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

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 $max\left(p,p∕\left(p-1\right)\right)$. The approach we present allows us to avoid the use of both differential forms and general spectral multipliers.