Download pdf - Holomorphic functional calculus of Hodge-Dirac operators in Lp

J. Evol. Equ. 11 (2011), 71–105© 2010 Springer Basel AG1424-3199/11/010071-35, published onlineSeptember 24, 2010DOI 10.1007/s00028-010-0082-y

Journal of EvolutionEquations

Holomorphic functional calculus of Hodge-Dirac operators in L p

Tuomas Hytonen, Alan McIntosh and Pierre Portal

Abstract. We study the boundedness of the H∞ functional calculus for differential operators acting inL p(Rn; CN ). For constant coefficients, we give simple conditions on the symbols implying such bounded-ness. For non-constant coefficients, we extend our recent results for the L p theory of the Kato square rootproblem to the more general framework of Hodge-Dirac operators with variable coefficients�B as treatedin L2(Rn; CN ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that �Bhas a bounded H∞ functional calculus, in terms of randomized boundedness conditions of its resolvent.This allows us to deduce stability under small perturbations of this functional calculus.

1. Introduction

A variety of problems in PDE’s can be solved by establishing the boundedness,and stability under small perturbations, of the H∞ functional calculus of certain dif-ferential operators. In particular, Axelsson et al. [10] have recovered and extendedthe solution of the Kato square root problem [5] by showing that Hodge-Dirac oper-ators with variable coefficients of the form �B = � + B1�

∗B2 have a boundedH∞ functional calculus in L2(Rn; CN ), when � is a homogeneous first-order differ-ential operator with constant coefficients, and B1, B2 ∈ L∞(Rn;L (CN )) are strictlyaccretive multiplication operators. Recently, Auscher et al. [4] have used related per-turbation results to show the openness of some sets of well-posedness for boundaryvalue problems with L2 boundary data.

In this paper, we first consider homogeneous differential operators with constant(matrix-valued) coefficients. For such operators, the boundedness of the H∞ func-tional calculus is established using Mikhlin’s multiplier theorem. However, the esti-mates on the symbols may be difficult to check in practice, especially when the nullspaces of the symbols are non-trivial. Here we provide a simple condition (invertibilityof the symbols on their ranges and inclusion of their eigenvalues in a bisector) thatgives such estimates. We then turn to operators with coefficients in L∞(Rn; C) ofthe form �B = � + B1 �B2, where � and �are nilpotent homogeneous first-orderoperators with constant (matrix-valued) coefficients, and B1, B2 ∈ L∞(Rn;L (CN ))

are multiplication operators satisfying some L p coercivity condition. For such oper-ators, we aim at perturbation results which give, in particular, the boundedness of the

Mathematics Subject Classification (2010): 42B37, 47A60, 47F05

72 T. Hytonen et al. J. Evol. Equ.

H∞ functional calculus when B1, B2 are small perturbations of constant-coefficientmatrices.

This presents two main difficulties. First of all, even in L2, the H∞ functional cal-culus of a (bi)sectorial operator is in general not stable under small perturbations in thesense that there exist a self-adjoint operator D and bounded operators A with arbitrarysmall norm such that D(I + A) does not have a bounded H∞ functional calculus(see [24]). Subtle functional analytic perturbation results exist (see [16] and [20]), butdo not give the estimates needed in [4] or [10]. To obtain such estimates, one needsto take advantage of the specific structure of differential operators using harmonicanalytic methods. Then, the problem of moving from the L2 theory to an L p theoryis substantial. Indeed, the operators under consideration fall outside the Calderón-Zygmund class and cannot be handled by familiar methods based on interpolation.A known substitute, pioneered by Blunck and Kunstmann in [12], and developed byAuscher and Martell [2,6–8], consists in establishing an extrapolation method adaptedto the operator, which allows to extend results from L2 to L p for p in a certain range(p1, p2) containing 2. In [19], we started another approach, which combines probabi-listic tools from functional analysis with the aforementioned L2 methods, and allowsL p results which do not rely on some L2 counterparts.

However, our goal in [19] was the Kato problem, and we did not reach the general-ity of [10] which has recently proven particularly useful in connection with boundaryvalue problems [4]. Here we close this gap and, in fact, reach a further level of gen-erality. Roughly speaking, for quite general differential operators, we show that theboundedness of the H∞ functional calculus coincides with the R-(bi)sectoriality (seeSect. 2 for relevant definitions). This then allows perturbation results, in contrast to thegeneral theory of sectorial operators, where R-sectoriality and bounded H∞ calculusare two distinct properties, and perturbation results are much more restricted.

For the operators with variable coefficients, the core of the argument is contained in[19], so the reader might want to have a copy of this paper handy. Here we focus on thepoints where [19] needs to be modified and develop some adaptation of the techniquesto generalized Hodge-Dirac operators which may be of interest in other problems. Tomake the paper more readable, we choose not to work in the Banach-space valuedsetting of [19], but the interested reader will soon realize that our proof carries overto that situation provided that, as in [19], the target space is a UMD space, and boththe space and its dual have the RMF property.

The paper is organized as follows. In Sect. 2, we recall the essential definitions.In Sect. 3, we present our setting and state the main results. In Sect. 4, we deal withconstant coefficient operators and obtain appropriate estimates on their symbols. InSect. 5, we use these estimates to establish an L p theory for operators with constant(matrix-valued) coefficients. In Sect. 6, we show that a certain (Hodge) decomposi-tion, crucial in our study, is stable under small perturbations. In Sect. 7, we give simpleproofs of general operator theoretic results on the functional calculus of bisectorialoperators. In Sect. 8, we prove our key results on operators with variable coefficients,referring to [19] when arguments are identical, and explaining how to modify them

Vol. 11 (2011) Functional calculus of Hodge-Dirac operators 73

using the results of the preceding sections when they are not. Finally, in Sect. 9, wederive from Sect. 8 Lipschitz estimates for the functional calculus of these operators.

2. Preliminaries

Fix some numbers n, N ∈ Z+. We consider functions u : Rn → CN , or A : Rn →L (CN ). The Euclidean norm in both Rn and CN , as well as the associated operatornorm in L (CN ), are denoted by | · |. To express the typical inequalities “up to a con-stant” we use the notation a � b to mean that there exists C < ∞ such that a ≤ Cb,and the notation a � b to mean that a � b � a. The implicit constants are meant tobe independent of other relevant quantities. If we want to mention that the constant Cdepends on a parameter p, we write a �p b.

Let us briefly recall the construction of the H∞ functional calculus (see [1,15,17,22,23] for details).

DEFINITION 2.1. A closed operator A acting in a Banach space Y is called bisec-torial with angle θ if its spectrum σ(A) is included in a bisector:

σ(A) ⊆ Sθ := �θ ∪ (−�θ), where

�θ := {z ∈ C\{0} ; | arg(z)| ≤ θ} ∪ {0},and outside the bisector it verifies the following resolvent bounds:

∀θ ′ ∈(θ,π

2

)∃C > 0 ∀λ ∈ C\Sθ ′ ‖λ(λI − A)−1‖L (Y ) ≤ C. (2.2)

We often omit the angle and say that A is bisectorial if it is bisectorial with someangle θ ∈ [0, π2

).

For 0 < ν < π/2, let H∞(Sν) be the space of bounded functions on Sν , whichare holomorphic on the interior of Sν . Note, in particular, that these functions aredefined at the origin, although they need not be continuous there. We also considerthe following subspace of functions with decay at zero and infinity:

H∞0 (Sν) :=

{φ ∈ H∞(Sν) : ∃α,C ∈ (0,∞) ∀z ∈ Sν |φ(z)| ≤ C

∣∣∣∣z

1 + z2

∣∣∣∣α}.

For a bisectorial operator A with angle θ < ω < ν < π/2, and ψ ∈ H∞0 (Sν), we

define

ψ(A)u := 1

2iπ

∫

∂Sωψ(λ)(λ− A)−1u dλ,

where ∂Sω is directed anti-clockwise around Sω.

DEFINITION 2.3. A bisectorial operator A, with angle θ , is said to admit a boundedH∞ functional calculus with angle μ ∈ [θ, π2 ) if, for each ν ∈ (μ, π2

),

∃C < ∞ ∀ψ ∈ H∞0 (Sν) ‖ψ(A)y‖Y ≤ C‖ψ‖∞‖y‖Y .


In this case, and if Y is reflexive, one can define a bounded operator f (A) forf ∈ H∞(Sν) by

f (A)u := f (0)P0u + limn→∞ψn(A)u,

where P0 denotes the projection on the null space of A corresponding to the decom-position Y = N(A)⊕ R(A), which exists for R-bisectorial operators, and (ψn)n∈N ⊂H∞

0 (Sν) is a bounded sequence which converges locally uniformly to f . See [1,15,17,22,23] for details.

DEFINITION 2.4. A family of operators T ⊂ L (Y ) is called R-bounded if forall M ∈ N, all T1, . . . , TM ∈ T , and all u1, . . . , uM ∈ Y ,

E

∥∥∥M∑

k=1

εk Tkuk

∥∥∥Y

� E

∥∥∥M∑

k=1

εkuk

∥∥∥Y,

where E is the expectation which is taken with respect to a sequence of independentRademacher variables εk , i.e., random signs with P(εk = +1) = P(εk = −1) = 1

2 .A bisectorial operator A is called R-bisectorial with angle θ in Y if the collection

{λ(λI − A)−1 : λ ∈ C\Sθ ′ }is R-bounded for all θ ′ ∈ (θ, π/2). The infimum of such angles θ is called the angleof R-bisectoriality of A.

Again, we may omit the angle and simply say that A is R-bisectorial if it is R-bisectorial with some angle θ ∈ (0, π/2). Notice that, by a Neumann series argument,this is equivalent to the R-boundedness of {(I + i t A)−1 : t ∈ R}. The reader unfamil-iar with R-boundedness and the derived notions can consult [19] and the referencestherein.

REMARK 2.5. On subspaces of L p, 1 < p < ∞, an operator with a boundedH∞ functional calculus, is R-bisectorial. The proof (stated for sectorial rather thanbisectorial operators) can be found in [21, Theorem 5.3].

3. Main results

We consider three types of operators. First, we look at differential operators of arbi-trary order with constant (matrix-valued) coefficients and provide simple conditionson their Fourier multiplier symbols to ensure that such operators are bisectorial and, infact, have a bounded H∞ functional calculus. Then, we focus on first-order operatorswith a special structure, the Hodge-Dirac operators, and prove that under an additionalcondition on the symbols, they give a specific (Hodge) decomposition of L p. Finallywe turn to Hodge-Dirac operators with (bounded measurable) variable coefficients andshow that the boundedness of the H∞ functional calculus is preserved under smallperturbation of the coefficients.


We work in the Lebesgue spaces L p := L p(Rn; CN ) with p ∈ (1,∞) and denoteby S (Rn; CN ) the Schwartz class of rapidly decreasing functions with values in CN ,and by S ′(Rn; CN ) the corresponding class of tempered distributions.

3.1. General constant-coefficient operators

In this subsection, we consider kth order homogeneous differential operators of theform

D = (−i)k∑

θ∈Nn :|θ |=k

Dθ ∂θ

acting on S ′(Rn; CN ) as a Fourier multiplier with symbol D(ξ) = ∑|θ |=k Dθ ξ θ ,

where Dθ ∈ L (CN ).

