# Endpoint Estimates For Riesz Transform And Hardy-Hilbert Type Inequalities

## TL;DR

This paper establishes endpoint boundedness of the Riesz transform on certain non-doubling manifolds, completing the understanding of its behavior at critical Lebesgue space exponents.

## Contribution

It proves the Riesz transform is bounded from Lorentz space L^{n^*,1} to itself on non-doubling manifolds, extending previous L^p results to endpoint cases.

## Key findings

- Riesz transform is weak type (1,1) on the manifolds.
- Bounded on L^p for 1<p<n^*.
- Bounded on Lorentz space L^{n^*,1}.

## Abstract

We consider a class of non-doubling manifolds $\mathcal{M}$ defined by taking connected sum of finite Riemannian manifolds with dimension N which has the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$ and the Euclidean dimension $n_i$ are not necessarily all the same. In arXiv:1805.00132v3 [math.AP], Hassell and Sikora proved that the Riesz transform on $\mathcal{M}$ is weak type $(1,1)$, bounded on $L^{p}(\mathcal{M})$ for all $1<p<n^*$ where $n^* = \min_k n_k$ and is unbounded for $p \ge n^*$. In this note we show that the Riesz transform is bounded from Lorentz space $L^{n^* ,1}(\mathcal{M})$ to $L^{n^*,1}(\mathcal{M})$. This complete the picture by obtaining the end point results for $p=n^*$. Our approach is based on parametrix construction described in arXiv:1805.00132v3 [math.AP] and a generalisation of Hardy-Hilbert type inequalities first studied by Hardy, Littlewood and P\'olya.