AI 中文总结
本文建立了分数积分与Sobolev不等式的定量矩阵加权估计，其加权常数指数与最优标量情形一致，从而证明了最优性。
AI 中文摘要
设$\alpha\in(0,n)$，$p\in(1,\frac{n}{\alpha})$，$q:=\frac{np}{n-\alpha p}$，且$W\in\mathscr A_{p,q}$。我们建立了分数积分$I_\alpha$的如下定量矩阵加权估计：对任意$\vec{f}\in L^p(W^p)$，有\begin{align*} \left\\|I_\alpha\vec f\right\\|_{L^q(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{ (1-\frac{\alpha}{n})\max\{1,\frac{p'}{q}\}} \left\\|\vec f\right\\|_{L^p(W^p)}, \end{align*}其中隐含的正常数不依赖于$W$和$\vec{f}$。设$n\geq2$为整数，$p\in[1,n)$，$q:=\frac{np}{n-p}$，且$W\in\mathscr A_{p,q}$。我们还证明了如下矩阵加权Sobolev不等式：对任意具有紧支撑的光滑$\mathbb{C}^d$值函数$\vec{f}$，有\begin{align*} \left\\|\vec{f}\right\\|_{L^{q}(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{\frac{n-1}{n}} \left\\|WD\vec{f}\right\\|_{L^p(\mathbb R^n,\mathbb C^{d\times n})}, \end{align*}其中隐含的正常数不依赖于$W$和$\vec{f}$，且$D\vec{f}$是$\vec{f}$的雅可比矩阵。在这两个估计中，$[W]_{\mathscr A_{p,q}}$的指数与相应的最优标量指数一致，因此是最优的。
英文摘要
For $α\in(0,n)$, $p\in(1,\frac{n}α)$, $q:=\frac{np}{n-αp}$, and $W\in\mathscr A_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_α$: for any $\vec{f}\in L^p(W^p)$, \begin{align*} \left\|I_α\vec f\right\|_{L^q(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{ (1-\fracα{n})\max\{1,\frac{p'}{q}\}} \left\|\vec f\right\|_{L^p(W^p)}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$. For integer $n\geq2$, $p\in[1,n)$, $q:=\frac{np}{n-p}$, and $W\in\mathscr A_{p,q}$, we establish the following matrix-weighted Sobolev inequality: for any smooth $\mathbb{C}^d$-valued function $\vec{f}$ with compact support, \begin{align*} \left\|\vec{f}\right\|_{L^{q}(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{\frac{n-1}{n}} \left\|WD\vec{f}\right\|_{L^p(\mathbb R^n,\mathbb C^{d\times n})}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$ and where $D\vec{f}$ is the Jacobian matrix of $\vec{f}$. Surprisingly, in both bounds above the exponents of $[W]_{\mathscr A_{p,q}}$ coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón--Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.