ASSUMPTION 3.1. The Fourier multiplier symbol D(ξ) satisfies

κ|ξ |k |e| ≤ |D(ξ)e| for all ξ ∈ Rn, all e ∈ R(D(ξ)), and some κ > 0,

(D1)

there exists ω ∈[0,π

2

)such that for all ξ ∈ Rn : σ(D(ξ)) ⊆ Sω.

(D2)

In each L p, let D act on its natural domain Dp(D) := {u ∈ L p ; Du ∈ L p}. InTheorem 5.1 we prove:

THEOREM 3.2. Let 1 < p < ∞. Under the assumptions (D1) and (D2), theoperator D is bisectorial in L p with angle ω, and has a bounded H∞ functionalcalculus in L p with angle ω.

REMARK 3.3. (a) In (D2), the bisector Sω can be replaced by the sector �ωwhere 0 ≤ ω < π . The operator D is then sectorial (with angle ω) and has abounded H∞ functional calculus (with angle ω) in the sectorial sense, i.e. f (D)is bounded for functions f ∈ H∞(�θ ) with any θ ∈ (ω, π).

(b) Assuming that (D1) holds for all e ∈ CN would place us in a more classical con-text, in which proofs are substantially simpler. We insist on this weaker ellipticitycondition since the operators we want to handle have, in general, a non-trivialnull space.

(c) Using Bourgain’s version of Mikhlin’s multiplier theorem [13] , the above theo-rem extends to function with values in X N , where X is a UMD Banach space.

3.2. Hodge-Dirac operators with constant coefficients

We now turn to first-order operators of the form

� = � + �,


where � = −i∑n

j=1 � j∂ j , acts on S ′(Rn; CN ) as a Fourier multiplier with symbol

� = �(ξ) =n∑

j=1

� jξ j , � j ∈ L (CN ),

the operator �is defined similarly, and both operators are nilpotent in the sense that�(ξ)2 = 0 and ˆ�(ξ)2 = 0 for all ξ ∈ Rn .

DEFINITION 3.4. We call � = � + �a Hodge-Dirac operator with constantcoefficients if its Fourier multiplier symbol � = � + ˆ�satisfies the followingconditions:

κ|ξ ||e| ≤ |�(ξ)e| for all e ∈ R(�(ξ)), all ξ ∈ Rn, and some κ > 0,

(�1)

σ(�(ξ)) ⊆ Sω for some ω ∈[0,π

2

), and all ξ ∈ Rn, (�2)

N(�(ξ)) = N(�(ξ)) ∩ N( ˆ�(ξ)) for all ξ ∈ Rn . (�3)

REMARK 3.5. The “Hodge-Dirac” terminology has its origins in applications ofthis formalism to Riemannian geometry where � would be the exterior derivative dand �= d∗. See [10] for details. Note that we are working here in a more general set-ting than [10], where the operator �was assumed to be the adjoint of �. In particular,our operator � does not need to be self-adjoint in L2(Rn; CN ).

In each L p, we let the operators ϒ ∈ {�, �,�} act on their natural domains

Dp(ϒ) := {u ∈ L p : ϒu ∈ L p},

where ϒu is defined in the distributional sense. Each ϒ is a densely defined, closedunbounded operator in L p with this domain. The formal nilpotence of� and �transfersinto the operator-theoretic nilpotence

Rp(�) ⊆ Np(�), Rp( �) ⊆ Np( �).

where Rp(�),Np(�) denote the range and kernel of � as an operator on L p.In Section 5, we show that the identity � = � + �is also true in the sense of

unbounded operators in L p. Moreover, in Theorem 5.5 we prove:

THEOREM 3.6. The operator � has a bounded H∞ functional calculus in L p

with angle ω and satisfies the Hodge decomposition

L p = Np(�)⊕ Rp(�)⊕ Rp( �).

REMARK 3.7. As in the previous subsection, the above theorem extends tofunctions with values in X N , where X is a UMD Banach space.


3.3. Hodge-Dirac operators with variable coefficients

We finally turn to Hodge-Dirac operators with variable coefficients. The study ofsuch operators is motivated by [4,10] and [19].

DEFINITION 3.8. Let 1 < p < ∞ and p′ denote the dual exponent of p. Let

B1, B2 ∈ L∞(Rn;L (CN )),

and identify these functions with bounded multiplication operators on L p in the naturalway. Also let � = � + �be a Hodge-Dirac operator. Then the operator

�B := � + �B, where �B := B1 �B2, (3.9)

is called a Hodge-Dirac operator with variable coefficients in L p if the followinghold:

〈v, �B2 B1 �u〉 := 〈 �∗v, B2 B1 �u〉 = 0 for u, v ∈ S (Rn; CN ), (B1)

‖u‖p � ‖B1u‖p ∀u ∈ Rp( �) and ‖v‖p′ � ‖B∗2v‖p′ ∀v ∈ Rp′( �∗). (B2)

Note that the operator equality (3.9), involving the implicit domain conditionDp(�B) := Dp(�) ∩ Dp( �B), was a proposition for Hodge-Dirac operators withconstants coefficients, but is taken as the definition for Hodge-Dirac operators withvariable coefficients.

The following simple consequences will be frequently applied. Their proofs are leftto the reader. First, the nilpotence condition (B1), a priori formulated for test functions,self-improves to

�B2 B1 �= 0 on Dp( �); hence Rp( �B) ⊆ Np( �B).

Second, the coercivity condition (B2) implies that

Rp( �B) = B1Rp( �B2) = B1Rp( �),

and

B1 : Rp( �) → Rp( �B) is an isomorphism.

Sometimes, we also need to assume that the related operator �B = �+ B2�B1 isa Hodge-Dirac operator with variable coefficients in L p, i.e.

�B1 B2� = 0 on S (Rn; CN ),

‖u‖p � ‖B2u‖p ∀u ∈ Rp(�) and ‖v‖p′ � ‖B∗1v‖p′ ∀v ∈ Rp′(�∗).

With the same proof as in [10, Lemma 4.1], one can show:


PROPOSITION 3.10. Assuming�B = �+ �B is a Hodge-Dirac operator with var-iable coefficients, then the operators �B := B1 �B2 and �∗B := B∗

2 �∗ B∗1 are closed,

densely defined, nilpotent operators in L p and L p′, respectively, and �∗B = ( �B)

∗.

However, the Hodge-decomposition and resolvent bounds, which in the context of[10] (and the first-mentioned one even in [19]) could be established as propositions,are now properties which may or may not be satisfied:

DEFINITION 3.11. We say that �B Hodge-decomposes L p if

L p = Np(�B)⊕ Rp(�)⊕ Rp( �B).

REMARK 3.12. We will mostly be interested in a Hodge-Dirac operator �B withthe property that it is R-bisectorial in L p and Hodge-decomposes L p. If this prop-erty holds for two exponents p ∈ {p1, p2}, then it holds for the intermediate valuesp ∈ (p1, p2) as well, and hence the set of exponents p, for which the mentioned prop-erty is satisfied, is an interval.

The proof that R-bisectoriality interpolates in these spaces can be found in [20,Corollary 3.9], where it is formulated for R-sectorial operators. As for the Hodge-decomposition, observe first that if a Hodge-Dirac operator�B is R-bisectorial in L p

and Hodge-decomposes L p, then the projections onto the three Hodge subspaces aregiven by

P0 = limt→∞(I + t2�2

B)−1, P� = lim

t→∞ t2��B(I + t2�B)−1,

P �B = limt→∞ t2 �B�B(I + t2�B)

−1,

where the limits are taken in the strong operator topology. In particular, if�B has theseproperties in two different L p spaces, then the corresponding Hodge subspaces havecommon projections, and one deduces the boundedness of these projection operatorsalso in the interpolation spaces.

The following main result concerning the operators �B gives a characterization ofthe boundedness of their H∞ functional calculus. It will be proven as Corollary 8.12to Theorem 8.1.

THEOREM 3.13. Let 1 ≤ p1 < p2 ≤ ∞, and let �B be a Hodge-Dirac operatorwith variable coefficients in L p which Hodge-decomposes L p for all p ∈ (p1, p2).Assume also that �B is a Hodge-Dirac operator with variable coefficients in L p.Then �B has a bounded H∞ functional calculus (with angle μ) in L p(Rn; CN ) forall p ∈ (p1, p2) if and only if it is R-bisectorial (with angle μ) in L p(Rn; CN ) for allp ∈ (p1, p2).

This characterization leads to perturbation results such as the following, proven inCorollary 8.16, thanks to the perturbation properties of R-bisectoriality.


COROLLARY 3.14. Let 1 ≤ p1 < p2 ≤ ∞, and let �A be a Hodge-Dirac oper-ator with variable coefficients, which is R-bisectorial in L p and Hodge-decomposesL p for all p ∈ (p1, p2). Then for each p ∈ (p1, p2), there exists δ = δp > 0such that, if�B and�B are Hodge-Dirac operators with variable coefficients, and if‖B1 − A1‖∞ + ‖B2 − A2‖∞ < δ, then �B has a bounded H∞ functional calculusin L p and Hodge-decomposes L p.

REMARK 3.15. The results in this paper concerning Hodge-Dirac operators withvariable coefficients can be extended to the Banach space valued setting, providedthe target space has the so-called UMD and RMF properties, and also its dual hasRMF. The UMD property, which passes to the dual automatically, is a well-knownnotion in the theory of Banach spaces, cf. [14]. We introduced the RMF property in[19] in relation with our Rademacher maximal function. It holds in (commutative ornot) L p spaces for 1 < p < ∞ and in spaces with type 2, and fails in L1. We donot know whether it holds in every UMD space. This paper, especially in Sect. 8,uses extensively the techniques from [19]. We choose not to formulate the results ina Banach space valued setting to make the paper more readable, but all proofs arenaturally suited to this more general context.

4. Properties of the symbols

In this section, we consider the symbols of the Fourier multipliers defined in Sub-sections 3.1 and 3.2. As a consequence of the assumptions made in these subsections,we obtain the various estimates which we need in the next sections to establish an L p

theory.In what follows, we denote D(0, r) = {z ∈ C ; |z| < r} and A(a, b) = {z ∈

C ; a ≤ |z| ≤ b}.LEMMA 4.1. Let D be a k-th order homogeneous differential operator with

constant matrix coefficients, satisfying (D1) and (D2). Then, denoting M =sup|ξ |=1 |D(ξ)|, we have that

(a) σ(D(ξ)) ⊂ (Sω ∩ A(κ|ξ |k,M |ξ |k)) ∪ {0},

(b) CN = N(D(ξ))⊕ R(D(ξ)),(c) ∀μ ∈ (ω, π2

) |(ζ I − D(ξ))−1| � |ζ |−1

∀ξ ∈ Rn ∀ζ ∈ C\(Sμ ∩ A( 1

2κ|ξ |k, 2M |ξ |k)).Note that claim (c) is void for ζ = 0, since it merely asserts that a certain norm

is at most ∞. Using compactness for |ξ | = 1 and homogeneity for |ξ | �= 1, this is aconsequence of the following lemma.

LEMMA 4.2. Let T ∈ L (CN ), κ > 0, and ω ∈ [0, π2), and suppose that

(i) κ|e| ≤ |T e| for all e ∈ R(T ), and(ii) σ(T ) ⊂ Sω.


