UDC 513.881

MATHEMATICS

I. V. Shragin

SPACES GENERATED BY GENFUNCTIONS

(Presented by Academician A. N. Tikhonov on 25 XII 1969)

1°. Let \(X\) be a space with a \(\sigma\)-finite complete measure \(\mu\) \((^{1})\), with \(0 \leq \mu X \leq \infty\); let \(S\) be the set of all measurable functions on \(X\) with values in \(\overline{R}=[-\infty,\infty]\); let \(S_f\) be the set of almost everywhere (a.e.) finite functions in \(S\). As usual, functions that coincide a.e. are regarded as equal.

Definition. A function \(M(u,x)\), \(0 \leq u \leq \infty\), \(x \in X\), with values in \([0,\infty]\), is called a pregenfunction if it is measurable in \(x\) for each \(u\), nondecreasing and left-continuous (in the topology of \(\overline{R}\)) in \(u\) for almost every \(x\), and the function \(M(0,x)\) is summable on \(X\).

Let \(M\) be a pregenfunction. Then, if \(\varphi \in S\), we also have \(M(|\varphi(\cdot)|,\cdot)\in S\).

Set

\[ I_M\varphi=\int_X M[|\varphi(x)|,x]\,d\mu,\qquad P_M=\{\varphi\in S_f:I_M\varphi<\infty\},\qquad P_M^\alpha= \]

\[ =\{\varphi\in S_f:\alpha\varphi\in P_M\},\qquad L^M=\bigcup_{\alpha>0}P_M^\alpha,\qquad L_M^f=\bigcap_{\alpha>0}P_M^\alpha. \]

Since the set \(P_M\) is convex and symmetric with respect to the zero point \(\theta\) (\(\theta(x)=0\) a.e.), \(L^M\) is a vector space, and \(L_M^f\) is its subspace. It is obvious that \(L_M^f=L^M\) if and only if \(L^M=P_M\), i.e., when \(P_M\) itself is a vector space.

Set \(d_M(x)=\sup\{u:M(u,x)<\infty\}\). It is known \((^{5})\) that \(d_M\in S\). One can show that \(L^M=\{\theta\}\) if and only if \(d_M=0\); \(L_M^f=\{\theta\}\) if and only if \(d_M\in S_f\).

2°. In their works \((^{10,11})\), Musielak and Orlicz introduced on the space \(L^M\) (which they considered as an example of an abstract modular space) the \(F\)-norm

\[ \|\varphi\|=\inf\{\varepsilon>0:I_M(\varepsilon^{-1}\varphi)\leq \varepsilon\}, \tag{1} \]

satisfying the condition: \(\lim\|\varphi_n\|=0\) if and only if \(\lim I_M(\alpha\varphi_n)=0\) for every \(\alpha>0\) (any \(F\)-norm in \(L^M\) possessing this property will be called normal). In \((^{10,11})\) it is assumed that the pregenfunction \(M\) takes finite values for all \((u,x)\in[0,\infty)\times X\), is continuous in \(u\) for each \(x\), and \(M(u,x)=0\) if and only if \(u=0\). It turns out that these conditions can be weakened. To this end we give the following

Definition. A pregenfunction \(M\) is called a genfunction if \(M(0,x)=0\) and \(M(\infty,x)>0\) a.e. on \(X\), while \(M(+0,x)=0\) a.e. on \(\{x:d_M(x)>0\}\).

Theorem 1. The space \(L^M\) admits the introduction of a normal \(F\)-norm if and only if \(M\) is a genfunction. In particular, if \(M\) is a genfunction, then formula (1) defines in \(L^M\) a normal \(F\)-norm, called the Musielak–Orlicz \(F\)-norm.

Let us note that \(\|\varphi\|\) in (1) is meaningful for every \(\varphi\in S\), but if \(\varphi\in S\setminus L^M\), then \(\|\varphi\|\) may be infinite. It is not difficult to see that the \(F\)-norm (1) has the property of monotonicity (if a.e. \(|\varphi(x)|\leq|\psi(x)|\), then \(\|\varphi\|\leq\|\psi\|\)) and of left monotone continuity (if a.e. \(|\varphi_n(x)|\uparrow|\varphi(x)|\), then \(\|\varphi_n\|\uparrow\|\varphi\|\)). Further, for any normal \(F\)-norm in \(L^M\), convergence in the \(F\)-norm implies convergence in measure on every subset of finite measure (but not necessarily in measure on all of \(X\), as is the case when the genfunction \(M\) is a function only of \(u\)). With the aid of this property one proves

Theorem 2. Let \(M\) be a genfunction. Then the space \(L^{M}\), with respect to every normal \(F\)-norm, is complete, i.e., is an \(F\)-space, and any two normal \(F\)-norms in \(L^{M}\) are topologically equivalent.

