Abstract:
We investigate operators of the form M_a H, HM_a, and H_b, where M_a is the multiplication operator on the function a, H is the integral convolution operator with the kernel h, and H_b 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 M_a H, wherein it is assumed that the function a approaches zero at infinity. Next, the commutator of the operators M_a 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 HM_a. In particular, we prove that the operators P_X H and HP_X are compact, where P_X is the multiplication operator on the characteristic function of a bounded measurable set X. Finally, the integral operator H_b is considered. It is shown that if characteristic b has a given behavior at infinity, then the operator H_b is compact.
First page:
PDF פאיכ