HOLOMORPHIC FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS IN Lp TUOMAS HYTÖNEN, ALAN MCINTOSH, AND PIERRE PORTAL Abstract. We study the boundedness of the H ∞ functional calculus for differential operators acting in Lp (Rn ; CN ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the Lp theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients ΠB as treated in L2 (Rn ; CN ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that ΠB has 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 differential operators. In particular, Axelsson, Keith, and McIntosh [10] have recovered and extended the solution of the Kato square root problem [5] by showing that Hodge-Dirac operators with variable coefficients of the form ΠB = Γ+B1 Γ∗ B2 have a bounded H ∞ functional calculus in L2 (Rn ; CN ), when Γ is a homogeneous first order differential operator with constant coefficients, and B1 , B2 ∈ L∞ (Rn ; L (CN )) are strictly accretive multiplication operators. Recently, Auscher, Axelsson, and McIntosh [4] have used related perturbation results to show the openness of some sets of well-posedness for boundary value 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 ∞ functional calculus is established using Mikhlin’s multiplier theorem. However, the estimates on the symbols may be difficult to check in practice, especially when the null spaces of the symbols are non-trivial. Here we provide a simple condition (invertibility of the symbols on their ranges and inclusion of their eigenvalues in a bisector), that gives such estimates. We then turn to operators with coefficients in L∞ (Rn ; C) of the form ΠB = Γ + B1 ΓB2 , where Γ and Γ are nilpotent homogeneous first order operators with constant (matrix-valued) coefficients, and B1 , B2 ∈ L∞ (Rn ; L (CN )) are multiplication operators satisfying some Lp coercivity condition. For such operators, we aim at perturbation results which give, in particular, the boundedness of the H ∞ functional calculus when B1 , B2 are small pertubations of constant-coefficient matrices. This presents two main difficulties. First of all, even in L2 , the H ∞ functional calculus of a (bi)sectorial operator is in general not stable under small perturbations in the sense that there exist a self-adjoint operator D and bounded operators A with arbitrary small norm such that D(I + A) does not have a bounded H ∞ functional calculus (see [23]). Subtle functional analytic perturbation results exist (see [16] and [19]), but do not give the estimates needed in [4] or [10]. To obtain such estimates, one needs to take advantage of the specific structure of differential operators using harmonic analytic methods. Then, the problem of moving from the L2 theory to an Lp theory is 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 by Auscher and Martell [2, 6, 7, 8], Date: July 10, 2009. 2000 Mathematics Subject Classification. 47A60, 47F05. 1
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consists in establishing an extrapolation method adapted to the operator, which allows to extend results from L2 to Lp for p in a certain range (p1 , p2 ) containing 2. In [18] we started another approach, which combines probabilistic tools from functional analysis with the aforementioned L2 methods, and allows Lp results which do not rely on some L2 counterparts. However, our goal in [18] was the Kato problem, and we did not reach the generality of [10] which has recently proven particularly useful in connection with boundary value problems [4]. Here we close this gap and, in fact, reach a further level of generality. Roughly speaking, for quite general differential operators, we show that the boundedness of the H ∞ functional calculus coincides with the R-(bi)sectoriality (see Section 2 for relevant definitions). This then allows perturbation results, in contrast with the general theory of sectorial operators, where R-sectoriality and bounded H ∞ calculus are two distinct properties, and perturbation results are much more restricted. For the operators with variable coefficients, the core of the argument is contained in [18], so the reader might want to have a copy of this paper handy. Here we focus on the points where [18] needs to be modified, and develop some adaptation of the techniques to generalized Hodge-Dirac operators which may be of interest in other problems. To make the paper more readable, we choose not to work in the Banach-space valued setting of [18], but the interested reader will soon realize that our proof carries over to that situation provided that, as in [18], the target space is a UMD space, and both the space and its dual have the RMF property. The paper is organized as follows. In Section 2, we recall the essential definitions. In Section 3, we present our setting and state the main results. In Section 4, we deal with constant coefficient operators and obtain appropriate estimates on their symbols. In Section 5, we use these estimates to establish an Lp theory for operators with constant (matrix-valued) coefficients. In Section 6, we show that a certain (Hodge) decomposition, crucial in our study, is stable under small perturbations. In Section 7, we give simple proofs of general operator theoretic results on the functional calculus of bisectorial operators. In Section 8, we prove our key results on operators with variable coefficients, referring to [18] when arguments are identical, and explaining how to modify them using the results of the preceding sections when they are not. Finally, in Section 9, we derive from Section 8 Lipschitz estimates for the functional calculus of these operators. Acknowledgments. This work advanced through visits of T.H. and P.P. at the Centre for Mathematics and its Applications at the Australian National University, and of P.P. at the University of Helsinki. Thanks go to these institutions for their outstanding support. The research was supported by the Australian Government through the Australian Research Council, and by the Academy of Finland through the project 114374 “Vector-valued singular integrals”. 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 operator norm in L (CN ), are denoted by | · |. To express the typical inequalities “up to a constant” we use the notation a . b to mean that there exists C < ∞ such that a ≤ Cb, and the notation a h b to mean that a . b . a. The implicit constants are meant to be independent of other relevant quantities. If we want to mention that the constant C depends on a parameter p, we write a .p b. Let us briefly recall the construction of the H ∞ functional calculus (see [1, 15, 17, 21, 22] for details). 2.1. Definition. A closed operator A acting in a Banach space Y is called bisectorial with angle θ if its spectrum σ(A) is included in a bisector: σ(A) ⊆ Sθ := Σθ ∪ (−Σθ ),
where
Σθ := {z ∈ C ; | arg(z)| ≤ θ},
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and outside the bisector it verifies the following resolvent bounds: π (2.2) ∀θ0 ∈ (θ, ) ∃C > 0 ∀λ ∈ C \ Sθ0 kλ(λI − A)−1 kL (Y ) ≤ C. 2 We often omit the angle, and say that A is bisectorial if it is bisectorial with some angle θ ∈ [0, π2 ). For 0 < ν < π/2, let H ∞ (Sν ) be the space of bounded functions on Sν , which are holomorphic on the interior of Sν , and consider the following subspace of functions with decay at zero and infinity: o n z α | . H0∞ (Sν ) := φ ∈ H ∞ (Sν ) : ∃α, C ∈ (0, ∞) ∀z ∈ Sν |φ(z)| ≤ C| 1 + z2 For a bisectorial operator A with angle θ < ω < ν < π/2, and ψ ∈ H0∞ (Sν ), we define Z 1 ψ(A)u := ψ(λ)(λ − A)−1 u dλ, 2iπ ∂Sω where ∂Sω is directed anti-clockwise around Sω . 2.3. Definition. A bisectorial operator A with angle θ, is said to admit a bounded H ∞ functional calculus with angle µ ∈ [θ, π2 ) if, for each ν ∈ (µ, π2 ), ∃C < ∞
∀ψ ∈ H0∞ (Sν ) kψ(A)ykY ≤ Ckψk∞ kykY .
In this case, and if Y is reflexive, one can define a bounded operator f (A) for f ∈ H ∞ (Sν ) by f (A)u := f (0)P0 u + lim ψn (A)u, n→∞
where P0 denotes the projection on the null space of A corresponding to the decomposition Y = N(A) ⊕ R(A), which exists for R-bisectorial operators, and (ψn )n∈N ⊂ H0∞ (Sν ) is a bounded sequence which converges locally uniformly to f . See [1, 15, 17, 21, 22] for details. 2.4. Definition. A family of operators T ⊂ L (Y ) is called R-bounded if for all M ∈ N, all T1 , . . . , TM ∈ T , and all u1 , . . . , uM ∈ Y , M M
X
X
εk uk , εk Tk uk . E E k=1
Y
k=1
Y
where E is the expectation which is taken with respect to a sequence of independent Rademacher variables εk , i.e., random signs with P(εk = +1) = P(εk = −1) = 21 . A bisectorial operator A is called R-bisectorial with angle θ in Y if the collection {λ(λI − A)−1 : λ ∈ C \ Sθ0 } is R-bounded for all θ0 ∈ (θ, π/2). The infimum of such angles θ is called the angle of Rbisectoriality 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 + itA)−1 : t ∈ R}. The reader unfamiliar with R-boundedness and the derived notions can consult [18] and the references therein. 2.5. Remark. On subspaces of Lp , 1 < p < ∞, an operator with a bounded H ∞ functional calculus is R-bisectorial. The proof (stated for sectorial rather than bisectorial operators) can be found in [20, Theorem 5.3]. 3. Main results We consider three types of operators. First, we look at differential operators of arbitrary order with constant (matrix valued) coefficients, and provide simple conditions on their Fourier multiplier symbols to ensure that such operators are bisectorial and, in fact, have a bounded H ∞ functional calculus. Then, we focus on first order operators with a special structure, the Hodge-Dirac operators, and prove that, under an additional condition on the symbols, they give a specific (Hodge) decomposition of Lp . Finally we turn to Hodge-Dirac operators with (bounded
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measurable) variable coefficients, and show that the boundedness of the H ∞ functional calculus is preserved under small perturbation of the coefficients. We work in the Lebesgue spaces Lp := Lp (Rn ; CN ) with p ∈ (1, ∞), and denote by S (Rn ; CN ) the Schwartz class of rapidly decreasing functions with values in CN , and by S 0 (Rn ; CN ) the corresponding class of tempered distributions. 3.A. General constant-coefficient operators. In this subsection, we consider kth order homogeneous differential operators of the form X ˆ θ∂θ D = (−i)k D θ∈Nn :|θ|=k
P ˆ ˆ θ ˆ acting on S 0 (Rn ; CN ) as a Fourier multiplier with symbol D(ξ) = |θ|=k Dθ ξ , where Dθ ∈ N L (C ). ˆ 3.1. Assumption. The Fourier multiplier symbol D(ξ) satisfies (D1) (D2)
ˆ κ|ξ|k |e| ≤ |D(ξ)e| for all ξ ∈ Rn , there exists ω ∈ [0,
ˆ all e ∈ R(D(ξ)),
π ) such that for all ξ ∈ Rn : 2
and some
κ > 0,
ˆ σ(D(ξ)) ⊆ Sω .
