Integral operators commuting with dilations and rotations in generalized Morrey-type spaces

Abstract

We find conditions for the boundedness of integral operators <i><b>K</i></b> commuting with dilations and rotations in a local generalized Morrey space. We also show that under the same conditions, these operators preserve the subspace of such Morrey space, known as vanishing Morrey space. We also give necessary conditions for the boundedness when the kernel is non‐negative. In the case of classical Morrey spaces, the obtained sufficient and necessary conditions coincide with each other. In the one‐dimensional case, we also obtain similar results for global Morrey spaces. In the case of radial kernels, we also obtain stronger estimates of <i><b>K</i></b> <i><b>f</i></b> via spherical means of <i><b>f</i></b>. We demonstrate the efficiency of the obtained conditions for a variety of examples such as weighted Hardy operators, weighted Hilbert operator, their multidimensional versions, and others.