Weighted $L_q(L_p)$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

We present a weighted $L_q(L_p)$-theory ($p,q\in (1,\mathrm{\infty })$) with Muckenhoupt weights for the equation $${\mathrm{\partial }}_t^{\alpha }u(t,x)=\mathrm{\Delta }u(t,x)+f(t,x),t>0,x\in {\mathbb{R}}^d\mathrm{.}$$ Here, $\alpha \in (0,2)$ and ${\mathrm{\partial }}_t^{\alpha }$ is the Caputo fractional derivative of order $\alpha$. In particular we prove that for any $p,q\in (1,\mathrm{\infty })$, $w_1(x)\in A_p$ and $w_2(t)\in A_q$, $${\int }_0^{\mathrm{\infty }}{\left({\int }_{{\mathbb{R}}^d}\mathrm{\mid }{u}_{xx}{\mathrm{\mid }}^p{w}_{1}dx\right)}^{q\mathrm{/}p}{w}_{2}dt\le N{\int }_0^{\mathrm{\infty }}{\left({\int }_{{\mathbb{R}}^d}\mathrm{\mid }f{\mathrm{\mid }}^p{w}_{1}dx\right)}^{q\mathrm{/}p}{w}_{2}dt,$$ 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.