, u) ^ N ~ l t, 1 К

(u-h(t))J

for almost every t e T and for every и ^ 0, where constant К and the function h are as above. By (1.1), we have

N ~ 1( ^ t , ^ ( u - h ( t ) ) Sj

Thus, the proof is finished.

R e m a rk 1.7. Lemma 1.6 holds also for M , N from any class of ^-functions which are considered in this paper.

Lemma 1.8. I f M -3N , where M , N еУЛ% and 0 ^ ^ t2 ^ 1, then Mti -3 M X2.

P roof. We shall prove that there exist a positive constant K, a set

< M l \ t , - u - h { t )

= M x

^ M f 1 t, к

T0 с Т of measure zero and a non-negative function h e E ^{ T ) such that (1.2) M Xl{t,u) ^ M X2(t,Ku) + h{t), V t e T\T0, V o O .

The last inequality is equivalent to (1.3)

M X2l (t, u)

M " 1 (t, u + h{t)) \ f t e T \ T 0, V u ^ O .

So, in order to prove that Mtl -3 Mt2 it suffices to show (1.3). From the condition M -3 N follows the existence of a constant К > 0 and a non­

negative function h e L \ ( T ) such that

(1.4) N 1(t,u)

M -1 (t, u + h(t)) V £e T\T0, V u ^ O . We have

M - 1^ ) _ [М - 1 (£, ц)]1 ~T2 [N - 1 (f, u)]t2 M " 1 (t, u + h(t)) [M -1 (t, u + /i(£))]1_T1 [ N -1 (t, u + /i(t))]T1

<

Thus (1.3) holds with K X2~X1 instead of K, and the proof is finished.

Co r o l l a r y 1.9. I f M, N M -3 N and 0 < т < 1, then M -3 Mx -3 N.

P ro o f. It suffices to put x1 — 0, x2 = x in Lemma 1.8.

Co r o l l a r y 1.10. Lemma 1.8 holds for every pair of (p-functions M, N, M -3 N from any (but the same) class of (p-functions, which are defined in Section 0.

Le m m a 1.11. Let M k( t , x ) be a sequence of (p-functions on an arbitrary Banach space X with parameter t such that the sequence of numbers M k (t , x) is non-decreasing for every x e X and for almost every t e T . Let, moreover,

[M ~1 (r, и + h (£))]1 -'2 I N ~1 (t, и)У2 [ M - l (t, u + h(t))Y~^ [T V 1 (t, u)]ri

N 1(t,u) M ' 1 (t, u + h(t))

T2 ~T1

< K'2-Ч

M ( t , x ) = lim M k( t , x ) < oo

k-* oo

for every x e X and for almost every t e T . Then M ( t , x ) is a (p-function on X with parameter t and for every we have

1М1мк.Л as k Î oo.

P roof. It is obvious that M ( t , x ) is a «^-function on X with parameter t.

Since M k(t, x)] M (t, x) as fcf oo for any x e X and for almost every t e T , so M k (t, y (£)) î M (t, y (£)) as /с Î oo for almost every t e T , and hence

в^{^}мЛу⁾ = J M k(t,y{t) )dnî J M(t,y(t))dfi = e^{$ ,}(y) as /с î 00.

T T

Since the sequence {efok Су)}*°= i is non-decreasing, i.e. ohk{y) ^ Q4iitk + l (y) for every уъМ' х, к — 1 ,2 ,..., so we have

QÏik(yM < QMk+l(y/u) < £&(у/м) for every u > 0. Hence, we have

{u > 0: QÏtk{у/и) < 1} => {и > 0: 6 ^ fc+1Cy/«) ^ 1} = {и > 0: &(у/ и) < 1}, and thus

1М1мк,р ^ 1М1л#к+1,д ^ 1М1м,„-

Thus, the sequence {1М1мк>/Лк°=1 is non-decreasing and bounded by ЦуЦм.д, so convergent. Denoting

9 = li m \\у\\мкФ,

k-> oo

we have g ^ ||у||м,^- Let us suppose for a contrary that g < v ^ ||>’||м,д- Then v > \\у\\МкФ for every /с = 1 ,2 ,..., so

6%ik{yM ^ 1, к — 1 ,2 ,...

