$L^p$-Analysis of the Hodge--Dirac operator associated with Witten Laplacians on complete Riemannian manifolds

We prove $R$-bisectoriality and boundedness of the $H^\infty$-functional calculus in $L^p$ for all $1<p<\infty$ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on $k$-forms.


Introduction
The Witten Laplacian was introduced by Witten [55] as a deformation of the Hodge Laplacian on a complete Riemannian manifold M and has been subsequently studied by many authors; see [9,13,15,23,26,29,30,44,45,46,56] and the references cited therein. The Witten Laplacian associated with a smooth strictly positive function ρ : M → R is the operator , where ∆ = ∇ * ∇ is the (negative) Laplace-Beltrami operator and ∇ is the gradient. Identifying functions with 0-forms, we have where d ρ is the L 2 -realisation of the exterior derivative d with respect to the measure m(dx) = ρ(x) dx on M , and d * ρ is the adjoint operator. The representation (1.1) can be used to define the Witten Laplacian for k-forms for k = 0. In the special case M = R n and ρ(x) = exp(− 1 2 |x| 2 ), L ρ corresponds to the Ornstein-Uhlenbeck operator.
Let m(dx) = ρ(x) dx is the weighted volume measure on M . Generalising the celebrated Meyer inequalities for the Ornstein-Uhlenbeck operator, Bakry [9] proved boundedness of the Riesz transform ∇L −1/2 ρ on L p (M, m) for all 1 < p < ∞ under a curvature condition on M . An extension of this result to the corresponding L p -spaces of k-forms is contained in the same paper. These results have been subsequently extended into various directions. As a sample of the extensive literature on this topic we mention [15,44,45,46,56] (for the Witten Laplacian); see also [3,4,10,19,37,42,47,49,52,54] (for the Laplace-Beltrami operator), [17,31,51] (for the Hodge-de Rham Laplacian), and [11] (for sub-elliptic operators).
The aim of the present paper is to develop Bakry's result along a different line by analysing the Hodge-Dirac operator from the point of view of its functional calculus properties. Our main result can be stated as follows (the relevant definitions are given in the main body of the paper). Theorem 1.1. If M has non-negative Bakry-Emery Ricci curvature on k-forms for all 1 ≤ k ≤ n, then the Hodge-Dirac operator D ρ is R-bisectorial and admits a bounded H ∞ -calculus in L p (ΛT M, m) for all 1 < p < ∞.
By standard arguments (cf. [8]), the boundedness of the H ∞ -calculus of D ρ implies (by considering the operator sgn(D ρ ), which is then well defined through the functional calculus) the boundedness of the Riesz transform D ρ L −1/2 ρ = sgn(D ρ ). As such our results may be thought of as a strengthening of those in [9].
In the unweighted case ρ ≡ 1, the second assertion of Theorem 1.1 is essentially known, although we are not aware of a place where it is formulated explicitly or in some equivalent form. It can be pieced together from known results as follows. Firstly, [6,Theorem 5.12] asserts that the unweighted Hodge-Dirac operator D has a bounded H ∞ -calculus on the Hardy space H p (ΛT M ), even for 1 ≤ p ≤ ∞, provided the volume measure has the so-called doubling property. By the Bishop comparison theorem (see [12]), this property is always satisfied if M has nonnegative Ricci curvature. Secondly, for 1 < p < ∞, this Hardy space is subsequently identified in [6,Theorem 8.5] to be the closure in L p (ΛT M ) of the range of D, provided the heat kernel associated with L satisfies Gaussian bounds on k-forms for all 0 ≤ k ≤ n. When M has non-negative Ricci curvature, such bounds were proved in [43] for 0-forms, i.e., for functions on M . The bounds for k-forms then follow, under the curvature assumptions in the present paper, via pointwise domination of the heat kernel on k-forms by the heat kernel for 0-forms (cf. (3.7) below). Modulo the kernel-range decomposition decomposition L p (ΛT M, m) = N(D) ⊕ R(D) (which follows from R-bisectorialy proved in the present paper, but could also be established on the basis of other known results), this gives the boundedness of the H ∞ -calculus in L p (ΛT M, m) in the unweighted case.
