# Southeastern Analysis Meeting XXVIII

## Abstracts

### Austin Amaya

Virginia Tech

Title: Zero-pole interpolation, Beurling-Lax representations for shift-invariant subspaces, and transfer function realizations: half-plane/continuous time versions

Abstract: Given a full-range simply-invariant shift-invariant subspace $\mathcal{M}$ of the vector-valued $L^{2}$ space $L^{2}_{\mathcal{U}}(\mathbb{T})$ over the unit circle, the classical Beurling-Lax-Halmos Theorem obtains a unitary operator-valued function $W$ on $\mathbb{T}$ so that $\mathcal{M} = WH^{2}_{\mathcal{U}}$; in this case necessarily $\mathcal{M}^\perp = W\left( H^{2}_{\mathcal{U}} \right)^\perp$. The Beurling-Lax-Halmos Theorem of Ball-Helton (1984) obtains such a representation for the case of a pair of shift-invariant subspaces $(\mathcal{M},\mathcal{M}^{cross})$---with $\mathcal{M}$ forward full-range simply-invariant and $\mathcal{M}^{cross}$ backward full-range simply-invariant---forming a direct-sum decomposition of $L^{2}_{\mathcal{U}}(\mathbb{T})$ with a new almost everywhere invertible $W$ on $\mathbb{T}$. For the case where $(\mathcal{M},\mathcal{M}^{cross})$ is a finite-dimensional perturbation of the model pair $(H^{2}_{\mathcal{U}}(\mathbb{T}),H^{2}_{\mathcal{U}}(\mathbb{T})^\perp)$, Ball-Gohberg-Rodman (1990) obtained a transfer function realization formule for the representer $W$, parameterized from zero-pole data computed from $\mathcal{M}$ and $\mathcal{M}^{cross}$. Later work by Ball-Raney (2007) extended this analysis to the nonrational case where the zero-pole data is taken in an appropriate infinite-dimensional operator-theoretic sense. Our current work obtains the analogue of these results for the case of a pair of subspaces $(\mathcal{M},\mathcal{M}^{cross})$ of $L^{2}_{\mathcal{U}}(\mathbb{R})$ invariant under the forward and backard translation groups. These results rely on recent advances in the understanding of continuous-time infinite-dimensional input-state-output linear systems now codified in the book of Staffans (2005). In this talk, we present a more definitive version of our results presented at SEAM 27.

### Valentin Andreev

Lamar University

Title: Estimating the Error in the Koebe Construction

Abstract: In 1912, Paul Koebe proposed an iterative method, the Koebe construction, to construct a conformal mapping of a non-degenerate, finitely connected domain D onto a circular domain C. In 1959, Gaier provided a convergence proof of the construction which depends on prior knowledge of the circular domain. We demonstrate that it is possible to compute the convergence rate solely from information about D.

### Joseph Ball

Virginia Tech

Title: The spectral set question and extreme points for the normalized Herglotz class over planar domains

Abstract: Given a domain $\Omega$ in the complex plane, the spectral set question asks whether a Hilbert-space operator $T \in {\mathcal L}({\mathcal H})$ with spectrum contained in $\Omega$ with a contractive $A(\Omega)$-functional calculus ($A(\Omega)$ equal to continuous functions on the closure of $\Omega$ which are holomorphic on $\Omega$) has a $\partial \Omega$-normal dilation. An equivalent reformulation due to Arveson asks whether any contractive representation of $A(\Omega)$ is in fact completely contractive. The question is known to have a positive answer if $\Omega$ is the unit disk (by the Sz.-Nagy dilation theorem) or if $\Omega$ an an annulus (Agler), and is now known to have a negative answer for $\Omega$ equal to certain triply-connected planar domains (Dritschel-McCullough). We discuss implications of this negative result for the extreme point structure of the convex normalized Herglotz class over $\Omega$.

