Rafeiro, H.Samko, StefanUmarkhadzhiev, S.2023-12-072023-12-0720230133-3852http://hdl.handle.net/10400.1/20195The approach to "locally" aggrandize Lebesgue spaces, previously suggested by the authors and based on the notion of "aggrandizer", is combined with the usual "global" aggrandization. We study properties of such spaces including embeddings, dependence of the choice of the aggrandizer and, in particular, we discuss the question when these spaces are not new, coinciding with globally aggrandized spaces, and when they proved to be new. We study the boundedness of the maximal, singular, and maximal singular operators in the introduced spaces.engGrand lebesgue spaceMaximal functionSingular integralMaximal singular integraljournal article10.1007/s10476-023-0243-1