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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 ultraspherical polynomials, Calderón--Zygmund operators
MSC Classifications: 42C05, 42C15frcs show english descriptions Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
unknown classification 42C15frcs
42C05 - Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
42C15frcs - unknown classification 42C15frcs
 

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