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

With Δ→j≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\Delta }_j\ge 0$$\end{document} is the uniquely determined self-adjoint realization of the Laplace operator acting on j-forms on a geodesically complete Riemannian manifold M and ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} the Levi-Civita covariant derivative, we prove among other things a Gaussian heat kernel bound for ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document}, if the curvature tensor of M and its covariant derivative are bounded, an exponentially weighted Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-bound for the heat kernel of ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document}, if the curvature tensor of M and its covariant derivative are bounded, that ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document} is bounded in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} for all 1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document}, if the curvature tensor of M and its covariant derivative are bounded, a second order Davies-Gaffney estimate (in terms of ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} and Δ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\Delta }_j$$\end{document}) for e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document} for small times, if the j-th degree Bochner-Lichnerowicz potential Vj=Δ→j-∇†∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$\end{document} of M is bounded from below (where V1=Ric\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1=\mathrm {Ric}$$\end{document}), which is shown to fail for large time, if Vj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_j$$\end{document} is bounded. Based on these results, we formulate a conjecture on the boundedness of the covariant local Riesz-transform ∇(Δ→j+κ)-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla (\vec {\Delta }_j+\kappa )^{-1/2}$$\end{document} in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} for all 1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document} (which we prove for 1≤p≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p\le 2$$\end{document}), and explain its implications to geometric analysis, such as the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-Calderón-Zygmund inequality. Our main technical tool is a Bismut derivative formula for ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document}. a Gaussian heat kernel bound for ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document}, if the curvature tensor of M and its covariant derivative are bounded, an exponentially weighted Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}-bound for the heat kernel of ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document}, if the curvature tensor of M and its covariant derivative are bounded, that ∇e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document} is bounded in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document} for all 1≤p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le p<\infty $$\end{document}, if the curvature tensor of M and its covariant derivative are bounded, a second order Davies-Gaffney estimate (in terms of ∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla $$\end{document} and Δ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {\Delta }_j$$\end{document}) for e-tΔ→j\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {e}^{ -t\vec {\Delta }_j }$$\end{document} for small times, if the j-th degree Bochner-Lichnerowicz potential Vj=Δ→j-∇†∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_j=\vec {\Delta }_j-\nabla ^{\dagger }\nabla $$\end{document} of M is bounded from below (where V1=Ric\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1=\mathrm {Ric}$$\end{document}), which is shown to fail for large time, if Vj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_j$$\end{document} is bounded.


Introduction
Let M be a smooth connected geodesically complete Riemannian m-manifold 1 .The geodesic distance will be denoted by ̺(x, y) and the induced open balls with B(x, r).Given a smooth vector bundle E → M carrying a canonically given metric and a canonically given covariant derivative, we denote its fiberwise metric simply by (•, •), with | • | = (•, •) and its covariant derivative simply with These remarks apply in particular to T M → M, T * M → M or tensor products thereof.We equip M with the Riemannian volume measure dµ; sometimes we will use the following local volume doubling property for the measure dµ, which writes: there exists C > 0 such that for all 0 < r ≤ R < +∞ and all z ∈ M, 1 All manifolds are understood to be without boundary, unless otherwise stated.By the Bishop-Gromov comparison theorem and the well-known formula for the volume of balls in the hyperbolic space, (LVD) holds if Ric ≥ −A 2 for some A ≥ 0 (and then the constant C in (LVD) only depends on m and A).A well-known consequence of (LVD) is the following volume comparison inequality: there is a constant C > 0 such that for all t > 0, x 1 , x 2 ∈ M and ε > 0, Indeed, letting r = ̺(x 1 , x 2 ), Upon using the elementary inequalities e C r ≤ e Given a smooth metric vector bundle E → M we define the Banach spaces Γ L p (M, E ) given by equivalence classes of Borel sections ψ in E → M such that where |ψ| p denotes the norm of the function |ψ| with respect to L p (M).Then Γ L 2 (M, E ) canonically becomes a Hilbert space with scalar product ψ 1 , ψ 2 = (ψ 1 , ψ 2 )dµ.