Operator-valued Hardy And BMO Spaces

This thesis is concerned with the theory of operator-valued Hardy spaces and BMO spaces on the Euclidean space Rⁿ.The main results we obtained in this thesis can be summarized as follows:（i）.For the operator-valued Hardy spaces introduced by Tao Mei[50],we prove that these operator-valued Hardy spaces can be characterized by using several square functions involving wavelets,which corresponds to Meyer’s wavelet characteriza- tions of the classical Hardy space when p=1;（ii）.We introduce and develop the theory of operator-valued harmonic function spaces, and characterize it by the Poisson integral of functions in non-commutative BMO spaces;（iii）.Inspired by Bownik’s[2]and Tao Mei’s[50]works,we develop the theory of operator- valued Hardy spaces and BMO spaces in this anisotropic setting,in terms of atoms, dualities,and interpolations.This thesis is divided into six chapters:In the first chapter,we recall some former results on the research progress of operator-valued Hardy spaces and BMO spaces,and classical Hardy spaces and B-MO spaces associated with the anisotropic dilations in recent years,list the problems we want to study,and then outline briefly our main results.In the second chapter,we recall some basic definitions of non-commutative L_p-spaces,including the Hilbert-valued non-commutative L_p-spaces andl_∞-valued non-commutative L_p-spaces.In the third chapter,we establish several equivalent characterizations of operator-valued Hardy spaces in terms of wavelets.For such a problem,in the context of non-commutative analysis,the first result on wavelet characterizations of operator-valued Hardy spaces was provided by Guixiang Hong and Zhi Yin[31],but they only proved an equivalent characterization of operator-valued Hardy spaces in terms of wavelets in this setting.The main new difficulty lies in the fact that a lot of real variable tools such as the maximal functions and the stopping time arguments are not available in the non-commutative setting.In the fourth chapter,we introduce and develop the operator-valued harmonic function spaces,and obtain a representation of operator-valued harmonic functions.To be precise,we prove that Carleson condition can characterize all operator-valued harmonic function u（x,t）on R₊ⁿ⁺¹with boundary value in non-commutative BMO spaces.Chapter 5 is devoted to the study of the theory of operator-valued Hardy spaces and BMO spaces associated with anisotropic dilations.For the anisotropic Hardy spaces and BMO spaces introduced by Bownik[2],a natura and interesting question is:is it possible to extend to the non-commutative setting?We give a positive answer to such a problem.To be more precise,we introduce the operator-valued anisotropic Hardy spaces and BMO spaces,and establish the theory of dualities,atomic decompositions and interpolations.When it comes back to the isotropic setting,i.e.,A:=2I_n×n,all of these results coincide with those of Tao Mei[50],where I_n×ndenotes the n×n unit matrix.As an important application,we obtain the boundedness of anisotropic Calderón-Zygmund operators on BMO spaces.In the sixth chapter,we list briefly some problems to be studied.