# Trimester Seminar Series

Zoom data will be provided to all long-term participants of the trimester program.

**April 11, 2022, 17:00 CEST**

**Helge Samuelsen**

## Fourier restriction in quantum harmonic analysis

Quantum Harmonic analysis was introduced by Werner in his 1984 article "Quantum harmonic analysis on phase space". Werner introduced a notion of harmonic analysis for operators, where the Schatten p-class operators replaces the $L^p$ spaces. In this talk I will give a brief introduction to quantum harmonic analysis, and present some ongoing work related to Fourier restriction in the setting of quantum harmonic analysis. This is joint work with F. Luef.

**March 31, 2022, 17:00 CEST**

**Alexander Volberg**

## One phase problem for two positive harmonic functions: many questions and a couple of answers

While working on a complex dynamical result I stumbled on the following question. Let $u, v$ be two harmonic positive functions in $\Omega$, an arbitrary domain (maybe infinitely connected). Let them vanish on the open part of the boundary. Then it is known, that if the domain is not too bad (say, with Lipschitz boundary), then $u/v$ makes sense on this part of the boundary and is Holder continuous. Here is a free boundary problem: suppose that $u/v$ is much better than Holder, e.g. let $u/v$ be in fact equal to a restriction of non-constant real analytic function to this part of the boundary. What can be said about this part of the boundary? Is it necessarily real analytic itself? I will consider some cases when the answer is "yes", but the answer in general eludes me.

**March 29, 2022, 17:00 CEST**

**Banhirup Sengupta**

## Pointwise descriptions of nearly incompressible vector fields with bounded curl

In this talk I will explain a recent work, where we provide a pointwise characterisation of nearly incompressible vector fields $b:\mathbb R^n \to\, \mathbb R^n$ with $|x|\, \log|x|$ growth at infinity for which $curl b=Db- D^{t}b$ is bounded. In the plane we can go further and describe still in pointwise sense, the vector fields $b:\mathbb R^2 \to\, \mathbb R^2$ for which $|div b| +|curl b|\, \in L^\infty.$ As an application, one can think of describing pointwise Yudovich solutions to Euler equations. This is a joint work with Albert Clop.

**March 24, 2022, 17:00 CET**

**Moritz Egert**

## The Kato problem on open subsets

I am interested in solving the Kato problem for elliptic operators L in divergence form with Dirichlet/Neumann boundary conditions on open subsets of Euclidean space. This means proving that the domain of the square root of L is the first-order L2-Sobolev space. I will give a brief overview on the history of the problem and try to explain how two geometric conditions near the boundary are sufficient to solve it: (i) Ahlfors-David regularity of the Dirichlet boundary part and (ii) a Jones condition close to where Neumann conditions are prescribed. This is joint work with Sebastian Bechtel and Robert Haller.

**March 23, 2022, 17:00 CET**

**Pascal Auscher**

## Non-local Gehring lemmas.

We shall describe results obtained in works with S. Bortz, M. Egert, and O. Saari. In PDE’s or conformal geometry, Gehring lemmas are about higher integrability for solutions or their gradient. The original lemma follows from the self-improvement of reverse Hölder inequalities on balls. These are local estimates. In parabolic PDE’s or in fractional PDE’s, a local reverse Hölder inequality may fail as the right hand-side may show non-local quantities such as weighted series of averages over concentric balls in geometric progression. Does self-improvement still hold ? Under appropriate conditions on the non-local terms, one can show that self-improvement does occur locally and/or globally. The results can be set-up in spaces of homogenous type.

**March 22, 2022, 17:00 CET**

**Giorgios Dosidis**

## The uncentered spherical maximal function and Nikodym sets

In this talk we consider the uncentered spherical maximal function, which is an analogue of the operator introduced by Stein, that also includes translations. The existence of Nikodym sets associated with spheres indicates that maximal operators given by translations of spherical averages are unbounded on Lp for all finite p. However, for lower-dimensional sets of translations, we obtain Lp boundedness for the associated maximally translated-dillated spherical averages for a certain range of p that depends on the Minkowski dimension of the set of translations. This is joint work with A. Chang and J. Kim.

