Endpoint estimates for riesz transform on manifolds with ends

We consider a class of non-doubling manifolds \(\mathcal {M}\) $\mathcal{M}$ consisting of finite many “Euclidean” ends, where the Euclidean dimensions at infinity are not necessarily all the same. In [17], Hassell and Sikora proved that the Riesz transform on \(\mathcal {M}\) $\mathcal{M}$ is of weak type (1, 1), bounded on \(L^{p}\) $L^p$ if and only if \(1<p<n_*\) $1<p<n_{\ast }^{}$ , where \(n_* = \min _k n_k\) $n_{\ast }^{}={min}_k{n}_k$ . In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space \(L^{n_*,1}\) $L_{}^{n_{\ast }^{},1}$ and unbounded from \(L^{n_*,p}\rightarrow L^{n_*,q}\) $L_{}^{n_{\ast }^{},p}\to L_{}^{n_{\ast }^{},q}$ for all \(1<p<\infty \) $1<p<\infty$ and \(p\le q\le \infty \) $p\le q\le \infty$ .