The classical Besicovitch projection theorem states that if a planar set E with finite length is purely unrectifiable, then almost all orthogonal projections of E have zero length. We prove a quantitative version of this result: if a planar set E is …
We show that for any compact set the visible part of has Hausdorff dimension at most for almost every direction. This improves recent estimates of Orponen and Matheus. If is -Ahlfors regular, where , we …
In this work we obtain a geometric characterization of the measures in with polynomial upper growth of degree such that the -dimensional Riesz transform …
Let be a Radon measure on the -th Heisenberg group . In this note we prove that if the -dimensional (Heisenberg) Riesz transform on is -bounded, and if for all Borel sets with …
Let be the Grassmannian manifold of -dimensional subspaces of , and let be the orthogonal projection. We prove that if is a compactly supported Radon measure on …
Let be a Radon measure on . We define and study conical energies , which quantify the portion of lying in the cone with vertex , direction , and aperture $\alpha\in …
Our main result marks progress on an old conjecture of Vitushkin. We show that a compact set in the plane with plenty of big projections (PBP) has positive analytic capacity, along with a quantitative lower bound. A higher dimensional counterpart is …