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

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.

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Riesz transform, Bellman function, orthogonal expansion
Mathematical Subject Classification 2010
Primary: 42C10, 42A50, 33C50
Received: 23 January 2017
Revised: 31 July 2017
Accepted: 23 September 2017
Published: 22 November 2017
Błażej Wróbel
Mathematical Institute
Universität Bonn
Instytut Matematyczny
Uniwersytet Wrocławski