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

Abstract

With $$\vec {\Delta }_j\ge 0$$ Δ → 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 $$\nabla $$ ∇ the Levi-Civita covariant derivative, we prove among other things a Gaussian heat kernel bound for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, an exponentially weighted $$L^p$$ L p -bound for the heat kernel of $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j , if the curvature tensor of M and its covariant derivative are bounded, that $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j is bounded in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ , if the curvature tensor of M and its covariant derivative are bounded, a second order Davies-Gaffney estimate (in terms of $$\nabla $$ ∇ and $$\vec {\Delta }_j$$ Δ → j ) for $$\mathrm {e}^{ -t\vec {\Delta }_j }$$ e - t Δ → j for small times, if the j-th degree Bochner-Lichnerowicz potential $$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$ V j = Δ → j - ∇ † ∇ of M is bounded from below (where $$V_1=\mathrm {Ric}$$ V 1 = Ric ), which is shown to fail for large time, if $$V_j$$ V j is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform $$\nabla (\vec {\Delta }_j+\kappa )^{-1/2}$$ ∇ ( Δ → j + κ ) - 1 / 2 in $$L^p$$ L p for all $$1\le p<\infty $$ 1 ≤ p < ∞ (which we prove for $$1\le p\le 2$$ 1 ≤ p ≤ 2 ), and explain its implications to geometric analysis, such as the $$L^p$$ L p -Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for $$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$ ∇ e - t Δ → j .

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

“…Thus to investigate whether these L p -Calderón-Zygmund inequalities hold are reduced to the study of conditions for boundedness of the classical Riesz transform d(∆ µ + σ) −1/2 on functions and boundedness of the covariant Riesz transform ∇(∆ (1) µ + σ) −1/2 on one-forms in L p -sense. Therefore, in [5] combining this argument with the result in [29] yields that the L p -Calderón-Zygmund inequalities hold for 1…”

“…Note that the approach in [29] is of stochastic nature and derivative estimates for the heat kernel are deduced from derivative formulae for semigroups on vector bundles, by means of the methodology of Driver and the second named author [15]. In [15] estimates of certain functionals of Brownian motion with respect to the Wiener measure are required; nevertheless pointwise estimate for the heat kernel e −t∆ (k) (x, y) and the derivative estimate of heat kernel ∇e −t∆ (k) (x, y) can be obtained from such general derivative formulas, where ∆ (k) is the unique self-adjoint realization of the Hodge-de Rham Laplacian acting on k-forms under explicit curvature condition, see [5]. By following a similar approach, Baumgarth, Devyver and Güneysu [5] studied the covariant Riesz transform on j-forms, removing the doubling volume property and using uniformly boundedness conditions of the curvature and the derivative of the curvature on differential forms.…”

“…In [15] estimates of certain functionals of Brownian motion with respect to the Wiener measure are required; nevertheless pointwise estimate for the heat kernel e −t∆ (k) (x, y) and the derivative estimate of heat kernel ∇e −t∆ (k) (x, y) can be obtained from such general derivative formulas, where ∆ (k) is the unique self-adjoint realization of the Hodge-de Rham Laplacian acting on k-forms under explicit curvature condition, see [5]. By following a similar approach, Baumgarth, Devyver and Güneysu [5] studied the covariant Riesz transform on j-forms, removing the doubling volume property and using uniformly boundedness conditions of the curvature and the derivative of the curvature on differential forms. These results are further used to establish Calderón-Zygmund inequalities for 1 < p ≤ 2, where for ϕ ∈ C ∞ c (M), the set of smooth functions of compact support, if there exist positive constants C 1 and C 2 such that | Hess(ϕ)| p ≤ C 1 ϕ p + C 2 ∆ϕ p then the Calderón-Zygmund inequalities holds.…”

“…These results are further used to establish Calderón-Zygmund inequalities for 1 < p ≤ 2, where for ϕ ∈ C ∞ c (M), the set of smooth functions of compact support, if there exist positive constants C 1 and C 2 such that | Hess(ϕ)| p ≤ C 1 ϕ p + C 2 ∆ϕ p then the Calderón-Zygmund inequalities holds. Note that the argument in [5] for establishing Calderón-Zygmund inequalities is from Güneysu and Pigola's paper [20], where the Calderón-Zygmund inequalities are equivalent to the L p boundedness of the operator Hess(∆ + σ) −1 for some constant σ > 0 and this operator further can be rewritten as ∇(∆ (1)…”

“…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. This work is devoted to investigating this problem on the L p boundedness of covariant Riesz transform for 1 < p ≤ 2 again.…”