In the weighted case, this approach cannot be pursued due to the absence of doubling and Gaussian bounds. Instead, our approach exploits the fact, proved in [56], that the non-negativity of the Bakry-Emery Ricci curvature implies, among other things, square function estimates on k-forms.
The analogue of Theorem 1.1 for the Hodge-Dirac operator associated with the Ornstein-Uhlenbeck operator has been established, in a more general formulation, in [48]. The related problem of the L p -boundedness of the H ∞ -calculus of Hodge-Dirac operators associated with the Kato square root problem was initiated by the influential paper [8] and has been studied by many authors [7,24,32,33,34,51].
The organisation of the paper is as follows. After a brief introduction to R-(bi)sectorial operators and H ∞ -calculi in Section 2, we introduce the Witten Laplacian L ρ in Section 3 and recall some of its properties. Among others we prove that it is R-sectorial of angle less than 1 2 π and admits a bounded H ∞ -calculus in L p for 1 < p < ∞. In Section 4 this result, together with the identity D 2 ρ = L ρ , is used to prove the corresponding assertions for the Hodge-Dirac operator D ρ .
On some occasions we will use the notation a b to signify that there exists a constant C such that a ≤ Cb. To emphasise the dependence of C on parameters p 1 , p 2 , . . . , we shall write a p1,p2,... b. Finally we write (respectively, p1,p2,... ) if both a b and b a (respectively, a p1,p2,... b and b p1,p2,... a) hold.

R-(Bi)sectorial operators and the H ∞ -functional calculus
In this section we present a brief overview of the various notions from operator theory used in this paper.
2.1. R-boundedness. Let X and Y be Banach spaces and let (r j ) j≥1 be a sequence of independent Rademacher variables defined on a probability space (Ω, P), i.e., P(r j = 1) = P(r j = −1) = 1 2 for each j. A collection of bounded linear operators T ⊆ L (X, Y ) is said to be R-bounded if there exists a C ≥ 0 such that for all M = 1, 2, . . . and all choices of x 1 , . . . , x M ∈ X and T 1 , . . . , T M ∈ T we have where E denotes the expectation with respect to P. By considering the case M = 1 one sees that every R-bounded family of operators is uniformly bounded. In Hilbert spaces the converse holds, as is easy to see by expanding the square of the norm as an inner product and using that Er m r n = δ mn .
Motivated by certain square function estimates in harmonic analysis, the theory of R-boundedness was initiated in [18] and has found widespread use in various areas of analysis, among them parabolic PDE, harmonic analysis and stochastic analysis. We refer the reader to [21,35,36,40] for detailed accounts.

Sectorial operators.
For σ ∈ (0, π) we consider the open sector . The least angle of sectoriality is denoted by ω + (A). If A is sectorial of angle σ ∈ (0, π) and the set {λ(λ − A) −1 : λ / ∈ Σ + ϑ } is R-bounded for all ϑ ∈ (σ, π), then A is said to be R-sectorial of angle σ. The least angle of R-sectoriality is denoted by ω + R (A). Remark 2.1. We wish to point out that most authors (including [21,36,40]) impose the additional requirements that A be injective and have dense range. In the setting considered here this would be inconvenient: already in the special case of the Ornstein-Uhlenbeck operator, the kernel is non-empty. It is worth noting, however, (see [28, Proposition 2.1.1(h)]) that a sectorial operator A on a reflexive Banach space X induces a direct sum decomposition The part of A in R(A) is sectorial and injective and has dense range. Thus, A decomposes into a trivial part and a part that is sectorial in the more restrictive sense of [21,36,40]. Since we will be working with L p -spaces in the reflexive range 1 < p < ∞ the results of [21,36,40] can be applied along this decomposition.
