Sub-Gaussian heat kernel estimates and quasi Riesz transforms for $1\leq   p\leq 2$
  On a complete non-compact Riemannian manifold $M$, we prove that a so-called quasi Riesz transform is always $L^p$ bounded for $1<p\leq 2$. If $M$ satisfies the doubling volume property and the sub-Gaussian heat kernel estimate, we prove that the quasi Riesz transform is also of weak type $(1,1)$. 
Li Chen
math.AP math.CA