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Two examples related to conical energies

In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for …

Cones, rectifiability, and singular integral operators

Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $\alpha\in …

Sufficient condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of supp$\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for …

An $\alpha$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets

We give a characterization of $L^{p}(\sigma)$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1

Necessary condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $n$-dimensional Hausdorff measure and $\mu$-almost all of supp$\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$⁠. In this paper, we give a necessary …

Characterization of Sobolev-Slobodeckij spaces using curvature energies

We give a new characterization of Sobolev-Slobodeckij spaces $W^{1+s,p}(\Omega)$ for $pn$ and $\frac{n}{p}