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About this book

Covering a range of subjects from operator theory and classical harmonic analysis to Banach space theory, this book contains survey and expository articles by leading experts in their corresponding fields, and features fully-refereed, high-quality papers exploring new results and trends in spectral theory, mathematical physics, geometric function theory, and partial differential equations. Graduate students and researchers in analysis will find inspiration in the articles collected in this volume, which emphasize the remarkable connections between harmonic analysis and operator theory. Another shared research interest of the contributors of this volume lies in the area of applied harmonic analysis, where a new notion called chromatic derivatives has recently been introduced in communication engineering.
The material for this volume is based on the 13th New Mexico Analysis Seminar held at the University of New Mexico, April 3-4, 2014 and on several special sections of the Western Spring Sectional Meeting at the University of New Mexico, April 4-6, 2014. During the event, participants honored the memory of Cora Sadosky—a great mathematician who recently passed away and who made significant contributions to the field of harmonic analysis. Cora was an exceptional mathematician and human being. She was a world expert in harmonic analysis and operator theory, publishing over fifty-five research papers and authoring a major textbook in the field. Participants of the conference include new and senior researchers, recent doctorates as well as leading experts in the area.

Table of Contents

Frontmatter

Erratum to: Remembering Cora Sadosky

Without Abstract
Maria Cristina Pereyra, Stefania Marcantognini, Alexander M. Stokolos, Wilfredo Urbina

Cora

Frontmatter

Cora Sadosky: Her Mathematics, Mentorship, and Professional Contributions

Abstract
We present some snapshots of Cora Sadosky’s career focusing on her intertwined roles as mathematician, mentor, and leader in the profession. We recount some of her contributions to specific areas of mathematics as well as her broader impact on the mathematical profession.
Rodolfo H. Torres

Cora’s Scholarly Work: Publications According to MathSciNet

Abstract
Cora’s Bibliography According to MathSciNET
Maria Cristina Pereyra, Stefania Marcantognini, Alexander M. Stokolos, Wilfredo Urbina

Remembering Cora Sadosky

Without Abstract
Maria Cristina Pereyra, Stefania Marcantognini, Alexander M. Stokolos, Wilfredo Urbina

Harmonic and Complex Analysis, Banach and Metric Spaces, and Partial Differential Equations

Frontmatter

Higher-Order Elliptic Equations in Non-Smooth Domains: a Partial Survey

Abstract
Recent years have brought significant advances in the theory of higher-order elliptic equations in non-smooth domains. Sharp pointwise estimates on derivatives of polyharmonic functions in arbitrary domains were established, followed by the higher-order Wiener test. Certain boundary value problems for higher-order operators with variable non-smooth coefficients were addressed, both in divergence form and in composition form, the latter being adapted to the context of Lipschitz domains. These developments brought new estimates on the fundamental solutions and the Green function, allowing for the lack of smoothness of the boundary or of the coefficients of the equation. Building on our earlier account of history of the subject (published in Concrete operators, spectral theory, operators in harmonic analysis and approximation). Operator Theory: Advances and Applications, vol. 236, Birkhäuser/Springer, Basel, 2014, pp. 53–93), this survey presents the current state of the art, emphasizing the most recent results and emerging open problems.
Ariel Barton, Svitlana Mayboroda

Victor Shapiro and the Theory of Uniqueness for Multiple Trigonometric Series

Abstract
In 1870, Georg Cantor proved that if a trigonometric series converges to 0 everywhere, then all its coefficients must be 0. In the twentieth century this result was extended to higher dimensional trigonometric series when the mode of convergence is taken to be spherical convergence and also when it is taken to be unrestricted rectangular convergence. We will describe the path to each result. An important part of the first path was Victor Shapiro’s seminal 1957 paper, Uniqueness of multiple trigonometric series. This paper also was an unexpected part of the second path.
J. Marshall Ash

A Last Conversation with Cora

Abstract
I cannot contribute to this volume without speaking of Cora and what her friendship meant to me. But I know that, would she be here, she would ask: “Raconte-moi tes maths,” that is, “what are you doing right now?” Because, first of all, she was a mathematician. My mathematical contribution tends to answer her question.
Aline Bonami

