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Quantitative Besicovitch projection theorem for irregular sets of directions

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 …

Visible parts and slices of Ahlfors regular sets

We show that for any compact set ERd the visible part of E has Hausdorff dimension at most d1/6 for almost every direction. This improves recent estimates of Orponen and Matheus. If E is s-Ahlfors regular, where sd1, we …

Structure of sets with nearly maximal Favard length

Let EB(1)R2 be an H1 measurable set with H1(E)0.

How much can heavy lines cover?

One formulation of Marstrand's slicing theorem is the following. Assume that t(1,2], and BR2 is a Borel set with Ht(B)t1? A positive answer for t-regular sets BR2 was …

The measures with L2-bounded Riesz transform and the Painlevé problem

In this work we obtain a geometric characterization of the measures μ in Rn+1 with polynomial upper growth of degree n such that the n-dimensional Riesz transform Rμ(x)=xy|xy|n+1dμ(y)

An α-number characterization of Lp spaces on uniformly rectifiable sets

We give a characterization of Lp(σ) for uniformly rectifiable measures σ using Tolsa's α-numbers, by showing, for $1

Necessary condition for the L2 boundedness of the Riesz transform on Heisenberg groups

Let μ be a Radon measure on the n-th Heisenberg group Hn. In this note we prove that if the (2n+1)-dimensional (Heisenberg) Riesz transform on Hn is L2(μ)-bounded, and if μ(F)=0 for all Borel sets with …

Integrability of orthogonal projections, and applications to Furstenberg sets

Let G(d,n) be the Grassmannian manifold of n-dimensional subspaces of Rd, and let πV:RdV be the orthogonal projection. We prove that if μ is a compactly supported Radon measure on …

Cones, rectifiability, and singular integral operators

Let μ be a Radon measure on Rd. We define and study conical energies E(x,V,α), which quantify the portion of μ lying in the cone with vertex xRd, direction VG(d,dn), and aperture $\alpha\in …

Analytic capacity and dimension of sets with plenty of big projections

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 …