# Posts

## 2022-2023

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.

If you are interested to topics of mathematical analysis related to operator theory, take a look to the Analysis and Operator Theory Seminar, which is hosted by Jonathan or Jan.

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 TBD

### 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.

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.

TBD