Proposition 5

The space $\mathscr{L}^{p'}(X\times Y)$ is contained in $\mathscr{N}^{p,p}(X\times Y)$. The linear mapping from $\mathscr{L}^{p'}(X\times Y)$ into $\mathscr{L}(L^p(X); L^{p'}(Y))$ defined by $k\mapsto u_k$ induces, after passing to the quotient, a continuous injective linear mapping from $L^{p'}(X\times Y)$ into $\mathscr{L}(L^p(X); L^{p'}(Y))$.

Let $k\in \mathscr{L}^{p'}(X\times Y)$. The mapping $k$ is measurable (INT, IV, p. 84, § 5, No.$^o6$, Theorem 5). Let $f\in \mathscr{L}^p(X)$ and $g\in \mathscr{L}^p(Y)$. The function $h: (x, y)\mapsto f(x)g(y)$ belong to $\mathscr{L}^p(X\times Y)$ and $\|h\|_p=\|f\|_p\|g\|_p$ (INT, V, p. 95, § 8, No.$^o3$, Corollary 2). The function $hk$ is therefore integrable on $X\times Y$ and we have

by Hölder's inequality (INT, IV, p. 208, § 6, No.$^o4$, Theorem 2), which proves that $k\in \mathscr{N}^{p,p}(X\times Y)$ with $\|k\|_{p,p}\leqslant \|k\|_{p'}$.

Since $p'\not = +\infty$, every function $k\in \mathscr{L}^{p'}(X\times Y)$ is moderate (INT, V, p. 9, § 1, No.$^o3$, Corollary), and the last assertion then results from Proposition 3, b) and c).

For every function $k: X\times Y\rightarrow \mathbf{C}$ and every $y\in Y$, we denote by $k_y$ the mapping of X into $\mathbf{C}$ defined by $k_y(x) =k(x, y)$, and we denote by $k_{\circ}$ the mapping of Y into $\mathscr{F}(X;\mathbf{C})$ defined by $y\mapsto k_y$. Analogously, we shall denote by the same symbol the mapping of Y into the space of functions defined $\mu$-almost everywhere on X which associates to $y$ the class of $k_y$.

For every function $k$ of $X\times Y$ into $\mathbf{C}$, we put

It is an element of $[0,+\infty ]$.