Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces

Abstract

Schur's test states that if $K:X\times Y\to \mathbb{C}$ satisfies ${\int }_Y|K(x,y)|d\nu (y)\le C$ and ${\int }_X|K(x,y)|d\mu (x)\le C$, then the associated integral operator acts boundedly on $L^p$ for all $p\in [1,\infty ]$. We derive a variant of this result ensuring boundedness on the (weighted) mixed-norm Lebesgue spaces $L_w^{p,q}$ for all $p,q\in [1,\infty ]$. For non-negative integral kernels our criterion is sharp; i.e., it is satisfied if and only if the integral operator acts boundedly on all of the mixed-norm Lebesgue spaces. Motivated by this criterion, we introduce solid Banach modules ${\mathcal{B}}_m(X,Y)$ of integral kernels such that all kernels in ${\mathcal{B}}_m(X,Y)$ map $L_w^{p,q}(\nu )$ boundedly into $L_v^{p,q}(\mu )$ for all $p,q\in [1,\infty ]$, provided that the weights $v,w$ are $m$-moderate. Conversely, if $\mathbf{A}$ and $\mathbf{B}$ are solid Banach spaces for which all kernels $K\in {\mathcal{B}}_m(X,Y)$ map $\mathbf{A}$ into $\mathbf{B}$, then $\mathbf{A}$ and $\mathbf{B}$ are related to mixed-norm Lebesgue-spaces; i.e., ${\left(L^1\cap L^{\infty }\cap L^{1,\infty }\cap L^{\infty ,1}\right)}_v\hookrightarrow \mathbf{B}$ and $\mathbf{A}\hookrightarrow {\left(L^1+L^{\infty }+L^{1,\infty }+L^{\infty ,1}\right)}_{1/w}$ for certain weights $v,w$ depending on the weight $m$. The kernel algebra ${\mathcal{B}}_m(X,X)$ is particularly suited for applications in (generalized) coorbit theory: Usually, a host of technical conditions need to be verified to guarantee that coorbit space theory is applicable for a given continuous frame $\Psi$ and a Banach space $\mathbf{A}$. We show that it is enough to check that certain integral kernels associated to $\Psi$ belong to ${\mathcal{B}}_m(X,X)$; this ensures that the coorbit spaces ${\mathrm{Co}}_{\Psi }(L_{\kappa }^{p,q})$ are well-defined for all $p,q\in [1,\infty ]$ and all weights $\kappa$ compatible with $m$.

Added appendix on sharpness for complex-valued integral kernels