The typical example of a sectorial operator is the realisation of the Laplace operator ∆ in L p (R n ), 1 ≤ p < ∞, and this operator is R-sectorial if 1 < p < ∞.
Typical examples of bisectorial operators are ±i d/ dx in L p (R) and the Hodge-

2.4.
The H ∞ -functional calculus. In a Hilbert space setting, the H ∞ -functional calculus was introduced in [50]. It was extended to the more general setting of Banach spaces in [20]. For detailed treatments we refer the reader to [21,28,36,40].
Let H ∞ (Σ + σ ) be the space of all bounded holomorphic functions on Σ + σ , and let H 1 (Σ + σ ) denote the space of all holomorphic functions ψ : If A is a sectorial operator and ψ is a function in H 1 (Σ + σ ) with 0 < ω + (A) < σ < π, we may define the bounded operator ψ(A) on X by the Dunford integral where ω + (A) < ν < σ and ∂Σ + ν is parametrised counter-clockwise. By Cauchy's theorem this definition does not depend on the choice of ν.

The Witten Laplacian
Let us begin by introducing some standard notations from differential geometry. For unexplained terminology we refer to [27,41].
Throughout this paper we work on a complete Riemannian manifold (M, g) of dimension n. The exterior algebra over the tangent bundle T M is denoted by Smooth sections of Λ k T M are referred to as k-forms. We set where C ∞ c (Λ k T M ) denotes the vector space of smooth, compactly supported kforms. The inner product of two k-forms dx i1 ∧ · · · ∧ dx i k and dx j1 ∧ · · · ∧ dx j k is defined, in a coordinate chart (U, x), as This definition extends to general k-forms by linearity. For smooth sections ω, η of ΛT M , say ω = n k=0 ω k and η = n k=0 η k , we define and we write |ω| := (ω · ω) 1/2 .
We now fix a strictly positive function ρ ∈ C ∞ (M ) and consider the measure to be the Banach space of all measurable k-forms for which the norm is finite, identifying two such forms when they agree m-almost everywhere on M . Equivalently, we could define this space as the completion of C ∞ c (Λ k T M ) with respect to the norm · p . Finally, we define and endow this space with the norm · p defined by ω p = n k=0 ω k p p , where ω = n k=0 ω k for k-forms ω k . In the case of p = 2, we will denote the Here, the subscript ρ indicates the dependence of the inner product on the function ρ. When considering the L 2 (Λ k T M, dx) inner product, we will simply write ·, · .
The exterior derivative, defined a priori only on C ∞ c (ΛT M ), is denoted by d. Its restriction as a linear operator from As a densely defined operator from L 2 (Λ k T M, m) to L 2 (Λ k+1 T M, m), d k is easily checked to be closable. With slight abuse of notation, its closure will again be denoted by d k . Its adjoint is well defined as a closed densely defined operator from L 2 (Λ k+1 T M, m) to L 2 (Λ k T M, m). We will denote this adjoint operator by It would perhaps be more accurate to follow the notation used in the Introduction and denote the operators d, d k and δ k by d ρ , d ρ,k and d * ρ,k respectively, to bring out their dependence on ρ, but this would unnecessarily burden the notation.
In Lemma 3.3 below we will state an identity relating δ k to the operator d * k , the adjoint of d k with respect to the volume measure dx. For this purpose we need the following definition. Let k ∈ {1, . . . , n}. Let ω be a k-form and X a smooth vector field. We define ι(X)ω as the (k − 1)-form given by for smooth vector fields Y 1 , . . . , Y k−1 . We refer to ι as the contraction on the first entry with respect to X. The next two lemmas are implicit in [9]; we include proofs for the reader's convenience.
where df * is the smooth vector field associated to the 1-form df by duality with respect to the Riemannian metric g.
