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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 …

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}