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

## TL;DR

This paper proves dimension-free $L^p$ bounds for vector Riesz transforms linked to orthogonal expansions, notably Jacobi polynomials, using a Bellman function approach that avoids differential forms and spectral multipliers.

## Contribution

Introduces a Bellman function method to establish dimension-free $L^p$ estimates for vector Riesz transforms associated with orthogonal expansions.

## Key findings

- Dimension-free $L^p$ bounds for Riesz transforms in orthogonal expansions.
- Linear dependence of bounds on $ ext{max}(p,p/(p-1))$.
- Applicable to Jacobi polynomial expansions.

## Abstract

An explicit Bellman function is used to prove a bilinear embedding theorem for operators associated with general multi-dimensional orthogonal expansions on product spaces. This is then applied to obtain $L^p,$ $1<p<\infty,$ boundedness 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.