Abstracts

• Sergei Treil, Brown University.
  Perturbations, two weight estimates, and Clark model.
  • An introduction: stationary processes and weighted estimates for the Hilbert transform.
  • Rank one perturbations, singular integral operators and Clark measures.
  • Regularization of singular integral operators.
  
     The theory of stationary random processes was one of the initial motivations for studying weighted estimates of the Hilbert transform (Helson-Szego and Helson-Sarason theorems). Two weight estimates might look just like a generalization for the sake of generalization, but it turns out that they appear naturally in the perturbation theory.
     In this series of talks I explain the connections of two weight estimates with rank one perturbations of self-adjoint and unitary operators, and Clark model (the latter will be explained in details in the talk by C. Liaw).
     I will also discuss some results about regularizations of singular integral operators, as well as the connection of the Sarason's conjecture with the problem of similarity to a normal operator.
  
  
• Constanze Liaw, Baylor University.
  Rank one perturbations of unitary operators and the Clark model in the general situation.
     For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter \(\gamma\), \(|\gamma|=1\). Namely all such unitary perturbations are the operators \(U_\gamma:=U+(\gamma-1) ( . , b_1 )_{\mathcal{H}} b\), where \(b\in \mathcal{H}\), \(\|b\|=1\), \(b_1=U^{-1} b\), \(|\gamma|=1\). For \(|\gamma|<1\) the operators \(U_\gamma\) are contractions with one-dimensional defects.
     Restricting our attention on the non-trivial part of perturbation we assume that \(b\) is a cyclic vector for \(U\), i.e. that \(\mathcal{H}=\overline{span\{U^n b : n\in\mathbb{Z}\}}\). In this case the operator \(U_\gamma\), \(|\gamma|<1\) is a completely non-unitary contraction, and thus unitarily equivalent to its functional model \(\mathcal{M}_\gamma\), which is the compression of the multiplication by the independent variable \(z\) onto the model space \(\mathcal{K}_{\theta_\gamma}\), where \(\theta_\gamma\) here is the characteristic function of the contraction \(U_\gamma\).
     The Clark operator \(\Phi_\gamma\) is a unitary operator intertwining the operator \(U_\gamma\), \(|\gamma|<1\) (in the spectral representation of the operator \(U\)) and its model \(\mathcal{M}_\gamma\), \(\mathcal{M}_\gamma \Phi_\gamma = \Phi_\gamma U_\gamma\). In the case when the spectral measure of \(U\) is purely singular (equivalently, the characteristic function \(\theta_\gamma\) is inner) the operator \(\Phi_\gamma\) was described from a slightly different point of view by D. Clark. The case when \(\theta_\gamma\) is an extreme point of the unit ball in \(H^\infty\) was treated by D. Sarason using the sub-Hardy spaces \(\mathcal{H}(\theta)\) introduced by L. de Branges.
     We treat the general case and give a systematic presentation of the subject. We first find a formula for the adjoint operator \(\Phi^*_\gamma\) which is represented by a singular integral operator, generalizing in a sense the normalized Cauchy transform studied by A. Poltoratskii. We first present a "universal'' representation that works for any transcription of the functional model. We then give the formulas adapted for specific transcriptions of the model, such as Sz.-Nagy-Foias and the de Branges-Rovnyak transcriptions, and finally obtain the representation of \(\Phi_\gamma\).
  
  
• Carlos Perez Moreno, University of Seville and Ikerbasque.
  Optimality of exponents and Yano's condition in weighted estimates and endpoint estimates.
     In the first part of the lecture I will explain why the exponents of known weighted \(L^p\) bounds are optimal. A natural condition appears in the scenario related to the blow up of the unweighted \(L^p\) norm of the operator as \(p\) gets close to \(1\). This condition was considered by Yano in his classical extrapolation theorem. In the second part of the lecture I will be discussing the following estimate \[ \|T^*f\|_{L^{1,\infty}(w)} \leq \frac{c_T}{\varepsilon} \int_{R^n} |f(x)|\,M_{L(\log L)^{\varepsilon}} (w)(x)\,dx \qquad w\geq 0 \] where \(T^*\) is any maximal singular integral operator \(T^*\) and \(M_{L(\log L)^{\varepsilon}}\) is a logarithmic perturbation of the classical maximal function. As an application of last estimate we will derive a quantitative \(A_1-A_{\infty}\) estimate improving some previous work with A. Lerner and S. Ombrosi. The first part is a joint work with T. Luque and E. Rela and the second with T. Hytonen.
  
  
• Rodolfo Torres, University of Kansas.
  Cora Sadosky: Her mathematics and mentorship.
     In this talk we will present some snapshots of Cora Sadosky's career focusing on her intertwined roles as mathematician and mentor. We will present some of her contributions to specific areas of mathematics as well as her broader impact on the profession.
  
  
• Svitlana Mayboroda, University of Minnesota.
  Localization of eigenfunctions and associated free boundary problems.
     The phenomenon of wave localization permits acoustics, quantum physics, elasticity, energy engineering. It was used in construction of the noise abatement walls, LEDs, optical devices. Anderson localization of quantum states of electrons has become one of the prominent subjects in quantum physics, as well as harmonic analysis and probability. Yet, no methods predict specific spatial location of the localized waves.
     In this talk I will present recent results revealing a universal mechanism of spatial localization of the eigenfunctions of an elliptic operator and emerging operator theory/analysis/geometric measure theory approaches and techniques. We prove that for any operator on any domain there exists a "landscape" which splits the domain into disjoint subregions and indicates location, shapes, and frequencies of the localized eigenmodes. In particular, the landscape connects localization to a certain multi-phase free boundary problem, regularity of minimizers, and geometry of free boundaries.
     This is joint work with D. Arnold, G. David, M. Filoche, and D. Jerison.
  
  
• Jill Pipher, Brown University. - CANCELLED
  Multiparameter commutators and BMO spaces.
     Cora Sadosky's interest in Toeplitz operators led her to make some important con- tributions in the area of multiparameter theory of singular integrals, BMO spaces, and com- mutators. In this talk we'll focus on commutators, beginning with some of Sadosky's results in this area and then describing some more recent results on iterated commutators of Riesz transforms and even more general multiparameter CZOs.
  
  
• Alex Stokolos, Georgia Southern University.
  Complex and Harmonic Analysis in Non-linear Dynamics.
     We will discuss some applications of complex and harmonic analysis to the problem of stabilizing the chaotic solutions of a non-linear discrete autonomous dynamical system. This is a joint work with D.Dmitrishin, A.Khamitova, A.Korenovskiy and A.Solyanik.
  
  
• Sergei Treil, Brown University.
  Two weight estimates following Arocena-Cotlar-Sadosky.
     I will explain the theory of generalized Toeplitz kernels by Arocena-Cotlar-Sadosky and its applications to two weight estimates of the Hilbert transform.
  
  
• Gustavo Ponce, University of California-Santa Barbara.
  The IVP for the Benjamin-Ono equation in weighted Sobolev spaces.
     We shall discuss results concerning the well posedness and some optimal uniqueness properties of the solutions to the IVP associated to the Benjamin-Ono equation. These results include some persistence property of the solution flow in weighted Sobolev spaces and some uniqueness properties of these solutions under conditions involving two and three different times. These results have been established in joint works with G. Fonseca and F. Linares, and in a recent work of Cynthia Flores.