  In this paper we consider an SPDE where the leading term is a second order
operator with periodic boundary conditions, coefficients which are measurable
in $(t,\omega)$, and H\"older continuous in space. Assuming stochastic
parabolicity conditions, we prove $L^p((0,T)\times \Omega, t^{\kappa}\,
\mathrm{d} t;H^{\sigma,q}(\mathbb{T}^d))$-estimates. The main novelty is that
we do not require $p=q$. Moreover, we allow arbitrary $\sigma\in \mathbb{R}$
and weights in time. Such mixed regularity estimates play a crucial role in
applications to nonlinear SPDEs which is clear from our previous work. To prove
our main results we develop a general perturbation theory for SPDEs. Moreover,
we prove a new result on pointwise multiplication in spaces with fractional
smoothness.
