Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces - Guliyev, Vagif S.; Hasanov, Javanshir J.; Samko, Stefan G. (MATHEMATICA SCANDINAVICA, 2010)
We consider generalized Morrey spaces ${\mathcal M}^{p(·),ω}(Ω)$ with variable exponent $p(x)$ and a general function $ω (x,r)$ defining the Morrey-type norm. In case of bounded sets $Ω \subset {\mathsf R}^n$ 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-Adams type ${\mathcal M}^{p(·),ω} (Ω)→ {\mathcal M}^{q(·),ω} (Ω)$-theorem for the potential operators $I^{α(·)}$, also of variable order. The conditions for the boundedness are given it terms of Zygmund-type integral inequalities on $ω(x,r)$, which do not assume any assumption on monotonicity of $ω(x,r)$ in $r$.