In each Lp , let D act on its natural domain Dp (D) := {u ∈ Lp ; Du ∈ Lp }. In Theorem 5.1 we prove: 3.2. Theorem. Let 1 < p < ∞. Under the assumptions (D1) and (D2), the operator D is bisectorial in Lp with angle ω, and has a bounded H ∞ functional calculus in Lp with angle ω. 3.3. Remark. (a) In (D2), the bisector Sω can be replaced by the sector Σω where 0 ≤ ω < π. The operator D is then sectorial (with angle ω) and has a bounded H ∞ functional calculus (with angle ω) in the sectorial sense, i.e. f (D) is bounded for functions f ∈ H ∞ (Σθ ) with any θ ∈ (ω, π). (b) Assuming that (D1) holds for all e ∈ CN would place us in a more classical context, in which proofs are substantially simpler. We insist on this weaker ellipticity condition since the operators we want to handle have, in general, a non-trivial null space. (c) Using Bourgain’s version of Mikhlin’s multiplier theorem [13] , the above theorem extends to functions with values in X N , where X is a UMD Banach space. 3.B. Hodge-Dirac operators with constant coefficients. We now turn to first order operators of the form Π = Γ + Γ, Pn ˆ 0 n where Γ = −i Γj ∂j , acts on S (R ; CN ) as a Fourier multiplier with symbol j=1
ˆ = Γ(ξ) ˆ Γ =
n X
ˆ j ξj , Γ
ˆ j ∈ L (CN ), Γ
j=1
ˆ 2=0 the operator Γ is defined similarly, and both operators are nilpotent in the sense that Γ(ξ) 2 n ˆ and Γ(ξ) = 0 for all ξ ∈ R . 3.4. Definition. We call Π = Γ+ Γ a Hodge-Dirac operator with constant coefficients if its Fourier ˆ =Γ ˆ+Γ ˆ satisfies the following conditions: multiplier symbol Π (Π1) (Π2) (Π3)
ˆ ˆ κ|ξ||e| ≤ |Π(ξ)e| for all e ∈ R(Π(ξ)), ˆ σ(Π(ξ)) ⊆ Sω
all ξ ∈ Rn ,
for some ω ∈ [0,
π ), 2
and all
and some ξ ∈ Rn ,
ˆ ˆ ˆ N(Π(ξ)) = N(Γ(ξ)) ∩ N(Γ(ξ)) for all ξ ∈ Rn .
κ > 0,
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3.5. Remark. The “Hodge-Dirac” terminology has its origins in applications of this formalism to Riemannian geometry where Γ would be the exterior derivative d and Γ = d∗ . See [10] for details. Note that we are working here in a more general setting than [10], where the operator Γ was assumed to be the adjoint of Γ. In particular, our operator Π does not need to be self-adjoint in L2 (Rn ; CN ). In each Lp , we let the operators Υ ∈ {Γ, Γ, Π} act on their natural domains Dp (Υ) := {u ∈ Lp : Υu ∈ Lp }, where Υu is defined in the distributional sense. Each Υ is a densely defined, closed unbounded operator in Lp with this domain. The formal nilpotence of Γ and Γ transfers into the operatortheoretic nilpotence Rp (Γ) ⊆ Np (Γ), Rp (Γ) ⊆ Np (Γ). where Rp (Γ), Np (Γ) denote the range and kernel of Γ as an operator on Lp . In Section 5 we show that the identity Π = Γ+ Γ is also true in the sense of unbounded operators in Lp . Moreover, in Theorem 5.5 we prove: 3.6. Theorem. The operator Π has a bounded H ∞ functional calculus in Lp with angle ω, and satisfies the Hodge decomposition Lp = Np (Π) ⊕ Rp (Γ) ⊕ Rp (Γ). 3.7. Remark. As in the previous subsection, the above theorem extends to functions with values in X N , where X is a UMD Banach space. 3.C. Hodge-Dirac operators with variable coefficients. We finally turn to Hodge-Dirac operators with variable coefficients. The study of such operators is motivated by [4], [10] and [18]. 3.8. Definition. Let 1 < p < ∞ and p0 denote the dual exponent of p. Let B1 , B2 ∈ L∞ (Rn ; L (CN )), and identify these functions with bounded multiplication operators on Lp in the natural way. Also let Π = Γ + Γ be a Hodge-Dirac operator. Then the operator ΠB := Γ + ΓB ,
(3.9)
where ΓB := B1 ΓB2 ,
is called a Hodge-Dirac operator with variable coefficients in Lp if the following hold: ΓB2 B1 Γ = 0 on S (Rn ; CN ),
(B1) (B2)
kukp . kB1 ukp
∀u ∈ Rp (Γ) and kvkp0 . kB2∗ vkp0
∀v ∈ Rp0 (Γ∗ ).
Note that the operator equality (3.9), involving the implicit domain condition Dp (ΠB ) := Dp (Γ) ∩ Dp (ΓB ), was a proposition for Hodge-Dirac operators with constants coefficients, but is taken as the definition for Hodge-Dirac operators with variable coefficients. The following simple consequences will be frequently applied. Their proofs are left to the reader. First, the nilpotence condition (B1), a priori formulated for test functions, self-improves to ΓB2 B1 Γ = 0 on Dp (Γ);
hence Rp (ΓB ) ⊆ Np (ΓB ).
Second, the coercivity condition (B2) implies that Rp (ΓB ) = B1 Rp (ΓB2 ) = B1 Rp (Γ), and B1 : Rp (Γ) → Rp (ΓB ) is an isomorphism. Sometimes, we also need to assume that the related operator ΠB = Γ+B2 ΓB1 is a Hodge-Dirac operator with variable coefficients in Lp , i.e. ΓB1 B2 Γ = 0 kukp . kB2 ukp
on S (Rn ; CN ),
∀u ∈ Rp (Γ) and kvkp0 . kB1∗ vkp0
With the same proof as in [10, Lemma 4.1], one can show:
∀v ∈ Rp0 (Γ∗ ).
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3.10. Proposition. Assuming ΠB = Γ + ΓB is a Hodge-Dirac operator with variable coefficients, then the operators ΓB := B1 ΓB2 and Γ∗B := B2∗ Γ∗ B1∗ are closed, densely defined, nilpotent opera0 tors in Lp and Lp repectively, and Γ∗B = (ΓB )∗ . However, the Hodge-decomposition and resolvent bounds, which in the context of [10] (and the first-mentioned one even in [18]) could be established as propositions, are now properties which may or may not be satisfied: 3.11. Definition. We say that ΠB Hodge-decomposes Lp if Lp = Np (ΠB ) ⊕ Rp (Γ) ⊕ Rp (ΓB ). 3.12. Remark. We will mostly be interested in a Hodge-Dirac operator ΠB with the property that it is R-bisectorial in Lp and Hodge-decomposes Lp . If this property holds for two exponents p ∈ {p1 , p2 }, then it holds for the intermediate values p ∈ (p1 , p2 ) as well, and hence the set of exponents p, for which the mentioned property is satisfied, is an interval. The proof that R-bisectoriality interpolates in these spaces can be found in [19, Corollary 3.9], where it is formulated for R-sectorial operators. As for the Hodge-decomposition, observe first that if a Hodge-Dirac operator ΠB is R-bisectorial in Lp and Hodge-decomposes Lp , then the projections onto the three Hodge subspaces are given by P0 = lim (I + t2 Π2B )−1 , t→∞
PΓ = lim t2 ΓΠB (I + t2 ΠB )−1 , t→∞
PΓB = lim t2 ΓB ΠB (I + t2 ΠB )−1 , t→∞
where the limits are taken in the strong operator topology. In particular, if ΠB has these properties in two different Lp spaces, then the corresponding Hodge subspaces have common projections, and one deduces the boundedness of these projection operators also in the interpolation spaces. The following main result concerning the operators ΠB gives a characterization of the boundedness of their H ∞ functional calculus. It will be proven as Corollary 8.12 to Theorem 8.1. 3.13. Theorem. Let 1 ≤ p1 < p2 ≤ ∞, and let ΠB be a Hodge-Dirac operator with variable coefficients in Lp which Hodge-decomposes Lp for all p ∈ (p1 , p2 ). Assume also that ΠB is a Hodge-Dirac operator with variable coefficients in Lp . Then ΠB has a bounded H ∞ functional calculus (with angle µ) in Lp (Rn ; CN ) for all p ∈ (p1 , p2 ) if and only if it is R-bisectorial (with angle µ) in Lp (Rn ; CN ) for all p ∈ (p1 , p2 ). This characterization leads to perturbation results such as the following, proven in Corollary 8.16, thanks to the perturbation properties of R-bisectoriality. 3.14. Corollary. Let 1 ≤ p1 < p2 ≤ ∞, and let ΠA be a Hodge-Dirac operator with variable coefficients, which is R-bisectorial in Lp and Hodge-decomposes Lp for all p ∈ (p1 , p2 ). Then for each p ∈ (p1 , p2 ), there exists δ = δp > 0 such that, if ΠB and ΠB are Hodge-Dirac operators with variable coefficients, and if kB1 − A1 k∞ + kB2 − A2 k∞ < δ, then ΠB has a bounded H ∞ functional calculus in Lp and Hodge-decomposes Lp . 3.15. Remark. The results in this paper concerning Hodge-Dirac operators with variable coefficients can be extended to the Banach space valued setting, provided the target space has the so-called UMD and RMF properties, and also its dual has RMF. The UMD property, which passes to the dual automatically, is a well-known notion in the theory of Banach spaces, cf. [14]. We introduced the RMF property in [18] in relation with our Rademacher maximal function. It holds in (commutative or not) Lp spaces for 1 < p < ∞ and in spaces with type 2, and fails in L1 . We do not know whether it holds in every UMD space. This paper, especially in Section 8, uses extensively the techniques from [18]. We choose not to formulate the results in a Banach space valued setting to make the paper more readable, but all proofs are naturally suited to this more general context. 4. Properties of the symbols In this section we consider the symbols of the Fourier multipliers defined in Subsections 3.A and 3.B. As a consequence of the assumptions made in these subsections, we obtain the various
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estimates which we need in the next sections to establish an Lp theory. In what follows, we denote A(a, b) = {z ∈ C ; a ≤ |z| ≤ b}. 4.1. Lemma. Let D be a k-th order homogeneous differential operator with constant matrix coefˆ ficients, satisfying (D1) and (D2). Then, denoting M = sup |D(ξ)|, we have that |ξ|=1
ˆ (a) σ(D(ξ)) ⊂ (Sω ∩ A(κ|ξ|k , M |ξ|k )) ∪ {0}, N ˆ ˆ (b) C = N(D(ξ)) ⊕ R(D(ξ)), π −1 ˆ (c) ∀µ ∈ (ω, 2 ) |(ζI − D(ξ)) | . |ζ|−1 ∀ξ ∈ Rn
∀ζ ∈ C \ (Sµ ∩ A( 21 κ|ξ|k , 2M |ξ|k )).
