  We prove weighted mixed-norm $L^q_t(W^{2,p}_x)$ and $L^q_t(C^{2,\alpha}_x)$
estimates for $1<p,q<\infty$ and $0<\alpha<1$, weighted mixed weak-type
estimates for $q=1$, $L^\infty_{t}(L^p_x)-BMO_t(W^{2,p}_x)$, and
$L^\infty_{t}(C^\alpha_x)-BMO_t(C^{2,\alpha}_x)$, and a.e.~pointwise formulas
for derivatives, for solutions $u=u(t,x)$ to parabolic equations of the form
$$\partial_tu-a^{ij}(t)\partial_{ij}u+u=f\quad
t\in\mathbb{R},~x\in\mathbb{R}^n$$ and for the Cauchy problem $$\begin{cases}
\partial_tv-a^{ij}(t)\partial_{ij}v+v=f&\hbox{for}~t >0,~x\in\mathbb{R}^n \\
v(0,x)=g&\hbox{for}~x\in\mathbb{R}^n. \end{cases}$$ The coefficients
$a(t)=(a^{ij}(t))$ are just bounded, measurable, symmetric and uniformly
elliptic. Furthermore, we show strong, weak type and $BMO$-Sobolev estimates
with parabolic Muckenhoupt weights. It is quite remarkable that most of our
results are new even for the classical heat equation $$\partial_tu-\Delta
u+u=f.$$
