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hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorMAGNIEZ, Jocelyn
dc.date.accessioned2024-04-04T03:20:25Z
dc.date.available2024-04-04T03:20:25Z
dc.date.created2014
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/194586
dc.description.abstractEnLet $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property. Let $\overrightarrow{\Delta}$ be the Hodge-de Rham Laplacian acting on $1$-differential forms. According to the Bochner formula, $\overrightarrow{\Delta}=\nabla^*\nabla+R_+-R_-$ where $R_+$ and $R_-$ are respectively the positive and negative part of the Ricci curvature and $\nabla$ is the Levi-Civita connection. We study the boundedness of the Riesz transform $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^*M)$. We prove that, if the heat kernel on functions $p_t(x,y)$ satisfies a Gaussian upper bound and if the negative part $R_-$ of the Ricci curvature is $\epsilon$-sub-critical for some $\epsilon\in[0,1)$, then $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^* M)$ for $p\in(p_0',2]$ where $p_0>2$ depends on $\epsilon$ and on a constant appearing in the doubling volume property. A duality argument gives the boundedness of the Riesz transform $d(\Delta)^{-\frac{1}{2}}$ from $L^p(M)$ to $L^p(\Lambda^1T^*M)$ for $p\in [2,p_0)$ where $\Delta$ is the non-negative Laplace-Beltrami operator. We also give a condition on $R_-$ to be $\epsilon$-sub-critical under both analytic and geometric assumptions.
dc.language.isoen
dc.subject.enRiemannian manifolds
dc.subject.enRiesz transforms
dc.subject.enHodge- de Rham Laplacian
dc.subject.enRiemannian manifolds.
dc.title.enRiesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.identifier.arxiv1410.0034
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-01070228
hal.version1
hal.audienceNon spécifiée
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01070228v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=MAGNIEZ,%20Jocelyn&rft.genre=preprint


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