UDC 513.88

MATHEMATICS

V. R. PORTNOV

ON THE THEORY OF ORLICZ SPACES GENERATED BY VARIABLE \(N\)-FUNCTIONS

(Presented by Academician S. L. Sobolev on 24 IX 1966)

\(1^\circ\). In the present article we describe some properties of Orlicz classes \(K_M(G)\) and Orlicz spaces \(L_M(G)\), which are constructed with the aid of so-called variable \(N\)-functions \(M(x,w)\). The theorems proved here are a generalization of the corresponding results for Orlicz spaces \(L_M(G)\) in the case when \(\operatorname{mes} G<\infty\) and \(M(x,w)=M(w)\) (see \((^1)\)), as well as of the results of the note \((^3)\); here the known \(\Delta_2\)-condition (\((^1)\), p. 35) for \(N\)-functions and the \(\widetilde{\Delta}_2\)-condition introduced in \((^3)\) are particular cases of the more general \(\Delta_2'\)-condition (see Definition 2). We note that a complete exposition of the theory of \(N\)-functions and 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 available in \((^1)\) (there one also finds detailed references to the literature on this question). The case \(\operatorname{mes}G=\infty\) and \(M(x,w)=M(w)\) was studied in the works \((^{4-6})\).

\(2^\circ\). Everywhere in what follows, by \(G\) we denote some space of points \(x\) with a \(\sigma\)-finite measure. We shall call \(G\) a space with a continuous measure if, for every set \(\mathcal E\subset G\) of finite measure and every \(\varepsilon>0\), there exists a partition of \(\mathcal E\) into such mutually disjoint sets \(\mathcal E_1,\mathcal E_2,\ldots,\mathcal E_R\) \((R\geq 2)\) that \(\operatorname{mes}\mathcal E_j<\varepsilon\) \((j=1,2,\ldots,R)\). By \(L_1^+(G)\) we shall denote the set of nonnegative functions from \(L_1(G)\).

Definition 1. A function \(M(x,w)\), defined on the set \(G\times(-\infty,\infty)\), will be called a variable \(N\)-function if, for each fixed \(x\in G\), it is an \(N\)-function of the argument \(w\) on the interval \((-\infty,\infty)\) (see \((^1)\), p. 16), and, for each fixed \(w\in(-\infty,\infty)\), it is measurable on \(G\) as a function of the argument \(x\). For every variable \(N\)-function \(M(x,w)\), by \(M^*(x,w)\) we shall denote the function which, for each fixed \(x\in G\), is the complementary function with respect to the \(N\)-function \(M(x,w)\) (see \((^1)\), p. 22). It is easy to prove that \(M^*(x,w)\) is also a variable \(N\)-function and that \(M^{**}(x,w)=M(x,w)\). By \(\mu(x,w)\) we shall denote the function which, for each fixed \(x\in G\), is the right derivative of the \(N\)-function \(M(x,w)\) with respect to the argument \(w\), and by \(\mu^*(x,w)\) the function which, for each fixed \(x\in G\), is the right derivative of the \(N\)-function \(M^*(x,w)\) with respect to the argument \(w\).

Definition 2. We shall say that a variable \(N\)-function \(M(x,w)\) satisfies on \(G\) the \(\Delta_2'\)-condition if the inequality
\[ M(x,2w)\leq f(x)+CM(x,w)\qquad (x\in G,\ -\infty<w<\infty), \]
is fulfilled, where \(f(x)\in L_1^+(G)\), and \(C\) is some positive number.

Theorem 1. Let \(M(x,w)\) be a variable \(N\)-function. Then the following three assertions a), b), and c) are equivalent: a) the inequality
\[ w\mu(x,w)\leq f_0(x)+C_0M(x,w)\qquad (w\geq 0,\ x\in G), \]
holds, where \(f_0(x)\in L_1^+(G)\), \(C_0\) is some positive number; b) the inequality
\[ (1+\gamma)M^*(x,w)\leq \psi_0(x)+w\mu^*(x,w)\qquad (w\geq 0,\ x\in G), \]
holds, where \(\psi_0(x)\in L_1^+(G)\), \(\gamma\) is some positive number; c) \(M(x,w)\) satisfies on \(G\) the \(\Delta_2'\)-condition.

