Schur's test for integral operators states that if a kernel $K:X\times Y\to \mathbb{C}$ 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 is bounded from ${\mathbf{L}}^p(\nu )$ into ${\mathbf{L}}^p(\mu )$, simultaneously for all $p\in [1,\infty ]$. We derive a variant of this result which ensures that the integral operator acts boundedly on the (weighted) mixed-norm Lebesgue spaces ${\mathbf{L}}_w^{p,q}$, simultaneously for all $p,q\in [1,\infty ]$. For non-negative integral kernels our criterion is sharp; that is, the integral operator satisfies our criterion if and only if it acts boundedly on all of the mixed-norm Lebesgue spaces.

Motivated by this new form of Schur's test, we introduce solid Banach modules ${\mathcal{B}}_m(X,Y)$ of integral kernels with the property that all kernels in ${\mathcal{B}}_m(X,Y)$ map the mixed-norm Lebesgue spaces ${\mathbf{L}}_w^{p,q}(\nu )$ boundedly into ${\mathbf{L}}_v^{p,q}(\mu )$, for arbitrary $p,q\in [1,\infty ]$, provided that the weights $v,w$ are m-moderate. Conversely, we show that if A and B are non-trivial solid Banach spaces for which all kernels $K\in {\mathcal{B}}_m(X,Y)$ define bounded maps from A into B, then A and B are related to mixed-norm Lebesgue-spaces, in the sense that ${({\mathbf{L}}^1\cap {\mathbf{L}}^{\infty }\cap {\mathbf{L}}^{1,\infty }\cap {\mathbf{L}}^{\infty ,1})}_v\hookrightarrow \mathbf{B}$ and $\mathbf{A}\hookrightarrow {({\mathbf{L}}^1+{\mathbf{L}}^{\infty }+{\mathbf{L}}^{1,\infty }+{\mathbf{L}}^{\infty ,1})}_{1/w}$ for certain weights $v,w$ depending on the weight m used in the definition of ${\mathcal{B}}_m$.

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 the coorbit space ${\mathrm{Co}}_{\mathrm{\Psi }}(\mathbf{A})$ associated to a continuous frame Ψ and a solid Banach space A are well-defined and that the discretization machinery of coorbit theory is applicable. As a simplification, we show that it is enough to check that certain integral kernels associated to the frame Ψ belong to ${\mathcal{B}}_m(X,X)$; this ensures that the spaces ${\mathrm{Co}}_{\mathrm{\Psi }}({\mathbf{L}}_{\kappa }^{p,q})$ are well-defined for all $p,q\in [1,\infty ]$ and all weights κ compatible with m. Further, if some of these integral kernels have sufficiently small norm, then the discretization theory is also applicable.

Schur 对积分运算符的检验表明，如果一个核 $钾：X\times 是\to \mathbb{C}$ 满足 ${\int }_{是}|钾(X,是)|d\nu (是)\le C$ 和 ${\int }_X|钾(X,是)|d\mu (X)\le C$，那么关联的积分运算符的边界是 ${\mathbf{升}}^{磷}(\nu )$ 进入 ${\mathbf{升}}^{磷}(\mu )$, 同时对所有 $磷\in [1,\infty ]$. 我们推导出这个结果的一个变体，它确保积分运算符有界地作用于（加权）混合范数勒贝格空间${\mathbf{升}}_{瓦}^{磷,q}$, 同时对所有 $磷,q\in [1,\infty ]$. For non-negative integral kernels our criterion is sharp; that is, the integral operator satisfies our criterion if and only if it acts boundedly on all of the mixed-norm Lebesgue spaces.

Motivated by this new form of Schur's test, we introduce solid Banach modules ${\mathcal{B}}_m(X,Y)$ of integral kernels with the property that all kernels in ${\mathcal{B}}_m(X,Y)$ map the mixed-norm Lebesgue spaces ${\mathbf{L}}_w^{p,q}(\nu )$ boundedly into ${\mathbf{L}}_v^{p,q}(\mu )$, for arbitrary $p,q\in [1,\infty ]$, provided that the weights $v,w$ are m-moderate. Conversely, we show that if A and B are non-trivial solid Banach spaces for which all kernels $K\in {\mathcal{B}}_m(X,Y)$ define bounded maps from A into B, then A and B are related to mixed-norm Lebesgue-spaces, in the sense that ${({\mathbf{L}}^1\cap {\mathbf{L}}^{\infty }\cap {\mathbf{L}}^{1,\infty }\cap {\mathbf{L}}^{\infty ,1})}_v\hookrightarrow \mathbf{B}$ and $\mathbf{A}\hookrightarrow {({\mathbf{L}}^1+{\mathbf{L}}^{\infty }+{\mathbf{L}}^{1,\infty }+{\mathbf{L}}^{\infty ,1})}_{1/w}$ for certain weights $v,w$ depending on the weight m used in the definition of ${\mathcal{B}}_m$.

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 the coorbit space ${\mathrm{Co}}_{\mathrm{\Psi }}(\mathbf{A})$ associated to a continuous frame Ψ and a solid Banach space A are well-defined and that the discretization machinery of coorbit theory is applicable. As a simplification, we show that it is enough to check that certain integral kernels associated to the frame Ψ belong to ${\mathcal{B}}_m(X,X)$; 这确保了空间${\mathrm{公司}}_{\mathrm{\Psi }}({\mathbf{升}}_{\kappa }^{磷,q})$ 对所有人都有明确的定义 $磷,q\in [1,\infty ]$和所有与m兼容的权重κ。此外，如果这些积分核中的一些具有足够小的范数，则离散化理论也适用。