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- Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spacesPublication . Guliyev, Vagif S.; Hasanov, Javanshir J.; Samko, StefanWe consider local "complementary" generalized Morrey spaces M-c({x0})p(.).omega (Omega) in which the p-means of function are controlled over Omega \ B(x(0), r) instead of B(x(0), r), where Omega subset of R-n is a bounded open set, p(x) is a variable exponent, and no monotonicity type condition is imposed onto the function omega(r) defining the "complementary" Morrey-type norm. In the case where omega is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type M-c({x0})p(.).omega (Omega) -> M-c({x0})p(.).omega (Omega)-theorem for the potential operators I-alpha(.), also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities-on omega(r), which do not assume any assumption on monotonicity of omega(r).
- Vanishing generalized Orlicz-Morrey spaces and fractional maximal operatorPublication . Deringoz, Fatih; Guliyev, Vagif S.; Samko, StefanWe find sufficient conditions for the non-triviality of the generalized Orlicz-Morrey spaces M-Phi,M-phi (R-n), and prove the boundedness of the fractional maximal operator and its commutators with BMO-coefficients in vanishing generalized Orlicz-Morrey spaces VM Phi,phi (R-n) including weak versions of these spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators involving the Young functions Phi(u), Psi(u) and the function phi(x,r) defining the space. No kind of monotonicity condition on phi(x,r) in r is imposed.