# \(L^p\)-Analysis of the Hodge–Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

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## Abstract

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.

## Keywords

Witten Laplacian Hodge–Dirac operator*R*-bisectoriality \(H^\infty \)-functional calculus Bakry–Emery Ricci curvature

## Mathematics Subject Classification

Primary 47A60 Secondary 58A10, 58J35, 58J60## 1 Introduction

*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 \(\rho :M\rightarrow \mathbb {R}\) is the operator

*M*, and \(\,\mathrm {d}_\rho ^*\) is the adjoint operator. The representation (1.1) can be used to define the Witten Laplacian for

*k*-forms for \(k \ne 0\). In the special case \(M = \mathbb {R}^n\) and \(\rho (x) = \exp (-\,\frac{1}{2}|x|^2)\), \(L_\rho \) corresponds to the Ornstein-Uhlenbeck operator.

Let \(m(\mathrm{d}x) = \rho (x)\,\mathrm {d}x\) be the weighted volume measure on *M*. Generalising the celebrated Meyer inequalities for the Ornstein-Uhlenbeck operator, Bakry [9] proved boundedness of the Riesz transform \(\nabla L_\rho ^{-\,1/2}\) on \(L^p(M,m)\) for all \(1<p<\infty \) 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).

### Theorem 1.1

If *M* has non-negative Bakry–Emery Ricci curvature on *k*-forms for all \(1\le k\le n\), then the Hodge–Dirac operator \(D_\rho \) is *R*-bisectorial and admits a bounded \(H^\infty \)-calculus in \(L^p(\Lambda TM,m)\) for all \(1<p<\infty \).

By standard arguments (cf. [8]), the boundedness of the \(H^\infty \)-calculus of \(D_\rho \) implies (by considering the operator sgn\((D_\rho )\), which is then well defined through the functional calculus) the boundedness of the Riesz transform \(D_\rho L_\rho ^{-1/2} = \mathrm{sgn}(D_\rho )\). As such our results may be thought of as a strengthening of those in [9].

In the unweighted case \(\rho \equiv 1\), 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 \(H^\infty \)-calculus on the Hardy space \(H^p(\Lambda TM)\), even for \(1\le p\le \infty \), 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 non-negative Ricci curvature. Secondly, for \(1<p<\infty \), this Hardy space is subsequently identified in [6, Theorem 8.5] to be the closure in \(L^p(\Lambda TM)\) of the range of *D*, provided the heat kernel associated with *L* satisfies Gaussian bounds on *k*-forms for all \(0\le k\le n\). 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 \(L^p(\Lambda TM,m) = \mathsf {N}(D) \oplus \overline{\mathsf {R}(D)}\) (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 \(H^\infty \)-calculus in \(L^p(\Lambda TM,m)\) in the unweighted case.

In the weighted case, this approach cannot be pursued due to the absence of the doubling property 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 \(H^\infty \)-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 \(H^\infty \)-calculi in Sect. 2, we introduce the Witten Laplacian \(L_\rho \) in Sect. 3 and recall some of its properties. Among others we prove that it is *R*-sectorial of angle less than \(\frac{1}{2}\pi \) and admits a bounded \(H^\infty \)-calculus in \(L^p\) for \(1<p<\infty \). In Sect. 4 this result, together with the identity \(D_\rho ^2 = L_\rho \), is used to prove the corresponding assertions for the Hodge–Dirac operator \(D_\rho \).

On some occasions, we will use the notation \(a\lesssim b\) to signify that there exists a constant *C* such that \(a \le Cb\). To emphasise the dependence of *C* on parameters \(p_1\), \(p_2\), ..., we shall write \(a \lesssim _{p_1,p_2,\ldots } b\). Finally, we write \(\eqsim \) (respectively, \(\eqsim _{p_1,p_2,\ldots }\)) if both \(a\lesssim b\) and \(b\lesssim a\) (respectively, \(a \lesssim _{p_1,p_2,\ldots } b\) and \(b \lesssim _{p_1,p_2,\ldots } a\)) hold.

## 2 *R*-(Bi)sectorial Operators and the \(H^\infty \)-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 \((r_j)_{j\ge 1}\) be a sequence of independent *Rademacher variables* defined on a probability space \((\Omega ,\mathbb {P})\), i.e. \(\mathbb {P}(r_j= 1) = \mathbb {P}(r_j=-\,1) = \tfrac{1}{2}\) for each *j*.