### Snehalathe Ballamoole

Mississippi State University

Title: Spectral properties of Cesàro-like operators on weighted Bergman spaces

Abstract: We consider operators \begin{equation*} C_{\nu}f(z):= \frac{1}{z^\nu}\int_{0}^{z}\frac{f(\omega)\omega^{\nu-1}}{1-\omega}d\omega , \hspace{2cm} (f\in \mathcal{H}(\mathbb{D}), z\in \mathbb{D}). \end{equation*} We obtain spectral properties of $C_{\nu}$ on the Bergman spaces $L_{a}^{p,\alpha}$, $\alpha>-1$, $p\ge1$, by computing resolvent estimates. This work is closely related to recent work of Dahlner, Aleman and Persson. This is joint work with Len Miller and Vivien Miller.

### Kelly Bickel

Washington University in St. Louis

Title: Fundamental Agler Decompositions

Abstract: It is well-known that every holomorphic function bounded by one on the bidsk possesses an Agler decomposition. In general, such decompositions are difficult to write down explicitly. In this talk, we present a constructive, elementary proof of the existence of Agler decompositions using shift-invariant subspaces of the Hardy space on the bidisk. We then use these constructed decompositions to analyze properties about general Agler decompositions.

### Miriam Castillo Gil

University of Florida

Title: Functions of Positive Real Part on the Polydisk.

Abstract: We study some classes of holomorphic functions of positive real part on certain kinds of domains $\Omega \subset \mathbb C^n$, and characterize these classes through operator-valued Herglotz formulas and through von Neumann-type inequalities. Inspired on the work of J.E. McCarthy and M Putinar we extend results over the unit ball, due to M.T. Jury, to the polydisk and other domains by defining a family of Fantappi? pairings on $\Omega$ to establish duality relations between certain pairs of classes.

### Raphael Clouatre

Indiana University

Title: Similarity results for operators of class $C_0$

Abstract: By virtue of the classification theorem, it is known that any multiplicity-free operator of class $C_0$ is quasisimilar to a Jordan block. In case the minimal function of the operator is a Blaschke product with roots forming a Carleson sequence, we will discuss a condition under which the relation above can be strengthened to similarity. We will also explain how this gives a new interpretation of Carleson's classical interpolation theorem in the setting of $C_0$ operators.

### David Cruz-Uribe

Trinity College

Title: $A_p$ bump conditions for two-weight norm inequalities for classical operators

Abstract: $A_p$ bump conditions are generalizations of the Muckenhoupt A_p condition. There is a longstanding conjecture that these conditions are sufficient for Calderon-Zygmund singular integrals to map $L^p(v)$ into $L^p(u)$. In this talk we will review recent work on this conjecture, including its connections with two conjectures of Muckenhoupt and Wheeden that were recently disproved. This work is joint with Martell and Perez and also with Volberg and Reznikov.

### Raul Curto

University of Iowa

Title:  Hyponormality and subnormality of block Toeplitz operators

Abstract: I will discuss hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space $H^2_{C^n}$ of the unit circle. In joint work with I.S. Hwang and W.Y. Lee, we first establish a tractable and explicit criterion to determine the hyponormality of block Toeplitz operators having bounded type symbols; we do this via the triangularization theorem for compressions of the shift operator. Secondly, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if $\Phi$ is a matrix-valued rational function whose co-analytic part has a coprime decomposition then every hyponormal Toeplitz operator $T_{\Phi}$ whose square is also hyponormal must be either normal or analytic. Next, we apply our results to solve the following Toeplitz completion problem: Find the unspecified Toeplitz entries of the partial block Toeplitz matrix $A:=\begin{pmatrix} U^* & ? \\ ? & U^* \end{pmatrix}$ so that A becomes subnormal, where $U$ is the unilateral shift on $H^2$.

### Francesco Di Plinio

Indiana University

Title: $L^p$ bounds for singular integrals along N directions in $R^2$

Abstract: Let $K$ be a Calderon-Zygmund convolution kernel on $R$. We are concerned with the $L^p$-boundedness of the maximal directional singular integral $$T_{V} f (x)= \sup_{v \in V} \Big| \int_R f(x+t v) K(t) \, dt \Big|$$ where $V$ is a finite set of $N$ directions. This is a discrete version of Stein's conjecture on the boundedness of the Hilbert transform along smooth vector fields in the plane. For this problem, we are able to prove sharp (in terms of $\log N$) $L^p$ and weak $L^2$ bounds for lacunary and Vargas sets of directions. The latter include the case of uniformly distributed directions and the finite truncations of the Cantor set. We make use of both classical harmonic analysis methods and product-BMO based time-frequency analysis techniques. In addition to presenting our results, we plan on discussing possible applications of these techniques to the solution of the full conjecture by Stein. This is joint work with Ciprian Demeter.

