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Abstracts and Talk Materials
Workshop for Women in Analysis and PDE
May 30 - June 2, 2012

Keywords of the presentation: Convex bodies, Bodies of revolution, Sections

There are many open problems related to the reconstruction of an origin-symmetric convex body K in Rn from its lower dimensional information (areas of sections or projections, perimeters of sections or projections, length of cords, etc.)

In this talk we will survey known results on the determination of a convex body from its central sections, parallel sections, or maximal sections. Some of these problems have been long-time open, and a few very recent results will be presented.

Keywords of the presentation: partial differential equations

Consider the Dirichlet problem in a Lipschitz domain in the plane. Suppose that the boundary data is in BMO. I will show that, if the coefficients have small imaginary part and are independent of one of the coordinates, then solutions to the Dirichlet problem satisfy a Carleson-measure condition.

We look at the summation conditions of the Buckley type for the Reverse Holder and Muckenhoupt weights and deduce them from an intrinsic lemma which gives a summation representation of the bumped average of a weight. We also obtain summation conditions for continuous Reverse Holder and Muckenhoupt classes and both continuous and dyadic weak Reverse Holder classes. We also show that the constant in each summation inequality of Buckley's type is comparable to the corresponding Muckenhoupt and Reverse Holder constant. To prove our results we use Bellman function method.

It is a joint work with Sasha Reznikov.

Poster: Laminates Meet Burkholder Functions
December 31, 1969

We will explain how to compute the exact Lp operator norm of a "quadratic perturbation" of the real part of the Ahlfors--Beurling operator. For the lower bound estimate we use a new approach of constructing a sequence of laminates (probability measures for which Jensen's inequality holds, but for rank one concave functions) to give an almost extremal sequence to approximate the operator.

We discuss the boundedness of the maximal directional singular integrals in the plane along a finite set V of N directions. This maximal directional operator can be thought of as a discrete model of the Hilbert transform along a smooth vector field in the plane, whose boundedness has been conjectured by Stein. Logarithmic dependence on N of the L^p bounds (for p greater than 2) are established for a set of arbitrary structure.

Sharp bounds are proved 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.

The $k$-plane transform is a bounded operator from $L^p(mathbb{R}^d)$ to $L^q$ of the Grassmann manifold of all affine hyperplanes in $R^n$ for certain exponents depending on $k$ and $n$. When $k=1$ this is the X-ray transform, and when $k=n-1$ it is the Radon transform. In the endpoint case, $q=n+1$. We show that all extremizers have the form $c(1+|phi(x)|^2)^{-(k+1)/2}$ where $phi$ an affine transformation of $R^n$.

Poster - Regularity of solutions of degenerate quasilinear equations
December 31, 1969

In their paper of 1983 C. Fefferman and D. H. Phong have characterized subellipticity of linear second order PDEs. The characterization is given in terms of subunit metric balls associated to the differential operator. An extension of this result to the case of the operator with non-smooth coefficients has been given by E. Sawyer and R. Wheeden in 2006. We extend the result to the case of non-doubling metric spaces. Our main result is continuity of weak solutions of infinitely degenerate quasilinear equations. This together with a recent result by Rios, Sawyer and Wheeden completes the proof of hypoellipticity of a certain class of infinitely degenerate quasilinear operators.

Keywords of the presentation: Favard length, Cantor sets, geometric measure theory

The Favard length of a planar set E is the average length of its one-dimensional projections. In a joint project with Bond and Volberg, we prove new upper bounds on the decay of the Favard length of finite iterations of 1-dimensional planar Cantor sets with a rational product structure. This improves on the earlier work of Nazarov-Peres-Volberg, Bond-Volberg, and Laba-Zhai, and introduces new algebraic and number-theoretic methods to this area of research. The estimates are of interest in geometric measure theory, ergodic theory and analytic function theory.

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The classical Cauchy integral is a fundamental object of complex analysis whose analytic properties are intimately related to the geometric properties of its supporting curve.

In this talk I will begin by reviewing the most relevant features of the classical Cauchy integral. I will then move on to the (surprisingly more involved) construction of the Cauchy integral for a hypersurface in Euclidean complex space. I will conclude by presenting new results joint with E. M. Stein concerning the regularity properties of this integral and their relations with the geometry of the hypersurface. (Time permitting) I will discuss applications of these results to the SzegH o and Bergman projections (that is, the orthogonal projections of the Lebesgue space L^2 onto, respectively, the Hardy and Bergman spaces of holomorphic functions).

Keywords of the presentation: linear and bilinear pseudodifferential operators, Hormander classes

Pseudodifferential operators (PDOs) stand as the centerpiece of the Fourier (or time-frequency) method in the study of PDEs. They extend the class of translation-invariant operators since "multipliers" are replaced by "symbols". The quantitative behavior of these symbols, primarily illustrated by the well-known Hormander classes, allows for a complete picture (largely based on the work of Calderon, Fefferman, Hormander, Kohn, Nirenberg, Stein, Vaillancourt, and Wainger) of the mapping properties of PDOs in Lebesgue spaces. Moreover, this classical theory establishes connections between PDOs and the Calderon-Zygmund theory of singular integrals as well as a description of the interplay between the classes of symbols and the operations of transposition and composition of PDOs.