Theorem 2. If the variable \(N\)-function \(M(x,w)\) satisfies on \(G\) the \(\Delta_2'\)-condition, then the inequality
\[ M^*(x,kw)\leq k^{1+\gamma}M^*(x,w)+ \]

\[ +\psi(x)\quad (w\geq 0,\ x\in G,\ 0\leq k\leq 1), \]
where \(\psi(x)\in L_1^+(G)\), \(\gamma\) is some positive number.

For the case when \(M(x,w)=M(w)\), \(\operatorname{mes} G<\infty\), and \(M(w)\) satisfies the \(\Delta_2\)-condition, Theorem 2 was formulated in the article of Ya. B. Rutitskii \((^2)\).

Definition 3. Let \(M(x,w)\) be a variable \(N\)-function. By \(K_M(G)\) we shall denote the totality of all such real measurable functions \(w(x)\), defined on \(G\), for which

\[ \rho(w;M)=\int_G M(x,w(x))\,dx<\infty . \]

The set \(K_M(G)\) will be called an Orlicz class.

Theorem 3. In order that the variable \(N\)-function \(M(x,w)\) satisfy on \(G\) the \(\Delta_2\)-condition, it is sufficient that at least one of the following inequalities hold:

\[ \text{a) }\ \operatorname{vrai}\sup_{x\in G}\left(\sup_{w>f_0(x)}\frac{w\mu(x,w)}{M(x,w)}\right)<\infty, \quad \text{where } f_0(x)\geq 0 \text{ and } f_0(x)\in K_M(G); \]

\[ \text{b) }\ \operatorname{vrai}\inf_{x\in G}\left(\inf_{w>\psi_0(x)}\frac{w\mu^*(x,w)}{M^*(x,w)}\right)>1, \quad \text{where } \psi_0(x)\geq 0 \text{ and } \psi_0(x)\in K_{M^*}(G); \]

\[ \text{c) }\ \operatorname{vrai}\inf_{x\in G}\left(\inf_{n>\varphi_0(x)}\frac{w\mu^*(x,w)}{M^*(x,w)}\right)>1, \quad \text{where } \varphi_0(x)\geq 0 \text{ and } \mu^*(x,\varphi_0(x))\in K_M(G). \]

Definition 4. Let \(M(x,w)\) be a variable \(N\)-function. By \(L_M(G)\) we shall denote the totality of all such real measurable functions \(w(x)\), defined on \(G\), for which

\[ \|w\|_{L_M(G)}=\sup_{\rho(v;M^*)\leq 1}\int_G |w(x)|\,|v(x)|\,dx<\infty . \tag{1} \]

The set \(L_M(G)\) will be called an Orlicz space. As was noted in \((^3)\), \(L_M(G)\), with the norm (1) defined on it, is a Banach space.

Definition 5. We shall say that a sequence of functions \(w_n(x)\in L_M(G)\) \((n=1,2,\ldots)\) converges in mean to the function \(w_0(x)\in L_M(G)\) if

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

A set \(\mathfrak{M}\subset K_M\) will be called bounded in mean if

\[ \sup_{w(x)\in \mathfrak{M}}\rho(w;M)<\infty . \]

Definition 6. Let \(M(x,w)\) be a variable \(N\)-function. Since \(G\) has \(\sigma\)-finite measure, there exists a sequence \(\widetilde G_k\) \((k=1,2,\ldots)\) of measurable sets of finite measure, nested one in another, such that

\[ \bigcup_{k=1}^{\infty}\widetilde G_k=G. \]

For any pair of natural numbers \(m\) and \(l\), evidently, there is a measurable set \(G_{ml}\subset \widetilde G_l\) such that

\[ \operatorname{mes}(\widetilde G_l-G_{ml})\leq 2^{-(m+l)} \]

and on the set \(G_{ml}\) the function \(M(x,m)\) is bounded. Put

\[ G_k=\bigcup_{l=1}^{k}\left(\bigcap_{m=1}^{\infty}G_{ml}\right). \]

