In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted ℓ^p-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 ℓ^p-boundedness properties of transplantation operators associated with the system \(\{\varphi _{n}^{\lambda } \}_{n\in \mathbb {N}}\) of ultraspherical functions, a family of eigenfunctions of Δ_λ. In order to show our results we previously establish a vector-valued local Calderón-Zygmund theorem in our discrete setting.

中文翻译：

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与超球面扩张相关的离散谐波分析

在本文中，我们研究与超球面正交函数相关的离散谐波分析。我们建立加权ℓ ^p最大运营商和的Littlewood-佩利的有界性质克-functions由泊松定义并由差分算子产生的热半群$$ {\德尔塔} _ {\拉姆达} F（N）：= A_ {N} ^ {\ lambda} f（n + 1）-2f（n）+ a_ {n-1} ^ {\ lambda} f（n-1），\ quad n \ in \ mathbb {N}，\ lambda> 0 ，$$其中\（a_ {n} ^ {\ lambda}：= \ {（2 \ lambda + n）（n + 1）/ [（n + \ lambda）（n + 1 + \ lambda）] \} ^ {1/2} \），\（n在\ mathbb {N} \）和\（a _ {-1} ^ {\ lambda}：= 0 \）中。我们也证明了加权ℓ ^p移植运营商有界特性与系统相关联\（\ {\ varphi _ {N} ^ {\拉姆达} \} _ {N \在\ mathbb {N}} \）的超球的功能，一个家庭的Δ的本征函数_λ。为了显示我们的结果，我们先前在离散环境中建立了矢量值的局部Calderón-Zygmund定理。