3

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 …

The measures with $L^2$-bounded Riesz transform satisfying a subcritical Wolff-type energy condition

In this work we obtain a geometric characterization of the measures $\mu$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}d\mu(y)$ …

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 …

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

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 …

Necessary condition for the $L^2$ boundedness of the Riesz transform on Heisenberg groups

Let $\mu$ be a Radon measure on the $n$-th Heisenberg group $\mathbb{H}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on $\mathbb{H}^n$ is $L^2(\mu)$-bounded, and if $\mu(F)=0$ for all Borel sets with …