Vol. 11, No. 3, 2018

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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
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(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