Harmonic analysis & several complex variables seminar

Harmonic analysis & several complex variables seminar
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August 27, 2025
March 30, 2026
kunz.83
The regular time and place for the seminar is Tuesday, 4:10 – 5:10 pm in Math Tower (MW) 154. Everyone is welcome to attend. If you are interested in giving a talk, please contact
Dusty Grundmeier
and/or
Valentin Kunz
If you are interested to topics of mathematical analysis related to operator theory, take a look to the
Analysis and Operator Theory Seminar
date
speaker
institution
title
Sep 23
Samantha Sandberg-Clark
Ohio State University
Generalized Non-Autonomous Parabolic Implosion
Oct 7
Debraj Chakrabarti
Central Michigan University
Restricted-type estimates on the Bergman projection
Oct 21
Yunus Zeytuncu
University of Michigan, Dearborn
Spectral Theory of the Kohn Laplacian on Quotient Manifolds
Oct 28
Abdullah Al Helal
Oklahoma State University
The first gap interval for proper holomorphic maps between annuli
Nov 4
Adam Christopherson
Baylor University
Lower endpoint blowup of the Bergman projection on non-smooth domains in $\mathbb{C}^n$
Nov 18
Vanessa Matus de la Parra
Stony Brook University
Correspondences in complex dynamics
Dec 9
Alex Lee
Brigham Young University
Signatures of Group-Invariant Hyperquadric Maps
Feb 10
Yeonwook Jung
University of California Irvine
Interior of pinned distance trees over thin Cantor sets
Feb 17
Alper Balci
Ohio State University
Harmonic Bergman Kernel and Projection on Annular Regions
Feb 24
Valentin Kunz
Ohio State University
Contour integral representations for singular solutions to the Helmholtz Equation on the sphere
Mar 3
Dong-June Choi
Ohio State University
Sharp $L^p$ regularity of the Szegö projection on the Hartogs triangle
Mar 24
Željko Čučković
University of Toledo
Compactness of composition operators on the Bergman space of the bidisc
TBD
Joseph Paugh
Ohio State University
TBA
Continue reading
August 19, 2024
March 19, 2025
Liding Yao
The regular time and place for the seminar is Tuesday at 3:00 – 4:00 pm in Math Tower (MW) 154. If you are interested, please contact
Dusty Grundmeier
and Valentin Kunz.
If you are interested to topics of mathematical analysis related to operator theory, take a look to the
Analysis and Operator Theory Seminar
date
speaker
institution
title
Sep 10
Joe Rosenblatt
UIUC
(On 2pm, at MW 152, joint with
ergodic
seminar) Approximate identity using singular measures
Sep 17
Taeyong Ahn
Inha University (Korea)
Equidistribution of inverse images of analytic subsets for holomorphic endomorphisms of compact K\”ahler manifolds
Sep 24
Song-Ying Li
UC Irvine
Supnorm Estimates for Solution of Cauchy-Riemann Equations
Oct 1
Mayuresh Londhe
Indiana Bloomington
An effective version of Fekete’s theorem
Oct 8
Yifan Jing
OSU
Sidon sets and sum-product phenomenon
Oct 15
Valentin Kunz
OSU
Several Complex Variables and the Quarter-Plane Problem: An Overview
Nov 12
John D’Angelo
UIUC
Positivity Conditions in Complex Analysis
Nov 14
Armin Schikorra
Pittsburgh
(Joint with
AOTS
seminar) On Calderon-Zygmund theory for the p-Laplacian
Nov 19
Tanya Firsova
Kansas State
Critical loci for automorphisms of C^2
Dec 3
Samantha Sandberg
OSU
Arithmetic Progressions in Fractal Sets of Sufficient Thickness
Dec 10
Liding Yao
OSU
The Cauchy-Riemann problem on distributions
Jan 6
David Cruz-Uribe
UAlabama
(Joint with
AOTS
seminar) The fine properties of Muckenhoupt weights in the variable Lebesgue spaces
Feb 4
Gabriel Coloma Irizarry
OSU
Resurgence functions: the lost tomes
Apr 1
Nathan Wagner
Brown
Boundedness and compactness of Bergman projection commutators in two-weight setting
Joe Rosenblatt
Note: Joe’s talk is on 2-3 pm, and the location is MW 152.