Then we have that

(a) σ(T ) ⊂ (Sω ∩ A (κ, |T |)) ∪ {0},(b) CN = N(T )⊕ R(T ),(c) ∀μ ∈ (ω, π2

) |(ζ I − T )−1| � |ζ |−1 ∀ζ ∈ C\ (Sμ ∩ A( 1

2κ, 2|T |)).Proof. Let us first remark that, for a non zero eigenvalue λwith eigenvector e, we havethat |λ||e| = |T e| ≥ κ|e|. This gives (a). Moreover, (i) also gives that N(T 2) = N(T ).Thus, writing T in Jordan canonical form, we have the splitting CN = N(T )⊕ R(T ).The resolvent bounds hold on N(T ). On R(T ), the function ζ �→ ζ(ζ I − T )−1 is con-tinuous from the closure of C\ (Sμ ∩ A

( 12κ, 2|T |)) to L (R(T ),CN ) and is bounded

at ∞, and thus is bounded on C\ (Sμ ∩ A( 1

2κ, 2|T |)). �

Assuming (D1) and (D2), we thus have that for all θ ∈ (0, π2 − ω

)and for all

ξ ∈ Rn ,

∃C > 0 ∀τ ∈ Sθ |(I + iτ D(ξ))−1|L (CN ) ≤ C.

For τ ∈ Sθ , we use the following notation:

Rτ (ξ) := (I + iτ D(ξ))−1,

Pτ (ξ) := 1

2(Rτ (ξ)+ R−τ (ξ)) = (I + τ 2 D(ξ)2)−1,

Qτ (ξ) := i

2(Rτ (ξ)− R−τ (ξ)) = τ D(ξ)Pτ (ξ).

If λ /∈ σ(D(ξ)) for some ξ ∈ Rn , then also λ /∈ σ(D(ξ ′)) for all ξ ′ in someneighborhood of ξ . One checks directly from the definition of the derivative that

∂ξ j (λ− D(ξ))−1 = (λ− D(ξ))−1(∂ξ j D)(ξ)(λ− D(ξ))−1.

By induction it follows that (λ− D(ξ))−1 is actually C∞ in a neighborhood of ξ forλ /∈ σ(D(ξ)). In particular, for τ ∈ Sθ , the function Rτ (ξ) is C∞ in ξ ∈ Rn , and

∂ξ j Rτ (ξ) = Rτ (ξ)(−iτ∂ξ j D(ξ))Rτ (ξ). (4.3)

PROPOSITION 4.4. Given the splitting CN = N(D(ξ))⊕ R(D(ξ)), the comple-mentary projections PN(D(ξ)) and PR(D(ξ)) = I −PN(D(ξ)) are infinitely differentiablein Rn\{0} and satisfy the Mikhlin multiplier conditions

|∂αξ PN(D(ξ))| �α |ξ |−|α|, ∀α ∈ Nn .

Proof. The projections PN(D(ξ)) are obtained by the Dunford–Riesz functional cal-culus by integrating the resolvent around a contour, which circumscribes the origincounterclockwise, and does not circumscribe any other point of the spectrum of D(ξ).By Lemma 4.1, for ξ in a neighborhood of the unit sphere

(say 3

4 < |ξ | < 43

), we may

choose

PN(D(ξ)) = 1

2π i

∫

∂D(0,2−k−1κ)

(λ− D(ξ))−1 dλ.


Using the smoothness of (λ− D(ξ))−1 discussed before the statement of the lemma,differentiation of arbitrary order in ξ under the integral sign may be routinely justified.This shows that PN(D(ξ)) is C∞ in a neighborhood of the unit sphere.

To complete the proof, it suffices to observe that D(tξ) = tk D(ξ) for t ∈ (0,∞).Hence N(D(ξ)) and R(D(ξ)), and therefore the associated projections, are invari-ant under the scalings ξ �→ tξ . It is a general fact that smooth functions, which arehomogeneous of order zero, satisfy the Mikhlin multiplier conditions. Indeed, for anyξ ∈ Rn\{0} and t ∈ (0,∞), we have

∂αξ PN(D(ξ)) = ∂αξ (PN(D(tξ))) = t |α|(∂αPN(D( · )))(tξ),

and setting t = |ξ |−1 and using the boundedness of the continuous function ∂αξ PN(D(ξ))on the unit sphere, the Mikhlin estimate follows. �

Notice that, by (D1), D(ξ) is an isomorphism of R(D(ξ)) onto itself for eachξ ∈ Rn\{0}. We denote by D−1

R (ξ) its inverse.

LEMMA 4.5. The function D−1R (ξ)PR(D(ξ)) is smooth in Rn\{0} and satisfies the

Mikhlin-type multiplier condition

∂αξ (D−1R (ξ)PR(D(ξ))) �α |ξ |−k−|α|, ∀α ∈ Nn .

Proof. By the Dunford–Riesz functional calculus, we have

m(ξ) := D−1R (ξ)PR(D(ξ)) = 1

2π i

∫

γ

(λ− D(ξ))−1 dλ

λ,

where γ is any path oriented counter-clockwise around the non-zero spectrum ofD(ξ). Since D(rξ) = rk D(ξ), a change of variables and Cauchy’s theorem showsthat m(rξ) = r−km(ξ), and the smoothness of m in Rn\{0} is checked as in the previ-ous proof by differentiating under the integral sign. The modified Mikhlin conditionfollows from this in a similar way as in the previous proof. �

PROPOSITION 4.6. Given ν ∈ (0, π2 − ω), the following Mikhlin conditions hold

uniformly in τ ∈ Sν :

|∂αξ Rτ (ξ)| �α |ξ |−|α|, ∀α ∈ Nn .

Similar estimates hold for Pτ (ξ) and Qτ (ξ).

Proof. We use induction to establish the desired bounds. For α = 0, this was provenin Lemma 4.1. In order to make the induction step, we will need the identity

∂αξ Rτ =∑

0�θ≤α

(α

θ

)(∂α−θ Rτ

) (−iτ∂θξ D

)Rτ , α � 0. (4.7)


We use the multi-index notation as follows: the binomial coefficients are(αθ

) :=∏ni=1

(αiθi

), the order relation θ ≤ αmeans that θi ≤ αi for every i = 1, . . . , n, whereas

θ � α means that θ ≤ α but θ �= α; finally, it is understood that(αθ

) = 0 if θ �≤ α.Let us prove the identity (4.7) by induction. For |α| = 1, the formula was already

established in (4.3). Assuming (4.7) for some α � 0, we prove it for α+ e j , where e j

is the j th standard unit vector. Indeed,

∂α+e jξ Rτ = ∂ξ j ∂

αξ Rτ

=∑

0�θ≤α

(α

θ

)[(∂α+e j −θ Rτ )(−iτ∂θξ D)Rτ + (∂α−θ Rτ )(−iτ∂

θ+e jξ D)Rτ

+ (∂α−θ Rτ )(−iτ∂θξ D)Rτ (−iτ∂ξ j D)Rτ]

=⎡⎣ ∑

0�θ≤α

(α

θ

)+

∑

e j �θ≤α+e j

(α

θ − e j

)⎤⎦

×(∂α+e j −θ Rτ )(−iτ∂θξ D)Rτ + (∂α Rτ )(−iτ∂e jξ D)Rτ

=∑

0�θ≤α+e j

[(α

θ

)+(

α

θ − e j

)](∂α+e j −θ Rτ )(−iτ∂θξ D)Rτ ,

and the proof of (4.7) is completed by the binomial identity(αθ

) + ( αθ−e j

) = (α+e jθ

).

Notice that the induction hypothesis was used twice: first to expand ∂αξ Rτ in the sec-ond step, and then to evaluate the summation over the last of the three terms in thethird one.

We then pass to the inductive proof of the assertion of the lemma. Let α � 0,and assume the claim proven for all β � α. We first consider (∂αξ Rτ )PR(D). By the

induction assumption, we know that the factors ∂α−θξ Rτ in 4.7 satisfy

|∂α−θξ Rτ (ξ)| � |ξ ||θ |−|α|.

Furthermore,

(−iτ∂θξ D

)RτPR(D) =

(∂θξ D

)Rτ(−iτ D

)D−1

R PR(D) =(∂θξ D

)(Rτ − I )D−1

R PR(D),

where the different factors are bounded by

|∂θξ D| � |ξ |k−|θ |, |Rτ − I | � 1,∣∣∣D−1

R (ξ)PR(D(ξ))

∣∣∣ � |ξ |−k .

(The second inequality is just the uniform boundedness of the resolvent symbols Rτ ,established in Lemma 4.1). Multiplying all these estimates, we get |(∂αξ Rτ )PR(D)| �|ξ |−|α|, as required.


It remains to estimate (∂αξ Rτ )PN(D). We have

(∂αξ Rτ

)PN(D) = ∂αξ

(RτPN(D)

)−∑

0≤β�α

(∂βξ Rτ

)∂α−β

PN(D)

= ∂αξ PN(D) −∑

0≤β�α

O(|ξ |−|β|) O

(|ξ |−|α−β|) = O

(|ξ |−|α|) ,

where the O-bounds follow from the induction assumption and the Mikhlin boundsfor PN(D(ξ)) established in Proposition 4.4. The proof is complete. �

PROPOSITION 4.8. Letμ∈ (ω, π2), and let f ∈ H∞

0 (Sμ). Then f (D) is a Mikhlin

multiplier, and more precisely its symbol f (D)(ξ) = f (D(ξ)) satisfies, for ξ �= 0,

∣∣∣∂αξ f (D(ξ))∣∣∣ �μ |ξ |−|α| sup

{| f (λ)| : λ ∈ Sμ,

12κ|ξ |k ≤ |λ| ≤ 2M |ξ |k

},

where M = sup|ξ |=1 |D(ξ)|.Proof. Pick θ ∈ (ω,μ). By the definition of the functional calculus, we have

∂αξf (D) = 1

2π i

∫

∂Sθf (ζ )∂αξ (ζ − D)−1 dζ = ∂αξ f (D). (4.9)

From 4.7 one sees that ∂αξ (ζ − D)−1 = ζ−1∂αξ Ri/ζ has poles at ζ ∈ σ(D). ByLemma 4.1, the non-zero spectrum satisfies

σ(D(ξ))\{0} ⊂ {ζ ∈ Sω : κ|ξ |k ≤ |ζ | ≤ M |ξ |k}.

Hence, for a fixed ξ ∈ Rn , we may deform the integration path in(4.9) to

∂[Sθ ∩ D(0, 2M |ξ |k)\D(0, 12κ|ξ |k)] ∪ ∂[Sθ ∩ D(0, ε)] =: γ1 ∪ γ2.

On γ1, there holds |ζ | � |ξ |k , while the length of the path is also �(γ1) � |ξ |k . Hence

∣∣∣∫

γ1

f (ζ )∂αξ (ζ − D(ξ))−1 dζ∣∣∣

≤ sup{| f (ζ )| : ζ ∈ Sμ ∩ D(0, 2M |ξ |k)\D(0, 12κ|ξ |k)}

∫

γ1

|∂αξ Ri/ζ (ξ)| | dζ ||ζ |

� sup{| f (λ)| : λ ∈ Sμ,12κ|ξ |k ≤ |λ| ≤ 2M |ξ |k} · |ξ |−|α|,

which has a bound of the desired form. On the other hand, the integral over γ2 vanishesin the limit as ε ↓ 0. �

We shall make use of operators with the following particular symbols:


COROLLARY 4.10. For t ∈ R and θ ∈ Nk with |θ | = k, the symbols

σt (ξ) := t2ξθ D(ξ)(I + t2 D(ξ)2)−1

are C∞ away from the origin and satisfy the Mikhlin multiplier estimates

|∂αξ σt (ξ)| �α |ξ |−|α| ∀α ∈ Nn .