Theorem 3. If \(M\) is a genfunction, then, for any normal \(F\)-norm, the subspace \(L_{M}^{f}\) is closed and coincides with the set of elements of the space \(L^{M}\) having absolutely continuous \(F\)-norms.

We give some examples of spaces \(L^{M}\).

1. If \(M(u,x)=u^{p}\rho(x)\), where \(0<p<\infty\), \(\rho(x)\) is a positive measurable function, then \(L^{M}\) is the space \(L^{p}\) with weight \(\rho(x)\).

2. Let \(M(u,x)=\varphi(u)\), where the function \(\varphi\) is continuous, nondecreasing, vanishes only at zero, and \(\varphi(u)\to\infty\) as \(u\to\infty\). Then \(L^{M}\) is the generalized Orlicz space \((^{9})\).

3. If a.e. \(0<d_{M}(x)<\infty\) and \(M(u,x)=0\) for \(0\le u\le d_{M}(x)\), then
   \[ L^{M}=\{\varphi\in S_{f}:\operatorname{vrai\,sup}|\varphi(x)|\cdot(d_{M}(x))^{-1}<\infty\}. \]
   Moreover, \(\|\varphi_{n}\|\to0\) if and only if
   \[ |\varphi_{n}(x)|\cdot(d_{M}x)^{-1}\to0 \]
   uniformly a.e.

4. If \(M\) is a pregenfunction for which the function \(M(\infty,x)\) is summable on \(X\), then \(L^{M}=S_{f}\); if, in addition, \(M\) is a genfunction, then convergence in \(L^{M}\) with respect to the normal \(F\)-norm is equivalent to convergence in measure on every subset of finite measure.

3°. If the pregenfunction \(M\) is a function only of \(u\) and \(d_{M}>0\) \((d_{M}=\infty)\), then every measurable function bounded on \(X\), distinct from zero on a set of finite measure, is contained in \(L^{M}\) (respectively in \(L_{M}^{f}\)). If, however, \(M\) is a function of \(u\) and \(x\), then this fact, generally speaking, does not hold. For example, if \(X=(0,1)\), \(\mu\) is Lebesgue measure, \(M(u,x)=ux^{-1}\), then nonzero constants do not belong to \(L^{M}=P_{M}\). However, every measurable bounded function that vanishes in a neighborhood of zero belongs to this space. It turns out that, also in the general case, as follows from what follows, there exist classes of bounded functions contained in \(L^{M}\) and \(L_{M}^{f}\).

We shall call a chain \((\pi)\) a nondecreasing sequence of measurable sets \(\pi_{n}\subset X\) for which \(\mu\pi_{n}<\infty\), \(n=1,2,\ldots\).

Lemma 1 (cf. \((^{8})\)). Let \(\mathfrak A\) be a nonempty family of measurable subsets of a measurable set \(Y\subset X\), and suppose the following conditions are fulfilled:
1) if \(E_{1},E_{2}\in\mathfrak A\), then \(E_{1}\cup E_{2}\in\mathfrak A\);
2) if \(E\subset Y\) and \(\mu E>0\), then there exists an \(F\subset E\) such that \(F\in\mathfrak A\) and \(\mu F>0\).
Then there exists a chain \((\pi)\) such that all \(\pi_{n}\in\mathfrak A\) and
\[ \mu\bigl(Y\setminus \lim \pi_{n}\bigr)=0. \]

For a given chain \((\pi)\), we shall call a function \(\varphi\) measurable on \(X\) \((\pi)\)-bounded if
\[ \operatorname{vrai\,sup}|\varphi(x)|<\infty \]
and \(\varphi(x)=0\) a.e. on \(X\setminus\pi_{n}\), beginning with some \(n\). With the help of Lemma 1 one proves

Theorem 4. If \(M\) is a pregenfunction, then there exists a chain \((\pi)\) such that
\[ \lim \pi_{n}=\{x:d_{M}(x)>0\} \quad \bigl(\lim \pi_{n}=\{x:d_{M}(x)=\infty\}\bigr), \]
and all \((\pi)\)-bounded functions are contained in \(L^{M}\) (respectively in \(L_{M}^{f}\)).

Let now \(M\) be a genfunction, and let \(L_{M}^{\pi}\) be the closure of the set of all \((\pi)\)-bounded functions contained in \(L^{M}\), with respect to any normal \(F\)-norm. Obviously, \(L_{M}^{\pi}\) is a subspace of \(L^{M}\). It is not difficult to show that if
\[ \mu\bigl(\{x:d_{M}(x)=\infty\}\setminus \lim \pi_{n}\bigr)=0, \]
then \(L_{M}^{f}\subset L_{M}^{\pi}\). Moreover, as follows from Theorem 4, there always exists a chain \((\pi)\) such that
\[ \lim \pi_{n}=\{x:d_{M}(x)=\infty\} \]
and \(L_{M}^{f}=L_{M}^{\pi}\). Relying on this assertion, one can show that if the measure \(\mu\) has a countable basis \((^{1})\), then \(L_{M}^{f}\) is separable.