Using compactness for |ξ| = 1 and homogeneity for |ξ| = 6 1, this is a consequence of the following lemma. 4.2. Lemma. 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 ) |(ζI − T )−1 | . |ζ|−1
∀ζ ∈ C \ (Sµ ∩ A( 21 κ, 2|T |)).
Proof. Let us first remark that, for a non zero eigenvalue λ with eigenvector e, we have that |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 7→ ζ(ζI − T )−1 is continuous from the closure of C \ (Sµ ∩ A( 21 κ, 2|T |)) to L (R(T ), CN ) and is bounded at ∞, and thus is bounded on C \ (Sµ ∩ A( 12 κ, 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
−1 ˆ |(I + iτ D(ξ)) |L (CN ) ≤ C.
For τ ∈ Sθ , we use the following notation: −1 ˆ τ (ξ) := (I + iτ D(ξ)) ˆ R , 1 ˆ 2 ˆ 2 −1 ˆ Pˆτ (ξ) := (R , τ (ξ) + R−τ (ξ)) = (I + τ D(ξ) ) 2 ˆ τ (ξ) := i (R ˆ τ (ξ) − R ˆ −τ (ξ)) = τ D(ξ) ˆ Pˆτ (ξ). Q 2 ˆ ˆ 0 )) for all ξ 0 in some neighbourhood of ξ. If λ ∈ / σ(D(ξ)) for some ξ ∈ Rn , then also λ ∈ / σ(D(ξ One checks directly from the definition of the derivative that −1 −1 −1 ˆ ˆ ˆ ˆ ∂ξj (λ − D(ξ)) = (λ − D(ξ)) (∂ξj D)(ξ)(λ − D(ξ)) . −1 ˆ ˆ By induction it follows that (λ − D(ξ)) is actually C ∞ in a neighbourhood of ξ for λ ∈ / σ(D(ξ)). ∞ n ˆ In particular, for τ ∈ Sθ , the function Rτ (ξ) is C in ξ ∈ R , and
(4.3)
ˆ τ (ξ) = R ˆ τ (ξ)(−iτ ∂ξ D(ξ)) ˆ ˆ τ (ξ). ∂ξj R R j
ˆ ˆ 4.4. Proposition. Given the splitting CN = N(D(ξ)) ⊕ R(D(ξ)), the complementary projections PN(D(ξ)) and PR(D(ξ)) = I − PN(D(ξ)) are infinitely differentiable in Rn \ {0} and satisfy the Mikhlin ˆ ˆ ˆ multiplier conditions |∂ξα PN(D(ξ)) | .α |ξ|−|α| , ∀ α ∈ Nn . ˆ Proof. The projections PN(D(ξ)) are obtained by the Dunford–Riesz functional calculus by inteˆ grating the resolvent around a contour, which circumscribes the origin and no other point of the
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HYTÖNEN, MCINTOSH, AND PORTAL
ˆ spectrum of D(ξ). By Lemma 4.1, for ξ in a neighbourhood of the unit sphere (say we may choose Z 1 −1 ˆ PN(D(ξ)) = (λ − D(ξ)) dλ. ˆ 2πi ∂D(0,2−k−1 κ)
3 4
< |ξ| < 43 ),
−1 ˆ Using the smoothness of (λ − D(ξ)) 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)) ˆ is C ∞ in a neighbourhood of the unit sphere. ˆ ˆ ˆ To complete the proof, it suffices to observe that D(tξ) = tk D(ξ) for t ∈ (0, ∞). Hence N(D(ξ)) ˆ and R(D(ξ)), and therefore the associated projections, are invariant under the scalings ξ 7→ tξ. It is a general fact that smooth functions, which are homogeneous of order zero, satisfy the Mikhlin multiplier conditions. Indeed, for any ξ ∈ Rn \ {0} and t ∈ (0, ∞), we have
∂ξα PN(D(ξ)) = ∂ξα (PN(D(tξ)) ) = t|α| (∂ α PN(D( ˆ ˆ ˆ · )) )(tξ), and setting t = |ξ|−1 and using the boundedness of the continuous function ∂ξα PN(D(ξ)) on the unit ˆ sphere, the Mikhlin estimate follows. ˆ ˆ Notice that, by (D1), D(ξ) is an isomorphism of R(D(ξ)) onto itself for each ξ ∈ Rn \ {0}. We −1 ˆ denote by DR (ξ) its inverse. ˆ −1 (ξ)PR(D(ξ)) is smooth in Rn \ {0} and satisfies the Mikhlin-type 4.5. Lemma. The function D R multiplier condition ˆ −1 (ξ)P ˆ ∂ξα D .α |ξ|−k−|α| , ∀ α ∈ Nn . R
R(D(ξ))
Proof. By the Dunford–Riesz functional calculus, we have Z 1 −1 −1 dλ ˆ ˆ (λ − D(ξ)) , m(ξ) := DR (ξ)PR(D(ξ)) = ˆ 2πi γ λ ˆ where γ is any path oriented counter-clockwise around the non-zero spectrum of D(ξ). Since −k k ˆ ˆ D(rξ) = r D(ξ), a change of variables and Cauchy’s theorem shows that m(rξ) = r m(ξ), and the smoothness of m in Rn \ {0} is checked as in the previous proof by differentiating under the integral sign. The modified Mikhlin condition follows from this in a similar way as in the previous proof. 4.6. Proposition. Given ν ∈ (0, π2 −ω), the following Mikhlin conditions hold uniformly in τ ∈ Sν : ˆ τ (ξ)| .α |ξ|−|α| , |∂ξα R
∀ α ∈ Nn .
ˆ τ (ξ). Similar estimates hold for Pˆτ (ξ) and Q Proof. We use induction to establish the desired bounds. For α = 0, this was proven in Lemma 4.1. In order to make the induction step, we will need the identity X α ˆτ = ˆ τ )(−iτ ∂ξθ D) ˆ R ˆτ , (4.7) ∂ξα R (∂ α−θ R α 0. θ 0θ≤α
n Y α αi We use the multi-index notation as follows: the binomial coefficients are := , the θ θi i=1 order relation θ ≤ α means that θi ≤ αiforevery i = 1, . . . , n, whereas θ α means that θ ≤ α α but θ 6= α; finally, it is understood that = 0 if θ 6≤ α. θ 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 + ej , where ej is the jth standard unit
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
9
vector. Indeed, α+ej
ˆ τ = ∂ξ ∂ξα R ˆτ R j X α h ˆ R ˆτ ˆ τ )(−iτ ∂ θ D) ˆ R ˆ τ + (∂ α−θ R ˆ τ )(−iτ ∂ θ+ej D) = (∂ α+ej −θ R ξ ξ θ
∂ξ
0θ≤α
ˆ τ )(−iτ ∂ξθ D) ˆ R ˆ τ (−iτ ∂ξ D) ˆ R ˆτ + (∂ α−θ R j
i
i h X α X α ˆ τ )(−iτ ∂ξθ D) ˆ R ˆ τ + (∂ α R ˆ τ )(−iτ ∂ ej D) ˆ R ˆτ + (∂ α+ej −θ R ξ θ θ − ej 0θ≤α ej θ≤α+ej X hα α i ˆ τ )(−iτ ∂ξθ D) ˆ R ˆτ , + (∂ α+ej −θ R = θ θ − ej =
0θ≤α+ej
and the proof of (4.7) is completed by the binomial identity
α θ
+
α θ−ej
=
α+ej θ
. Notice that
ˆτ ∂ξα R
the induction hypothesis was used twice: first to expand in the second step, and then to evaluate the summation over the last of the three terms in the third one. We then pass to the inductive proof of the assertion of the lemma. Let α 0, and assume ˆ τ )P ˆ . By the induction assumption, we the claim proven for all β α. We first consider (∂ξα R R(D) ˆ τ in (4.7) satisfy know that the factors ∂ α−θ R ξ
ˆ τ (ξ)| . |ξ||θ|−|α| . |∂ξα−θ R Furthermore, ˆ R ˆ τ P ˆ = (∂ θ D) ˆ R ˆ τ (−iτ D) ˆ D ˆ −1 P ˆ = (∂ θ D)( ˆ R ˆ τ − I)D ˆ −1 P ˆ , (−iτ ∂ξθ D) ξ ξ R R R(D) R(D) R(D) where the different factors are bounded by ˆ . |ξ|k−|θ| , ˆ τ − I| . 1, |∂ξθ D| |R
ˆ −1 (ξ)P ˆ | . |ξ|−k . |D R R(D(ξ))
ˆ τ )P ˆ | . |ξ|−|α| , as required. Multiplying all these estimates, we get |(∂ξα R R(D) ˆ τ )P ˆ . We have It remains to estimate (∂ξα R N(D) X αˆ α ˆ ˆ τ ) ∂ α−β P ˆ (∂ξ Rτ )PN(D) (∂ξβ R ˆ = ∂ξ (Rτ PN(D) ˆ )− N(D) 0≤βα
= ∂ξα PN(D) ˆ −
X
O(|ξ|−|β| )O(|ξ|−|α−β| ) = O(|ξ|−|α| ),
0≤βα
where the O-bounds follow from the induction assumption and the Mikhlin bounds for PN(D(ξ)) ˆ established in Proposition 4.4. The proof is complete. 4.8. Proposition. Let µ ∈ (ω, π2 ), and let f ∈ H0∞ (Sµ ). Then f (D) is a Mikhlin multiplier, and ˆ more precisely its symbol f[ (D)(ξ) = f (D(ξ)) satisfies, for ξ 6= 0, α −|α| ˆ |∂ξ f (D(ξ))| .µ |ξ| sup |f (λ)| : λ ∈ Sµ , 12 κ|ξ|k ≤ |λ| ≤ 2M |ξ|k , ˆ where M = sup |D(ξ)| . |ξ|=1
Proof. Pick θ ∈ (ω, µ). By the definition of the functional calculus, we have Z 1 ˆ −1 dζ = ∂ξα f (D). ˆ f (ζ)∂ξα (ζ − D) (4.9) ∂ξα f[ (D) = 2πi ∂Sθ ˆ −1 = ζ −1 ∂ α R ˆ ˆ From (4.7) one sees that ∂ξα (ζ − D) ξ i/ζ has poles at ζ ∈ σ(D). By Lemma 4.1, the non-zero spectrum satisfies ˆ σ(D(ξ)) \ {0} ⊂ {ζ ∈ Sω : κ|ξ|k ≤ |ζ| ≤ M |ξ|k }.