Proof. The symbols are

σt (ξ) = ξθ D−1R (ξ)t2 D2(ξ)(I + t2 D(ξ)2)−1 =

(ξθ D−1

R (ξ)PR(D(ξ))

) (I − Pt (ξ)

),

where both factors are smooth in Rn\{0}, and the first satisfies the Mikhlin multipliercondition by Lemma 4.5 and the second by Proposition 4.6. �

5. L p theory for operators with constant coefficients

In this section, we consider the Fourier multipliers in L p, which correspond to thesymbols studied in Sect. 4. We denote by Rt , Pt , Qt the multiplier operators with thesymbols Rt , Pt , Qt . We start with the operators D from Subsection 3.1. The estimatesobtained in the preceding section give Theorem 3.2, restated here for convenience:

THEOREM 5.1. Let 1 < p < ∞. Under the assumptions (D1) and (D2), the oper-ator D is bisectorial in L p with angle ω, and has a bounded H∞ functional calculusin L p with angle ω.

Proof. By the Mikhlin multiplier theorem, the bisectoriality follows from Prop-osition 4.6, while the boundedness of the H∞ functional calculus follows fromProposition 4.8. �

The coercivity condition (D1) for the symbol has the following reincarnation on thelevel of operators:

PROPOSITION 5.2. For all u ∈ Rp(D) ∩ Dp(D), there holds u ∈ Dp(∇k), and

‖∇ku‖p � ‖Du‖p.

Proof. For u ∈ Dp(D) ∩ Rp(D), we have (for real t)

ut := t2 D Pt (Du) = t2 D2 Pt u = (I − Pt )u → PRp(D)u = u, t → ∞.

The operators t2∂θ D Pt , |θ | = k, are bounded on L p by Corollary 4.10. It follows thatut ∈ Dp(∂

θ ), and

‖∂θut‖p = ‖t2∂θ D Pt (Du)‖p � ‖Du‖p.


Let w be a test function in the dual space. Then

|〈∂θu, w〉| = |〈u, ∂θw〉| = limt→∞ |〈ut , ∂

θw〉| = limt→∞ |〈∂θut , w〉|

≤ lim inft→∞ ‖∂θut‖p‖w‖p′ � ‖Du‖p‖w‖p′ .

Thus, u ∈ Dp(∂θ ) and ‖∂θu‖p � ‖Du‖p for all |θ | = k. �

We turn to the Hodge-Dirac operators � = � + �, which satisfy the hypotheses atthe start of Subsection 3.2, and note that they then satisfy the conditions on D withk = 1. In particular, Theorem 5.1 and Proposition 5.2 hold for D = � and k = 1.Moreover, there is an operator version of the symbol condition (�3):

LEMMA 5.3. There holds Np(�) = Np(�) ∩ Np( �).

Proof. The inclusion ⊇ is clear. Let u ∈ Np(�). Then �u = 0 in the sense of distri-butions. It follows from Lemma 4.5 that the function ϒ(ξ) := �(ξ)�−1

R (ξ)PR(�(ξ))is C∞ away from the origin. Hence, ifψ ∈ C∞

c (Rn\{0}), then alsoψϒ is in the same

class, and the product ψϒ · �u is well-defined and vanishes as a distribution. But

ϒ(ξ)�(ξ) = �(ξ)�−1R (ξ)PR(�(ξ))�(ξ) = �(ξ)PR(�(ξ)) = �(ξ)

by (�3). Thus, we have shown that ψ�u = ψϒ�u = 0 for every ψ ∈ C∞c (R

n\{0}).This means that the distribution �u is at most supported at the origin, and hence�u = P , a polynomial. But also �u = −i

∑nj=1 � j∂ j u = ∑n

j=1 ∂ j u j , where u j =−i � j u ∈ L p. Let φ ∈ S (Rn) be identically one in a neighborhood of the origin.Then P = Pφ and hence P = P ∗ φ. But

P ∗ φ(y) = 〈P, φ(y − ·)〉 =n∑

j=1

〈u j , (∂ jφ)(y − ·)〉 → 0

as |y| → ∞ (using just the fact that u j ∈ L p and ∂ jφ ∈ L p′), and a polynomial with

this property must vanish identically. This shows that �u = P = 0 and then also�u = �u − �u = 0. �

This then implies:

PROPOSITION 5.4. The operator identity � = � + �holds in L p, in the sensethat Dp(�) = Dp(�) ∩ Dp( �) and �u = �u + �u for all u ∈ Dp(�).

Proof. It is clear that Dp(�) ∩ Dp( �) ⊆ Dp(�). Since � is bisectorial in L p, thereis the topological decomposition L p = Np(�) ⊕ Rp(�). Write u ∈ Dp(�) asu = u0 + u1 in this decomposition. Then u0 ∈ Np(�) = Np(�) ∩ Np( �), andu1 = u −u0 ∈ Dp(�)∩Rp(�). By Proposition 5.2, u1 ∈ Dp(∇) ⊆ Dp(�)∩Dp( �).Thus also Dp(�) ⊆ Dp(�) ∩ Dp( �). The coincidence of �u and �u + �u is clearfrom the distributional definition. �

We are ready to prove Theorem 3.6, restated here:


THEOREM 5.5. Let� be a Hodge-Dirac operator with constant coefficients, andlet 1 < p < ∞. Then the operator � has a bounded H∞ functional calculus in L p

with angle ω, and satisfies the following Hodge decomposition

L p = Np(�)⊕ Rp(�)⊕ Rp( �),

where Rp(�) = Rp(�)⊕ Rp( �).

Proof. The fact that� is a bisectorial operator with a bounded H∞ functional calculusis a particular case of Theorem 5.1. The bisectoriality already implies the decomposi-tion L p = Np(�)⊕ Rp(�), which we now want to refine.

We first check that Rp(�) ⊆ Rp(�) + Rp( �). If u ∈ Rp(�), then u =lim j→∞�y j for some y j ∈ Dp(�) ∩ Rp(�) ⊆ Dp(∇). Then ‖�(y j − yk)‖p �‖∇(y j − yk)‖p � ‖�(y j − yk)‖p → 0 (using Proposition 5.2), and hence �y j

converges to some v ∈ Rp(�) with ‖v‖p � ‖u‖p. Similarly, �y j converges tow ∈ Rp( �), and u = v + w ∈ Rp(�)+ Rp( �).

Next we show that Rp(�) ⊆ Rp(�). Indeed, Rp(�) = �(Dp(�)) = �(Dp(�) ∩Rp(�)) (by the decomposition in the first paragraph and Lemma 5.3) ⊆ �(Dp(�) ∩(Rp(�) + Rp( �))) (by the previous paragraph) = �(Dp(�) ∩ Rp( �)) (because �is nilpotent) = �(Dp(�) ∩ Rp( �)) (because �is nilpotent) ⊆ Rp(�). ThereforeRp(�) ⊆ Rp(�). In a similar way, we see that Rp( �) ⊆ Rp(�).

On combining these two results with that in the preceding paragraph, we obtainRp(�)+ Rp( �) = Rp(�). To show that the sum is direct, observe that

Rp(�) ∩ Rp( �) ⊆ Rp(�) ∩ Np(�) ∩ Np( �) = Rp(�) ∩ Np(�) = {0}

by nilpotence, Lemma 5.3 and the decomposition L p = Np(�)⊕ Rp(�). �

We conclude this section with an analogue, in our matrix-valued context, ofBourgain’s [13, Lemma 10]. It is an important property of Hodge–Dirac operatorswith constant coefficients which we use to study Hodge–Dirac operators with variablecoefficients in Sect. 8.

PROPOSITION 5.6. For all z ∈ Rn, there holds

E

∥∥∥∑

k

εkτ2k z Q2k u∥∥∥

p� (1 + log+ |z|)‖u‖p.

where τz denotes the translation operator τzu(x) := u(x − z).

Proof. Let us fix a test function ϕ ∈ D(Rn) such that 1B(0,2−1) ≤ ϕ ≤ 1B(0,1),and write ψ(ξ) := ϕ(ξ) − ϕ(2ξ), ψm(ξ) := ψ(2−mξ), and ϕm(ξ) := ϕ(2−mξ) form ∈ Z+. Let �m and �m , m ∈ Z, be the corresponding Fourier multiplier operatorswith symbols ϕm and ψm . Then we have the partition of unity ϕk +∑∞

m=1 ψk+m ≡ 1,and the corresponding operator identity �k +∑∞

m=1�m+k = I for all k ∈ Z.


Since the support of the Fourier transform of Q2k�m−k is contained in B(0, 2m−k),by Bourgain’s [13, Lemma 10], there holds

E

∥∥∥∑

k

εkτ2k z Q2k�m−ku∥∥∥

p� (1 + log+(2m |z|))E

∥∥∥∑

k

εk Q2k�m−ku∥∥∥

p. (5.7)

The same reasoning applies with � in place of �.We now estimate the right side of (5.7) as a Fourier multiplier transformation. The

symbol of the operator acting on u is given by

σ(ξ) =∑

k

εk f (2k�(ξ))ψ(2k−mξ), f (τ ) = τ(1 + τ 2)−1.

For every α ∈ {0, 1}n , a computation shows that

∂ασ (ξ) =∑

k

εk

∑β≤α

∂β f (2k�(ξ))(∂α−βψ)(2k−mξ)2(k−m)|α−β|.

By the support property of ψ , the series in k reduces to at most two non-vanishingterms for which 2k−m |ξ | � 1. By Proposition 4.8, there moreover holds

|∂β f (2k�(ξ))| � 2k |ξ |1 + (2k |ξ |)2 |ξ |−|β|, (5.8)

which shows that, for m ∈ Z+,

|∂ασ (ξ)| � 2−m |ξ |−|α|.

Hence, the associated Fourier multiplier is bounded with norm � 2−m .A similar computation can be made with�−k in place of �m−k , but the estimation

of the symbol then involves an infinite series of terms:

|∂ασ (ξ)| �∑β≤α

∑k

|∂β f (2k�(ξ))||(∂α−βϕ)(2kξ)|2k|α−β|

�∑β≤α

∑

k:|2kξ |≤1

2k(1+|α−β|)|ξ |1−|β| � |ξ |−|α|,

where (5.8) was used again in the second estimate. We conclude that the operatoracting on u in (5.7), with �−k in place of �m−k , is also bounded.

Collecting all the estimates, we have shown that

E

∥∥∥∑

k

εkτ2k z Q2k u∥∥∥

p

� (1 + log+ |z|)∥∥∥∑

k

εk Q2k�−ku∥∥∥

p+

∞∑m=1

(m + log+ |z|)∥∥∥∑

k

εk Q2k�m−ku∥∥∥

p

�∞∑

m=0

(max{1,m} + log+ |z|)2−m‖u‖p � (1 + log+ |z|)‖u‖p,

which is the asserted bound. �


6. Properties of Hodge decompositions

In this section, we collect various results concerning the Hodge decomposition.These include duality results, a relation of the Hodge decompositions of the operator�B and its variant �B , and finally some stability properties of the Hodge decompo-sition under small perturbations of the coefficient matrices B1 and B2. These will beneeded in proving the stability of the functional calculus of the Hodge–Dirac operatorsunder small perturbations later on.

