Reverse Riesz Inequality on Manifolds with Ends

TL;DR

This paper proves the boundedness of the reverse Riesz transform on manifolds with ends across all L^p spaces, revealing nuanced differences from the classical Riesz transform and providing new insights into their distinct behaviors.

Contribution

It establishes the boundedness of the reverse Riesz transform on manifolds with ends for all p, contrasting with the known limited range for the Riesz transform, thus highlighting their non-equivalence.

Findings

Reverse Riesz transform is bounded on all L^p spaces for 1<p<∞.

The Riesz transform is only bounded within a specific p-range, typically for 1<p<n_*.

The study provides a counterexample to the presumed equivalence between Riesz and reverse Riesz transforms.

Abstract

In our investigation, we focus on the reverse Riesz transform within the framework of manifolds with ends. Such manifolds can be described as the connected sum of finite number of Cartesian products $\mathbb{R}^{n_i} \times \mathcal{M}_i$, where $\mathcal{M}_i$ are compact manifolds. We rigorously establish the boundedness of this transform across all $L^p$ spaces for $1<p<\infty$. Notably, existing knowledge indicates that the Riesz transform in such a context demonstrates boundedness solely within a specific range of $L^p$ spaces, typically observed for $1<p<n_*$, where $n_*$ signifies the smallest dimension of the manifold's ends on a large scale. This observation serves as a significant counterexample to the presumed equivalence between the Riesz and reverse Riesz transforms. Our study illuminates the nuanced behaviour of these transforms within the setting of manifolds with ends, providing valuable insights into their distinct properties. Although the lack of equivalence has been previously noted in relevant literature, our investigation contributes to a deeper understanding of the intricate interplay between the Riesz and reverse Riesz transforms.