UDC 513.88 : 513.881

MATHEMATICS

I. V. SHRAGIN

ON THE THEORY OF ORLICZ SPACES

(Presented by Academician A. N. Tikhonov on 21 XII 1966)

1°. Introduction. In this note several new results from the theory of Orlicz spaces in the sense of Zaanen are formulated. We give the necessary information about these spaces \((^{1,2})\).

A function \(M(u)\), \(0 \le u \le \infty\), \(0 \le M(u) \le \infty\), is called a Young function if it satisfies the following conditions: \(M(u)\) is nondecreasing on \([0,\infty]\), with \(M(0)=0\), \(M(\infty)=\infty\); \(0<d_M\le\infty\), where \(d_M=\sup\{u\mid M(u)<\infty\}\); \(0\le c_M<\infty\), where \(c_M=\sup\{u\mid M(u)=0\}\); \(M(u)\) is continuous for every \(u<d_M\); \(M(u)\) is continuous from the left at \(u=d_M\); \(M(u)\) is convex on the interval \([0,d_M)\).

Definition. \(M(u)\) satisfies the \(\Delta_\beta\)-condition (where \(1<\beta<\infty\)) on \([a,b]\), \(0\le a<b\le\infty\), if there exists a \(K\), \(0\le K<\infty\), such that \(M(\beta u)\le K M(u)\) for \(a\le u\le b\).

Lemma. If \(d_M<\infty\), then the following two conditions are equivalent:
1) \(M(d_M)<\infty\); 2) there exist \(\beta>1\) and \(u_0\in[0,\beta^{-1}d_M)\) such that \(M(u)\) satisfies the \(\Delta_\beta\)-condition on \([u_0,\beta^{-1}d_M]\).

Let \(X\) be a nonempty set; \(R\), some \(\sigma\)-algebra of its subsets; \(\mu\), a \(\sigma\)-finite measure \((^3)\) defined on \(R\), with \(0<\mu X\le\infty\). Fix an increasing sequence of sets \(X_n\in R\) such that \(0<\mu X_n<\infty\), \(n=1,2,\ldots\), and

\[ X=\bigcup_{n=1}^{\infty} X_n. \]

Following \((^2)\), we shall call a set \(E\subset X\) bounded if \(E\subset X_n\) starting from some \(n\). We shall assume that the measure \(\mu\) is complete (i.e., if \(\mu A=0\) and \(B\subset A\), then \(B\in R\)) and continuous (i.e., if \(\mu A>0\) and \(0<\delta<\mu A\), then there exists a \(B\in R\) such that \(B\subset A\) and \(\mu B=\delta\)).

Let \(S\) be the set of all functions measurable on \(X\) with respect to the measure \(\mu\), taking values in the extended real line or in the extended complex plane. Functions which coincide almost everywhere on \(X\) are regarded as identical.

The Young class is the set

\[ P_M=\left\{u\in S\mid \int_X M[|u(x)|]\,dx<\infty\right\}. \]

Obviously, \(P_M\) is contained in the set of almost everywhere finite functions from \(S\) and is a convex set.

The Orlicz space \(L_M\) is the union of all sets \(aP_M\), \(0<a<\infty\). \(L_M\) is a vector space (under certain conditions the Young class \(P_M\) itself is a vector space; then \(L_M=P_M=aP_M\), \(0<a<\infty\)), and after the introduction of the norm

\[ \|u\|_1=\inf\left\{\alpha>0\mid \int_X M[\alpha^{-1}|u(x)|]\,dx\le 1\right\} \]

or

\[ \|u\|_2=\sup\left\{\int_X |u(x)v(x)|\,dx\mid \int_X N[|v(x)|]\,dx\le 1\right\} \]

(where \(N(v)=\max\{uv-M(u)\mid 0\le u<\infty\}\), \(0\le v\le\infty\)) it becomes a Banach space.

The norms \(\|\ \|_1\) and \(\|\ \|_2\) are equivalent; more precisely,
\(\|u\|_1\le \|u\|_2\le 2\|u\|_1\). Both norms are defined on the whole set \(S\), but if \(u\in S\setminus L_M\), then \(\|u\|_k=\infty,\ k=1,2\). We note that

\[ \{u\mid \|u\|_1\le 1\}= \left\{u\ \middle|\ \int_X M[|u(x)|]\,dx\le 1\right\}\subset P_M. \]

An important role in the theory of the space \(L_M\) is played by its closed subspace \(L_M^f\), defined as the intersection of all the sets \(aP_M\), \(0<a<\infty\). Here \(L_M^f=\{\theta\}\) (where \(\theta\) is the zero function) if and only if \(d_M<\infty\). On the other hand, \(L_M^f=L_M\) if and only if \(L_M=P_M\).

2°. Topological and metric properties of the Young class as a subset of an Orlicz space. In the following Theorems 1–3, certain results obtained in \((^4)\) for a narrower class of Orlicz spaces are generalized and supplemented (in particular, in \((^4)\) it is assumed that \(c_M=0,\ d_M=\infty\), and \(\mu X<\infty\)).

Theorem 1. The following three conditions are pairwise equivalent:

A. The Young class \(P_M\) is a closed set in the Orlicz space \(L_M\).

B. \(P_M=\{u\in L_M\mid |u(x)|\le d_M\) almost everywhere on \(X\}\).

C. The function \(M(u)\) satisfies the \(\Delta_\beta\)-condition on \([u_0,\beta^{-1}d_M]\) for some \(\beta>1\) and \(u_0\in[0,\beta^{-1}d_M)\), and, if \(\mu X=\infty\), then \(u_0=0\).

Corollary 1. If \(d_M=\infty\), then \(P_M\) is closed if and only if \(L_M=P_M\).

