Work on this project will concentrate on three problems in mathematical analysis. The first, arising from the Calderon- Zygmund theory of operators, synthesizes the traditional operator theory with methods of real variables and harmonic analysis to give sharp criteria for the power-boundedness of pseudo-differential operators. Particular emphasis is to be placed on the so-called Kato conjecture which contends that the domain of the square root of a general divergence-type operator is the fundamental Sobolev space H. This conjecture has been shown to be connected with on-going research on the problem of boundedness of the Cauchy-kernel along curves. Work will be done on the corresponding multilinear question where very little is known at this time except that the one-dimensional proof of the conjecture will not carry over. Applications to boundedness of broad classes of differential operators will be developed. The second line of investigation is related to the idea that a function which is infinitely differentiable in vertical and horizontal directions is itself infinitely differentiable. In the study of Anosov diffeomorphisms or flows one wants to show similarly that certain functions defined on a manifold are smooth. This work will consider the restriction of functions on stable and unstable foliations in an effort to expand present results from infinitely differentiable to the real-analytic version. Results of this nature should have application to the study of the Livsic equation for systems with non-zero Lyapounov exponents. A third thrust will be concerned with the time evolution of a system in quantum theory, if the evolution is determined by a one-parameter group. This work deals with the problem of giving an infinitesimal characterization of the cocyles, for a given one-parameter automorphism group, that is, a structure theorem playing for cocyles, the role of Stone's theorem for one- parameter unitary groups. This investigation should provide insight into the relationships between methods of classical and quantum probabilities.