Endpoint Estimates For Riesz Transform And Hardy-Hilbert Type Inequalities

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 $1<p<n^{\ast }$ where $n^{\ast }={\min }_k{n}_k$ and is unbounded for $p\ge n^{\ast }$. In this note we show that the Riesz transform is bounded from Lorentz space $L^{n^{\ast },1}(\mathcal{M})$ to $L^{n^{\ast },1}(\mathcal{M})$. This complete the picture by obtaining the end point results for $p=n^{\ast }$. 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 Pólya.

24 pages, 2 figures