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Michele Villa

Publications and source records attributed to Michele Villa.

At least 19 recordsLinked to original sources
arXiv2025

Quantitative differentiability on uniformly rectifiable sets

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-type theorem holds in this context: the $L^p$ norm of the gradient of a Sobolev function $f: E \to \mathbb{R}$ is comparable to the $L^p$ norm of a new square function measuring both the affine deviation of $f$ and how flat the subset $E$ is. A corollary dealing with extensions and traces of Sobolev functions may be found in a companion article.

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arXiv2025

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^p$ Hajlasz-Sobolev space on the boundary from the space of functions on the domain with $L^p$ norm involving the non-tangential maximal function of the gradient and the conical square function of the Hessian. This fundametally uses the Dorronsoro theorem for UR sets proven in a companion paper.

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arXiv2025

Quantitative Carleson's conjecture for Ahlfors regular domains

In this article, we prove a quantitative version of Carleson's $\varepsilon^2$ conjecture in higher dimension: we characterise those Ahlfors-David regular domains in $\mathbb{R}^{n+1}$ for which the Carleson's coefficients satisfy the so-called strong geometric lemma.

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arXiv2024

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

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 stability for the Faber-Krahn inequality in terms of Newtonian capacities and Hausdorff contents of positive codimension, thus providing an answer to a question posed by De Philippis and Brasco. One of our results asserts that for any bounded domain $Ω\subset\mathbb R^n$, $n\geq3$, with Lebesgue measure equal to that of the unit ball $B_0$ and whose first eigenvalue is $λ_Ω$, denoting by $λ_{B_0}$ the first eigenvalue for the unit ball, for any $a\in (0,1)$ it holds $$λ_Ω- λ_{B_0} \geq C(a) \,\inf_B \bigg(\sup_{t\in (0,1)} \frac1{H^{n-1}(\partial ((1-t) B))} \int_{\partial ((1-t) B)} \frac{\operatorname{Cap}_{n-2}(B(x,atr_B)\setminus Ω)}{(t\,r_B)^{n-3}}\,dH^{n-1}(x)\bigg)^2,$$ where the infimum is taken over all balls $B$ with the same Lebesgue measure as $Ω$ and $\operatorname{Cap}_{n-2}$ is the Newtonian capacity of homogeneity $n-2$. In fact, this holds for bounded subdomains of the sphere and the hyperbolic space, as well. In a second result, we also apply the new Faber-Krahn type inequalities to quantify the Hayman-Friedland inequality about the characteristics of disjoint domains in the unit sphere. Thirdly, we propose a natural extension of Carleson's $\varepsilon^2$-conjecture to higher dimensions in terms of a square function involving the characteristics of certain spherical domains, and we prove the necessity of the finiteness of such square function in the tangent points via the Alt-Caffarelli-Friedman monotonicity formula. Finally, we answer in the negative a question posed by Allen, Kriventsov and Neumayer in connection to rectifiability and the positivity set of the ACF monotonicity formula.

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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

Carleson's $\varepsilon^2$ conjecture in higher dimensions

In this paper we prove a higher dimensional analogue of Carleson's $\varepsilon^2$ conjecture. Given two arbitrary disjoint open sets $Ω^+,Ω^-\subset \mathbb{R}^{n+1}$, and $x\in\mathbb{R}^{n+1}$, $r>0$, we denote $$\varepsilon_n(x,r) := \frac{1}{r^n}\, \inf_{H^+} \mathcal{H}^n \left( ((\partial B(x,r)\cap H^+) \setminus Ω^+) \cup ((\partial B(x,r)\cap H^-) \setminus Ω^-)\right),$$ where the infimum is taken over all open affine half-spaces $H^+$ such that $x \in \partial H^+$ and we define $H^-= \mathbb{R}^{n+1} \setminus \overline {H^{+}}$. Our first main result asserts that any Borel subset of $$\left\{x\in\mathbb{R}^{n+1}\, :\, \int_0^1 \varepsilon_n(x,r)^2 \, \frac{dr}{r}<\infty\right\}$$ is $n$-rectifiable. For our second main result we assume that $Ω^+, Ω^-$ are open and that $Ω^+\cupΩ^-$ satisfies the capacity density condition. For each $x \in \partial Ω^+ \cup \partial Ω^-$ and $r>0$, we denote by $α^\pm(x,r)$ the characteristic constant of the (spherical) open sets $Ω^\pm \cap \partial B(x,r)$. We show that, up to a set of $\mathcal{H}^n$ measure zero, $x$ is a tangent point for both $\partial Ω^+$ and $ \partial Ω^-$ if and only if\begin{equation*} \int_0^{1} \min(1,α^+(x,r) + α^-(x,r) -2) \frac{dr}{r} < \infty. \end{equation*} The first result is new even in the plane and the second one improves and extends to higher dimensions the $\varepsilon^2$ conjecture of Carleson.

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arXiv2023

Ricci curvature bounded below and uniform rectifiability

We prove that Ahlfors-regular RCD spaces are uniformly rectifiable. The same is shown for Ahlfors regular boundaries of non-collapsed RCD spaces. As an application we deduce a type of quantitative differentiation for Lipschitz functions on these spaces.