Remark. From Theorem 1, in particular, there follows the well-known criterion for the coincidence of \(L_M\) with \(P_M\): \(L_M=P_M\) if and only if \(d_M=\infty\) and \(M(u)\) satisfies the \(\Delta_2\)-condition on \([u_0,\infty]\), where \(u_0=0\) if \(\mu X=\infty\).

Theorem 2. The Young class \(P_M\) is an open set in the space \(L_M\) if and only if \(L_M=P_M\).

For \(u\in L_M\), put
\[ \rho_k(u,L_M^f)=\inf\{\|u-v\|_k\mid v\in L_M^f\},\qquad \Pi_k=\{u\in L_M\mid \rho_k(u,L_M^f)<1\}, \]
\(k=1,2\).

Theorem 3. Let \(d_M=\infty\). Then \(\Pi_k=\operatorname{int} P_M\), \(\overline{\Pi}_k=\overline{P}_M=\{u\in L_M\mid \rho_k(u,L_M^f)\le 1\}\), \(k=1,2\).

Corollary 2. If \(d_M=\infty\), then
\[ \rho_1(u,L_M^f)=\rho_2(u,L_M^f)= \inf\{a>0\mid u\in aP_M\} \]
for every \(u\in L_M\).

Corollary 2 means that if \(d_M=\infty\), then the norms \(\|\ \|_1\) and \(\|\ \|_2\) induce one and the same norm in the factor space \(L_M/L_M^f\).

Corollary 3. If \(d_M=\infty\) and \(L_M\ne P_M\), then the maximal radius of balls contained in \(P_M\) is equal to one both for the norm \(\|\ \|_1\) and for the norm \(\|\ \|_2\).

Theorem 4.
\[ \sup\{\|u\|_k\mid u\in P_M\}=\|d_M\|_k,\qquad k=1,2. \]
Here \(d_M\) denotes the function taking the value \(d_M\) at all \(x\in X\).

Corollary 4. \(P_M\) is a bounded set in the space \(L_M\) if and only if \(d_M\in L_M\).

From Corollary 4 there follows the following boundedness criterion for \(P_M\): if \(d_M=\infty\), then \(P_M\) is unbounded; if \(d_M<\infty\), then \(P_M\) is bounded if and only if \(\mu X<\infty\), or \(\mu X=\infty\) and \(c_M>0\).

Theorem 5. Let \(d_M<\infty\). Then
\[ \max\{r\mid D_r\subset P_M\}=1, \]
where \(D_r=\{u\mid \|u\|_1\le r\}\). Moreover, if \(M(d_M)\mu X\le 1\), then \(\|d_M\|_1=1\) and \(P_M=D_1\); if \(M(d_M)\mu X>1\), then \(\|d_M\|_1>1\) and both inclusions
\[ D_1\subset P_M\subset \{u\mid \|u\|_1\le \|d_M\|_1\} \]
are strict.

3°. On some subspaces of an Orlicz space. Following \((^2)\), denote by \(L_M^b\) the closure in the space \(L_M\) of the set of bounded functions \(u(x)\) for which the set \(\{x\in X\mid u(x)\ne 0\}\) is bounded. In addition, denote by \(E_M^f\) the closure in \(L_M\)

sets of bounded functions \(u(x)\) for which \(\mu\{x\in X\mid u(x)\ne 0\}<\infty\), and by \(E_M\) \((^4)\) the closure in \(L_M\) of the set of bounded functions contained in \(L_M\) (one should take into account that if \(\mu X=\infty\) and \(c_M=0\), then, for example, identically constant functions different from zero do not belong to \(L_M\)).

As shown in \((^2)\), \(L_M^f\subset L_M^b\), and \(L_M^f=L_M^b\) if and only if \(d_M=\infty\). On the other hand, it is obvious that \(L_M^b\subset E_M^f\subset E_M\subset L_M\).

Theorem 6. If \(d_M=\infty\), then \(L_M^b=E_M^f\) \((^2)\). If \(d_M<\infty\), then \(L_M^b=E_M^f\) if and only if \(\mu(X\setminus X_n)=0\), starting from some \(n\).

Theorem 7. If \(\mu X<\infty\), then \(E_M^f=E_M\). If \(\mu X=\infty\), then \(E_M^f=E_M\) if and only if \(c_M=0\) and \(M(u)\) satisfies the \(\Delta_2\)-condition on \([0,b]\) for some \(b>0\).

Theorem 8. If \(d_M<\infty\), then \(E_M=L_M\). If \(d_M=\infty\), then \(E_M=L_M\) if and only if \(M(u)\) satisfies the \(\Delta_2\)-condition on \([u_0,\infty]\) for some \(u_0>0\).

4°. Convergence in mean. It is said \((^4)\) that a sequence \(u_n\in L_M\) converges in mean to \(u\in L_M\) if

\[ \lim_{n\to\infty}\int_X M[|u_n(x)-u(x)|]\,dx=0. \]

From the definition of the norm \(\|\ \|_1\) it follows easily that convergence in norm entails convergence in mean.

Theorem 9. Convergence in norm in the space \(L_M\) is equivalent to convergence in mean if and only if \(L_M=P_M\) and \(c_M=0\).

It should be noted that Theorems 8 and 9 generalize the corresponding results from \((^4)\).

Kishinev State
University

Received
12 XII 1966

REFERENCES

\(^1\) A. C. Zaanen, Linear Analysis, Amsterdam—N. Y., 1953.
\(^2\) W. A. J. Luxemburg, Banach Function Spaces, Thesis Delft Techn. Univ., 1955.
\(^3\) B. Z. Vulikh, A Short Course in the Theory of Functions of a Real Variable. Moscow, 1965.
\(^4\) M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces. Moscow, 1958.