Interpolation réelle des espaces de Sobolev sur les espaces métriques mesurés et applications aux inégalités fonctionnelles

Abstract : In this Thesis, we study the real interpolation of Sobolev spaces and its applications. It is composed of two parts. In the first part, we prove in the first chapter that the non homogenous (resp.homogeneous) Sobolev spaces W^1_p defined on a complete Riemannian manifold satisfying the doubling property and admitting a Poincare inequality, form a real interpolation scale on an interval of values of p. We extend this result to other geometric frame. In a second chapter, we compare different Sobolev spaces defined on the euclidean cone and see how they behave with repect to the interpolation. With this example, we show tha Poincaré inequality is not a necessary condition to interpolate sobolev spaces. In the last chapter of this part, we define the non homogeneous (resp. homogeneous) Sobolev spaces W^1_p,V associated to positif potential V on a Riemmanian manifold. We prove that if the manifold satisfies the doubling property and admits a Poincaré inequality and if V belongs to a Reverse Holder class, these Sobolev spaces form a real interpolation scale on an interval of values of p. We also extend this result to the case of Lie groups. In the second part, in the first chapter in collaboration with E. Russ, we sutdy on a graph satisfying the doubling property and Poincaré inequality, the L^p boundedness of the Riesz transform for p>2 and its reverse inequality for p<2. For our purpose, we also prove Littlewood-Paley inequalities and interpolation results for Sobolev spaces in this context, which are of independent interest. In the last chapter, using our interpolation result, we prove Gagliardo-Nirenberg inequalities on a Riemannian manifold satisying the doubling property and admitting Poincaré and Pseudo-Poincaré inequalities. This result also applies in the case of Lie groups and on graphs.
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Nadine Badr. Interpolation réelle des espaces de Sobolev sur les espaces métriques mesurés et applications aux inégalités fonctionnelles. Géométrie métrique [math.MG]. Université Paris Sud - Paris XI, 2007. Français. ⟨tel-00736066⟩

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