Sharp Quantitative Matrix-Weighted Estimates for Fractional Integrals and Sobolev Inequalities

Abstract

Let $\alpha\in(0,n)$, $p\in(1,\frac{n}{\alpha})$, $q:=\frac{np}{n-\alpha p}$, and $W\in\mathscr A_{p,q}$. We establish the following quantitative matrix-weighted estimate for the fractional integral $I_\alpha$: for any $\vec{f}\in L^p(W^p)$, \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*} where the implicit positive constant is independent of $W$ and $\vec{f}$. Let $n\geq2$ be an integer, $p\in[1,n)$, $q:=\frac{np}{n-p}$, and $W\in\mathscr A_{p,q}$. We also prove 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 $D\vec{f}$ is the Jacobian matrix of $\vec{f}$. In both estimates, the exponent of $[W]_{\mathscr A_{p,q}}$ coincides with the corresponding optimal scalar exponent, and hence is optimal.