The Hodge–de Rham Laplacian and Lp-boundedness of Riesz transforms on non-compact manifolds

International audience; Let M be a complete non-compact Riemannian manifold satisfying the doubling volume property as well as a Gaussian upper bound for the corresponding heat kernel. We study the boundedness of the Riesz transform d∆ − 1 2 on both Hardy spaces H p and Lebesgue spaces L p under two different conditions on the negative part of the Ricci curvature R − . First we prove that if R − is α-subcritical for some α ∈ [0, 1), then the Riesz transform d * − → ∆ − 1 2 on differential 1-forms is bounded from the associated Hardy space H p − → ∆ (Λ 1 T * M) to L p (M) for all p ∈ [1, 2]. As a consequence, d∆ − 1 2 is bounded on L p for all p ∈ (1, p 0) where p 0 > 2 depends on α and the constant appearing in the doubling property. Second, we prove that if 1 0 |R − | 1 2 v(·, √ t) 1 p 1 p1 dt √ t + ∞ 1 |R − | 1 2 v(·, √ t) 1 p 2 p2 dt √ t < ∞, for some p 1 > 2 and p 2 > 3, then the Riesz transform d∆ − 1 2 is bounded on L p for all 1 < p < p 2 . In the particular case where v(x, r) ≥ Cr D for all r ≥ 1 and |R − | ∈ L D/2−η ∩ L D/2+η for some η > 0, then d∆ − 1 2 is bounded on L p for all 1 < p < D. Furthermore, we study the boundedness of the Riesz transform of Schrödinger operators A = ∆ + V on L p for p > 2 under conditions on R − and the potential V . We prove both positive and negative results on the boundedness of dA − 1 2 on L p .