In this lecture we will review the classical linear theory of PDOs, introduce and motivate the bilinear theory and report on the current status of the known mapping properties of bilinear PDOs in products of Lebesgue spaces. We will draw analogies between the linear and bilinear theory as well as stress some differences between them. Applications and motivations for the bilinear theory include the study of paraproducts, commutators, and fractional Leibniz-type rules.

Keywords of the presentation: Laplacian, Lame system, boundary value problems

In this talk I will discuss recent results on the mixed boundary value problem in Lipschitz domains. Consider a bounded Lipschitz domain with boundary decomposed into two disjoint sets. On one portion of the boundary Neumann data is prescribed. On the remainder of the boundary Dirichlet data is prescribed. I will discuss the existence and uniqueness of solutions of the mixed problem for the Laplacian and the Lame system of elastostatics with boundary data taken from L^p, where p is greater than or equal to 1. I will highlight how reverse Holder estimates play a key role in obtaining estimates on the non-tangential maximal function of the gradient of solutions to the mixed problem.

In this minicourse we will introduce some basic ideas in harmonic analysis via simpler dyadic models and dyadic tools such us Haar shift operators and Bellman functions. We will show how they can be used to describe important continuous objects such as the Hilbert transform or more generally singular integral operators. We will illustrate the power of this approach with the successful resolution of the $A_2$ conjecture by Hytonen.

The minicourse is aimed at graduate students with a basic background in analysis, including a first course in measure theory and functional analysis, but also to recent graduates with an interest in the area.

We will carefully describe the dyadic setting, introduce Haar functions, and basic dyadic operators such as: martingale transform, dyadic maximal function, dyadic square function, dyadic paraproducts and Haar shift operators. We will present Petermichl's representation theorem for the Hilbert transform in terms of Haar shift operators, and time permitting Hyt"onen's representation theorem for Calder'on-Zygmund singular integral operators.

In our journey, we will illustrate the use of some classical techniques (Cotla'rs Lemma, Calder'on-Zygmund decomposition, Covering Lemmas, interpolation) and some not that classical (extrapolation, stopping time techniques, Carleson Lemma, and Bellman functions) to study boundedness of these operators on $L^p$ spaces and on weighted Lebesgue spaces, $L^p(w)$ for weights in the $A_p$ class.

L. Carleson introduced the measures which bear his name to solve an interpolation problem for analytic functions (Ann. of Math.,1962), establishing their relationship with the existence of nontangential limits at the boundary. These measures were subsequently understood within the larger context of duality of tent spaces. Carleson measures have played a fundamental role in the theory of elliptic boundary value problems, especially in determining solvability of boundary value problems in the context of non-smooth real or complex coecient operators. The appearance of Carleson measures in this theory is quite natural: solvability in Lp of an elliptic Dirichlet problem is determined by the property of the weight" or elliptic measure, and weight classes are closely connected to the function space BMO and even have Carleson type characterizations. However, there is an extraordinary variety of ways in which the subtelty of the Carleson measure characterization emerges in elliptic theory. In these lectures, we describe some classical and some modern results in this subject which illustrate this theme.

Keywords of the presentation: Resolution of singularities, critical integrability index

The structure of the zero set of a multivariate polynomial is a topic of wide interest, in view of its ubiquity in problems of analysis, algebra, partial differential equations, probability and geometry. The study of such sets, known in algebraic geometry literature as resolution of singularities, originated in the pioneering work of Jung, Abhyankar and Hironaka and has seen substantial recent advances, albeit in an algebraic setting.

In this talk, I will discuss a few situations in analysis where the study of polynomial zero sets play a critical role, and discuss prior work in this analytical framework in two dimensions. Our main result (joint with Tristan Collins and Allan Greenleaf) is a formulation of an algorithm for resolving singularities of a multivariate real-analytic function with a view to applying it to a class of problems in harmonic analysis.

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The Klein-Gordon equation was originally introduced as a relativistic model of the Schrodinger equation, but in some ways, its solutions exhibit behavior that is more like that of solutions to the wave equation. In this talk we will discuss recent results establishing upper bounds on the growth rate of finite time blowup solutions of the focusing nonlinear Klein-Gordon equation. All of the results that we will mention have analogues for the nonlinear wave equation.

This is joint work with Rowan Killip and Monica Visan

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Recently, it has been shown that the unimodular Fourier multipliers $e^{it|Delta |^{frac{alpha }{2}}}$ are bounded on all modulation spaces. In this paper, using the almost orthogonality of projections and some techniques on oscillating integrals, we obtain asymptotic estimates for the unimodular Fourier multipliers $e^{it|Delta |^{frac{alpha }{2}}}$ on the modulation spaces. As applications, we give the grow-up rates of the solutions for the Cauchy problems for the free Schr% "{o}dinger equation, the wave equation and the Airy equation with the initial data in a modulation space. We also obtain a quantitative form about the solution to the Cauchy problem of the nonlinear dispersive equations.

We apply the discrete Littlewood-Paley-Stein analysis to establish the duality theorem of weighted multi-parameter Hardy spaces H^p_Z(w) associated with Zygmund dilations, i.e., (H^p_Z(w))^*= CMO^p_Z(w) for 0

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