Discrete Harmonic Analysis Associated with Ultraspherical Expansions

Abstract

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted lp-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator $$ {\Delta}_{\lambda} f(n):=a_{n}^{\lambda} f(n+1)-2f(n)+a_{n-1}^{\lambda} f(n-1),\quad n\in \mathbb{N}, \lambda >0, $$ where $a_{n}^{\lambda } :=\{(2\lambda +n)(n+1)/[(n+\lambda )(n+1+\lambda )]\}^{1/2}$ , $n\in \mathbb {N}$ , and $a_{-1}^{\lambda }:=0$ . We also prove weighted lp-boundedness properties of transplantation operators associated with the system $\{\varphi _{n}^{\lambda } \}_{n\in \mathbb {N}}$ of ultraspherical functions, a family of eigenfunctions of Δλ. In order to show our results we previously establish a vector-valued local Calderon-Zygmund theorem in our discrete setting.