Hence, taking oo , we obtain gfaiy/v) < 1, so ||y||Mj/1 ^ v, a contra­

diction. Thus, g = \\у\\м,ц.

Le m m a 1.12. Let M ( t , x) be a (p-function on an arbitrary Banach space X with parameter t and let condition (e) hold. Then there exists a sequence M ke $ (with X instead of R + ) such that M k(t, x ) ] M ( t , x) as k] oo for every x e X and for almost every t e T, and

(L5) 1М1мк>Л Ы1м,д as к î oo

for every y e I ?M(T).

P ro o f. We assume that M ( t , x ) is a non-decreasing function of t for every x e X . Let us denote

M k{ t , x ) = м ( ^ а + - ^ - ( Ь ~ а ) , х ^ for a + - ^ ~ ( b - a )

< t < а + ^ ф - а ) , i = l , . . . , 2 k and Mk(b,x) = M(b, x).

For a non-increasing «p-function M ( t , x ) of parameter t for every x e X , we put

i — 1 i

for a + -k— (b - a) ^ t < а + ^ ф - а ) , i = 1,...,2*

and M k(b,x) = M{b,x) . From the definition of (e) and from the definition of M k it follows that M k{t,x) is a non-decreasing sequence and M k(t, x )|M (r, x) as /сI oo for every x e X and for almost every t e T . Moreover, it is obvious that М кеШ^ (with X instead of R +) for к = 1, 2 ,... Condition (1.5) follows from Lemma 1.11.

Le m m a 1.13. I f M e ® § (with an arbitrary Banach space X instead of R +) and satisfies condition (f) or (g), then there exists a sequence М кеШц (with X instead of R +) for k = 1 ,2 ,... such that M k( t , x ) ] M( t, x) as к Î oo for every x e X and for almost every t e T and

(1-6) Ы1мк,д î as к Î oo

for every yeLfM(T).

P ro o f. 1° Let M satisfy condition (f). Then there exists a sequence bkî °°, bk > a, such that 0 ^ M ( t , x ) — M(x) < 1/k for t > bk, x e X ,

bk+j — a

k = 1 ,2 ,... Moreover, we may assume that —---= ck are integers bk a

for k = 1 ,2 ,... We divide the interval [a, bk] in 2k subintervals in the same way as Lemma 1.12. These partial intervals we denote by Tljk, T2fjk,..., Moreover, we denote Tmk<k = [bk, oo). We define

Mk(f, x ) = min M (t, x) if t e Tik, i = 1 ,..., m k —1,

^Ti,k

Mk(t,x ) = M(x) if t e T mbk.

It is obvious that МкеЩ1д for к = 1 ,2 ,... and that M k( t , x ) < M k+l(t,x)

^ M ( f , x ) for every t e T ^ x e X , k = 1 ,2 ,... Moreover, by condition (f), M k(t, x ) t M (t, x) as k ] oo for every x e X and for almost every t e T .

2° Let M satisfy condition (g). Then, we apply case 1° for a = 0 and we define M k( — t , x ) = M k(t,x) for every x e X and for almost every t e [ 0, oo). Condition (1.6) follows from Lemma 1.11.

Le m m a 1.14. Let condition (e) or (f) or (g) be satisfied for (p-functions (р е Ш^1, M efflft2 (with X instead of R + ) and let a function x e be fixed. Let P be a linear operation from L ^(T ) into L ^ ( T ) such that

Let e > 0 be arbitrary positive number. Then for sufficiently large к de­

pending on x and £, we have

WP x h k,n2 ^ (ï+ e J K i И1мк,Д1,

where cpk and M k are <p-functions with parameter defined as in Lemmas 1.12 and 1.13.

