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- Grand lebesgue spaces with mixed local and global aggrandization and the maximal and singular operatorsPublication . Rafeiro, H.; Samko, Stefan; Umarkhadzhiev, S.The 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.
- Herz spaces meet Morrey type spaces and complementary Morrey type spacesPublication . Rafeiro, Humberto; Samko, StefanWe introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complementary spaces, based on the mentioned inclusion.
- Coincidence of variable exponent Herz spaces with variable exponent Morrey type spaces and boundedness of sublinear operators in these spacesPublication . Rafeiro, Humberto; Samko, StefanWe introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized variable Herz spaces with application to variable Morrey type spaces and their complementary spaces, based on the mentioned inclusion.
- Local grand Lebesgue spaces on quasi-metric measure spaces and some applicationsPublication . Rafeiro, Humberto; Samko, Stefan; Umarkhadzhiev, SalaudinWe introduce local grand Lebesgue spaces, over a quasi-metric measure space (X, d, mu), where the Lebesgue space is "aggrandized" not everywhere but only at a given closed set F of measure zero. We show that such spaces coincide for different choices of aggrandizers if their Matuszewska-Orlicz indices are positive. Within the framework of such local grand Lebesgue spaces, we study the maximal operator, singular operators with standard kernel, and potential type operators. Finally, we give an application to Dirichlet problem for the Poisson equation, taking F as the boundary of the domain.