arXiv subjects

Damian Dąbrowski

Publications and source records attributed to Damian Dąbrowski.

18 recordsLinked to original sources

arXiv2025

The measures with $L^2$-bounded Riesz transform and the Painlevé problem

In this work we provide a geometric characterization of the measures $μ$ in $\mathbb R^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $Rμ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$. More precisely, it is shown that $$\|Rμ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{2,μ}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset\mathbb R^{n+1}$. As a corollary, we obtain a characterization of the removable sets for Lipschitz harmonic functions in terms of a metric-geometric potential and we deduce that the class of removable sets for Lipschitz harmonic functions is invariant by bilipschitz mappings.

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arXiv2025

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 $E\subset\mathbb{R}^2$ is AD-regular and there exists a set of direction $G\subset \mathbb{S}^1$ with $\mathcal{H}^1(G)\gtrsim 1$ such that for every $θ\in G$ we have $\|π_θ\mathcal{H}^1|_E\|_{L^{\infty}}\lesssim 1$, then a big piece of $E$ can be covered by a Lipschitz graph $Γ$ with $\mathrm{Lip}(Γ)\lesssim 1$. The main novelty of our result is that the set of good directions $G$ is assumed to be merely measurable and large in measure, while previous results of this kind required $G$ to be an arc. As a corollary, we obtain a result on AD-regular sets which avoid a large set of directions, in the sense that the set of directions they span has a large complement. It generalizes the following easy observation: a set $E$ is contained in some Lipschitz graph if and only if the complement of the set of directions spanned by $E$ contains an arc.

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arXiv2025

On the logarithmic equilibrium measure on curves

Let $μ$ be the logarithmic equilibrium measure on a compact set $γ\subset \mathbb{R}^{d}$. We prove that $μ$ is absolutely continuous with respect to the length measure on the part of $γ$ which can be locally expressed as the graph of a $C^{1,α}$-function $\mathbb{R} \to \mathbb{R}^{d - 1}$, $α> 0$. For $d = 2$, at least in the case where $γ$ is a compact $C^{1,α}$-graph, our result can also be deduced from the classical fact that $μ$ coincides with the harmonic measure of $Ω=\mathbb{R}^{2} \, \setminus \, γ$ with pole at $\infty$. For $d \geq 3$, however, our result is new even for $C^{\infty}$-graphs. In fact, up to now it was not even known if the support of $μ$ has positive dimension.

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arXiv2024

Visible parts and slices of Ahlfors regular sets

We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where $s>d-1$, we prove a much better estimate. In that case for almost every direction the Hausdorff dimension of the visible part is at most $s - α(s-d+1),$ where $α>0.183$ is absolute. The estimate is new even for self-similar sets satisfying the open set condition. Along the way, we prove a refinement of the Marstrand's slicing theorem for Ahlfors regular sets.

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arXiv2024

Favard length and quantitative rectifiability

The Favard length of a Borel set $E\subset\mathbb{R}^2$ is the average length of its orthogonal projections. We prove that if $E$ is Ahlfors 1-regular and it has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of the Besicovitch projection theorem. As a corollary, we answer questions of David and Semmes and of Peres and Solomyak. We also make progress on Vitushkin's conjecture.

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arXiv2024

On the dimension of $s$-Nikodým sets

Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.

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arXiv2024

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 also proved for capacities related to the Riesz kernel, including the Lipschitz harmonic capacity. The proof uses a construction of a doubling Frostman measure on a lower content regular set, which may be of independent interest. Our second main result is the Analyst's Traveling Salesman Theorem for sets with plenty of big projections. As a corollary, we obtain a lower bound for the Hausdorff dimension of uniformly wiggly sets with PBP. The second corollary is an estimate for the capacities of subsets of sets with PBP, in the spirit of the quantitative solution to Denjoy's conjecture.

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arXiv2023

How much can heavy lines cover?

