MATHEMATICS

NICOLAE DINCULEANU

ON THE INTEGRAL REPRESENTATION OF LINEAR OPERATORS IN ORLICZ SPACES

(Presented by Academician I. M. Vinogradov on 19 V 1962)

1. Let \(Z\) be a locally compact space; \(\nu\) a positive Radon measure on \(Z\); \(\mathscr E=(E(z))_{z\in Z}\) a family of Banach spaces and \(\mathscr E'=(E'(z))_{z\in Z}\) a family of spaces dual to \(E(z)\). Denote by \(\mathscr C(\mathscr E)\) (respectively by \(\mathscr C(\mathscr E')\)) the set of all vector fields* \(x\) (respectively functional fields \(x'\)) defined on \(Z\) such that \(x(z)\in E(z)\) (respectively \(x'(z)\in E'(z)\)) for every \(z\in Z\).

We assume that there exists a family \(\mathscr A\subset \mathscr C(\mathscr E)\) of continuous vector fields and a family \(\mathscr A'\subset \mathscr C(\mathscr E')\) of continuous functional fields, and that the following condition is satisfied:

For every \(x\in\mathscr A\) and every \(x'\in\mathscr A'\) the scalar function \(z\to \langle x(z),x'(z)\rangle\) is continuous.

Let \(\varphi\) be a positive, increasing and left-continuous function defined on \([0,+\infty]\) such that \(\varphi(0)=0\) and \(0<\varphi(t)<+\infty\) for \(0<t<+\infty\); let \(\psi\) be the function “inverse” to \(\varphi\); \(\Phi\) and \(\Psi\) are the functions defined on \([0,+\infty]\) by the equalities

\[ \Phi(u)=\int_0^u \varphi(t)\,dt,\qquad \Psi(v)=\int_0^v \psi(s)\,ds . \]

Consider the Orlicz space*** \(\mathscr L_{\mathscr A}^{\Phi}(\nu)\). For each linear mapping \(f\) of the space \(\mathscr L_{\mathscr A}^{\Phi}(\nu)\) into the Banach space \(F\), put

\[ |\!|\!|f|\!|\!|=\sup \sum_i |f(\varphi_{A_i}x_i)|,\qquad \|f\|=\sup \left|\sum_i f(\varphi_{A_i}x_i)\right|, \]

where the supremum is taken over all finite families \((A_i)_{1\le i\le n}\) of disjoint, relatively bicompact, Borel subsets of the space \(Z\) and over all finite families \((x_i)_{1\le i\le n}\) of elements of the basic family \(\mathscr A\) such that \(\left\|\sum_i \varphi_{A_i}x_i\right\|_{\Phi}\le 1\).

We have \(\|f\|\le |\!|\!|f|\!|\!|\le +\infty\). If \(F=C\), then \(\|f\|=|\!|\!|f|\!|\!|\). If there exists a constant \(M>0\) such that \(\Phi(2u)\le M\Phi(u)\) for \(u>0\), then

\[ \|f\|=\sup |f(x)|,\quad x\in\mathscr L_{\mathscr A}^{\Phi},\quad \|x\|_{\Phi}\le 1 . \]

In \((^4)\) we proved the following theorem:

Theorem 1. Let \(f\) be a linear mapping of \(\mathscr L_{\mathscr A}^{\Phi}\) into \(F\). Suppose that: 1) there exists a constant \(M>0\) such that \(\Phi(2u)\le M\Phi(u)\) for \(u>0\); 2) \(\mathscr A\) satisfies axiom (G)**; 3) \(F\) is a space dual to a separable Banach space \(S\).

* For the theory of integration see \((^1)\).
* For the theory of vector fields see \((^2)\).
* For the definition and properties of Orlicz spaces see \((^3)\).
*** (G). There exists a countable set \(\mathscr A_0\subset\mathscr A\) such that for every \(z\in Z\) the set \(\{x(z)\mid x\in\mathscr A_0\}\) is dense in \(E(z)\).