**March 21, 2022, 17:00 CET**

**Andrew Morris**

## A first-order approach to solvability for singular Schrödinger equations

We will first give a brief overview of the first-order approach to boundary value problems, which factorises second-order divergence-form equations into Cauchy-Riemann systems. The advantage is that the holomorphic functional calculus for such systems can provide semigroup solution operators in tremendous generality, extending classical harmonic measure and layer potential representations. We will then show how recent developments now allow for the incorporation of singular perturbations in the associated quadratic estimates. This allows us to solve Dirichlet and Neumann problems for Schrödinger equations with potentials in reverse Hölder spaces. This is joint work with Andrew Turner.

**March 18, 2022, 16:00 CET**

**Murat Akman**

## Perturbations of elliptic operators on rough domains

In this talk, we study perturbations of elliptic operators on domains with rough boundaries. In particular, we focus on the following problem: suppose that we have "good estimates" for the Dirichlet problem for a uniformly elliptic operator $L_0$ (with corresponding elliptic measure $\omega_{L_0}$), under what optimal conditions, are those good estimates transferred to the Dirichlet problem for uniformly elliptic operator $L$ (with corresponding elliptic measure $\omega_{L}$) which is a "perturbation" of $L_0$?

When the domain is 1-sided NTA satisfying the capacity density condition, we show that if the discrepancy of the corresponding matrices satisfies a natural Carleson measure condition with respect to $\omega_{L_0}$ then $\omega_L\in A_\infty(\omega_{L_0})$. Moreover, we obtain that $\omega_L\in RH_q(\omega_{L_0})$ for any given $1<q<\infty$ if the Carleson measure condition is assumed to hold with a sufficiently small constant.

This is a joint work with Steve Hofmann, Jose Maria Martell, and Tatiana Toro.

**March 17, 2022, 17:00 CET**

**Antoine Julia**

## Sets with unit Hausdorff density and the isodiametric problem in

homogeneous groups

A important property of rectifiable sets in Euclidean spaces is that their Hausdorff measure has unit density. This property is shared by rectifiable sets in any metric space. The topic of my talk is the converse question: does unit Hausdorff density imply rectifiability? An affirmative answer was given in 1975 by Mattila for subsets of the euclidean space, but the question is still open in many metric spaces. I will show that the question is related to the isodiametric problem and answer it for subsets of some homogeneous groups. The new results I will present are part of a joint work with Andrea Merlo.

**March 15, 2022, 17:00 CET**

**Andrea Merlo**

## The density problem in the parabolic space

In this talk I will introduce the Density Problem in metric spaces and present its solution, in codimension 1, in any parabolic space. This answers completely, in codimension 1, a question posed by Mattila. We will also give counterexamples showing that the boundedness of the square function associated to the density does not imply rectifiability of the measure by means of parabolic regular graphs. This is a joint project with M. Mourgoglou and C. Puliatti.

**March 10, 2022, 17:00 CET**

**Giovanni Alberti**

## The vanishing mass conjecture and its geometric interpretation

G. Bouchitté formulated the "Vanishing Mass Conjecture" about twenty years ago in the context of optimization of light structures. Since then the only progress towards a proof of this conjecture has been obtained by J.F. Babadjian, F. Iurlano and F. Rindler in 2021. In this talk I will illustrate this conjecture placing the emphasis on the underlying geometric nature, and explore the connections to other results and questions of geometric (and measure-theoretic) flavor.

**March 8, 2022, 17:00 CET**

**Farid Bozorgnia**

## Multi-phase segregations systems: Existence, Numeric, and Applications

In this talk, we consider some models of Reaction-Diffusion Systems with high competition rates. We review different aspects and properties of these models such as existence and uniqueness of the solution for each model and their singular limit to phase segregating system. Moreover, we use properties of limiting problems to construct efficient numerical simulations for given systems. As an application, we implement ideas from segregation models and Poisson learning in graph-based semi-supervised learning.

**March 3, 2022, 17:00 CET**

**Yannick Sire**

## The generalized thin one-phase free boundary problem

I'll consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in $\mathbb{R}^{n+1}_+$ plus the area of the positivity set of that function in $\mathbb{R}^n$. I will describe full regularity of the free boundary for dimensions $n \leq 2$, prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While the results are typical for the calculus of variations, the approach does not follow the standard one first introduced by Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE. I will describe several possible lines of research and open problems.