Proof. Working in a coordinate chart (U, x), by linearity it suffices to prove the Here the third line follows by recalling that the inner product can be seen as the determinant of a matrix, and that we can develop this determinant to the row of df . The last equality follows by simply expanding ι(df * )ω.
where we used that k-forms are linear over C ∞ functions to arrive at the second line. The last equality follows from the previous lemma. The claim now follows.
In the special case that ρ ≡ 1, we recover the Hodge-de Rham Laplacian Using Lemma 3.3 for 1-forms, we obtain the following identity for the Witten Laplacian on functions: where the second identity follows by duality via the Riemannian inner product. The Bochner-Lichnérowicz-Weitzenböck formula (cf. [9, Section 5]) asserts that where Q k is a quadratic form which depends on the Ricci curvature tensor (see [9,Section 5]). Notice that in [9] there is an additional term 1 k! , which comes from the fact that we define |∇ω| 2 in a similar way as for k-forms, while [9] defines it in the sense of tensors.
An analogue of (3.1) may be derived for the Witten Laplacian as follows. Firstly, if we expand the above definitions using Lemma 3.3, we can express L k in terms of ∆ k : Obviously, when k = 0 the second term on the right-hand side vanishes, while for k = n the last term vanishes. Inserting (3.2) into equation (3.1) we obtain the following variant of the Bochner-Lichnérowicz-Weitzenböck formula: As Q k only depends on the Ricci curvature tensor, we see that Q k only depends on the Ricci curvature tensor and the positive function ρ. One has Q 0 = 0, while for k = 1 one has Q 1 (ω, ω) = Ric(ω * , ω * ) − ∇∇ log ρ(ω * , ω * ) (see [9]). The latter is usually referred to as the Bakry-Emery Ricci curvature. In what follows, we will refer to Q k as the Bakry-Emery Ricci curvature on k-forms.
3.1. The main hypothesis. We are now ready to state the key assumption, which is a special case of the one in Bakry [9]: We assume non-negativity of the Bakry-Emery Ricci curvature, rather than its boundedness from below (as done in [9]), as in the case of (negative) lower bounds one obtains inhomogeneous Riesz estimates only (see [9, Theorem 4.1,5.1]). Also note (see [9]) that to obtain boundedness of the Riesz transform on k-forms, not only does one need non-negativity of Q k , but also of Q k−1 and Q k+1 .
As an example, we will show what this assumption means in the case of M = R n . The result of our computation is likely to be known, but for the reader's convenience we provide the details of the computation. Note that the case k = 1 is much easier due to the simple coordinate free expression for the Bakry-Emery Ricci curvature Q 1 . In particular, we will see that this assumption is satisfied in the case of the Ornstein-Uhlenbeck operator on R n . Example 3.6. Let M = R n with its usual Euclidean metric and consider a smooth strictly positive function ρ on R n . Let k ∈ {1, 2 . . . , n}. We will derive a sufficient on ρ in order that Q k (ω, ω) ≥ 0 for all k-forms ω.
As to the first term, from |ω| 2 = f 2 we obtain Turning to the second term, Hence Computing the final term, we have From this it follows that Noting that only the terms with i = j can contribute a non-zero contribution to the inner product with ω, we obtain Collecting everything, we find that We thus see that Q k (ω, ω) ≥ 0 precisely when k i=1 ∂ 2 i (log ρ) ≤ 0. Recalling the simplification for notational purposes, we conclude that Q k (ω, ω) ≥ 0 for all k-forms ω precisely if for all 1 ≤ i 1 < · · · < i k ≤ n it holds that In the special case ρ(x) = e − 1 2 |x| 2 which corresponds to the Ornstein-Uhlenbeck operator this condition is clearly satisfied. Indeed, for any j = 1, . . . , n we have ∂ 2 j (log ρ) = −1.
We can use the previous example to consider a more general situation.
