Analysis Seminar Fall 2023

9/11

Adam Chistopherson

Ohio State University

Title: Weak-type regularity of the Bergman projection on generalized Hartogs triangles

Abstract: In this talk, we give a complete characterization of the weak-type regularity of the Bergman projection on rational Hartogs triangles in \(\mathbb C^2\) and and power-generalized Hartogs triangles in \(\mathbb C^3\). In particular, we show that the Bergman projection satisfies a weak-type estimate only at the upper endpoint of \(L^p\) boundedness on both of these domains. A similar result has been observed by Huo-Wick and Koenig-Wang for the classical Hartogs triangle and punctured unit ball, respectively. This work is joint with K.D. Koenig.

Brandon Sweeting

University of Alabama

Title: Multiplier Weak-Type Inequalities for Maximal Operators and Singular Integrals

Abstract: We discuss a kind of weak type inequality for the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior. In this talk, I will discuss quantitative estimates obtained for \(A_p\) weights, \(p > 1\), that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Ríos for \(p = 1\). I will also discuss an endpoint result for the Riesz potentials.

9/25

William Blair

University of Arkansas

Title: An atomic representation for Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations

Abstract: We develop a representation of the second kind for certain Hardy classes of solutions to nonhomogeneous Cauchy-Riemann equations and use it to show that boundary values in the sense of distributions of these functions can be represented as the sum of an atomic decomposition and an error term. We use the representation to show continuity of the Hilbert transform on this class of distributions and use it to show that solutions to a Schwarz-type boundary value problem can be constructed in the associated Hardy classes.

Kenta Kojin

Nagoya University, Japan

Title: Some relations between Schwarz-Pick type inequality and von Neumann's inequality

Abstract: We define a new pseudo-distance on a polynomial polyhedron and study a Schwarz-pick type inequality for the Schur-Agler class. In our operator theoretical approach, von Neumann’s inequality for a class of diagonalizable \(2 \times 2\) matrices plays an important role rather than holomorphy. Therefore, we introduce a new class that contains the Schur-Agler class and prove a Schwarz-Pick type inequality. Moreover, we use our pseudo-distance to consider dilation theory for \(2 \times 2\) matrices.

Fall Break

No Seminar

Zane Li

North Carolina State University

Title: Interpreting a classical argument for Vinogradov's Mean Value Theorem into decoupling language

Abstract: There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT ''asymptotically'' and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

Greg Knese

Title: Local square integrability of rational functions in two variables

Abstract: Given a polynomial \(p(x,y)\) with no zeros in the bidisk \(\{ |x|,|y| <1\}\) we are interested in boundary singularities of rational functions \(Q(x,y)/p(x,y)\). There are many ways to study the nature of a boundary singularity but in this talk we will discuss square-integrability on the two torus \(\{ |x|=|y|=1\} \) of the rational function. This problem converts to a local square integrability question on \(\mathbb R^2\) near \( (0,0)\) after applying conformal maps. After doing this we are able to give a complete characterization of the locally square integrable rational functions in two variables (with denominator non-vanishing on the product upper half plane). Time permitting we will discuss an application to sums of squares decompositions for stable polynomials.

Neeraja Kulkarni

California Institute of Technology

Title: An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes

Abstract: A Kakeya set is defined as a compact subset of \(\mathbb{R}^n\) which contains a line segment of length \(1\) in every direction. The Kakeya conjecture says that every Kakeya set has Minkowski and Hausdorff dimensions equal to \(n\). Interest in this conjecture began around 1971, when Fefferman used Kakeya sets to construct a counterexample to the ball multiplier conjecture in Fourier analysis. Fefferman's work shows that if the Kakeya conjecture is false, other large conjectures in Fourier analysis, the Fourier restriction conjecture and the Bochner-Riesz conjecture, would be false as well.
In this talk, I will discuss an improved Minkowski dimension estimate for Kakeya sets in dimensions \(n\geq 5\). The improved estimate comes from using a geometric argument called a ''\(k\)-planebrush'', which is a higher dimensional analogue of Wolff's ''hairbrush'' argument from 1995. The \(k\)-planebrush argument is used in conjunction with a previously known ''\(k\)-linear'' result on Kakeya sets proved by Hickman-Rogers-Zhang (and concurrently by Zahl) in 2019 along with an x-ray transform estimate which is a corollary of Hickman-Rogers-Zhang (and Zahl). The x-ray transform estimate is used to deduce that the Kakeya set has a structural property called ''stickiness,'' which was first introduced in a paper by Katz-Laba-Tao in 1999. Sticky Kakeya sets exhibit a self-similar structure which is exploited by the \(k\)-planebrush argument.