10
HYTÖNEN, MCINTOSH, AND PORTAL
Hence, for a fixed ξ ∈ Rn , we may deform the integration path in (4.9) to ∂[Sθ ∩ D(0, 2M |ξ|k ) \ D(0, 12 κ|ξ|k )] ∪ ∂[Sθ ∩ D(0, ε)] =: γ1 ∪ γ2 . On γ1 , there holds |ζ| h |ξ|k , while the length of the path is also `(γ1 ) h |ξ|k . Hence Z −1 ˆ f (ζ)∂ξα (ζ − D(ξ)) dζ γ1
≤ sup{|f (ζ)| : ζ ∈ Sµ ∩ D(0, 2M |ξ|k ) \ D(0, 12 κ|ξ|k )}
Z
ˆ i/ζ (ξ)| |∂ξα R
γ1
| dζ| |ζ|
. sup |f (λ)| : λ ∈ Sµ , 12 κ|ξ|k ≤ |λ| ≤ 2M |ξ|k · |ξ|−|α| , which has a bound of the desired form. On the other hand, the integral over γ2 vanishes in the limit as ε ↓ 0. We shall make use of operators with the following particular symbols: 4.10. Corollary. For t ∈ R and θ ∈ Nk with |θ| = k, the symbols ˆ ˆ 2 )−1 σt (ξ) := t2 ξ θ D(ξ)(I + t2 D(ξ) are C ∞ away from the origin and satisfy the Mikhlin multiplier estimates |∂ξα σt (ξ)| .α |ξ|−|α|
∀ α ∈ Nn .
Proof. The symbols are ˆ −1 (ξ)t2 D ˆ 2 (ξ)(I + t2 D(ξ) ˆ 2 )−1 = ξ θ D ˆ −1 (ξ)P ˆ ˆ σt (ξ) = ξ θ D R R R(D(ξ)) I − Pt (ξ) , where both factors are smooth in Rn \ {0}, and the first satisfies the Mikhlin multiplier condition by Lemma 4.5 and the second by Proposition 4.6. 5. Lp theory for operators with constant coefficients In this section, we consider the Fourier multipliers in Lp , which correspond to the symbols ˆ t , Pˆt , Q ˆt. studied in Section 4. We denote by Rt , Pt , Qt the multiplier operators with the symbols R We start with the operators D from Subsection 3.A. The estimates obtained in the preceding section give Theorem 3.2, restated here for convenience: 5.1. Theorem. Let 1 < p < ∞. Under the assumptions (D1) and (D2), the operator D is bisectorial in Lp with angle ω, and has a bounded H ∞ functional calculus in Lp with angle ω. Proof. By the Mikhlin multiplier theorem, the bisectoriality follows from Proposition 4.6, while the boundedness of the H ∞ functional calculus follows from Proposition 4.8. The coercivity condition (D1) for the symbol has the following reincarnation on the level of operators: 5.2. Proposition. For all u ∈ Rp (D) ∩ Dp (D), there holds u ∈ Dp (∇k ), and k∇k ukp . kDukp . Proof. For u ∈ Dp (D) ∩ Rp (D), we have (for real t) ut := t2 DPt (Du) = t2 D2 Pt u = (I − Pt )u → PRp (D) u = u,
t → ∞.
The operators t2 ∂ θ DPt , |θ| = k, are bounded on Lp by Corollary 4.10. It follows that ut ∈ Dp (∂ θ ), and k∂ θ ut kp = kt2 ∂ θ DPt (Du)kp . kDukp . Let w be a test function in the dual space. Then |h∂ θ u, wi| = |hu, ∂ θ wi| = lim |hut , ∂ θ wi| = lim |h∂ θ ut , wi| t→∞
t→∞
θ
≤ lim inf k∂ ut kp kwkp0 . kDukp kwkp0 . t→∞
θ
θ
Thus u ∈ Dp (∂ ) and k∂ ukp . kDukp for all |θ| = k.
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
11
We turn to the Hodge-Dirac operators Π = Γ + Γ which satisfy the hypotheses at the start of Subsection 3.B, and note that they then satisfy the conditions on D with k = 1. In particular, Theorem 5.1 and Proposition 5.2 hold for D = Π and k = 1. Moreover there is an operator version of the symbol condition (Π3): 5.3. Lemma. There holds Np (Π) = Np (Γ) ∩ Np (Γ). ˆ u = 0 in the sense of distributions. Proof. The inclusion ⊇ is clear. Let u ∈ Np (Π). Then Πˆ ∞ ˆ ˆ −1 (ξ)P ˆ It follows from Lemma 4.5 that the function Υ(ξ) := Γ(ξ)Π away from the R R(Π(ξ)) is C ∞ n ˆu origin. Hence, if ψ ∈ Cc (R \ {0}), then also ψΥ is in the same class, and the product ψΥ · Πˆ is well-defined and vanishes as a distribution. But ˆ ˆ Π ˆ −1 (ξ)P ˆ Π(ξ) ˆ ˆ ˆ Υ(ξ)Π(ξ) = Γ(ξ) = Γ(ξ)P = Γ(ξ) ˆ R R(Π(ξ)) R(Π(ξ)) ˆ u = ψΥΠˆ ˆ u = 0 for every ψ ∈ C ∞ (Rn \ {0}). This means by (Π3). Thus we have shown that ψ Γˆ c ˆ u is at most supported at the origin, and hence Γu = P , a polynomial. But that the distribution Γˆ Pn Pn ˆ p n ˆ ˆ also Γu = −i j=1 Γ j ∂j u = j=1 ∂j uj , where uj = −iΓj u ∈ L . Let φ ∈ S (R ) be identically ˆ one in a neighbourhood of the origin. Then Pˆ = Pˆ φ and hence P = P ∗ φ. But P ∗ φ(y) = hP, φ(y − ·)i =
n X huj , (∂j φ)(y − ·)i → 0 j=1 0
as |y| → ∞ (using just the fact that uj ∈ Lp and ∂j φ ∈ Lp ), and a polynomial with this property must vanish identically. This shows that Γu = P = 0, and then also Γu = Πu − Γu = 0. This then implies: 5.4. Proposition. The operator identity Π = Γ + Γ holds in Lp , in the sense that Dp (Π) = Dp (Γ) ∩ Dp (Γ) and Πu = Γu + Γu for all u ∈ Dp (Π). Proof. It is clear that Dp (Γ) ∩ Dp (Γ) ⊆ Dp (Π). Since Π is bisectorial in Lp , there is the topological decomposition Lp = Np (Π) ⊕ Rp (Π). Write u ∈ Dp (Π) as u = u0 + u1 in this decomposition. Then u0 ∈ Np (Π) = Np (Γ) ∩ Np (Γ), and u1 = u − u0 ∈ Dp (Π) ∩ Rp (Π). By Proposition 5.2, u1 ∈ Dp (∇) ⊆ Dp (Γ) ∩ Dp (Γ). Thus also Dp (Π) ⊆ Dp (Γ) ∩ Dp (Γ). The coincidende of Πu and Γu + Γu is clear from the distributional definition. We are ready to prove Theorem 3.6, restated here: 5.5. Theorem. Let Π be a Hodge-Dirac operator with constant coefficients, and let 1 < p < ∞. Then the operator Π has a bounded H ∞ functional calculus in Lp with angle ω, and satisfies the following Hodge decomposition Lp = Np (Π) ⊕ Rp (Γ) ⊕ Rp (Γ), where Rp (Π) = Rp (Γ) ⊕ Rp (Γ). Proof. The fact that Π is a bisectorial operator with a bounded H ∞ functional calculus is a particular case of Theorem 3.2. The bisectoriality already implies the decomposition Lp = Np (Π)⊕ Rp (Π), which we now want to refine. We first check that Rp (Π) ⊆ Rp (Γ) + Rp (Γ). If u ∈ Rp (Π), then u = limj→∞ Πyj for some yj ∈ Dp (Π) ∩ Rp (Π) ⊆ Dp (∇). Then kΓ(yj − yk )kp . k∇(yj − yk )kp . kΠ(yj − yk )kp → 0 (using Proposition 5.2), and hence Γyj converges to some v ∈ Rp (Γ) with kvkp . kukp . Similarly, Γyj converges to w ∈ Rp (Γ), and u = v + w ∈ Rp (Γ) + Rp (Γ). Next we show that Rp (Γ) ⊆ Rp (Π). Indeed, Rp (Γ) = Γ(Dp (Γ)) = Γ(Dp (Γ) ∩ Rp (Π)) (by the decomposition in the first paragraph and Lemma 5.3) ⊆ Γ(Dp (Γ) ∩ (Rp (Γ) + Rp (Γ))) (by the previous paragraph) = Γ(Dp (Γ) ∩ Rp (Γ)) (because Γ is nilpotent) = Π(Dp (Π) ∩ Rp (Γ)) (because Γ is nilpotent) ⊆ Rp (Π). Therefore Rp (Γ) ⊆ Rp (Π). In a similar way, we see that Rp (Γ) ⊆ Rp (Π).
12
HYTÖNEN, MCINTOSH, AND PORTAL
On combining these two results with that in the preceding paragraph, we obtain Rp (Γ)+Rp (Γ) = Rp (Π). To show that the sum is direct, observe that Rp (Γ) ∩ Rp (Γ) ⊆ Rp (Π) ∩ Np (Γ) ∩ Np (Γ) = Rp (Π) ∩ Np (Π) = {0} by nilpotence, Lemma 5.3 and the decomposition Lp = Np (Π) ⊕ Rp (Π).
We conclude this section with an analogue, in our matrix-valued context, of Bourgain’s [13, Lemma 10]. It is an important property of Hodge-Dirac operators with constant coefficients which we use to study Hodge-Dirac operators with variable coefficients in Section 8. 5.6. Proposition. For all z ∈ Rn , there holds
X
E εk τ2k z Q2k u . (1 + log+ |z|)kukp . p
k
where τz denotes the translation operator τz u(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−m ξ), and ϕm (ξ) := ϕ(2−m ξ) for m ∈ Z+ . Let Φm and Ψm , m ∈ Z, be the corresponding FourierPmultiplier operators with symbols ϕm and ψm . Then we have ∞ the P∞ partition of unity ϕk + m=1 ψk+m ≡ 1, and the corresponding operator identity Φk + m=1 Ψm+k = I for all k ∈ Z. Since the support of the Fourier transform of Q2k Ψm−k is contained in B(0, 2m−k ), by Bourgain’s [13, Lemma 10], there holds
X
X
(5.7) E εk τ2k z Q2k Ψm−k u . (1 + log+ (2m |z|))E εk Q2k Ψm−k u . p
k
k
p
The same reasoning applies with Φ in place of Ψ. 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 X k−m ˆ σ(ξ) = εk f (2k Π(ξ))ψ(2 ξ), f (τ ) = τ (1 + τ 2 )−1 . k n
For every α ∈ {0, 1} , a computation shows that X X α−β ˆ ∂ α σ(ξ) = εk ∂ β f (2k Π(ξ))(∂ ψ)(2k−m ξ)2(k−m)|α−β| . k
β≤α
By the support property of ψ, the series in k reduces to at most two non-vanishing terms for which 2k−m |ξ| h 1. By Proposition 4.8, there moreover holds (5.8)
ˆ |∂ β f (2k Π(ξ))| .
