Endpoint estimates for riesz transform on manifolds with ends

We consider a class of non-doubling manifolds M 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 M M is of weak type (1, 1), bounded on Lᵖ L p if and only if 1<p<n_* 1 < p < n ∗ , where n_* = min ₖ nₖ n ∗ = min k n k . In this note, we complete the picture by giving an endpoint estimate: Riesz transform is bounded on Lorentz space Ln_*,1 L n ∗ , 1 and unbounded from Ln_*,p→ Ln_*,q L n ∗ , p → L n ∗ , q for all 1<p<∞ 1 < p < ∞ and p≤ q≤ ∞ p ≤ q ≤ ∞ .