# 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.

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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. Sets with topology, the Analyst's TST, and applications.**

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

- 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

- 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!