*R*

*-bounded*if there exists a \(C\ge 0\) such that for all \(M=1,2,\ldots \) and all choices of \(x_1, \ldots , x_M \in X\) and \(T_1, \ldots , T_M \in \mathscr {T}\) we have

*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 \(\mathbb {E}r_mr_n = \delta _{mn}\).

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.

### 2.2 Sectorial Operators

*X*is said to be

*sectorial*of angle \(\sigma \in (0,{\pi })\) if \(\sigma (A) \subseteq \overline{\Sigma _\sigma ^+}\) and the set \(\{ \lambda (\lambda -A)^{-1} : \lambda \notin \overline{\Sigma _\vartheta ^+}\}\) is bounded for all \(\vartheta \in (\sigma ,{\pi })\). The least angle of sectoriality is denoted by \(\omega ^+(A)\). If

*A*is sectorial of angle \(\sigma \in (0,\pi )\) and the set \(\{ \lambda (\lambda -A)^{-1} : \lambda \notin \overline{\Sigma _\vartheta ^+}\}\) is

*R*-bounded for all \(\vartheta \in (\sigma ,{\pi })\), then

*A*is said to be

*R*-

*sectorial*of angle \(\sigma \). The least angle of

*R*-sectoriality is denoted by \(\omega _R^+(A).\)

### Remark 2.1

*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

*A*in \(\overline{\mathsf {R}(A)}\) 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 \(1<p<\infty \) 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 \(\Delta \) in \(L^p(\mathbb {R}^n)\), \(1\le p<\infty \), and this operator is *R*-sectorial if \(1<p<\infty \). More general examples, including the Laplace-Beltrami operator, are discussed in [21, 36, 40].

### 2.3 Bisectorial Operators

*bisector*of angle \(\sigma \). A closed densely defined linear operator \((A,\mathsf {D}(A))\) acting in a complex Banach space

*X*is called

*bisectorial*of angle \(\sigma \) if \(\sigma (A) \subseteq \overline{\Sigma _\sigma ^\pm }\) and the set \(\{ \lambda (\lambda -A)^{-1} : \lambda \notin \overline{\Sigma _\vartheta ^\pm }\}\) is bounded for all \(\vartheta \in (\sigma ,\frac{1}{2}{\pi })\). The least angle of bisectoriality is denoted by \(\omega ^\pm (A)\). If

*A*is bisectorial and the set \(\{ \lambda (\lambda -A)^{-1} : \lambda \notin \overline{\Sigma _\vartheta ^\pm }\}\) is

*R*-bounded for all \(\vartheta \in (\sigma ,\frac{1}{2}{\pi })\), then

*A*is said to be

*R*-

*bisectorial*of angle \(\sigma \in (0,\frac{1}{2}{\pi })\). The least angle of

*R*-bisectoriality is denoted by \(\omega _R^\pm (A).\)

### Remark 2.2

If *A* is bisectorial (of angle \(\vartheta \)), then *iA* is sectorial (of angle \(\frac{1}{2}\pi +\vartheta \)), and therefore Remark 2.1 applies to bisectorial operators as well.

Typical examples of bisectorial operators are \(\pm i \,\mathrm {d}/\,\mathrm {d}x\) in \(L^p(\mathbb {R})\) and the Hodge–Dirac operator \(\Bigl (\begin{matrix} 0 &{} \nabla ^*\\ \nabla &{} 0 \end{matrix}\Bigr )\) on \(L^p(\mathbb {R}^n)\oplus L^p(\mathbb {R}^n;\mathbb {C}^n)\), \(1\le p<\infty \). These operators are *R*-bisectorial if \(1<p<\infty \).

### 2.4 The \(H^\infty \)-Functional Calculus

In a Hilbert space setting, the \(H^\infty \)-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].

*A*is a sectorial operator and \(\psi \) is a function in \(H^1(\Sigma _\sigma ^+)\) with \(0< \omega ^+(A)< \sigma <\pi \), we may define the bounded operator \(\psi (A)\) on

*X*by the Dunford integral

*A*on

*X*is said to admit a

*bounded*\(H^\infty (\Sigma _{\sigma }^+)\)

*-functional calculus*, or a

*bounded*\(H^\infty \)

*-calculus of angle*\(\sigma \), if there exists a constant \(C_\sigma \ge 0\) such that for all \(\psi \in H^1(\Sigma _\sigma ^+)\cap H^\infty (\Sigma _\sigma ^+)\) and all \(x\ \in X\), we have

*C*exists is denoted by \(\omega _{H^\infty }^+(A).\) We say that a sectorial operator

*A*admits a

*bounded*\(H^\infty \)

*-calculus*if it admits a bounded \(H^\infty (\Sigma _{\sigma }^+)\)-calculus for some \(0<\sigma <\pi \).

