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

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 operator is bounded from Lp(ν) into Lp(μ), simultaneously all p∈[1,∞]. We derive variant of this result which ensures acts boundedly on (weighted) mixed-norm Lebesgue spaces Lwp,q, p,q∈[1,∞]. For non-negative kernels our criterion sharp; is, only it spaces. Motivated by new form test, we introduce solid Banach modules Bm(X,Y) with property in map Lwp,q(ν) Lvp,q(μ), arbitrary p,q∈[1,∞], provided weights v,w are m-moderate. Conversely, show A B non-trivial K∈Bm(X,Y) define maps B, related to Lebesgue-spaces, sense (L1∩L∞∩L1,∞∩L∞,1)v↪B A↪(L1+L∞+L1,∞+L∞,1)1/w certain depending weight m used definition Bm. The algebra Bm(X,X) particularly suited applications (generalized) coorbit theory. Usually, host technical conditions need be verified guarantee space CoΨ(A) continuous frame Ψ well-defined discretization machinery theory applicable. As simplification, enough check belong Bm(X,X); CoΨ(Lκp,q) p,q∈[1,∞] κ compatible m. Further, some these have sufficiently small norm, also