# Fall 2020 and Spring 2021 Analysis Seminars

## Contents

- 1 Seminar Schedule
- 2 Abstracts
- 2.1 Alexei Poltoratski
- 2.2 Polona Durcik and Joris Roos
- 2.3 Andrew Zimmer
- 2.4 Hong Wang
- 2.5 Kevin Luli
- 2.6 Niclas Technau
- 2.7 Terence Harris
- 2.8 Yuval Wigderson
- 2.9 Oscar Dominguez
- 2.10 Tamas Titkos
- 2.11 Shukun Wu
- 2.12 Jonathan Hickman
- 2.13 Hanlong Fang
- 2.14 Bingyang Hu
- 2.15 Krystal Taylor
- 2.16 Dominique Maldague
- 2.17 Diogo Oliveira e Silva
- 2.18 Oleg Safronov
- 2.19 Ziming Shi
- 2.20 Xiumin Du
- 2.21 Etienne Le Masson
- 2.22 Theresa Anderson
- 2.23 Nathan Wagner
- 2.24 David Beltran
- 2.25 Yumeng Ou

# Seminar Schedule

# Abstracts

### Alexei Poltoratski

Title: Dirac inner functions

Abstract: My talk will focus on some new (and old) complex analytic objects arising from Dirac systems of differential equations. We will discuss connections between problems in complex function theory, spectral and scattering problems for differential operators and the non-linear Fourier transform.

### Polona Durcik and Joris Roos

Title: A triangular Hilbert transform with curvature, I & II.

Abstract: The triangular Hilbert is a two-dimensional bilinear singular originating in time-frequency analysis. No Lp bounds are currently known for this operator. In these two talks we discuss a recent joint work with Michael Christ on a variant of the triangular Hilbert transform involving curvature. This object is closely related to the bilinear Hilbert transform with curvature and a maximally modulated singular integral of Stein-Wainger type. As an application we also discuss a quantitative nonlinear Roth type theorem on patterns in the Euclidean plane. The second talk will focus on the proof of a key ingredient, a certain regularity estimate for a local operator.

### Andrew Zimmer

Title: Complex analytic problems on domains with good intrinsic geometry

Abstract: In this talk, I will describe a new class of domains in complex Euclidean space which is defined in terms of the existence of a Kaehler metric with good geometric properties. This class is invariant under biholomorphism and includes many well-studied classes of domains such as strongly pseudoconvex domains, finite type domains in dimension two, convex domains, homogeneous domains, and embeddings of Teichmuller spaces. Further, certain analytic problems are tractable for domains in this family even when the boundary is non-smooth. In particular, it is possible to characterize the domains in this family where the dbar-Neumann operator on (0, q)-forms is compact (which generalizes an old result of Fu-Straube for convex domains).

### Hong Wang

Title: Improved decoupling for the parabola

Abstract: In 2014, Bourgain and Demeter proved the $l^2$ decoupling estimates for the paraboloid with constant $R^{\epsilon}$. We prove an $(l^2, L^6)$ decoupling inequality for the parabola with constant $(\log R)^c$. This is joint work with Larry Guth and Dominique Maldague.

### Kevin Luli

Title: Smooth Nonnegative Interpolation

Abstract: Suppose E is an arbitrary subset of R^n. Let f: E \rightarrow [0, \infty). How can we decide if f extends to a nonnegative function C^m function F defined on all of R^n? Suppose E is finite. Can we compute a nonnegative C^m function F on R^n that agrees with f on E with the least possible C^m norm? How many computer operations does this take? In this talk, I will explain recent results on these problems. Non-negativity is one of the most important shape preserving properties for interpolants. In real life applications, the range of the interpolant is imposed by nature. For example, probability density, the amount of snow, rain, humidity, chemical concentration are all nonnegative quantities and are of interest in natural sciences. Even in one dimension, the existing techniques can only handle nonnegative interpolation under special assumptions on the data set. Our results work without any assumptions on the data sets.

### Niclas Technau

Title: Number theoretic applications of oscillatory integrals

Abstract: We discuss how the analysis of oscillatory integrals can be used to solve number theoretic problems. More specifically, the focus will be on understanding fine-scale statistics of sequences on the unit circle. Further, we shall briefly explain a connection to quantum chaos.

### Terence Harris

Title: Low dimensional pinned distance sets via spherical averages

Abstract: An inequality is derived for the average t-energy of weighted pinned distance measures, where 0 < t < 1, in terms of the L^2 spherical averages of Fourier transforms of measures. This generalises the result of Liu (originally for Lebesgue measure) to pinned distance sets of dimension smaller than 1, and strengthens Mattila's result from 1987, originally for the full distance set.

