A characterization of two-weight norm inequalities for multidimensional Hausdorff operators on Lebesgue spaces

Abstract

In this paper we give necessary conditions and sufficient conditions for the boundedness of multidimensional Hausdorff operator on weighted Lebesgue spaces. In particular, we establish necessary and sufficient conditions for the boundedness of special type multidimensional Hausdorff operator on weighted Lebesgue spaces for monotone radial weight functions. Also, we get similar results for important operators of harmonic analysis which are special cases of the multidimensional Hausdorff operator. The results are illustrated by a number of examples.