We have \(\|||f\||<+\infty\) if and only if there exists an operator field \(z\to U(z)\in \mathcal L(E(z),F)\) such that \(\|U\|_{\Psi}<+\infty\) and

\[ \langle s,f(\mathbf x)\rangle=\int \langle s,U(z)\mathbf x(z)\rangle\,d\nu(z),\qquad \mathbf x\in\mathscr L_{\mathcal A}^{\Phi},\quad s\in S. \tag{1} \]

In this case

\[ {}^{1}\!/_{2}\,\|U\|_{\Psi}\leq \|||f\||\leq \|U\|_{\Psi}. \tag{2} \]

In the present article, starting from a given operator field, we construct, without any assumptions concerning \(\mathcal A\), \(F\), and \(\Phi\), a linear mapping \(f:\mathscr L_{\mathcal A}^{\Phi}\to F\) satisfying equation (1), and seek a condition on \(\mathcal A\) and \(F\) less restrictive than in the preceding theorem, such that \(f\) satisfy relation (2).

1. We first prove the following theorem:

Theorem 2. If \(z\to U(z)\in\mathcal L(E(z),F)\) is an operator field such that, for every \(\mathbf x\in\mathscr L_{\mathcal A}^{\Phi}\) and every \(y'\in F'\), the scalar function \(z\to\langle U(z)\mathbf x(z),y'\rangle\) is \(\nu\)-measurable and the function \(z\to |U(z)\mathbf x(z)|\) is \(\nu\)-integrable, then there exists a linear mapping \(f:\mathscr L_{\mathcal A}^{\Phi}\to F''\) such that

\[ \langle f(\mathbf x),y'\rangle=\int \langle U(z)\mathbf x(z),y'\rangle\,d\nu(z),\qquad \mathbf x\in\mathscr L_{\mathcal A}^{\Phi},\quad y'\in F'; \tag{1'} \]

moreover, \(\|||f\||\leq \|U\|_{\Psi}\leq +\infty\).

Proof. For \(\mathbf x\in\mathscr L_{\mathcal A}^{\Phi}\) and \(y'\in F'\) the function \(z\to\langle U(z)\mathbf x(z),y'\rangle\) is \(\nu\)-integrable, since it is \(\nu\)-measurable and

\[ \int^{*} |\langle U(z)\mathbf x(z),y'\rangle|\,d\nu(z) \leq |y'|\int^{*}|U(z)\mathbf x(z)|\,d\nu(z)<+\infty. \]

The mapping

\[ f(\mathbf x):\quad y'\to \int \langle U(z)\mathbf x(z),y'\rangle\,d\nu(z) \]

is linear and continuous:

\[ \left|\int \langle U(z)\mathbf x(z),y'\rangle\,d\nu(z)\right| \leq |y'|\int |U(z)\mathbf x(z)|\,d\nu(z). \]

Therefore \(f(\mathbf x)\in F''\),

\[ |f(\mathbf x)|\leq \int |U(z)\mathbf x(z)|\,d\nu(z), \]

\[ \langle f(\mathbf x),y'\rangle=\int \langle U(z)\mathbf x(z),y'\rangle\,d\nu(z),\qquad \mathbf x\in\mathscr L_{\mathcal A}^{\Phi},\quad y'\in F'. \]

Obviously, the mapping \(f\) from the space \(\mathscr L_{\mathcal A}^{\Phi}\) into \(F''\) is linear.

Let \(\sum_{i=1}^{n}\varphi_{A_i}\mathbf x_i\) be a vector field such that the \(A_i\) are pairwise disjoint, relatively bicompact sets of the space \(Z\); \(\mathbf x_i\) are elements of the basic family \(\mathcal A\), and

\[ \left\|\sum_{i=1}^{n}\varphi_{A_i}\mathbf x_i\right\|_{\Phi}\leq 1. \]

Then

\[ \sum_{i=1}^{n} |f(\varphi_{A_i}\mathbf x_i)| \leq \sum_{i=1}^{n}\int |U(z)\varphi_{A_i}(z)\mathbf x_i(z)|\,d\nu(z) = \]

\[ = \int\left|U(z)\sum_{i=1}^{n}\varphi_{A_i}(z)\mathbf x_i(z)\right|\,d\nu(z) \leq \|U\|_{\Psi} \left\|\sum_{i=1}^{n}\varphi_{A_i}\mathbf x_i\right\|_{\Phi} \leq \|U\|_{\Psi}; \]

therefore \(\|||f\||\leq \|U\|_{\Psi}\); this completes the proof.