Title: Approximate identity using singular measures
Abstract: We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures $\mµ_n$ on $\Bbb R$. Assume that this sequence of operators constitutes an $L^p$-norm approximate identity for some $1\le p<\infty$. We ask, under what additional conditions do we have almost everywhere convergence for all $f\in L^p(\Bbb R)$.
We focus on the particular case of a sequence of contractions $C_{t_n}\mu$ of a single Borel probability measure $\mu$, with $t_n\to0$, so that that the sequence of operators is an $L^p$-norm approximate identity. If $\mu$ is discrete, then no sequence of such contractions can give a.e. convergence for all of $L^p(\Bbb R)$. If $\mu$ is absolutely continuous with respect to Lebesgue measure, then there is a sequence $(t_n)$ such that a.e. convergence holds on all of $L^1(\Bbb R)$.
But when the measure µ is continuous and singular to Lebesgue measure, obtaining a.e. results for some sequence $(t_n)$, is more challenging. Such results can always be obtained on $L^2(\Bbb R)$ when $\mu$ is a Rajchman measure. For non-Rajchman measures obtaining a.e. results on $L^2(\Bbb R)$ is sometimes possible, but not easy. In fact, it may be the case that there is a continuous, singular probability measure $\mu$ for which there is no sequence $(t_n)$ tending to zero with $C_{t_n}\mu\ast f\to f$ a.e., even just for all $f\in L^\infty(\Bbb R)$.
Taeyong Ahn
Title: Equidistribution of inverse images of analytic subsets for holomorphic endomorphisms of compact K\”ahler manifolds
Abstract: In this talk, we discuss the limit of inverse images of analytic subsets under a given holomorphic endomorphism of a compact K\”ahler manifold. We expect that the limit sits inside the Julia set of the given map in a reasonable sense under reasonable conditions. To this end, we will briefly talk about currents and superpotentials as methods. We will investigate a sufficient condition for this convergence. If time permits, we will discuss the difference between the complex projective space and a general compact K\”ahler manifold in this topic and in particular, the regularization of a positive closed currents and semi-regular transforms of currents.
Song-Ying Li
Title: Supnorm estimates for solution of Cauchy-Riemann equations
Abstract: In this talk, I will present some recent developments on the estimate of the solutions of Cauchy-Riemann equation $\overline\partial u=f$ on bounded domains $\Omega$ in $\mathbf C^n$ with non-smooth boundary, in particular for the product case $\Omega=D^n$ where $D\subset\mathbf C$. This include my recent result on the supnorm estimate when $D$ is has $C^{1,\alpha}$ boundary.
I will also present the most recent joint work with Long and Luo. Assuming that $D\subset\mathbf C$ is a bounded Lipschitz domain, we construct a new solution operator $Tf$ on $(0,1)$-form $f$ being $\overline\partial$-closed and $T:L^p_{(0,1)}(D^n)\to L^p(D^n)$ for $1Mayuresh Londhe
Title: An effective version of Fekete’s theorem
Abstract: A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. In this talk, we discuss an effective version of Fekete’s theorem in terms of a height function. As an application, we give a lower bound on the growth of the leading coefficient of certain polynomial sequences, generalizing a result by Schur. Lastly, if time permits, we discuss an upper bound on minimal asymptotics of height over sequences of algebraic numbers. This talk is based on a joint work with Norm Levenberg.
Yifan Jing
Title:
Sidon sets and sum-product phenomenon
Abstract:
Given natural numbers s and k, we say that a finite set X of integers is an additive B_s[k] set if for any integer n, the number of solutions to the equation n = x_1+x_2+ … +x_s, with x_1, x_2, …, x_s lying in X, is at most k. Here we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative B_s[k] set analogously. These sets have been studied thoroughly from various different perspectives in combinatorial and additive number theory. In this talk, we will discuss sum-product phenomena for these sets. We show that there is either an exceptionally large additive B_s[k] set, or an exceptionally large multiplicative B_s[k] set. This is joint work with Akshat Mudgal.
