Dimension-free Lp estimates for vectors of Riesz transforms associated with orthogonal expansions

Wróbel, Błażej

doi:10.2140/apde.2018.11.745

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

Błażej Wróbel

Vol. 11 (2018), No. 3, 745–773

DOI: 10.2140/apde.2018.11.745

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 $L^{p}$ boundedness, $1 < p < \infty$ , of appropriate vectorial Riesz transforms, in particular in the case of Jacobi polynomials. Our estimates for the $L^{p}$ norms of these Riesz transforms are both dimension-free and linear in $max (p, p ∕ (p - 1))$ . The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.

Keywords

Riesz transform, Bellman function, orthogonal expansion

Mathematical Subject Classification 2010

Primary: 42C10, 42A50, 33C50

Milestones

Received: 23 January 2017

Revised: 31 July 2017

Accepted: 23 September 2017

Published: 22 November 2017

Authors

Błażej Wróbel
Mathematical Institute Universität Bonn Bonn Germany	Instytut Matematyczny Uniwersytet Wrocławski Wrocław Poland

Vol. 11, No. 3, 2018