Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

Abstract

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.