Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${Vmathcal{L}^{p,varphi}_Pi (Omega), Omega subseteq mathbb{R}^n}$ 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 ${Vmathcal{L}^{p,varphi}_Pi (mathbb{R}^n)}$ 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 ${Vmathcal{L}^{p,varphi}_Pi (mathbb{R}^n) rightarrow Vmathcal{L}^{q,varphi^frac{q}{p}}_Pi (mathbb{R}^n)}$ -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