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 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.

27 pages, to appear in Analysis & PDE

References in corpus (2)

• Dimension free $L^p$ estimates for Riesz transforms via an $H^{\infty }$ joint functional calculus
• Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

Cited by in corpus (5)

• New Gaussian Riesz transforms on variable Lebesgue spaces
• On derivatives, Riesz transforms and Sobolev spaces for Fourier-Bessel expansions
• On Some Operators Associated with Non-Degenerate Symmetric $\alpha$-Stable Probability Measures
• Trilinear embedding for divergence-form operators with complex coefficients
• Bellman Functions and Dimension Free $L^p$estimates for the Riesz Transforms in Bessel settings