On Convolution-Type Operators with Kernels from Modified Morrey Spaces

We investigate operators of the form MaH, HMa, and Hb, where Ma is the multiplication operator on the function a, H is the integral convolution operator with the kernel h, and Hb is the integral operator with the bounded characteristic b(x,y). It is assumed that the kernel h belongs to the intersection of the Lebesgue and Morrey spaces, and the operators themselves act from the Lebesgue space to the Morrey space. First, using conditions for the precompactness of a set in the Morrey space, we prove the compactness of the operator Ma H, wherein it is assumed that the function a approaches zero at infinity. Next, the commutator of the operators Ma and H is considered. It is shown that if the function a belongs to a certain class of functions with a given behavior at infinity, then the commutator is the compact operator. This, in turn, allows us to establish the compactness of the operator HMa. In particular, we prove that the operators PX H and HPX are compact, where PX is the multiplication operator on the characteristic function of a bounded measurable set X. Finally, the integral operator Hb is considered. It is shown that if characteristic b has a given behavior at infinity, then the operator Hb is compact.