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 $\int_Y |K(x,y)|d\nu(y)\leq C$ and $\int_X |K(x,y)|d\mu(x)\leq C$, 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 $\operatorname{Co}_\Psi (L_\kappa^{p,q})$ are well-defined for all $p,q\in [1,\infty]$ and all weights $\kappa$ compatible with $m$.