Typical examples of operators having a bounded \(H^\infty \)-calculus include the sectorial operators mentioned in Sect. 2.2. In fact, it requires quite some effort to construct sectorial operators without a bounded \(H^\infty \)-calculus, and to this date only rather artificial constructions of such examples are known.

Replacing the role of sectors by bisectors, the above definitions can be repeated for bisectorial operators. The examples of bisectorial operators mentioned in Sect. 2.3 have a bounded \(H^\infty \)-calculus.

### 2.5 *R*-(bi)sectorial Operators and Bounded \(H^\infty \)-functional Calculi

The following result is a straightforward generalisation of [5, Proposition 8.1] and [1, Sect. H] (see [36, Chapter 10] for the present formulation):

### Proposition 2.3

*A*is an

*R*-bisectorial operator on a Banach space of finite cotype. Then \(A^2\) is

*R*-sectorial, and for each \(\omega \in (0,\frac{1}{2}\pi )\) the following assertions are equivalent:

- (1)
*A*admits a bounded \(H^\infty (\Sigma _\omega ^\pm )\)-calculus; - (2)
\(A^2\) admits a bounded \(H^\infty (\Sigma _{2\omega }^+)\)-calculus.

## 3 The Witten Laplacian

Let us begin by introducing some standard notations from differential geometry. For unexplained terminology, we refer to [27, 41].

*M*,

*g*) of dimension

*n*. The exterior algebra over the tangent bundle

*TM*is denoted by

*k*

*-forms*. We set

*k*-forms. The inner product of two

*k*-forms \(\,\mathrm {d}x^{i_1} \wedge \cdots \wedge \mathrm {d}x^{i_k}\) and \(\,\mathrm {d}x^{j_1} \wedge \cdots \wedge \mathrm {d}x^{j_k}\) is defined, in a coordinate chart (

*U*,

*x*), as

*g*in the chart (

*U*,

*x*). This definition extends to general

*k*-forms by linearity. For smooth sections \(\omega ,\eta \) of \(\Lambda TM\), say \(\omega = \sum _{k=0}^n \omega ^k\) and \(\eta = \sum _{k=0}^n \eta ^k\), we define

*M*, where \(\,\mathrm {d}x\) is the volume measure. For \(1 \le p < \infty \), we define \(L^p(\Lambda ^kTM,m)\) to be the Banach space of all measurable

*k*-forms for which the norm

*m*-almost everywhere on

*M*. Equivalently, we could define this space as the completion of \(C_\mathrm{c}^\infty (\Lambda ^kTM)\) with respect to the norm \(\Vert \cdot \Vert _p\). Finally, we define

*k*-forms \(\omega ^k\). In the case of \(p = 2\), we will denote the \(L^2(\Lambda ^kTM,m)\) inner product of two

*k*-forms \(\omega ,\eta \in L^2(\Lambda ^kTM,m)\) by

The exterior derivative, defined a priori only on \(C_\mathrm{c}^\infty (\Lambda TM)\), is denoted by \(\mathrm {d}\). Its restriction as a linear operator from \(C_\mathrm{c}^\infty (\Lambda ^kTM)\) to \(C_\mathrm{c}^\infty (\Lambda ^{k+1}TM)\) is denoted by \(\mathrm {d}_k\). As a densely defined operator from \(L^2(\Lambda ^kTM,m)\) to \(L^2(\Lambda ^{k+1}TM,m)\), \(\mathrm {d}_k\) is easily checked to be closable. With slight abuse of notation, its closure will again be denoted by \(\mathrm {d}_k\). Its adjoint is well defined as a closed densely defined operator from \(L^2(\Lambda ^{k+1}TM,m)\) to \(L^2(\Lambda ^{k}TM,m)\). We will denote this adjoint operator by \(\delta _k\). It maps \(C_\mathrm{c}^\infty (\Lambda ^{k+1}TM)\) into \(C_\mathrm{c}^\infty (\Lambda ^kTM)\).

### Remark 3.1

It would perhaps be more accurate to follow the notation used in the Introduction and denote the operators \(\mathrm {d}\), \(\mathrm {d}_k\) and \(\delta _k\) by \(\mathrm {d}_\rho \), \(\mathrm {d}_{\rho ,k}\) and \(\mathrm {d}^*_{\rho ,k}\), respectively, to bring out their dependence on \(\rho \), but this would unnecessarily burden the notation.

*k*-form and

*X*a smooth vector field. We define \(\iota (X)\omega \) as the \((k-1)\)-form given by

*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.

