Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds

Abstract

We address some fundamental questions concerning geometric analysis on Riemannian manifolds. It has been asked whether the L p -Calderón-Zygmund inequalities extend to a reasonable class of non-compact Riemannian manifolds without the assumption of a positive injectivity radius. In the present paper, we give a positive answer for 1 < p < 2 under the natural assumption of a lower bound on the Ricci curvature. For p > 2, we complement the study in Güneysu-Pigola (2015) and derive sufficient geometric criteria for the validity of the Calderón-Zygmund inequality by adding Kato class bounds on the Riemann curvature tensor and the covariant derivative of Ricci curvature. Probabilistic tools, like Hessian formulas and Bismut type representations for heat semigroups, play a significant role throughout the proofs.

Covariant Riesz transform on differential forms for $$1<p\le 2$$

“…Comparing Theorem 1.6 with existing results on CZ µ (p), it should be pointed out that the result is valid without any injectivity radius assumptions and boundedness of R ∞ and ∇R ∞ as in [20]. Our result extends [7] to the weighted manifold by only requiring the curvature-dimension condition.…”

“…Let us now present the main steps of the proof of Theorem 1.2 and Theorem 1.3, following closely the approach of [11, Theorems 1.1 and 1.2]. Some of the arguments have been used already in [7] and can be taken from there. For the convenience of the reader and for the sake of completeness we give details here.…”

“…On the other hand, very recently, Cao, Cheng and Thalmaier [7] established the L p -Calderón-Zygmund inequality for 1 < p < 2 by only using the natural assumption of a lower Ricci curvature bound. Since the conditions for the L p boundedness of the classical Riesz transform d(∆ µ + σ) −1/2 for 1 < p ≤ 2 are quite weak, it arises us to wondering whether the condition for the L p boundedness of the covariant Riesz transform on one-forms in [5] is too strong.…”

Gradient estimates under integral Ricci bounds

“…By an integration by parts, the validity of (1.2) implies the corresponding L p -gradient estimate, see [13,Corollary 3.11] as well as [14,Theorem 2]. It should be noted that the converse is generally false: in [20] the first and third author were able to construct an example of a manifold which supports L p -gradient estimates but where the Calderón-Zygmund inequalities fail for large p. Conditions which ensure the validity of L p -Calderón-Zygmund inequality can be found in [13,17,18,24,21] or the very recent [4,6] In this paper, we establish L p -gradient estimates under integral Ricci lower bounds, that is, the Ricci curvature is allowed some explosion at −∞ as long as it is controlled in a mean integral sense. These bounds appear naturally in some isospectral and geometric variational problems as well as in Ricci and Kähler-Ricci flows, [7,25,26,2,3,1].…”

Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

“…The study of the Hessian of a Feynman-Kac semigroup generated by a Schrödinger operator of the type ∆ + V where V is a potential, has been pushed forward by Li [11,10]. Very recently, Bismut-type Hessian formulas have been used for new applications, see for instance Cao-Cheng-Thalmaier [4] where L p Calderón-Zygmund inequalities on Riemannian manifolds have been established under natural geometric assumptions for indices p > 1.…”