Schur multipliers on $\mathcal{B}(L^p,L^q)$

Let $({\mathrm{\Omega }}_1,{\mathcal{F}}_1,{\mu }_1)$ and $({\mathrm{\Omega }}_2,{\mathcal{F}}_2,{\mu }_2)$ be two measure spaces and let $1\le p,q\le +\mathrm{\infty }$. We give a definition of Schur multipliers on $\mathcal{B}(L^p({\mathrm{\Omega }}_1),L^q({\mathrm{\Omega }}_2))$ which extends the definition of classical Schur multipliers on $\mathcal{B}({\mathrm{\ell }}_p,{\mathrm{\ell }}_q)$. Our main result is a characterization of Schur multipliers in the case $1\le q\le p\le +\mathrm{\infty }$. When $1 < q \leq p < +\infty$,$\phi \in L^{\infty}(\Omega_1 \times \Omega_2)$ is a Schur multiplier on $\mathcal{B}(L^p({\mathrm{\Omega }}_1),L^q({\mathrm{\Omega }}_2))$ if and only if there are a measure space (a probability space when $p\mathrm{\ne }q$) $(\mathrm{\Omega },\mu )$, $a\in L^{\infty}(\mu_1, L^{p}(\mu))$and$b\in L^{\infty}(\mu_2, L^{q’}(\mu))$ such that, for almost every $(s,t)\in {\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2$, $$\varphi (s,t)=⟨a(s),b(t)⟩\mathrm{.}$$ Here, $L^{\infty}(\mu_1, L^{r}(\mu))$denotestheBochnerspaceon$\Omega_1$valuedin$L^r(\mu)$. This result is new, even in the classical case. As a consequence, we give new inclusion relationships between the spaces of Schur multipliers on $\mathcal{B}({\mathrm{\ell }}_p,{\mathrm{\ell }}_q)$.