Classes of operators on weighted function spaces in Dunkl analysis

TL;DR

This paper establishes weighted norm inequalities for operators in Dunkl analysis, including Riesz potentials and fractional maximal operators, and proves a weighted Sobolev inequality, extending classical harmonic analysis results to the Dunkl setting.

Contribution

It provides new sufficient conditions for weighted inequalities of operators in Dunkl analysis, including Riesz potentials and fractional maximal operators, and introduces a weighted Sobolev inequality.

Findings

Weighted (Lp, Lq) boundedness of Riesz potentials in Dunkl analysis.

Weighted boundedness of fractional maximal operators for the Dunkl transform.

A new weighted generalized Sobolev inequality in the Dunkl setting.

Abstract

For indices p and q, 1 < p <= q < infini and a linear operator L satisfying some weak-type boundedness conditions on suitable function spaces, we give in the Dunkl setting sufficient conditions on nonnegative pairs of weight functions to obtain weighted norm inequalities for the operator L. We apply our results to obtain weighted (Lp, Lq) boundedness of the Riesz potentials and of the related fractional maximal operators for the Dunkl transform. Finally, we prove a weighted generalized Sobolev inequality.