P roof. We may assume that x ( t ) is not equal zero for almost every t e T . From Lemma 1.11, we have

\\Px\\w < I|F*IU>/12 ^ K i

But ||x||Mfc;Mlî ||х||м>Д1, so there exists kXyB such that ||х||м>Д1 ^ (l + e)x

X М1мк>М1 for k > k x, E- Thus

\\Рх \\(рк,и2 ^ (1 + е)1М1мк,Д1 for к ^ kx<E.

Interpolation theorems. The case of functions with values in X.

Th e o r e m 1.15. I f M, N M -3 N, (р,ф еУЯ22 and a linear operation P satisfies the conditions:

P: E Ï Î ( T ) ^ K 2(T), P: EÜ*(T )-> E j2(T) and

\\Px \\<p,H2 ^ K l WPxh.,2 ^ K 2 ||х||№>Д1

for х е £ м ( Т ) and x e E ^ i T ) , respectively, then P : EfâfiT) E~2 (T) and (1-7) ||Лс|и,,М2 < 4 К Г ТК Г2 И1м,Д1

for every xeEgjfiT), 0 ^ т ^ 1, where фх amd M x are the x-intermediate N -functions between M, N and (p, ф, respectively.

P roof. Let the N -functions determining the IV-functions M, N, cp, ф be M l5 IVi еЯИ'"1 and (р1, ф 1е ШЦ2, respectively. We define for every z e C with 0 ^ Re z ^ 1 the functions

(1.8) K , { t , u ) =

(1.9) Q J t , u ) = t < p ; 4 t , u ) y - * l f ; 4 t , u ) Y .

Let x e l j 1, y e J ( ? be such that IW U ,,^ = 1Ы1г,,„2 = 1- Further, we define the following functions for every z e C with 0 ^ Re z ^ 1 and for every 0 ^ t ^ 1 :

and

We have

r K 2[ t , M u (t, 1И0Ц)]

0z(O = < \\x(t)\\ X[) lo

if x(t) Ф 0, if x(f) = 0

K(v)

Qz[v><Pu(v, II? Mil)] , s

\\y(v)\\ nV)

0

iïy{v) ф 0, if y (у) = 0.

p A

x{t) = X ^aiXAi(t), y(ü) = £ ^bjXBjiv)

i=l j = 1

for some at, bj e X , At e , B} e I 2. Hence, we have 9z(t)

д x z I k l l f l

À I N I

M «) = X j= i

Q zfo y n fo » llbjll)]

IIM bjXBj(v),

in case if at = 0 or bj = 0, we take here zero as the respective term. Further, by the form of the iV-functions M lt N l t (pl , ÿ 1, we have

(1.10)

(1.11)

^ £ £ K hz [M 1#,T(M )1 _

в М ||a,.|| ' a<XAB T*(t)’

ш = х X 6|,㹄 Т г — ] » л ь а г,о> (3)- й=1 j=l llbjl

We define the function

/( * ) = J <{Pgz)(v), hz (v))dfi2

T

for every z from the strip 0 ^ Re z ^ 1. By (1.10) and (1.11), we have К н Л м ш ( м т о к21 Ф ш ( щ т

M ЩИ

X J <[F(«i Xa^tJ] («0, bj x B j n T h(»)> dii2.

m p k

/ 00 = X X X Ji= 1 i= 1 j = 1

Since the functions а,-;Цпгя belong to Е*м(Т), so the functions F ^ x ^ n r * ) belong to £jj2 (T). Moreover, we have Ь}Хв}птк б (T). Applying the Holder’s inequality with iV-functions q> and ф, we obtain that all integrals in the definition of / (z) are finite. Thus, / (z) is an analytic and bounded function in the strip 0 ^ Re z ^ 1. Now, we shall prove the conditions

(1.12) I / (z)| ^ 2 K X for Re z '= 0 and |/(z)| ^ 1K2 for Re z = 1.

1° Let Re z = 0. Then we have

(1.13) ||0z(t)ll = M ï l [t, M u (t, ||x(t)||)], \\hz {v)\\ = ( p îl [v, (pu {v, lly(u)ll)].