### Son Duong

University of California at San Diego

Title: Transversality in CR Geometry

Abstract: We consider the transversality of holomorphic mappings between CR submanifolds of complex spaces. In equidimension case, we show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Ebenfelt and Rothschild six years ago. In different dimensions, the situation is more delicate as examples show. We will show that under certain restrictions on the dimensions and the rank of Levi forms, the mappings whose set of degenerate rank is of codimension at least 2 is transversal to the target. In addition, we show that under more restrictive conditions on the manifolds, finite holomorphic mappings are transversal. This is a joint work with Peter Ebenfelt.

### Matthew Gamel

Nicholls State University

Title: High Order Derivatives of Blaschke Products in $H^p$ Spaces

Abstract: In this paper, we consider higher order derivatives of Blaschke products in $H^p$ spaces given conditions on the zeros. Specifically, we prove that if $B(z)$ is a Blaschke product with zeros $(a_n)$ and $m \in \mathbb{N}$, then:

• if $$\sum_{j=1}^\infty (1-|a_j|^2)^\alpha< \infty$$ for some $\alpha$ with $0 < \alpha < \tfrac{1}{m+1}$, then $B^{(m)} \in H^{\frac{1-\alpha}{m}}$.
• if $$\sum_{j=1}^\infty (1-|a_j|^2)^{\frac{1}{m+1}} \log \left(\frac{1}{1-|a_j|^2} \right) < \infty$$ then $B^{(m)} \in H^{\frac{1}{m+1}}$.
These results generalize some of the work done by D. Protas in 1973 for $m=1$. This is joint work with Manfred Stoll.

### Jarod Hart

University of Kansas

Title:  Bilinear Vector Valued Calderon-Zygmund, Square Functions and Littlewood-Paley estimates

Abstract: In this work, an extension of the bilinear Calderon-Zygmund theory to the vector valued setting is developed and used together with interpolation arguments to derive new bilinear square function and Litlewood-Paley estimates.

### Mike Jury

University of Florida

Title: "Noncommutative" Aleksandrov-Clark measures

Abstract: We consider deBranges-Rovnyak type subspaces of the Drury-Arveson space; these are reproducing kernel spaces $\mathcal{H}(b)$ in the unit ball with kernel of the form $(1-b(z)b(w)^*)(1-zw*)^{-1}$. When this kernel is positive, the Cayley transform $(1+b)(1-b)^{-1}$ is represented as a "noncommutative" Herglotz integral of a state on the Cuntz-Toeplitz operator system. We prove that many of the known connections between $\mathcal{H}(b)$ spaces and Aleksandrov-Clark measures have analogs in this setting (e.g. expressing $\mathcal{H}(b)$ functions as Cauchy transforms, Clark's theorem on rank-one perturbations of the backward shift, etc.)

### Ilya Krishtal

Northern Illinois University

Title: Gabor frames in amalgam spaces

Abstract: We discuss several results on convergence of multiwindow Gabor frames in Wiener amalgam spaces. The talk is based on the joint work with R. Balan, J. Christensen, K.Okoudjou, andd J.-L. Romero.

### Hyun Kwon

Seoul National University

Title: Similarity of Operators in the Bergman Space Setting

Abstract: We give a necessary and sufficient condition for an n-hypercontraction to be similar to the adjoint of the operator of multiplication by the independent variable in a weighted Bergman space. The description is a generalization of the one given in the Hardy space setting where the geometry of the eigenvector bundles of the operators involved are considered. This talk is based on joint work with Ronald G. Douglas and Sergei Treil.

### Michael Lacey

Georgia Tech

Title: On the two weight inequality for the Hilbert transform.

Abstract: The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two $L^2$ to weak-$L^2$ inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero. The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two $L^2$ to weak-$L^2$ inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions. Joint work with Eric Sawyer, Chun-Yun Shen, and Ignacio Uriate-Tuero.

