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Autour de l'analyse géométrique. 1) Comportement au bord des fonctions harmoniques 2) Rectifiabilité dans le groupe de Heisenberg

Abstract : In this thesis, we are interested in two topics of geometric analysis. The first one is concerned with the asymptotic behaviour of harmonic functions in connection with geometry on graphs and manifolds. We study criteria for convergence at boundary of harmonic functions such as non-tangential boundedness, finiteness of non-tangential energy or finiteness of the energy density. We deal with Gromov hyperbolic manifolds, Gromov hyperbolic graphs, Diestel-Leader graphs and with an abstract frame to obtain criteria at minimal Martin boundary points. The methods, coming from probability theory and metric geometry, use the relation between harmonic functions and martingales. The second topic concerns the rectifiability properties of 1-dimensional sets in the Heisenberg group in connection with the boundedness of singular integral operators. We extend to this sub-Riemannian setting parts of the theory of uniformly rectifiable sets due to David and Semmes. In particular, we obtain a geometric traveling salesman theorem which provides a condition for an Ahlfors regular set of the first Heisenberg group to be contained in an Ahlfors regular curve.
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Camille Petit. Autour de l'analyse géométrique. 1) Comportement au bord des fonctions harmoniques 2) Rectifiabilité dans le groupe de Heisenberg. Mathématiques générales [math.GM]. Université de Grenoble, 2012. Français. ⟨NNT : 2012GRENM041⟩. ⟨tel-00744491⟩

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