2k |ξ| |ξ|−|β| , 1 + (2k |ξ|)2
which shows that, for m ∈ Z+ , |∂ α σ(ξ)| . 2−m |ξ|−|α| . Hence the associated Fourier multiplier is bounded with norm . 2−m . A similar computation can be made with Φ−k in place of Ψm−k , but the estimation of the symbol then involves an infinite series of terms: XX α−β ˆ |∂ α σ(ξ)| . |∂ β f (2k Π(ξ))||(∂ ϕ)(2k ξ)|2k|α−β| β≤α k
.
X
X
2k(1+|α−β|) |ξ|1−|β| . |ξ|−|α| ,
β≤α k:|2k ξ|≤1
where (5.8) was used again in the second estimate. We conclude that the operator acting on u in (5.7), with Φ−k in place of Ψm−k , is also bounded.
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
13
Collecting all the estimates, we have shown that
X
E εk τ2k z Q2k u p
k
∞
X
X X
(m + log+ |z|) εk Q2k Ψm−k u . (1 + log+ |z|) εk Q2k Φ−k u + p
k
.
∞ X
m=1
k
p
(max{1, m} + log+ |z|)2−m kukp . (1 + log+ |z|)kukp ,
m=0
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 ΠB and its variant ΠB , and finally some stability properties of the Hodge decomposition under small perturbations of the coefficient matrices B1 and B2 . These will be needed in proving the stability of the functional calculus of the Hodge-Dirac operators under small perturbations later on. 6.1. Lemma. Let 1 < p < ∞ and let ΠB be a Hodge-Dirac operator with variable coefficients in Lp . The following assertions are equivalent: p (1) Π (B Hodge-decomposes L . p L = Np (Γ) ⊕ Rp (ΓB ), (2) Lp = Np (ΓB ) ⊕ Rp (Γ). Proof. (1) ⇒ (2). We first show that Np (ΠB ) = Np (Γ) ∩ Np (ΓB ). If u ∈ Np (ΠB ) then Γu = −ΓB u ∈ Rp (Γ) ∩ Rp (ΓB ) = {0}. It follows that Np (ΠB ) ⊕ Rp (Γ) ⊆ Np (Γ). Also Np (Γ) ∩ Rp (ΓB ) ⊆ Np (ΠB ) ∩ Rp (ΓB ) = {0}. Hence Np (ΠB ) ⊕ Rp (Γ) = Np (Γ), and thus Lp = Np (Γ) ⊕ Rp (ΓB ). The second part of (2) is similarly proven. (2) ⇒ (1). By (2), u ∈ Lp can be decomposed as u = v0 + v1 + u1 where v0 + v1 ∈ Np (Γ), v0 ∈ Np (ΓB ), v1 ∈ Rp (Γ), and u1 ∈ Rp (ΓB ). Then ΠB v0 = Γ(v0 +v1 −v1 ) = 0, and kv0 kp +kv1 kp . kv0 + v1 kp . kukp . Moreover Np (ΠB ) ∩ Rp (Γ) ⊆ Np (ΓB ) ∩ Rp (Γ) = {0}, Np (ΠB ) ∩ Rp (ΓB ) ⊆ Np (Γ) ∩ Rp (ΓB ) = {0}, Rp (ΓB ) ∩ Rp (Γ) ⊆ Np (Γ) ∩ Rp (ΓB ) = {0}.
The proof is complete. 6.2. Lemma. Let D0 and D1 be closed, densely defined operators in Lp . Then Lp = Np (D0 ) ⊕ Rp (D1 )
if and only if
0
Lp = Np0 (D1∗ ) ⊕ Rp0 (D0∗ ).
Proof. Assuming the first decomposition, we have that 0
⊥
Lp = Rp (D1 ) ⊕ Np (D0 )⊥ = Np0 (D1∗ ) ⊕ Rp0 (D0∗ ).
The other implication follows by symmetry.
6.3. Lemma. Let ΠB be a Hodge-Dirac operator with variable coefficients in Lp which Hodge 0 decomposes Lp . Then Π∗B is a Hodge-Dirac operator with variable coefficients in Lp which Hodge 0 decomposes Lp . 0
Proof. We have to show that B ∗ = (B1∗ , B2∗ ) satisfy (B2) in Lp . Let us remark that B1 is an isomorphism from Rp (Γ) onto Rp (ΓB ). By Lemma 6.1, this means that B1 is an isomorphism from 0 Rp (Γ) onto Lp /Np (Γ), and thus B1∗ is an isomorphism from Rp0 (Γ∗ ) onto Lp /Np0 (Γ∗ ). This gives the part of the result concerning B1∗ thanks to Lemma 6.2. The case of B2∗ is handled in the same
14
HYTÖNEN, MCINTOSH, AND PORTAL
way. Condition (B1) is obtained by duality, and the proof is concluded by applying Lemma 6.1 and Lemma 6.2. Recall that ΠB := Γ + B2 ΓB1 . 6.4. Lemma. Let 1 < p < ∞ and suppose that ΠB and ΠB are both Hodge-Dirac operator with variable coefficients in Lp , and that ΠB Hodge-decomposes Lp . Then ΠB also Hodge-decomposes Lp . If, moreover, ΠB is an R-bisectorial operator in Lp , then so is ΠB . Proof. By Lemmas 6.1 and 6.2, the assumption that ΠB Hodge-decomposes Lp is equivalent to Lp = Np (Γ) ⊕ Rp (ΓB ),
(6.5)
0
Lp = Np0 (Γ∗ ) ⊕ Rp0 (Γ∗B ),
whereas the claim that ΠB Hodge-decomposes Lp is equivalent to Lp = Np (B2 ΓB1 ) ⊕ Rp (Γ),
(6.6)
0
Lp = Np0 (B1∗ Γ∗ B2∗ ) ⊕ Rp0 (Γ∗ ).
We show that the first decomposition in (6.5) implies the first one in (6.6). Let u ∈ Lp and write B1 u = v + w where v ∈ Np (Γ) and w ∈ Rp (ΓB ). Let w = B1 x for x ∈ Rp (Γ). Then u − x ∈ Np (B2 ΓB1 ) and kxkp . kB1 xkp = kwkp . kB1 ukp . kukp . The deduction of the second decomposition in (6.6) from the second one in (6.5) is analogous. We now turn to the second statement. Let us denote by P1 the projection on Rp (Γ), by P2 the projection on Rp (B2 ΓB1 ), by P1 the projection on Rp (Γ), and by P2 the projection on Rp (B1 ΓB2 ). Let (tk )k∈N ⊂ R and (uk )k∈N ⊂ Lp , and remark first that (I + itk ΠB )−1 = I − (tk ΠB )2 (I + (tk ΠB )2 )−1 − itk ΠB (I + (tk ΠB )2 )−1 . The R-bisectoriality of ΠB will thus follow once we have proven that
X
X
εk itk ΠB (I + (tk ΠB )2 )−1 uk . E εk uk (6.7) E p
k
and
p
k
X
X
E εk (tk ΠB )2 (I + (tk ΠB )2 )−1 uk . E εk uk .
(6.8)
p
k
p
k
To do so we note that, since ΠB and ΠB are Hodge-Dirac operators with variable coefficients, we have: ΓB1 (I + (tk ΠB )2 )−1 P1 = Γ(I + (tk ΠB )2 )−1 B1 P1 , Γ(I + (tk ΠB )2 )−1 B2 P1 = ΓB2 (I + (tk ΠB )2 )−1 P1 . We can now proceed with the estimates, using the R-bisectoriality of ΠB .
X
X
E εk itk ΠB (I + (tk ΠB )2 )−1 P1 uk = E εk itk B2 ΓB1 (I + (tk ΠB )2 )−1 P1 uk , k
p
p
k
X
= E εk itk B2 Γ(I + (tk ΠB )2 )−1 B1 P1 uk , p
k
X
= E εk itk B2 ΠB (I + (tk ΠB )2 )−1 B1 P1 uk , k
X
X
. E εk B1 P1 uk . E εk uk . k
p
k
p
p
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
15
Introducing vk such that B2 P1 vk = P2 uk , we also get:
X
X
E εk itk ΠB (I + (tk ΠB )2 )−1 P2 uk = E εk itk Γ(I + (tk ΠB )2 )−1 B2 P1 vk , p
k
p
k
X
= E εk itk ΓB2 (I + (tk ΠB )2 )−1 P1 vk , p
k
X
. E εk itk ΠB (I + (tk ΠB )2 )−1 P1 vk , k
X
X
εk uk . . E εk P1 vk . E k
p
k
p
p
The estimate (6.8) is proven in the same way.
6.9. Lemma. If a Banach space splits as X = X0 ⊕ X1 = P0 X ⊕ P1 X, then there is δ > 0 such ˜0X ⊕ P ˜ 1 X, that for all T ∈ L (X1 , X) with kT k < δ, it also splits as X = X0 ⊕ (I − T )X1 = P ˜ ˜ with kP0 − P0 k + kP1 − P1 k . δ. Proof. For δ :=
1 2kP1 k ,
I − T P1 is invertible. Let U := (I − T P1 )−1 , and observe the identity U = I + U T P1 .
Define the operators ˜ 0 := P0 U, P
˜ 1 := (I − T P1 )P1 U. P
˜ 0 ) = X0 , R(P ˜ 1 ) = (I − T )X1 , and Then R(P ˜0 + P ˜ 1 = (I − P1 + (I − T P1 )P1 )U = (I − T P1 )U = I. P ˜ 0 (and then also P ˜ 1 ) is a projection. This follows from It remains to show that P ˜0P ˜ 0 = P0 U P0 U = P0 (I + U T P1 )P0 U = P0 U = P ˜0, P where we used P1 P0 = 0 and P20 = I. ˜ 0 − P0 = P0 U T P1 , P ˜ 1 − P1 = P1 U T P1 − T P1 U , and kU k ≤ 2, we also have that Since P ˜ ˜ kP0 − P0 k + kP1 − P1 k . δ. 6.10. Proposition. Let p ∈ (1, ∞), and ΠA be a Hodge-Dirac operator with variable coefficients in Lp which Hodge-decomposes Lp . There exists δ > 0 such that, if ΠB is another Hodge-Dirac operator with variable coefficients in Lp with kB1 − A1 k∞ + kB2 − A2 k∞ < δ, then ΠB Hodgedecomposes Lp . Moreover, the associated Hodge-projections satisfy B B A B A kPA 0 − P0 k + kPΓ − PΓ k + kPΓA − PΓB k . δ.