### Lemma 3.2

*k*-forms \(\omega \) and \((k-1)\)-forms \(\epsilon \) and compactly supported smooth functions

*f*on

*M*, we have

*g*.

### Proof

*U*,

*x*), by linearity it suffices to prove the claim for \(\omega = g\mathrm {d}x^{i_1} \wedge \cdots \wedge \mathrm {d}x^{i_k}\) where \(1 \le i_1< \cdots < i_k \le n\) and \(\epsilon = h\mathrm {d}x^{j_1} \wedge \cdots \wedge \mathrm {d}x^{j_{k-1}}\) where \(1 \le j_1< \cdots < j_{k-1} \le n\). In that case, we find

### Lemma 3.3

*k*-form, then

*g*.

### Proof

*k*-form. For any \((k-1)\)-form \(\epsilon \), we have

*k*-forms are linear over \(C^\infty \) functions to arrive at the second line. The last equality follows from the previous lemma. The claim now follows. \(\square \)

### Definition 3.4

*Witten Laplacian on*

*k*

*-forms*associated with \(\rho \) is the operator \(L_k\) defined on \(C_\mathrm{c}^\infty (\Lambda ^kTM)\) as

*Bochner-Lichnérowicz-Weitzenböck formula*(cf. [9, Sect. 5]) asserts that

*k*-forms, while [9] defines it in the sense of tensors.

*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]:

### Hypothesis 3.5

(Non-negative curvature condition) For all \(k = 1,\ldots ,n\) the Bakry–Emery Ricci curvature on *k*-forms is non-negative, i.e. we have \(Q_k(\omega ,\omega ) \ge 0\) for all *k*-forms \(\omega \).

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 \(M=\mathbb {R}^n\). 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 \(\mathbb {R}^n\).

### Example 3.6

Let \(M = \mathbb {R}^n\) with its usual Euclidean metric and consider a smooth strictly positive function \(\rho \) on \(\mathbb {R}^n\). Let \(k \in \{1,2\ldots ,n\}\). We will derive a sufficient condition on \(\rho \) so that \(Q_k(\omega ,\omega ) \ge 0\) for all *k*-forms \(\omega \).

*k*-forms \(\omega \). Focussing on the remaining terms in (3.4), we will first show that \(Q_k\) has the ‘Pythagorean’ property described in (3.5) below. Suppose

*N*.

*k*-forms \(\omega _1 = f \mathrm {d}x^{i_1} \wedge \cdots \wedge \mathrm {d}x^{i_k}\), where \(1 \le i_1< \cdots < i_k \le n\) and \(\omega _2 = g\mathrm {d}x^{j_1} \wedge \cdots \wedge \mathrm {d}x^{j_k}\), where \(1 \le j_1< \cdots < j_k \le n\) and suppose that \((i_1,\ldots ,i_k) \ne (j_1,\ldots ,j_k)\). Now consider \(\omega = \omega _1 + \omega _2\). Since the set of ‘elementary’

*k*-forms

*X*,

Now consider a *k*-form \(\omega \) of the form \(f \mathrm {d}x^{i_1} \wedge \cdots \wedge \mathrm {d}x^{i_k}\) with \(1 \le i_1< \cdots < i_k \le n\). To simplify notations a bit, we shall suppose that \((i_1,\ldots ,i_k) = (1,\ldots ,k)\). We compute the three last terms on the right-hand side of (3.4).

*k*-forms \(\omega \) precisely if for all \(1 \le i_1< \cdots < i_k \le n\) it holds that

We can use the previous example to consider a more general situation.

### Example 3.7

*M*,

*g*) be a complete Riemannian manifold. Suppose the quadratic form \(\tilde{Q}_k\) depending solely on the Ricci curvature is bounded from below for all \(1\le k\le n\), i.e. there exist constants \(a_1,\ldots ,a_n\) such that for all

*k*-forms \(\omega \), we have

*p*reduces to the one of the previous examples. Consequently, \(Q_k(\omega ,\omega ) \ge 0\) for any

*k*-form \(\omega \) if for any \(p \in M\) and any \(1 \le i_1< \cdots < i_k \le n\) one has \(\sum _{r=1}^k \partial _{i_r}^2(\log \rho )(p) \le a_k\), where the last expression is in normal coordinates around

*p*.