**March 1, 2022, 17:00 CET**

**Francesco Di Plinio**

## Maximal subspace averages

We study maximal operators associated to singular averages along finite subsets of the Grassmannian of $d$-dimensional subspaces of the $n-$dimensional Euclidean space. These are singular versions of the $(d,n)$ Kakeya and Nikodym maximal functions, whose Lebesgue space bounds relate to dimensionality questions for respectively Kakeya and Nikodym sets.

The well studied $d = 1$ case corresponds to the usual directional maximal function. We provide a systematic study of all cases $1 ≤ d < n$ and prove essentially sharp $L_2$ bounds for the maximal subspace averaging operator in terms of the cardinality of the finite subset without any assumption on the structure. In the codimension $1$ case, that is $n = d + 1$, we prove the precise critical weak $(2, 2)$-bound.

Our estimates rely on Fourier analytic almost orthogonality principles, combined with polynomial partitioning, but we also use spatial analysis based on the precise calculation of intersections of $d$-dimensional plates.

Joint work with Ioannis Parissis (University of Basque Country)

**February 24, 2022, 17:00 CET**

**Pablo Hidalgo**

## Carleson measure estimates, corona decompositions and perturbation of elliptic operators without connectivity

During the last decades, many researchers have made progress in their understanding of the solvability of the Dirichlet problem for elliptic operators in rough domains. These works show that the connectivity of the set plays an important role since, under strong connectivity conditions, the good behavior of the elliptic measure is equivalent to certain Carleson measure estimates for weak null-solutions. In this talk, we will review these results and present some extensions to them, done in collaboration with M. Cao and J.M. Martell. Concretely, we extend the theory since we no longer assume any connectivity, and still get characterizations of some weak Carleson measure estimates for bounded solutions in terms of a Corona decomposition for the elliptic measure. As a consequence of the developed techniques, we also obtain Fefferman-Kenig-Pipher perturbation results working without connectivity.

**February 21, 2022, 17:00 CET**

**Josep Gallegos**

## Unique continuation at the boundary for solutions of elliptic PDEs

In 1991 Fang-Hua Lin posed the following question. Let $\Omega\subset\mathbb R^d$ be a Lipschitz domain and $\Sigma$ be an open subset of its boundary $\partial\Omega$. Let $u$ be a harmonic function in $\Omega$, continuous in $\overline \Omega$, that vanishes on $\Sigma$, and that its normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure. Is it true that $u$ must be identically zero? Last year Xavier Tolsa showed a positive answer to the previous question in the case $\Omega$ is a Lipschitz domain with small Lipschitz constant. In this talk, I will explain a recent work where we show that (in the same setting as Tolsa) we can find a family of balls $(B_i)_i$ centered on $\Sigma$ such that $u|_{B_i\cap\Omega}$ does not change sign and that $K\backslash \cup_i B_i$ has positive Minkowski codimension for any compact $K\subset\Sigma$. We also prove the previous result for solutions of divergence form elliptic PDEs with Lipschitz coefficients, although we will only focus on the harmonic case during the talk. I will also try to motivate why the set of points of $\Sigma$ where $u$ does not change sign nearby is interesting by comparing it with the usual singular set on the boundary.

**February 18, 2022, 16:00 CET**

**Matthew Hyde**

## A d-dimensional Analyst's Travelling Salesman Theorem for arbitrary subsets

Abstract: In the early nineties Peter Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$ numbers, which measure flatness at a given scale and location. This characterization is the so-called Analyst's Travelling Salesman Theorem. Several analogous results for higher dimensional subsets have since appeared, but typically one requires some extra assumptions e.g. Ahlfors regularity. In this talk we discuss a recent $d$-dimensional Travelling Salesman Theorem for arbitrary subsets of Euclidean space, stated in terms of a new $\beta$-type number.

