  We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) with
Muckenhoupt weights for the equation $$ \partial_{t}^{\alpha}u(t,x)=\Delta
u(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $\alpha\in (0,2)$ and
$\partial_{t}^{\alpha}$ is the Caputo fractional derivative of order $\alpha$.
In particular we prove that for any $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ and
$w_{2}(t)\in A_q$, $$ \int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p
\,w_{1} dx \right)^{q/p}\,w_{2}dt \leq N
\int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx
\right)^{q/p}\,w_{2}dt, $$ where $A_p$ is the class of Muckenhoupt $A_p$
weights. Our approach is based on the sharp function estimates of the
derivatives of solutions.
