Generalization of the notion of grand Lebesgue space

Abstract

We introduce families of weighted grand Lebesgue spaces which generalize weighted grand Lebesgue spaces (known also as Iwaniec-Sbordone spaces). The generalization admits a possibility of expanding usual (weighted) Lebesgue spaces to grand spaces by various ways by means of additional functional parameter. For such generalized grand spaces we prove a theorem on the boundedness of linear operators under the information of their boundedness in ordinary weighted Lebesgue spaces. By means of this theorem we prove boundedness of the Hardy-Littlewood maximal operator and the Calderon-Zygmund singular operators in the weighted grand spaces.