On Abel summability of Jacobi polynomials series, the Watson Kernel and applications

TL;DR

This paper advances the understanding of Abel summability for Jacobi polynomial series by developing new estimation techniques using Poisson kernels and weight theory, extending results to broader parameter ranges and measures.

Contribution

It introduces novel techniques for estimating the Watson kernel, generalizes Zygmund's theorem to Borel measures, and applies A_p weight theory to Jacobi expansions with negative exponents.

Findings

Established new estimates for the Watson kernel handling 0<α and -1<α<0 cases.

Proved Jacobi measure is doubling and developed A_p weight theory for Jacobi expansions.

Derived L^p bounds for the maximal Abel summability operator in the Jacobi setting.

Abstract

In this paper we return to the study of the Watson kernel for the Abel summabilty of Jacobi polynomial series. These estimates have been studied for over more than 30 years. The main innovations are in the techniques used to get the estimates that allow us to handle the case 0<\alpha as well as -1< \alpha <0, with essentially the same method; using an integral superposition of Poisson type kernel and Muckenhoupt A_p-weight theory. We consider a generalization of a theorem due to Zygmund in the context to Borel measures. The proofs are therefore different from the ones given in previous papers by several authors. We will also discuss in detail the Calder\'on-Zygmund decomposition for non-atomic Borel measures in the real line. Then, we prove that the Jacobi measure is doubling and therefore, following a work of A. P. Calder\'on, we study the corresponding A_p weight theory in the setting of Jacobi expansions, considering power weights of the form (1-x)^{\bar{\alpha}}, (1+x)^{\bar{\beta}}, -1 < {\bar{\alpha}}<0,\, -1 < {\bar{\beta}}<0 with negative exponents. Finally, as an application of the weight theory we obtain L^p estimates for the maximal operator of Abel summability of Jacobi function expansions for suitable values of p.