  We present a maximal $L_{q}(L_{p})$-regularity theory with Muckenhoupt
weights for the equation \begin{equation}\label{eqn 01.26.16:00}
\partial^{\alpha}_{t}u=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad
t>0,x\in\mathbb{R}^{d}. \end{equation} Here, $\partial^{\alpha}_{t}$ is the
Caputo fractional derivative of order $\alpha\in(0,2)$ and $a^{ij}$ are
functions of $(t,x)$. Precisely, we show that \begin{equation*} \begin{aligned}
&\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-\Delta)^{\gamma/2}u_{xx}(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt
\\ &\quad \leq N
\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-\Delta)^{\gamma/2}f(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt,
\end{aligned} \end{equation*} where $1<p,q<\infty$, $\gamma\in\mathbb{R}$, and
$w_{1}$ and $w_{2}$ are Muckenhoupt weights, and this implies that we prove
maximal regularity theory. Our approach is based on Fourier multiplier and
singular integral theory with Muckenhoupt weights. Also, we use abstract
interpolation theory to find interpolation inequality in weighted Sobolev
space.
