MATHEMATICS

Ya. B. Rutitskii

On a Property of the Orlicz Norm

(Presented by Academician V. I. Smirnov on 24 XII 1960)

Let \(L_M^*\) be some Orlicz space of functions defined on a bounded closed set \(G\) of finite-dimensional Euclidean space. As I. V. Gelman showed \({}^{1}\), in the study of certain problems of mathematical physics it is convenient to use Orlicz spaces in which the norm has the following property:

\[ \lim_{\|u\|_M \to \infty} \frac{1}{\|u\|_M}\int_G M[\,u(x)\,]\,dx=\infty . \tag{1} \]

For the spaces \(L^p\) (\(p>1\)), which are a special case of the Orlicz spaces \(L_M^*\) determined by the function \(M(u)=|u|^p/p\) (\(p>1\)), property (1) is evidently satisfied. It is known, however, that not all properties of \(L^p\) spaces remain valid for arbitrary Orlicz spaces. Therefore the question arises under what conditions the Orlicz norm in the space \(L_M^*\) has property (1). In the present note an answer to this question is given. (For all definitions and assertions from the general theory of Orlicz spaces used in this note, see \({}^{2}\).)

Lemma. In order that relation (1) be satisfied, it is necessary that for every \(k>0\) the function \(M(u)\) satisfy the condition

\[ \lim_{u\to\infty}\frac{M(ku)}{uN^{-1}(u)}=\infty, \tag{2} \]

where \(N^{-1}(u)\) is the function inverse to the function \(N(v)\), complementary to \(M(u)\).

Proof. Suppose that condition (2) is not satisfied. Then there will be a \(k_0>0\), a sequence of numbers \(u_n\) (\(n=1,2,\ldots\)) tending to infinity, and a \(C>0\) such that

\[ \frac{M(k_0u_n)}{u_nN^{-1}(u_n)}\le C \quad (n=1,2,\ldots). \tag{3} \]

Consider sets \(G_n\subset G\) with measure \(\operatorname{mes} G_n=1/u_n\) and put

\[ u_n(x)= \begin{cases} k_0u_n, & \text{if } x\in G_n,\\ 0, & \text{if } x\bar{\in}G_n. \end{cases} \quad (n=1,2,\ldots) \]

Using the formula for the norm of a characteristic function, we obtain

\[ \|u_n\|_M = k_0u_n\,\operatorname{mes}G_n\,N^{-1}\!\left(\frac{1}{\operatorname{mes}G_n}\right) = k_0N^{-1}(u_n). \]

Since \(\lim_{n\to\infty}N^{-1}(u_n)=\infty\), it follows that

\[ \lim_{n\to\infty}\|u_n\|_M=\infty. \]

On the other hand, by virtue of (3),

\[ \frac{\displaystyle \int_G M[u_n(x)]\,dx}{\|u_n\|_M} = \frac{M(k_0u_n)\operatorname{mes}G_n}{\|u_n\|_M} = \frac{M(k_0u_n)}{k_0u_nN^{-1}(u_n)} \leq \frac{C}{k_0} \qquad (n=1,2,\ldots), \]

i.e., for the constructed sequence of functions \(u_n(x)\) relation (1) is not satisfied. The lemma is proved.

The assertion of the lemma makes it possible to indicate such Orlicz spaces \(L_M^*\) for which relation (1) is not satisfied. Such spaces, for example, will be the spaces \(L_M^*\) defined by \(N\)-functions \(M(u)\) complementary to \(N\)-functions \(N(v)\) satisfying the \(\Delta_3\)-condition, since for such functions, for large values of \(u\) and some \(k\), the inequality
\[ M(ku)\leq uN^{-1}(u) \]
holds.

Remark. In checking whether condition (2) is satisfied, instead of the function \(N^{-1}(u)\) one may consider the function \(\widetilde N^{-1}(u)\), inverse to any function \(\widetilde N(v)\) equivalent to \(N(v)\).

Theorem. In order that relation (1) hold, it is necessary and sufficient that there exist a function \(f(u)\) \((0\leq u<\infty)\) satisfying the condition

\[ \lim_{u\to\infty} f(u)=\infty \tag{4} \]

and such that, for every \(v\) and all sufficiently large values of \(u\), the inequality

\[ M(uv)\geq uf(u)M(v). \tag{5} \]

holds.