Example 3.7. Let (M, g) be a complete Riemannian manifold. Suppose the quadratic formQ k depending solely on the Ricci curvature is bounded from below for all 1 ≤ k ≤ n, i.e., there exist constants a 1 , . . . , a n such that for all k-forms ω we have Q k (ω, ω) ≥ a k |ω| 2 .
Fix k ∈ {1, . . . , n}. In normal coordinates around a point p ∈ M , the expression for Q k (ω, ω) at p reduces to the one of the previous example. Consequently, Q k (ω, ω) ≥ 0 for any k-form ω if for any p ∈ M and any 1 ≤ i 1 < · · · < i k ≤ n one has k r=1 ∂ 2 ir (log ρ)(p) ≤ a k , where the last expression is in normal coordinates around p.

3.2.
The heat semigroup generated by −L k . We return to the general setting described at the beginning of this section. For each k = 0, 1, . . . , n the operator L k is essentially self-adjoint on L 2 (Λ k T M, m) (see [9,54] for the case ρ ≡ 1 and [56]) and satisfies L k ω, ω ρ = |d k ω| 2 + |δ k−1 ω| 2 ≥ 0 for all smooth k-forms ω. Consequently, its closure is a self-adjoint operator on L 2 (Λ k T M, m). With slight abuse of notation we shall denote this closure by L k again. By the spectral theorem, −L k generates a strongly continuous contraction semigroup From now on we assume that Hypothesis 3.5 is satisfied. As was shown in [9,56], under this assumption the restriction of (P k t ) t≥0 to L p (Λ k T M, m) ∩ L 2 (Λ k T M, m) extends to a strongly continuous contraction semigroup on L p (Λ k T M, m) for any p ∈ [1, ∞). These extensions are consistent, i.e., the semigroups (P k t ) t≥0 on L pi (Λ k T M, m), i = 1, 2, agree on the intersection L p1 (Λ k T M, m) ∩ L p2 (Λ k T M, m).
The infinitesimal generator of the semigroup (P k t ) t≥0 in L p (Λ k T M, m) will be denoted (with slight abuse of notation) by −L k and its domain by D p (L k ).
As an operator acting in L 2 (Λ k T M, m), L k is the closure of an operator defined a priori on C ∞ c (Λ k T M ) and therefore the inclusion C ∞ c (Λ k T M ) ⊆ D 2 (L k ) trivially holds. The definition of the domain D p (L k ) is indirect, however, and based on the fact that L k generates a strongly continuous semigroup on L p (Λ k T M, m). Nevertheless we have: Proof. We follow the idea of [48,Lemma 4.8]. Pick an arbitrary k-form ω ∈ C ∞ c (Λ k T M, m). Then ω ∈ D 2 (L k ) (by definition of L k on L 2 (Λ k T M, m)) and also ω ∈ L p (Λ k T M, m). Since L p (Λ k T M, m) is a reflexive Banach space, a standard result in semigroup theory states that in order to show that ω ∈ D p (L k ) it suffices to show that lim sup (see, e.g., [14]). Note that 1 , we can interpret the integral on the right-hand side as a Bochner integral in the Banach space L p (Λ k T M, m) (see [35,Chapter 1]). Consequently we may estimate But then lim sup t↓0 1 t P k t ω − ω p ≤ L k ω p < ∞. This proves the claim.
By the Stein interpolation theorem [53, Theorem 1 on p.67], for p ∈ (1, ∞) and k = 0, 1, . . . , n the mapping t → P k t extends analytically to a strongly continuous L (L p (Λ k T M, m))-valued mapping z → P k z defined on the sector Σ ωp with ω p = π 2 (1 − |2/p − 1|). On this sector the operators P k z are contractive. This implies that L k is sectorial of angle ω p .