### 3.2 The Heat Semigroup Generated by \(-L_k\)

*k*-forms \(\omega \). Consequently, its closure is a self-adjoint operator on \(L^2(\Lambda ^kTM,m)\). 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 \((P_t^k)_{t\ge 0}\) to \(L^p(\Lambda ^kTM,m)\cap L^2(\Lambda ^kTM,m)\) extends to a strongly continuous contraction semigroup on \(L^p(\Lambda ^kTM,m)\) for any \(p \in [1,\infty )\). These extensions are consistent, i.e. the semigroups \((P_t^k)_{t\ge 0}\) on \(L^{p_i}(\Lambda ^kTM,m)\), \(i=1,2\), agree on the intersection \(L^{p_1}(\Lambda ^kTM,m) \cap L^{p_2}(\Lambda ^kTM,m)\).

The infinitesimal generator of the semigroup \((P_t^k)_{t\ge 0}\) in \(L^p(\Lambda ^kTM,m)\) will be denoted (with slight abuse of notation) by \(-L_k\) and its domain by \({\mathsf D}_p(L_k)\).

As an operator acting in \(L^2(\Lambda ^kTM,m)\), \(L_k\) is the closure of an operator defined a priori on \(C_\mathrm{c}^\infty (\Lambda ^kTM)\) and therefore the inclusion \(C_\mathrm{c}^\infty (\Lambda ^kTM)\subseteq {\mathsf D}_2(L_k)\) trivially holds. The definition of the domain \({\mathsf D}_p(L_k)\) is indirect, however, and based on the fact that \(L_k\) generates a strongly continuous semigroup on \(L^p(\Lambda ^kTM,m)\). Nevertheless we have:

### Lemma 3.8

\(C_\mathrm{c}^\infty (\Lambda ^kTM)\) is contained in \({\mathsf D}_p(L_k)\) for all \(1< p < \infty \).

### Proof

*k*-form \(\omega \in C_\mathrm{c}^\infty (\Lambda ^kTM,m)\). Then \(\omega \in {\mathsf D}_2(L_k)\) (by definition of \(L_k\) on \(L^2(\Lambda ^kTM,m)\)) and also \(\omega \in L^p(\Lambda ^kTM,m)\). Since \(L^p(\Lambda ^kTM,m)\) is a reflexive Banach space, a standard result in semigroup theory states that in order to show that \(\omega \in {\mathsf D}_p(L_k)\) it suffices to show that

By the Stein interpolation theorem [53, Theorem 1 on p.67], for \(p\in (1,\infty )\) and \(k=0,1,\ldots ,n\) the mapping \(t \mapsto P_t^k\) extends analytically to a strongly continuous \(\mathscr {L}(L^p(\Lambda ^kTM,m))\)-valued mapping \(z \mapsto P_z^k\) defined on the sector \(\Sigma _{\omega _p}\) with \(\omega _p = \frac{\pi }{2}(1 - |2/p - 1|)\). On this sector, the operators \(P_z^k\) are contractive. This implies that \(L_k\) is sectorial of angle \(\omega _p\).

As a consequence of (3.6), the operators \(P_t^0\) 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 \(< \frac{1}{2}\pi \).

### Lemma 3.9

*R*-sectoriality via pointwise domination) Let

*M*be a Riemannian manifold of dimension

*n*equipped with a measure

*m*. Let \(k \in \{0,1\ldots ,n\}\) and suppose

*A*and

*B*are sectorial operators of angle \(<\frac{1}{2}\pi \) on the space \(L^p(M,m)\) and \(L^p(\Lambda ^kTM,m)\), respectively, with \(1\le p<\infty \). Suppose the bounded analytic \(C_0\)-semigroups \((S_t)_{t\ge 0}\) and \((T_t)_{t\ge 0}\) generated by \(-A\) and \(-B\) satisfy the pointwise bound

*C*is a constant. If the set \(\{(I+sA)^{-1}:\, s>0\}\) is

*R*-bounded (in particular, if

*A*is

*R*-sectorial), then

*B*is

*R*-sectorial of angle \(<\frac{1}{2}\pi \).

For the proof of this lemma, we need the following result.

### Lemma 3.10

*M*,

*g*) be a Riemannian manifold of dimension

*n*equipped with a measure

*m*. For all \(\omega _1,\ldots ,\omega _N \in L^p(\Lambda ^kTM,m)\), we have

*p*.

### Proof

*Step 1*—First we assume that \(\omega _1,\ldots ,\omega _N\) are supported in a single coordinate chart (*U*, *x*). With slight abuse of notation, we will identify each \(\omega _i|_U \) with the corresponding \(\mathbb {C}^{d_k}\)-valued function on *U*; here, \(d_k = \left( {\begin{array}{c}n\\ k\end{array}}\right) \) is the dimension of \(\Lambda ^k TU\).