**February 10, 2022, 17:00 CET**

**Olli Tapiola**

## Two-sided local John condition implies Harnack chains and boundary Poincaré inequalities.

In a very recent work, Mourgoglou and Tolsa considered the solvability of the Dirichlet and regularity problems and their connections in corkscrew domains with codimension 1 Ahlfors--David regular boundaries. Some of their results were proven in the presence of the 2-sided local John condition or weak boundary Poincaré inequalities. We revisit these geometric assumptions and prove the following two results. First, we show that a 2-sided corkscrew domain satisfying the local John condition is a chord-arc domain. Second, we show that the boundary of a 2-sided chord-arc domain supports a weak $(1,p)$-Poincaré inequality of Heinonen--Koskela type for any $1 < p < \infty$. Our proofs are based on significant advances in e.g. harmonic measure, uniform rectifiability and metric Poincaré theories. This is a joint work with Xavier Tolsa.

**January 26, 2022, 15:15 CET**

**Guy David**

## A story of the landscape function and its relations with eigenvectors of a Schrödinger operator $L = -\Delta V$.

Let $L = - \Delta V$ be a Schrödinger operator on a nice domain (like a box) in Rn. We think of V as being random, but a priori this is not needed. The landcape function is the solution of $Lu = 1$ (and, typically, Dirichlet boundary conditions, or periodic). This function, and the corresponding "effective potential" $W = 1/u$, can be used to describe the eigenfunctions of L, their localization and their decay in the places where W is large, and the IDOS (number of eigenfunctions with eigenvalues less than E > 0). We try te explain why and maybe mention connected questions.

**January 20, 2022, 17:00 CET**

**Joris Roos**

## Lp improving for spherical maximal functions on Heisenberg groups

We consider maximal averages on codimension two spheres in Heisenberg groups and prove Lp to Lq estimates that are sharp in the Lebesgue exponents, up to endpoints. Key ingredients include a surprising application of known Fourier integral operator bounds of Mockenhaupt-Seeger-Sogge which relies on a certain cone having the maximal number of nonvanishing principal curvatures, as well as a new counterexample. The upper bounds extend to the case of Métivier groups, which can have arbitrarily high codimension. This is joint work with Andreas Seeger and Rajula Srivastava.

**January 19, 2022, 17:00 CET**

**Polona Durcik**

## Local bounds for singular Brascamp-Lieb forms with cubical structure

Singular Brascamp-Lieb forms arise when one of the functions in a Brascamp-Lieb form is replaced by a singular integral kernel. In this talk we discuss a range of Lp bounds for singular Brascamp-Lieb forms with cubical structure. We pass through local and sparse bounds. This is a joint work with L. Slavíková and C. Thiele.

**January 17, 2022, 17:00 CET**

**David Beltran**

## Variation bounds for spherical averages

Variational estimates are refinements of maximal function estimates, in which the $\ell^\infty$ norm is replaced by a larger $V^r$ norm, $1 \leq r \leq \infty$. In 2008, Jones, Seeger and Wright proved that the r-variation operator associated to the spherical averages $\{f \ast \sigma_t \}_{t >0}$ is bounded on $L^p(\mathbb{\R}^d)$ if $d/(d-1)<p$ $\leq$ $2d$ and $r>2$ or $p>2d$ and $r>p/d$, and this is sharp except for the endpoint case $r=p/d$, which remains open. In this talk I will present $L^p(\R^d) \to L^q(\R^d)$ bounds for the local r-variation operator, that is, the associated with $\{f \ast \sigma_t\}_{1 \leq t \leq 2}$. The bounds are sharp up to endpoints (except in dimension 3), and some positive results also hold in some endpoints cases. In particular, it can be established the interesting endpoint bound $L^{q/d}(\R^d) \to L^q(\R^d)$ for $r=q/d$, $q>(d^2+1)/(d-1)$ if $d\geq 3$. This is joint work with Richard Oberlin, Luz Roncal, Andreas Seeger and Betsy Stovall.

**January 13, 2022, 17:00 CET**

**Andreas Seeger**

## Lp improving bounds for spherical maximal operators

Consider families of spherical means where the radii are restricted to a given subset of a compact interval. One is interested in the Lp improving estimates for the associated maximal operators and related objects. I will mostly report on recent joint work with Joris Roos; this also relates to our earlier paper with Theresa Anderson and Kevin Hughes. Results depend on several notions of fractal dimension of the dilation set, or subsets of it. There are some unexpected statements on the shape of the possible type sets.