As explained in [56, p. 625] it follows from the general theory of Dirichlet forms [25] that there exists a Markov process (X t ) t≥0 such that Here E x denotes expectation under the law of the process (X t ) t≥0 starting almost surely in x ∈ M . Using this together with Hypothesis 3.5 (this corresponds to the assumption made in [56, eq. (1.2)], see the explanation preceding the proof of theorem 3.12), it is then shown in [56, Proposition 2.3] that there exists a Markov process (V t ) t≥0 such that Here, E v denotes expectation under the law of the process (V t ) t≥0 starting almost surely in v ∈ M .
As a consequence of (3.6) the operators P 0 t are positive, in the sense that they send non-negative functions to non-negative functions. This, together with the following lemma, allows us to show that L k is in fact R-sectorial of angle < 1 2 π.
, where (r i ) i is a Rademacher sequence; the implicit constant only depends on p. Proof.
Step 1 -First we assume that ω 1 , . . . , ω N are supported in a single coordinate chart (U, x). With slight abuse of notation we will identify each ω i | U with the corresponding C d k -valued function on U ; here, for p ∈ U . By using the Kahane-Khintchine inequality, .
Next, by the square function characterisation of Rademacher sums for C d k -valued functions, .
Step 2 -We now turn to the general case. Let (φ U ) U∈U be a partition of unity subordinate to a collection of coordinate charts U covering M . Then, using Fubini's theorem and the result of Step 1, .
Here, R denotes the R-bound of the set {(I + sA) −1 : s > 0}. This gives the R-boundedness of the set {(I + λB) −1 : Re λ > 0}. A standard Taylor expansion argument allows us to extend this to the R-boundedness of the set {(I + λB) −1 : λ ∈ Σ ν } for some ν > 1 2 π. We now return to the setting considered at the beginning of this section. Combining the preceding lemmas we arrive at the following result.
Proposition 3.11 (R-sectoriality of L k ). Let Hypothesis 3.5 be satisfied. For all 1 < p < ∞ and k = 0, 1, . . . , n, the operator L k is R-sectorial on L p (Λ k T M, m) with angle ω + R (L k ) < 1 2 π. Proof. Fix 1 < p < ∞. As we have already noted, −L k generates a strongly continuous analytic contraction semigroup on L p (Λ k T M ). By [9,56], these semigroups satisfy the pointwise bound for all ω ∈ L p (Λ k T M, m). Since the semigroup generated by −L 0 is positive, L 0 is R-sectorial by [38,Corollary 5.2]. Lemma 3.9 then implies that L k is R-sectorial, of angle < 1 2 π. We are now ready to state our first main result.
Theorem 3.12 (Bounded H ∞ -calculus for L k ). Let Hypothesis 3.5 be satisfied. For all 1 < p < ∞ and all k = 0, 1, . . . , n, the operator L k has a bounded H ∞ -calculus on L p (Λ k T M, m) of angle < 1 2 π. For k = 0 the proposition is an immediate consequence of [38, Corollary 5.2]; see [16] for a more detailed quantitative statement. For k = 1, . . . , n this argument cannot be used and instead we shall apply the square function estimates of [56]. To make the link between the definitions used in that paper and the ones used here, we need to make some preliminary remarks.
In [56], the Hodge Laplacian on k-forms is defined as This is motivated by the fact that on functions this operator agrees with ∆ k (see [27]). Similarly in [56] one defines Actually, the definition in [56] there differs notationally from (3.8) in that e −ρ is written for the strictly positive function that we denote by ρ. Define [56, eq. (1. 2)], recalling our convention of considering the negative Laplacian). We will show in a moment that so that Hypothesis 3.5 can be rephrased as assuming that ω · V k ω ≥ 0. This corresponds to the assumption made in [56,Eq. (1.4)]. Thus, the results from [56] may be applied in the present situation.
This can be simplified to Indeed, in a coordinate chart one has Noting that L 0 = L 0 , combining (3.3) and (3.10) gives ω · V k ω = Q k (ω, ω) as desired.
