# Difference between revisions of "Analysis Seminar"

The 2021-2022 Analysis Seminar will be organized by David Beltran and Andreas Seeger. Some of the talks will be in person (room Van Vleck B139) and some will be online. The regular time for the Seminar will be Tuesdays at 4:00 p.m. (in some cases we will schedule the seminar at different times, to accommodate speakers).

Zoom links will be sent to those who have signed up for the Analysis Seminar List. 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. If you are from an institution different than UW-Madison, please send as well as an additional email to David and Andreas (dbeltran, seeger at math (dot) wisc (dot) edu) to notify the request.

If you'd like to suggest speakers for the fall semester please contact David and Andreas.

# Analysis Seminar Schedule

 date speaker title host(s) institution September 21, VV B139 Dóminique Kemp UW-Madison Decoupling by way of approximation September 28, VV B139 Jack Burkart UW-Madison Transcendental Julia Sets with Fractional Packing Dimension October 5, Online Giuseppe Negro University of Birmingham Stability of sharp Fourier restriction to spheres October 12, VV B139 Rajula Srivastava UW Madison Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups October 19, Online Itamar Oliveira Cornell University A new approach to the Fourier extension problem for the paraboloid October 26, VV B139 Changkeun Oh UW Madison Decoupling inequalities for quadratic forms and beyond October 29, Colloquium Alexandru Ionescu Princeton University Polynomial averages and pointwise ergodic theorems on nilpotent groups November 2, VV B139 Liding Yao UW Madison An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains November 9, VV B139 Lingxiao Zhang UW Madison Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition November 12, Colloquium Kasso Okoudjou Tufts University An exploration in analysis on fractals November 16, VV B139 Rahul Parhi UW Madison (EE) On BV Spaces, Splines, and Neural Networks November 30, VV B139 Alexei Poltoratski UW Madison Title December 7 TBA TBA Title December 14 Tao Mei Baylor University Title February 1 Person Institution Title February 8 Person Institution Title February 15 Person Institution Title February 22 Person Institution Title March 1 Person Institution Title March 8 Brian Street UW Madison Title March 15: No Seminar Person Institution Title March 23 Person Institution Title March 30 Person Institution Title April 5 Person Institution Title April 12 Person Institution Title April 19 Person Institution Title April 22, Colloquium Detlef Müller University of Kiel Title April 29 Person Institution Title May 3 Person Institution Title Date Person Institution Title

# Abstracts

### Dóminique Kemp

Decoupling by way of approximation

Since Bourgain and Demeter's seminal 2017 decoupling result for nondegenerate hypersurfaces, several attempts have been made to extend the theory to degenerate hypersurfaces $M$. In this talk, we will discuss using surfaces derived from the local Taylor expansions of $M$ in order to obtain "approximate" decoupling results. By themselves, these approximate decouplings do not avail much. However, upon considerate iteration, for a specifically chosen $M$, they culminate in a decoupling partition of $M$ into caps small enough either as originally desired or otherwise genuinely nondegenerate at the local scale. A key feature that will be discussed is the notion of approximating a non-convex hypersurface $M$ by convex hypersurfaces at various scales. In this manner, contrary to initial intuition, non-trivial $\ell^2$ decoupling results will be obtained for $M$.

### Jack Burkart

Transcendental Julia Sets with Fractional Packing Dimension

If f is an entire function, the Julia set of f is the set of all points such that f and its iterates locally do not form a normal family; nearby points have very different orbits under iteration by f. A topic of interest in complex dynamics is studying the fractal geometry of the Julia set.

In this talk, we will discuss my thesis result where I construct non-polynomial (transcendental) entire functions whose Julia set has packing dimension strictly between (1,2). We will introduce various notions of dimension and basic objects in complex dynamics, and discuss a history of dimension results in complex dynamics. We will discuss some key aspects of the proof, which include a use of Whitney decompositions of domains as a tool to calculate the packing dimension, and some open questions I am thinking about.

### Giuseppe Negro

Stability of sharp Fourier restriction to spheres

In dimension $d\in\{3, 4, 5, 6, 7\}$, we establish that the constant functions maximize the weighted $L^2(S^{d-1}) - L^4(R^d)$ Fourier extension estimate on the sphere, provided that the weight function is sufficiently regular and small, in a proper and effective sense which we will make precise. One of the main tools is an integration by parts identity, which generalizes the so-called "magic identity" of Foschi for the unweighted inequality with $d=3$, which is exactly the classical Stein-Tomas estimate.

Joint work with E.Carneiro and D.Oliveira e Silva.