Valentin Kunz
Title: Several Complex Variables and the Quarter-Plane Problem: An Overview
Abstract: The ‘quarter-plane problem’ refers to the boundary value problem which models the interaction of a monochromatic plane-wave, e.g., an acoustic pressure field, with the tip of an infinitely thin corner in three spatial dimensions, e.g., the tip of a turbofan blade. In this talk, we will outline how the theory of several complex variables can help us gain a better understanding of such physical phenomena. We are particularly interested in finding an asymptotic formula for the field observed at a great distance from the quarter-plane, which, in turn, requires knowledge of the singularity structure of some two-complex-variable spectral functions.
John D’Angelo
Title: Positivity Conditions in Complex Analysis
Abstract: This talk surveys some positivity conditions in Several Complex Variables and CR geometry by way of Hermitian symmetric functions. Most of the talk will be accessible to graduate students in all fields of mathematics. After discussing a decisive example in detail, we make some connections to Complex Geometry and to Kohn’s approach to subelliptic multipliers.
Armin Schikorra
Note: Armin’s talk is on Thursday 3-4pm, with usual location.
Title: On Calderon-Zygmund theory for the p-Laplacian
Abstract: I will discuss how to disprove a conjecture by Iwaniec from 1983 about Calderon-Zygmund estimates for L^r where r is close to p-1.
Tanya Firsova
Title:
Critical loci for automorphisms of C^2
Abstract:
For one-dimensional holomorphic maps, the dynamics of the map is largely determined by the orbits of the critical points. Automorphisms of C^2 are invertible and, as such, do not have critical points.
Critical loci, as defined by E. Bedford, J. Smillie, and J. Hubbard, are the sets where dynamically defined foliations or laminations exhibit tangencies. They often serve as a good analog of the critical points. We’ll introduce critical loci within various dynamically significant regions, explaining their interactions and relationship to the system’s dynamics. We’ll describe the known topological models of the critical locus in the escape region. In particular, the model for complex Hénon maps in an HOV region, the first one developed in a non-perturbative setting. This is a joint work with R. Radu and R. Tanase.
Samantha Sandberg
Title: Arithmetic Progressions in Fractal Sets of Sufficient Thickness
Abstract: We consider the conditions required on a set that guarantee it contains arithmetic progressions. Szemeredi proved the existence of arithmetic progressions in subsets of the natural numbers with positive upper density. In the fractal setting, it is known by Maga and Keleti that full Hausdorff dimension is not enough to guarantee the existence of a 3-term arithmetic progression in subsets of d-dimensional Euclidean space; however, it turns out that Fourier decay coupled with nearly full Hausdorff dimension is sufficient for the existence of arithmetic progressions, as shown by Laba and Pramanik. In this talk, we consider another notion of size: Newhouse thickness. It is known that thickness larger than 1 is enough in the real line to guarantee the existence of a 3-term arithmetic progression. In higher dimensions, Yavicoli showed that it takes thickness larger than 10^8, along with some additional assumptions, to guarantee a 3-point configuration. We give the first result in higher dimensions showing the existence of 3-term arithmetic progressions in sets of thickness larger than 2/(1-2r), where r is a constant dependent on the set.
Liding Yao
Title: The Cauchy-Riemann problem on distributions
Abstract: The Cauchy-Riemann problem, also known as the ∂¯¯¯-problem, is a central problem in several complex variables. It concerns the regularity estimates to the equation ∂¯¯¯u=f on forms in a certain bounded domain Ω⊂C^n. We will talk about some background of the ∂¯¯¯-regularity theory, and the obstructions on solving the ∂¯¯¯ equation when f is a generic distributions. No prerequisite on complex analysis is required in this talk.
David Cruz-Uribe
Note: David’s talk is on Monday.
Title: The fine properties of Muckenhoupt weights in the variable Lebesgue spaces
Abstract: The theory of Muckenhoupt A_p weights, 1≤p<∞, has been an important area of harmonic analysis for more than 50 years. As part of this, a rich structure theory for these weights has been developed, including the reverse Hölder inequality, left-openness, Jones factorization, and Rubio de Francia extrapolation.
