UDC 513.88

MATHEMATICS

V. R. PORTNOV

SOME PROPERTIES OF ORLICZ SPACES GENERATED BY FUNCTIONS (M(x,w))

(Presented by Academician S. L. Sobolev, January 14, 1966)

Let (G) be an arbitrary space of points (x) with a (\sigma)-finite measure; let (M(x,w)) be a function defined on the set (G \times (-\infty,\infty)), and suppose that for each (w \in (-\infty,\infty)) it is measurable on (G) and for almost all (a.a.) (x \in G) is an (N)-function on the interval ((-\infty,\infty)) (see ((^1)), p. 16). By (M^*(x,w)) we shall denote the function which, for a.a. (x \in G), is complementary with respect to (M(x,w)) (((^1)), p. 22). This function has the same properties as the function (M(x,w)), namely: it is measurable on (G) for each (w \in (-\infty,\infty)), and for a.a. (x \in G) is an (N)-function on the interval ((-\infty,\infty)).

Definition 1. We shall say that a real-valued function (w(x)) belongs to the Orlicz class (K_M(G)) if it is measurable on (G) and

[
g(w;M)=\int_G M(x,w(x))\,dx<\infty .
\tag{1}
]

Definition 2. We shall say that a sequence of functions (w_n(x)) from (K_M(G)) ((n=1,2,\ldots)) converges in the mean to the function (w_0(x) \in K_M(G)) if

[
\lim_{n\to\infty}\int_G M(x,w_n(x)-w_0(x))\,dx=0 .
\tag{2}
]

Definition 3. The Orlicz space (L_M(G)) is the collection of measurable real-valued functions (w(x)) satisfying the condition

[
|w|M=\sup\int_G |w(x)|\,|v(x)|\,dx<\infty .
\tag{3}
]

Definition 4. We shall say that the function (M(x,w)) satisfies the (\widetilde{\Delta}_2)-condition if, for some nonnegative function (\psi_0(x)\in K_M(G)),

[
\operatorname*{vrai\,sup}{x\in G}\ \sup
\frac{M(x,2w)}{M(x,w)}<\infty .
\tag{4}
]

Definition 5. We shall say that the function (M(x,w)) satisfies the (\Lambda_0)-condition if there exists a nonnegative function (\delta_0(x)\in L_M(G)) such that

[
\lim_{\alpha\to +0}
\left(
\operatorname*{vrai\,sup}{x\in G}
\left(
\sup
\frac{M(x,\alpha w)}{\alpha M(x,w)}
\right)
\right)=0 .
\tag{5}
]

We now formulate a number of theorems characterizing properties of Orlicz spaces and classes.

Theorem 1. a) (L_M(G)=K_M(G)); b) convergence in norm implies convergence in the mean; c) the functional (3) is a norm on (L_M(G)), and (L_M(G)) is complete with respect to this norm.

Theorem 2. The inequalities

[
\int_G |w(x)|\,|v(x)|\,dx \leq |w|M |v|
\tag{6}
]

hold for any pair of functions (w(x)), (v(x)) from the Orlicz spaces (L_M(G)), (L_{M^*}(G)), respectively, and

[
g(w;M) \leq |w|_M
\tag{7}
]

for any function (w(x)) from (L_M(G)) for which (|w|_M \leq 1).

Theorem 3. If the function (M(x,w)) satisfies the (\widetilde{\Delta}_2)-condition, then:
a) (L_M(G)=K_M(G)); b) convergence in the mean is equivalent to convergence in norm; c) the general form of a linear functional (f(w(x))) on (L_M(G)) is given by the formula

[
f(w(x))=\int_G w(x)v(x)\,dx,
\tag{8}
]

where (v(x)\in L_{M^*}(G)) and is uniquely determined by (f).

Theorem 4. If the functions (M(x,w)) and (M^*(x,w)) satisfy the (\widetilde{\Delta}_2)-condition, then the space (L_M(G)) is reflexive.