(1.1)In particular, if Riem denotes the Riemann curvature tensor, seen for instance as a (0, 4)tensor, then we can consider ||Riem|| ∞ to be the || • || ∞ norm of Riem ∈ Γ C ∞ (M, T 0 4 M), where T p q M → M is by definition the vector bundle of tensors of type (p, q).Likewise, seeing ∇Riem as a (0, 1 + 4)-tensor, we can consider ||∇Riem|| ∞ .Given another smooth metric bundle F → M, the operator norm of a linear map A : Γ L p (M, E ) −→ Γ L q (M, F ) will be denoted by A p,q = sup Af q : Given a smooth linear partial differential operator its formal adjoint with respect to the scalar products •, • is denoted by be the exterior differential.Then one defines the Laplace-Beltrami operator acting on 0-forms, and the Hodge Laplacian acting on j-forms, respectively, by The induced direct-sum data will be denoted by Note the commutation rules d j ∆ j = ∆ j d j and d † j−1 ∆ j = ∆ j−1 d † j−1 .In the case j = m and if M is oriented, the Hodge Laplacian ∆ m is just the conjugate of the scalar Laplacian ∆ = ∆ 0 by the Hodge star operator The Bochner-Lichnerowicz formula for the Hodge Laplacian writes

where
V j ∈ Γ C ∞ (M, End(Λ j T * M)) is a fiberwise self-adjoint 0-th order operator, which satisfies |V j | ≤ C|Riem|, where C = C(m) > 0 is a constant that only depends on m. (1.2) It is worth noting that for j = 1 one has V 1 = Ric tr , where the Ricci curvature is read as a section Ric ∈ Γ C ∞ (M, End(T M)), and its transpose Ric tr ∈ Γ C ∞ (M, End(T * M)) is defined by duality using the Riemannian metric.Given a Borel function Ψ : R → R and a self-adjoint operator A, then Ψ(A) is the selfadjoint operator which is induced by the spectral calculus.Its domain of definition is then denoted by Dom(Ψ(A)).As M is geodesically complete, for any j ∈ {0, • • • , n}, ∆ j is essentially self-adjoint [12] in Γ L 2 (M, Λ j T * M) when initially defined on Γ C ∞ c (M, Λ j T * M).By a usual abuse of notation, the corresponding self-adjoint realizations will be denoted by ∆ ≥ 0, resp., ∆ j ≥ 0 again.By local parabolic regularity, for all square-integrable j-forms α ∈ Γ L 2 (M, Λ j T * M), the time dependent 1-form (0, ∞) × M ∋ (t, x) −→ e −t ∆ j α(x) ∈ Λ j T * x M has a smooth representative, which extends smoothly to [0, ∞) × M, if α is smooth.Moreover, there exists a uniquely determined smooth map the heat kernel of ∆ j , such that for all α as above, t > 0, x ∈ M one has

Note in particular that
and that the heat kernel satisfies the usual semigroup identity (cf.Theorem II.1 in [16]).
Estimates for the heat kernel of the Hodge Laplacian and its derivatives under curvature assumptions, or more generally estimates of the heat kernel of covariant Schrödinger operators of the form ), with a smooth metric vector bundle E → M, have already been studied for a long time.When H V is bounded from below in the sense of quadratic forms, then H V is essentially self-adjoint [7]; moreover, if the potential V , a pointwise self-adjoint smooth section of End(E ) → M, satisfies V ≥ −a 2 for some constant a ∈ R (meaning that all x ∈ M all eigenvalues of V (x) : E x → E x are bounded from below by −a 2 ), then semigroup domination [4] states that for every α ∈ Γ L 2 (M, E ), t > 0, one has |e −tH V α| ≤ e a 2 t e −t∆ |α|.
As, by the Li-Yau heat kernel estimate [25], the assumption Ric ≥ −A 2 for some constant A ∈ R implies the existence of constants in this case semigroup domination implies In particular, assuming ||Riem|| ∞ ≤ A for some A > 0 (cf.(1.2)) and using the above result for where C, D > 0 depend only on A and m.The commutation rules d j ∆ j = ∆ j+1 d j and d † j ∆ j = ∆ j+1 d † j can then be used in order to prove that similar pointwise estimates hold for the kernels of d j e −t ∆ j and d † j e −t ∆ j : Proposition 1.1.Assume that there is a constant A > 0 such that ||Riem|| ∞ ≤ A. Then there exist constants C = C(A, m) > 0, D = D(A, m) > 0, such that for all j ∈ {1, . . ., m}, x, y ∈ M, t > 0 one has Above and in the sequel, we understand d j to act on the first variable of the heat kernel, so d j e −t ∆ j (x, y) := d j e −t ∆ j (•, y)(x), and likewise for d † j−1 , and in similar situations such as ∇e −t ∆ j (x, y).Note here that for fixed y, the map x → e −t ∆ j (x, y) becomes a section of a bundle of the form with W a fixed finite dimensional linear space, which explains the action of these differential operators on the heat kernel.Although we expect this result to be well-known to the experts, for the sake of completeness, we will provide a proof of Proposition 1.1 in the appendix.In this article, our main goal is to prove the analogous estimates for the covariant derivative of the heat kernel of the Hodge Laplacian: more precisely, we wish to obtain pointwise estimates of the form: For j = 0 we have ∇ = d 0 , so (dUE) and (∇UE) are equivalent, and can thus be obtained with assuming merely that ||Riem|| ∞ < ∞.The same is true for j = m by Hodge duality (if M is