On The Hardy Spaces Of A Kind Of Quaternionic Regular Functions And Their Cauchy-Szeg(?) Kernel

The notation of 0-regular functions is one of quaternionic counterparts of the notation of holomorphic functions of several complex variables.In this paper,we study the Hardy space of 0-regular functions on the quaternionic Siegel upper half space,which is a noncommutative generalization of holomorphic Hardy space.In Chapter 1,we introduce the holomorphic Hardy space and its noncommutative gener-alization,and state main results of this thesis.In Chapter 2,we discuss the invariance of the Siegel upper half space under the action of a subgroup of Sp(9),1),which consists of dilations,a kind of rotations and Heisenberg translations.And we show the preservation of 0-regularity under the action of this subgroup.In Chapter 3,we show that the 0-regular Hardy space is a nontrivial infinite dimensional space,and that 0-regular Hardy space is also preserved under the action of this subgroup.We also establish the existence of boundary values of elements of 0-regular Hardy space,and proved that this Hardy space is isometric to a Hilbert subspace of the space of~2-integrable functions defined on the boundary.Therefore there exist an orthogonal projection to this subspace,called Cauchy-Szeg(?)projection.In Chapter 4,we characterize the kernel of this projection operator,i.e.Cauchy-Szeg(?)kernel,and give the explicit formula of this kernel.