In the past 15 years this theory has been extended to the variable Lebesgue spaces, with the introduction of the A_{p(⋅)} weights by Cruz-Uribe, Fiorenza, and Neugebauer. Given an exponent function p(⋅), we say that a weight w is in A_{p(⋅)} if [w]_{A_{p(⋅)}}=\sup_Q |Q|^{-1}\|w\|_{L^{p(⋅)}(Q)}\|w^{-1}\|_{L^{p'(⋅)}(Q)} is finite.
These weights give sufficient (and in some cases necessary) conditions for maximal operators, singular integrals, and other operators of classical harmonic analysis to be bounded on the weighted variable Lebesgue space L^{p(⋅)}(w).
However, unlike in the constant exponent case, very little is known about the structural properties of A_{p(⋅)} weights, beyond a theory of extrapolation developed by Cruz-Uribe and Wang. In this talk I will discuss recent progress in this area, proving a reverse Hölder inequality for weights in A_{p(⋅)}. As an application, I will use it to prove left and right-openness of these weight classes, a result which we can also prove for matrix weights. If time permits I will also discuss other partial results on the structure of A_{p(⋅)} weights.
This research is joint with my PhD student, Michael Penrod.
Gabriel Coloma Irizarry
Title: Resurgence functions: the lost tomes
Note: Gabriel’s talk is on 4:30-5:30.
Abstract: The first half of the 1980s marked the birth of resurgence analysis with Ecalle’s monumental triple tomes
Fonctions Resurgentes
(with 6 planned total). The slow build of resurgence overtime, initially localized to complex dynamics and holomorphic invariants, is now felt deeply in vast corners of mathematical-physics and the theories of numbers & representations. In this talk, principally from the perspective of quantum resurgence we look at zero-range Hamiltonian interactions, and proceed from there to shine a light on buried themes of harmonic synthesis & alien representations with new homotopy conceptions of Borel plane.
Nathan Wagner
Title:
Boundedness and compactness of Bergman projection commutators in two-weight setting
Note: Nathan’s talk is on 4:30-5:30.
Abstract:
The Bergman projection is a fundamental operator in complex analysis with connections to singular integral theory, and it is of interest to study the commutator operator of the Bergman projection with multiplication by a measurable function b. In particular, we study the boundedness and compactness of the Bergman projection commutators in two weighted settings via weighted BMO (bounded mean oscillation) and VMO (vanishing mean oscillation) spaces, respectively. The novelty of our work lies in the distinct treatment of the symbol b in the commutator, depending on whether it is analytic or not, which turns out to be quite different. In particular, we show that an additional weight condition due to Aleman, Pott, and Reguera is necessary to study the commutators when b is not analytic, while it can be relaxed when b is analytic. Complete characterizations of two weight boundedness and compactness are obtained in the analytic case, which parallel results of S. Bloom for the Hilbert transform. Our work initiates a study of the commutators acting on complex function spaces with different symbols. In this talk, we will discuss our main results, as well as the principal ideas of the proofs. This talk is based on joint work with Bingyang Hu and Ji Li.
August 29, 2023
August 19, 2024
Liding Yao
The regular time and place for the seminar is Tuesday at 3:30 – 4:30 pm in Math Tower 154. If you are interested, please contact
Liding Yao
date
speaker
institution
title
Sep 5
Alex McDonald
OSU
The Newhouse gap lemma and patterns in products of Cantor sets (part I)
Sep 12
Alex McDonald
OSU
The Newhouse gap lemma and patterns in products of Cantor sets (part II)
Sep 19
Adam Christopherson
OSU
Weak-type regularity of the Bergman projection on generalized Hartogs triangles in C^3
Oct 20
Ben Bruce (on Zoom)
UBC
Hausdorff dimension and patterns determined by curves
Oct 31
Gennady Uraltsev
University of Arkansas
Multilinear and uniform bounds in harmonic analysis
Nov 7
Terence Harris
UW-Madison
Horizontal
Besicovitch
sets of measure zero and some related problems
Nov 14
Eyvindur Ari Palsson
Virginia Tech
A restricted Falconer distance problem
Nov 28
Robert Fraser
Wichita State
Oscillatory integrals arising from algebraic number fields
Apr 1
Michael Roysdon
Case Western
Comparison Problems for Radon Transforms (Monday, at EA 160)
Apr 15
Yuan Zhang
Purdue Fort Wayne
Optimal $\bar\partial$ regularity on product domains and its application to the Hartogs triangle
July 18
Zhenghui Huo
Duke Kunshan University
Weighted estimates of the Bergman projection and some applications
Alex McDonald
Title: The Newhouse gap lemma and patterns in products of Cantor sets
Abstract: Many important problems throughout mathematics can be summarized as follows: how large must a set be in order to ensure that it exhibits some type of structure?  A classic example of importance in geometric measure theory and harmonic analysis is the Falconer distance problem, which asks how large the Hausdorff dimension of a compact set must be to ensure it determines a positive measure worth of distances.  More generally, one may ask for an abundance of more complicated point configurations.  There has been considerable recent progress on variants of these problems where Hausdorff dimension is replaced by a quantity called thickness, which provides an alternative way to quantify the “size” of a compact set.  In these talks, I will give an overview of thickness from the ground up, and discuss some applications to finding point configurations in Cantor sets.