oriented).However, for j ∈ {1, • • • , m − 1}, the corresponding covariant derivative estimates are significantly stronger than (dUE) and (d † UE), and are harder to prove as well, as we shall see.In fact, in order to prove these, we will not only need a uniform bound on the Riemannian curvature tensor, but also on its covariant derivative.We can now state our main result: Then (∇UE) holds; more precisely, there exist constants C = C(A, m), D = D(A, m) > 0, such that for all j ∈ {1, . . ., m}, t > 0, x, y ∈ M one has The proof of Theorem 1.2 is given in Section 3 and is based on a probabilistic representation of ∇e − t 2 ∆ j (x, y) in terms of the Brownian bridge, namely a so called Bismut derivative formula, which should be of independent interest and which is proved based on the methods from [14] (see also [32]) in Section 2. In fact, we first prove a local Bismut derivative formula for ∇e − t 2 ∆ j α(x) in terms of Brownian motion for α ∈ Γ C ∞ ∩L 2 (M, Λ j T * M), which does not require any assumptions on the geometry.Then we use this formula to obtain global L ∞ estimates under (1.3) for ∇e − t 2 ∆ j , which are then used to prove a global Bismut derivative formula for ∇e − t 2 ∆ j α(x) in terms of Brownian motion.The reason for this rather technical procedure is that, unlike its global counterpart, the local Bismut derivative formula contains a first exit time of Brownian motion from a ball B around its starting point x, which is why this formula cannot be controlled well in terms of Brownian bridge (which is conditioned to be in a fixed point y at its terminal time, which need not be in in B).At this point, let us mention the recent paper [24], where, with completely different methods, L p → L q estimates for the covariant derivates of heat kernels of covariant Schrödinger operators have been considered for so called asymptotically locally Euclidean Riemannian manifolds.In addition to the fact a very special form of the geometry is required, the estimates from [24] are also different in their nature than ours: in [24] one needs an additional decay at ∞ of the potentials in order to obtain a damping effect in the constants also for for large t > 0, while our estimates do not require any decay at ∞ of the potentials, while they only damp for small t > 0. As a consequence of the pointwise estimates from Theorem 1.2 and local volume doubling, one obtains L p → L p bounds for ∇e −t ∆ j , as well as weighted L p -estimates for the kernel of ∇e −t ∆ j : Then: I) For all 1 ≤ p < ∞ there exists a constant C = C(A, m, p) > 0, such that for all j ∈ {1, . . ., m}, t > 0 one has II) There exists a constant γ = γ(A, m) > 0, and for all 1 ≤ p < ∞ a constant C = C(A, m, p) > 0, such that for all j ∈ {1, . . ., m} and t > 0 one has It is a well-known principle that stems from the work of Coulhon and Duong [9], as well as later works by Auscher, Coulhon, Duong and Hofmann [1], that (at least for the scalar Laplacian) estimates for the spatial derivative of the heat kernel should have consequences for the corresponding Riesz transform.In this respect, applying (1.5) with p = 2, we are going to establish the following result concerning the covariant Riesz transform: Then there exists a κ 0 = κ 0 (A, m) > 0, which only depends on A, m, such that for all κ ≥ κ 0 , and all j ∈ {1, . . ., m}, the operator ∇( ∆ j + κ) −1/2 is of weak (1, 1) type with a bound only depending on A, m, κ; in other words, there exist a constant D = D(A, m, κ) > 0, which only depends on A, m and κ, such that for all j ∈ {1, . . ., m}, λ > 0, In particular, for all 1 < p ≤ 2 there exists a constant C = C(A, m, p, κ) > 0, which only depends on A, m, p, κ, such that for all j ∈ {1, . . ., m} one has Corollary 1.4 is proved in Section 5 (where we show that the (1,1) property indeed implies the L p -boundedness through).This results improves a result by Thalmaier and Wang [32, Theorem D]: more precisely, in [32,Theorem D] the same conclusion for the covariant Riesz transform is obtained, however an additional assumption on the volume growth of M is made.This volume assumption excludes in particular hyperbolic geometries (see [27]), while such geometries are covered by our Corollary 1.4.In light of the our main result, Theorem 1.2, and the results in [1] for the scalar Riesz transform, it is natural to expect that a uniform bound on Riem and ∇Riem implies that the covariant Riesz transform is bounded on L p for all 1 < p < ∞; specifically, we make the following conjecture: There exists a κ 0 = κ 0 (A, m) > 0, which only depends on A, m, such that for all κ ≥ κ 0 , < ∞, with bound only depending on A, m, κ, p.
However, the case of Einstein manifold is very special, because for Einstein manifolds there is a nice commutation formula beween ∇ and ∆ j (cf.[3, formula (6.1)]).We currently do not know whether the assumption on ∇Riem is really necessary in Conjecture 1.5; however, it is known that the curvature hypotheses cannot be weakened to merely boundedness from below of the sectional curvature: in fact a recent result of Marini and Veronelli [26] shows that there exist manifolds with positive sectional curvature, for which the covariant Riesz transform is not bounded on L p for all p ∈ (1, ∞).
