Boundedness of Integral Operators with Homogeneous Kernels in Weighted Lp-spaces with Radial Weights

In the weighted space Lp(Rn, w), we consider multidimensional integral operators with kernels homogeneous of degree (-n), where the weight function w is assumed to be radial. We obtain sufficient conditions for the boundedness of integral operators with homogeneous kernels in these spaces. In the formulation of these conditions, we make essential use of the dilation function associated with the weight w. These results generalize a theorem by S. M. Umarkhadzhiev, which was previously proved under the additional assumption that the kernel is invariant under all rotations of Rn. We single out the case where the weight function is semi-multiplicative. In addition, we consider separately the case of integral operators with homogeneous kernels acting in the space L∞(Rn, w). Furthermore, we derive necessary conditions on the kernel for the boundedness of integral operators in Lp(Rn, w). As a consequence, we establish necessary conditions for boundedness in the case of kernels invariant under all rotations of Rn, for which the conditions take a significantly simpler form. It is shown that, in general, the sufficient conditions for boundedness do not coincide with the necessary ones. All obtained results are compared with known (classical) results.