4°. Here we shall consider some metric properties of the space \(L^{M}\) endowed with the \(F\)-norm (1). First of all, note that if \(\varphi\in P_{M}\), then
\[ \|\varphi\|\le \|d_{M}\|, \]
as follows from the monotonicity of the \(F\)-norm (1).

Lemma 2. If \(M\) is a pregenfunction, then there exists a nondecreasing sequence of nonnegative functions \(\varphi_{n}\in P_{M}\) such that
\[ \lim \varphi_{n}(x)=d_{M}(x)\quad \text{a.e.} \]

Theorem 5. If \(M\) is a genfunction, then
\[ \|d_{M}\|=\sup\{\|\varphi\|:\varphi\in P_{M}\}. \]

Remark 1. From Lemma 2 there also follows the following proposition, more general than Theorem 5. Let \(f\) be a functional on \(S\) with values

on \([0,\infty]\), possessing the property of monotone left-continuity. Then, if \(M\) is a pregenfunction, then

\[ f(d_M)=\sup \{f(\varphi):\varphi\in P_M\}. \]

A natural supplement to Theorem 5 is

Theorem 6. Let \(M\) be a genfunction. Then
\[ \{\varphi\in L^M:\|\varphi\|\leq 1\}\subset \{\varphi\in L^M:I_M\varphi\leq 1\}\subset P_M; \]
moreover,
\[ P_M=\{\varphi\in L^M:\|\varphi\|\leq 1\} \]
if and only if \(I_M(d_M)\leq 1\).

Let us now consider the question of the position of the set \(P_M\) relative to the subspace \(L_M^f\). To this end put
\[ \rho(\varphi,L_M^f)=\inf \{\|\varphi-\psi\|:\psi\in L_M^f\}. \]

Theorem 7. Let \(M\) be a genfunction. Then
\[ \{\varphi\in L^M:\rho(\varphi,L_M^f)<1\}\subset P_M. \]
Moreover, if \(\mu\{x:0<d_M(x)<\infty\}=0\), then
\[ P_M\subset \{\varphi\in L^M:\rho(\varphi,L_M^f)\leq 1\}. \]

5°. Let the genfunction \(M\) be convex in \(u\) for almost every \(x\). Then \(M(\infty,x)=\infty\) a.e. and the function \(M\) is continuous in \(u\) on \([0,d_M(x))\) for almost every \(x\) for which \(d_M(x)>0\). Such a genfunction is called a Young function, and the space \(L^M\) generated by the Young function will be called (5) an Orlicz–Nakano space, since this space was first described in (12), and a particular case of it (when \(M\) is a function only of \(u\)) is an Orlicz space, more precisely an Orlicz space in the sense of Zaanen (14, 15) (cf. (2, 3)).

The Orlicz–Nakano space is a Banach space with norm
\[ \|\varphi\|_1=\inf \{\varepsilon>0:I_M(\varepsilon^{-1}\varphi)\leq 1\} \]
or
\[ \|\varphi\|_2=\inf \{\alpha^{-1}(1+I_M(\alpha\varphi)):0<\alpha<\infty\} \]
(for \(\|\varphi\|_2\) there is also another expression (2, 5), using the complementary Young function). Both norms are normal and, consequently, topologically equivalent to the \(F\)-norm (1) (see also (10), where inequalities are established for the \(F\)-norm (1) and the norm \(\|\cdot\|_1\)). We also note that, as in Orlicz spaces,
\[ \|\varphi\|_1\leq \|\varphi\|_2\leq 2\|\varphi\|_1. \]

The basic facts of the theory of Orlicz spaces extend to Orlicz–Nakano spaces. Here we shall give several propositions generalizing some results from (4, 6).

Let \(M\) be a Young function;
\[ \rho_k(\varphi,L_M^f)=\inf \{\|\varphi-\psi\|_k:\psi\in L_M^f\}, \]
\[ \Pi_k=\{\varphi\in L^M:\rho_k(\varphi,L_M^f)<1\},\quad k=1,2. \]
It is easy to see that
\[ \overline{\Pi}_k=\{\varphi\in L^M:\rho_k(\varphi,L_M^f)\leq 1\},\quad k=1,2. \]

Theorem 8. \(\Pi_2\subset \Pi_1\subset P_M\).