Walton Green

Title: Local and global Sobolev regularity of quasiconformal maps

Abstract: One important inspiration for weighted Lebesgue space bounds of singular integral operators is their application to quasiconformal and quasiregular maps through the Beurling-Ahlfors transform. The original results along these lines in the 0th order case were due to Astala-Iwaniec-Saksman and Petermichl-Volberg at the turn of the millennium. We extend these results using our recently developed weighted Sobolev theory of singular integral operators and their compressions to domains.

Guo Chuan Thiang

Peking University

Title: Quantum kilogram and the quantization of commutator traces

Abstract: In the 70s, Helton-Howe uncovered intricate structure in the traces of commutators. In the 80s, physicists discovered macroscopic quantization in the quantum Hall effect, leading to a redefinition of the kilogram in 2019. I will explain how these are directly related, and offer some perspectives on the coarse geometry approach to index theory on noncompact manifolds

Calvin Reedy

Title: Finite element methods and superconvergence

Abstract: We consider various finite element methods for numerically solving PDE's, using several formulations of the Poisson problem as examples. All fall under the framework of Galerkin methods, which provide approximate solutions by solving discrete versions of the problem. Conforming methods may have limitations on account of the continuity requirements of the spaces involved. Discontinuous Galerkin (DG) methods yield additional flexibility but result in greater computational complexity. Hybridizable discontinuous Galerkin (HDG) methods address this issue using additional unknowns, which create a nicer structure for the discrete problems, allowing for an increase in efficiency via static condensation. For HDG methods for the Poisson problem, it has been shown that a property of the discrete spaces called an ''M-decomposition'' results in desirable properties for the methods, including ''superconvergence''-the ability to define a new approximation for one of the unknowns which converges as fast as the difference between two approximations in the discrete space. Current research focuses on whether these results apply in the setting of finite element exterior calculus (FEEC), a framework which encompasses many known methods for certain problems as well as methods which have yet to be studied in detail.

Ljupcho Petrov

Title: Weighted Estimates for the Martingale Transform and One-Sided Calderón-Zygmund Operators

Abstract: We prove sharp weighted bounds for the martingale transform using the disbalanced Haar functions. Our proof of the Bilinear Embedding Theorem relies on the Carleson Lemma, which results in a single condition that needs to be checked before applying the Bilinear Theorem. We examine the limitations of this method when applied to the one-sided martingale transform and explore the proof of its boundedness using testing conditions. We finally discuss how the \(A_p^+\) weights determine the boundedness of one-sided Calderón-Zygmund operators.

Alan Chang

Title: Embedding snowflakes of the Heisenberg group into Euclidean space

Abstract: One consequence of Assouad's embedding theorem is that the snowflaked Heisenberg group has a bi-Lipschitz embedding into Euclidean space. (Those terms will be defined in the talk.) Terence Tao improved this result by constructing an embedding which is in some sense optimal. His proof uses the Nash-Moser iteration scheme, Littlewood-Paley theory on the Heisenberg group, and quantitative homotopy lifting arguments. (Those terms will not be defined in the talk.) We present an alternative proof of Tao's result which relies primarily on the lattice structure of the Heisenberg group as well as one of the oldest tricks in harmonic analysis. This is joint work with Seung-Yeon Ryoo.