Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 3 (2018), 745-773.

Dimension-free $L^p$ estimates for vectors of Riesz transforms associated with orthogonal expansions

Błażej Wróbel

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Abstract

An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multidimensional orthogonal expansions on product spaces. This is then applied to obtain Lp boundedness, 1<p<, of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the Lp norms of these Riesz transforms are both dimension-free and linear in max(p,p(p1)). The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.

Article information

Source
Anal. PDE, Volume 11, Number 3 (2018), 745-773.

Dates
Received: 23 January 2017
Revised: 31 July 2017
Accepted: 23 September 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513774534

Digital Object Identifier
doi:10.2140/apde.2018.11.745

Mathematical Reviews number (MathSciNet)
MR3738261

Zentralblatt MATH identifier
1380.42022

Subjects
Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42A50: Conjugate functions, conjugate series, singular integrals 33C50: Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable

Keywords
Riesz transform Bellman function orthogonal expansion

Citation

Wróbel, Błażej. Dimension-free $L^p$ estimates for vectors of Riesz transforms associated with orthogonal expansions. Anal. PDE 11 (2018), no. 3, 745--773. doi:10.2140/apde.2018.11.745. https://projecteuclid.org/euclid.apde/1513774534


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