Adam Christopherson
Title: Weak-type regularity of the Bergman projection on generalized Hartogs triangles in C^3
Abstract: In this talk, we give a complete characterization of the weak-type regularity of the Bergman projection on rational  power-generalized Hartogs triangles in 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.
Ben Bruce
Note: Ben will give a Zoom talk on Zoom on Oct 20 11:30- 12:25, during the class Math 8210 at Scott Lab E103. Zoom link is
here
Title: Hausdorff dimension and patterns determined by curves
Abstract: In this talk, I will discuss joint work with Malabika Pramanik on the problem of locating patterns in sets of high Hausdorff dimension. More specifically, suppose Γ is a smooth curve in Euclidean space that passes through the origin. Is it true that every set with sufficiently high Hausdorff dimension must contain two distinct points x,y such that x-y \in Γ? We showed that if Γ is suitably curved then the answer is yes, while for certain flat curves the answer is no. This generalizes work of Kuca, Orponen, and Sahlsten, who answered this question affirmatively when Γ is the standard parabola in R^2.
Gennady Uraltsev
Title: Multilinear and uniform bounds in harmonic analysis
Abstract: See the pdf
here
Terence Harris
Title: Horizontal Besicovitch sets of measure zero and some related problems
Abstract: We show that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are discussed for the SL_2 Kakeya maximal function.
Eyvindur Ari Palsson
Note: Eyvi’s talk will be at Baker Systems Engineering, BE394. Also he will give a pre-talk on Nov 13 11:30-12:25 at E103, during the class Math 8210.
Title: A restricted Falconer distance problem
Abstract: The Falconer distance problem, a continuous analogue of the celebrated Erdos distance problem asks: How large does the Hausdorff dimension of a Borel set, in the plane or higher dimensions, need to be to ensure that the Lebesgue measure of its distance set is positive? This question has seen much progress in recent years, yet the conjectured threshold remains open. After a quick introduction of this problem I will give an overview of a number of variants leading to a formulation of a restricted Falconer distance problem with a particular example being a diagonally restricted distance set.
Robert Fraser
Title: Oscillatory integrals arising from algebraic number fields
Abstract: We introduce a new class of real-variable oscillatory integrals arising as the trace of a polynomial in the algebra (R ⊗_Q K) where K is an algebraic number field. We prove bounds for such oscillatory integrals that generalize a classical result of Arhipov, Cubarikov, and Karacuba for real polynomials and a recent result of Wright for complex polynomials.
Michael Roysdon
Note: Michael’s talk is on Monday, and the location is EA 160.
Title: Comparison Problems for Radon Transforms
Abstract: At the start of the 20th century J. Radon answered the following question: can one reconstruct a function based on its integral on lines? While this question is simple at its core, the methods involved have had a lasting effects in various scientific fields, and have even brought about two entire fields of mathematics: geometric tomography and analytic tomography. Each of these fields concern questions about determining information about of an object given lower-dimensional information; for example, information about the volume of the object knowing the volumes of sections and projections of that object onto planes.
