  Let $M$ be a complete Riemannian manifold satisfying the doubling volume
condition for geodesic balls and $L^q$ scaled Poincar\'e inequalities on
suitable remote balls for some $q<2$. We prove the inequality $\left\Vert
\Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ for all
$p\in (q,2]$, which generalizes previous results due to Auscher and Coulhon.
Our conclusion applies, in particular, when $M$ has a finite number of
Euclidean ends. The proof strongly relies on Hardy inequalities, which are also
new in this context and of independent interest.
  The second part of this work deals with analogous questions in fractal-like
cable systems. In this framework, it was already proved by Chen, Coulhon,
Feneuil and the second author that, in the Vicsek cable system, the inequality
$\left\Vert \Delta^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla
f\right\Vert_p$ may be false for all $p\in [1,2)$. Following a recent joint
work by the two authors and Yang, we examine the validity of inequalities of
the form $\left\Vert \Delta^{\gamma}e^{-\Delta}f\right\Vert_p\lesssim
\left\Vert \nabla f\right\Vert_p$. In the Vicsek case, we give the optimal
range of $p$ for which this inequality holds.
