Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces with a general function defining the Morrey-type norm. Here is an arbitrary subset in Omega including the extremal cases and I = Omega, which allows to unify vanishing local and global Morrey spaces. In the spaces 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 -theorem for the potential operator I (alpha) . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on . No monotonicity type condition is imposed on . In case 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 . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces.