Some Inequalities for Integral Operators, Associated with the Bessel Differential Operator

Abstract

In this paper we consider maximal functions, fractional maximal functions and fractional integrals which are generated by a generalized shift operator, associated with the Bessel differential operator \(B = (B_1 , \ldots ,B_n ),\;B_i = \frac{{\partial ^2 }} {{\partial x_i^2 }} + \frac{{\gamma _i }} {{x_i }}\frac{\partial } {{\partial x_i }},\;i = 1, \ldots ,n.\) We present inequalities for these operators in corresponding weightedL p -spaces.In a special case we have found necessary and sufficient conditions for pairs of weights ensuring the validity of strong type inequalities for fractional integrals.