LEMMA 6.1. Let 1 < p < ∞ and let�B be a Hodge-Dirac operator with variablecoefficients in L p. The following assertions are equivalent:

(1) �B Hodge-decomposes L p.

(2)

{L p = Np(�)⊕ Rp( �B),

L p = Np( �B)⊕ Rp(�).

Proof. (1) ⇒ (2). We first show that Np(�B) = Np(�)∩Np( �B). If u ∈ Np(�B) then�u = − �Bu ∈ Rp(�)∩ Rp( �B) = {0}. It follows that Np(�B)⊕ Rp(�) ⊆ Np(�).Also Np(�) ∩ Rp( �B) ⊆ Np(�B) ∩ Rp( �B) = {0}. Hence Np(�B) ⊕ Rp(�) =Np(�), and thus L p = Np(�)⊕ Rp( �B). The second part of 2 is similarly proven.

(2) ⇒ (1). By 2, u ∈ L p can be decomposed as u = v0 + v1 + u1 where v0 + v1 ∈Np(�), v0 ∈ Np( �B), v1 ∈ Rp(�), and u1 ∈ Rp( �B). Then �Bv0 = �(v0 + v1 −v1) = 0, and ‖v0‖p + ‖v1‖p � ‖v0 + v1‖p � ‖u‖p. Moreover,

Np(�B) ∩ Rp(�) ⊆ Np( �B) ∩ Rp(�) = {0},Np(�B) ∩ Rp( �B) ⊆ Np(�) ∩ Rp( �B) = {0},Rp( �B) ∩ Rp(�) ⊆ Np(�) ∩ Rp( �B) = {0}.

The proof is complete. �

LEMMA 6.2. Let D0 and D1 be closed, densely defined operators in L p. Then

L p = Np(D0)⊕ Rp(D1) if and only if L p′ = Np′(D∗1)⊕ Rp′(D∗

0).

Proof. Assuming the first decomposition, we have that

L p′ = Rp(D1)⊥ ⊕ Np(D0)

⊥ = Np′(D∗1)⊕ Rp′(D∗

0).

The other implication follows by symmetry. �

LEMMA 6.3. Let�B be a Hodge-Dirac operator with variable coefficients in L p

which Hodge decomposes L p. Then �∗B is a Hodge-Dirac operator with variable

coefficients in L p′which Hodge decomposes L p′

.

Proof. We have to show that B∗ = (B∗

1 , B∗2

)satisfy (B2) in L p′

. Let us remark that

B1 is an isomorphism from Rp( �) onto Rp( �B). By Lemma 6.1, this means that B1

is an isomorphism from Rp( �) onto L p/Np(�), and thus B∗1 is an isomorphism from


Rp′(�∗) onto L p′/Np′( �∗). This gives the part of the result concerning B∗

1 thanks toLemma 6.2. The case of B∗

2 is handled in the same way. Condition (B1) is obtainedby duality, and the proof is concluded by applying Lemma 6.1 and Lemma 6.2. �

Recall that �B := �+ B2�B1.

LEMMA 6.4. Let 1 < p < ∞ and suppose that�B and�B are both Hodge-Diracoperator with variable coefficients in L p, and that �B Hodge-decomposes L p. Then�B also Hodge-decomposes L p. If, moreover,�B is an R-bisectorial operator in L p,then so is �B.

Proof. By Lemmas 6.1 and 6.2, the assumption that �B Hodge-decomposes L p isequivalent to

L p = Np(�)⊕ Rp( �B), L p′ = Np′(�∗)⊕ Rp′( �∗B), (6.5)

whereas the claim that �B Hodge-decomposes L p is equivalent to

L p = Np(B2�B1)⊕ Rp( �), L p′ = Np′(B∗1�

∗B∗2 )⊕ Rp′( �∗). (6.6)

We show that the first decomposition in (6.5) implies the first one in (6.6). Letu ∈ L p and write B1u = v + w where v ∈ Np(�) and w ∈ Rp( �B). Let w = B1xfor x ∈ Rp( �). Then u − x ∈ Np(B2�B1) and

‖x‖p � ‖B1x‖p = ‖w‖p � ‖B1u‖p � ‖u‖p.

The deduction of the second decomposition in (6.6) from the second one in (6.5) isanalogous.

We now turn to the second statement. Let us denote by P1 the projection on Rp( �),by P2 the projection on Rp(B2�B1), by P1 the projection on Rp(�), and by P2 theprojection on Rp(B1 �B2). Let (tk)k∈N ⊂ R and (uk)k∈N ⊂ L p, and remark first that

(I + i tk�B)−1 = I − (tk�B)

2(I + (tk�B)2)−1 − i tk�B(I + (tk�B)

2)−1.

The R-bisectoriality of �B will thus follow once we have proven that

E

∥∥∥∑

k

εki tk�B(I + (tk�B)2)−1uk

∥∥∥p

� E

∥∥∥∑

k

εkuk

∥∥∥p

(6.7)

and

E

∥∥∥∑

k

εk(tk�B)2(I + (tk�B)

2)−1uk

∥∥∥p

� E

∥∥∥∑

k

εkuk

∥∥∥p. (6.8)

To do so we note that, since �B and �B are Hodge-Dirac operators with variablecoefficients, we have:

�B1(I + (tk�B)2)−1

P1 = �(I + (tk�B)2)−1 B1P1,

�(I + (tk�B)2)−1 B2P1 = �B2(I + (tk�B)

2)−1P1.


We can now proceed with the estimates, using the R-bisectoriality of �B .

E

∥∥∥∑

k

εki tk�B(I + (tk�B)2)−1

P1uk

∥∥∥p

= E

∥∥∥∑

k

εki tk B2�B1(I + (tk�B)2)−1

P1uk

∥∥∥p,

= E

∥∥∥∑

k

εki tk B2�(I + (tk�B)2)−1 B1P1uk

∥∥∥p,

= E

∥∥∥∑

k

εki tk B2�B(I + (tk�B)2)−1 B1P1uk

∥∥∥p,

� E

∥∥∥∑

k

εk B1P1uk

∥∥∥p

� E

∥∥∥∑

k

εkuk

∥∥∥p.

Introducing vk such that B2P1vk = P2uk , we also get:

E

∥∥∥∑

k


P2uk

∥∥∥p

= E

∥∥∥∑

k

εki tk �(I + (tk�B)2)−1 B2P1vk

∥∥∥p,

= E

∥∥∥∑

k

εki tk �B2(I + (tk�B)2)−1

P1vk

∥∥∥p,

� E

∥∥∥∑

k


P1vk

∥∥∥p,

� E

∥∥∥∑

k

εkP1vk

∥∥∥p

� E

∥∥∥∑

k

εkuk

∥∥∥p.

The estimate (6.8) is proven in the same way. �

LEMMA 6.9. If a Banach space splits as X = X0 ⊕ X1 = P0 X ⊕ P1 X, thenthere is δ > 0 such that for all T ∈ L (X1, X) with ‖T ‖ < δ, it also splits asX = X0 ⊕ (I − T )X1 = P0 X ⊕ P1 X, with ‖P0 − P0‖ + ‖P1 − P1‖ � δ.

Proof. For δ := 12‖P1‖ , I − T P1 is invertible. Let U := (I − T P1)

−1, and observethe identity

U = I + U T P1.

Define the operators

P0 := P0U, P1 := (I − T P1)P1U.

Then R(P0) = X0, R(P1) = (I − T )X1, and

P0 + P1 = (I − P1 + (I − T P1)P1)U = (I − T P1)U = I.


It remains to show that P0 (and then also P1) is a projection. This follows from

P0P0 = P0UP0U = P0(I + U T P1)P0U = P0U = P0,

where we used P1P0 = 0 and P20 = I .

Since P0 − P0 = P0U T P1, P1 − P1 = P1U T P1 − T P1U , and ‖U‖ ≤ 2, we alsohave that ‖P0 − P0‖ + ‖P1 − P1‖ � δ. �

PROPOSITION 6.10. Let p ∈ (1,∞), and �A be a Hodge-Dirac operator withvariable coefficients in L p which Hodge-decomposes L p. There exists δ > 0 suchthat, if �B is another Hodge-Dirac operator with variable coefficients in L p with‖B1 − A1‖∞ + ‖B2 − A2‖∞ < δ, then �B Hodge-decomposes L p. Moreover, theassociated Hodge-projections satisfy

‖PA0 − P

B0 ‖ + ‖P

A� − P

B�‖ + ‖P

A�A

− PB�B‖ � δ.

Proof. Consider the condition (6.5) equivalent to the Hodge-decomposition. Let usdefine T1 ∈ L (Rp( �A), L p) by T1 A1 �u := (A1 − B1) �u which, by (B2), gives awell-defined operator of norm ‖T1‖ � δ, and we have B1 �= (I − T1)A1 �.

On the dual side, we define T2 ∈ L (Rp′( �∗A), L p′) by T2 A∗

2 �∗v = (A∗2 − B∗

2 ) �v,which is similarly well-defined and satisfies ‖T2‖ � δ. Let then T2 := (T2P �∗A )

∗ ∈L (L p), where P �∗A is the projection in L p′

associated to the decomposition in (6.5).By duality, it follows that Rp(B2 − A2(I − T2)) ⊆ Np( �), which means that �B2 =�A2(I − T2). Since the operators I − T2 and I − T1P �A are invertible for δ small

enough, we then have that

R( �B) = R((I − T1) �A(I − T2)) = (I − T1)R( �A).

Similarly, with B∗2 �∗ = (I − T3)A∗

2 �∗ and �∗ B∗1 = �∗ A∗

1(I − T4), thereholds R( �∗B) = (I − T3)R( �∗A). Hence the claim follows from two applica-tions of Lemma 6.9 with (X0, X1, T ) = (Np(�),R( �A), T1) and (X0, X1, T ) =(Np′(�∗),R( �∗A), T3). �

By (B2), the restriction A1 : Rp( �) → Rp( �A) is an isomorphism, and we denoteby A−1

1 its inverse. Thus, the operator A−11 P �A is well-defined. We shall also need to

perturb such operators:

COROLLARY 6.11. Under the assumptions of Proposition 6.10, there also holds

‖A−11 P

A�A

− B−11 P

B�B‖ � δ.

Proof. We use the same notation as in the proof of Proposition 6.10 and observe fromthe identity B1 �= (I − T1)A1 �that I − T1 : Rp( �A) → B1Rp( �), and

B−11 (I − T1)P

A�A

= A−11 P

A�A. (6.12)


We further recall from the proof of Lemma 6.9 that the projection PB�B

related to the

splitting L p = Np(�)⊕ Rp( �B), where Rp( �B) = (I − T1)Rp( �A), is given by

PB�B

= (I − T1)PA�A(I − T1P

A�A)−1.

