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.

中文翻译：

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带有时间分数阶导数的扩散波方程的 Muckenhoupt 权重加权 L(L) 估计

我们提出了一个加权 $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) 与方程 $$ \partial_{t}^{\alpha 的 Muckenhoupt 权重}u(t,x)=\Delta u(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d。$$ 这里，$\alpha\in (0,2)$ 和 $\partial_{t}^{\alpha}$ 是 $\alpha$ 阶的 Caputo 分数阶导数。特别是我们证明对于任何 $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ 和 $w_{2}(t)\in A_q$, $$ \ int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p \,w_{1} dx \right)^{q/p}\,w_{2 }dt \leq N \int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx \right)^{q/p}\, w_{2}dt, $$ 其中 $A_p$ 是 Muckenhoupt $A_p$ 权重的类。我们的方法基于对解的导数的尖锐函数估计。