Thus, by Lemma 1.3, we have

(1.14) eU(e.) = eîijOI&ll) = t f l t (llx||) = eÿ'(x), (1.15) e; 2(K) = <?S*(||fcI||) = # , ( ||> ||) = e%(y),

II^

II

, ^ Il Il Æf T,^ j f ?

11^г11ф,Д2 Н.У 11<рт,Д2 = !•

(3) For simplicity of the proof we assume that the sets Th appearing in the definition of N-functions M j, N lf <p1; \p1 are the same.

By (1.14), (1.15) and by Holder’s inequality, we obtain the first part of (1.12).

2° Let Re z = 1. Then we have

\Kz (t,u)\ = N ^ i t , u), \Qz (t,u)\ = 1 (t, m).

Hence, we get

Il0z(f)ll =

||x(OII)),

IlkMIl))-

Thus, by Lemma 1.3, we have

(1-16) t f y = \\gz\\N,H = N liW ^ = 1 and

(1.17) Q $ 2 ( h t ) = Q-/x { y ) , ||M *.„2 ⁼ | | y | | ^ 2 ⁼ 1.

By (1.16), (1.17) and by Holder’s inequality, we obtain the second part of (1.12). Further, by Hadamard’s three lines theorem, we get

|/(z )| ^ 2K{~RezK2ez for every z e C with 0 ^ Re z < 1.

Moreover, putting z = т in the definitions of the functions gz and hz, we obtain

gx{t) = x(t), hz(v), V O ^ r ^ l . Hence we have

| / (t)| ^ 2 K \ ~ Z K z2 for every 0 ^ т ^ 1.

Hence, taking into account that ||х||&т#| = 1 and applying inequality (0.9), we obtain

l|P*llit,M2 = sup {| f ((Px){v),y(v)}dfi21: \\y\\~T>ll2 ^ 1}

. T

= 2 sup {I j <{Px)(v), y(v))dfi2\: \\у\\ф „2 = 1} < 4K \ ~ z K \ .

T

Hence, by the left-hand side inequality of (0.10), we obtain

\\Рх\\ф1Ф2 < 4K \ ~zK z2 for every ||х||мт>Д1 = 1, x e . # * 1. Hence it follows (1-18) l |P x ||^ 2 ^ 4 K \ ~ ZK z2 Цх||мт>Д1

for every x é J j 1 and 0 ^ т < 1, immediately. Now, we extend inequality (1.18) to Emx{T). Since the space is dense in E ÿ x(T), where 0 ^ т ^ 1 so the operator P may be extended to the whole space £ ^ T(T). We denote by P this extension. Then we have

d-19) \\ Рхих,Ц2^ 4 К \ ^ К \ \ \ х \ \ м х^

for every х е Е ^ ( Г ) . Thus, it suffices to prove that the operators P and P coincide on Estz(T). It is known (see [2]) that if p(T) < oo and if a sequence {xk}k= i is convergent to a function x with respect to the norm |H |M>#J, where M is an arbitrary (p-function with parameter, then {xk}®= t is convergent to x with respect to the measure p. Let x eE ^x{T). Then there exists a sequence xke for k — 1 , 2 , . . . such that

-►0 as к -> cc.

Since from the condition M -3 N follows that M -3 M x -3 N , so E^1 (T) c Émx{T) с= E*m(T), and every sequence which is convergent with respect to the norm || • ||я is also convergent with respect to the norm || • ||М,Д1- Thus, the sequence {P(x — xk)}*=1 is convergent with respect to the norm

|| • ||<p,^2-Hence, the sequence {P(x — х к)х л }к = 1 1S convergent to zero with respect to the measure p2 on every set A <= T of finite measure /t2(4).

