Dimension free boundedness of Riesz transforms for the Grushin operator

TL;DR

This paper proves that Riesz transforms linked to the Grushin operator are bounded on L^p spaces with bounds independent of the dimension, advancing understanding of harmonic analysis on degenerate operators.

Contribution

It establishes dimension-free boundedness of Riesz transforms for the Grushin operator, a significant result in analysis of degenerate elliptic operators.

Findings

Riesz transforms are bounded on L^p for 1<p<∞

Boundedness norms are independent of the dimension n

Extends harmonic analysis techniques to the Grushin operator

Abstract

Let $G = - \Delta_{\xi} - |\xi|^2 \frac{\partial^2}{\partial \eta^2}$ be the Grushin operator on $\R^n \times \R.$ We prove that the Riesz transforms associated to this operator are bounded on $L^p (\R^{n+1}), 1 < p < \infty$ and their norms are independent of the dimension $n$.