As we said, the L p -boundedness part of this result follows from [3, Theorem 5.1]; the weak (1, 1) part appears to be new in this generality.The latter is established using the estimates (dUE) and (d † UE), and Coulhon-Duong theory as in the proof of Corollary 1.4, yielding an alternative proof of the L p boundedness part of Theorem 1.7.This will be done in Section 5.
Let us stress that for applications in geometric analysis, the L p -boundedness of ∇( ∆ j + κ) −1/2 is more important than that of d j ( ∆ j + κ) −1/2 or d † j ( ∆ j + κ) −1/2 .For example, as shown in [18, Proof of Theorem 4.13], the former boundedness for j = 1 implies the L p -Calderón-Zygmund inequality where where C only depends on D CZ and any upper bound for Riem ∞ .Hence, Corollary 1.4 readily implies: Then for all 1 < p ≤ 2, there exists a constant and such that for every distributional solution Note that the CZ-inequality (1.6) improves Theorem D in [18] by getting rid of the volume assumption made there.
In addition to a priori estimates for the Poisson equation, the L p -Calderón-Zygmund inequality implies precompactness results for isometric immersions (cf.Theorem 1.1 in [8] and [19]).Moreover, the recent survey article [27] contains the state-of-the art for the L p -Calderón-Zygmund inequality for large p: it is explained therein that Riem ∞ < ∞ is enough for the L p -Calderón-Zygmund inequality to hold for all p > max(2, m/2).In this sense, Corollary 1.8 can be considered a complementary result for small p.
A fundamental tool in [1] for obtaining the boundedness of the Riesz transform on L p for p > 2 are the so-called Davies-Gaffney estimates for the gradient of the scalar Laplacian, that is to say L 2 off-diagonal estimates for e −t∆ and de −t∆ .At zeroth order, these estimates are equivalent to the finite speed of propagation of the associated wave equation, and they hold true for e −t ∆ j for all j = 0, • • • , m [28] (note that Davies-Gaffney estimates for covariant Schrödinger semigroups of the form e −tH V for unbounded V 's play a fundamental role in the context of essential self-adjointness of covariant Schrödinger operators [20]).One can ask more generally whether Davies-Gaffney hold for the covariant derivative of the heat kernel of the Hodge Laplacian.In this respect, we have the following result, which is proved in Section 6, and where χ A denotes the indicator function of a set A ⊂ M: Theorem 1.9.There exist universal constants c 1 , c 2 > 0 such that for all j ∈ {1, . . ., m} with V j ≥ −A for some constant A ≥ 0, all t > 0, all Borel subsets E, F ⊂ M with compact closure, and all Actually, the above Davies-Gaffney estimate for e −t ∆ j and t ∆ j e −t ∆ j , even without the extra √ t factor on the right-hand side, are already known (cf [2, Lemma 3.8]), but for the sake of completeness we will provide a proof.The novelty is the Davies-Gaffney bound for the gradient term √ t∇e −t ∆ j .Note also that the above Davies-Gaffney bounds implies that for all 0 < t < 1 one has which is ultimately what is needed for the machinery from [1].Remarkably, for j > 0 the latter form of the Davies inequality is false for large times, unless one makes additional geometric assumptions on M.This means that, contrary to what happens for the scalar Laplacian, even L 2 off-diagonal estimates for the covariant derivative of the heat operator of the Hodge Laplacian are non-trivial.This is the content of the following result, which is proved in Section 7: Theorem 1.10.Assume that M is noncompact, that there exists j ∈ {1, . . ., m} with V j ∞ < ∞, and that there exist constants c 1 , c 2 > 0 such that for all t > 0, all Borel subsets E, F ⊂ M with compact closure and all Then one has Ker L 2 ( ∆ j ) = {0}.
Acknowledgements: The authors are indepted to Stefano Pigola and Anton Thalmaier for very helpful discussions.

