Michele Villa

activity

20172023

most citedQuantitative Comparisons of Multiscale Geometric Properties

8 citations · 23 across the 13 of their papers we have counts for

collaborators

15 papers

math.CA2023

Quantitative differentiability on uniformly rectifiable sets

Jonas Azzam, Mihalis Mourgoglou, Michele Villa

We prove $L^{p}$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-typ…

math.AP2023

Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $ε^{2}$ conjecture in higher dimensions

Ian Fleschler, Xavier Tolsa, Michele Villa

The main aim of this article is to prove quantitative spectral inequalities for the Laplacian with Dirichlet boundary conditions. More specifically, we prove sharp quantitative sta…

math.CA2022★ 1 cited

Analytic capacity and dimension of sets with plenty of big projections

Damian Dąbrowski, Michele Villa

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…

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

A note on the Hausdorff dimension of uniformly non flat sets with plenty of big projections

Michele Villa

Using a recent result of Orponen (Invent. math. '21), we show that sets with plenty of big projections (PBP) admit an Analyst's Travelling Salesman Theorem. We then show that sets…

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.CA2021★ 1 cited

Cone and paraboloid points of arbitrary subsets of Euclidean space

Matthew Hyde, Michele Villa

In this paper we characterise cone points of arbitrary subsets of Euclidean space. Given $E \subset R^{n}$ , $x \in E$ is a cone point of $E$ if and only if \begin{align*} \i…

math.CA2020

Sub-elliptic boundary value problems in flag domains

Tuomas Orponen, Michele Villa

A flag domain in $R^{3}$ is a subset of $R^{3}$ of the form ${(x, y, t) : y < A (x)}$ , where $A : R \to R$ is a Lipschitz function. We solve…

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

A proof of Carleson's $ε^{2}$ -conjecture

Benjamin Jaye, Xavier Tolsa, Michele Villa

In this paper we provide a proof of the Carleson $ε^{2}$ -conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of…

math.CA2019★ 1 cited

A square function involving the center of mass and rectifiability

Michele Villa

For a Radon measure $μ$ on $R^{d}$ , define $C_{μ}^{n} (x, t) = (\frac{1}{t ^{n}} ∣ \int_{B (x, t)} \frac{x - y}{t} d μ (y) ∣)$ . This coefficient quantifies how symmetric the meas…

math.CA2019★ 7 cited

Higher dimensional Jordan curves

Michele Villa

We address the question of what is the correct higher dimensional analogue of Jordan curves from the point of view of quantitative rectifiability. More precisely, we show that 'top…

math.CA2019

$Ω$ -symmetric measures and related singular integrals

Michele Villa

Let $S \subset C$ be the circle in the plane, and let $Ω : S \to S$ be an odd bi-Lipschitz map with constant $1 + δ_{Ω}$ , where $δ_{Ω} > 0$ is small. Ass…

math.AP2019★ 8 cited

Quantitative Comparisons of Multiscale Geometric Properties

Jonas Azzam, Michele Villa

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For…

math.CA2017

Tangent points of lower content $d$ -regular sets and $β$ numbers

Michele Villa

Given a lower content $d$ -regular set in $R^{n}$ , we prove that the subset of points in $E$ where a certain Dini-type condition on the so-called Jones $β$ numbers holds coin…

Quantitative differentiability on uniformly rectifiable sets

Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's ε2 conjecture in higher dimensions

Analytic capacity and dimension of sets with plenty of big projections

Structure of sets with nearly maximal Favard length

A note on the Hausdorff dimension of uniformly non flat sets with plenty of big projections

Integrability of orthogonal projections, and applications to Furstenberg sets

Cone and paraboloid points of arbitrary subsets of Euclidean space

Sub-elliptic boundary value problems in flag domains

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

A proof of Carleson's ε2-conjecture

A square function involving the center of mass and rectifiability

Higher dimensional Jordan curves

Ω-symmetric measures and related singular integrals

Quantitative Comparisons of Multiscale Geometric Properties

Tangent points of lower content d-regular sets and β numbers

Faber-Krahn inequalities, the Alt-Caffarelli-Friedman formula, and Carleson's $ε^{2}$ conjecture in higher dimensions

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

A proof of Carleson's $ε^{2}$ -conjecture

$Ω$ -symmetric measures and related singular integrals

Tangent points of lower content $d$ -regular sets and $β$ numbers