# Analysis Seminar Spring 2023

Department of Mathematics and Statistics
Washington University in St. Louis
Cupples I Room 199
Mondays from 3:00-4:00pm CT

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Date Speaker Title and Abstract

1/30

(Zoom)

Bruno Poggi

Universitat Autonòma de Barcelona

Title: Solvability of the Poisson problem with interior data in $$L^p$$ Carleson spaces and its applications to the regularity problem

Abstract: On Corkscrew domains with Ahlfors-regular boundary, we prove the equivalence of the classically considered $$L^p$$-solvability of the (homogeneous) Dirichlet problem with a new concept which we introduce, of the solvability of the inhomogeneous Poisson problem with interior data in an $$L^p$$-Carleson space (with a natural bound on the $$L^p$$ norm of the non-tangential maximal function of the solution), and we study several applications. Our main application is towards the $$L^q$$ Dirichlet-regularity problem for second-order elliptic operators satisfying the Dahlberg-Kenig-Pipher condition (this is, roughly speaking, a Carleson measure condition on the square of the gradient of the coefficients), in the geometric generality of bounded Corkscrew domains with uniformly rectifiable boundaries. This solves an open problem from 2001. Other applications include: several new characterizations of the $$L^p$$-solvability of the Dirichlet problem, new non-tangential maximal function estimates for the Green's function, a new local $$T1$$-type theorem for the $$L^p$$ solvability of the Dirichlet problem, new estimates for eigenfunctions, and a bridge to the theory of the landscape function (also known as torsion function). This is joint work with Mihalis Mourgoglou and Xavier Tolsa.

2/6 Walton Green

Title: Weighted estimates for the Bergman projection in planar domains

Abstract: The Bergman projection, $$B_E$$, is the orthogonal projection from $$L^2(E)$$ onto the closed subspace of holomorphic functions on $$E$$. When $$E$$ is smooth enough, $$B_E$$ is a singular integral operator and estimates on $$L^p(E)$$ and even $$L^p(E,w)$$ can be obtained using standard harmonic analysis techniques. When $$E$$ is a simply connected domain in the complex plane, we can connect weighted estimates for $$B_E$$ to properties of a conformal map from $$E$$ to the unit disc. To do so, we study the properties of various weight classes under composition with conformal maps. This work is in progress with Nathan Wagner.

2/13

Greg Knese

Title: Bounded rational functions in new settings

Abstract: The admissible numerator problem is the following: given a polynomial $$p$$ with no zeros on some domain $$D$$, describe all polynomials $$q$$ such that $$q/p$$ is bounded on $$D$$. Joint work with Bickel, Pascoe, Sola along with a paper by Kollar solved the admissible numerator problem for the bidisk by converting it to a local problem in the context of the bi-upper half plane. A key tool was understanding Puiseux expansions for stable polynomials. Recently we have extended some elements of this theory to the unit ball in $$\mathbb C^2$$, where the Puiseux series behave very differently, as well as to special cases in the tri-disk where Puiseux series are unavailable. In particular, the case of $$p$$ having a smooth point of its zero set intersecting the 3-torus becomes tractable.

2/20

Laurent Baratchart

National Institute for Research in Digital Science and Technology (INRIA)

Title: Pseudo-holomorphic conjugate functions and the Dirichlet problem

Abstract: on a simply connected rectifiable planar domain, we consider pseudo-holomorphic functions $$w$$ satisfying $$\partial w=\alpha \bar{w}$$, where $$\alpha$$ is a square summable complex valued function. We introduce Hardy-Smirnov classes of exponent $$1 < p < \infty$$ together with their boundary values, and discuss the M. Riesz problem: given a real valued $$u$$ in $$L^p$$ of the boundary, to find a real valued $$v$$ such that $$u+iv$$ is the trace of a solution. For a large class of rectifiable domains, we characterize solvability in terms of the $$A_p$$ condition of the derivative of a conformal map onto the disk. This in turn impinges on the range of exponents for which the Dirichlet problem with $$L^p$$-boundary values is solvable for elliptic equations with sufficiently smooth (say Lipschitz) coefficients. This is joint work with E. Russ and E. Pozzi.