1. We now seek sufficient conditions in order that \(f\) take values in \(F\); that the given operator field \(U\) be unique, locally \(\nu\)-almost everywhere determined by equation (1) or (1′); and that \(f\) satisfy relation (2).

Proposition 1. \(f\) takes values in \(F\) in each of the following cases:

a) \(F\) is dual to a Banach space \(S\);

b) the operator field \(U\) is simply \(\nu\)-measurable, i.e., for every \(\mathbf{x}\in \mathscr{L}_{\mathscr{A}}^{\Phi}\) the function \(z\to U(z)\mathbf{x}(z)\) (with values in \(F\)) is \(\nu\)-measurable. In this case

\[ f(\mathbf{x})=\int U(z)\mathbf{x}(z)\,d\nu(z),\qquad \mathbf{x}\in \mathscr{L}_{\mathscr{A}}^{\Phi}; \tag{1″} \]

c) \(F\) is of countable type (in particular, \(F=C\)). In this case (1″) holds.

To prove a), we consider \(S\subset S''\subset F'\) and, for the proof, take \(y'\in S\).

For the proof of b), we note that the function \(z\to U(z)\mathbf{x}(z)\) is \(\nu\)-integrable for every \(\mathbf{x}\in \mathscr{L}_{\mathscr{A}}^{\Phi}\). Then

\[ \int \langle U(z)\mathbf{x}(z),\,y'\rangle\,d\nu(z) = \left\langle \int U(z)\mathbf{x}(z)\,d\nu(z),\,y'\right\rangle, \]

therefore

\[ f(\mathbf{x})=\int U(z)\mathbf{x}(z)\,d\nu(z). \]

c) is a consequence of b), since if \(F\) is of countable type, then \(U\) is simply \(\nu\)-measurable.

Proposition 2. In each of the following cases the operator field \(U\) is uniquely, locally \(\nu\)-almost everywhere, determined by the indicated equations:

a) \(\mathscr{A}\) satisfies axiom (G) and there exists a countable subset \(S\subset F'\) such that
\[ |y|=\sup_{s\in S}\frac{|\langle y,s\rangle|}{|s|} \]
for every \(y\in F\); equation (1′);

b) \(\mathscr{A}\) satisfies axiom (G) and \(F\) is of countable type; equation (1″);

c) \(\mathscr{A}\) satisfies axiom (G) and \(F\) is dual to a Banach space \(S\) of countable type; equation (1);

d) \(\mathscr{A}\) satisfies axiom (G) and \(U\) is simply \(\nu\)-measurable; equation (1″).

Cases b) and c) reduce to case a). Indeed, in case c) we may assume that \(S\subset F'\); in case b), if \((y_n)\) is a dense sequence in \(F\), then for each \(n\) there exists an element \(s_n\in F'\) such that \(|s_n|=1\) and \(\langle y_n,s_n\rangle=|y_n|\); if we take \(S=\{s_n\}\), then
\[ |y|=\sup_n \frac{|\langle y,s_n\rangle|}{|s_n|} \]
for every \(y\in F\).

To prove the uniqueness of \(U\) in case a), suppose that

\[ \int \langle U(z)\mathbf{x}(z),\,y'\rangle\,d\nu(z)=0 \]

for every \(\mathbf{x}\in \mathscr{L}_{\mathscr{A}}^{\Phi}\) and \(y'\in F'\). There exists a \(\nu\)-null set \(N(\mathbf{x},y')\) such that for \(z\notin N(\mathbf{x},y')\) we have \(\langle U(z)\mathbf{x}(z),y'\rangle=0\). The set
\[ N(\mathbf{x})=\bigcup_{y'\in S} N(\mathbf{x},y') \]
is locally \(\nu\)-null, and for \(z\notin N(\mathbf{x})\) we have \(U(z)\mathbf{x}(z)=0\).

