  The Stokes equation on a domain $\Omega \subset R^n$ is well understood in
the $L^p$-setting for a large class of domains including bounded and exterior
domains with smooth boundaries provided $1<p<\infty$. The situation is very
different for the case $p=\infty$ since in this case the Helmholtz projection
does not act as a bounded operator anymore. Nevertheless it was recently proved
by the first and the second author of this paper by a contradiction argument
that the Stokes operator generates an analytic semigroup on spaces of bounded
functions for a large class of domains. This paper presents a new approach as
well as new a priori $L^\infty$-type estimates to the Stokes equation. They
imply in particular that the Stokes operator generates a $C_0$-analytic
semigroup of angle $\pi/2$ on $C_{0,\sigma}(\Omega)$, or a non-$C_0$-analytic
semigroup on $L^\infty_\sigma(\Omega)$ for a large class of domains. The
approach presented is inspired by the so called Masuda-Stewart technique for
elliptic operators. It is shown furthermore that the method presented applies
also to different type of boundary conditions as, e.g., Robin boundary
conditions.