Inspired by the famed Busemann-Petty problem from convex geometry and geometric tomography (~1954), we address more general questions of this nature in the realm of analytic tomography. We ask the very simple question: given a pair of even, non-negative, continuous and integrable functions f and g, such that the Radon transform of f is pointwise smaller than the Radon transform of g, does it necessarily follow that the L^p-norm of f is smaller than the L^p norm of g when p>1? We address this question for two types of Radon transforms: the classical Radon transform and the spherical Radon transform. The solution to this question is quite subtle and requires techniques from Harmonic Analysis and Fourier Analysis. As it turns out, this question is intimately related to the slicing problem of Bourgain (a question from Asymptotic Geometric Analysis (Geometric Probability)), reverse estimates for the Radon transform due to Oberlin and Stein from the 1980s, and finally some very recent estimate on the Radon transform due to Bennett and Tao. If time permits, we will discuss a lower-dimensional analogue of this problem.
Based on a joint work with Alexander Koldobsky and Artem Zvavitch.
Yuan Zhang
Note: Yuan’s talk is on Monday.
Title: Optimal $\bar\partial$ regularity on product domains and its application to the Hartogs triangle
Abstract: The $\bar\partial$ problem is to study the existence and regularity of the Cauchy-Riemann equation $\bar\partial u = f$ on pseudoconvex domains. Since H\”ormander’s fundamental $L^2$ theory, there has been substantial investigation for domains exhibiting favorable geometry and/or regularity. In this talk, we shall first focus on the $\bar\partial$ problem on a specific type of Lipschitz domains — product domains, and discuss recent advancements regarding its optimal Sobolev and H\”older regularity. Then we explore its application to the optimal Sobolev regularity on the Hartogs triangle. Part of the talk is based on joint works with Yifei Pan.
Zhenghui Huo
Note: Zhenghui’s talk is on Thursday 2-3 pm at Math Tower 100A.
Title: Weighted estimates of the Bergman projection and some applications
Abstract: In harmonic analysis, the Muckenhoupt $A_p$ condition characterizes weighted spaces on which classical operators are bounded. An analogue $B_p$ condition for the Bergman projection on the unit ball was given by Bekolle and Bonami. As the development of the dyadic harmonic analysis techniques, people have made progress on weighted norm estimates of the Bergman projection for various settings. In this talk, I will discuss some of these results and outline the main ideas behind the proof. I will also mention the application of these results in analyzing the $L^p$ boundedness of the projection. This talk is based on joint work with Nathan Wagner and Brett Wick.
September 1, 2022
August 19, 2024
Liding Yao
The regular time and place for the seminar is Monday at 4:30 – 5:30 pm in Math Tower 154. If you are interested, please contact
Liding Yao
date
speaker
institution
title
Sep 12
Liding Yao
OSU
Sharp Hölder regularity for Nirenberg’s complex Frobenius theorem
Sep 29
Brian Street
UW-Madison
Colloquium talk
) Maximal Subellipticity
Oct 3
Adam Christopherson
OSU
Weak-type regularity of the Bergman projection on rational Hartogs triangles
Oct 10
Xianghong Gong
UW-Madison
On the regularity of d-bar solutions on domains in complex manifolds satisfying condition a_q
Nov 14
Ziming Shi
Rutgers
Solvability of d-bar equation in spaces of negative smoothness index and its applications to Newlander-Nirenberg theorem with boundary
Nov 21
Lingxiao Zhang
UW-Madison
Real Analytic Singular Radon Transforms With Product Kernels: necessity of the Stein-Street condition
Jan 23
Brett Wick
Washington – St. Louis
Wavelet Representation of Singular Integral Operators
Feb 27
Yunus Zeytuncu
Michigan-Dearborn
Spectral analysis of Kohn Laplacian on Spherical Manifolds
Liding Yao
Title: Sharp Hölder regularity for Nirenberg’s complex Frobenius theorem
Abstract: Nirenberg’s famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when it is equal to the span of some real and complex coordinate vector fields. In the talk I will discuss some differential complexes and how some of the notions make sense in the non-smooth setting. For a $C^{k,s}$ complex Frobenius structure, we show that there is a $C^{k,s}$ coordinate chart such that the structure is spanned by coordinate vector fields which are $C^{k,s-\varepsilon}$ for all $\varepsilon>0$. Here the $\varepsilon>0$ loss in the result is optimal.
