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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 …

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 …

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 $\mathcal{R}\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}d\mu(y)$ …