One formulation of Marstrand's slicing theorem is the following. Assume that $t \in (1,2]$, and $B \subset \mathbb{R}^{2}$ is a Borel set with $\mathcal{H}^{t}(B) < \infty$. Then, for almost all directions $e \in S^{1}$, $\mathcal{H}^{t}$ almost all of $B$ is covered by lines $\ell$ parallel to $e$ with $\dim_{\mathrm{H}} (B \cap \ell) = t - 1$. We investigate the prospects of sharpening Marstrand's result in the following sense: in a generic direction $e \in S^{1}$, is it true that a strictly less than $t$-dimensional part of $B$ is covered by the heavy lines $\ell \subset \mathbb{R}^{2}$, namely those with $\dim_{\mathrm{H}} (B \cap \ell) > t - 1$? A positive answer for $t$-regular sets $B \subset \mathbb{R}^{2}$ was previously obtained by the first author. The answer for general Borel sets turns out to be negative for $t \in (1,\tfrac{3}{2}]$ and positive for $t \in (\tfrac{3}{2},2]$. More precisely, the heavy lines can cover up to a $\min\{t,3 - t\}$ dimensional part of $B$ in a generic direction. We also consider the part of $B$ covered by the $s$-heavy lines, namely those with $\dim_{\mathrm{H}} (B \cap \ell) \geq s$ for $s > t - 1$. We establish a sharp answer to the question: how much can the $s$-heavy lines cover in a generic direction? Finally, we identify a new class of sets called sub-uniformly distributed sets, which generalise Ahlfors-regular sets. Roughly speaking, these sets share the spatial uniformity of Ahlfors-regular sets, but pose no restrictions on uniformity across different scales. We then extend and sharpen the first author's previous result on Ahlfors-regular sets to the class of sub-uniformly distributed sets.

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arXiv2022

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 $π_{V} \colon \mathbb{R}^{d} \to V$ be the orthogonal projection. We prove that if $μ$ is a compactly supported Radon measure on $\mathbb{R}^{d}$ satisfying the $s$-dimensional Frostman condition $μ(B(x,r)) \leq Cr^{s}$ for all $x \in \mathbb{R}^{d}$ and $r > 0$, then $$\int_{\mathcal{G}(d,n)} \|π_{V}μ\|_{L^{p}(V)}^{p} \, dγ_{d,n}(V) < \infty, \qquad 1 \leq p < \frac{2d - n - s}{d - s}.$$ The upper bound for $p$ is sharp, at least, for $d - 1 \leq s \leq d$, and every $0 < n < d$. Our motivation for this question comes from finding improved lower bounds on the Hausdorff dimension of $(s,t)$-Furstenberg sets. For $0 \leq s \leq 1$ and $0 \leq t \leq 2$, a set $K \subset \mathbb{R}^{2}$ is called an $(s,t)$-Furstenberg set if there exists a $t$-dimensional family $\mathcal{L}$ of affine lines in $\mathbb{R}^{2}$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ for all $\ell \in \mathcal{L}$. As a consequence of our projection theorem in $\mathbb{R}^{2}$, we show that every $(s,t)$-Furstenberg set $K \subset \mathbb{R}^{2}$ with $1 < t \leq 2$ satisfies $$\dim_{\mathrm{H}} K \geq 2s + (1 - s)(t - 1).$$ This improves on previous bounds for pairs $(s,t)$ with $s > \tfrac{1}{2}$ and $t \geq 1 + ε$ for a small absolute constant $ε> 0$. We also prove a higher dimensional analogue of this estimate for codimension-1 Furstenberg sets in $\mathbb{R}^{d}$. As another corollary of our method, we obtain a $δ$-discretised sum-product estimate for $(δ,s)$-sets. Our bound improves on a previous estimate of Chen for every $\tfrac{1}{2} < s < 1$, and also of Guth-Katz-Zahl for $s \geq 0.5151$.