2/27

Cristina Pereyra

The University of New Mexico

Title: Weighted Inequalities for $$t$$-Haar multipliers

Abstract: The $$t$$-Haar multipliers are dyadic operators analogue to pseudo-differential operators where the trigonometric functions have been replaced by the Haar functions and the symbol is now a function of the space variable and of the dyadic frequency (dyadic intervals) instead of the Fourier frequency. The symbols of the $$t$$-Haar multipliers depend on the real number $$t$$, a weight $$w$$, a choice of signs and the dyadic intervals. We will discuss necessary and sufficient conditions on the triple of weights $$(u,v,w)$$ (and the parameter $$t$$) for the uniform (with respect to the choice of signs ) boundedness of $$t$$-Haar multipliers from $$L^2(u)$$ into $$L^2(v)$$. Our result recovers all previous known weighted estimates for the martingale transform ($$t=0$$) including Wittwer's one-weight estimates as well as the celebrated two-weight estimates of Nazarov, Treil and Volberg. It also recovers and improves upon the known $$L^2$$-estimates for the unsigned $$t$$-Haar multipliers of Katz and Pereyra.

3/6

(Zoom)

Athanasios Kouroupis

Norwegian University of Science and Technology (NTNU)

Title: Beurling primes and Hardy spaces of Dirichlet series

Abstract: PDF

3/13 Spring Break No Seminar
3/20 Ana Colovic

Title: Hankel Operators, Commutators of the Hilbert transform and possible weighted generalizations

Abstract: It is known that commutator of the Hilbert transform and a multiplication operator $$M_b$$, $$[M_b,H]$$ is bounded if and only if $$b$$ is a function of bounded mean oscillation. Projections and restrictions of the commutator to the Hardy space and its orthogonal complement in $$L^2$$ give either Hankel operators or their adjoints. So, properties of Hankel operators are linked to properties of the commutator of the Hilbert transform. In particular, Hankel operators are bounded if and only if their antianalytic symbol, belong to the space of functions of bounded mean oscillation. We discuss known generalizations of the boundedness result of commutators of the Hilbert transform, and possible generalizations of the Hankel operator results.

3/27

Thomas Sinclair

Purdue University

Title: A notion of index for finite-dimensional operator systems

Abstract: Inspired by a well-known characterization of the Jones index of an inclusion of $$II_1$$ factors due to Pimsner and Popa, we define an index-type invariant for inclusions of finite-dimensional operator systems. We will compute examples of this invariant, and explain how it generalizes the Lovasz theta invariant to general matricial systems in a manner that is closely related to the quantum Lovasz theta invariant defined by Duan, Severini, and Winter.

4/3 Jesus Sanchez

Title: Heat Flow and the Geometry of Manifolds

Abstract: The heat equation is a famous equation first studied in detail by Fourier and is the poster child for any parabolic PDE. By considering the heat equation on a manifold, the study of heat flow has found numerous applications to the study of geometric structures on smooth manifolds. In this talk we will give a short introduction to this topic with a view towards the Atiyah-Singer index theorems and some recent developments in this area.

4/10

Sean Douglas

University of Missouri

4/17 Angel Roman
4/24 Shintaro

# Analysis Seminar Fall 2022

Date Speaker Title and Abstract

9/12

(Zoom)

Carmelo Puliatti

University of the Basque Country

Title: $$L^2$$-boundedness of gradients of single layer potentials for elliptic operators with coefficients of Dini mean oscillation-type

Abstract: We consider a uniformly elliptic operator $$L_A$$ in divergence form associated with a matrix $$A$$ with real, bounded, and possibly non-symmetric coefficients. If a proper $$L^1$$-mean oscillation of the coefficients of $$A$$ satisfies suitable Dini-type assumptions, we prove the following: if $$\mu$$ is a compactly supported Radon measure in $$\mathbb R^{n+1}$$, $$n \ge 2$$, the $$L^2(\mu)$$-operator norm of the gradient of the single layer potential $$T_\mu$$ associated with $$L_A$$ is comparable to the $$L^2$$-norm of the $$n$$-dimensional Riesz transform $$R_\mu$$, modulo an additive constant. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for the Riesz transform, which were recently extended to $$T_\mu$$ under a Hölder continuity assumption on the coefficients of the matrix $$A$$.
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.

9/19 Angel Roman

Title: Mackey Analogy in Unitary Representation Theory and in Reduced Group $$C^*$$-Algebras

Abstract: In the 1970's, George Mackey proposed an analogy between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group, known as the Cartan motion group. In this talk, I will introduce tempered unitary representations of a reductive Lie group to develop the Mackey analogy. Next, I will define reduced group $$C^*$$-algebras and show that the reduced group $$C^*$$-algebra of the Cartan motion group can be embedded into the reduced group $$C^*$$-algebra of the semisimple group. This can be used to characterize the Mackey analogy. This talk is based on joint work with Nigel Higson where we focused mostly on the complex group.

