Weighted maximal $$L_{q}(L_{p})$$-regularity theory for time-fractional diffusion-wave equations with variable coefficients

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, $$\partial }_{t}$$ is Caputo fractional derivative of order $$\alpha \in (0,2)$$ and $$a^{ij}$$ are functions (t, x). Precisely, we show that \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 _{\mathbb {R}^{d}}|(1-\varDelta /2} f(t,x)|^{p}w_{1}(x)\textrm{d}x\right) ^{q/p}w_{2}(t)\textrm{d}t, \end{aligned} where $$1<p,q<\infty $$ , $$\gamma \mathbb {R}$$ $$w_{1}$$ $$w_{2}$$ weights. This implies prove regularity theory, sharp solution according to f. To our main result, also proved complex interpolation weighted Sobolev spaces, $$\begin{aligned}{}[H^{\gamma _{0}}_{p_{0}}(w_{0}), H^{\gamma _{1}}_{p_{1}}(w_{1})]_{[\theta ]} = }_{p}(w), $$\theta (0,1)$$ _{0},\gamma _{1}\in $$p_{0},p_{1}\in (1,\infty )$$ $$w_{i}$$ ( $$i=0,1$$ ) arbitrary $$A_{p_{i}}$$ weight, \gamma =(1-\theta )\gamma _{0}+\theta _{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}.