It is clear that

\[ G_1\subset G_2\subset \cdots \subset G_k\subset \cdots,\qquad \operatorname{mes} G_k<\infty \]

\[ (k=1,2,\ldots)\quad \text{and}\quad \bigcup_{k=1}^{\infty}G_k=G. \]

We shall regard the sequence of sets \(G_k\) \((k=1,2,\ldots)\) as fixed. We shall say that a function \(w(x)\in \mathfrak{A}_M(G)\) if its values are only rational numbers, it is measurable, finite-valued, and there exists a natural number \(k_0\), depending, generally speaking, on \(w(x)\), such that \(w(x)=0\) if \(x\in G_{k_0}\). It is clear that

\[ \mathfrak{A}_M(G)\subset L_M(G). \]

By \(E_M(G)\) we shall denote the closure of the set \(\mathfrak{A}_M(G)\) in the space \(L_M(G)\) in the metric (1).

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

Theorem 4. The inclusions \(\mathfrak{A}_{M}(G) \subset E_{M}(G) \subset K_{M}(G) \subset L_{M}(G)\) hold, and the set \(\mathfrak{A}_{M}(G)\) is dense in the Orlicz class \(K_{M}(G)\) in the sense of convergence in the mean.

Theorem 5. For every functional \(f \in (E_{M}(G))^{*}\) there exists, and moreover is unique, a function \(v(x) \in L_{M^{*}}(G)\) such that the equality

\[ f(w(x))=\int_{G} v(x)w(x)\,dx \tag{2} \]

holds for all \(w(x) \in E_{M}(G)\).

Theorem 6. If there exists a not more than countable collection \(G'_{\nu}\) \((\nu=1,2,\ldots)\) of measurable subsets of the set \(G\) such that, for any set \(\mathscr{E}\subset G\) of finite measure,

\[ \inf_{\nu}\bigl(\operatorname{mes}((\mathscr{E}-G'_{\nu})\cup(G'_{\nu}-\mathscr{E}))\bigr)=0, \]

then the space \(E_{M}(G)\) is separable.

Theorem 7. Let the variable \(N\)-function \(M(x,w)\) satisfy the inequality
\[ M(x,kw)\le k\delta(k)M(x,w)+\psi(x) \]
\((x\in G,\ w>0,\ 0<k<k_{0})\), where \(\psi(x)\in L_{1}^{+}(G)\); \(k_{0}>0\); \(\delta(k)\) is a nonnegative function defined on the interval \((0,k_{0}]\) such that \(\lim_{k\to 0}\delta(k)=0\). Then the relation

\[ \lim_{\|w\|_{L_{M}(G)}\to\infty} \frac{\rho(w;M)}{\|w\|_{L_{M}(G)}}=\infty \tag{3} \]

holds.

In the case where \(\operatorname{mes}G<\infty\) and \(M(x,w)=M(w)\), a necessary and sufficient condition for (3) to hold was obtained by Ya. B. Rutitskii in \((^{2})\).

Theorem 8. Let the variable \(N\)-function \(M(x,w)\) satisfy on \(G\) the \(\Delta_{2}'\)-condition. Then: a) convergence in norm in the space \(L_{M}(G)\) coincides with convergence in the mean; b) \(E_{M}(G)=K_{M}(G)=L_{M}(G)\); c) the general form of a linear functional on \(L_{M}(G)\) is given by equality (2);

d)
\[ \lim_{\|w\|_{L_{M^{*}}(G)}\to\infty} \frac{\rho(w;M^{*})}{\|w\|_{L_{M^{*}}(G)}}=\infty; \]
e) every set \(\mathfrak{M}\subset L_{M}(G)\) bounded in norm is also bounded in the mean.

Theorem 9. Let \(G\) be a space with a nonatomic measure and let the variable \(N\)-function \(M(x,w)\) not satisfy on \(G\) the \(\Delta_{2}'\)-condition. Then: a) the Orlicz class \(K_{M}(G)\) is not a linear set (and consequently \(E_{M}(G)\ne K_{M}(G)\), \(K_{M}(G)\ne L_{M}(G)\), and \(E_{M}(G)\ne L_{M}(G)\)); b) there exists a functional \(f_{0}\in (L_{M}(G))^{*}\) for which there is no function \(v(x)\) such that for all \(w(x)\) from \(L_{M}(G)\), \(v(x)w(x)\in L_{1}(G)\) and the equality
\[ f_{0}(w(x))=\int_{G} v(x)w(x)\,dx \]
holds; in other words, (2) is not the general form of a linear functional on \(L_{M}(G)\).

