showing 2 related works from this author
Integrability of orthogonal projections, and applications to Furstenberg sets
2022
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 $\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $\mu(B(x,r)) \leq Cr^{s}$ for all $x \in \mathbb{R}^{d}$ and $r > 0$, then $$\int_{\mathcal{G}(d,n)} \|\pi_{V}\mu\|_{L^{p}(V)}^{p} \, d\gamma_{d,n}(V) \tfrac{1}{2}$ and $t \geq 1 + \epsilon$ for a small absolute constant $\epsilon > 0$. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in $\mathbb{R}^{d}$. As another corollary of our method,…
Two examples related to conical energies
2022
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 boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and M\"{o}rters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.