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Boundedness of projection operators and Cesàro means in weighted $ L^p$ space on the unit sphere


Authors: Feng Dai and Yuan Xu
Journal: Trans. Amer. Math. Soc. 361 (2009), 3189-3221
MSC (2000): Primary 33C50, 42B08, 42C10
DOI: https://doi.org/10.1090/S0002-9947-09-04846-6
Published electronically: January 28, 2009
MathSciNet review: 2485423
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Abstract: For the weight function $ \prod_{i=1}^{d+1}\vert x_i\vert^{2\kappa_i}$ on the unit sphere, sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the Cesàro $ (C,\delta)$ means in the weighted $ L^p$ space for $ \delta$ above the critical index. Similar results are also proved for corresponding weight functions on the unit ball and on the simplex.


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Additional Information

Feng Dai
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta. Edmonton, Alberta, Canada T6G 2G1
Email: dfeng@math.ualberta.ca

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: yuan@math.uoregon.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04846-6
Keywords: Projection operator, Ces\`aro means, weighted $L^p$ space, unit sphere
Received by editor(s): July 19, 2007
Published electronically: January 28, 2009
Additional Notes: The first author was partially supported by the NSERC Canada under grant G121211001
The second author was partially supported by the NSF under Grant DMS-0604056
Article copyright: © Copyright 2009 American Mathematical Society

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