Combining this with (6.12) shows that

B−11 P

B�B

= A−11 P

A�A

(I − T1P

A�A

)−1

= A−11 P

A�A

−(

A−11 P

A�A

)T1P

A�A

(I − T1P

A�A

)−1,

which proves the claim, since the second term contains the factor T1 of norm‖T1‖ � δ. �

7. Functional calculus

In this section, we collect some general facts about the functional calculus ofbisectorial operators in reflexive Banach spaces. We provide, in the context of thediscrete randomized quadratic estimates required in [19], versions of results origi-nally obtained in [15]. Lemma 7.1 can be seen as a discrete Calderón reproducingformula, and Lemma 7.2 as a Schur estimate, while Propositions 7.3 and 7.5 expressthe fundamental link between functional calculus and square function estimates. Suchresults are not new, and have been developed from [15] by various authors, most nota-bly Kalton and Weis (cf. [21]). Here we hope, however, to provide simpler versionsof both the statements and the proofs of these facts.

Let A denote a bisectorial operator in a reflexive Banach space, with angle ω, andlet θ ∈ (ω, π2

). We use the following notations.

r(A) = (I + iA)−1, p(A) = r(A)r(−A) = (I + A2)−1,

q(A) = i

2(r(A)− r(−A)) = A

I + A2 .

LEMMA 7.1. The following series converges in the strong operator topology:

3

2

∑k∈Z

q(2kA)q(2k+1A) = PR(A).

Proof. Observe first that p(tA) → PN(A) as t → ∞ and p(tA) → I as t → 0.Hence

PR(A) = I − PN(A) =∑

k

(p(2kA)− p(2k+1A))

=∑

k

p(2kA)[(I + 22(k+1)A2)− (I + 22kA2)

]p(2k+1A)

= 3

2

∑k

p(2kA)(2kA)(2k+1A)p(2k+1A) = 3

2

∑k

q(2kA)q(2k+1A),

as we wanted to show. �


LEMMA 7.2. Let A be R-bisectorial and let η(x) := min{x, 1/x}(1 +log max{x, 1/x}). Then the set

{η(s/t)−1q(tA) f (A)q(sA) : t, s > 0; f ∈ H∞0 (Sθ ), ‖ f ‖∞ ≤ 1}

is R-bounded.

Proof. Denoting qt (λ) := q(tλ), observe that

q(tA) f (A)q(sA) = (qt · f · qs)(A) = 1

2π i

∫

γ

(qt f qs)(λ)

(I − 1

λA)−1 dλ

λ,

where γ denotes (∂Sθ ′), for some θ ′ ∈ (θ, π2 ), parameterized by arclength and directedanti-clockwise around Sθ , and the resolvents (I − λ−1A)−1 belong to an R-boundedset. The operators q(tA) f (A)q(sA) are hence in a dilation of the absolute convexhull of this R-bounded set (see [22] for information on R-boundedness techniques).To evaluate the dilation factor, observe that

|qt f qs(λ)| � ‖ f ‖∞s|λ|

1 + (s|λ|)2t |λ|

1 + (t |λ|)2 ,

and splitting the integral into three regions (depending on the position of |λ| withrespect to min

( 1t ,

1s

)and max

( 1t ,

1s

)) it follows that

∫

γ

|qt f qs(λ)| | dλ||λ| � ‖ f ‖∞η(s/t),

which implies the asserted R-bound. �PROPOSITION 7.3. Let A be R-bisectorial (with angleμ) and satisfy the two-sided

quadratic estimate

‖u‖ � E

∥∥∥∑

k

εkq(2kA)u∥∥∥, u ∈ R(A). (7.4)

Then A has a bounded H∞ functional calculus (with angle μ).

Proof. Suppose (7.4) holds. Let u ∈ R(A), θ ∈ (μ, π2 ), and f ∈ H∞(Sθ ). Then

‖ f (A)u‖ � E

∥∥∥∑

k

εkq(2kA) f (A)u∥∥∥

� E

∥∥∥∑

k

εkq(2kA) f (A)∑

j

q(2 jA)q(2 j+1A)u∥∥∥

�∑

m

E

∥∥∥∑

k

εkq(2kA) f (A)q(2k+mA)q(2k+m+1A)u∥∥∥

�∑

m

η(2m)‖ f ‖∞∥∥∥∑

k

εkq(2k+m+1A)u∥∥∥

�∑

m

(1 + |m|)2−|m|‖ f ‖∞‖u‖ � ‖ f ‖∞‖u‖,


where we used � from (7.4), Lemma 7.1, the triangle inequality after relabelingj = k + m, Lemma 7.2, and � from (7.4). �

We also use the following variant.

PROPOSITION 7.5. Let A be R-bisectorial (with angle μ) and satisfy the twoquadratic estimates

E

∥∥∥∑

k

εkq(2kA)u∥∥∥

X� ‖u‖X , u ∈ X,

E

∥∥∥∑

k

εkq(2kA∗)v∥∥∥

X∗ � ‖v‖X∗ , v ∈ X∗,(7.6)

Then A has a bounded H∞ functional calculus (with angle μ).

Proof. Let u ∈ X , v ∈ X∗, θ ∈ (μ, π2), and f ∈ H∞(Sθ ). Then

|〈 f (A)u, v〉| ≤∑

k

|〈q(2kA) f (A)u, q(2k+1A∗)v〉|

� E

∥∥∥∑

k


XE

∥∥∥∑

k

εkq(2kA∗)v∥∥∥

X∗

� E

∥∥∥∑

k


X‖v‖X∗ .

The proof is then concluded as in Proposition 7.3. �

8. L p theory for operators with variable coefficients

In this section, we give the proofs of the results stated in Subsection 3.3, and somevariations. The core result, Theorem 8.1, is a generalization of [19, Theorem 3.1].The key ingredients of the proof are contained in [19]. Here we indicate where [19]needs to be modified, using the results from the preceding sections. Let us recall that� denotes a Hodge-Dirac operator with constant coefficients as defined in Definition3.4 and that�B denotes a Hodge-Dirac operator with variable coefficients as definedin Definition 3.8. We also use the following notation.

RBt := (I + i t�B)

−1 = r(t�B),

P Bt := (I + t2�2

B)−1 = p(t�B),

Q Bt := t�B P B

t = q(t�B),

and denote by Rt , Pt , Qt the corresponding functions of �.


THEOREM 8.1. Let 1 ≤ p1 < p2 ≤ ∞, and let �B be an R-bisectorial Hodge-Dirac operator with variable coefficients in L p which Hodge-decomposes L p for allp ∈ (p1, p2). Then

E

∥∥∥∑

k

εk Q B2k u∥∥∥

p� ‖u‖p, u ∈ Rp(�), (8.2)

and

E

∥∥∥∑

k

εk

(Q B

2k

)∗v

∥∥∥p′ � ‖v‖p′ , v ∈ Rp′(�∗). (8.3)

Proof. We first prove (8.2). Let us recall the following notation from [19]. Let

� :=⋃k∈Z

�2k , �2k := {2k([0, 1)n + m) : m ∈ Zn}.

denote a system of dyadic cubes, and

A2k u(x) := 〈u〉Q := 1

|Q|∫

Qu(y) dy, x ∈ Q ∈ �2k .

be the corresponding conditional expectation projections. Let

γ2k (x)w := Q B2k (w)(x) :=

∑Q∈�2k

Q B2k (w1Q)(x), x ∈ Rn, w ∈ CN . (8.4)

denote the principal part of Q B2k , which we also identify with the corresponding point-

wise multiplication operator.The proof of (8.2) now divides into the following four estimates:

E

∥∥∥∑

k

εk Q B2k (I − P2k )u

∥∥∥p

� ‖u‖p, u ∈ Rp(�). (8.5)

E

∥∥∥∑

k

εk(QB2k − γ2k A2k )P2k u

∥∥∥p

� ‖u‖p, u ∈ Rp(�). (8.6)

E

∥∥∥∑

k

εkγ2k A2k (I − P2k )u∥∥∥

p� ‖u‖p, u ∈ Rp(�). (8.7)

E

∥∥∥∑

k

εkγ2k A2k u∥∥∥

p� ‖u‖p, u ∈ Rp(�). (8.8)

Inequality (8.5) follows from the fact that �B Hodge-decomposes L p, and from theR-bisectoriality of �B : as in [19, Lemma 6.3], denoting by P �B the projection ontoRp( �B) in the Hodge-decomposition, we have

Q B2k (I − P2k )u = (I − P B

2k )P �B Q2k u, u ∈ Rp(�),

and


E

∥∥∥∑

k

εk

(I − P B

2k

)P �B Q2k u

∥∥∥p

� E

∥∥∥∑

k

εk Q2k u∥∥∥

p� ‖u‖p, u ∈ L p, (8.9)

where the last inequality follows from Theorem 3.6.To prove (8.6) and (8.7), we first note that commutators of the form [ηI, �]

are multiplication operators by an L∞ function bounded by ‖∇η‖∞. Then theR-bisectoriality of �B and [19, Proposition 6.4] give the following off-diagonal R-bounds: for every M ∈ N, every Borel subsets Ek, Fk ⊂ Rn , every uk ∈ L p, andevery (tk)k∈Z ⊆ {2 j } j∈Z such that dist(Ek, Fk)/tk > � for some � > 0 and all k ∈ Z,there holds

E

∥∥∥∑

k

εk1Ek Q Btk 1Fk uk

∥∥∥p

� (1 + �)−ME

∥∥∥∑

k

εk1Fk uk

∥∥∥p. (8.10)

This, in turn, gives that the family (γ2k A2k )k∈Z is R-bounded exactly as in [19,Lemma 6.5].

Now let us prove (8.6). This is similar to [19, Lemma 6.6] but, since a few mod-ifications need to be made, we include the proof. Letting vk = P2k u, and using theoff-diagonal R-bounds, we have:

E

∥∥∥∑

k

εk

(Q B

2k − γ2k A2k

)vk

∥∥∥p

= E

∥∥∥∑

k

εk

∑Q∈�2k

1Q Q B2k

(vk − 〈vk〉Q

) ∥∥∥p.

≤∑

m∈Zn

E

∥∥∥∑

k

εk

∑Q∈�2k

1Q Q B2k

(1Q−2k m(vk − 〈vk〉Q)

) ∥∥∥p

�∑

m∈Zn

(1 + |m|)−ME

∥∥∥∑

k

εk

∑Q∈�2k

1Q(vk − 〈vk〉Q+2k m)

∥∥∥p. (8.11)

A version of Poincaré’s inequality [19, Proposition 4.1] allows to majorize the lastfactor by

∫

[−1,1]n

∫ 1

0E

∥∥∥∑

k

εk2k(m + z) · ∇ τt2k (m+z)P2k u∥∥∥

pdt dz.

We then use Propositions 5.2 and 5.6 to estimate this term by

(1 + |m|)(1 + log+ |m|)‖u‖p,

and the proof is thus completed by picking M large enough.To prove (8.7), we first use the R-boundedness of (γ2k A2k ) and the idempotence of

A2k . We thus have to show that

E

∥∥∥∑

k

εk A2k (I − P2k )u∥∥∥

p� ‖u‖p, u ∈ Rp(�).


This is essentially like [19, Proposition 5.5]. We indicate the beginning of the argu-ment, where the operator-theoretic Lemma 7.1 replaces a Fourier multiplier trick usedin [19]. Indeed, with u ∈ Rp(�) ⊆ Rp(�), we have

E

∥∥∥∑

k

εk A2k (I − P2k )u∥∥∥

p� E

∥∥∥∑

k

εk A2k (I − P2k )∑

j

Q2 j Q2 j+1u∥∥∥

p

≤∑

m

E

∥∥∥∑

k

εk(

A2k (I − P2k )Q2k+m

)Q2k+m+1u

∥∥∥p.