Theorem 10. In order that the Orlicz space \(L_{M}(G)\) be reflexive, it is sufficient, and if \(G\) is a space with a nonatomic measure, also necessary, that each of the variable \(N\)-functions \(M(x,w)\) and \(M^{*}(x,w)\) satisfy on \(G\) the \(\Delta_{2}'\)-condition.

Theorem 11. Let \(M_{1}(x,w)\) and \(M_{2}(x,w)\) be variable \(N\)-functions satisfying on \(G\) the \(\Delta_{2}'\)-condition; let \(\varphi(x,w)\) be a function satisfying the Carathéodory condition, and suppose the inequality

\[ M_{2}(x,\varphi(x,w))\le f(x)+CM_{1}(x,w)\quad (x\in G,\ -\infty<w<\infty), \tag{4} \]

holds, where \(f(x)\in L_{1}^{+}(G)\), and \(C\) is some positive number. Then the operator \(\Phi\), defined by the equality \(\Phi w(x)=\varphi(x,w(x))\), maps \(L_{M_{1}}(G)\) into \(L_{M_{2}}(G)\) and is continuous.

Theorem 12. Let \(G\) be a space with a nonatomic measure; \(M_{1}(x,w)\) and \(M_{2}(x,w)\) be variable \(N\)-functions satisfying on \(G\) the \(\Delta_{2}'\)-condition; let \(\varphi(x,w)\) be a function satisfying the Carathéodory condition, and suppose \(\varphi(x,w(x))\in K_{M_{2}}(G)\) for every function \(w(x)\subset K_{M_{1}}(G)\). Then the opera-

the operator \(\Phi\), defined by the equality \(\Phi w(x)=\varphi(x,w(x))\), maps \(L_{M_1}(G)\) into \(L_{M_2}(G)\), is continuous, and is bounded on every ball of the space \(L_{M_1}(G)\), and for the function \(\varphi(x,w)\) the inequality
\[ M_2(x,\varphi(x,w))\leq f(x)+C M_1(x,w)\quad (x\in G,\ -\infty<w<\infty), \]
holds, where \(f(x)\in L_1^+(G)\), and \(C\) is some positive number.

Remark. Theorems 9 and 12 are proved on the basis of a result of M. A. Krasnosel’skii \(\bigl((^7),\) p. 26\bigr).

The following theorem is a generalization of the well-known Vitali theorem (see \((^8),\) p. 167).

Theorem 13. Let the variable \(N\)-function \(M(x,w)\) satisfy the \(\Delta_2'\)-condition on \(G\). Let, further, \(w_n(x)\) \((n=1,2,\ldots)\) be a sequence of functions from the space \(L_M(G)\), and let \(w_0(x)\) be some real-valued measurable and almost everywhere finite function on \(G\). In order that \(w_0(x)\in L_M(G)\) and
\[ \lim_{n\to\infty}\|w_n(x)-w_0(x)\|_{L_M(G)}=0, \]
it is necessary and sufficient that the following conditions be satisfied: a) the sequence \(w_n(x)\) \((n=1,2,\ldots)\) converges to the function \(w_0(x)\) in measure on every set of finite measure \(\mathcal E\subset G\); b) the equalities
\[ \inf_{\{\mathcal E:\operatorname{mes}\mathcal E<\infty\}} \left(\sup_{n\geq 1}\int_{G\setminus \mathcal E} M(x,w_n(x))\,dx\right)=0, \]
\[ \inf_{\delta>0}\left( \sup_{\{\mathcal E:\operatorname{mes}\mathcal E<\delta\}} \left(\sup_{n\geq 1}\int_{\mathcal E} M(x,w_n(x))\,dx\right) \right)=0. \]

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

Received
17 IX 1966

REFERENCES

1. M. A. Krasnosel’skii, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Moscow, 1958.
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