Proof. Consider the condition (6.5) equivalent to the Hodge-decomposition. Let us define T1 ∈ L (Rp (ΓA ), Lp ) by T1 A1 Γu := (A1 − B1 )Γu which, by (B2), gives a well-defined operator of norm kT1 k . δ, and we have B1 Γ = (I − T1 )A1 Γ. 0 On the dual side, we define T˜2 ∈ L (Rp0 (Γ∗A ), Lp ) by T˜2 A∗2 Γ∗ v = (A∗2 − B2∗ )Γv, which is similarly well-defined and satisfies kT˜2 k . δ. Let then T2 := (T˜2 PΓ∗A )∗ ∈ L (Lp ), where PΓ∗A 0 is the projection in Lp associated to the decomposition in (6.5). By duality, it follows that Rp (B2 − A2 (I − T2 )) ⊆ Np (Γ), which means that ΓB2 = ΓA2 (I − T2 ). Since the operators I − T2 and I − T1 PΓA are invertible for δ small enough, we then have that R(ΓB ) = R((I − T1 )ΓA (I − T2 )) = (I − T1 )R(ΓA ). Similarly, with B2∗ Γ∗ = (I − T3 )A∗2 Γ∗ and Γ∗ B1∗ = Γ∗ A∗1 (I − T4 ), there holds R(Γ∗B ) = (I − T3 )R(Γ∗A ). Hence the claim follows from two applications of Lemma 6.9 with (X0 , X1 , T ) = (Np (Γ), R(ΓA ), T1 ) and (X0 , X1 , T ) = (Np0 (Γ∗ ), R(Γ∗A ), T3 ) By (B2), the restriction A1 : Rp (Γ) → Rp (ΓA ) is an isomorphism, and we denote by A−1 its 1 inverse. Thus the operator A−1 P is well-defined. We shall also need to perturb such operators: Γ A 1
16
HYTÖNEN, MCINTOSH, AND PORTAL
6.11. Corollary. Under the assumptions of Proposition 6.10, there also holds −1 B A kA−1 1 PΓA − B1 PΓB k . δ.
Proof. We use the same notation as in the proof of Proposition 6.10 and observe from the identity B1 Γ = (I − T1 )A1 Γ that I − T1 : Rp (ΓA ) → B1 Rp (Γ), and A B1−1 (I − T1 )PΓAA = A−1 1 PΓA .
(6.12)
We further recall from the proof of Lemma 6.9 that the projection PΓBB related to the splitting Lp = Np (Γ) ⊕ Rp (ΓB ), where Rp (ΓB ) = (I − T1 )Rp (ΓA ), is given by PΓBB = (I − T1 )PΓAA (I − T1 PΓAA )−1 . Combining this with (6.12) shows that A A −1 B1−1 PΓBB = A−1 1 PΓA (I − T1 PΓA ) −1 A A A A −1 = A−1 , 1 PΓA − (A1 PΓA )T1 PΓA (I − T1 PΓA )
which proves the claim, since the second term contains the factor T1 of norm kT1 k . δ.
7. Functional calculus In this section we collect some general facts about the functional calculus of bisectorial operators in reflexive Banach spaces. We provide, in the context of the discrete randomized quadratic estimates required in [18], versions of results originally obtained in [15]. Lemma 7.1 can be seen as a discrete Calderón reproducing formula, and Lemma 7.2 as a Schur estimate, while Propositions 7.3 and 7.5 express the fundamental link between functional calculus and square function estimates. Such results are not new, and have been developed from [15] by various authors, most notably Kalton and Weis (cf. [20]). Here we hope, however, to provide simpler versions of both the statements and the proofs of these facts. Let A denote a bisectorial operator in a reflexive Banach space, with angle ω, and let 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 A (r(A) − r(−A)) = . 2 I + A2
7.1. Lemma. The following series converges in the strong operator topology: 3X q(2k A)q(2k+1 A) = PR(A) . 2 k∈Z
Proof. Observe first that p(tA) → PN(A) as t → ∞ and p(tA) → I as t → 0. Hence X PR(A) = I − PN(A) = (p(2k A) − p(2k+1 A)) k
=
X
p(2k A)[(I + 22(k+1) A2 ) − (I + 22k A2 )]p(2k+1 A)
k
3X 3X = p(2k A)(2k A)(2k+1 A)p(2k+1 A) = q(2k A)q(2k+1 A), 2 2 k
k
as we wanted to show.
7.2. Lemma. 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)−1 q(tA)f (A)q(sA) : t, s > 0; f ∈ H0∞ (Sθ ), kf k∞ ≤ 1} is R-bounded.
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
17
Proof. Denoting qt (λ) := q(tλ), observe that 1 q(tA)f (A)q(sA) = (qt · f · qs )(A) = 2πi 0
Z (qt f qs )(λ)(I − γ
1 −1 dλ A) , λ λ
π 2 ), −1
where γ denotes ∂Sθ0 , for some θ ∈ (θ, parameterized by arclength and directed anticlockwise around Sθ , and the resolvents (I − λ A)−1 belong to an R-bounded set. The operators q(tA)f (A)q(sA) are hence in a dilation of the absolute convex hull of this R-bounded set (see [21] for information on R-boundedness techniques). To evaluate the dilation factor, observe that |qt f qs (λ)| . kf k∞
t|λ| s|λ| , 1 + (s|λ|)2 1 + (t|λ|)2
and splitting the integral into three regions (depending on the position of |λ| with respect to min( 1t , 1s ) and max( 1t , 1s )) it follows that Z | dλ| |qt f qs (λ)| . kf k∞ η(s/t), |λ| γ which implies the asserted R-bound.
7.3. Proposition. Let A be R-bisectorial (with angle µ) and satisfy the two-sided quadratic estimate
X
εk q(2k A)u , u ∈ R(A). (7.4) kuk h E k
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
X
kf (A)uk . E εk q(2k A)f (A)u k
X X
εk q(2k A)f (A) q(2j A)q(2j+1 A)u h E j
k
X
X . εk q(2k A)f (A)q(2k+m A)q(2k+m+1 A)u E m
.
k
X
X
η(2 )kf k∞ εk q(2k+m+1 A)u m
m
k
X . (1 + |m|)2−|m| kf k∞ kuk . kf k∞ kuk, m
where we used . from (7.4), Lemma 7.1, the triangle inequality after relabelling j = k + m, Lemma 7.2, and & from (7.4). We also use the following variant. 7.5. Proposition. Let A be R-bisectorial (with angle µ) and satisfy the two quadratic estimates
X
E εk q(2k A)u . kukX , u ∈ X, X
k
(7.6)
X
E εk q(2k A∗ )v
X∗
k
v ∈ X ∗,
. kvkX ∗ ,
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 X |hf (A)u, vi| ≤ |hq(2k A)f (A)u, q(2k+1 A∗ )vi| k
X
X
. E εk q(2k A)f (A)u E εk q(2k A∗ )v k
X
k
X∗
X
. E εk q(2k A)f (A)u kvkX ∗ . k
X
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HYTÖNEN, MCINTOSH, AND PORTAL
The proof is then concluded as in Proposition 7.3. 8. Lp theory for operators with variable coefficients
In this section, we give the proofs of the results stated in Subsection 3.C, and some variations. The core result, Theorem 8.1, is a generalization of [18, Theorem 3.1]. The key ingredients of the proof are contained in [18]. Here we indicate where [18] needs to be modified, using the results from the preceding sections. Let us recall that Π denotes an Hodge-Dirac operator with constant coefficients as defined in Definition 3.4, and that ΠB denotes a Hodge-Dirac operator with variable coefficients as defined in Definition 3.8. We also use the following notation. RtB := (I + itΠB )−1 = r(tΠB ), PtB := (I + t2 Π2B )−1 = p(tΠB ), B QB t := tΠB Pt = q(tΠB ),
and denote by Rt , Pt , Qt the corresponding functions of Π. 8.1. Theorem. Let 1 ≤ p1 < p2 ≤ ∞, and let ΠB be an R-bisectorial Hodge-Dirac operator with variable coefficients in Lp which Hodge-decomposes Lp for all p ∈ (p1 , p2 ). Then
X
(8.2) E εk QB u u ∈ Rp (Γ),
. kukp , k 2 p
k
and (8.3)
X
∗ E εk (QB 2k ) v
p0
k
. kvkp0 ,
v ∈ Rp0 (Γ∗ ).
Proof. We first prove (8.2). Let us recall the following notation from [18]. Let [ 4 := 42k , 42k := 2k ([0, 1)n + m) : m ∈ Zn . k∈Z
denote a system of dyadic cubes, and A2k u(x) := huiQ :=
1 |Q|
Z x ∈ Q ∈ 42k .
u(y) dy, Q
be the corresponding conditional expectation projections. Let X (8.4) γ2k (x)w := QB QB x ∈ Rn , w ∈ CN . 2k (w)(x) := 2k (w1Q )(x), Q∈42k
QB 2k ,
denote the principal part of which we also identify with the corresponding pointwise multiplication operator. The proof of (8.2) now divides into the following four estimates:
X
(8.5) E εk QB (I − P )u u ∈ Rp (Γ). k
. kukp , k 2 2 p
k
(8.6)
X
E εk (QB − γ k A2k )P2k u . kukp , k 2 2 p
k
(8.7)
X
E εk γ2k A2k (I − P2k )u . kukp , p
k
(8.8)
X
E εk γ2k A2k u . kukp , k
p
u ∈ Rp (Γ). u ∈ Rp (Γ).
u ∈ Rp (Γ).
