Translation invariant maps on function spaces over locally compact groups

Abstract

We prove that under adequate geometric requirements, translation invariant mappings between vector-valued quasi-Banach function spaces on a locally compact group G have a bounded extension between Köthe–Bochner spaces L r ( G , E ) . The class of mappings for which our results apply includes polynomials and multilinear operators. We develop an abstract approach based on some new tools as abstract convolution and matching among Banach function lattices, and also on some classical techniques as Maurey–Rosenthal factorization of operators. As a by-product we show when Haar measures which appear in certain factorization theorems for nonlinear mappings are in fact Pietsch measures. We also give applications to operators between Köthe–Bochner spaces.