University of Toronto
                  PDE/Applied Math/Analysis Seminar
                    Friday 12 January, 3:10-4:00pm
                          6183 Bahen Center      



SPEAKER:  Ciprian Demeter (UCLA)

TITLE:    Breaking the duality in the Return Times Theorem

ABSTRACT: Let ${\bf X_1}=(X,\Sigma_1,m_1, \tau)$ be a dynamical system. Let 
	  also $1\le p,q\le \infty$ be such that $\frac1p+\frac1q\le 1$. 
	  The Return Times Theorem proved by Bourgain asserts the following: 
	  For each function $f\in L^{p}(X)$ there is a universal set 
	  $X_0\subseteq X$ with $m_1(X_0)=1$, such that for each second 
	  dynamical system ${\bf Y}=(Y,\Sigma_2,m_2,\sigma)$, 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 $m_2$ 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 signed averages, where no positive results were previously 
	  known. This is joint work with Michael Lacey, Terence Tao and 
	  Christoph Thiele.



------------------------------------------------------------------
University of Toronto PDE/Applied Math/Analysis Seminar 
http://www.math.toronto.edu/appmath/

2006-2007 organizers:
          Pieter Blue           pblue@ math.toronto.edu
          Almut Burchard        almut@ math.toronto.edu
          Robert McCann		mccann@ math.toronto.edu
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