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Yongsheng Han

2009-04-28
14:00:00 - 16:00:00

Some recent developments in Hardy $H^p$ theory

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

The Hardy $H^p$ theory originated during 1910's and 1920's in the setting of Fourier series and complex analysis in one variable, has long time history and has been transformed into a rich and multifaceted theory. The Hardy space $H^p$ provides a natural extension of $L^p$ in the sense that many important operators are bounded on $L^p, p>1,$ but not on $L^p, p\le 1.$ However, they are bounded on $H^p, p\le 1,$ or from $H^p to L^p, p\le 1.$ Moreoever, the Hardy spaces $H^p$ have been found many important applications in PDE. Recently, a lot of attentions are made to so-called multi-parameter Hardy space and $H^p$ defined on manifolds. In this talk, we will describe some developments of the Hardy space. To be precise, we will introduce the analytic Hardy space defined on the unit disk and then describe the breakthrough made by Stein-Weiss, which generalized the Hardy space to $R^n.$ In the seminal paper of Fefferman-Stein, several equivalent definitions of Hardy space were given and the $H^p$ boundedness of Calderon-Zygmund operatos was obtained. The evolution of Fefferman-Sten's results led to a constructive characterization of $H^p$ via the so-called atomic decomposition, obtained by Coifman. The advent of the atomic method enabled the extension of $H^p$ to a far more general setting, that of a space of homogeneous type intrduced by Coifman-Weiss. Space of homogeneous type includes $R^n,$ the Lipschitz domains in $R^n,$ particularly, any $C^\infty$ manifolds. We will also describe some developments of multiparameter Hardy space theory which are associated to several complex variables. As Stein pointed out that one expects a theory which is correcsponding to several complex variales as the classical Calderon-Zygmund theory which is correcsponding to complex of one variale.