Damian Dąbrowski

activity

20172026

most citedNecessary condition for rectifiability involving Wasserstein distance

W_{2}

10 citations · 27 across the 11 of their papers we have counts for

collaborators

12 papers

math.CA2026

Favard length of non-homogeneous random disc-like Cantor sets

Alan Chang, Damian Dąbrowski, Giacomo Del Nin

The Favard length of a planar set is the average length of its orthogonal projections. We provide sharp estimates for the Favard length decay of a random variant of the non-homogen…

math.CA2023

How much can heavy lines cover?

Damian Dąbrowski, Tuomas Orponen, Hong Wang

One formulation of Marstrand's slicing theorem is the following. Assume that $t \in (1, 2]$ , and $B \subset R^{2}$ is a Borel set with $H^{t} (B) < \infty$ . Then,…

math.CA2022★ 1 cited

Quantitative Besicovitch projection theorem for irregular sets of directions

Damian Dąbrowski

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 len…

math.CA2022★ 5 cited

Structure of sets with nearly maximal Favard length

Alan Chang, Damian Dąbrowski, Tuomas Orponen +1

Let $E \subset B (1) \subset R^{2}$ be an $H^{1}$ measurable set with $H^{1} (E) < \infty$ , and let $L \subset R^{2}$ be a line segment with $\mat…

math.CA2021

Integrability of orthogonal projections, and applications to Furstenberg sets

Damian Dąbrowski, Tuomas Orponen, Michele Villa

Let $G (d, n)$ be the Grassmannian manifold of $n$ -dimensional subspaces of $R^{d}$ , and let $π_{V} : R^{d} \to V$ be the orthogonal projection. We p…

math.CA2020★ 2 cited

Two examples related to conical energies

Damian Dąbrowski

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…

math.CA2020★ 1 cited

An $α$ -number characterization of $L^{p}$ spaces on uniformly rectifiable sets

Jonas Azzam, Damian Dąbrowski

We give a characterization of $L^{p} (σ)$ for uniformly rectifiable measures $σ$ using Tolsa's $α$ -numbers, by showing, for $1 < p < \infty$ and $f \in L^{p} (σ)$ , that \[ \lVert f\rVert_…

math.CA2020★ 3 cited

Cones, rectifiability, and singular integral operators

Damian Dąbrowski

Let $μ$ be a Radon measure on $R^{d}$ . We define and study conical energies $E_{μ, p} (x, V, α)$ , which quantify the portion of $μ$ lying in the cone with vertex $x\in…

math.CA2020

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

Damian Dąbrowski, Michele Villa

Let $μ$ be a Radon measure on the $n$ -th Heisenberg group $H^{n}$ . In this note we prove that if the $(2 n + 1)$ -dimensional (Heisenberg) Riesz transform on $H^{n}$ is $…

math.CA2019★ 5 cited

Sufficient condition for rectifiability involving Wasserstein distance $W_{2}$

Damian Dąbrowski

A Radon measure $μ$ is $n$ -rectifiable if it is absolutely continuous with respect to $H^{n}$ and $μ$ -almost all of $supp μ$ can be covered by Lipschitz images of $…

math.CA2019★ 10 cited

Necessary condition for rectifiability involving Wasserstein distance $W_{2}$

Damian Dąbrowski

A Radon measure $μ$ is $n$ -rectifiable if $μ ≪ H^{n}$ and $μ$ -almost all of $supp μ$ can be covered by Lipschitz images of $R^{n}$ . In this paper we give a n…

math.CA2017

Characterization of Sobolev-Slobodeckij spaces using curvature energies

Damian Dąbrowski

We give a new characterization of Sobolev-Slobodeckij spaces W^{1+s,p} for n/p<1+s, where n is the dimension of the domain. To achieve this we introduce a family of curvature energ…

Favard length of non-homogeneous random disc-like Cantor sets

How much can heavy lines cover?

Quantitative Besicovitch projection theorem for irregular sets of directions

Structure of sets with nearly maximal Favard length

Integrability of orthogonal projections, and applications to Furstenberg sets

Two examples related to conical energies

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

Cones, rectifiability, and singular integral operators

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

Sufficient condition for rectifiability involving Wasserstein distance W2​

Necessary condition for rectifiability involving Wasserstein distance W2​

Characterization of Sobolev-Slobodeckij spaces using curvature energies

An $α$ -number characterization of $L^{p}$ spaces on uniformly rectifiable sets

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

Sufficient condition for rectifiability involving Wasserstein distance $W_{2}$

Necessary condition for rectifiability involving Wasserstein distance $W_{2}$