Harmonic Bergman spaces, Hardy-type spaces and harmonic analysis of a symmetric diffusion semigroup on R^n

This thesis is divided into two parts, which deal with quite diverse subjects. The first part is, in turn, divided into two chapters. The first focuses on the development of new function spaces in $R^n$, called generalized Bergman spaces, and on their application to the Hardy space $H^1(R^n)$. The second is devoted to the theory of Bergman spaces on noncompact Riemannian manifolds which possess the doubling property and to its relationships with spaces of Hardy type. The latter are tailored to produce endpoint estimates for interesting operators, mainly related to the Laplace-Beltrami operator. The second part is devoted to the study of some interesting properties of the operator $A f = -1/2 \Delta f- x \cdot \nabla f \forall f \in C_c^\infty (R^n)$, which is essentially self-adjoint with respect to the measure $d \gamma_{-1}(x) = \pi^{n/2} \e^{|x|^2} d \lambda (x) \forall x \in R^n$, where $\lambda$ denotes the Lebesgue measure, and of the semigroup that $A$ generates.