  In this article we prove a maximal $L^p$-regularity result for stochastic
convolutions, which extends Krylov's basic mixed $L^p(L^q)$-inequality for the
Laplace operator on ${\mathbb{R}}^d$ to large classes of elliptic operators,
both on ${\mathbb{R}}^d$ and on bounded domains in ${\mathbb{R}}^d$ with
various boundary conditions. Our method of proof is based on McIntosh's
$H^{\infty}$-functional calculus, $R$-boundedness techniques and sharp
$L^p(L^q)$-square function estimates for stochastic integrals in $L^q$-spaces.
Under an additional invertibility assumption on $A$, a maximal space--time
$L^p$-regularity result is obtained as well.
