Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces $${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$$ with a general function $${\varphi(x, r)}$$ defining the Morrey-type norm. Here $${\Pi \subseteq \Omega}$$ is an arbitrary subset in Ω including the extremal cases $${\Pi=\{x_0\}, x_0 \in \Omega}$$ and Π=Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces $${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$$ 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 $${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$$ -theorem for the potential operator I α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on $${\varphi(x, r)}$$ . No monotonicity type condition is imposed on $${\varphi(x, r)}$$ . In case $${\varphi}$$ 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 $${\varphi}$$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces Copyright Springer Science+Business Media New York 2013