Maximal, potential and singular operators in vanishing generalized Morrey spaces
We introduce vanishing generalized Morrey spaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$$</EquationSource> </InlineEquation> with a general function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${\varphi(x, r)}$$</EquationSource> </InlineEquation> defining the Morrey-type norm. Here <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$${\Pi \subseteq \Omega}$$</EquationSource> </InlineEquation> is an arbitrary subset in Ω including the extremal cases <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\Pi=\{x_0\}, x_0 \in \Omega}$$</EquationSource> </InlineEquation> and Π=Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$$</EquationSource> </InlineEquation> we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$$</EquationSource> </InlineEquation> -theorem for the potential operator I <Superscript> α </Superscript>. The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$${\varphi(x, r)}$$</EquationSource> </InlineEquation>. No monotonicity type condition is imposed on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$${\varphi(x, r)}$$</EquationSource> </InlineEquation>. In case <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$${\varphi}$$</EquationSource> </InlineEquation> has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$${\varphi}$$</EquationSource> </InlineEquation>. The proofs are based on pointwise estimates of the modulars defining the vanishing spaces Copyright Springer Science+Business Media New York 2013