*D*(

*p*) is diagonal with positive diagonal entries. Now set

*Step 2*—We now turn to the general case. Let \((\phi _U)_{U\in \mathscr {U}}\) be a partition of unity subordinate to a collection of coordinate charts \(\mathscr {U}\) covering

*M*. Then, using Fubini’s theorem and the result of Step 1,

### Proof of Lemma 3.9

*R*denotes the

*R*-bound of the set \(\{(I+sA)^{-1}:\,s>0\}\). This gives the

*R*-boundedness of the set \(\{(I+\lambda B)^{-1}:\, {{\mathrm{Re}}}\lambda >0\}\). A standard Taylor expansion argument allows us to extend this to the

*R*-boundedness of the set \(\{(I+\lambda B)^{-1}:\, \lambda \in \Sigma _\nu \}\) for some \(\nu >\frac{1}{2}\pi \). \(\square \)

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 \(1< p < \infty \) and \(k=0,1,\ldots ,n\), the operator \(L_k\) is *R*-sectorial on \(L^p(\Lambda ^kTM,m)\) with angle \(\omega _R^+(L_k) < \frac{1}{2}\pi \).

### Proof

*R*-sectorial by [38, Corollary 5.2]. Lemma 3.9 then implies that \(L_k\) is

*R*-sectorial, of angle \(<\frac{1}{2}\pi \). \(\square \)

We are now ready to state our first main result.

### Theorem 3.12

[Bounded \(H^\infty \)-calculus for \(L_k]\) Let Hypothesis 3.5 be satisfied. For all \(1< p < \infty \) and all \(k= 0,1,\ldots ,n\), the operator \(L_k\) has a bounded \(H^\infty \)-calculus on \(L^p(\Lambda ^kTM,m)\) of angle \(<\frac{1}{2}\pi \).

For \(k=0\) the proposition is an immediate consequence of [38, Corollary 5.2]; see [16] for a more detailed quantitative statement. For \(k=1,\dots ,n\) 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.

*k*-forms is defined as

### Proof of Theorem 3.12

*R*-sectorial on \(L^p(\Lambda ^kTM,m)\) and \(\omega _R^+(L_k)<\frac{1}{2}\pi \). Pick \(\vartheta \in (\omega _R^+(L_k),\frac{1}{2}\pi )\). The function \(\psi (z) := \frac{1}{\sqrt{2}}\sqrt{z}e^{-\sqrt{z}}\) belongs to \(H^1(\Sigma _\vartheta ^+)\cap H^\infty (\Sigma _\vartheta ^+)\). Using the substitution \(t = s^2\), we see that

*k*-forms \(\omega \in L^p(\Lambda ^kTM,m)\).

Now it is well known that for an *R*-sectorial operator, the square function estimate (3.11) implies the operator having a bounded \(H^\infty \)-calculus of angle at most equal to its angle of *R*-sectoriality (see [39] or [36, Chapter 10]). \(\square \)

## 4 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, Sect. 5] proved for the special case \(\rho \equiv 1\) carry over to general strictly positive functions \(\rho \in C^\infty (M)\). Whenever we refer to results from [9], we bear this in mind.

### Definition 4.1

*Hodge–Dirac operator*associated with \(\rho \) is the linear operator

*D*on \(C_\mathrm{c}^\infty (\Lambda TM)\) defined by

As in Remark 3.1 it would be more accurate to denote this operator by \(D_\rho \), but again we prefer to keep the notation simple.

*D*can be represented by the \((n+1)\times (n+1)\)-matrix

### Lemma 4.2

For all \(1\le p<\infty \), the operator *D* is closable as a densely defined operator on \(L^p(\Lambda TM,m)\).

### Proof

For the reader’s convenience, we include the easy proof. Let \((\omega _n)_n\) be a sequence in \(C_\mathrm{c}^\infty (\Lambda TM)\) and suppose that \(\omega _n \rightarrow 0\) and \(D\omega _n \rightarrow \eta \) in \(L^p(\Lambda TM, m)\). Decomposing along the direct sum, we find that \(\omega _n^k \rightarrow \omega ^k\) in \(L^p(\Lambda ^kTM,m)\) for \(0\le k \le n\) and \(\,\mathrm {d}_{k-1}\omega _n^{k-1} + \delta _k\omega _n^{k+1} \rightarrow \eta ^k\) in \(L^p(\Lambda ^kTM,m)\) for \(1 \le k \le n-1\); for \(k=0\) we have \(\delta _0\omega _n^1 \rightarrow \eta ^0\) in \(L^p(\Lambda ^0TM,m)\) and for \(k = n\) we have \(\,\mathrm {d}_{n-1}\omega _n^{n-1} \rightarrow \eta ^n\) in \(L^p(\Lambda ^nTM,m)\).