Inequality (8.5) follows from the fact that ΠB Hodge-decomposes Lp , and from the R-bisectoriality of ΠB : as in [18, Lemma 6.3], denoting by PΓB the projection onto Rp (ΓB ) in the Hodgedecomposition, we have B QB 2k (I − P2k )u = (I − P2k )PΓB Q2k u,
u ∈ Rp (Γ),
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
19
and (8.9)
X
X
εk Q2k u . kukp , E εk (I − P2Bk )PΓB Q2k u . E p
k
u ∈ Lp ,
p
k
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, Γ] are multiplication operators by an L∞ function bounded by k∇ηk∞ . Then the R-bisectoriality of ΠB and [18, Proposition 6.4] give the following off-diagonal R-bounds: for every M ∈ N, every Borel subsets Ek , Fk ⊂ Rn , every uk ∈ Lp , and every (tk )k∈Z ⊆ {2j }j∈Z such that dist(Ek , Fk )/tk > % for some % > 0 and all k ∈ Z, there holds
X
X
−M E εk 1Ek QB E εk 1Fk uk . tk 1Fk uk . (1 + %) (8.10) p
k
p
k
This, in turn, gives that the family (γ2k A2k )k∈Z is R-bounded exactly as in [18, Lemma 6.5]. Now let us prove (8.6). This is similar to [18, Lemma 6.6] but, since a few modifications need to be made, we include the proof. Letting vk = P2k u, and using the off-diagonal R-bounds, we have:
X
εk (QB − γ E k A2k )vk k 2 2 p
k
X X
εk 1Q QB v − hv i ) = E
k k k Q 2 k
(8.11) ≤
X m∈Zn
.
X m∈Zn
Q∈4
p
k
X 2 X
εk 1Q QB 1 E k m (vk − hvk iQ ) k Q−2 2 k
p
Q∈42k
X
X
(1 + |m|)−M E εk 1Q (vk − hvk iQ+2k m ) . k
p
Q∈42k
A version of Poincaré’s inequality [18, Proposition 4.1] allows to majorize the last factor by Z Z 1 X
εk 2k (m + z) · ∇ τt2k (m+z) P2k u dt dz. E [−1,1]n
0
p
k
We then use Propositions 5.2 and 5.6 to estimate this term by (1 + |m|)(1 + log+ |m|)kukp , 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
X
E εk A2k (I − P2k )u . kukp , u ∈ Rp (Γ). p
k
This is essentially like [18, Proposition 5.5]. We indicate the beginning of the argument, where the operator-theoretic Lemma 7.1 replaces a Fourier multiplier trick used in [18]. Indeed, with u ∈ Rp (Γ) ⊆ Rp (Π), we have
X
X
X
E εk A2k (I − P2k )u h E εk A2k (I − P2k ) Q2j Q2j+1 u k
p
j
k
p
X
X
≤ E εk A2k (I − P2k )Q2k+m Q2k+m+1 u . m
k
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 (Lp ) is R-bounded with R-bound C2−δ|m| for some δ > 0. This is done by repeating the argument of [18, Proposition 5.5].
20
HYTÖNEN, MCINTOSH, AND PORTAL
Finally, the proof of (8.8) is done exactly as in [18, Theorem 8.2 and Proposition 9.1]. Notice that this last part is the only place where we need the assumptions for p in 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 ΠB ∗ = Γ∗ + B2 ∗ Γ∗ B1 ∗ is 0 0 a Hodge-Dirac operator with variable coefficients in Lp which Hodge-decomposes Lp . It is also R-bisectorial, as this property is preserved under duality (see [20, Lemma 3.1]). So the proof of ∗ ∗ 2 −1 ∗ (8.2) adapts to give the quadratic estimate (8.3) involving (QB . t ) = tΠB (I + (tΠB ) ) We now prove Theorem 3.13 as a corollary. 8.12. Corollary. Let 1 ≤ p1 < p2 ≤ ∞, µ ∈ (ω, π/2), and let ΠB be a Hodge-Dirac operator with variable coefficients in Lp which Hodge-decomposes Lp for all p ∈ (p1 , p2 ). Assume also that ΠB is a Hodge-Dirac operator with variable coefficients in Lp . Then ΠB has a bounded H ∞ functional calculus (with angle µ) in Lp (Rn ; CN ) for all p ∈ (p1 , p2 ) if and only if it is R-bisectorial (with angle µ) in Lp (Rn ; CN ) for all p ∈ (p1 , p2 ). Proof. The fact that a bounded H ∞ functional calculus implies R-bisectoriality is a general property (see Remark 2.5). To prove the other direction, assume that ΠB is R-bisectorial on Lp (Rn ; CN ) for all p ∈ (p1 , p2 ). By Theorem 8.1, we have that
X
(8.13) E εk QB u u ∈ Rp (Γ). 2k . kukp , p
k
Moreover, since ΠB also satisfies the assumptions of Theorem 8.1 (using Lemma 6.4), we have that
X
(8.14) E εk 2k ΠB (I + (2k ΠB )2 )−1 u . kukp , u ∈ Rp (Γ). p
k
For u ∈ Rp (Γ), there holds 2k ΠB (I + (2k ΠB )2 )−1 u = 2k B2 ΓB1 (I + (2k ΠB )2 )−1 u = 2k B2 Γ(I + (2k ΠB )2 )−1 B1 u = 2k B2 ΠB (I + (2k ΠB )2 )−1 B1 u. Thus by (B2), the estimate (8.14) implies
X
εk QB (8.15) E 2k u . kukp , p
k
u ∈ Rp (ΓB ).
Combining (8.13) and (8.15) with the Hodge-decomposition and the obvious fact that QB 2k annihilates Np (ΠB ), we arrive at
X
E εk QB u ∈ Lp . 2k u . kukp , k
p
In the same way, one gets the dual estimate
X
∗ E εk (QB ) u
0 . kukp0 , k 2 k
0
u ∈ Lp ,
p
where p0 denotes the conjugate exponent of p. The functional calculus then follows from Proposition 7.5. 8.16. Corollary. Let 1 ≤ p1 < p2 ≤ ∞, and let ΠA be a Hodge-Dirac operator with variable coefficients, which is R-bisectorial in Lp and Hodge-decomposes Lp for all p ∈ (p1 , p2 ). Then for each p ∈ (p1 , p2 ), there exists δ = δp > 0 such that, if ΠB and ΠB are Hodge-Dirac operators with variable coefficients such that kB1 − A1 k∞ + kB2 − A2 k∞ < δ, then ΠB has an H ∞ functional calculus in Lp and Hodge-decomposes Lp .
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
21
Proof. Let p ∈ (p1 , p2 ). By Proposition 6.10, we have that, for δ small enough, ΠB Hodgedecomposes Lp . We need to show that ΠB is R-bisectorial in Lp provided δ is sufficiently small. As in the proof of Proposition 6.10, let T1 ∈ L (Rp (ΓA ), Lp ) and T2 ∈ L (Lp ) be operators of norm kTi k . δ such that B1 Γ = (I − T1 )A1 Γ, ΓB2 = ΓA2 (I − T2 ). Then ΠB = Γ + B1 ΓB2 = Γ + (I − T1 )A1 ΓA2 (I − T2 ) = (I − T1 PΓA )ΠA (I − PΓ T2 ), where PΓ and PΓA are the Hodge-projections associated to ΠA , onto Rp (Γ) and Rp (ΓA ), respectively. Hence I + itΠB = (I − T1 PΓA )(I + itΠA )(I − PΓ T2 ) + (T1 PΓA + PΓ T2 ) = (I − T1 PΓA )(I + itΠA )(I − PΓ T2 ) × I + (I − PΓ T2 )−1 (I + itΠA )−1 (I − T1 PΓA )−1 (T1 PΓA + PΓ T2 )], where the inverses involving Ti exist for δ small enough. Hence (I + itΠB )−1 can be expressed as a Neumann series involving powers of the operators (I + itΠA )−1 , which are R-bounded by assumption, times powers of fixed bounded operators, including T1 PΓA + PΓ T2 which has norm at most Cδ. For δ small enough, the R-boundedness of (I + itΠB )−1 follows from this representation. Given ε ∈ (p − p1 , p2 − p) we thus have that there exists δp,ε such that ΠB is R-bisectorial in Lp−ε and in Lp+ε , and Hodge decomposes Lp−ε and Lp+ε . By interpolation (cf. Remark 3.12), ΠB is R-bisectorial in Lp˜ and Hodge decomposes Lp˜ for all p˜ ∈ (p − ε, p + ε). Now the conditions of Corollary 8.12 are verified for the operators ΠB and ΠB , so the mentioned result implies that ΠB has a bounded H ∞ functional calculus in Lp , as claimed. With potential applications to boundary value problems in mind (see [4]), we conclude this section with the following special case. This proof is essentially the same as in the L2 case [10, Theorem 3.1]. Pn ˆ 8.17. Corollary. Let 1 ≤ p1 < p2 ≤ ∞. Let D = −i j=1 D j ∂j be a first order differential ˆ j ∈ L (CN ), and A ∈ L∞ (Rn ; L (CN )) be such that operator with D ˆ |ξ||e| . |D(ξ)e| (H1)
ˆ σ(D(ξ)) ⊆ Sω
ˆ e ∈ R(D(ξ)), π for some ω ∈ (0, ) 2
for all
and and all
(H2)
kukp . kAukp
for all
u ∈ Rp (D),
(H3)
kukp0 . kA∗ ukp0
for all
u ∈ Rp0 (D∗ ),
ξ ∈ Rn ,
for all p ∈ (p1 , p2 ). Then we have the following: (1) The operator DA has an H ∞ functional calculus (with angle µ) in Lp for all p ∈ (p1 , p2 ) if and only if it is R-bisectorial (with angle µ) in Lp for all p ∈ (p1 , p2 ). (2) If the equivalent conditions of (1) hold, then for each p ∈ (p1 , p2 ), there exists δ = δp > 0 ˜ ∞ < δ, then DA˜ also has an such that, if another A˜ ∈ L∞ (Rn ; L (CN )) satisfies kA − Ak ∞ p H functional calculus in L . Proof. (1) The philosophy of the proof is to reduce the consideration of an operator of the form DA to the Hodge-Dirac operator with variable coefficients ΠB , which we already understand from the previous results. On CN ⊕ CN , consider the matrices ˆ ˆ j := 0 Dj , B1 := A 0 , B2 := 0 0 . ˆ j := 0 0 , Γ Γ ˆj 0 0 0 0 A D 0 0 We define the associated differential operators Γ, Γ and Π acting in Lp := Lp (Rn ; CN ⊕ CN ) as in Subsections 3.B, and the operators 0 ADA 0 D ΠB = , ΠB = , D 0 ADA 0
22
HYTÖNEN, MCINTOSH, AND PORTAL
as in Subsection 3.C. We claim that both ΠB and ΠB are then Hodge-Dirac operators with variable coefficients in Lp which Hodge-decompose Lp for all p ∈ (p1 , p2 ). Indeed, the hypotheses (H1), (H2) and (H3) guarantee the conditions (Π1), (Π2) and (B2). The remaining requirements (Π3) and (B1), as well as the Hodge decomposition (see [18, Lemma 3.5]), are satisfied because of the special form of ΠB . A computation shows that I −itADA I 0 I 0 −1 (I + itΠB ) = . 0 I 0 (I + t2 (DA)2 )−1 −itD I Assuming that DA is R-bisectorial, we check that so is ΠB . This amounts to verifying the Rboundedness of the families of operators (I + t2 (DA)2 )−1 ,
−itADA(I + t2 (DA)2 )−1 ,
−it(I + t2 (DA)2 )−1 D,
