Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds

Abstract

Let $M$ 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, $\overrightarrow{\Delta }={\nabla }^{\ast }\nabla +R_{+}-R_{-}$ 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 $d^{\ast }(\overrightarrow{\Delta }{)}^{-\frac{1}{/2}}$ from $L^p({\Lambda }^1{T}^{\ast }M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrow{\Delta }{)}^{-\frac{1}{2}}$ from $L^p({\Lambda }^1{T}^{\ast }M)$ to $L^p({\Lambda }^2{T}^{\ast }M)$. 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 $d^{\ast }(\overrightarrow{\Delta }{)}^{-\frac{1}{2}}$ is bounded from $L^p({\Lambda }^1{T}^{\ast }M)$ to $L^p(M)$ and $d(\overrightarrow{\Delta }{)}^{-\frac{1}{2}}$ is bounded from $L^p({\Lambda }^1{T}^{\ast }M)$ to $L^p({\Lambda }^2{T}^{\ast }M)$ for $p\in (p_0^{\prime },2]$ 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 $L^p({\Lambda }^1{T}^{\ast }M)$ 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.