### Constanze Liaw

Texas A&M

Title: Dilations and finite rank perturbations

Abstract: For a fixed natural number n, we consider a family of rank n perturbations of a completely non-unitary (cnu) contraction T. We allow the corresponding characteristic operator function of T to be non-inner. We relate the unitary dilation of T to its rank n unitary perturbations. Based on this construction, we prove that the spectra of the perturbed operators are purely singular if and only if the operator-valued characteristic function corresponding to the unperturbed operator is inner. In the case where n=1 the latter statement reduces to a well-known result in the theory of rank one perturbations. However, our method of proof via the theory of dilations extends to the case of arbitrary n. We also consider the analogous family of cnu contractions that arise as rank n perturbations of T.

### Issam Louhichi

KFUM

Title: Quasihomogeneous Toeplitz operators on the harmonic Bergman space

Abstract: This talk is about the product of Toeplitz operators on the harmonic Bergman space of the unit disk of the complex plane C. Mainly, we discuss when the product of two quasihomogeneous Toeplitz operators is also a Toeplitz operator, and when such operators commute.

### Neil Lyall

University of Georgia

Title:  Polynomial Patterns in Subsets of the Integers

Abstract: It is a striking and elegant fact (proved independently by Furstenberg and Sarkozy) that any subset of the integers of positive upper density necessarily contains two distinct elements whose difference is given by a perfect square. We will discuss recent quantitative extensions and generalizations of this result.

### Svitlana Mayboroda

University of Minnesota

Title: Singular integrals, perturbation problems, boundary regularity, and harmonic measure for elliptic PDEs in rough media

Abstract: Elliptic boundary value problems are well-understood in the case when the boundary, the data, and the coefficients exhibit smoothness. However, perfectly uniform smooth systems do not exist in nature, and every real object inadvertently possesses irregularities (a sharp edge of the boundary, an abrupt change of the medium, a defect of the construction). The analysis of general non-smooth elliptic PDEs gives rise to decisively new challenges: possible failure of maximal principle and positivity, breakdown of boundary regularity, lack of well-posedness in $L^2$, to mention just a few. Further progress builds on a powerful blend of harmonic analysis, potential theory and geometric measure theory techniques. In this talk we are going to discuss some highlights of the history, conjectures, paradoxes, and recent discoveries such as the higher-order Wiener criterion and maximum principle for higher order PDEs, solvability of rough elliptic boundary problems and perturbation in $L^p$, development of the new theory of Hardy spaces and analysis of singular integrals beyond the realm of the Calderon-Zygmund theory, as well as an intriguing phenomenon of localization of eigenfunctions.

### Andrew Morris

University of Missouri

Title: Finite Propagation Speed for First Order Systems and Huygens' Principle for Hyperbolic Equations

Abstract: We prove that strongly continuous groups generated by first order systems on Riemannian manifolds have finite propagation speed. Our procedure provides a new direct proof for self-adjoint systems, and allows an extension to operators on metric measure spaces. As an application, we present a new approach to the weak Huygens' principle for second order hyperbolic equations.

### Katharine Ott

University of Kentucky

Title: The mixed problem for the Lamé system of elastostatics

Abstract: In this talk I will discuss recent progress on the study of the mixed boundary value problem for the Lamé system of elastostatics in Lipschitz domains in two dimensions. This is joint work with Russell Brown.

### Jonathan Poelhuis

Indiana University

Title: Local Fractional Maximal Operators.

Abstract: Fractional maximal operators arise in the study of Riesz potentials and are discussed in the work of Hedberg (1972) and others. These operators have interesting properties that are distinct from those of the Hardy-Littlewood maximal operator, which they resemble. In another direction, many results have followed Stromberg's (1979) paper discussing local maximal operators. Local operators in this sense have the advantage that they are defined for measurable functions, not just integrable ones. We define local analogues for the fractional maximal operators and discuss several of their properties, including a John-Nirenberg type inequality.

### Alexei Poltoratski

Texas A&M

Title: The Gap and Type Problems.