where t ∈ R. For the first two, this is immediate from the R-bisectoriality of DA and the boundedness of A. For the third one, we need the stability of R-boundedness in the Lp spaces under adjoints, and hypothesis (H3) which allows to reduce the adjoint −itD∗ (I + t2 (A∗ D∗ )2 )−1 to a function of (DA)∗ = A∗ D∗ by composing with A∗ from the left. Thus, by Corollary 8.12, ΠB has an H ∞ functional calculus on Lp for all p ∈ (p1 , p2 ). But the resolvent formula above also gives Au Af (DA)u f (ΠB ) = , u f (DA)u for f ∈ H0∞ (Sθ ) and u ∈ Lp , and hence we find that DA has an H ∞ functional calculus, too. This completes the proof that the R-bisectoriality of DA implies functional calculus. The converse direction is a general property of the functional calculus in Lp (see Remark 2.5). (2) We turn to the second part and assume that DA is R-bisectorial. We first note that A˜ also satisfies the hypotheses (H2) and (H3) when δ is small enough. Indeed, for u ∈ Rp (D), we have ˜ p ≥ kAukp − kA − Ak ˜ ∞ kukp ≥ (c − δ)kukp , and (H3) is proven similarly. Moreover, the kAuk R-bisectoriality of DA implies the same property for DA˜ by a Neumann series argument, as ˜ = (I + itDA)[I − (I + itDA)−1 itD(A − A)], ˜ I + itDA˜ = I + itDA − itD(A − A) where the family {(I + itDA)−1 itD : t ∈ R} is R-bounded (by duality, (H3), and the R˜ ∞ < δ ensures convergence for δ small enough. bisectoriality of DA), and the factor kA − Ak ∞ The H calculus then follows from part (1) applied to A˜ in place of A. 9. Lipschitz estimates In this final section, we prove Lipschitz estimates of the form kf (ΠB )u − f (ΠA )ukp . max kAi − Bi k∞ kf k∞ kukp , i=1,2
for small perturbations of the coefficient matrices involved in the Hodge-Dirac operators. Such estimates are obtained via holomorphic dependence results for perturbations Bz depending on a complex parameter z. This technique can be seen as one of the original motivations for studying the Kato problem for operators with complex coefficients. We start with the operators studied in Corollary 8.17, and then deduce similar 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 open set of C and (Az )z∈U a family of multiplication operators such that the map z 7→ Az ∈ L∞ (Rn ; CN ) is holomorphic. Let 1 ≤ p1 < p2 ≤ ∞, z0 ∈ U , and assume that DAz0 has a bounded H ∞ functional calculus in Lp for all p ∈ (p1 , p2 ). By Corollary 8.17, for each p ∈ (p1 , p2 ), there then exists a δ = δp > 0 such that DAz has a bounded H ∞ functional calculus in Lp for all z ∈ B(z0 , δ). Moreover, we have the following. 9.1. Proposition. For θ ∈ (µ, π2 ) and f ∈ H ∞ (Sθ ), the function z 7→ f (DAz ) is holomorphic on D(z0 , δ).
FUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS
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Proof. This is entirely similar to [10, Theorem 6.1, Theorem 6.4]. Letting τ ∈ C \ Sθ , we have d (I + τ DAz )−1 = −(I + τ DAz )−1 τ DA0z (I + τ DAz )−1 . dz From (H1), (H3) and the bisectoriality of DAz , these operators are uniformly bounded for z ∈ U , and thus the functions z 7→ (I + τ DAz )−1 are holomorphic. The result is then obtained by passing to uniform limits in the strong operator toplogy. The Lipschitz estimates now follow. 9.2. Corollary. Let 1 ≤ p1 < p2 ≤ ∞, let D be a first order differential operator, and A ∈ L∞ (Rn ; CN ) a multiplication operator which satisfy the hypotheses (H1), (H2) and (H3) of Corollary 8.17. Let moreover DA be R-bisectorial. Then, for each p ∈ (p1 , p2 ), there exists δ = δp > 0 ˜ ∞ < δ, then DA˜ has a bounded H ∞ functional such that, if A˜ ∈ L∞ (Rn ; CN ) satisfies kA − Ak π p calculus in L with some angle ω ∈ (0, 2 ), and for θ ∈ (ω, π2 ), f ∈ H ∞ (Sθ ), and u ∈ Lp we have ˜ p . kA − Ak ˜ ∞ kf k∞ kukp . kf (DA)u − f (DA)uk Proof. Let Az := A + z(A˜ − A)/kA˜ − Ak∞ . Then A0 = A, Az1 = A˜ for z1 = kA˜ − Ak∞ , and z 7→ Az is holomorphic. For z ∈ D(0, δ), where δ is small enough, DAz has a bounded H ∞ functional calculus in Lp by Corollary 8.17, and z 7→ f (DAz ) is holomorphic for f ∈ H ∞ (Sθ ) by Proposition 9.1. By the Schwarz Lemma, kf (DA0 )u − f (DAz1 )ukp . |z1 | kf k∞ kukp ,
which gives the assertion.
We finally turn to the Lipschitz estimates for Hodge-Dirac operators with variable coefficients, using the same approach as in [3, Section 10.1]. 9.3. Corollary. Let 1 ≤ p1 < p2 ≤ ∞, and let ΠA and ΠA be Hodge-Dirac operators with variable coefficients, where ΠA is R-bisectorial in Lp and Hodge-decomposes Lp for all p ∈ (p1 , p2 ). Then, for each p ∈ (p1 , p2 ), there exists δ = δp > 0 such that, if ΠB and ΠB are also Hodge-Dirac operators with variable coefficients with kA1 −B1 k∞ +kA2 −B2 k∞ < δ, then both ΠA and ΠB have a bounded H ∞ functional calculus with some angle ω ∈ (0, π2 ), and for θ ∈ (ω, π2 ), f ∈ H ∞ (Sθ ), and u ∈ Lp there holds kf (ΠA )u − f (ΠB )ukp . max kAi − Bi k∞ kf k∞ kukp . i=1,2
Proof. The philosophy of the proof is analogous to that of Corollary 8.17 but goes in the opposite direction: we now deduce results for operators of the form ΠA from what we already know for the operators DA. To this end, consider the space Lp ⊕ Lp ⊕ Lp and the operators 0 0 0 0 0 0 A := 0 A1 0 . D := 0 0 Γ , 0 Γ 0 0 0 A2 A A Let us write PA 0 , PΓ and PΓA for the Hodge-projections associated to ΠA . By (B2), the restriction A1 : Rp (Γ) → Rp (ΓA ) is an isomorphism, and we write A−1 for its inverse. Then a 1 computation shows that I 0 0 A 2 2 −1 (I + itDA)−1 = 0 A−1 A1 −itΓA2 (I + t2 Π2A )−1 , 1 PΓA (I + t ΠA ) 2 2 −1 2 2 −1 0 −itΓ(I + t ΠA ) A1 PA Γ (I + t ΠA )
and one can check that the R-bisectoriality and the Hodge-decomposition of ΠA imply the Rbisectoriality of DA. Indeed, for the diagonal elements above it is immediate, and for the non−1 A A diagonal elements it follows after writing ΓA2 = A−1 1 ΓA = A1 PΓA ΠA and Γ = PΓ ΠA . p p p p p p p We next define operators SA : L → L ⊕ L ⊕ L and TA : L ⊕ L ⊕ L → Lp by −1 A A SA u := (PA 0 u, A1 PΓA u, PΓ u), A A TA (u, v, w) := PA 0 u + PΓ w + PΓA A1 v.
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One checks that TA SA = I and SA ΠA = (DA)SA . Hence SA (λ − ΠA )−1 = (λ − DA)−1 SA , and then by the definition of the functional calculus, f (ΠA )u = TA f (DA)SA u,
f ∈ H ∞ (Sθ ),
u ∈ Lp .
We repeat the above definitions and observations with B in place of A, and then f (ΠA )u − f (ΠB )u = [TA − TB ]f (DA)SA u + TB [f (DA) − f (DB)]SA u + TB f (DB)[SA − SB ]u. The asserted Lipschitz estimate then follows by using Corollary 9.2 for the middle term, and Proposition 6.10 and Corollary 6.11 for the other two terms. References [1] D. Albrecht, X. Duong, A. McIntosh, Operator theory and harmonic analysis. In Instructional Workshop 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 Lp estimates of Riesz transforms associated to 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 and boundary value problems. In Proceedings of the 8th International Conference on Harmonic Analysis 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 boundary 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 square root problem for second order elliptic operators on Rn . 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 I: General operator theory and weights, Adv. Math. 212 (2007), 225–276. [7] 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. [8] 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. [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 H 1 -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 geometry of 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 ∞ functional calculus, 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 Applications 169, Birkhäuser Verlag, Basel (2006) [18] T. Hytönen, A. McIntosh, P. Portal, Kato’s square root problem in Banach spaces. J. Funct. Anal. 254 (2008), no. 3, 675–726. [19] N. J. Kalton, P. C. Kunstmann, L. Weis, Perturbation and interpolation theorems for the H ∞ -calculus with applications to differential operators, Math. Ann, 336 (2006) no. 4, 747–801. [20] N. J. Kalton, L. Weis, The H ∞ -calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345. [21] P. C. Kunstmann, L. Weis, Maximal Lp regularity for parabolic problems, Fourier multiplier theorems and H ∞ -functional calculus, In Functional Analytic Methods for Evolution Equations (Editors: M. Iannelli, R. Nagel, S. Piazzera). Lect. Notes in Math. 1855, Springer-Verlag (2004). [22] A. McIntosh, Operators which have an H ∞ functional calculus. In Miniconference on operator theory and partial differential equations (North Ryde, 1986). Proc. Centre Math. Appl. Austral. Nat. Univ., 14 (1986), 210–231. [23] A. McIntosh, A. Yagi, Operators of type ω without a bounded H ∞ functional calculus. Proc. Centre Math. Appl. Austral. Nat. Univ., 24 (1990), 159–174.
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Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2b, FI-00014 Helsinki, Finland E-mail address:
[email protected] Centre for Mathematics and its Applications, Australian National University, Canberra ACT 0200, Australia E-mail address:
[email protected] Université Lille 1, Laboratoire Paul Painlevé, 59655 Villeneuve d’Ascq, France E-mail address:
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