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

Abstract

Schur's test for integral operators states that if a kernel K:X×Y→C satisfies ∫Y|K(x,y)|dν(y)≤C and ∫X|K(x,y)|dμ(x)≤C, then the associated integral operator is bounded from Lp(ν) into Lp(μ), simultaneously for all p∈[1,∞]. We derive a variant of this result which ensures that the integral operator acts boundedly on the (weighted) mixed-norm Lebesgue spaces Lwp,q, simultaneously for all p,q∈[1,∞]. 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 Bm(X,Y) of integral kernels with the property that all kernels in Bm(X,Y) map the mixed-norm Lebesgue spaces Lwp,q(ν) boundedly into Lvp,q(μ), for arbitrary p,q∈[1,∞], 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∈Bm(X,Y) define bounded maps from A into B, then A and B are related to mixed-norm Lebesgue-spaces, in the sense that (L1∩L∞∩L1,∞∩L∞,1)v↪B and A↪(L1+L∞+L1,∞+L∞,1)1/w for certain weights v,w depending on the weight m used in the definition of Bm.The kernel algebra Bm(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 CoΨ(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 Bm(X,X); this ensures that the spaces CoΨ(Lκp,q) are well-defined for all p,q∈[1,∞] 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.