Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincar\'e inequalities for the nonlocal nonconvex functionals $$ \Phi_{\lambda,p}^\gamma(g;Q_0) :=\lambda^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}\}} |x-y|^{\gamma-N}\,dx\,dy, $$ wher...
Feng Dai, Da-Chun Yang, Wen Yuan et al.· 0 citations
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...
Let $T$ be a convolution Calder\'on--Zygmund operator satisfying Hyt\"onen's non-degeneracy condition. We prove that, for every $p\in(1,\infty)$ and every matrix dimension $m\ge3$, there exists a positive constant $C$ such that, for any $t\in[1,\infty)$, \begin{equation*} \sup_{[W]_{\mathcal A_p}\le t} \|T\|_{L^p(W)\to...
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 2 citations
We establish two pointwise estimates for fractional difference operators, tracking explicitly the dependence of the constants on the smoothness index $s\in(0,1)$. Using these, within the framework of ball Banach function spaces we obtain two fractional Gagliardo--Nirenberg inequalities, including the BMO endpoint case....
Pingxu Hu, Yinqin Li, Da-Chun Yang et al.· 0 citations
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