Proof of Theorem 3.12. Fix 1 < p < ∞. By Proposition 3.11, L k is R-sectorial on Using the substitution t = s 2 we see that , where E k 0 denotes projection onto the kernel of L k . By a routine density argument (using that convergence in the mixed L p (L 2 )-norm implies almost everywhere convergence along a suitable subsequence) these inequalities extend to arbitrary k-forms ω ∈ L p (Λ k T M, m).
Now it is well known that for an R-sectorial operator, the square function estimate (3.11) implies the operator having a bounded H ∞ -calculus of angle at most equal to its angle of R-sectoriality (see [39] or [36,Chapter 10]).

The Hodge-Dirac operator
Throughout this section we shall assume that Hypothesis 3.5 is in force. Under this assumption one may check, using the Bochner-Lichnérowicz-Weitzenböck formula (3.3) instead of (3.1), that the results in [9, Section 5] proved for the special case ρ ≡ 1 carry over to general strictly positive functions ρ ∈ C ∞ (M ). Whenever we refer to results from [9] we bear this in mind. As in Remark 3.1 it would be more accurate to denote this operator by D ρ , but again we prefer to keep the notation simple.
With respect to the decomposition C ∞ c (ΛT M ) = n k=0 C ∞ c (Λ k T M ), D can be represented by the (n + 1) × (n + 1)-matrix Proof. For the reader's convenience we include the easy proof. Let (ω n ) n be a sequence in C ∞ c (ΛT M ) and suppose that ω n → 0 and Dω n → η in L p (ΛT M, m). Decomposing along the direct sum we find that ω k First consider 1 ≤ k ≤ n − 1, and pick φ ∈ C ∞ c (Λ k T M, m). By Hölder's inequality, This is justified since both ω k+1 n and φ are compactly supported and therefore belong to D q (δ k ), respectively D q (δ k−1 ), with 1 p + 1 q = 1. It follows that η k = 0 by density. The cases k = 0 and k = n are treated similarly. We conclude that η k = 0 for all k, so η = 0. With slight abuse of notation we will denote the closure again by D and write D p (D) for its domain in L p (ΛT M, m). The main result of this section asserts that, under Hypothesis 3.5, for all 1 < p < ∞ the operator D is R-bisectorial on L p (ΛT M, m) and has a bounded H ∞ -calculus on this space.
Since L k is sectorial on L p (Λ k T M, m), 1 < p < ∞, its square root is well defined and sectorial. Moreover we have By the choice of the sequence ω n the latter tends to 0 and consequently we have ω n → ω in D p (L 1/2 k ). The following result is essentially a restatement of [9,Theorem 5.1,Corollary 5.3] in the presence of non-negative curvature. The results in [9] are stated only for the case ρ ≡ 1 and given in the form of inequalities for smooth compactly supported k-forms.
and for all ω in this common domain we have Here, D k := d k + δ k−1 is the restriction of D as a densely defined operator acting from L p (Λ k T M, m) into L p (ΛT M, m).

Proof. We start by showing that
k ) by Lemma 4.3, we can find a sequence (ω i ) i of k-forms in this space converging to ω in D p (L 1/2 k ). By [9, Theorem 5.1] we then find, for all i, j, k ω j p which shows that (ω i ) i is Cauchy in D p (d k + δ k−1 ). By the closedness of d k + δ k−1 this sequence converges to some η ∈ D p (d k + δ k−1 ). Since both D p (L 1/2 k ) and D p (d k + δ k−1 ) are continuously embedded into L p (Λ k T M, m) we have ω i → ω and ω i → η in L p (Λ k T M, m), and therefore η = ω. This shows that ω ∈ D p (d k + δ k−1 ). To prove the estimate, by [9, Theorem 5.1] we obtain, for all i, The reverse inclusion and estimate may be proved in a similar manner. Now one uses that C ∞ c (Λ k T M ) is dense in D p (d k + δ k−1 ), d k + δ k−1 being the closure of its restriction to C ∞ c (Λ k T M ). One furthermore uses the estimate in [9, Corollary 5.3] which holds (with e = 0 in the notation of [9]) by Hypothesis 3.5. Finally, by definition of the norm on L p (ΛT M, m), for all ω ∈ C ∞ c (Λ k T M ) we have (4.1) Our proof of the R-bisectoriality of D will be based on R-gradient bounds to which we turn next. We begin with a lemma.