Brian Street
Title: Maximal Subellipticity
Abstract: The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general linear and fully nonlinear elliptic PDEs. Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs leans heavily on the Fourier transform and Riemannian geometry. Starting with work of Hormander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced: now known as maximal subellipticity or maximal hypoellipticity. In the intervening years, many authors have adapted results from elliptic PDEs to various special cases of maximally subelliptic PDEs. Where elliptic operators are connected to Riemannian geometry, maximally subelliptic operators are connected to sub-Riemannian geometry. The Fourier transform is no longer a central tool but can be replaced with more modern tools from harmonic analysis. In this talk, we present the sharp regularity theory of general linear and fully nonlinear maximally subelliptic PDEs.
Adam Christopherson
Title: Weak-type regularity of the Bergman projection on rational Hartogs triangles
Abstract: In this talk, we give a complete characterization of the weak-type regularity of the Bergman projection on the rational power-generalized Hartogs triangle. In particular, we expand on a result of Huo-Wick for the classical Hartogs triangle by showing that the Bergman projection satisfies a weak-type estimate only at the upper endpoint of L^p boundedness. This work is joint with K.D. Koenig.
Xianghong Gong
Title: On the regularity of d-bar solutions on domains in complex manifolds satisfying condition a_q
Abstract: We will start with some recent regularity results for the d-bar equation on strictly pseudoconvex domains with C^2 boundary in the complex Euclidean space C^n. These results have been proved by using homotopy formulas and estimates hold for forms that are not necessarily d-bar closed.
We will then describe new regularity results for the d-bar equation on a domain in a complex manifold when the boundary of the domain has either a sufficient number of positive or negative Levi eigenvalues. We will prove the same regularity result for given forms of type (0,1). For forms of type (0,q) with q>1, the same regularity for the d-bar solutions holds when the boundary is sufficiently smooth.
Ziming Shi
Title: Solvability of d-bar equation in spaces of negative smoothness index and its applications to Newlander-Nirenberg theorem with boundary
Abstract: This talk has two parts. In the first part, I will show the solvability of the d-bar equation in space of negative smoothness index, using some new harmonic analysis techniques that we recently developed. In the second part, I will show a subsequent application in Newlander-Nirenberg theorem on domains with boundary, which improves an earlier result of Gan-Gong. The talk is partially based on joint work with Liding Yao.
Lingxiao Zhang
Title: Real Analytic Singular Radon Transforms With Product Kernels: necessity of the Stein-Street condition
Abstract: We discuss operators of the form $Tf(x) = \psi(x) \int f(\gamma_t(x)) K(t)dt$, where $\psi(x) \in C_c^\infty(\mathbb{R}^n)$, $\gamma_t(x)$ 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 $\gamma_0(x) \equiv x$, and $K(t)$ is a product kernel with small support in $\mathbb{R}^N$. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_t(x)$, in the special case when $K(t)$ is a Calder\’on-Zygmund kernel. Street and Stein generalized their work to (for instance) the product kernel case, and gave sufficient conditions for the $L^p$ boundedness of such operators for all such kernels $K$. In this talk, we will state that when $\gamma_t(x)$ is real analytic, the Stein-Street condition is also necessary, and will also use several simple examples and graphs to illustrate this necessary and sufficient condition and explain the main ideas of the proof methods.
Brett Wick
Title: Wavelet Representation of Singular Integral Operators
Abstract: In this talk, we’ll discuss a novel approach to the representation of singular integral operators of Calderón-Zygmund type in terms of continuous model operators. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón-Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed.   Our representation formulas lead naturally to a new family of T1 theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the A_2 theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. As an additional application, it is possible to provide a proof of the commutator theorems of Calderó-Zygmund operators with BMO functions.
Yunus Zeytuncu
Title: Spectral analysis of Kohn Laplacian on Spherical Manifolds
Abstract: In this talk, we discuss the spectral analysis of Kohn Laplacian on spheres and the quotients of spheres. In particular, we obtain an analog of Weyl’s law for the Kohn Laplacian on lens spaces. We also show that two 3-dimensional lens spaces with fundamental groups of equal prime order are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.