Thanks to the second estimate in (8.9), it suffices to show that the operator family

{A2k (I − P2k )Q2k+m : k ∈ Z

} ⊂ L (L p)

is R-bounded with R-bound C2−δ|m| for some δ > 0. This is done by repeating theargument of [19, Proposition 5.5].

Finally, the proof of (8.8) is done exactly as in [19, Theorem 8.2 and Proposition9.1]. Notice that this last part is the only place where we need the assumptions for pin an open interval (p1, p2); all the other estimates work for a fixed value of p.

This completes the proof of (8.2).Let us turn our attention to (8.3). By Lemma 6.3, we have that �B

∗ = �∗ +B2

∗ �∗B1∗ is a Hodge-Dirac operator with variable coefficients in L p′

which Hodge-decomposes L p′

. It is also R-bisectorial, as this property is preserved under duality(see [21, Lemma 3.1]). So the proof of (8.2) adapts to give the quadratic estimate (8.3)involving

(Q B

t

)∗ = t�B∗(I + (t�B

∗)2)−1. �

We now prove Theorem 3.13 as a corollary.

COROLLARY 8.12. Let 1 ≤ p1 < p2 ≤ ∞, μ ∈ (ω, π/2), and let �B be aHodge-Dirac operator with variable coefficients in L p which Hodge-decomposes L p

for all p ∈ (p1, p2). Assume also that �B is a Hodge-Dirac operator with variablecoefficients in L p. Then �B has a bounded H∞ functional calculus (with angle μ)in L p(Rn; CN ) for all p ∈ (p1, p2) if and only if it is R-bisectorial (with angle μ) inL p(Rn; CN ) for all p ∈ (p1, p2).

Proof. The fact that a bounded H∞ functional calculus implies R-bisectoriality is ageneral property (see Remark 2.5). To prove the other direction, assume that �B isR-bisectorial on L p(Rn; CN ) for all p ∈ (p1, p2). By Theorem 8.1, we have that

E

∥∥∥∑

k


p� ‖u‖p, u ∈ Rp(�). (8.13)

Moreover, since�B also satisfies the assumptions of Theorem 8.1 (using Lemma 6.4),we have that

E

∥∥∥∑

k

εk2k�B(I + (2k�B)2)−1u

∥∥∥p

� ‖u‖p, u ∈ Rp( �). (8.14)


For u ∈ Rp( �), there holds

2k�B(I + (2k�B)2)−1u = 2k B2�B1(I + (2k�B)

2)−1u

= 2k B2�(I + (2k�B)2)−1 B1u

= 2k B2�B(I + (2k�B)2)−1 B1u.

Thus by (B2), the estimate (8.14) implies

E

∥∥∥∑

k


p� ‖u‖p, u ∈ Rp( �B). (8.15)

Combining (8.13) and (8.15) with the Hodge-decomposition and the obvious fact thatQ B

2k annihilates Np(�B), we arrive at

E

∥∥∥∑

k


p� ‖u‖p, u ∈ L p.

In the same way, one gets the dual estimate

E

∥∥∥∑

k

εk(QB2k )

∗u∥∥∥

p′ � ‖u‖p′ , u ∈ L p′,

where p′ denotes the conjugate exponent of p. The functional calculus then followsfrom Proposition 7.5. �

COROLLARY 8.16. Let 1 ≤ p1 < p2 ≤ ∞, and let �A be a Hodge-Dirac oper-ator with variable coefficients, which is R-bisectorial in L p and Hodge-decomposesL p for all p ∈ (p1, p2). Then for each p ∈ (p1, p2), there exists δ = δp > 0 suchthat, if �B and �B are Hodge-Dirac operators with variable coefficients such that‖B1 − A1‖∞ +‖B2 − A2‖∞ < δ, then�B has an H∞ functional calculus in L p andHodge-decomposes L p.

Proof. Let p ∈ (p1, p2). By Proposition 6.10, we have that, for δ small enough, �B

Hodge-decomposes L p. We need to show that �B is R-bisectorial in L p provided δis sufficiently small. As in the proof of Proposition 6.10, let T1 ∈ L (Rp( �A), L p)

and T2 ∈ L (L p) be operators of norm ‖Ti‖ � δ such that

B1 �= (I − T1)A1 �, �B2 = �A2(I − T2).

Then

�B = � + B1 �B2 = � + (I − T1)A1 �A2(I − T2) = (I − T1P �A)�A(I − P�T2),

where P� and P �A are the Hodge-projections associated to �A, onto Rp(�) andRp( �A), respectively. Hence

I + i t�B = (I − T1P �A )(I + i t�A)(I − P�T2)+ (T1P �A + P�T2)

= (I − T1P �A )(I + i t�A)(I − P�T2)

×[

I + (I − P�T2)−1(I + i t�A)

−1(I − T1P �A )−1(T1P �A + P�T2)

],


where the inverses involving Ti exist for δ small enough. Hence (I + i t�B)−1 can

be expressed as a Neumann series involving powers of the operators (I + i t�A)−1,

which are R-bounded by assumption, times powers of fixed bounded operators, includ-ing T1P �A +P�T2 which has norm at most Cδ. For δ small enough, the R-boundednessof (I + i t�B)

−1 follows from this representation.Given ε ∈ (p − p1, p2 − p) we thus have that there exists δp,ε such that �B is

R-bisectorial in L p−ε and in L p+ε, and Hodge decomposes L p−ε and L p+ε. By inter-polation (cf. Remark 3.12),�B is R-bisectorial in L p and Hodge decomposes L p forall p ∈ (p − ε, p + ε).

Now the conditions of Corollary 8.12 are verified for the operators�B and�B , sothe mentioned result implies that �B has a bounded H∞ functional calculus in L p,as claimed. �

With potential applications to boundary value problems in mind (see [4]), we con-clude this section with the following special case. This proof is essentially the sameas in the L2 case [10, Theorem 3.1].

COROLLARY 8.17. Let 1 ≤ p1 < p2 ≤ ∞. Let D = −i∑n

j=1 D j∂ j be a first-

order differential operator with D j ∈ L (CN ), and A ∈ L∞(Rn;L (CN )) be suchthat

|ξ ||e| � |D(ξ)e| for all e ∈ R(D(ξ)), and

σ(D(ξ)) ⊆ Sω for some ω ∈(

0,π

2

)and all ξ ∈ Rn, (H1)

‖u‖p � ‖Au‖p for all u ∈ Rp(D), (H2)

‖u‖p′ � ‖A∗u‖p′ for all u ∈ Rp′(D∗), (H3)

for all p ∈ (p1, p2). Then we have the following:

(1) The operator D A has an H∞ functional calculus (with angle μ) in L p forall p ∈ (p1, p2) if and only if it is R-bisectorial (with angle μ) in L p for allp ∈ (p1, p2).

(2) If the equivalent conditions of (1) hold, then for each p ∈ (p1, p2), there existsδ = δp > 0 such that, if another A ∈ L∞(Rn;L (CN )) satisfies ‖A− A‖∞ < δ,then D A also has an H∞ functional calculus in L p.

Proof. (1) The philosophy of the proof is to reduce the consideration of an operatorof the form D A to the Hodge-Dirac operator with variable coefficients�B , which wealready understand from the previous results. On CN ⊕ CN , consider the matrices

� j :=(

0 0D j 0

), ˆ�j :=

(0 D j

0 0

), B1 :=

(A 00 0

), B2 :=

(0 00 A

).

We define the associated differential operators �, �and � acting in L p :=L p(Rn; CN ⊕ CN ) as in Subsections 3.2, and the operators


�B =(

0 AD AD 0

), �B =

(0 DAD A 0

),

as in Subsection 3.3.We claim that both�B and�B are then Hodge-Dirac operators with variable coeffi-

cients in L p which Hodge-decompose L p for all p ∈ (p1, p2). Indeed, the hypotheses(H1), (H2) and (H3) guarantee the conditions (�1), (�2) and (B2). The remainingrequirements (�3) and (B1), as well as the Hodge decomposition (see [19, Lemma3.5]), are satisfied because of the special form of �B .

A computation shows that

(I + i t�B)−1 =

(I −i t AD A0 I

)(I 00 (I + t2(D A)2)−1

)(I 0−i t D I

).

Assuming that D A is R-bisectorial, we check that so is�B . This amounts to verifyingthe R-boundedness of the families of operators

(I + t2(D A)2)−1, −i t AD A(I + t2(D A)2)−1, −i t (I + t2(D A)2)−1 D,

where t ∈ R. For the first two, this is immediate from the R-bisectoriality of D A andthe boundedness of A. For the third one, we need the stability of R-boundedness inthe L p spaces under adjoints, and hypothesis (H3) which allows to reduce the adjoint−i t D∗(I + t2(A∗ D∗)2)−1 to a function of (D A)∗ = A∗ D∗ by composing with A∗from the left.

Thus, by Corollary 8.12, �B has an H∞ functional calculus on L p for all p ∈(p1, p2). But the resolvent formula above also gives

f (�B)

(Auu

)=(

A f (D A)uf (D A)u

),

for f ∈ H∞0 (Sθ ) and u ∈ L p, and hence we find that D A has an H∞ functional cal-

culus, too. This completes the proof that the R-bisectoriality of D A implies functionalcalculus. The converse direction is a general property of the functional calculus in L p

(see Remark 2.5).(2) We turn to the second part and assume that D A is R-bisectorial. We first note

that A also satisfies the hypotheses (H2) and (H3) when δ is small enough. Indeed, foru ∈ Rp(D), we have ‖ Au‖p ≥ ‖Au‖p − ‖A − A‖∞‖u‖p ≥ (c − δ)‖u‖p, and (H3)is proven similarly. Moreover, the R-bisectoriality of D A implies the same propertyfor D A by a Neumann series argument, as

I +i t D A= I +i t D A−i t D(A− A)=(I +i t D A)[I −(I +i t D A)−1i t D(A− A)],where the family {(I + i t D A)−1i t D : t ∈ R} is R-bounded (by duality, (H3), andthe R-bisectoriality of D A), and the factor ‖A − A‖∞ < δ ensures convergence forδ small enough. The H∞ calculus then follows from part (1) applied to A in placeof A. �


9. Lipschitz estimates

In this final section, we prove Lipschitz estimates of the form

‖ f (�B)u − f (�A)u‖p � maxi=1,2

‖Ai − Bi‖∞‖ f ‖∞‖u‖p,

for small perturbations of the coefficient matrices involved in the Hodge-Dirac oper-ators. Such estimates are obtained via holomorphic dependence results for perturba-tions Bz depending on a complex parameter z. This technique can be seen as one ofthe original motivations for studying the Kato problem for operators with complexcoefficients. We start with the operators studied in Corollary 8.17, and then deducesimilar estimates for general Hodge-Dirac operators with variable coefficients, as in[3, Section 10.1].

Let D be a first-order differential operator as in Corollary 8.17. Let U be an openset of C and (Az)z∈U a family of multiplication operators such that the map z �→ Az ∈L∞(Rn; CN ) is holomorphic. Let 1 ≤ p1 < p2 ≤ ∞, z0 ∈ U , and assume that D Az0

has a bounded H∞ functional calculus in L p for all p ∈ (p1, p2). By Corollary 8.17,for each p ∈ (p1, p2), there then exists a δ = δp > 0 such that D Az has a boundedH∞ functional calculus in L p for all z ∈ B(z0, δ). Moreover, we have the following.