### Yuval Wigderson

Title: New perspectives on the uncertainty principle

Abstract: The phrase ``uncertainty principle* refers to a wide array of results in several disparate fields of mathematics, all of which capture the notion that a function and its Fourier transform cannot both be ``very localized*. The measure of localization varies from one uncertainty principle to the next, and well-studied notions include the variance (and higher moments), the entropy, the support-size, and the rate of decay at infinity. Similarly, the proofs of the various uncertainty principles rely on a range of tools, from the elementary to the very deep. In this talk, I'll describe how many of the uncertainty principles all follow from a single, simple result, whose proof uses only a basic property of the Fourier transform: that it and its inverse are bounded as operators $L^1 \to L^\infty$. Using this result, one can also prove new variants of the uncertainty principle, which apply to new measures of localization and to operators other than the Fourier transform. This is joint work with Avi Wigderson.

### Oscar Dominguez

Title: New Brezis--Van Schaftingen--Yung inequalities via maximal operators, Garsia inequalities and Caffarelli--Silvestre extensions

Abstract: The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. The goal of this talk is twofold. Firstly, we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to ergodic theory, Fourier series, etc. In the second part of the talk, we shall investigate the fractional setting in the BvSY theorem. Our approach is based on new Garsia-type inequalities and an application of the Caffarelli--Silvestre extension. This is joint work with Mario Milman.

### Tamas Titkos

Title: Isometries of Wasserstein spaces

Abstract: Due to its nice theoretical properties and an astonishing number of applications via optimal transport problems, probably the most intensively studied metric nowadays is the p-Wasserstein metric. Given a complete and separable metric space $X$ and a real number $p\geq1$, one defines the p-Wasserstein space $\mathcal{W}_p(X)$ as the collection of Borel probability measures with finite $p$-th moment, endowed with a distance which is calculated by means of transport plans \cite{5}.

The main aim of our research project is to reveal the structure of the isometry group $\mathrm{Isom}(\mathcal{W}_p(X))$. Although $\mathrm{Isom}(X)$ embeds naturally into $\mathrm{Isom}(\mathcal{W}_p(X))$ by push-forward, and this embedding turned out to be surjective in many cases (see e.g. [1]), these two groups are not isomorphic in general. Kloeckner in [2] described the isometry group of the quadratic Wasserstein space $\mathcal{W}_2(\mathbb{R}^n)$, and it turned out that the case of $n=1$ is special in the sense that $\mathrm{Isom}(\mathcal{W}_2(\mathbb{R})$ is extremely rich. Namely, it contains a large subgroup of wild behaving isometries that distort the shape of measures. Following this line of investigation, in \cite{3} we described $\mathrm{Isom}(\mathcal{W}_p(\mathbb{R}))$ and $\mathrm{Isom}(\mathcal{W}_p([0,1])$ for all $p\geq 1$.

In this talk I will survey first some of the earlier results in the subject, and then I will present the key results of [3]. If time permits, I will also report on our most recent manuscript [4] in which we extended Kloeckner's multidimensional results. Joint work with Gy\"orgy P\'al Geh\'er (University of Reading) and D\'aniel Virosztek (IST Austria).

[1] J. Bertrand and B. Kloeckner, \emph{A geometric study of Wasserstein spaces: isometric rigidity in negative curvature}, International Mathematics Research Notices, 2016 (5), 1368--1386.

[2] B. Kloeckner, \emph{A geometric study of Wasserstein spaces: Euclidean spaces}, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Tome 9 (2010) no. 2, 297--323.

[3] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{Isometric study of Wasserstein spaces – the real line}, Trans. Amer. Math. Soc., 373 (2020), 5855--5883.

[4] Gy. P. Geh\'er, T. Titkos, D. Virosztek, \emph{The isometry group of Wasserstein spaces: The Hilbertian case}, submitted manuscript.

[5] C. Villani, \emph{Optimal Transport: Old and New,} (Grundlehren der mathematischen Wissenschaften) Springer, 2009.

### Shukun Wu

Title: On the Bochner-Riesz operator and the maximal Bochner-Riesz operator

Abstract: The Bochner-Riesz problem is one of the most important problems in the field of Fourier analysis. It has a strong connection to other famous problems, such as the restriction conjecture and the Kakeya conjecture. In this talk, I will present some recent improvements to the Bochner-Riesz conjecture and the maximal Bochner-Riesz conjecture. The main methods we used are polynomial partitioning and the Bourgain Demeter l^2 decoupling theorem.

