Estimates for the covariant derivative of the heat semigroup on differential forms, and covariant Riesz transforms

Abstract

With ∆ j ≥ 0 is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and ∇ the Levi-Civita covariant derivative, we prove amongst other things• a Li-Yau type heat kernel bound for ∇e −t ∆j , if the curvature tensor of M and its covariant derivative are bounded, • an exponentially weighted L p bound for the heat kernel of ∇e −t ∆j , if the curvature tensor of M and its covariant derivative are bounded, • that ∇e −t ∆j is bounded in L p for all 1 ≤ p < ∞, if the curvature tensor of M and its covariant derivative are bounded, • a second order Davies-Gaffney estimate (in terms of ∇ and ∆ j ) for e −t ∆j for small times, if the j-th degree Bochner-Lichnerowicz potential V j = ∆ j − ∇ † ∇ of M is bounded from below (where V 1 = Ric), which is shown to fail for large times if V j is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform ∇( ∆ j + κ) −1/2 in L p for all 1 ≤ p < ∞ (which we prove for 1 ≤ p ≤ 2), and explain its implications to geometric analysis, such as the L p -Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for ∇e −t ∆j .

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

“…Very recently, Baumgarth, Devyver and Güneysu [4] studied the covariant Riesz transform on j-forms and established a second order Davies-Gaffney estimate for the semigroup e −t ∆ j for small times, where ∆ j is the unique self-adjoint realization of the Hodge-de Rham Laplacian acting on jforms. These results can be applied to CZ(p) when 1 < p < 2 and allow to establish CZ(p) under strong curvature assumptions but without requirements on the injectivity radius.…”

“…Proof. Recall the following L 2 -Gaffney off-diagonal estimate on one-forms [4]. For α ∈ Γ L 2 (T * M) with support supp(α) ⊂ E and any s ∈ (0, 1), it holds…”

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].…”