Weighted LP estimates on Riemannian manifolds

Abstract : The topics addressed in this thesis lie in the field of harmonic analysis and more pre- cisely, weighted inequalities. Our main interests are the weighted Lp-bounds of the Riesz transforms on complete Riemannian manifolds and the sharpness of the bounds in terms of the power of the characteristic of the weights. We first obtain a linear and dimensionless result on non necessarily homogeneous spaces, when p = 2 and the Bakry-Emery curvature is non-negative. We use here an analytical approach by exhibiting a concrete Bellman function. Next, using stochastic techniques and sparse domination, we prove that the Riesz transforms are Lp-bounded for p ∈ (1, +∞) and obtain the previous result for free. Finally, we use an elegant change in the precedent proof to weaken the condition on the curvature and assume it is bounded from below.
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Kamilia Dahmani. Weighted LP estimates on Riemannian manifolds. Numerical Analysis [math.NA]. Université Paul Sabatier - Toulouse III, 2018. English. ⟨NNT : 2018TOU30188⟩. ⟨tel-02308920⟩

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