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

Kurzfassung/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.