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

Abstract

Let $({\Omega }_1,{\mathcal{F}}_1,{\mu }_1)$ and $({\Omega }_2,{\mathcal{F}}_2,{\mu }_2)$ be two measure spaces and let $1\le p,q\le +\infty$. We give a definition of Schur multipliers on $\mathcal{B}(L^p({\Omega }_1),L^q({\Omega }_2))$ which extends the definition of classical Schur multipliers on $\mathcal{B}({\ell }_p,{\ell }_q)$. Our main result is a characterization of Schur multipliers in the case $1\le q\le p\le +\infty$. When $1<q\le p<+\infty$, $\varphi \in L^{\infty }({\Omega }_1\times {\Omega }_2)$ is a Schur multiplier on $\mathcal{B}(L^p({\Omega }_1),L^q({\Omega }_2))$ if and only if there are a measure space (a probability space when $p\ne q$) $(\Omega ,\mu )$, $a\in L^{\infty }({\mu }_1,L^p(\mu ))$ and $b\in L^{\infty }({\mu }_2,L^{q^{\prime }}(\mu ))$ such that, for almost every $(s,t)\in {\Omega }_1\times {\Omega }_2$, $$\varphi (s,t)=⟨a(s),b(t)⟩.$$ Here, $L^{\infty }({\mu }_1,L^r(\mu ))$ denotes the Bochner space on ${\Omega }_1$ valued in $L^r(\mu )$. This result is new, even in the classical case. As a consequence, we give new inclusion relationships between the spaces of Schur multipliers on $\mathcal{B}({\ell }_p,{\ell }_q)$.

25 pages