Browsing by Author "Vuorinen, Juho Emil"

The thesis begins by giving a proof for the Riesz-Thorin interpolation theorem concerning bounded linear operators on complex valued Lp-spaces. This proof does not use duality of L^p-spaces nor maximum principles of holomorphic functions but instead the central idea relies on Jensen's inequality and maximum properties of subharmonic functions. One virtue of this proof as compared to the classical proof is that it works for exponents 0 < p < 1 also. The needed properties of subharmonic functions and the Jensen's inequality are proved in the first section. After the proof of the Riesz-Thorin a more general interpolation method called complex interpolation method is introduced. The complex method is a generalization of the original proof of the Riesz-Thorin and in the same spirit Banach valued holomorphic functions and their maximum principles play a central role in its construction. The complex method is a process of producing new spaces from a pair of compatible Banach spaces and interpolation of bounded linear operators follows from the construction of the complex method. As an example the complex method is applied after its construction to Banach valued L^p-spaces. The original Riesz-Thorin has an extension that deals with analytic families of operators and also the abstract complex method may be used in a similar situation. Two theorems are proved as an illustration of this kind of theorems. One concrete interpolation problem about analytic families of operators on complex L^p-spaces is discussed and the interpolation technique introduced in the proof of the Riesz-Thorin is applied again. The aim of the final section is to provide examples of situations where the abstract complex method may be applied. The section begins by introducing the Bergman spaces on the unit disc. Hilbert space structure of the Bergman space L^2_a(D, udA_α) is used to derive the Bergman kernel for the unit disc. This leads to investigate integral operators whose kernels resemble the Bergman kernel. The mapping properties of these new operators allow one to study the behavior of Bergman type projections on L^p-spaces over the unit disc with different standard weights. The Bloch and the Besov spaces come up as ranges of these Bergman type projections. Finally the complex interpolation method is applied to the analytic function spaces introduced.