Riesz transforms in one dimension

TL;DR

This paper investigates the boundedness of Riesz transforms associated with various operators on real lines and half-lines, analyzing how the dimension parameter influences $L^p$ boundedness and exploring related operators like the Hodge projector.

Contribution

It provides a detailed analysis of Riesz transform boundedness across all real dimensions greater than one, including special cases with potentials and boundary conditions, extending previous Euclidean results.

Findings

Boundedness thresholds depend on the dimension $d$, with a critical change at $d=2$.

The Riesz transform is bounded on $L^p$ only at the upper threshold when $d=2$.

Results connect to recent studies on asymptotically Euclidean manifolds and operators with potentials.

Abstract

We study the boundedness on $L^p$ of the Riesz transform $\nabla L^{-1/2}$, where $L$ is one of several operators defined on $\R$ or $\R_+$, endowed with the measure $r^{d-1} dr$, $d > 1$, where $dr$ is Lebesgue measure. For integer $d$, this mimics the measure on Euclidean $d$-dimensional space, and in this case our setup is equivalent to looking at the Laplacian acting on radial functions on Euclidean space or variations of Euclidean space such as the exterior of a sphere (with either Dirichlet or Neumann boundary conditions), or the connected sum of two copies of $\R^d$. In this way we illuminate some recent results on the Riesz transform on asymptotically Euclidean manifolds. We are however interested in all real values of $d > 1$, and another goal of our analysis is to study the range of boundedness as a function of $d$; it is particularly interesting to see the behaviour as $d$ crosses 2. For example, in one of our cases which models radial functions on the connected sum of two copies of $\R^d$, the upper threshold for $L^p$ boundedness is $p=d$ for $d \ge 2$ and $p=d/(d-1)$ for $d < 2$. Only in the case $d=2$ is the Riesz transform actually bounded on $L^p$ when $p$ is equal to the upper threshold. We also study the Riesz transform when we have an inverse square potential, or a delta function potential; these cases provide a simple model for recent results of the first author and Guillarmou. Finally we look at the Hodge projector in a slightly more general setup.