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

We consider local “complementary” generalized Morrey spaces ${\dual \cal M}{{x_0}}^{p(\cdot),\om}(\Om)$inwhichthe$p$-means of function are controlled over $\text{Om}\mathrm{\backslash }B(x_0,r)$ instead of $B(x_0,r)$, where $\Om \subset \Rn$isaboundedopenset,$p(x)$ is a variable exponent, and no monotonicity type conditio is imposed onto the function $\text{om}(r)$ defining the “complementary” Morrey-type norm. In the case where $\text{om}$ 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 ${\dual \cal M}{{x_0}}^{p(\cdot),\om} (\Om)\rightarrow {\dual \cal M}_{{x_0}}^{q(\cdot),\om} (\Om)$-theorem for the potential operators $I^{\text{al}(\cdot )},$ also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $\text{om}(r)$, which do not assume any assumption on monotonicity of $\text{om}(r)$.