k ) by Lemma 4.3, we can find a sequence (ω i ) i of k-forms in this space converging to ω in D p (L 1/2 k ). By [9, Theorem 5.1] we then find, for all i, j, which shows that (ω i ) i is Cauchy in D p (d k ). By the closedness of d k we then find that this sequence converges to some η ∈ D p (d k ). As in the proof of Theorem 4.4 we show that ω = η. It follows that ω ∈ D p (d k ).
This proves the inclusion D p (L Thanks to the lemma, the operators k ω → δ k−1 ω are well defined, and by Theorem 4.4 combined with the equivalence of norms (4.1) they are in fact L p -bounded.
It also follows from the lemma that the operators d k (I + t 2 L k ) −1 and δ k−1 (I + t 2 L k ) −1 are well defined and L p -bounded for all t ∈ R; indeed, just note that . The next proposition asserts that these operators form an R-bounded family: and Proof. We will only prove that the first set is R-bounded. The R-boundedness of the other set if proved in exactly the same way.
In order to prove the R-bisectoriality of the Hodge-Dirac operator we need one more lemma, which concerns commutativity rules used in the computation of the resolvents of the Hodge-Dirac operator.
Similar identities hold with (I + t 2 L k+1 ) −1 replaced by (I + t 2 L k+1 ) −1/2 or P k t . Proof. We will only prove the first identity; the second is proved in a similar manner. The corresponding results for P k t can be proved along the same lines, or deduced from the results for the resolvent using Laplace inversion, and in turn the identities involving (I + t 2 L k+1 ) −1/2 follow from this.
For k-forms ω ∈ C ∞ c (Λ k T M, m) we have P k+1 t d k ω = d k P k t ω (see [9]). Here, the right-hand side is well defined as P k t ω ∈ D p (L k ) ⊆ D p (d k ) (which holds by analyticity of P k t ). Now pick ω ∈ D p (d k ) and let ω n ∈ C ∞ c (Λ k T M ) be a sequence to compute the three diagonals, as the other elements of the product clearly vanish. It is easy to see that the k-th diagonal element becomes (4.5) t 2 d k−2 δ k−2 (I + t 2 L k−1 ) −1 + (I + t 2 L k−1 ) −1 + t 2 δ k−1 d k−1 (I + t 2 L k−1 ) −1 = (I + t 2 L k−1 )(I + t 2 L k−1 ) −1 = I using that L k−1 = −( d k−2 δ k−2 + δ k−1 d k−1 ); obvious adjustments need to be made for k = 1 and k = n. For the two other diagonals it is easy to see that one gets two terms which cancel.
If we multiply with I − itD from the right and use Lemma 4.7, we easily see that the product is again the identity.
It remains to show that the set {it(it − D) −1 : t = 0} = {(it − D) −1 : t = 0} is R-bounded. For this, observe that the diagonal entries are R-bounded by the R-sectoriality of L k . The R-boundedness of the other entries follows from the Rgradient bounds (Proposition 4.6). Since a set of operator matrices is R-bounded precisely when each entry is R-bounded, we conclude that D is R-bisectorial. This result may seem obvious by formal computation, but the issue is to rigorously justify the matrix multiplication involving products of unbounded operators.
Here we used that (I + t 2 L) −1 converges to I strongly as t → 0 by the general theory of sectorial operators. But then we find that L(I + t 2 L) −1 ω = D 2 (I + t 2 L) −1 ω → D 2 ω, t → 0.
We are now ready to prove that D has a bounded H ∞ -calculus on L p (ΛT M, m).