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arXiv2022

Structure of sets with nearly maximal Favard length

Let $E \subset B(1) \subset \mathbb R^{2}$ be an $\mathcal{H}^{1}$ measurable set with $\mathcal{H}^{1}(E) < \infty$, and let $L \subset \mathbb R^{2}$ be a line segment with $\mathcal{H}^{1}(L) = \mathcal{H}^{1}(E)$. It is not hard to see that $\mathrm{Fav}(E) \leq \mathrm{Fav}(L)$. We prove that in the case of near equality, that is, $$ \mathrm{Fav}(E) \geq \mathrm{Fav}(L) - δ, $$ the set $E$ can be covered by an $ε$-Lipschitz graph, up to a set of length $ε$. The dependence between $ε$ and $δ$ is polynomial: in fact, the conclusions hold with $ε= Cδ^{1/70}$ for an absolute constant $C > 0$.

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arXiv2021

Cones, rectifiability, and singular integral operators

Let $μ$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{μ,p}(x,V,α)$, which quantify the portion of $μ$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $α\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $μ$ has polynomial growth, we give a sufficient condition for $L^2(μ)$-boundedness of singular integral operators with smooth odd kernels of convolution type.

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arXiv2021

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 boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters, the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

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arXiv2021

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 $μ$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}μ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$, under the assumption that $μ$ satisfies the following Wolff energy estimate, for any ball $B\subset\mathbb{R}^{n+1}$: $$\int_B \int_0^\infty \left(\frac{μ(B(x,r))}{r^{n-\frac38}}\right)^2\,\frac{dr}r\,dμ(x)\leq M\,\bigg(\frac{μ(2B)}{r(B)^{n-\frac38}}\bigg)^2\,μ(2B).$$ More precisely, we show that $μ$ satisfies the following estimate: $$\|\mathcal{R}μ\|_{L^2(μ)}^2 + \|μ\|\approx \int\!\!\int_0^\infty β_{μ,2}(x,r)^2\,\frac{μ(B(x,r))}{r^n}\,\frac{dr}r\,dμ(x) + \|μ\|,$$ where $β_{μ,2}(x,r)^2 = \inf_L \frac1{r^n}\int_{B(x,r)} \left(\frac{\mathrm{dist}(y,L)}r\right)^2\,dμ(y),$ with the infimum taken over all affine $n$-planes $L\subset\mathbb{R}^{n+1}$. In a companion paper which relies on the results obtained in this work it is shown that the same result holds without the above assumption regarding the Wolff energy of $μ$. This result has important consequences for the Painlevé problem for Lipschitz harmonic functions.

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arXiv2020

Sufficient condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $μ$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $μ$-almost all of $\text{supp}\,μ$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $α$ and $β_2$ numbers. The second one involves $α_2$ numbers -- coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too -- the first one was shown to be necessary by Tolsa, while the necessity of the $α_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.

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arXiv2020

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

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 \[ \lVert f\rVert_{L^{p}(σ)}\sim \left\lVert\left(\int_{0}^{\infty} \left(α_{fσ}(x,r)+|f|_{x,r}α_σ(x,r)\right)^2\ \frac{dr}{r} \right)^{\frac{1}{2}}\right\rVert_{L^{p}(σ)}. \]

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arXiv2020

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

Let $μ$ 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 $L^2(μ)$-bounded, and if $μ(F)=0$ for all Borel sets with $\dim_H(F)\leq 2$, then $μ$ must have $(2n+1)$-polynomial growth. This is the Heisenberg counterpart of a result of Guy David from 1991.

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arXiv2019

Necessary condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $μ$ is $n$-rectifiable if $μ\ll\mathcal{H}^n$ and $μ$-almost all of $\text{supp}\,μ$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give a necessary condition for rectifiability in terms of the so-called $α_2$ numbers -- coefficients quantifying flatness using Wasserstein distance $W_2$. In a recent article we showed that the same condition is also sufficient for rectifiability, and so we get a new characterization of rectifiable measures.

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arXiv2018

Characterization of Sobolev-Slobodeckij spaces using curvature energies

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 energies inspired by the classical concept of integral Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij space if and only if it is in L^p and the appropriate energy is finite.

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