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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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arXiv2022

A square function involving the center of mass and rectifiability

For a Radon measure $μ$ on $\mathbb{R}^d$, define $C^n_μ(x, t)= \ (\frac{1}{t^n} \ |\int_{B(x,t)} \frac{x-y}{t} \, dμ(y)\ | \ )$. This coefficient quantifies how symmetric the measure $μ$ is by comparing the center of mass at a given scale and location to the actual center of the ball. We show that if $μ$ is $n$-rectifiable, then $ \int_0^\infty |C^n_μ(x,t)|^2 \frac{dt}{t} < \infty \, \, μ\mbox{-almost everywhere}. $ Together with a previous result of Mayboroda and Volberg, where they showed that the converse holds true, this gives a characterisation of $n$-rectifiability. To prove our main result, we also show that for an $n$-uniformly rectifiable measure, $|C_μ^n(x,t)|^2 dt/t dμ$ is a Carleson measure on $\mathrm{spt}(μ) \times (0,\infty)$. We also show that, whenever a measure $μ$ is $1$-rectifiable in the plane, then the same Dini condition as above holds for more general kernels. Moreover, we give a characterisation of uniform 1-rectifiability in the plane in terms of a Carleson measure condition.

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arXiv2021

Cone and paraboloid points of arbitrary subsets of Euclidean space

In this paper we characterise cone points of arbitrary subsets of Euclidean space. Given $E \subset \mathbb{R}^n$, $x \in E$ is a cone point of $E$ if and only if \begin{align*} \int_{0}^1 β_{E}^{d,2}(B(x,r))^2 \frac{dr}{r} < \infty, \end{align*} up to a set of zero $d$-measure. The coefficients $β_E^{d,2}$ are a variation of the Jones coefficients. This is a high dimensional counterpart of a theorem of Bishop and Jones from 1994. We also prove similar results for $α$-paraboloid points, which are the $C^{1,α}$ rectifiability counterparts to cone points: $x \in E$ is an $α$-paraboloid point if and only if \begin{align*} \int_0^1 \frac{\overlineβ_{E}^{d,2}(B(x,r))^2}{r^{2α}} \, \frac{dr}{r} < \infty \end{align*} up to a set of zero $d$-measure. Here, $\overlineβ^{d,2}_E$ is another variant of the Jones coefficients, introduced by Azzam and Schul.

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arXiv2020

Higher dimensional Jordan curves

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 'topologically stable' sets can be used as covering objects in Analyst's Travelling Salesman Theorem-type theorems: if $E$ is lower $d$-regular (in a certain suitable sense), then we show that there exists a topologically stable surface $Γ$ so that $E \subset Γ$ and $$ \mathrm{diam}(E)^d + β^d(E) \approx \mathcal{H}^d(Γ), $$ where $β^d$ is a term quantifying the curvature of $E$. A corollary of the main result of this paper and a construction by Hyde, is a higher dimensional analogue of Peter Jones TST, valid for \textit{any} subset of Euclidean space.

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arXiv2020

$Ω$-symmetric measures and related singular integrals

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $Ω: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+δ_Ω$, where $δ_Ω>0$ is small. Assume also that $Ω$ is twice continuously differentiable. Motivated by a question raised by Mattila and Preiss in [MP95], we prove the following: if a Radon measure $μ$ has positive lower density and finte upper density almost everywhere, and the limit $$ \lim_{ε\downarrow 0} \int_{\mathbb{C} \setminus B(x,ε)} \frac{Ω\left((x-y)/|x-y|\right)}{|x-y|} \, dμ(y) $$ exists $μ$-almost everywhere, then $μ$ is $1$-rectifiable. To achieve this, we prove first that if an Ahlfors-David 1-regular measure $μ$ is symmetric with respect to $Ω$, that is, if $$ \int_{B(x,r)} |x-y|Ω\left(\frac{x-y}{|x-y|}\right) \, dμ(y) = 0 \mbox{ for all } x \in \mbox{spt}(μ) \mbox{ and } r>0, $$ then $μ$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $μ= c \mathcal{H}^{1}|_{L}$.

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arXiv2020

Quantitative Comparisons of Multiscale Geometric Properties

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 example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \lesssim ({\rm diam} R)^{d}.\] We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if $β_{E}(R)$ denotes the square sum of $β$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then \[ \mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim β_{E}(R). \] We prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.

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arXiv2020

Sub-elliptic boundary value problems in flag domains

A flag domain in $\mathbb{R}^{3}$ is a subset of $\mathbb{R}^{3}$ of the form $\{(x,y,t) : y < A(x)\}$, where $A \colon \mathbb{R} \to \mathbb{R}$ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn-Laplacian $\bigtriangleup^{\flat} = X^{2} + Y^{2}$ in flag domains $Ω\subset \mathbb{R}^{3}$, with $L^{2}$-boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order $L^{2}$-Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries.

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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

A proof of Carleson's $\varepsilon^2$-conjecture

In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon^2$-square function.

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