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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 volume doubling property. Let Δ⃗ be the Hodge–de Rham Laplacian acting on 1‐differential forms. According to the Bochner formula, Δ⃗=∇*∇+R+−R− where R+ and R− are respectively the positive and negative part of the Ricci curvature and ∇ is the Levi–Civita connection. We study the boundedness of the Riesz transform d*Δ⃗−12 from Lp(Λ1T*M) to Lp(M) and of the Riesz transform dΔ⃗−12 from Lp(Λ1T*M) to Lp(Λ2T*M) . We prove that, if the heat kernel on functions pt(x,y) satisfies a Gaussian upper bound and if the negative part R− of the Ricci curvature is ε‐sub‐critical for some ε∈[0,1) , then d*Δ⃗−12 is bounded from Lp(Λ1T*M) to Lp(M) and dΔ⃗−12 is bounded from Lp(Λ1T*M) to Lp(Λ2T*M) for p∈(p0′,2] where p0>2 depends on ε and on a constant appearing in the volume doubling property. A duality argument gives the boundedness of the Riesz transform d(Δ)−12 from Lp(M) to Lp(Λ1T*M) for p∈[2,p0) 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.