Maximal, potential, and singular operators in the generalized variable exponent Morrey spaces on unbounded sets (Q376513)

The authors consider generalized Morrey spaces \(\mathcal{M}^{p(\cdot),\omega(\cdot)}(\Omega)\) with a variable exponent \(p(x)\) and a general function \(\omega(x,r)\) for bounded sets \(\Omega \subset \mathbb{R}^n\). They prove the boundedness of the Hardy-Littlewood maximal operator and singular integral operators for generalized Morrey spaces \(\mathcal{M}^{p(\cdot),\omega(\cdot)}(\Omega)\). They also prove both Spanne type and Adams type theorems of fractional maximal operators \(M^{\alpha}\) and potential type operators \(I^{\alpha}\) of variable order \(\alpha(x)\) for generalized Morrey space \(\mathcal{M}^{p(\cdot),\omega(\cdot)}(\Omega)\) even if \(\Omega\) is unbounded. The boundedness conditions are formulated either in terms of Zygmund type integral inequalities on \(\omega(x,r)\) or in terms of supremal operators.