MATHEMATICS

I. V. SHRAGIN

ON THE CONTINUITY OF THE NEMYTSKII OPERATOR IN ORLICZ SPACES

(Presented by Academician A. N. Kolmogorov on 9 V 1961)

In papers \((^{1-6})\) certain sufficient conditions were established for the continuity of the Nemytskii operator in Orlicz spaces. However, even the identity transformation of an arbitrary Orlicz space (such a transformation is a special case of a Nemytskii operator) does not satisfy these conditions.

Here we establish necessary and sufficient conditions for the continuity of the Nemytskii operator in Orlicz spaces, as well as a number of other propositions.

\(1^\circ\). Some information from the theory of Orlicz spaces in the sense of Zaanen—Luxemburg \((^{7,8})\). Let a nonnegative function \(\varphi(t)\), \(0 \le t < \infty\) (which may assume infinite values, but does not become identically on \([0,+\infty)\) either zero or infinity), be nondecreasing, with \(\varphi(t+0)=\varphi(t)\) for every \(t\).

The function
\[ \Phi(u)=\int_0^u \varphi(t)\,dt,\quad u\geqslant 0, \]
is called a Young function. Put, for \(\alpha>0\),
\[ L_\Phi^\alpha=\left\{u(x):\int_B \Phi[\alpha |u(x)|]\,dx<\infty\right\}, \]
where \(B\) is some subset of finite-dimensional Euclidean space of positive (finite or infinite) Lebesgue measure. By definition, the Orlicz space \(L^\Phi\) is the union of all classes \(L_\Phi^\alpha\). We shall consider it normed by means of the norm
\[ \|u\|_\Phi=\inf\left\{\rho>0:\int_B \Phi[\rho^{-1}|u(x)|]\,dx\leqslant 1\right\}. \]

By \(L_\Phi^f\) is denoted the subspace of the space \(L^\Phi\) that is the intersection of all classes \(L_\Phi^\alpha\). We note that: 1) \(L_\Phi^f\) consists only of the zero element if and only if \(d_\Phi=\sup\{u:\Phi(u)<\infty\}<\infty\); 2) \(L_\Phi^f=L^\Phi\) if and only if \(d_\Phi=\infty\) and \(\Phi(u)\) satisfies the \(\Delta_2\)-condition: \(\Phi(2u)\leqslant C\Phi(u)\) for \(u\geqslant u_0\geqslant 0\) \((C=\text{const})\), where \(u_0=0\) if \(\operatorname{mes} B=\infty\).

In this note we shall also consider Orlicz spaces whose elements are vector functions defined on the set \(B\). Namely, let \(M_1(u), M_2(u),\ldots,M_s(u)\) be arbitrary Young functions. Denote by \(\vec L_M^{\,\vec \alpha}\), where \(\vec\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_s)\), the class of vector functions \(\mathbf u(x)=(u_1(x),u_2(x),\ldots,u_s(x))\), for which \(u_k(x)\in L_{M_k}^{\alpha_k}\), \(k=1,2,\ldots,s\). Put further,
\[ L^M=\{u(x): u_k(x)\in L^{M_k},\ k=1,2,\ldots,s\},\quad L_M^f=\{u(x): u_k(x)\in L_{M_k}^f,\ k=1,2,\ldots,s\}. \]
In the space \(L^M\) we introduce the norm
\[ \|u\|_M=\max \|u_k\|_{M_k},\quad k=1,2,\ldots,s. \]

2°. Let the real-valued function \(g(u_1,u_2,\ldots,u_s,x)\equiv g(\mathbf u,x)\), where \(x\in B\), \(u_k\in(-\infty,+\infty)\), \(k=1,2,\ldots,s\), be measurable in \(x\) for each \(\mathbf u\) and continuous in \(\mathbf u\) for almost every \(x\in B\). Such a function generates a Nemytskii operator \(h\), \(h\mathbf u(x)=g(\mathbf u(x),x)\).

In the present section we shall establish several propositions on the action of the operator \(h\) in Orlicz spaces. The proofs of these propositions use the construction applied in \((^3,^4)\).

Theorem 1. In order that the operator \(h\) act from \(L_M^{\vec\alpha}\) to \(L_\Phi^\beta\), it is necessary and sufficient that there exist a \(\gamma>0\) and an integrable function \(f(x)\) on \(B\) such that, for all \(x\in B\) and \(\mathbf u\in R^s\),

\[ \Phi[\beta |g(\mathbf u,x)|]\leq \gamma\sum_{k=1}^{s} M_k[\alpha_k |u_k|]+f(x). \]

Put \(\mathbf T_r=\{\mathbf u\in L^M:\|\mathbf u\|_M\leq r\}\). It is known \((^6,^8)\) that \(\mathbf T_r\subset L_M^{\vec\alpha}\), where \(\vec\alpha=(r^{-1},r^{-1},\ldots,r^{-1})\).

Theorem 2. If \(h\) maps the ball \(\mathbf T_r\) into the class \(L_\Phi^\beta\) (in the space \(L^\Phi\)), then it maps \(L_M^{\vec\alpha}\), where \(\vec\alpha=(r^{-1},r^{-1},\ldots,r^{-1})\), into \(L_\Phi^\beta\) (respectively, into some class \(L_\Phi^\mu\)).

Theorem 3. If \(d_{M_k}=\infty\), \(k=1,2,\ldots,s\), and \(h\) maps the ball \(\{\mathbf u\in L_M^f:\|\mathbf u\|_M\leq r\}\) into the class \(L_\Phi^\beta\) (in the space \(L^\Phi\)), then it maps some class \(L_M^{\vec\lambda}\) into \(L_\Phi^\beta\) (respectively, into some class \(L_\Phi^\mu\)).