Fourier Multipliers of the Homogeneous Sobolev Space Ẇ 1,1

Abstract
We prove that the restriction to an affine subspace of such a Fourier multiplier is still a Fourier multiplier, generalizing a celebrated theorem of de Leeuw for Fourier multipliers of L p . This may be seen as a complement to the spectacular result that such Fourier multipliers are continuous, which has been recently proved by Kazaniecki and Wojciechowski.
Aline Bonami

A Note on Nonhomogenous Weighted Div-Curl Lemmas

Abstract
We prove some nonhomogeneous versions of the div-curl lemma in the context of weighted spaces. Namely, assume the vector fields \(\mathbf{V},\mathbf{W}\!\!: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), along with their distributional divergence and curl, respectively, lie in L μ p and L ν q , \(\frac{1} {p} + \frac{1} {q} = 1\), where μ and ν are in certain Muckenhoupt weight classes. Then the resulting scalar product V ⋅ W is in the weighted local Hardy space \(h_{\omega }^{1}(\mathbb{R}^{n})\), for \(\omega =\mu ^{\frac{1} {p} }\nu ^{\frac{1} {q} }\) in \(A_{1+ \frac{1} {n} }\).
Der-Chen Chang, Galia Dafni, Hong Yue

A Remark on Bilinear Square Functions

Abstract
We provide some remarks concerning a bilinear square function formed by products of Littlewood–Paley operators over arbitrary intervals. For 1 < p 1, p 2 <  with 1∕p = 1∕p 1 + 1∕p 2, we show that this square function is bounded from \(L^{p_{1}}(\mathbf{R}) \times L^{p_{2}}(\mathbf{R})\) to L p (R) when p > 2∕3 and unbounded when p < 2∕3.
Loukas Grafakos

Unique Continuation for the Elasticity System and a Counterexample for Second-Order Elliptic Systems

Abstract
In this paper we study the global unique continuation property for the elasticity system and the general second-order elliptic system in two dimensions. For the isotropic and the anisotropic systems with measurable coefficients, under certain conditions on coefficients, we show that the global unique continuation property holds. On the other hand, for the anisotropic system, if the coefficients are Lipschitz, we can prove that the global unique continuation is satisfied for a more general class of media. In addition to the positive results, we also present counterexamples to unique continuation and strong unique continuation for general second elliptic systems.
Carlos Kenig, Jenn-Nan Wang

Hardy Spaces of Holomorphic Functions for Domains in ℂ n with Minimal Smoothness

Abstract
We prove various representations and density results for Hardy spaces of holomorphic functions for two classes of bounded domains in \(\mathbb{C}^{n}\), whose boundaries satisfy minimal regularity conditions (namely the classes C 2 and C 1, 1, respectively) together with naturally occurring notions of convexity.
Loredana Lanzani, Elias M. Stein

On the Preservation of Eccentricities of Monge–Ampère Sections

Abstract
A study on the preservation of eccentricities of Monge–Ampère sections under an integral Dini-type condition on the Monge–Ampère measure is presented. The approach is based solely on C 2, α -estimates for solutions to the Monge–Ampère equation. The main results are then related to the local quasi-conformal Jacobian problem and to a priori estimates for solutions to the linearized Monge–Ampère equation.
Diego Maldonado

BMO: Oscillations, Self-Improvement, Gagliardo Coordinate Spaces, and Reverse Hardy Inequalities

Abstract
A new approach to classical self improving results for BMO functions is presented. “Coordinate Gagliardo spaces” are introduced and a generalized version of the John-Nirenberg Lemma is proved. Applications are provided.
Mario Milman

Besov Spaces, Symbolic Calculus, and Boundedness of Bilinear Pseudodifferential Operators

Abstract
Mapping properties of bilinear pseudodifferential operators with symbols of limited smoothness in terms of Besov norms are proved in the context of Lebesgue spaces. Techniques used include the development of a symbolic calculus for certain classes of symbols considered.
Jodi Herbert, Virginia Naibo

Metric Characterizations of Some Classes of Banach Spaces

Abstract
The main purpose of the paper is to present some recent results on metric characterizations of superreflexivity and the Radon–Nikodým property.
Mikhail Ostrovskii

On the IVP for the k-Generalized Benjamin–Ono Equation

Abstract
We shall study special properties of solutions to the IVP associated to the k-generalized Benjamin–Ono equation. We shall compare them with those for the k-generalized Korteweg-de Vries equation and for the k-generalized dispersive Benjamin–Ono equation. Also we shall discuss some open questions appearing in this subject.
Gustavo Ponce
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