Generalization of a notion of grand Lebesgue space

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 the usual weighted Lebesgue spaces. By means of this theorem we prove the boundedness of the Hardy–Littlewood maximal operator and Calderon–Zygmund singular operators win the weighted grand spaces under consideration.