Bismut derivative formula
Fix j ∈ {1, . . ., m}.The following endomorphisms are built from the curvature and its first derivative and will play a crucial role in the probabilistic formula for ∇e − t 2 ∆ j , the main result of this section.In this section, we read the Riemannian curvature as a section Then the section where e j is any smooth local orthonormal basis for T x M, and the section For the formulation of the probabilistic results of this section, we will assume that the reader is familiar with stochastic analysis on manifolds.Classical references in this context are e.g.[23,22,15,21] (see also [5] for a very brief summary the notions relevant in the sequel).Let (Ω, F , F * , P) be a filtered probability space which satisfies the usual conditions and which for every x ∈ M carries an adapted Brownian motion denotes the lifetime of X x (noting that ζ x = ∞ a.s., if for example Ric ≥ −a for some a > 0).Given a metric vector bundle E → M with metric connection, let denote the (pathwise orthogonal) parallel transport with respect to ∇ along X x .We define continuous adapted processes with paths having a locally finite variation by x M .In addition, for every r > 0 let be the first exit time of X x from B(x, r).Note that ζ x > τ x r > 0 P-a.s.Let Here, •d denotes the Stratonovich stochastic differential, where Itô stochastic differentials will be denoted by d.The following definitions will be very convenient in the sequel: We define a set of processes A j (x, r, t, ξ) to be given by all bounded adapted process x M ⊗ Λ j T x M with locally absolutely continuous paths such that For every ℓ ∈ A j (x, r, t, ξ) we define the continuous semimartingale The proof of the following result follows the arguments of Theorem 4.1 from [14]: The following two well-known facts will be used in the proof of Theorem 2.2: Lemma 2.3.Let τ be a P-a.s.finite stopping time and Lemma 2.4 (Burkholder-Davis-Gundy inequality).For all 0 < q < ∞ there exists a constant C(q) < ∞ with the following property: if τ is a P-a.s.finite stopping time and is a continuous local martingale taking values in a finite dimensional Hilbert space H and staring from 0, then one has We start by noting that for all s ≥ 0 one has by Gronwall's lemma, and as Q x j and Q x j are invertible with x M , we also have (2.3) Using Itô's formula one shows that [14].Using (2.2), (2.3), the assumptions on ℓ, the Burkholder-Davis-Gundy inequality (the latter to estimate U (ℓ) ) and that X x takes values in a compact set on [0, t ∧ τ x r ], the process Y is in fact a true martingale by Lemma 2.3, in particular, Y has a constant expectation.Evaluating Y s at the times s = 0 and s = t ∧ τ x r and taking expectations, we get x M there exists a process ℓ ∈ A (x, r, t, ξ) such that for all 1 ≤ q < ∞ and all constants a ≥ 0 with Ric ≥ −a in B(x, r) one finds constants C q,m , C a,q,m < ∞ satisfying Proof.It is well-known (cf. the proof of Corollary 5.1 in [31]) how to construct a bounded adapted process Thus we may simply set ℓ s := k s ξ.
Proof.As already noted in the introduction, this semigroup domination is a well-known fact [4].Much more general statements, which do not require constant lower bounds on the potential, can be found in [20] and are referred to as Kato-Simon inequality there.
The following covariant Feynman-Kac formula is well-known in much more general situations [16,14] to hold a.e. in M; the point of the proof below (which is the usual one for compact M's) is that it identifies the smooth representative of e − t 2 ∆ j α pointwise on M: Proof.Note that the statement of above formula includes that the right-hand side coincides for all x ∈ M and not only for µ-a.e.x ∈ M with the smooth representative of e − s 2 ∆ j α.To prove the formula, we can assume that t > 0. Then the process ) is a continuous local martingale.Under the stated assumptions, using Lemma 2.6 and |Q x j (s)| ≤ e −as P-a.s.(by Gronwall's lemma), one finds so that Y is in fact a martingale by Lemma 2.3.Evaluating Y s at s = 0 and s = t and taking expectations proves the claim.
Lemma 2.8.Assume (1.3).Then there exists a constant C = C(A, m) > 0, such that for all j ∈ {1, . . ., m}, t > 0, Proof.In the sequel, C(a, . . . ) will denote a constant that only depends on a, . . ., and which may differ from line to line.Let t > 0, r > 0, x ∈ M, ξ ∈ T * x M ⊗ Λ j T * x M be arbitrary and pick ℓ ∈ A (x, r, t, ξ) as in Lemma 2.5.We set It follows from the covariant Feynman-Kac formula, the fact that the (inverse) damped parallel transport Q j // −1 is a multiplicative functional (cf.equation (61) in [16]) and the strong Markov property of Brownian motion, that and so since ℓ r is F t∧τ x r -measurable, the law of total expectation gives Thus (2.1) implies and we have Taking r → ∞, we have managed to construct C(A, m) < ∞, such that for all x ∈ M, t > 0, one has Being equipped with the latter a priori L ∞ bound, we can now prove the global Bismut derivative formula.To this end, for fixed Theorem 2.9 (Global Bismut derivative formula).Assume (1.3).For every t > 0, x ∈ M , ξ ∈ T * x M ⊗ Λ j T x M , and every α ∈ Γ L 2 ∩C ∞ ∩L ∞ (M, Λ j T * M ) one has As before, the process [14]).Using Lemma 2.8 we have and using the Burkholder-Davis-Gundy inequality as well as (2.2) and ( 2.3), one easily finds showing that Y is a martingale, and the global Bismut derivative formula follows from

Proof of Theorem 1.2
With E x,y t [•] denoting integration with respect to the Brownian bridge measures [17], the global Bismut derivative formula together with the disintegration property valid for all Borel-measurable (vector-valued) functions Ψ on the space of continuous paths C([0, t], M ), one has Furthermore, using the time reversal property and the defining relation of the Brownian bridge [17] we have and using Cauchy-Schwarz and estimating the stochastic integral using Burkholder-Davis-Gundy, keeping in mind that X x | [0,t] is a semimartingale under the Brownian bridge measure [16], where we have used the Li-Yau estimate e C(A,m)s for all s > 0, x 1 , x 2 ∈ M , twice, and e − t 2 ∆ (x, z)e − t 2 ∆ (z, y)dµ(z) = e −t∆ (x, y).
