International Conference on
Harmonic Analysis and Ergodic Theory

December 2-4, 2005
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Contributed Talks

Speakers
W. AbuShammala Indiana University
P. Avramidou Ohio State University
M. Bekker University of Missouri, Rolla
L. DeCarli Florida International University
C. Demeter UCLA
S. Episkoposian State University of Yerevan, Armenia
A.E. Gatto DePaul University
M. Kulshrestha Anand Agricultural University, Anand, India
T. Mei Texas A&M University
Biao Ou University of Toledo
C. Rios Trinity College
P. Roy Cornell University
G. Sampson Auburn University
C. Silva Williams College
Yue Wu Rice University



W. AbuShammala - Indiana University
Title: The Hardy-Lorentz spaces H^{p,q}(R^n)

Abstract


P. Avramidou - Ohio State University
Title: Certain sequences of reals and convergence, recurrence

Abstract: For a one-parameter continuous flow $\{S^t\}_{t\in\mathbb{R}}$, and sequences $x_n=p(n)\lfloor q(n)\rfloor,\ n\in\mathbb{N}$, where $p,q$ are of the form $\sum a_i x^{\alpha_i}$ we study the convergence (in norm, pointwise) of the averages $\frac{1}{N} \sum_{n=1}^N f\circ S^{x_n}$. This is closely related to the question of recurrence along the sequences $\lfloor x_n \rfloor$ for discrete flows. This is a joint work with V. Bergelson (Ohio State University), I.J. Knutson (Agder University College, Norway)


M. Bekker - University of Missouri, Rolla
Title: Integral Representations of Positive Definite Generalized Toeplitz Kernels

Abstract


L. DeCarli - Florida International University
Title: Reverse Holder inequalities for polynomials on (0,1)

Abstract: We prove new weighted reverse Holder inequalities for the classical ultraspherical polynomials and, more in general, for polynomials on [0,1].


C. Demeter - UCLA
Title: Breaking the duality in the Return Times Theorem

Abstract: Bourgain proved the following Return Times theorem:
Let ${\bf X}=(X,\Sigma,\mu, \tau)$ be a dynamical system. Let also $1\le p,q\le \infty$ be such that $\frac1p+\frac1q\le 1$. For each function $f\in L^{p}(X)$ there is a universal set $X_0\subseteq X$ with $\mu(X_0)=1$, such that for each second dynamical system ${\bf Y}=(Y,\F,\nu,\sigma)$, each $g\in L^{q}(Y)$ and each $x\in X_0$, the averages $$\frac1{N}\sum_{n=0}^{N-1}f(\tau^nx)g(\sigma^ny)$$ converge $\nu$- almost everywhere.
We show how to break the duality in this theorem. More precisely, we prove that the result remains true if $p>1$ and $q\ge 2$. We emphasize the strong connections between this result and the Carleson-Hunt theorem on the convergence of the Fourier series. We also prove similar results for the analog of Bourgain's theorem for series, where no positive results were previously known. This is joint work with Michael Lacey, Terence Tao and Christoph Thiele.


S. Episkoposian - State University of Yerevan, Armenia
Title: Double universal series by Walsh-Paley system in weighted L_\mu^1([0,1]^2) spaces

Abstract


A.E. Gatto - DePaul University
Title: Lipschitz Spaces and Calderon-Zygmund Operators associated to non-doublin g measures

Abstract: In the setting if a metric measure space (X,d,\mu) with an n-dimensional Radon measure \mu, we give a necessary and sufficient condition for the boundedness of CZ operators associated to the measure \mu on Lipschitz space on the support of \mu. ALso, for the Eucleadean space R^d with an arbitrary Radon measure \mu, we give several characterization of Liptschitz spaces on the support of \mu, Lip(\alpha, \mu) in terms of mean oscillations involving \mu. This allows us to view the "regular" BMO space of X.Tolsa as a limiting case for \alpha\to 0 of the space Lip(\alpha, \mu). This is a joint work with Jose Garsia-Cuerva.


M. Kulshrestha - Anand Agricultural University, Anand, India
Title: Estimation of soil Temperatures By Harmonic Analysis

Abstract


T. Mei - Texas A&M University
Title: Matrix (Operator) valued Hardy Spaces

Abstract: We studied Hardy spaces for matrix valued functions(more general, for functions with values in the non-commutative L^p spaces with a semifinite von Neuman algebra). This is motivated by recent study on matrix valued harmonic anlaysis and non-commutative martingale inequalities. Besides several non-commutative analogues of classical results, we proved our matrix valued H^1 is a predual of a matrix valued BMO spaces raised from these two directions. Our matrix valued Hardy spaces are defined by non-commutative Littlewood-Paley G-functions.


Biao Ou - University of Toledo
Title: Radial Symmetry and Monotonicity of Positive Regular Solutions to an Integral Equation

Abstract: We use the method of moving planes to prove the radial symmetry and monotonicity of positive regular solutions to the integral equation $$ u(x) = \int_{R^n} \frac{1}{ |x-y|^{n-\alpha} } u(y)^{(n+\alpha)/(n-\alpha) } dy $$ where $0<\alpha< n$. It follows that the solutions are a constant multiple of functions of form $$(\frac{t}{t^2 + |x-x_{0}|^2})^{(n-\alpha)/2 }$$ where $ t>0, x_{0} \in R^n.$ This is a joint work with Wenxiong Chen and Congming Li.


C. Rios - Trinity College
Title: Harmonic measure for elliptic operators with singular drift

Abstract: Given two elliptic operators in nondivergence form with singular drift terms in a Lipschitz domain, it is shown that if the difference of their coefficients satisfy certain Carleson measure condition, then their respective harmonic measures belong the same A_\infty class. This is an analog result to known theory for divergence form operators. As an application of this and a new approximation argument we are able to extend known results for divergence form operators while obtaining totally new theorems for nondivergence form operators. The theorems are sharp in all cases.


P. Roy - Cornell University
Title: Hopf Decomposition of Nonsingular Z^d-Actions and Long Range Dependence of Stationary S\alphaS Random Fields

Abstract: Jan Rosinski (2000) obtained a decomposition of a stationary symmetric \alpha-stable random field into three independent parts. This work establishes the connection between this decomposition and Hopf decomposition of nonsingular Z^d-actions. With the help of this connection, we study a partial maxima sequence of the stable random field. Depending on the ergodic theoretical structures of the underlying group action, we observe different kinds of asymptotic behavior of this partial maxima sequence and argue that this may indicate a phase transition from short memory to long memory. This is a joint work with Gennady Samorodnitsky.


G. Sampson - Auburn University
Title: Mapping Properties for Oscillatory Integrals in d-Dimensions

Abstract


C. Silva - Williams College
Title: Mixing Rank-One Actions of Locally Compact Abelian Groups

Abstract


Yue Wu - Rice University
Title: Whirly automorphisms

Abstract: Borel lift from a group action defined on the measure algebra to a Borel action was recently studied by E. Glasner, B. Tsirelson, B. Weiss. In the same paper, they defined "whirly action' and showed that whirly action has no spatial factor. In my thesis, it is proved that almost all three interval exchange transformations(i.e.t.) with permutation (321) are whirly (with respect to weak topology). The whirly property on all the cylinder sets is also gained for del Junco-Rudolph's map.