# Research

My area of interest is geometric measure theory. More precisely, I focus on quantitative aspects of rectifiability. Also, geometric problems from harmonic analysis, for example boundedness of singular integral operators on non-smooth sets.

## Papers and preprints

Here is a list of papers, some published, some in preprint version.

In this paper we apply the method of layer potential to the singular data Dirichlet problem for the Kohn Laplacian. In this setting, we are naturally lead to conduct our analysis in the framework of Heisenberg geometry.

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** 7. Necessary condition for the $L^2$ boundedness of the Riesz transform in
Heisenberg group.** With Damian Dabrowski.

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

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** 6. A proof of Carleson $\epsilon^2$ conjecture.** With Ben Jaye and Xavier Tolsa.

We give a proof of Carleson $\epsilon^2$ conjecture.

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** 5. A square function involving the center of mass and rectifiability. **

The most `popular' coefficients used in GMT to measure the local regularity of sets and measures usually quantify how different the set or measure is from being a plane. See for example the $\beta$ numbers of Peter Jones or the $\alpha$ numbers of Xavier Tolsa. But one could think of different ways to quantify local regularity. For example, for a measure $\mu$, put $$ C_\mu(x,r):=\frac{1}{r} \int_{B(x,r)} \frac{x-y}{r} \, d\mu(y). $$ This coefficient measures how symmetric a measure is: indeed, $\int_{B(x,r)}y \, d\mu(y)$ is the center of mass, and so we are effectively quantifying how far is the center of mass from the center of the ball. Mattila gave a complete characterisation of what he calls symmetric measures in the plane. He proves that if a measure $\mu$ satisfies $$\int_{B(x,r)} x-y \, d\mu(y) = 0 $$ for all $r>0$ and all $x \in \text{support}(\mu)$, then either $\mu$ is singular or it is continuous. In the latter case, it is either the 2 dimensional Lebesgue measure, or it is a sum of planes with equal distance. In this paper, I show that if $\mu$ is rectifiable, then it is `quasi' symmetric in the sense that $$ \int_0^\infty C_\mu(x,r) \, \frac{dt}{t} $$ is finite for $\mu$-almost all $x$'s. The converse is also true and was proved by Mayboroda and Volberg. Putting together the two results one gets a new characterisation of rectifiability in terms of center of mass.

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** 4. Higher dimensional Jordan curves.**

The starting question of this paper was: what sort of condition should I impose on a higher dimensional subset $E$ to guarantee that the traveling salesman theorem of Peter Jones holds? In fact, what I wanted this hypotetical condition to guarantee, is the quantitative estimate $$ C_0^{-1} \leq \frac{ \text{diameter}(E)+ \beta_E^2(E)}{\mathcal{H}^d(E)} \leq C_0. $$ This is the quantitative estimate given by Peter Jones. Here let us just say that the $\beta$ quantity in the inequalities above measures how flat a set is, and it does so at all scales and location. Quite recently, J. Azzam and R. Schul proved a similar estimate for a higher dimensional set: $$ C_0^{-1} \leq \frac{ \text{diameter}(E)+ \beta_E^2(E)}{\mathcal{H}^d(E)+ \Theta_E} \leq C_0. $$ Note the difference between the first and the second expression: the term $\Theta_E$ appears in the theorem by Azzam and Schul. With this in mind, I rephrase the initial question: what sort of condition on $E$ would guarantee an estimate of the first type (rather than of the second type)? It turns out that a topological condition firstly introduced by Guy David does the job. This is the main result of this paper. David's condition guarantees a very robust lower bound on the dimension of the set $E$. It says that no matter how we (locally) deform and bend $E$, we will not be able to lower its dimension. In a way, this is true for curves (imagine bending a little bit an metal thread).

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** 3. Quantitative comparisons of multiscale geometric properties. **
With Jonas Azzam.