PROPOSITION 9.1. For θ ∈ (μ, π2 ) and f ∈ H∞(Sθ ), the function z �→ f (D Az)

is holomorphic on D(z0, δ).

Proof. This is entirely similar to [10, Theorem 6.1, Theorem 6.4]. Letting τ ∈ C\Sθ ,we have

d

dz(I + τD Az)

−1 = −(I + τD Az)−1τD A′

z(I + τD Az)−1.

From (H1), (H3) and the bisectoriality of D Az , these operators are uniformly boundedfor z ∈ U , and thus the functions z �→ (I + τD Az)

−1 are holomorphic. The result isthen obtained by passing to uniform limits in the strong operator toplogy. �

The Lipschitz estimates now follow.

COROLLARY 9.2. Let 1 ≤ p1 < p2 ≤ ∞, let D be a first-order differential oper-ator, and A ∈ L∞(Rn; CN ) a multiplication operator which satisfy the hypotheses(H1), (H2) and (H3) of Corollary 8.17. Let moreover D A be R-bisectorial. Then, foreach p ∈ (p1, p2), there exists δ = δp > 0 such that, if A ∈ L∞(Rn; CN ) satisfies‖A − A‖∞ < δ, then D A has a bounded H∞ functional calculus in L p with someangle ω ∈ (0, π2 ), and for θ ∈ (ω, π2 ), f ∈ H∞(Sθ ), and u ∈ L p we have

‖ f (D A)u − f (D A)u‖p � ‖A − A‖∞‖ f ‖∞‖u‖p.

Proof. Let Az := A + z( A − A)/‖ A − A‖∞. Then A0 = A, Az1 = A for z1 =‖ A−A‖∞, and z �→ Az is holomorphic. For z ∈ D(0, δ), where δ is small enough, D Az


has a bounded H∞ functional calculus in L p by Corollary 8.17, and z �→ f (D Az) isholomorphic for f ∈ H∞(Sθ ) by Proposition 9.1. By the Schwarz Lemma,

‖ f (D A0)u − f (D Az1)u‖p � |z1| ‖ f ‖∞‖u‖p,

which gives the assertion. �

We finally turn to the Lipschitz estimates for Hodge-Dirac operators with variablecoefficients, using the same approach as in [3, Section 10.1].

COROLLARY 9.3. Let 1 ≤ p1 < p2 ≤ ∞, and let �A and �A be Hodge-Diracoperators with variable coefficients, where �A is R-bisectorial in L p and Hodge-decomposes L p for all p ∈ (p1, p2). Then, for each p ∈ (p1, p2), there exists δ =δp > 0 such that, if�B and�B are also Hodge-Dirac operators with variable coeffi-cients with ‖A1−B1‖∞+‖A2−B2‖∞ < δ, then both�A and�B have a bounded H∞functional calculus with some angle ω ∈ (0, π2 ), and for θ ∈ (ω, π2 ), f ∈ H∞(Sθ ),and u ∈ L p there holds

‖ f (�A)u − f (�B)u‖p � maxi=1,2

‖Ai − Bi‖∞‖ f ‖∞‖u‖p.

Proof. The philosophy of the proof is analogous to that of Corollary 8.17 but goes inthe opposite direction: we now deduce results for operators of the form�A from whatwe already know for the operators D A. To this end, consider the space L p ⊕ L p ⊕ L p

and the operators

D :=⎛⎝

0 0 00 0 �

0 � 0

⎞⎠ , A :=

⎛⎝

0 0 00 A1 00 0 A2

⎞⎠.

Let us write PA0 ,P

A� and P

A�A

for the Hodge-projections associated to�A. By (B2),

the restriction A1 : Rp( �) → Rp( �A) is an isomorphism, and we write A−11 for its

inverse. Then a computation shows that

(I + i t D A)−1 =⎛⎝

I 0 00 A−1

1 PA�A(I + t2�2

A)−1 A1 −i t �A2(I + t2�2

A)−1

0 −i t�(I + t2�2A)

−1 A1 PA�(I + t2�2

A)−1

⎞⎠,

and one can check that the R-bisectoriality and the Hodge-decomposition of�A implythe R-bisectoriality of D A. Indeed, for the diagonal elements above it is immediate, andfor the non-diagonal elements it follows after writing �A2 = A−1

1 �A = A−11 P

A�A�A

and � = PA��A.

We next define operators SA : L p → L p ⊕ L p ⊕ L p and TA : L p ⊕ L p ⊕ L p → L p

by


SAu :=(P

A0 u, A−1

1 PA�A

u,PA�u),

TA(u, v, w) := PA0 u + P

A�w + P

A�A

A1v.

One checks that TA SA = I and SA�A = (D A)SA. Hence SA(λ − �A)−1 = (λ −

D A)−1SA, and then by the definition of the functional calculus,

f (�A)u = TA f (D A)SAu, f ∈ H∞(Sθ ), u ∈ L p.

We repeat the above definitions and observations with B in place of A, and then

f (�A)u − f (�B)u = [TA − TB] f (D A)SAu + TB[ f (D A)

− f (DB)]SAu + TB f (DB)[SA − SB]u.The asserted Lipschitz estimate then follows by using Corollary 9.2 for the middleterm, and Proposition 6.10 and Corollary 6.11 for the other two terms. �

10. Addendum

Since the submission of this paper, two of the authors have shown that the assump-tions of the main theorems concerning an interval of exponents p ∈ (p1, p2) may berelaxed: If the assumed R-bisectoriality conditions hold in L p for one p = p0, theyautomatically hold in L p for all p in some open interval (p1, p2) � p0. See [18] fordetails.

Acknowledgments

This work advanced through visits of T.H. and P.P. at the Centre for Mathematics and itsApplications at the Australian National University, and of P.P. at the University of Hel-sinki. Thanks go to these institutions for their outstanding support. The research wassupported by the Australian Government through the Australian Research Council,and by the Academy of Finland through the project 114,374 “Vector-valued singularintegrals”.

REFERENCES

[1] D. Albrecht, X. Duong, A. McIntosh, Operator theory and harmonic analysis. In Instructional Work-shop on Analysis and Geometry, Part III (Canberra 1995), Proc. Centre Math. Appl. Austral. Nat.Univ., 34 (1996), 77–136.

[2] P. Auscher, On necessary and sufficient conditions for L p estimates of Riesz transforms associatedto elliptic operators on Rn and related estimates. Mem. Amer. Math. Soc. 871 (2007).

[3] P. Auscher, A. Axelsson, A. McIntosh, On a quadratic estimate related to the Kato conjecture andboundary value problems. In Proceedings of the 8th International Conference on Harmonic Anal-ysis and PDE’s (El Escorial 2008), Contemp. Math., Amer. Math. Soc., Providence, RI, to appear(math.CA/0810.3071).


[4] P. Auscher, A. Axelsson, A. McIntosh, Solvability of elliptic systems with square integrable bound-ary data, Ark. Mat., to appear (math.AP/0809.4968).

[5] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, Ph. Tchamitchian, The solution of the Kato squareroot problem for second order elliptic operators on R

n . Ann. of Math. (2) 156 (2002), no. 2, 633–654.[6] P. Auscher, J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators.

Part III: Harmonic Analysis of elliptic operators, J. Funct. Anal. 241 (2006), 703–746.[7] P. Auscher, J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators.

Part I: General operator theory and weights, Adv. Math. 212 (2007), 225–276.[8] P. Auscher, J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators.

Part II: Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007), 265–316.[9] P. Auscher, A. McIntosh, E. Russ, Hardy spaces of differential forms and Riesz transforms on

Riemannian manifolds. J. Geom. Anal., 18 (2008), 192–248.[10] A. Axelsson, S. Keith, A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac

operators. Invent. Math. 163 (2006), no. 3, 455–497.[11] A. Axelsson, S. Keith, A. McIntosh, The Kato square root problem for mixed boundary value

problems, J. London Math. Soc. 74 (2006), 113–130.[12] S. Blunck, P. C. Kunstmann, Calderón-Zygmund theory for non-integral operators and the H∞-

functional calculus, Rev. Mat. Iberoamericana 19 (2003), no. 3, 919–942.[13] J. Bourgain, Vector-valued singular integrals and the H1-BMO duality. In Probability theory and

harmonic analysis (Cleveland, Ohio, 1983). Monogr. Textbooks Pure Appl. Math., 98, Dekker,New York (1986), 1–19.

[14] D. L. Burkholder, Martingales and singular integrals in Banach spaces. In Handbook of the geometryof Banach spaces, Vol. I, 233–269, North-Holland, Amsterdam, 2001.

[15] M. Cowling, I. Doust, A. McIntosh, A. Yagi, Banach space operators with a bounded H∞ functionalcalculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.

[16] R. Denk, G. Dore, M. Hieber, J. Prüss, A. Venni, New thoughts on old results of R. T. Seeley, Math.Ann., 328 (2004) 545–583.

[17] M. Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Appli-cations 169, Birkhäuser Verlag, Basel (2006)

[18] T. Hytönen, A. McIntosh, Stability in p of the H∞-calculus of first-order systems in L p . In TheAMSI-ANU workshop on spectral theory and harmonic analysis (Canberra, 2009). Proc. CentreMath. Appl. Austral. Nat. Univ., 44 (2010), 167–181.

[19] T. Hytönen, A. McIntosh, P. Portal, Kato’s square root problem in Banach spaces. J. Funct. Anal.254 (2008), no. 3, 675–726.

[20] N. J. Kalton, P. C. Kunstmann, L. Weis, Perturbation and interpolation theorems for the H∞-cal-culus with applications to differential operators, Math. Ann, 336 (2006) no. 4, 747–801.

[21] N. J. Kalton, L. Weis, The H∞-calculus and sums of closed operators, Math. Ann. 321 (2001),no. 2, 319–345.

[22] P. C. Kunstmann, L. Weis, Maximal L p regularity for parabolic problems, Fourier multiplier theo-rems and H∞-functional calculus, In Functional Analytic Methods for Evolution Equations (Edi-tors: M. Iannelli, R. Nagel, S. Piazzera). Lect. Notes in Math. 1855, Springer-Verlag (2004).

[23] A. McIntosh, Operators which have an H∞ functional calculus. In Miniconference on operatortheory and partial differential equations (North Ryde, 1986). Proc. Centre Math. Appl. Austral.Nat. Univ., 14 (1986), 210–231.

[24] A. McIntosh, A. Yagi, Operators of typeωwithout a bounded H∞ functional calculus. Proc. CentreMath. Appl. Austral. Nat. Univ., 24 (1990), 159–174.

T. HytönenDepartment of Mathematics andStatistics, University of Helsinki,Gustaf Hällströmin katu 2b,FI-00014 Helsinki, FinlandE-mail: [email protected]


A. McIntoshCentre for Mathematics and itsApplications, Australian NationalUniversity, Canberra, ACT 0200,AustraliaE-mail: [email protected]

P. PortalUniversité Lille 1, Laboratoire PaulPainlevé, 59655 Villeneuve d’Ascq,FranceE-mail: [email protected]