### Jonathan Hickman

Title: Sobolev improving for averages over space curves

Abstract: Consider the averaging operator given by convolution with arclength measure on compact piece of a smooth curve in R^n. A simple question is to precisely quantify the gain in regularity induced by this averaging, for instance by studying the L^p-Sobolev mapping properties of the operator. This talk will report on ongoing developments towards understanding this problem. In particular, we will explore some non-trivial necessary conditions on the gain in regularity. Joint with D. Beltran, S. Guo and A. Seeger.

### Hanlong Fang

Title: Canonical blow-ups of Grassmann manifolds

Abstract: We introduce certain canonical blow-ups \mathcal T_{s,p,n}, as well as their distinct submanifolds \mathcal M_{s,p,n}, of Grassmann manifolds G(p,n) by partitioning the Plücker coordinates with respect to a parameter s. Various geometric aspects of \mathcal T_{s,p,n} and \mathcal M_{s,p,n} are studied, for instance, the smoothness, the holomorphic symmetries, the (semi-)positivity of the anti-canonical bundles, the existence of Kähler-Einstein metrics, the functoriality, etc. In particular, we introduce the notion of homeward compactification, of which \mathcal T_{s,p,n} are examples, as a generalization of the wonderful compactification.

### Bingyang Hu

Title: Some structure theorems on general doubling measures.

Abstract: In this talk, we will first several structure theorems about general doubling measures. Secondly, we will include some main idea to prove one of these results. More precisely, we will focus on the construction of an explicit family of measures that are p-adic doubling for any finite set of primes, however, not doubling. This part generalizes the work by Boylan, Mills and Ward in 2019 in a highly non-trivial way. As some application, we apply these results (that is, the same construction) to show analogous statements for Muckenhoupt Ap weights and reverse Holder weights. This is a joint work with Tess Anderson.

### Krystal Taylor

Title: Quantifications of the Besicovitch Projection theorem in a nonlinear setting

Abstract: There are several classical results relating the geometry, dimension, and measure of a set to the structure of its orthogonal projections. It turns out that many nonlinear projection-type operators also have special geometry that allows us to build similar relationships between a set and its "projections", just as in the linear setting. We will discuss a series of recent results from both geometric and probabilistic vantage points. In particular, we will see that the multi-scale analysis techniques of Tao, as well as the energy techniques of Mattila, can be strengthened and generalized to projection-type operators satisfying a transversality condition. As an application, we address the Buffon curve problem, which is to find upper and lower bounds for the rate of decay of the Favard curve length of the four-corner Cantor set.

### Dominique Maldague

Title: A new proof of decoupling for the parabola

Abstract: Decoupling has to do with measuring the size of functions with specialized Fourier support (in our case, in a neighborhood of the truncated parabola). Bourgain and Demeter resolved the l^2 decoupling conjecture in 2014, using ingredients like the multilinear Kakeya inequality, L^2 orthogonality, and induction-on-scales. I will present the ideas that go into a new proof of decoupling and make some comparison between the two approaches. This is related to recent joint work with Larry Guth and Hong Wang, as well as forthcoming joint work with Yuqiu Fu and Larry Guth.

### Diogo Oliveira e Silva

Title: Global maximizers for spherical restriction

Abstract: We prove that constant functions are the unique real-valued maximizers for all $L^2-L^{2n}$ adjoint Fourier restriction inequalities on the unit sphere $\mathbb{S}^{d-1}\subset\mathbb{R}^d$, $d\in\{3,4,5,6,7\}$, where $n\geq 3$ is an integer. The proof uses tools from probability theory, Lie theory, functional analysis, and the theory of special functions. It also relies on general solutions of the underlying Euler--Lagrange equation being smooth, a fact of independent interest which we discuss. We further show that complex-valued maximizers coincide with nonnegative maximizers multiplied by the character $e^{i\xi\cdot\omega}$, for some $\xi$, thereby extending previous work of Christ & Shao (2012) to arbitrary dimensions $d\geq 2$ and general even exponents. This talk is based on results obtained with René Quilodrán.

### Oleg Safronov

Title: Relations between discrete and continuous spectra of differential operators

Abstract: We will discuss relations between different parts of spectra of differential operators. In particular, we will see that negative and positive spectra of Schroedinger operators are related to each other. However, there is a stipulation: one needs to consider two operators one of which is obtained from the other by flipping the sign of the potential at each point x. If one knows only that the negative spectra of the two operators are discrete, then their positive spectra do not have gaps. If one knows more about the rate of accumulation of the discrete negative eigenvalues to zero, then one can say more about the absolutely continuous component of the positive spectrum.

