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

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property. Let $\overrightarrowΔ$ be the Hodge-de Rham Laplacian acting on 1-differential forms. According to the Bochner formula, $\overrightarrowΔ=\nabla^*\nabla+R_+-R_-$ 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 $d^*(\overrightarrowΔ)^{-\frac{1}{/2}}$ from $L^p(Λ^1T^*M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrowΔ)^{-\frac{1}{2}}$ from $L^p(Λ^1T^*M)$ to $L^p(Λ^2T^*M)$. 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 $ε$-sub-critical for some $ε\in[0,1)$, then $d^*(\overrightarrowΔ)^{-\frac{1}{2}}$ is bounded from $L^p(Λ^1T^*M)$ to $L^p(M)$ and $d(\overrightarrowΔ)^{-\frac{1}{2}}$ is bounded from $L^p(Λ^1T^*M)$ to $L^p(Λ^2T^* M)$ for $p\in(p_0',2]$ where $p_0>2$ depends on $ε$ and on a constant appearing in the doubling volume property. A duality argument gives the boundedness of the Riesz transform $d(Δ)^{-\frac{1}{2}}$ from $L^p(M)$ to $L^p(Λ^1T^*M)$ for $p\in [2,p_0)$ where $Δ$ is the non-negative Laplace-Beltrami operator. We also give a condition on $R_-$ to be $ε$-sub-critical under both analytic and geometric assumptions.