# Spring 2023 Analysis Seminar

Organizer: Shaoming Guo

Email: shaomingguo (at) math (dot) wisc (dot) edu

Time: Tuesdays, 4-5pm

Room: Van Vleck B139

All talks will be in-person unless otherwise specified.

In some cases the seminar may be scheduled at different time to accommodate speakers.

If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+join (at) g-groups (dot) wisc (dot) edu

# Abstracts

### Shaoming Guo

Title: Hormander's generalization of the Fourier restriction problem

Abstract: Hörmander 1973 proposed to study a generalized Fourier extension operator, and asked whether the generalized operator satisfies the same L^p bounds as that of the standard Fourier extension operator. Surprisingly, Bourgain 1991 gave a negative answer to Hörmander’s question. In this talk, I will discuss a modification of Hörmander’s question whose answer may be affirmative. This is a joint work with Hong Wang and Ruixiang Zhang.

### Diogo Oliveira e Silva

Title: The Stein-Tomas inequality: three recent improvements

Abstract: The Stein-Tomas inequality dates back to 1975 and is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. The goal of this talk is three-fold: we present a recent proof of the sharp endpoint Stein-Tomas inequality in three space dimensions; we present a variational refinement and withdraw some consequences; and we discuss how to improve the Stein-Tomas inequality in the presence of certain symmetries.

### Jack Burkart

Title: Sobolev Spaces for General Metric Spaces

Abstract: Sobolev spaces are classically defined in Euclidean space as L^p functions possessing weak derivatives (of some order). Recently, there has been interest in doing analysis and developing a theory of calculus on general metric spaces. A natural question one might ask is how can one define Sobolev spaces in an arbitrary metric space? In this talk, I'll discuss some ways we can generalize concepts like the Poincare inequalty to an arbitrary metric space and showcase some alternative definitions that can be used in more general settings. After discussing some known results in this area, I'll spend the latter part of the talk discussing some of my own ongoing research involving establishing Poincare inequalities in domains in Euclidean space that are not necessarily W^{1,p} extension domains and some other questions we are currently considering. This talk features joint and ongoing work with Ryan Alvarado, Lisa Naples, and Benham Esmayli.

### Shengwen Gan

Title: Exceptional set estimates in finite field

Abstract: Let $A\subset \mathbb{F}^3_p$ with $\# A=p^a$. For any direction $\theta$ in $\mathbb{F}^3_p$, define $\pi_{\theta}(A)$ to be the set of lines in direction $\theta$ and passing through $A$. Define the exceptional set $E_s(A):=\{\theta: \# \pi_\theta (A)<p^s \}$. Falconer-type estimate gives $\# E_s(A)\lesssim p^{2+s-a} $. I will talk about a new result: If $s<\frac{a+1}{2}$, then $\# E_s(A)\lesssim p^{2+2s-2a}$.

### Yuqiu Fu

Title: Incidence estimates for tubes and balls with dimensional spacing condition in R^2.

Abstract: We will discuss essentially sharp incidence estimates in R^2 for a collection of tubes of dimension \delta \times 1 and a collection of balls of radius \delta, which satisfy some dimensional spacing condition. An application of these estimates is a new lower bound on the Hausdorff dimension of a (s,t) – Furstenberg set in R^2 when t > 1 + \epsilon(s,t) and s + t/2 \geq 1, where \epsilon is small depending on (s,t). This is joint work with Kevin Ren.

### Zhiren Wang

Title: Classification of smooth actions by higher rank lattices in critical dimensions.

Abstract: The Zimmer program asks how lattices in higher rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SL(n,R) with n\geq 3 or other higher rank R-split semisimple Lie groups, the action is trivial up to a finite group action. In this talk, we will explain what happens in the critical dimension for higher rank R-split semisimple Lie groups. For example, non-trivial actions by lattices in SL(n,R), n\geq 3, on (n-1)-dimensional manifolds are isomorphic to the standard action on RP^{n-1} up to a finite quotient group and a finite covering. This is a joint work with Aaron Brown and Federico Rodriguez Hertz.

### Jaehyeon Ryu

Title: Endpoint eigenfunction bounds for the Hermite operator

Abstract: We study the problem of obtaining a sharp $L^2$--$L^q$ bound on the spectral projection operator for the Hermite operator at $q = 2(d+3)/(d+1)$. The point is called the endpoint because in previous related work of Koch-Tataru, the authors obtained sharp $L^2$--$L^q$ bounds except for $q = 2(d+3)/(d+1)$. As for the endpoint, they also obtained a bound involving a logarithmic term, but they did not expect that this bound would be optimal and instead conjectured that the logarithmic term can be removed. In this talk, we prove that this conjecture is true in dimensions greater or equal to 3. This talk is based on a joint work with Eunhee Jeong, Sanghyuk Lee.

### Liding Yao

Title: Sobolev and H\"older Estimates for Homotopy Operators of $\overline\partial$-Equations on Convex Domains of Finite Multitype

Abstract: For a bounded smooth convex domain $\Omega\subset\mathbb C^n$ that has finite type $m$, we construct a $\overline\partial$ solution operator $\mathcal T_q$ on $(0,q)$-forms that has (fractional) Sobolev boundedness $\mathcal T_q:H^{s,p}\to H^{s+1/m,p}$ for all $1<p<\infty$ and $s\in\mathbb R$. In the talk I will briefly repeat the basic materials of $\overline\partial$-Equations (from Spring 2022 Math 921); review the so-called “integral representations” construction; and a new aspect of extension operators on solving $\overline\partial$.

### Dominique Maldague

Title: A sharp square function estimate for the moment curve in R^3

Abstract: I will present recent work which proves a sharp L^7 square function estimate for the moment curve (t , t^2, t^3) in R^3 using ideas from decoupling theory. In the context of restriction theory, in which we consider functions with specialized (curved) Fourier support, this is the only known sharp square function estimate with a non-even L^p exponent (p=7). The basic set-up is to consider a function f with Fourier support in a small neighborhood of the moment curve. Then partition the neighborhood into box-like subsets and form a square function in the Fourier projections of f onto these box-like regions. We will use a combination of recent tools including the "high-low" method and wave envelope estimates to bound f in L^7 by the square function of f in L^7.

### David Beltran

Title: On sharp isoperimetric inequalities on the hypercube

Abstract: The classical edge-isoperimetric inequality on the hypercube states that $|\nabla A| \geq |A| \log_2 (1/|A|)$ for any set $A \subseteq \{0,1\}^d$, where $\nabla A$ is the set of edges between A and its complement. This is sharp, since the inequality saturates on any subcube. Extensions and variants of this inequality have been studied by several authors, but so far none of them has the property of saturating on all sucubes. In this talk, we will present such an inequality, as well as improved versions of existing estimates. We will also discuss some applications. This is joint work with Paata Ivanisvili and José Madrid

### Lisa Naples

Title: Radon measures and Lipschitz graphs

Abstract: One way to understand the geometry of a measure is by exploring how that measure assigns weight to a collection of sets, all of which satisfy a particular geometric property. In this talk, we will broadly discuss the Identification Problem as a means of establishing a geometric relationship between a particular class of measures and a particular collection of sets. We will then present necessary and sufficient conditions under which a measure is rectifiable in the sense that there exist countably many m-Lipschitz graphs that carry the measure. In doing so, we provide a solution to the Identification Problem for Radon measures and Lipschitz graphs. This is joint work with Matthew Badger.

[1] Previous Analysis Seminars

[2] Fall 2022 Analysis Seminar