Moreover, by (1.19), the operator P also maps the sequence {xk}£°=1 onto the sequence {Pxk}£°=1, which is also convergent to Px with respect to the measure p2 on every set A <= T, p 2(A) < oo. Since the measure ц2 is fj-finite, the operators P and P coincide on the dense subset M'fif in £mt(T) and since the limit of a convergent sequence with respect to the measure p2 on sets of finite measure p2 is unique, so P and P coincide on the whole space E ^ ( T ) , and the proof is finished.

Theorem 1.16. I f M , N еШ^1 , (р,ф еШц52 and if all other assumptions of Theorem 1.15 are satisfied, then P: Egfx( T ) - +E~2 (T) and there holds ine­

quality (1.7) for every x eE ^x(T), 0 < т < 1.

Proof. This theorem follows from Theorem 1.15 and from Lemmas 1.1 and 1.13, immediately (5),

Theorem 1.17. I f in Theorems 1.15 and 1.16 all assumptions with the exception of M -3 N are satisfied, then these theorems are true with the extension operator P (see the proof of Theorem 1.15) instead of the oper­

ator P.

Interpolation theorems. The case of functions with values in C".

Theorem K18. Let M, N gSRj1, q>, ф еШ\2 and P: Е*м (T) -*■ £^2(T), P: E^1 (T) -> Еф2 (T), simultaneously. Moreover, let

1|Лс||,.,2 < K 1 IMIa u j, V x e Е*м (T),

||P x ||^ 2 ^ K 2 ||x||WfM1, V x e Ен1 (T).

Then P: E$ix(T) - E^2 (T) and

(4) See [2].

(5) We apply here also the fact that the conditions M k(t, x) | M( t , x) and N k(t, x) Î N( t , x) for every x e X and for almost every re Г imply that Mtr(r, x ) | Mt (r, x) for every 0 ^ t ^ 1, x e l and almost every r e T.

(1.201 ЦЛс||ф,,„2 ^ 4 / С Г ’ К У х Ц я ^ ,, ЧхеЕГн\ ( Т )

for every 0 ^ т ^ 1, where the x-intermediate N -functions with parameter, M x( t , x ) and q>t (t,x), are defined by formula (0.5).

P ro o f. We define

П n

K z (t,x) = Z K iz(t,Xi), Q z ( t , X) = Z Q i z ( t , X i)

i = 1 i = 1

for every z e C with 0 ^ Re z ^ 1, where Kiz(f,x f) and Qiz(t,Xi) are defined by the formulas

(1.21) K iz(t, W ) = [ M f ^ t , i X i D y - ^ N r U t , W )]z, (1.22) Qiz(t, W) = [ ÿ f 1 (t, W )1-z [«АГ1 (*, W )]z,

where the inverse function Mf_1(t,u), JVf_1(t, м), <рг_1(г,и), \jifl (t, u) are taken with respect to the variable w for fixed t. Let us take functions x z J ^ y e J l ^ such that ||х||Ят,М1 = IMIÿt,„2 = 1- We have x(t) = (x.-(0)7=i and y{v) — (уг(и))"=1, where x t and yt are functions from J t c 1 and J t ç 2, respectively. Let us assume that

Xi(t) = Z afxAjit) and y f t ) = Z ь? Х в ^ ) ,

i j=i

where a(p , b f e C and Aj e I l , B j e I 2 are pairwise disjoint, respectively.

Now, we define

9z (t) = (9iz (t))U i and hz (r) = (hiz (r))?= !, where

g,At) = K j t , Mh (t, M î)l))e ,*rg,,'w, М » ) = Qlz(v, ÿ h(ii, 1У,М1))еы‘ ”‘м . Let

f ( z ) = 1 <(Pgz)(v), hz(v))dp2 = Z J (P9z)i(v)-hiz(v)dp2.

T i = 1 T

In the same way as in Theorem 1.15, we may prove that the function / (z) is analytic and bounded in the strip 0 ^ Re z ^ 1. Now, we shall prove that

(1.23) I / (z)| ^ 2K x for Re z = 0 and \f(z)\ ^ 1K2 for Re z = 1.