Likewise, we have By interpolation, it is enough to prove (4.1) for p = 1 and p = ∞.However, and likewise, The volume comparison inequality (VC ǫ ) with small enough ǫ implies that there exist constants C 1 , C 2 > 0 such that for all x, y ∈ M and all t > 0, Hence, (4.1) will follow from the estimate: there is a constant C > 0 such that Then, using (LVD), one gets Using the elementary inequality e C2 i+1 √ t = e C ǫ t e −ǫ4 i with ǫ = c 2 , we arrive to which completes the proof of (4.2).

Proof of Corollary 1.4 and Theorem 1.7
In this section, we explain how one can use the heat kernel estimates (UE), (∇UE), (dUE) and (d † UE) in order to get results for the Riesz transforms ∇( ∆ j +λ) −1/2 and (d j +d † j−1 )( ∆ j +λ) −1/2 (i.e.prove Corollary 1.4 and Theorem 1.7 respectively).The idea of the proof is to follow closely the proof of [9,Theorem 1.2], where a result for the localized scalar Riesz transform d(∆+λ) −1/2 is proved.The proof is based on the Calderón-Zygmund decomposition and kernel estimates, which follow from the assumed heat kernel estimates (UE), (∇UE), (dUE) and (d † UE).However we feel that in the proof of [9,Theorem 1.2] the issue of localization may have been overlooked a little: there, it is wrongly asserted that (LVD) implies that every open ball of radius 1 in M is a doubling space, with a doubling constant that can be chosen independently of the ball; actually, this property depends on the geometry of balls, and not only on the validity of (LVD) in the whole M , and we don't see why it should hold in the context of [9,Theorem 1.2].In order to clarify the matter, we decided to give full proofs for the localization procedure that we use.The first ingredient needed in our proof is a localized Calderón-Zygmund decomposition f = g + b for a smooth section f ∈ Γ(M, Λ j T * M ) which has support inside a ball B = B(x, 1).This decomposition holds thanks to the local doubling assumption (LVD).More precisely, the version of the Calderón-Zygmund decomposition we need is the following: Lemma 5.1.Let E → M be a Riemannian vector bundle, where M is locally doubling.Then there is a constant C > 0, which depends only on the local doubling constant, with the following property: for every ball B = B(x, 1), every u ∈ Γ C ∞ (M, E ) with support inside B, and every 0 < λ < C µ(B) B |u|, there exists a countable collection of balls (B i ) i∈I , of integrable sections (2) the balls (B i ) i∈I have the finite intersection property: there is N ∈ N such that for every i ∈ N, ( ) For all i ∈ I, b i has support inside B i , and Furthermore, as a consequence of ( 2), ( 3) and ( 5), there holds for some constant C: The proof of this version of the Calderón-Zygmund decomposition closely follows the classical one, with three differences: firstly, since one has only local doubling but not doubling, one has to use a modified maximal function M, defined as follows: where r(B) denotes the radius of the ball B. The particular value 8 in the definition of M is chosen for later technical purposes (see the proof of Lemma 5.14).Note that local doubling implies that M is weak type (1, 1) and bounded on L p for all p ∈ (1, ∞], as follows from a careful inspection of the proof of [29, Theorem 1 p. 13] and the fact that the definition of M involves only balls with bounded radii.Secondly, in the Calderón-Zygmund decomposition localized in the ball B, the balls B i do not have to be included inside the ball B, only inside 2B.Lastly, the fact that we are dealing here with sections of a vector bundle instead of mere functions: this does not create any real difficulty and the standard arguments apply mutatis mutandis if one puts norms instead of absolute values everywhere it is needed.A detailed proof of Lemma 5.1 is presented in Appendix C. Let us now present the main steps of the proof of Corollary 1.4 and Theorem 1.7, following closely the approach of [9, Theorem 1.2].Let T be either ∇( ∆ j + λ) −1/2 or (d j + d † j−1 )( ∆ j + λ) −1/2 .We start with boundedness of T on L 2 : Lemma 5.2.For all κ > 0 the operator where C only depends on λ, A, m.
