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Calderón–Zygmund Operators Associated to Ultraspherical Expansions

Published online by Cambridge University Press:  20 November 2018

Dariusz Buraczewski ,
Teresa Martinez  and
José L. Torrea
Dariusz Buraczewski
Affiliation:
Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland email: dbura@math.uni.wroc.pl
Teresa Martinez
Affiliation:
Departamento de Matemáticas, Faculdad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain email: teresa.martinez@uam.es, joseluis.torrea@uam.es
José L. Torrea
Affiliation:
Departamento de Matemáticas, Faculdad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain email: teresa.martinez@uam.es, joseluis.torrea@uam.es
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Abstract

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We define the higher order Riesz transforms and the Littlewood-Paley $g$-function associated to the differential operator ${{L}_{\lambda }}f(\theta )\,=\,-{f}''(\theta )-2\lambda \cot \theta {f}'(\theta )+{{\lambda }^{2}}f(\theta )$. We prove that these operators are Calderón–Zygmund operators in the homogeneous type space $((0,\,\pi ),\,{{(\sin t)}^{2\lambda }}dt)$. Consequently, ${{L}^{p}}$ weighted, ${{H}^{1}}\,-\,{{L}^{1}}$ and ${{L}^{\infty }}\,-\,BMO$ inequalities are obtained.

Keywords

42C0542C15ultraspherical polynomialsCalderón–Zygmund operators
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 6 , 01 December 2007 , pp. 1223 - 1244
Copyright
Copyright © Canadian Mathematical Society 2007

References

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