9/26 Walton Green

Title: Sobolev Regularity of the Truncated Beurling Transform

Abstract: I will introduce the Lebesgue and Sobolev Theory of quasiconformal and quasiregular maps in the complex plane. An analogous theory for subdomains motivates our investigation of Sobolev boundedness of truncated Calderón-Zygmund operators. We will introduce certain Carleson measures and give a complete weighted Sobolev theory in some special situations.

10/3

(Zoom)

Mihalis Mourgoglou

University of the Basque Country

Title: The Dirichlet problem with Sobolev boundary data for the Laplace equation in rough domains

Abstract: In this talk I will present some recent advances on Boundary Value Problems for the Laplace operator with rough boundary data in a bounded corkscrew domain in $$\mathbb{R}^{n+1}$$ whose boundary is uniformly $$n$$-rectifiable. In particular, I will discuss the equivalence between solvability of the Dirichlet problem for the Laplacian with boundary data in $$L^{p'}$$ and solvability of the regularity problem for the Laplacian with boundary data in an appropriate Sobolev space $$W^{1,p}$$, where $$p \in (1,2+\epsilon)$$ and $$1/p+1/p'=1$$. As chord-arc domains satisfy the aforementioned geometric assumptions, our result answers a question posed by Carlos Kenig in 1991. This is joint work with Xavier Tolsa

10/10 Fall Break No Seminar
10/17

Kaifeng Bu

Harvard University

Title: Wasserstein distance and its application in quantum circuit complexity

Abstract: Quantum circuit complexity—a measure of the minimum number of gates needed to implement a given unitary transformation—fundamental concept in quantum computation, with widespread applications ranging from determining the running time of quantum algorithms to understanding the physics of black holes. In this talk, I will introduce our recent results on quantum circuit complexity via quantum resource and Wasserstein distance. Moreover, I will also talk about the connection between quantum version of wassersetin distance and quantum relative entropy, the quantum version of Fourier entropy influence inequality, and their applications in quantum computation.

10/24

(Zoom)

Marcu-Antone Orsoni

University of Toronto Mississauga

Title: Dominating sets, spectral estimates and null-controllability

Abstract: Let $$(\Omega, \mu)$$ be a measure space and let $$\mathcal{F} \subset L^p(\Omega, \mu)$$ be a subspace of holomorphic functions. A measurable set $$E$$ is said to be dominating for $$\mathcal{F}$$ if there exists a constant $$C_E > 0$$ such that $$\int_\Omega |f|^p d\mu \le C_E \int_E |f|^p d\mu, \forall f \in \mathcal{F}.$$ In this talk, I will start giving an overview of the results concerning dominating sets for classical spaces of holomorphic functions. Then, I will explain how this question is related to certain spectral inequalities that play a central role in the null controllability of parabolic equations.

10/31 Jeremy Cummings

Title: History and Future Directions of Wavelet Representation

Abstract: In 2007, Petermichl proved a sharp $$A_p$$ bound for the Hilbert transform via a decomposition into dyadic shifts, which act on the basis of Haar functions. This sparked an interest in such representations of operators, leading to the Hytönen representation theorem for Calderón-Zygmund operators, again in terms of averages of dyadic shifts. In this talk I will outline the history of such representations, including both the Haar case and the smooth representation of Di Plinio, Wick, and Williams. I will further examine several avenues for generalizations of the latter theorem, particularly ways of extending the notion of wavelet representation to spaces of homogeneous type. In the course of this discussion we will review notions of wavelet bases in such spaces as well possible notions of Sobolev spaces.

11/7

Matthew Lorentz

Michigan State University

Title: The Hochschild Cohomology of Roe Type Algebras

Abstract: Many times in analysis we focus on the small scale structure of a metric space, e.g., continuity, derivations, etc. However, to examine the large scale structure of a metric space we turn to coarse geometry. To help us study the coarse geometry of a space we look at invariants, one such invariant is the uniform Roe algebra of the space. Indeed, if a metric space $$(X,d_X)$$ is coarsely equivalent to $$(Y, d_Y)$$ then their uniform Roe algebras are isomorphic. Originally looked at as a method compute higher index theory, uniform Roe algebras are a highly tractable $$C^*$$-algebra contained in the bounded operators on square summable sequences indexed by a metric space $$X$$ (note that purely topological definitions exist).

In this talk we will look at the Hochschild cohomology of uniform Roe algebras. Hochschild cohomology can be thought of as a noncommutative analog of multivector fields. We will first give the relevant definitions and look at a few examples. We will then explore the Hochschild cohomology of uniform Roe algebras with coefficients in various uniform Roe bimodules.