Let $\H$ denote the discrete Heisenberg group, equipped with a word metric
$d_W$ associated to some finite symmetric generating set. We show that if
$(X,\|\cdot\|)$ is a $p$-convex Banach space then for any Lipschitz function
$f:\H\to X$ there exist $x,y\in \H$ with $d_W(x,y)$ arbitrarily large and
\begin{equation}\label{eq:comp abs} \frac{\|f(x)-f(y)\|}{d_W(x,y)}\lesssim
\left(\frac{\log\log d_W(x,y)}{\log d_W(x,y)}\right)^{1/p}. \end{equation}

In a previous note, the author proved that the algebra of Schur operators on
l^2 is not inverse-closed. When l^2=l^2(X) where X is a metric space, we can
consider elements of the Schur algebra with certain decay at infinity. For
instance if X has the doubling property, then Q. Sun has proved that the
weighted Schur algebra for a strictly polynomial weight is inverse-closed.
Here, we prove a result dealing with left-invertibility.

The Schur algebra is the algebra of operators which are bounded on l^1 and on
l^{\infty}. Q. Sun conjectured that the Schur algebra is inverse-closed. In
this note, we disprove this conjecture. Precisely, we exhibit an operator in
the Schur algebra, invertible in l^2, whose inverse is not bounded on l^1 nor
on l^{\infty}.