  Let $(X, P)$ be a toric variety. In this note, we show that the $C^0$-norm of
the Calabi flow $\varphi(t)$ on $X$ is uniformly bounded in $[0, T)$ if the
Sobolev constant of $\varphi(t)$ is uniformly bounded in $[0, T)$. We also show
that if $(X, P)$ is uniform $K$-stable, then the modified Calabi flow converges
exponentially fast to an extremal K\"ahler metric if the Ricci curvature and
the Sobolev constant are uniformly bounded. At last, we discuss an extension of
our results to a quasi-proper K\"ahler manifold.
