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

Abstract

For $\alpha \in (0,n)$, $p\in (1,\frac{n}{\alpha })$, $q:=\frac{np}{n-\alpha p}$, and $W\in {\mathcal{A}}_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_{\alpha }$: 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 $n\ge 2$, $p\in [1,n)$, $q:=\frac{np}{n-p}$, and $W\in {\mathcal{A}}_{p,q}$, 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 $[W{]}_{{\mathcal{A}}_{p,q}}$ 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.