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

Abstract

In this paper, we return to the study of the Watson kernel for the Abel summability of Jacobi polynomial series. These estimates have been studied for over more than 40 years. The main innovations are in the techniques used to get the estimates that allow us to handle the cases $0<\alpha$ as well as $-1<\alpha<0$, with essentially the same methods. To that effect, we use an integral superposition of Natanson kernels, and the A. P. Calderón-Kurtz, B. Muckenhoupt $A_{p}$-weight theory. We consider also a generalization of a theorem due to Zygmund in the context of Borel measures. The proofs are different from the ones given in (Sobre la conjugación y sumabilidad de series de Jacobi (1971) Universidad de Buenos Aires, Studia Math. 49 (1974) 217–224, Colloq. Math. 30 (1974) 277–288 and Illinois J. Math. 41 (1997) 237–265). We will discuss in detail the Calderón–Zygmund decomposition for nonatomic Borel measures in $\mathbb{R}$. We prove that the Jacobi measure is doubling and following (Studia Math. 57 (1976) 297–306), we study the $A_{p}$ weight theory in the context of Abel summability of Jacobi expansions. We consider power weights of the form $(1-x)^{\overline{\alpha}}$, $(1+x)^{\overline{\beta}}$, $-1<{\overline{\alpha}}<0$, $-1<{\overline{\beta}}<0$. 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 $p$.