The first estimate is a simple consequence of the functional calculus: since the Dirac operator D := d + d † acting on smooth, compactly supported differential forms, is essentially self-adjoint on M , it follows that for all g in the domain D 2 = ∆ (which is included in the domain of D), Apply the above inequality to g = (D 2 + κ) −1/2 f , which is the domain of D 2 by functional calculus, we obtain (5.2) with C = 1.Let us now prove (5.3).Recall that since M is complete, the operator ∇ † ∇ acting on smooth compactly supported differential forms is essentially self-adjoint, associated with the quadratic form (u, v) → ∇u, ∇v .In particular, if Hence, for such a g, using that ||V j || ∞ ≤ A ′ , where Take g = ( ∆ j + κ) −1/2 f , which is in the domain of ∆ j : indeed, writing which can be done, since being smooth and compactly supported, f is in the domain of ∆ j , one has g = ( ∆ j + 1) −1 ( ∆ j + 1) −1/2 ( ∆ j + 1)f, so that g is in the domain of ∆ j by functional calculus.It follows that where we have used that ||( ∆ j + κ) −1/2 || 2,2 ≤ κ −1/2 by functional calculus.This proves (5.3).
Let us now come to the actual proof of Corollary 1.4 and Theorem 1.7: given the result of Lemma 5.2 and using interpolation, one sees that it is enough to prove that T is bounded from Γ L 1 (M, Λ j T * M ) to the space of weakly integrable sections Γ L 1 w (M, Λ j T * M ), that is: one can find a constant C > 0 such that for all f ∈ Γ L 1 (M, Λ j T * M ) and all λ > 0, (5.4) By a density argument, it is enough to prove it for f smooth with compact support.So, take such an f , and fix λ > 0. Take (x j ) j∈N a maximal 1-separated subset, hence the balls B(x j , 1) cover M , while the balls B(x j , 1 2 ) are disjoint.Local doubling then implies that the balls B(x j , 1) have the finite intersection property.Let (φ j ) j∈N be a smooth partition of unity associated to the covering of M by the balls B(x j , 1), and let f j := φ j f .The fact that the covering has the finite intersection property implies that for some constant C > 0, Hence, it is enough to prove (5.4) for f j (with a constant independent of j and f ).In what follows, we therefore assume that j ∈ N is fixed, and let u = f j and B = B(x j , 1).We have two cases, according to whether or not (here, C is the constant in Lemma 5.1).We first treat the case where λ ≤ C µ(B) B |u|, for which there are two steps: first, show that (5.5) µ({x and then show that For (5.5), notice that {x ∈ 2B : where we have used successively (LVD) and the assumption on λ.This proves (5.5).Now let us prove (5.6).By the Markov inequality, we see that (5.6) follows from the L 1 estimate: (5.7) In turns, (5.7) can be proved as in [9, p. 1163], using the heat kernel estimates (∇UE), (dUE) and (d † UE) respectively.Now, we deal with the case λ > C µ(B) B |u|.In this case one can use the Calderón-Zygmund decomposition u = g + i∈I b i from Lemma 5.1.Let r i be the radius of B i , and let t i = r 2 i .Then, write where we recall that χ A denotes the indicator function of the Borel set A. The weak L 1 estimate (5.4) will follow from the four estimates: Firstly, the fact that |g| ≤ Cλ a.e. and that T is bounded on L 2 leads to which shows (5.8).Concerning (5.9), the same argument using the L 2 boundedness of T shows that (5.9) will follow from the L 2 estimate: (5.12) Let B i = B(y i , r i ).Point (5) of the Calderón-Zygmund decomposition together with the heat kernel estimate (UE) and the fact that t i ≤ 2 (since B i ⊂ 2B) imply that where in the one before last line we have used local doubling (LVD) with t i ≤ 2, and where we have set V (z, r) := µ(B(z, r)), z ∈ M, r > 0.
In order to prove (5.12), it is then enough to prove that To estimate the above L 2 norm, we dualize against v ∈ Γ L 2 (M, Λ j T * M ) with support inside 3B; we have by Fubini, Next we proof that for every i ∈ I and y ∈ B i one has Also, let N ∈ N be the smallest integer so that 2 N +1 √ t i ≥ 4.Then, By definition of N , we have for every k ≤ N , 2 k+1 √ t i ≤ 8, therefore by local doubling, and it follows by definition of M that 2 km e −c2 k Mv(y) ≤ CMv(y), and (5.14) is proved.
According to the remark made immediately after the definition of M, (LVD) implies that the operator M is bounded on L 2 , so using Hölder, (3) from the Calderón-Zygmund decomposition, and (5.14) we get that Dividing by ||v|| 2 2 and taking the sup over all non-zero v, we obtain which proves (5.13), hence (5.9).It thus remains to prove (5.10) and (5.11).It relies on the following lemma: Then there is a constant C = C(A, m) > 0, such that for every t > 0, s > 0 and y ∈ M , Then there is a constant C = C(A, m) > 0, such that for every t > 0, s > 0 and y ∈ M , Proof.For the integral involving ∇e −s ∆ j (x, y), it is an immediate consequence of Corollary 1.3, part II with the choice p = 1.The proof for the second integral follows along the same lines, using (dUE) and (d † UE) instead of (∇UE) for the proof of the weighted estimate analogous to Corollary 1.3, part II.