Proof of necessity. If condition (5) is not satisfied, then this means that for every function \(f(u)\) satisfying condition (4), there will be found a \(v_0\) and a sequence of numbers \(u_n\to\infty\) such that

\[ M(u_nv_0)<u_nf(u_n)M(v_0). \]

In particular, such numbers will also be found for the function \(f(u)=N^{-1}(u)\). But this means that condition (2), necessary for relation (1) to hold, is not satisfied.

Proof of sufficiency. Suppose that the conditions of the theorem are satisfied. We shall prove first of all that relation (1) is satisfied for any sequence of functions \(u_n(x)\) from the subspace \(E_M\), which is the closure in \(L_M^*\) of the set of bounded functions.

By \(\|u\|_{(M)}\) we denote the norm defined for \(u(x)\in L_M^*\) by the equality

\[ \|u\|_{(M)}=\inf k, \]

where the infimum is taken over all such \(k>0\) for which
\[ \rho\!\left(\frac{u}{k};M\right) = \int_G M\!\left[\frac{u(x)}{k}\right]\,dx\leq 1. \]
At the same time, for functions \(u(x)\in E_M\) the equality

\[ \int_G M\!\left[\frac{u(x)}{\|u\|_M}\right]\,dx=1. \]

holds.

This norm is equivalent to the ordinary Orlicz norm:

\[ \|u\|_{(M)}\leq \|u\|_M\leq 2\|u\|_{(M)}. \]

Let \(u_n(x)\in E_M\) \((n=1,2,\ldots)\) and \(\lim_{n\to\infty}\|u_n\|_M=\infty\). Then also \(\lim_{n\to\infty}\|u_n\|_{(M)}=\infty\). Since

\[ \frac{\displaystyle \int_G M[u_n(x)]\,dx}{\|u_n\|_M} \geq \frac{\displaystyle \int_G M[u_n(x)]\,dx}{2\|u_n\|_{(M)}} , \]

it is enough to prove that

\[ \lim_{n\to\infty} \frac{\displaystyle\int_G M[u_n(x)]\,dx}{\|u_n\|_{(M)}}=\infty . \]

The last relation follows directly from inequality (5). Indeed, for large values of \(\|u_n\|_{(M)}\), by (5),

\[ \frac{\displaystyle\int_G M[u_n(x)]\,dx}{\|u_n\|_{(M)}} \ge \frac{\|u_n\|_{(M)}\, f(\|u_n\|_{(M)}) \displaystyle\int_G M\!\left[\frac{u_n(x)}{\|u_n\|_{(M)}}\right]\,dx} {\|u_n\|_{(M)}} = f(\|u_n\|_{(M)}). \]

Now let \(u_n(x)\) \((n=1,2,\ldots)\) be an arbitrary sequence of functions from \(L_M^*\) \((\|u_n\|_M\to\infty)\). Without loss of generality, we may assume that all \(u_n(x)\in L_M\). Since every function from \(L_M\) is at distance not exceeding one from \(E_M\), for each \(u_n(x)\) one can indicate a function \(\tilde u_n(x)\in E_M\) such that \(\|u_n-\tilde u_n\|_M\le 2\). Moreover, the functions \(\tilde u_n(x)\) can be chosen so that, for every \(x\in G\), the inequality \(|\tilde u_n(x)|\le |u_n(x)|\) is satisfied. Then

\[ \frac{\displaystyle\int_G M[u_n(x)]\,dx}{\|u_n\|_M} \ge \frac{\displaystyle\int_G M[\tilde u_n(x)]\,dx}{\|\tilde u_n\|_M+2}, \]

since \(\|u_n\|_M\le \|\tilde u_n\|_M+2\). From the obtained inequality relation (1) follows, since it has already been proved for functions \(\tilde u_n(x)\in E_M\). The theorem is proved.

Taking in condition (5), as the function \(f(u)\), various concrete functions, we obtain various sufficient conditions for the validity of relation (1). Thus, for example, it is easy to verify that the functions
\(M_1(u)=|u|^\alpha (|\ln |u||+\beta)\)
\(\left(\alpha>1,\ \beta\ge \dfrac{2\alpha-1}{\alpha(\alpha-1)}\right)\),
\(M_2(u)=e^{|u|}-|u|-1\),
\(M_3(u)=e^{u^2}-1\)
satisfy condition (5), where as the function \(f(u)\) the function \(|u|^r\) is taken for some \(r>0\).

Voronezh Civil Engineering Institute

Received
20 XII 1960

References

1. I. V. Gelman, Dokl. Akad. Nauk SSSR, 120, No. 3, 454 (1958).
2. M. A. Krasnosel’skii, Ya. B. Rutitskii, Convex Functions and Orlicz Spaces, 1958.