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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.

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