Let \((\mathbf{x}_n)\) be a sequence of elements of \(\mathscr{A}\) such that for each \(z\in Z\) the sequence \((\mathbf{x}_n(z))\) is dense in \(E(z)\). For every bicompact set \(K\subset T\), the set
\[ N_K=\bigcup_{n=1}^{\infty} N(\mathbf{x}_n\varphi_K) \]
is \(\nu\)-null, and for \(z\notin N_K\) we have \(U(z)=0\). Hence it follows that \(\nu\)-almost everywhere on \(K\), \(U(z)=0\); therefore locally \(\nu\)-almost everywhere \(U(z)=0\).

Proposition 3. The relations
\[ {}^{1}\!/_{2}\,\|U\|_{\Psi}\le \|f\|\le \|U\|_{\Psi} \]
hold in each of the following cases:

a) \(\mathscr{A}\) satisfies axiom (G) and \(F\) is of countable type;

b) \(F=C\) and the operator field \(U=\mathbf{x}'\in \mathscr{C}(\mathscr{E}')\) is measurable with respect to \(\mathscr{A}'\) and \(\nu\).

It remains for us to prove only the inequality \(\frac12\|U\|_{\Psi}\leq |||f|||\).

In case a), let \((x_n)\) be a sequence dense in \(F\). Construct a sequence \((s_n)\) in \(F'\) such that \(|s_n|=1\) and \(\langle x_n,s_n\rangle=|x_n|\) for every \(n\). Let \(S\) be the closed subspace in \(F'\) generated by the sequence \((s_n)\). Then \(S\) is of countable type, and \(F\) may be regarded as a subspace of the space \(S'\) dual to \(S\); therefore, for every \(z\in Z\), we may regard \(U(z)\in\mathcal L(E(z),S')\). The inequality \(\frac12\|U\|_{\Psi}\leq |||f|||\) now follows from Theorem 1.

To prove b), we note that for \(U=x'\) we have

\[ \|x'\|_{\Psi}=\sup_{|x|_{\Phi}\leq 1}\int |\langle x(z),x'(z)\rangle|\,d\nu(z), \]

where \(x\in O_{\mathfrak A}^{\Phi}\) (\(x\) is measurable with respect to \(\mathfrak A\) and \(\nu\)), and

\[ |x|_{\Phi}=\int \Phi(|x(z)|)\,d\nu(z)<+\infty). \]

Since for every \(x\in O_{\mathfrak A}^{\Phi}\) the function \(z\to\langle x(z),x'(z)\rangle\) is integrable, we have

\[ \|x'\|_{\Psi} =\sup_{|x|_{\Phi}\leq 1}\left|\int \langle x(z),x'(z)\rangle\,d\nu(z)\right| = \]

\[ =\sup_{|x|_{\Phi}\leq 1}|f(x)| \leq \sup_{|x|_{\Phi}\leq 1}\|f\|\,\|x\|_{\Phi} \leq \sup_{|x|_{\Phi}\leq 1}\|f\|(|x|_{\Phi}+1) \leq 2\|f\|=2|||f|||. \]

It follows from this that

\[ \frac12\|U\|_{\Psi}=\frac12\|x'\|_{\Psi}\leq |||f|||; \]

this completes the proof.

Received
14 V 1962

REFERENCES

¹ N. Bourbaki, Éléments de mathématique, Livre 6, Intégration, ch. I–IV, Paris, 1952; M. A. Naimark, Normed Rings, Moscow, 1956. ² R. Godement, Ann. of Math., 53, 68 (1951). ³ N. Dinculeanu, Rend. Accad. Lincei, 22, 135 (1957); M. A. Krasnosel’skii, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Moscow, 1958. ⁴ N. Dinculeanu, Studia Math., 19, 321 (1960).