We present a maximal $$L_{q}(L_{p})$$ -regularity theory with Muckenhoupt weights for the equation 0.1 $$\begin{aligned} \partial ^{\alpha }_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in {\mathbb {R}}^{d}. \end{aligned}$$ 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 $$\begin{aligned} \begin{aligned}&\int _{0}^{T}\left( \int _{{\mathbb {R}}^{d}}|(1-\varDelta )^{\gamma /2}u_{xx} (t,x)|^{p}w_{1}(x)\textrm{d}x\right) ^{q/p}w_{2}(t)\textrm{d}t \\&\quad \le N \int _{0}^{T}\left( \int _{\mathbb {R}^{d}}|(1-\varDelta )^{\gamma /2} f(t,x)|^{p}w_{1}(x)\textrm{d}x\right) ^{q/p}w_{2}(t)\textrm{d}t, \end{aligned} \end{aligned}$$ where $$1<p,q<\infty $$ , $$\gamma \in \mathbb {R}$$ , and $$w_{1}$$ and $$w_{2}$$ are Muckenhoupt weights. This implies that we prove maximal regularity theory, and sharp regularity of solution according to regularity of f. To prove our main result, we also proved the complex interpolation of weighted Sobolev spaces, $$\begin{aligned}{}[H^{\gamma _{0}}_{p_{0}}(w_{0}), H^{\gamma _{1}}_{p_{1}}(w_{1})]_{[\theta ]} = H^{\gamma }_{p}(w), \end{aligned}$$ where $$\theta \in (0,1)$$ , $$\gamma _{0},\gamma _{1}\in \mathbb {R}$$ , $$p_{0},p_{1}\in (1,\infty )$$ , $$w_{i}$$ ( $$i=0,1$$ ) are arbitrary $$A_{p_{i}}$$ weight, and $$\begin{aligned} \gamma =(1-\theta )\gamma _{0}+\theta \gamma _{1}, \quad \frac{1}{p}=\frac{1-\theta }{p_{0}} + \frac{\theta }{p_{1}},\quad w^{1/p}=w^{\frac{(1-\theta )}{p_{0}}}_{0}w^{\frac{\theta }{p_{1}}}_{1}. \end{aligned}$$