A T(b) theorem for the Poincare upper half-space and hyperbolic geometry

The dissertation consists of two parts: harmonic analysis and hyperbolic geometry.; Harmonic analysis. The theory of operators on function spaces is important in analysis, and the most important operators are bounded operators. Function spaces studied in this theory are often the {dollar}Lsp{lcub}p{rcub}{dollar}-spaces and their generalizations. One of the first examples of a non-convolution operator associated with a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the Hilbert transform is the first Calderon commutator. That commutator was shown to be bounded on {dollar}Lsp2{dollar}-spaces in 1965 by A. P. Calderon. The {dollar}Lsp2{dollar} boundedness of those operators whose kernels satisfy the size and smoothness alluded to implies the {dollar}Lsp2{dollar} boundedness for all p's in the interval (1, {dollar}infty{dollar}). Many operators in analysis, such as certain classes of pseudo-differential operators and the Cauchy integral on a curve, are associated with kernels having these properties. For these operators, one of the major questions is whether or not they are bounded on the {dollar}Lsp2{dollar}-spaces. In 1984, Guy David and Jean-Lin Journe gave the necessary and sufficient conditions for such an operator to be bounded on {dollar}Lsp2(IRsp{lcub}n{rcub}).{dollar} This theorem is called the T(1) theorem. Together with the circle of ideas associated with it and with its proof, this theorem is fundamental to the entire subject.; In 1985 the T(b) theorem, which generalizes the T(1) theorem for the n-dimensional Euclidean space {dollar}IRsp{lcub}n{rcub}{dollar}, was proved. Mathematicians want to prove a similar theorem for hyperbolic spaces (e.g., the Poincare upper half-space). One of the difficulties is the fact that the doubling property fails for hyperbolic spaces. In my dissertation a T(b) theorem for the Poincare upper half-space is proved.; Hyperbolic geometry. A metric which is similar to the Kobayashi metric on complex manifolds is introduced on real hyperbolic manifolds. In this dissertation the new metric is shown to be equal to the hyperbolic metric. This enables us to look at the hyperbolic metric in a new way. As a consequence the family of distance decreasing maps from the Poincare unit ball to a hyperbolic manifold is normal.