Function Spaces Over The Siegel Upper Half-space

The theme of this thesis is the function theory on the Siegel upper half-space.We attempt to establish fundamental results and basic tools to the function spaces on the Siegel upper half-space,and lay a foundation for subsequent research on this area.Al-though a homogeneous Siegel domain of type two is biholomorphically equivalent to a bounded homogeneous domain,there are different aspects between their functional spaces and the associated operators.This motivates us to study the function spaces on the symmetric Siegel domain of type two and try to reveal more new phenomenons.We begin this work with the case of Siegel upper half-space,which is biholomorphically equivalent to the unit ball through the Cayley transform.In this paper we study the Bergman spaces,Bloch space,Besov spaces and BMO over the Siegel upper half-space.Also,we characterize the Toeplitz operators and Han-kel operators on the Bergman spaces.In the course we find some new phenomenons in contrast to the setting of the unit ball.The main results for Bergman spaces are:the reproducing formulae of functions in A^p related to its n-th partial derivative;the Bergman norm is equivalent to a "derivative nor";the cancellation property of A¹ functions;the dense subspaces of A^p;the atomic decomposition of A^p functions.In order to introduce the notion of the Besov spaces on the Siegel upper half-space and determine its dual space,we need to consider a more natural and reasonable defini-tion of the Bloch space rather than the one given by Bekolle in[12].The Bloch space denoted by B,consisting of all holomorphic functions satisfy sup|▽f（z）|<∞ and vanish at a point（0’,i）.The main results include:the repro-ducing formulae of Bloch functions（related to the n-th partial derivative）;the norm of Bloch functions in terms of its "derivative";the dual space of B is A¹;the little Bloch space B₀ is the predual space of A¹;the relationship of Bloch space and BMO with respect to the Bergman metric.We introduce a definition for Besov spaces over the Siegel upper half-space.The Besov spaces denoted by B_p consist of functions f in B such that the functions ρ^<k>L^kf all belong to L^p（dτ）,where<k>is any positive integer satisfying<k>>n/p.Our mains results are:the dense subspaces of the Besov spaces;the dual space of B_p is B_q,where 1/p+1/q=1;the Mobius invariance of functions in Besov spaces.The boundedness and compactness of positive Toeplitz operators on Bergman spaces over the Siegel upper half-space are concerned.Positive Borel measures be the sym-bols of Toeplitz operators are confined in a special set M₊,then the associated Toeplitz operators are densely defined.We obtain characterizations of boundedness and com-pactness of Toeplitz operators with symbols in M₊ in terms of Carleson measures and vanishing Carleson measures respectively.We characterize the boundedness,compactness and membership in Schatten classes of Hankel operators on A².In the last,we characterize the L^p-L^q boundedness of Bergman-type operators over the Siegel upper half-space.This extends a result of Cheng et al.（2017）[17]to higher dimensions.