We have for Re z = 0

IQiz(t, u)\ = ФГ 1(t,u) and |hiz{v)\ = ФГ1 [v, фк (у, |у,-(v)\)].

Hence, we obtain

(1-24) Qïf(hiz) = в%?х{уд if Re z = 0.

Now, applying Lemma 1.2, we get

вф2(К) = Ql2t iy) if Re z = 0.

Similarly we may show that

Qm (0 z ) = вмх(х) if Re z = 0.

From the last two inequalities we conclude that

1/(2» « 2 WPgAv.,t |1М»„2 2K t if Re z = 0.

Now, we shall show the second part of (1.22). Let Re z = 1. Then

\Kiz(t, u)\ = \Qiz(t,u)\ = ф г Ч ^ и ) and

\giz(t)\ = MOI)), hiz(v) = IАГ1^ , Vh(v, LM*0i))- Hence, we get

eViSz) = 6m z{x), ôï2(K) = Ôïz {y) if Re z = 1 and

1 = II^zIIn,^^ » 1 H^zlltL)J2 11у11<»>т,д2 ^ ^ f • By the last inequalities and by Holder inequality, we obtain

|/(z )| ^ 2 \\РдЛф,ц2 \ Ы ф « 2 < 2 K 2 if Re z = 1.

By Hadamard’s three lines theorem, we obtain

I / (z)| ^ 2K[~Rez K T if 0 ^ Re z ^ 1.

Putting in the last inequality т instead of z, where 0 ^ т < 1, we get

| / (t) K 2 K \ ~ ' K \ .

But taking z = x, we have gz{t) = x(r), hz(v) = y(y), where 0 ^ t ^ 1.

Hence, for 0 < t ^ 1 and with ЦхЦ#^ = 1, we obtain l/(*)l = II <(Px)(v),y(v)}dii2\ ^ 2K \ ~ xK r2.

T

Thus, similarly as in the proof of Theorem 1.15, we obtain (1-25) ' HP*IU,„2 ^ 4 К ГтК 2 ||х||Йт,Д1

for every 0 ^ т < 1 and for every x e i j * . In the same way as in the proof of Theorem 1.15 we may extend inequality (1.25) to the whole space E$iz{T). The proof is finished.

R e m a r k 1.19. Replacing the classes ЯК?1, Ш*2 by Ш^2, respect­

ively, there hold theorems analogous to Theorems 1.16 and 1.17.

R e m a r k 1.20. Let ^-functions M ( t , x ) , N( t, x), x), il/(t,x) satisfy

the condition A2. Replacing the spaces £ ^ ( T ) , Е ^ ( Т ) , e£2(T), £ j 2(T), E b l ( T l E ^ {T) by Li ï( T) , L V( T ), К 2( П K 2 (T), L%\{T), L%(T), re­

spectively, there hold all interpolation theorems of Section 1. This follows from Lemma 1.6 and hence that Lftf(T) = E ^ ( T ) for every ^-function M ( t , x ) with parameter t considered in this paper and satisfying condition A 2 and for every cr-finite non-negative, complete and atomless measure ц.

References

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[2] —, —, Some remarks on convergence in Orlicz spaces, Comment. Math. 21 (1979), p. 81-88.

[3] J. Ish ii, On equivalence of modular spaces, Proc. Japan Acad. 35 (1959), p. 551-556.

[4] A. K o zek , Orlicz spaces of functions with values in Banach spaces, Comment. Math.

19(1977). p. 259-288.

[5] M. A. K r a s n o s e l’s k it and Y a. B. R u tic k il, Convex functions and Orlicz spaces (trans­

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[6] W. K ra y n ek , Interpolation o f sublinear operators on generalized Orlicz and Hardy-Orlicz spaces, Studia Math. 43 (1972), p. 93-123.

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INSTITUTE O F MATHEMATICS, A. M ICKIEW ICZ UNIVERSITY MATHEMATICAL INSTITUTE, POLISH ACADEMY O F SCIENCES POZNAN