*Analysis and PDEs.*

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** 2. $\Omega$-symmetric measures and related singular integrals. **

*Revista Matematica Iberoamericana*

Let $S$ be the 1-sphere in the plane, and let \(\Omega: S \to S\) be bi-Lipschitz with constant $1+\delta_\Omega$, where $\delta_\Omega$ should be thought to be small (it will bounded above by a universal constant smaller than 1). In this note we prove that if an Ahlfors-David 1-regular measure $\mu$ is symmetric with respect to $\Omega$, that is, if $$ \int_{B(x,r)} |x-y|\Omega\left(\frac{x-y}{|x-y|}\right) \, d\mu(y) = 0 \mbox{ for all } x \in \spt(\mu) \mbox{ and } r>0, $$ then $\mu$ is flat, or, in other words, there exists a constant $c>0$ and a line $L$ so that $\mu= c \mathcal{H}^{1}|_{L}$. This result will be applied in a future companion paper to give a characterisation of rectifiability in terms of finitness of a certain square function involving this type of kernels.

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 $\beta$ numbers holds coincides with the set of tangent points of $E$, up to a set of $H^d$-measure zero. The main point of our result is that $H^d|_E$ is not $\sigma$-finite; because of this, we use a certain variant of the $\beta$ coefficient, firstly introduced by Azzam and Schul, which is given in terms of integration with respect to the Hausdorff content.

## Talks

- October 2020. Talk at the Jyvaskyla Geometry Seminar.
- June 2020. Talk at the Virtual Harmonic Analysis Seminar.
- Nov. 2019. Seminar talk at the Universtiy of Edinburgh.
- Oct. 2019.
*A proof of the Carleson $\epsilon^2$-conjecture.*Talk at the workshop on Geometry and Analysis at IM PAN (Warsaw) in October 2019. - June 2019.
*$\Omega$-symmetric measures and related singular integrals*. Contributed talk at BAC2019, June 2019. - June 2019. Contributed talk at the HAPDE conference in Helsinki (June 2019).
- May 2019.
*A family of analyst's travelling salesman theorems*. Seminar talk at the joint Analysis Seminar of UAB-UB in Barcelona, 20/05/2019. - May 2017.
*Towards a new characterisation of uniform rectifiability*. Talk given at the MIGSAA 2017 symposium. - April 2017.
*Non-tangential behaviour and Carleson measures*. Talk given for the SMSTC (Scottish Mathematical Training Center) course of Harmonic Analysis. - June 2016.
*Classification of three dimensional steady flow*. Talk given at the Topological fluid dynamics seminar at the University of Dundee following VI Arnold's work on the subject.

## Conferences, Workshops, Summer schools

See the CV for a more up-to-date list.- Workshop on Geometry and Analysis at IM PAN (Warsaw). October 2019.
- Simons Semester at IM PAN (Warsaw) on geometry and analysis in function and mapping theory on Euclidean and metric measure spaces.
- IAS/PCMI Graduate School in Harmonic Analysis. July 2018, Park City, Utah (US).
- Harmonic Analysis and Geometric measure theory. A conference at CIRM, Marseilles, France. October 2017.
- Neurogeometry. This was a summer school organised by SMI (Scuola matematica interuniversitaria) in Cortona, at the Palazzone della Normale. July 2017.
- New trends in Analysis and Geometry in Matric Spaces. A series of short courses organised by CIME-CIRM in Levico Terme, Italy. June 2017.
- Geometric PDEs at Warwick. A week-long workshop at the Mathematics Institute at Warwick University, UK. December 2017.
- SMI Summer School. A summer school for just-graduated students. Perugia, Italy. August 2016.

# Teaching

- Spring '19: Workshops for Honours Complex Analysis (HCV, MATH10067, one-hour tutorial + two-hour presentation skills).
- Fall '18: Workshops for Honours Analysis (HA, MATH10068, one-hour tutorial + two-hour presentation skills)
- Spring '18: Tutorials for Proofs and Problem Solving (PPS, MATH08059, two-hour tutorial)
- Fall '17: Tutorials for Measure and Integration, a graduate course of the SMSTC.

# Extra

Under construction!