  Reduced cohomology motivates to look at harmonic functions which satisfy
certain gradient conditions. If $G$ is a direct product of two infinite groups
or a (FC-central)-by-cyclic group, then there are no harmonic functions with
gradient in $c_0$ on its Cayley graphs. From this, it follows that a metabelian
group $G$ has no harmonic functions with gradient in $\ell^p$. Furthermore,
under a radial isoperimetric condition, groups whose isoperimetric profile is
bounded by power of logarithms also have no harmonic functions with gradient in
$\ell^p$.