Abstract: One of the basic problems of Harmonic Analysis is to determine if a given collection of functions spans a given Hilbert space. A classical theorem by Beurling and Malliavin solved such a problem in the case when the space is $L^2$ on an interval and the collection consists of complex exponentials. In my talk I will discuss two classical problems closely related to the Beurling-Malliavin theorem, the so-called Gap and Type Problems, that remained open until recently.

### Alex Rice

University of Georgia

Title: Sarkozy's Theorem for P-intersective Polynomials

Abstract: We will discuss improvements and generalizations of two theorems of Sarkozy, the qualitative versions of which state that any subset of the natural numbers of positive upper density necessarily contains two distinct elements which differ by a perfect square, as well as two elements which differ by one less than a prime number, confirming conjectures of Lovasz and Erdos, respectively. Specifically, we will present a new generalized hybrid of these two results, giving a strong quantitative bound on the size of the largest subset of {1,2,...,N} which contains no nonzero differences of the form $h(p)$ for any prime $p$, where $h$ lies in the largest possible class of polynomials.

### Sonmez Sahutoglu

University of Toledo

Title: Localization of compactness of Hankel operators on pseudoconvex domains in C^n

Abstract: We prove the following localization for compactness of Hankel operators on Bergman spaces. Assume that $D$ is a bounded pseudoconvex domain in $C^n$,p is a boundary point of $D$, and $B(p,r)$ is a ball centered at p with radius r so that $U=D\cap B(p,r)$ is a domain. We show that if the Hankel operator $H_f$ with symbol $f\in C^1(\overline{D})$ is compact on $A^2(D)$ then $H_f$ is compact on $A^2(U)$ where $A^2(D)$ and $A^2(U)$ denote the Bergman spaces on D and U, respectively.

### Prabath Silva

Indiana University Bloomington

Title: Bilinear Hilbert transform tensor product with paraproduct

Abstract: C. Muscalu, J. Pipher, T. Tao and C. Thiele showed that bi-parameter paraproduct maps $L^p \times L^q$ into $L^r$ when $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $p,q> 1$ and the much more singular operator double bilinear Hilbert transform does not satisfy any $L^p$ bounds and raised the question about $L^p$ bounds for the operator bilinear Hilbert transform tensor product with paraproduct, which is more singular than bi-parameter paraproduct and less singular that double bilinear Hilbert transform. We give a positive answer by showing the bilinear Hilbert transform tensor product with paraproduct maps $L^p \times L^q$ into $L^r$ when $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $p,q, r> 1.$

### Esma Yildiz Ozkan

Gazi University

Title: A bivariate generalization of Meyer-König and Zeller type operators and its Approximation properties

Abstract: A bivariate generalization of a general sequence of Meyer-König and Zeller operators based on q-integers is constructed. Approximation properties of these operators are obtained by using Volkov-type convergence theorem for bivariate functions.Furthermore, rates of convergence by means of modulus of continuity and the elements of Lipschitz class functionals are also established.An r-th order generalization of these operators is also defined and its approximation properties are observed.

### Sawano Yoshihiro

Kyoto University

Title: Hardy spaces with variable exponents and generalized Campanato spaces

Abstract: Hardy spaces play an important role not only in harmonic analysis but also in partial differential equations because singular integral operators are bounded on Hardy spaces. The Hardy space $H^1$, which substitute for $L^1$, and the Hardy spaces $H^p$ with $p \in (0,1)$, are different in that the latter contains non-regular distributions. Although it will turn out to be an equivalent expression of $L^p$, for $1< p <\infty$, we can define the Hardy space $H^p$. To have a unified understanding of these situations, we consider and define Hardy spaces with variable exponents on ${\mathbb R}^n$. We will connect harmonic analysis with function spaces with variable exponents. We then obtain the atomic decomposition and the molecular decomposition.

### Abdelrahman Yousef

University of Jordan

Title: Commutants of a Toeplitz operator with a certain harmonic symbol.

Abstract: In this talk, we show that if an operator $T$ in the norm closed subalgebra, generated by Toeplitz operators with bounded symbols of the form $\displaystyle f(re^{i\theta})=\sum_{k=-\infty}^{N} e^{ik\theta}f_k(r)$, commutes with $T_{z+\bar{z}}$, then $T$ must be a polynomial of $T_{z+\bar{z}}$.