*k*, so \(\eta = 0\). \(\square \)

With slight abuse of notation, we will denote the closure again by *D* and write \({\mathsf D}_p(D)\) for its domain in \(L^p(\Lambda TM, m)\). The main result of this section asserts that, under Hypothesis 3.5, for all \(1< p < \infty \) the operator *D* is *R*-bisectorial on \(L^p(\Lambda TM, m)\) and has a bounded \(H^\infty \)-calculus on this space.

Since \(L_k\) is sectorial on \(L^p(\Lambda ^kTM,m)\), \(1<p<\infty \), its square root is well defined and sectorial. Moreover, we have \( C_\mathrm{c}^\infty (\Lambda ^kTM) \subseteq {\mathsf D}_p(L_k)\subseteq {\mathsf D}_p(L_k^{1/2})\) (cf. Lemma 3.8).

### Lemma 4.3

For all \(1< p < \infty \) and \(k=0,1,\ldots ,n\), \(C_\mathrm{c}^\infty (\Lambda ^kTM)\) is dense in \({\mathsf D}_p(L_k^{1/2})\).

### Proof

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 \(\rho \equiv 1\) and given in the form of inequalities for smooth compactly supported *k*-forms.

### Theorem 4.4

Here, \(D_k := \mathrm {d}_k + \delta _{k-1}\) is the restriction of *D* as a densely defined operator acting from \(L^p(\Lambda ^kTM,m)\) into \(L^p(\Lambda TM,m)\).

### Proof

*k*-forms in this space converging to \(\omega \) in \({\mathsf D}_p(L_k^{1/2})\). By [9, Theorem 5.1] we then find, for all

*i*,

*j*,

*i*,

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.

### Lemma 4.5

For all \(1< p < \infty \) and \(k=0,1,\ldots ,n\), we have \({\mathsf D}_p(L_k^{1/2})\subseteq {\mathsf D}_p(\mathrm {d}_k)\cap {\mathsf D}_p(\delta _{k-1})\).

### Proof

*k*-forms in this space converging to \(\omega \) in \({\mathsf D}_p(L_k^{1/2})\). By [9, Theorem 5.1] we then find, for all

*i*,

*j*,

This proves the inclusion \({\mathsf D}_p(L_k^{1/2}) \subseteq {\mathsf D}_p(\mathrm {d}_k)\). The inclusion \({\mathsf D}_p(L_k^{1/2}) \subseteq {\mathsf D}_p(\delta _k)\) is proved in the same way. \(\square \)

It also follows from the lemma that the operators \(\,\mathrm {d}_k(I + t^2L_k)^{-1}\) and \(\delta _{k-1}(I + t^2L_k)^{-1}\) are well defined and \(L^p\)-bounded for all \(t \in \mathbb {R}\); indeed, just note that \({\mathsf D}_p(L_k) \subseteq {\mathsf D}_p(L_k^{1/2})\subseteq {\mathsf D}_p(\mathrm {d}_k) \cap {\mathsf D}_p(\delta _{k-1})\). The next proposition asserts that these operators form an *R*-bounded family:

### Proposition 4.6

*R*-gradient bounds) Let Hypothesis 3.5 hold. For all \(1< p < \infty \) and \(k =0,1,\ldots ,n\) the families of operators

*R*-bounded.

### Proof

We will only prove that the first set is *R*-bounded. The *R*-boundedness of the other set is proved in exactly the same way.

*R*-bounded in \(\mathscr {L}(L^p(\Lambda ^kTM,m))\). Since \(\,\mathrm {d}_kL_k^{-1/2}\) is bounded, it follows that the set

*R*-bounded in \(\mathscr {L}(L^p(\Lambda ^kTM,m),L^p(\Lambda ^{k+1}TM,m))\). This concludes the proof.

\(\square \)

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.

### Lemma 4.7

### Proof

We will only prove the first identity; the second is proved in a similar manner. The corresponding results for \(P_t^{k+1}\) can be proved along the same lines, or deduced from the results for the resolvent using Laplace inversion, and in turn the identities involving \((I + t^2L_{k+1})^{-1/2}\) follow from this.

