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- W2073627447 abstract "A probabilistic proof is given to identify the complex interpolation space of H1 (T) and H? (T) as HP(T). INTRODUCTION In this note, a soft propabilistic proof of P.W. Jones's theorem on the complex interpolation space between H1 and HOc) is given. We shall work with N.Th. Varopoulos's space of holomorhpic martingales. The observation presented in this article is that the use of a stopping time decomposition simplifies constructions of Serguei V. Kislyakov and Quanhua Xu to obtain the following Theorem. The complex interpolation space [TH'(T), H(T)]0, o<0 e< 1, coincides with HP(T) provided that 1 = 1 03. P As is well known this result has been obtained by P.W. Jones using L? estimates to the a problem (see [J]). His work also contains the description of the real interpolation spaces for the couple (Hl, H??) . At about the same time Jean Bourgain obtained a Marcinkiewicz type decomposition, using completely different techniques (see [B1]). Recently in a series of papers Jean Bourgain [B2], Serguei Kislyakov [Kl], [K2], [K-X], Gilles Pisier [P] and Quanhua Xu [Xl], [X2] obtained deep results concerning real and complex interpolation methods between vector-valued, weighted Hardy spaces of analytic functions. S. Kislyakov's paper [Kl] contains the following idea to approximate the characteristic function 1{1f1<2} by analytic functions on the circle T: He starts a=max{l,itl} and then considers 1 a +iHa where H denotes the Hilbert transform on T. Received by the editors August 11, 1992 and, in revised form, February 1, 1994; originally communicated to the Proceedings of the AMS by Dale Alspach. 1991 Mathematics Subject Classification. Primary 60G46, 42B30." @default.
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- W2073627447 date "1995-05-01" @default.
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- W2073627447 title "Holomorphic martingales and interpolation between Hardy spaces: the complex method" @default.
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