### Ziming Shi

Title: Sharp Sobolev $1/2$-estimate for $\bar\partial$ equations on strictly pseudoconvex domains with $C^2$ boundary

Abstract: We give a solution operator for $\bar\partial$ equation that gains the sharp $1/2$-derivative in the Sobolev space $H^{s,p}$ on any strictly pseudoconvex domain with $C^2$-boundary, for all $1< p < \infty$ and $s>1/p$. We also show that the same solution operator gains a $1/2$-derivative in the H\"older-Zygmund space $\Lambda^s$ for any $s>0$, where previously it was known for $s>1$ by work of X. Gong. The main ingredients used in our proof are a Hardy-Littlewood lemma of Sobolev type and a new commutator estimate. Joint work with Liding Yao.

### Xiumin Du

Title: Falconer's distance set problem

Abstract: A classical question in geometric measure theory, introduced by Falconer in the 80s is, how large does the Hausdorff dimension of a compact subset in Euclidean space need to be to ensure that the Lebesgue measure of its set of pairwise Euclidean distances is positive. In this talk, I'll report some recent progress on this problem, which combines several ingredients including Orponen's radial projection theorem, Liu's L^2 identity obtained using a group action argument, and the refined decoupling theory. This is based on joint work with Alex Iosevich, Yumeng Ou, Hong Wang, and Ruixiang Zhang.

### Etienne Le Masson

Title: Quantum ergodicity for Eisenstein series on large genus hyperbolic surfaces

Abstract: We will present a delocalisation result for eigenfunctions of the Laplacian on finite area hyperbolic surfaces of large genus. This is a quantum ergodicity result analogous to a theorem of Zelditch showing that the mass of most L2 eigenfunctions and Eisenstein series (eigenfunctions associated with the continuous spectrum) equidistributes when the eigenvalues tend to infinity. Here we will fix a bounded spectral window and look at a similar equidistribution phenomenon when the area/genus goes to infinity (more precisely the surfaces Benjamini-Schramm converge to the plane). The conditions we require on the surfaces are satisfied with high probability in the Weil-Petersson model of random surfaces introduced by Mirzakhani. They also apply to congruence covers of the modular surface, where we recover a result of Nelson on the equidistribution of Maass forms (with weaker convergence rate). The proof is based on ergodic theory methods. Joint work with Tuomas Sahlsten.

### Theresa Anderson

Title: Dyadic analysis (virtually) meets number theory

Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first, which we will spend the most time motivating and discussing, involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. If time remains, secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.

### Nathan Wagner

Title: Weighted Estimates for the Bergman and Szegő projections on strongly pseudoconvex domains with near minimal smoothness

Abstract: The Bergman and Szegő projections are fundamental operators in complex analysis in one and several complex variables. Consequently, the mapping properties of these operators on L^p and other function spaces have been extensively studied. In this talk, we discuss some recent results for these operators on strongly pseudoconvex domains with near minimal smoothness. In particular, weighted L^p estimates are obtained, where the weight belongs to a suitable generalization of the Békollé-Bonami or Muckenhoupt class. For these domains with less boundary regularity, we use an operator-theoretic technique that goes back to Kerzman and Stein. We also obtain weighted estimates for the endpoint p=1, including weighted weak-type (1,1) estimates. Here we use a modified version of singular-integral theory and a generalization of the Riesz-Kolmogorov characterization of precompact subsets of Lebesgue spaces. This talk is based on joint work with Brett Wick and Cody Stockdale.

### David Beltran

Title: Sobolev improving for averages over curves in $\mathbb{R}^4$

Abstract: Given a smooth non-degenerate space curve (that is, a smooth curve whose n-1 curvature functions are non-vanishing), it is a classical question to study the smoothing properties of the averaging operators along a compact piece of such a curve. This question can be quantified, for example, by studying the $L^p$-Sobolev mapping properties of those operators. These are well understood in 2 and 3 dimensions, and in this talk, we present a new sharp result in 4 dimensions. We focus on the positive results; the non-trivial examples which show that our results are best possible were presented by Jonathan Hickman in December 1st. This is joint work with Shaoming Guo, Jonathan Hickman and Andreas Seeger.

### Yumeng Ou

Title: On the multiparameter distance problem

Abstract: In this talk, we will describe some recent progress on the Falconer distance problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{|x-y|: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent work on this problem, which also includes some new results on the multiparameter radial projection theory of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.