We note that the last two theorems contain assertions close to Theorem 17.2 of \((^6)\).

3°. Criterion for continuity of the Nemytskii operator at a point. In the present section we assume throughout that \(h\mathbf u_0\in L^\Phi\), where \(\mathbf u_0\) is a fixed point of the space \(L^M\). By continuity of the operator \(h\) at the point \(\mathbf u_0\) we shall mean the following: \(h\) maps some neighborhood \(U\) (in the space \(L^M\)) of the point \(\mathbf u_0\) into the space \(L^\Phi\), and \(\lim\|h\mathbf u-h\mathbf u_0\|_\Phi=0\) when \(\|\mathbf u-\mathbf u_0\|_M\to0\), where \(\mathbf u\in U\).

Lemma 1. For continuity of \(h\) at the point \(\mathbf u_0\) it is necessary and sufficient that, for every \(\mu>0\), there exist a \(\vec\lambda\) such that the operator \(h_1\), \(h_1\mathbf v=h(\mathbf u_0+\mathbf v)-h\mathbf u_0\), maps \(L_M^{\vec\lambda}\) into \(L_\Phi^\mu\).

Lemma \(1'\). For continuity of \(h\) at the point \(\mathbf u_0\) it is necessary, and if \(d_{M_k}=\infty\), \(k=1,2,\ldots,s\), then also sufficient, that the operator \(h_1\) map \(L_M^f\) into \(L_\Phi^f\).

Theorem 4. For continuity of \(h\) at the point \(\mathbf u_0\) it is necessary and sufficient that, for every \(\mu>0\), there exist a \(\rho>0\) and an integrable function \(f(x)\) on \(B\) such that

\[ \Phi\!\left[\mu\left|g(\mathbf u_0(x)+\mathbf v,x)-g(\mathbf u_0(x),x)\right|\right] \leq \sum_{k=1}^{s} M_k[\rho |v_k|]+f(x) \]

for all \(x\in B\) and \(\mathbf v\in R^s\).

Remark. Let \(d_{M_k}=\infty\), \(k=1,2,\ldots,s\). Then continuity (in the sense indicated above) of the operator \(h\) at the point \(\mathbf u_0\) follows from continuity in the “narrow” sense: \(\lim\|h(\mathbf u_0+\mathbf v)-h\mathbf u_0\|_\Phi=0\) as \(\|\mathbf v\|_M\to0\), where \(\mathbf v\in L_M^f\).

4°. With the aid of the criterion for continuity of \(h\) at a point established above, the following propositions are obtained.

Theorem 5. If \(h\) is continuous at the point \(\mathbf u_0\in L^M\), then it is continuous at every point \(\mathbf u=\mathbf u_0+\mathbf u_1\), where \(\mathbf u_1\in L^f_M\).

Corollary 1. If \(L^M=L^f_M\) and \(h\) is continuous at some point \(\mathbf u_0\in L^M\), then it is continuous at every point \(\mathbf u\in L^M\).

Theorem 6. If \(h\) is continuous at every point of the ball \(T_r\), then it is continuous at every point of the class \(L^{\vec\alpha}_M\), where \(\vec\alpha=(r^{-1},r^{-1},\ldots,r^{-1})\).

5°. Criterion for the continuity of the Nemytskii operator on the subspace \(L^f_M\).

Theorem 7. Let \(g(0,x)\in L^\Phi\). Then, for \(h\) to be continuous at every point \(\mathbf u_0\in L^f_M\), it is necessary and sufficient that for every \(\mu>0\) there exist \(\rho>0\) and a function \(f(x)\), integrable on \(B\), such that for all \(x\in B\) and \(\mathbf v\in R^s\)

\[ \Phi\,[\mu\,|g(\mathbf v,x)-g(0,x)|]\geq \sum_{k=1}^{s} M_k[\rho\,|v_k|]+f(x). \]

Theorem \(7'\). Let \(g(0,x)\in L^\Phi\). Then, for \(h\) to be continuous at every point \(\mathbf u_0\in L^f_M\), it is necessary, and if \(d_{M_k}=\infty,\ k=1,2,\ldots,s\), also sufficient, that the operator \(\widetilde h\),

\[ \widetilde h\mathbf v(x)=g(\mathbf v(x),x)-g(0,x), \]

act from \(L^f_M\) into \(L^f_\Phi\).

6°. Criterion for the continuity of the Nemytskii operator on the class \(L^{\vec\alpha}_M\) and the space \(L^M\).

Theorem 8. Let \(h\) act from \(L^{\vec\alpha}_M\) into \(L^\Phi\) (from \(L^M\) into \(L^\Phi\)). Then, for \(h\) to be continuous at every point \(\mathbf u_0\in L^{\vec\alpha}_M\) (respectively, \(\mathbf u_0\in L^M\)), it is necessary and sufficient that for every \(\mu>0\) (respectively, for any positive \(\alpha_1,\alpha_2,\ldots,\alpha_s\) and \(\mu\)) there exist \(\gamma>0\), \(\rho>0\), and a function \(f(x)\), integrable on \(B\), such that

\[ \Phi\,[\mu\,|g(\mathbf u+\mathbf v,x)-g(\mathbf u,x)|]\leq \gamma\sum_{k=1}^{s} M_k[\alpha_k\,|u_k|] +\sum_{k=1}^{s} M_k[\rho\,|v_k|]+f(x) \]

for all \(x\in B\) and \(\mathbf u,\mathbf v\in R^s\).

Kostroma State Pedagogical Institute
named after N. A. Nekrasov

Received
5 V 1961

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