Endpoint estimates for riesz transform on manifolds with ends

We consider a class of non-doubling manifolds \(\mathcal {M}\) 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 \(\mathcal {M}\) is of weak type (1, 1), bounded on \(L^{p}\) if and only if \(1<p<n_*\), where \(n_* = \min _k n_k\). In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space \(L^{n_*,1}\) and unbounded from \(L^{n_*,p}\rightarrow L^{n_*,q}\) for all \(1<p<\infty \) and \(p\le q\le \infty \).