Put
\[ X^0=\{x:0<d_M(x)<\infty\}. \]

Theorem 9. If \(\mu X^0=0\), then
\[ \Pi_1=\Pi_2=\operatorname{int} P_M,\qquad \overline{\Pi}_1=\overline{\Pi}_2=\overline{P}_M. \]

Corollary. If \(\mu X^0=0\), then
\[ \rho_1(\varphi,L_M^f)=\rho_2(\varphi,L_M^f)=\inf \{\alpha>0:\alpha^{-1}\varphi\in P_M\} \]
for every \(\varphi\in L^M\), i.e. the norms \(\|\cdot\|_1\) and \(\|\cdot\|_2\) generate one and the same norm in the quotient space \(L^M/L_M^f\) (cf. (7), p. 8).

Theorem 10. If \(\mu X^0>0\), then
\[ \max \{r:D_r\subset P_M\}=1, \]
where
\[ D_r=\{\varphi:\|\varphi\|_1\leq r\}. \]
Moreover, if \(I_M(d_M)\leq 1\), then \(\|d_M\|_1=1\) and \(P_M=D_1\); if \(I_M(d_M)>1\), then \(\|d_M\|_1>1\) and both inclusions
\[ D_1\subset P_M\subset \{\varphi:\|\varphi\|_1\leq \|d_M\|_1\} \]
are proper.

Remark 2. Since the norms \(\|\cdot\|_1\) and \(\|\cdot\|_2\) have the property of monotone left-continuity, by Remark 1,
\[ \|d_M\|_k=\sup \{\|\varphi\|_k:\varphi\in P_M\},\quad k=1,2. \]
It follows that the set \(P_M\) is bounded in the Orlicz–Nakano space \(L_M\) if and only if \(d_M\in L^M\).

In conclusion we note that a number of properties of the spaces \(L_M\) (conditions for the embedding of one space in another, criteria for closedness and openness of the set \(P_M\), etc.) can be obtained in the form of simple consequences of certain propositions on the Nemytskii operator, whose investigation in spaces generated by genfunctions is carried out in (13).

I sincerely thank M. M. Vainberg for valuable advice concerning this work.

Proof correction note. After the manuscript had been submitted for publication, the author became acquainted with the work \(^{(16)}\), in which spaces generated by genfunctions are considered under the condition that \(d_M(x)>0\) for all \(x\).

Tambov Institute
of Chemical Machine-Building

Received
16 XI 1969

REFERENCES

\(^{1}\) A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Moscow, 1968.
\(^{2}\) V. R. Portnov, DAN, 170, No. 6 (1966).
\(^{3}\) V. R. Portnov, DAN, 175, No. 2 (1967).
\(^{4}\) I. V. Shragin, Scientific Notes of Kishinev Univ., 91, 81 (1967).
\(^{5}\) I. V. Shragin, ibid., 91, 91 (1967).
\(^{6}\) I. V. Shragin, DAN, 179, No. 5 (1968).
\(^{7}\) T. Andô, Nieuw Arch. Wiskunde, 8, 1 (1960).
\(^{8}\) W. A. J. Luxemburg, A. C. Zaanen, Proc. Nederl. Akad. Wet., A59, No. 1, 110 (1956).
\(^{9}\) W. Matuszewska, Bul. Acad. Pol. sci., Sér. sci. math., astr., phys., 8, No. 6, 349 (1960).
\(^{10}\) J. Musielak, W. Orlicz, ibid., 7, No. 11, 661 (1959).
\(^{11}\) J. Musielak, W. Orlicz, Stud. math., 18, No. 1, 49 (1959).
\(^{12}\) H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo, 1950.
\(^{13}\) I. V. Shragin, DAN, 189, No. 1 (1969)*.
\(^{14}\) A. C. Zaanen, Linear Analysis, N. Y., 1953.
\(^{15}\) A. C. Zaanen, Proc. Nederl. Akad. Wet., A52, No. 5–6, 488 (1949).
\(^{16}\) T. Itô, J. Fac. Sci. Hokkaido Univ., ser. I, Math., 15, No. 3, 4 (1961).

* Correction. In article \(^{(13)}\) the following corrections must be made:

p. 64, formula (1) should read

\[ \Phi[\beta|g(u,x)|,x]\leq \gamma M[\alpha|u|,x]+f(x). \tag{1} \]

p. 64, line 25 from the bottom: printed \(h[P_M^\alpha(\Delta)L^\Phi\), should read \(h[P_M^\alpha(\Delta)]\subset L^\Phi\).

p. 64, line 21 from the bottom: printed “то,” should read “что.”

p. 66, line 4, should read

\[ \Phi[\beta|g(u'',x)-g(u',x)|]\leq \gamma[M(|u'|,x)+M(\alpha|u''-u'|,x)]+f(x). \]

p. 66, line 5: printed \(\alpha\), should read \(a\).