Riesz transform and related inequalities on non‐compact Riemannian manifolds

Let M be a complete, noncompact Riemannian manifold, μ the Riemannian measure, ∇ the Riemannian gradient, and 1 the (positive) Laplace-Beltrami operator on M . Denote by | · | the length in the tangent space, and by ‖ · ‖p the norm in L p(M, μ), 1 ≤ p ≤ +∞. If one wants to define homogeneous Sobolev spaces of order one on M , that is, spaces of functions with one derivative in L p(M, μ), 1 < p < +∞, there are two obvious candidates for the seminorm: ‖|∇ f |‖p and ‖11/2 f ‖p. The former is local and of geometric nature, the latter is nonlocal and more analytic. When M is the Euclidean space, these two seminorms are equivalent for all p ∈ ]1, +∞[: C−1 p ‖1 f ‖p ≤ ‖|∇ f |‖p ≤ Cp‖1 f ‖p ∀ f ∈ C∞ 0 (R) . This relies on singular integral theory (see [40, 42]); indeed, the second inequality above is nothing but the L p-boundedness of the so-called Riesz transforms ∂ ∂xi 1−1/2, i = 1, . . . , n in Rn , and the first one follows by duality. It is, of course, a basic issue (which was raised in [43]) to ask for which complete, noncompact Riemannian manifolds M and which p ∈ ]1, +∞[ one has (1.1) C−1 p ‖1 f ‖p ≤ ‖|∇ f |‖p ≤ Cp‖1 f ‖p ∀ f ∈ C∞ 0 (M). For p = 2, on any complete Riemannian manifold, one has the equality