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 ${L}^{p}$ boundedness, $1, 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 $max\left(p,p∕\left(p-1\right)\right)$. 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