Maximal, potential and singular operators in the local "complementary" variable exponent Morrey type spaces

Guliyev, Vagif S.; Hasanov, Javanshir J.; Samko, Stefan

http://hdl.handle.net/10400.1/11724

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Authors

Guliyev, Vagif S.

Hasanov, Javanshir J.

Samko, Stefan

Abstract(s)

We 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).