Maximal and singular integral operators in weighted grand variable exponent Lebesgue spaces

Abstract

Weighted inequalities with power-type weights for operators of harmonic analysis, such as maximal and singular integral operators, and commutators of singular integrals in grand variable exponent Lebesgue spaces are derived. The spaces and operators are defined on quasi-metric measure spaces with doubling condition (spaces of homogeneous type). The proof of the result regarding the Hardy–Littlewood maximal operator is based on the appropriate sharp weighted norm estimates with power-type weights. To obtain the results for singular integrals and commutators we prove appropriate weighted extrapolation statement in grand variable exponent Lebesgue spaces. The extrapolation theorem deals with a family of pairs of functions (f,\xa0g). One of the consequences of the latter result is the weighted extrapolation for sublinear operators S acting in these spaces. As one of the applications of the main results we present weighted norm estimates for the Hardy–Littlewood maximal function, Cauchy singular integral operator, and its commutators in grand variable exponent Lebesgue spaces defined on rectifiable regular curves.