The estimates (5.10) and (5.11) follow from Lemma 5.3, in a fashion that is identical to the proof of [9, Theorem 1.2], and thus whose details will be omitted.Finally, all four estimates (5.8), (5.9), (5.10) and (5.11) are proved.This concludes the proof of Theorem 1.7 and Corollary 1.4.

6.
Proof of Theorem 1.9 We prepare the proof with the following estimate from complex analysis can be found in [10]: valid for all α 1 with support in E and α 2 with support in F has been proved in [20].If we apply Phragmen-Lindelöf's inequality with noting that one may pick A = α 1 2 α 2 2 because of ∆ j ≥ 0 so that e −z ∆ j is a contraction, we get the bound The latter inequality is equivalent to the statement of step 1.
Step 2: One has where C < ∞ is a universal constant.Proof of step 2: The asserted estimate is equivalent to To see (6.5), we first note that by applying the Phragmen-Lindelöf estimate to the estimate (6.4) we get the bound valid for all z with ℜz > 0. By Cauchy's integral formula we have Since by (6.6) we have this proves step 2.
Step 3: One has , and so Then we have Using step 1 and step 2 and

Proof of Theorem 1.10
The assumption is equivalent to the following operator norm bound (7.1)

Ct
for disjoint Borel subsets E, F ⊂ M with compact closure and every t > 0. Let A t = √ t∇e −t ∆ j .Fix an arbitrary E as above, fix t > 0, and define So, we have that the operator A t 1 E is bounded in Γ L 2 (M, Λ j T * M ), uniformly with respect to t > 0 and the set E. Taking an exhaustion of M by compacts E n ր M , we obtain that A t is bounded in Γ L 2 (M, Λ j T * M ), uniformly in t > 0. Let α ∈ Γ L 2 (M, Λ j T * M ).We compute, using ∆ j = ∇ † ∇ + V j , 0 ≤ || √ t∇e −t ∆ j α|| 2 2 = t∇ † ∇e −t ∆ j α, e −t ∆ j α = t ∆ j e −t ∆ j α, e −t ∆ j α − t M (V j e −t ∆ j α, e −t ∆ j α)dµ.
The first term on the right is bounded, uniformly in t > 0 by the spectral theorem.Also, the left hand side is non-negative and bounded.Let us take α ∈ Ker L 2 ( ∆ j ), then e −t ∆ j α = α, and we get that t M (V j α, α)dµ is non-positive and bounded, which can happen only if M (V j α, α)dµ = 0.
We now deal with the sum ∞ k=0 .For every k ≥ 0 and i ∈ A k , one has by definition of A k that  This concludes the proof of (B.1), and the proof of Lemma A.5.
Taking the supremum over all balls B with radius r( B) ≤ 8 containing x, one gets Mu(x) ≤ λ, and consequently x / ∈ Ω.Therefore, we have proved that Ω ⊂ 2B.For all x ∈ Ω, let r x := 1  10 ̺(x, M \ Ω) and B x := B(x, r x ), so that B x ⊂ Ω, and Ω = x∈Ω B x .Since the radii of the balls B x are uniformly bounded, there exists a denumerable collection of points (x i ) i≥1 ∈ Ω such that the balls B x i are pairwise disjoint and Ω = i≥1 5B x i .For all i, write s i := 5r x i ≤ 1 and let B i = B(x i , s i ).Notice that B i ⊂ 2B for all i.Furthermore, the balls 1  5 B i being disjoint together with local doubling entail that the covering by balls B i has the finite intersection property (property (2) of the Calderón-Zygmund decomposition).And by construction also, 3B i ∩ F = ∅ for every i.Let (χ i ) i≥1 be a partition of unity of Ω, subordinated to the covering (B i ) i≥1 .Then, define b i = uχ i , so that b i has support in B i .We also let (the above sum in fact contains at every point only a finite number of terms, thanks to the finite intersection property of the covering).The Lebesgue differentiation theorem implies that |g| ≤ λ a.e. on F , proving point (4) of the Calderón-Zygmund decomposition.Next, since r(3B i ) ≤ 3 ≤ 8, the fact that 3B i ∩ F = ∅ implies that 1 µ(3B i ) 3B i |u| ≤ λ.
Local doubling then implies that where in the last line we have used the fact that M is weak type (1,1).This proves point (3) of the Calderón-Zygmund decomposition, and this concludes the proof of Lemma 5.1.

Ct 8ǫ e 2ǫr 2 t
, e C √ t ≤ C ′ e Ct , one easily gets (VC ǫ ) (with a different value of the constant C).