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# Calderón--Zygmund Operators Associated to Ultraspherical Expansions

Published:2007-12-01
Printed: Dec 2007
• Dariusz Buraczewski
• Teresa Martinez
• José L. Torrea
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## Abstract

We define the higher order Riesz transforms and the Littlewood--Paley $g$-function associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta f'(\theta)+\l^2f(\theta)$. We prove that these operators are Calder\'{o}n--Zygmund operators in the homogeneous type space $((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted, $H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.
 Keywords: ultraspherical polynomials, Calderón--Zygmund operators
 MSC Classifications: 42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45] 42C15frcs - unknown classification 42C15frcs