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- Riesz fractional integrals in grand lebesgue spaces on ℝnPublication . Samko, Stefan; Umarkhadzhiev, SalaudinWe introduce conditions on the construction of grand Lebesgue spaces on R-n which imply the validity of the Sobolev theorem for the Riesz fractional integrals I-alpha and the boundedness of the maximal operator, in such spaces. We also give an inversion of the operator I-alpha by means of hypersingular integrals, within the frameworks of the introduced spaces. We also proof the denseness of C-0(infinity)(R-n) in a subspace of the considered grand space.
- Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline casesPublication . Rafeiro, Humberto; Samko, Stefan; Umarkhadzhiev, SalaudinWe define the grand Lebesgue space corresponding to the case p=infinity$ p = \infty$ and similar grand spaces for Morrey and Morrey type spaces, also for p=infinity$ p = \infty$, on open sets in Rn$ \mathbb {R}<^>n$. We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases alpha p=n$ \alpha p = n$ for Lebesgue spaces and alpha p=n-lambda$ \alpha p = n-\lambda$ for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.
- On Grand Lebesgue spaces on sets of infinite measurePublication . Samko, Stefan; Umarkhadzhiev, SalaudinWe consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so-called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
- 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.