### Rajula Srivastava

Lebesgue space estimates for Spherical Maximal Functions on Heisenberg groups

We discuss $L^p\to L^q$ estimates for local maximal operators associated with dilates of codimension two spheres in Heisenberg groups, sharp up to endpoints. The proof shall be reduced to estimates for standard oscillatory integrals of Carleson-Sj\"olin-H\"ormander type, relying on the maximal possible number of nonvanishing curvatures for a cone in the fibers of the associated canonical relation. We shall also discuss a new counterexample which shows the sharpness of one of the edges in the region of boundedness. Based on joint work with Joris Roos and Andreas Seeger.

### Itamar Oliveira

A new approach to the Fourier extension problem for the paraboloid

An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $L^{2+\frac{2}{d}}([0,1]^{d})$ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1})$ for every $\varepsilon>0$. It has been fully solved only for $d=1$ and there are many partial results in higher dimensions regarding the range of $(p,q)$ for which $L^{p}([0,1]^{d})$ is mapped to $L^{q}(\mathbb{R}^{d+1})$. One can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $g$ of the form $g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d})$. We will present this theorem as a proof of concept of a more general framework and set of techniques that can also address multilinear versions of this problem and get similar results. This is joint work with Camil Muscalu.

### Changkeun Oh

Decoupling inequalities for quadratic forms and beyond

In this talk, I will present some recent progress on decoupling inequalities for some translation- and dilation-invariant systems (TDI systems in short). In particular, I will emphasize decoupling inequalities for quadratic forms. If time permits, I will also discuss some interesting phenomenon related to Brascamp-Lieb inequalities that appears in the study of a cubic TDI system. Joint work with Shaoming Guo, Pavel Zorin-Kranich, and Ruixiang Zhang.

### Alexandru Ionescu

Polynomial averages and pointwise ergodic theorems on nilpotent groups

I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.

### Liding Yao

An In-depth Look of Rychkov's Universal Extension Operators for Lipschitz Domains

Given a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, Rychkov showed that there is a linear extension operator $\mathcal E$ for $\Omega$ which is bounded in Besov and Triebel-Lizorkin spaces. We introduce a class of operators that generalize $\mathcal E$ which are more versatile for applications. We also derive some quantitative blow-up estimates of the extended function and all its derivatives in $\overline{\Omega}^c$ up to boundary. This is a joint work with Ziming Shi.

### Lingxiao Zhang

Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition

We study operators of the form $Tf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt$ where $\gamma_t(x)$ is a real analytic function of $(t,x)$ mapping from a neighborhood of $(0,0)$ in $\mathbb{R}^N \times \mathbb{R}^n$ into $\mathbb{R}^n$ satisfying $\gamma_0(x)\equiv x$, $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, and $K(t)$ is a `multi-parameter singular kernel' with compact support in $\mathbb{R}^N$; for example when $K(t)$ is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the single-parameter case when $K(t)$ is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the $L^p$-boundedness of such operators. This paper shows that when $\gamma_t(x)$ is real analytic, the sufficient conditions of Street and Stein are also necessary for the $L^p$-boundedness of $T$, for all such kernels $K$.

### Kasse Okoudjou

An exploration in analysis on fractals

Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.

### Rahul Parhi

On BV Spaces, Splines, and Neural Networks

Many problems in science and engineering can be phrased as the problem of reconstructing a function from a finite number of possibly noisy measurements. The reconstruction problem is inherently ill-posed when the allowable functions belong to an infinite set. Classical techniques to solve this problem assume, a priori, that the underlying function has some kind of regularity, typically Sobolev, Besov, or BV regularity. The field of applied harmonic analysis is interested in studying efficient decompositions and representations for functions with certain regularity. Common representation systems are based on splines and wavelets. These are well understood mathematically and have been successfully applied in a variety of signal processing and statistical tasks. Neural networks are another type of representation system that is useful in practice, but poorly understood mathematically.

In this talk, I will discuss my research which aims to rectify this issue by understanding the regularity properties of neural networks in a similar vein to classical methods based on splines and wavelets. In particular, we will show that neural networks are optimal solutions to variational problems over BV-type function spaces defined via the Radon transform. These spaces are non-reflexive Banach spaces, generally distinct from classical spaces studied in analysis. However, in the univariate setting, neural networks reduce to splines and these function spaces reduce to classical univariate BV spaces. If time permits, I will also discuss approximation properties of these spaces, showing that they are, in some sense, "small" compared to classical multivariate spaces such as Sobolev or Besov spaces.

This is joint work with Robert Nowak.