  We prove the following sharp upper bound for the gradient of the Neumann
semigroup $P_t$ on a $d$-dimensional compact domain $\OO$ with boundary either
$C^2$-smooth or convex:
  $$\|\nn P_t\|_{1\to \infty}\le \ff{c}{t^{(d+1)/2}},\ \ t>0,$$ where $c>0$ is
a constant depending on the domain and $\|\cdot\|_{1\to\infty}$ is the operator
norm from $L^1(\OO)$ to $L^\infty(\OO)$. This estimate implies a Gaussian type
point-wise upper bound for the gradient of the Neumann heat kernel, which is
applied to the study of the Hardy spaces, Riesz transforms, and regularity of
solutions to the inhomogeneous Neumann problem on compact convex domains.