Theorem 5. If the function (M(x,w)) satisfies the (\Delta_0)-condition, then

[
\lim_{|w|_M\to\infty}\frac{g(w;M)}{|w|_M}=\infty .
\tag{9}
]

Theorem 6. Let the function (M(x,w)) satisfy the (\widetilde{\Delta}2)-condition and suppose the following conditions are fulfilled: a) the sequence of functions ({w_n(x)}) ((n=1,2,\ldots)) converges a.e. on (G) to the function (w_0(x)); b) (w_n(x)\in K_M(G)) for all (n=1,2,\ldots); c) for every (\varepsilon>0) there exists a measurable set (E\varepsilon\subset G) such that (\operatorname{mes} E_\varepsilon<\infty) and

[
\sup_{n\geq 1}\int_{G\setminus E_\varepsilon} M(x,w_n(x))\,dx<\varepsilon;
]

d) for every (\varepsilon>0) there exists (\delta(\varepsilon)>0) such that

[
\sup_{{E:\operatorname{mes} E<\delta(\varepsilon)}}
\left[
\sup_{n\geq 1}\int_E M(x,w_n(x))\,dx
\right]<\varepsilon .
]

Then (w_0(x)\in K_M(G)) and

[
\lim_{n\to\infty} g(w_n(x)-w_0(x);M)=0.
]

Theorem 7. Let the function (M(x,w)) satisfy the (\widetilde{\Delta}_2)-condition and

[
\lim_{n\to\infty} g(w_0(x)-w_n(x);M)=0,
]

where (w_0(x), w_1(x), w_2(x),\ldots) are functions from (K_M(G)). Then: a) for every (\varepsilon>0) there exists a measurable set (E_\varepsilon\subset G) such that (\operatorname{mes} E_\varepsilon<\infty) and

[
\sup_{n\geq 1}\int_{G\setminus E_\varepsilon} M(x,w_n(x))\,dx<\varepsilon;
]

b) for every (\varepsilon>0) there exists (\delta(\varepsilon)>0) such that

[
\sup_{{E:\operatorname{mes} E<\delta(\varepsilon)}}
\left(
\sup_{n\geq 1}\int_E M(x,w_n(x))\,dx
\right)<\varepsilon .
]

We formulate some conditions equivalent to the (\widetilde{\Delta}_2)-condition.

Theorem 8. In order that the function (M(x,w)) satisfy the (\widetilde{\Delta}_2)-condition, it is necessary and sufficient that there exist such a nonnegative

a function (\psi_0(x)\in K_M(G)) such that

[
\operatorname{vrai\,sup}{x\in G}\ \sup
\frac{w p(x,w)}{M(x,w)}<\infty,
]

where (p(x,w)), for a.e. (x\in G), is the right derivative of the function (M_x(w)=M(x,w)) with respect to the argument (w).

Theorem 9. In order that the function (M^(x,w)) satisfy the (\widetilde{\Delta}2)-condition, it is necessary and sufficient that there exist a measurable nonnegative function (\gamma_0(x)) on (G) such that (p(x,\gamma_0(x))\in K{M^}(G)) ((p(x,w)) has the same meaning as in Theorem 8) and

[
\operatorname{vrai\,inf}{x\in G}
\left(
\inf
\frac{w p(x,w)}{M(x,w)}
\right)>1.
]

Theorem 10. Suppose: 1) (M(x,w)) satisfies the (\widetilde{\Delta}2)-condition; 2) in (G) there exists a collection of measurable sets ({G_n}) ((G_n\subset G,\ n=1,2,\ldots)), no more than countable, such that for any set (E\subset G) of finite measure and any (\varepsilon>0) there is a finite number of sets (G) ((n_1,n_2,\ldots,n_R,R) depend on (E) and (\varepsilon)) satisfying the inequality},G_{n_2},\ldots,G_{n_R

[
\operatorname{mes}\left(
\left(E\setminus \bigcup_{k=1}^{R}G_{n_k}\right)
\cup
\left(\left(\bigcup_{k=1}^{R}G_{n_k}\right)\setminus E\right)
\right)<\varepsilon.
]

Then the space (L_M(G)) is separable.

We now give several examples; throughout below the functions (b(x)), (p(x)), (q(x)), (c(x)), (\beta(x)), and (\lambda(x)) are measurable on (G), a.e. finite and a.e. positive. Further,

[
\operatorname{vrai\,inf}{x\in G} p(x)>1,\qquad
\operatorname{vrai\,sup} p(x)<\infty.
]

1) (M_1(x,w)=b(x)|w|^{p(x)}) and (M_1^*(x,w)) satisfy the (\widetilde{\Delta}_2)-condition and the (\Delta_0)-condition.

