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Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces

Abstract : This thesis is devoted to results in real harmonic analysis in discrete (graphs) or continuous (Lie groups) geometric contexts.Let Gamma be a graph (a set of vertices and edges) equipped with a discrete laplacian Delta=I-P, where P is a Markov operator.Under suitable geometric assumptions on Gamma, we show the Lp boundedness of fractional Littlewood-Paley functionals. We introduce H1 Hardy spaces of functions and of 1-differential forms on Gamma, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in Gamma. As a consequence, we derive the H1 boundedness of the Riesz transform. Assuming furthermore pointwise upper bounds for the kernel (Gaussian of subgaussian upper bounds) on the iterates of the kernel of P, we also establish the Lp boundedness of the Riesz transform for 10, 1leq pleq+infty and 1leq qleq +infty.These results hold for polynomial as well as for exponential volume growth of balls.
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Joseph Feneuil. Harmonic analysis on graphs and Lie groups : quadratic functionals, Riesz transforms and Besov spaces. Functional Analysis [math.FA]. Université Grenoble Alpes, 2015. English. ⟨NNT : 2015GREAM040⟩. ⟨tel-01280100⟩

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