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

摘要

Schur's test for integral operators states that if a kernel K: XxY. C satisfies integral(Y) K( x, y) d.(y) X K( x, y) d mu(x) = C, then the associated integral operator is bounded from Lp(nu) into L-p(mu), simultaneously for all p is an element of1, infinity. We derive a variant of this result which ensures that the integral operator acts boundedly on the (weighted) mixed-normLebesgue spaces L-w(p,q), simultaneously for all p, q.1, infinity. For non-negativeintegral kernels our criterion is sharp; that is, the integral operator satisfies our criterion if and only ifit 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 mixednorm Lebesgue spaces Lp, qw(.) boundedly into Lp, qv( mu), for arbitrary p, q.1, 8, provided that the weights v, ware m-moderate. Conversely, we show that if Aand Bare non-trivial solid Banach spaces for which all kernels K. Bm( X, Y) define bounded maps from Ainto B, then Aand Bare related to mixed-norm Lebesgue-spaces, in the sense that (L1n L8n L1,8n L8,1) v. Band A(sic)(L1+ L8+ L1,8+ L8,1) 1/wfor certain weights v, wdepending on the weight mused 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 Co.(A) associated to a continuous frame.and a solid Banach space Aare 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.belong to Bm( X, X); this ensures that the spaces Co.(Lp, q.) are welldefined for all p, q.1, 8 and all weights.compatible with m. Further, if some of these integral kernels have sufficiently small norm, then the discretization theory is also applicable. (C) 2021 Elsevier Inc. All rights reserved.