2) (M_2(x,w)=b(x)M(x)), where (M(x)) is an (N)-function satisfying the (\Delta_2)-condition for all (w) (see [1], p. 35). (M_2(x,w)) satisfies the (\widetilde{\Delta}_2)-condition. It will also satisfy the (\Delta_0)-condition if (M^*(w)) satisfies the (\Delta_2)-condition for large (w) (see [6], Theorem 4).

3) (M_3(x,w)=b(x)M(w)), where (M(w)) is an (N)-function satisfying the (\Delta_2)-condition for large (w), and moreover (\int_G b(x)\,dx<\infty). The function (M_3(x,w)) satisfies the (\widetilde{\Delta}_2)-condition. It will also satisfy the (\Delta_0)-condition if (M^(w)) satisfies the (\Delta_2)-condition for large (w) (see [6], Theorem 4). Further, the function (M^(x,w)) satisfies the (\Delta_0)-condition; moreover, it will satisfy the (\widetilde{\Delta}_2)-condition in the case when (M^*(w)) satisfies the (\Delta_2)-condition for large (w).

4) (M_4(x,w)=b(x)|w|^{p(x)}\exp(-c(x)|w|^{-q(x)})), where

[
2p(x)-1\ge q(x),
]

[
\int_G b(x)\exp(-q^{-1}(x))(C(x)q(x))^{p(x)/q(x)}\,dx<\infty.
]

The functions (M_4(x,w)) and (M_4^*(x,w)) satisfy the (\widetilde{\Delta}_2)-condition; moreover, (M(x,w)) satisfies the (\Delta_0)-condition.

5) (M_5(x,w)=b(x)|w|^{1+\beta(x)}(1+|w|^{\beta(x)})^{1/\beta(x)}), where (\beta(x)\le 1) and

[
\int_G b(x)2^{1/\beta(x)}\,dx<\infty.
]

The functions (M_5(x,w)) and (M_5^*(x,w)) satisfy the (\widetilde{\Delta}_2)-condition.

6) (M_6(x,w)=b(x)|w|\ln^{-1}(1+\lambda(x)/|w|)), where

[
\int_G b(x)\lambda(x)\,dx<\infty.
]

The function (M_6(x,w)) satisfies the (\widetilde{\Delta}_2)-condition and the (\Delta_0)-condition. The function (M_6^*(x,w)) satisfies the (\widetilde{\Delta}_2)-condition.

Remark. A. A complete exposition of the theory of Orlicz spaces in the case when (M(x,w)=M(w)), and (G) is a closed set of finite measure situated in (n)-dimensional Euclidean space, is given in ((^1)). References to the literature on this question are also given there.

B. The case (\operatorname{mes} G=\infty) and (M(x,w)=M(w)) is considered in the works ((^{2-4})).

C. Orlicz spaces generated by the functions (M(x,w)=|w|^{p(x)}), (p(x)>1), were studied in ((^5)).

D. In the case when (\operatorname{mes} G<\infty), (M(x,w)=M(w)), the necessary and sufficient condition for the fulfillment of (9) was obtained for the first time by Ya. B. Rutitskii; the (\Delta_0)-condition in the case when (M(x,w)=M(w)), (\delta_0(x)=w_0), (\operatorname{mes} G<\infty), is a condition of Ya. B. Rutitskii ((^6)).

E. Theorems 6 and 7 for the case of Lebesgue spaces (M(x,w)=|w|^p) are contained in ((^7)).

F. Properties of Orlicz spaces generated by functions (M(x,w)), different from the properties expressed by Theorems 2–9, have been considered by various authors; here we note only the works ((^8,^9)).

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
6 I 1966

REFERENCES

(^1) M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.
(^2) V. I. Sobolev, Tr. Seminar on Functional Analysis, Voronezh State Univ., issue 277 (1956).
(^3) Yu. I. Gribanov, ibid., 6, 29 (1958).
(^4) W. Orlicz, Bull. Intern. Acad. Polon. ser. A, Cracovie, (1936).
(^5) Z. Birnbaum, W. Orlicz. Studia math., 3, 1 (1931).
(^6) Ya. B. Rutitskii, Dokl. AN UkrSSR, No. 10, 1290 (1962).
(^7) N. Dunford, J. Schwartz, Linear Operators, IL, 1962.
(^8) H. Nakano, Modulated Semiorordered Linear Spaces, Tokio, 1950.
(^9) S. Yamamuro, Fac. Sci. Hokkaido Univ., 1, 12, 211 (1953).