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Operator algebras and Poisson semigroups on a space of homogeneous type. (Algèbres d’opérateurs et semi-groupes de Poisson sur un espace de nature homogène.) (French) Zbl 0598.58045
Publ. Math. Orsay 84/3, 130 p. (1984).
This thesis deals with singular integral operators (SIO) in a very general setting, namely in a space of homogeneous type (in the spirit of Coifman-Weiss). It encompasses the following themes. (a) Spaces of homogeneous type, corresponding positively indexed Lipschitz, Beppo Levi and Sobolev spaces and SIO therein. (b) Invariance of smoothness (in various senses) under a SIO. (c) A method to define and to study Beppo Levi and Sobolev spaces with an arbitrary real index via an analogue of the Poisson kernel. (d) The \(L^2\)-estimate of SIO as a corollary of estimates in “smooth” spaces and in their duals. (e) A new approach to (and a generalization of) the theorem of G. David and J.-L. Journé on the \(L^2\)-continuity of a SIO [C. R. Acad. Sci., Paris, Sér. I 296, 761–764 (1983; Zbl 0523.45009)]. (f) Conditions ensuring that the product of two Calderon-Zygmund SIO’s is again a Calderon-Zygmund SIO. (g) An illustration of the above theory in the context of nilpotent stratified Lie groups.

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
22E25 Nilpotent and solvable Lie groups
45P05 Integral operators