For *k*-forms \(\omega \in C_\mathrm{c}^\infty (\Lambda ^kTM,m)\), we have \(P_t^{k+1}\mathrm {d}_k\omega = \mathrm {d}_kP_t^k\omega \) (see [9]). Here, the right-hand side is well defined as \(P_t^k\omega \in {\mathsf D}_p(L_k) \subseteq {\mathsf D}_p(\mathrm {d}_k)\) (which holds by analyticity of \(P_t^k\)). Now pick \(\omega \in {\mathsf D}_p(\mathrm {d}_k)\) and let \(\omega _n \in C_\mathrm{c}^\infty (\Lambda ^kTM)\) be a sequence converging to \(\omega \in {\mathsf D}_p(\mathrm {d}_k)\). Such a sequence exists by the definition of \(\,\mathrm {d}_k\) as a closed operator. Thus \(\omega _n \rightarrow \omega \) and \(\mathrm {d}_k\omega _n \rightarrow \mathrm {d}_k\omega \) in \(L^p(\Lambda ^kTM,m)\) respectively \(L^p(\Lambda ^{k+1}TM,m)\). The boundedness of \(P_t^k\) and \(P_t^{k+1}\) then implies that \(P_t^k\omega _n \rightarrow P_t^k\omega \) and \(P_t^{k+1}\mathrm {d}_k\omega _n \rightarrow P_t^{k+1}\mathrm {d}_k\omega \) in \(L^p(\Lambda ^kTM,m)\) respectively \(L^p(\Lambda ^{k+1}TM,m)\). As \(P_t^{k+1}\mathrm {d}_k\omega _n = \mathrm {d}_kP_t^k\omega _n\) for every *n*, and as the left-hand side converges, we obtain that \(\,\mathrm {d}_kP_t^k\omega _n\) converges in \(L^p(\Lambda ^{k+1}TM,m)\). The closedness of \(\mathrm {d}_k\) shows that \(P_t^k\omega \in {\mathsf D}_p(\mathrm {d}_k)\) and that \(P_t^{k+1}\mathrm {d}_k\omega = \mathrm {d}_kP_t^k\omega \).

### Remark 4.8

We now obtain the following result.

### Theorem 4.9

(*R*-bisectoriality of *D*) Let Hypothesis 3.5 hold. For all \(1< p < \infty \) the Hodge–Dirac operator *D* is *R*-bisectorial on \(L^p(\Lambda TM, m)\).

### Proof

*D*. We will do this by showing that \(I - itD\) has a two-sided bounded inverse given by

*R*-sectoriality of \(L_k\) (Proposition 3.11) and the

*R*-gradient bounds (Proposition 4.6) all entries are bounded. It only remains to check that this matrix defines a two-sided inverse of \(I-itD\). Let us first multiply with \(I - itD\) from the left. It suffices 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

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 \(\{it(it - D)^{-1}: t \ne 0\} = \{(it - D)^{-1}: t\ne 0\}\) 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 *R*-gradient 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. \(\square \)

### Proposition 4.10

Let \(1< p < \infty \). Then \(D^2 = L\) as densely defined closed operators on \(L^p(\Lambda TM, m)\).

This result may seem obvious by formal computation, but the issue is to rigorously justify the matrix multiplication involving products of unbounded operators.

### Proof

It suffices to show that \(\mathsf {D}_p(L) \subset \mathsf {D}_p(D^2)\) and \(D^2(I+t^2L)^{-1} = L(I+t^2L)^{-1}\), or equivalently, \( (\mathrm {d}_{k-1}\delta _{k-1} + \delta _k\,\mathrm {d}_k)(I+t^2L_k)^{-1} = L_k(I+t^2L_k)^{-1}\) for all \(k=0,1,\ldots ,n\). The rigorous justification of the equivalent identity (4.5) has already been given in the course of the above proof.

*I*strongly as \(t\rightarrow 0\) by the general theory of sectorial operators. But then we find that

*L*gives \(\omega \in \mathsf {D}(L)\) and \(L\omega = D^2\omega \). \(\square \)

We are now ready to prove that *D* has a bounded \(H^\infty \)-calculus on \(L^p(\Lambda TM, m)\).

### Theorem 4.11

(Bounded \(H^\infty \)-functional calculus for *D*) Let Hypothesis 3.5 hold. For all \(1< p < \infty \) the Hodge–Dirac operator *D* on \(L^p(\Lambda TM, m)\) has a bounded \(H^\infty \)-calculus on a bisector.

## Notes

### Acknowledgements

The authors thank Alex Amenta and Pierre Portal for helpful comments. Jan van Neerven acknowledges financial support from the ARC Discovery Grant DP 160100941. Rik Versendaal is supported by the Peter Paul Peterich Foundation via TU Delft University Fund.

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