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Integrability of orthogonal projections, and applications to Furstenberg sets

Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of $\mathbb{R}^{d}$, and let $\pi_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $\mu$ is a compactly supported Radon measure on …

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}