Martingale Inequalities On(Noncommutative) Symmetric Spaces

Martingale theory plays important roles in many fields of mathematics,especially in space theory.Martingale theory is inspired by the development of space theory and vice versa.On the frame of martingale theory,a series of martingale inequalities has been discovered,which widely applies in fourier analysis,stochastic process,geometric analysis on Banach spaces and functional analysis.As improved in martingale inequality,some space theories are further developed.To begin with,the background of symmetric space and the current situation of development at home and abroad are introduced.Then we emphasize on the basic theory and tools of classical symmetric spaces and noncommutative symmetric spaces,on which we study several classical martingale inequalities and non-commutative martingale inequality.The first goal of this thesis is to extend Lorentz-Shimogaki result,which characterizes interpolation spaces between noncommutative symmetric space.Let ε be a symmetrically Δ-normed ideal in B(H).For 0<p<q<∞,q≥1,we give a necessary and sufficient condition for ε to be an interpolation space between the couple of Schatten-von Neumann classes(Sp,Sq),which extends classical result due to Lorentz and Shimogaki and recent result due to Cadilhac,Sukochev and Zanin.To prove martingale inequalities associated with Orlicz function in symmetric spaces is our second main work.Let Φ be an Orlicz function and X be a symmetric space.According to the idea of Davis decomposition,we construct a distributive estimate of Burkholder inequalities.Hence,by Simonenko indices of Φ and Boyd indices of X,applied by new Boyd interpolation theorem,we obtain Burkholder inequalities on symmetric space with Φ moment.And hence,this extends some important results of Kikuchi.The last aim is to investigate martingale inequalities in noncommutative symmetric spaces.Let M be a semifinite von Neumann algebra and let {Rk}k≥1 be either a Rademacher sequence or a sequence of Fermions.For the series of the form ∑k=1∞ xk?Rk,we find their distributive estimate.By this,we are able to get Khintchine inequalities in more generalized space.And hence we obtain distributive estimates of Burkholder-Gundy inequalites.110 references...