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

摘要

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property. Let (A) over right arrow be the Hodge-de Rham Laplacian acting on 1-differential forms. According to the Bochner formula, (A) over right arrow = del*del + 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*((Delta) over right arrow)(-1/2) from L-p(A(1)T*M) to L-p(M) and of the Riesz transform d((Delta) over right arrow)(-1/2) from L-p(Lambda T-1*M) to L-p(A(2)T*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 epsilon [0, 1), then d*(((Delta) over right arrow)(-1/2) is bounded from L-p(A(1)T*M) to L-p(M) and d((Delta) over right arrow)(-1/2) is bounded from L-p(Lambda T-1*M) to L-p(A(2)T*M) for p epsilon (p(0)', 2] where p(0) > 2 depends on epsilon and on a constant appearing in the volume doubling property. A duality argument gives the boundedness of the Riesz transform d(((Delta) over right arrow))(-1/2) from L-p(M) to L-p(Lambda T-1*M) for p is an element of [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. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim