Lp-Mapping Properties of a Class of Spherical Integral Operators

Abstract

In this paper, we study a class of spherical integral operators IΩf. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator IΩf for some 1<p<2 whenever Ω belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on Ω in order to obtain the L2-boundedness of IΩ. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators.