Holomorphic functional calculus of Hodge-Dirac operators in Lp
aa r X i v : . [ m a t h . F A ] J u l HOLOMORPHIC FUNCTIONAL CALCULUS OF HODGE-DIRACOPERATORS IN L p TUOMAS HYTÖNEN, ALAN MCINTOSH, AND PIERRE PORTALAbstra t. We study the boundedness of the H ∞ fun tional al ulus for di(cid:27)erential operatorsa ting in L p ( R n ; C N ) . For onstant oe(cid:30) ients, we give simple onditions on the symbolsimplying su h boundedness. For non- onstant oe(cid:30) ients, we extend our re ent results for the L p theory of the Kato square root problem to the more general framework of Hodge-Dira operators with variable oe(cid:30) ients Π B as treated in L ( R n ; C N ) by Axelsson, Keith, andM Intosh. We obtain a hara terization of the property that Π B has a bounded H ∞ fun tional al ulus, in terms of randomized boundedness onditions of its resolvent. This allows us todedu e stability under small perturbations of this fun tional al ulus.1. Introdu tionA variety of problems in PDE's an be solved by establishing the boundedness, and stabilityunder small perturbations, of the H ∞ fun tional al ulus of ertain di(cid:27)erential operators. In par-ti ular, Axelsson, Keith, and M Intosh [10℄ have re overed and extended the solution of the Katosquare root problem [5℄ by showing that Hodge-Dira operators with variable oe(cid:30) ients of theform Π B = Γ+ B Γ ∗ B have a bounded H ∞ fun tional al ulus in L ( R n ; C N ) , when Γ is a homo-geneous (cid:28)rst order di(cid:27)erential operator with onstant oe(cid:30) ients, and B , B ∈ L ∞ ( R n ; L ( C N )) are stri tly a retive multipli ation operators. Re ently, Aus her, Axelsson, and M Intosh [4℄ haveused related perturbation results to show the openness of some sets of well-posedness for boundaryvalue problems with L boundary data.In this paper, we (cid:28)rst onsider homogeneous di(cid:27)erential operators with onstant (matrix-valued) oe(cid:30) ients. For su h operators the boundedness of the H ∞ fun tional al ulus is established usingMikhlin's multiplier theorem. However, the estimates on the symbols may be di(cid:30) ult to he kin pra ti e, espe ially when the null spa es of the symbols are non-trivial. Here we provide asimple ondition (invertibility of the symbols on their ranges and in lusion of their eigenvalues ina bise tor), that gives su h estimates. We then turn to operators with oe(cid:30) ients in L ∞ ( R n ; C ) of the form Π B = Γ + B Γ B , where Γ and Γ are nilpotent homogeneous (cid:28)rst order operators with onstant (matrix-valued) oe(cid:30) ients, and B , B ∈ L ∞ ( R n ; L ( C N )) are multipli ation operatorssatisfying some L p oer ivity ondition. For su h operators, we aim at perturbation results whi hgive, in parti ular, the boundedness of the H ∞ fun tional al ulus when B , B are small pertu-bations of onstant- oe(cid:30) ient matri es.This presents two main di(cid:30) ulties. First of all, even in L , the H ∞ fun tional al ulus of a(bi)se torial operator is in general not stable under small perturbations in the sense that thereexist a self-adjoint operator D and bounded operators A with arbitrary small norm su h that D ( I + A ) does not have a bounded H ∞ fun tional al ulus (see [23℄). Subtle fun tional analyti perturbation results exist (see [16℄ and [19℄), but do not give the estimates needed in [4℄ or [10℄. Toobtain su h estimates, one needs to take advantage of the spe i(cid:28) stru ture of di(cid:27)erential operatorsusing harmoni analyti methods. Then, the problem of moving from the L theory to an L p the-ory is substantial. Indeed, the operators under onsideration fall outside the Calderón-Zygmund lass, and annot be handled by familiar methods based on interpolation. A known substitute,pioneered by Blun k and Kunstmann in [12℄, and developed by Aus her and Martell [2, 6, 7, 8℄,Date: O tober 10, 2018.2000 Mathemati s Subje t Classi(cid:28) ation. 47A60, 47F05.1 HYTÖNEN, MCINTOSH, AND PORTAL onsists in establishing an extrapolation method adapted to the operator, whi h allows to extendresults from L to L p for p in a ertain range ( p , p ) ontaining . In [18℄ we started anotherapproa h, whi h ombines probabilisti tools from fun tional analysis with the aforementioned L methods, and allows L p results whi h do not rely on some L ounterparts.However, our goal in [18℄ was the Kato problem, and we did not rea h the generality of [10℄whi h has re ently proven parti ularly useful in onne tion with boundary value problems [4℄.Here we lose this gap and, in fa t, rea h a further level of generality. Roughly speaking, forquite general di(cid:27)erential operators, we show that the boundedness of the H ∞ fun tional al ulus oin ides with the R-(bi)se toriality (see Se tion 2 for relevant de(cid:28)nitions). This then allows per-turbation results, in ontrast with the general theory of se torial operators, where R-se torialityand bounded H ∞ al ulus are two distin t properties, and perturbation results are mu h morerestri ted.For the operators with variable oe(cid:30) ients, the ore of the argument is ontained in [18℄, sothe reader might want to have a opy of this paper handy. Here we fo us on the points where [18℄needs to be modi(cid:28)ed, and develop some adaptation of the te hniques to generalized Hodge-Dira operators whi h may be of interest in other problems. To make the paper more readable, we hoose not to work in the Bana h-spa e valued setting of [18℄, but the interested reader will soonrealize that our proof arries over to that situation provided that, as in [18℄, the target spa e is aUMD spa e, and both the spa e and its dual have the RMF property.The paper is organized as follows. In Se tion 2, we re all the essential de(cid:28)nitions. In Se tion 3,we present our setting and state the main results. In Se tion 4, we deal with onstant oe(cid:30) ientoperators and obtain appropriate estimates on their symbols. In Se tion 5, we use these estimatesto establish an L p theory for operators with onstant (matrix-valued) oe(cid:30) ients. In Se tion6, we show that a ertain (Hodge) de omposition, ru ial in our study, is stable under smallperturbations. In Se tion 7, we give simple proofs of general operator theoreti results on thefun tional al ulus of bise torial operators. In Se tion 8, we prove our key results on operatorswith variable oe(cid:30) ients, referring to [18℄ when arguments are identi al, and explaining how tomodify them using the results of the pre eding se tions when they are not. Finally, in Se tion 9,we derive from Se tion 8 Lips hitz estimates for the fun tional al ulus of these operators.A knowledgments. This work advan ed through visits of T.H. and P.P. at the Centre for Mathe-mati s and its Appli ations at the Australian National University, and of P.P. at the Universityof Helsinki. Thanks go to these institutions for their outstanding support. The resear h wassupported by the Australian Government through the Australian Resear h Coun il, and by theA ademy of Finland through the proje t 114374 (cid:16)Ve tor-valued singular integrals(cid:17).2. PreliminariesFix some numbers n, N ∈ Z + . We onsider fun tions u : R n → C N , or A : R n → L ( C N ) .The Eu lidean norm in both R n and C N , as well as the asso iated operator norm in L ( C N ) , aredenoted by | · | . To express the typi al inequalities (cid:16)up to a onstant(cid:17) we use the notation a . b tomean that there exists C < ∞ su h that a ≤ Cb , and the notation a h b to mean that a . b . a .The impli it onstants are meant to be independent of other relevant quantities. If we want tomention that the onstant C depends on a parameter p , we write a . p b .Let us brie(cid:29)y re all the onstru tion of the H ∞ fun tional al ulus (see [1, 15, 17, 21, 22℄ fordetails).2.1. De(cid:28)nition. A losed operator A a ting in a Bana h spa e Y is alled bise torial with angle θ if its spe trum σ ( A ) is in luded in a bise tor: σ ( A ) ⊆ S θ := Σ θ ∪ ( − Σ θ ) , where Σ θ := { z ∈ C ; | arg( z ) | ≤ θ } , UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 3and outside the bise tor it veri(cid:28)es the following resolvent bounds:(2.2) ∀ θ ′ ∈ ( θ, π ∃ C > ∀ λ ∈ C \ S θ ′ k λ ( λI − A ) − k L ( Y ) ≤ C. We often omit the angle, and say that A is bise torial if it is bise torial with some angle θ ∈ [0 , π ) .For < ν < π/ , let H ∞ ( S ν ) be the spa e of bounded fun tions on S ν , whi h are holomorphi on the interior of S ν , and onsider the following subspa e of fun tions with de ay at zero andin(cid:28)nity: H ∞ ( S ν ) := n φ ∈ H ∞ ( S ν ) : ∃ α, C ∈ (0 , ∞ ) ∀ z ∈ S ν | φ ( z ) | ≤ C | z z | α o . For a bise torial operator A with angle θ < ω < ν < π/ , and ψ ∈ H ∞ ( S ν ) , we de(cid:28)ne ψ ( A ) u := 12 iπ Z ∂S ω ψ ( λ )( λ − A ) − u d λ, where ∂S ω is dire ted anti- lo kwise around S ω .2.3. De(cid:28)nition. A bise torial operator A with angle θ , is said to admit a bounded H ∞ fun tional al ulus with angle µ ∈ [ θ, π ) if, for ea h ν ∈ ( µ, π ) , ∃ C < ∞ ∀ ψ ∈ H ∞ ( S ν ) k ψ ( A ) y k Y ≤ C k ψ k ∞ k y k Y . In this ase, and if Y is re(cid:29)exive, one an de(cid:28)ne a bounded operator f ( A ) for f ∈ H ∞ ( S ν ) by f ( A ) u := f (0) P u + lim n →∞ ψ n ( A ) u, where P denotes the proje tion on the null spa e of A orresponding to the de omposition Y = N ( A ) ⊕ R ( A ) , whi h exists for R-bise torial operators, and ( ψ n ) n ∈ N ⊂ H ∞ ( S ν ) is a boundedsequen e whi h onverges lo ally uniformly to f . See [1, 15, 17, 21, 22℄ for details.2.4. De(cid:28)nition. A family of operators T ⊂ L ( Y ) is alled R-bounded if for all M ∈ N , all T , . . . , T M ∈ T , and all u , . . . , u M ∈ Y , E (cid:13)(cid:13)(cid:13) M X k =1 ε k T k u k (cid:13)(cid:13)(cid:13) Y . E (cid:13)(cid:13)(cid:13) M X k =1 ε k u k (cid:13)(cid:13)(cid:13) Y , where E is the expe tation whi h is taken with respe t to a sequen e of independent Radema hervariables ε k , i.e., random signs with P ( ε k = +1) = P ( ε k = −
1) = .A bise torial operator A is alled R-bise torial with angle θ in Y if the olle tion { λ ( λI − A ) − : λ ∈ C \ S θ ′ } is R-bounded for all θ ′ ∈ ( θ, π/ . The in(cid:28)mum of su h angles θ is alled the angle of R-bise toriality of A .Again, we may omit the angle and simply say that A is R-bise torial if it is R-bise torial withsome angle θ ∈ (0 , π/ . Noti e that, by a Neumann series argument, this is equivalent to theR-boundedness of { ( I + itA ) − : t ∈ R } . The reader unfamiliar with R-boundedness and thederived notions an onsult [18℄ and the referen es therein.2.5. Remark. On subspa es of L p , < p < ∞ , an operator with a bounded H ∞ fun tional al ulusis R-bise torial. The proof (stated for se torial rather than bise torial operators) an be found in[20, Theorem 5.3℄. 3. Main resultsWe onsider three types of operators. First, we look at di(cid:27)erential operators of arbitrary or-der with onstant (matrix valued) oe(cid:30) ients, and provide simple onditions on their Fouriermultiplier symbols to ensure that su h operators are bise torial and, in fa t, have a bounded H ∞ fun tional al ulus. Then, we fo us on (cid:28)rst order operators with a spe ial stru ture, theHodge-Dira operators, and prove that, under an additional ondition on the symbols, they givea spe i(cid:28) (Hodge) de omposition of L p . Finally we turn to Hodge-Dira operators with (bounded HYTÖNEN, MCINTOSH, AND PORTALmeasurable) variable oe(cid:30) ients, and show that the boundedness of the H ∞ fun tional al ulusis preserved under small perturbation of the oe(cid:30) ients.We work in the Lebesgue spa es L p := L p ( R n ; C N ) with p ∈ (1 , ∞ ) , and denote by S ( R n ; C N ) the S hwartz lass of rapidly de reasing fun tions with values in C N , and by S ′ ( R n ; C N ) the orresponding lass of tempered distributions.3.A. General onstant- oe(cid:30) ient operators. In this subse tion, we onsider k th order ho-mogeneous di(cid:27)erential operators of the form D = ( − i ) k X θ ∈ N n : | θ | = k ˆ D θ ∂ θ a ting on S ′ ( R n ; C N ) as a Fourier multiplier with symbol ˆ D ( ξ ) = P | θ | = k ˆ D θ ξ θ , where ˆ D θ ∈ L ( C N ) .3.1. Assumption. The Fourier multiplier symbol ˆ D ( ξ ) satis(cid:28)es(D1) κ | ξ | k | e | ≤ | ˆ D ( ξ ) e | for all ξ ∈ R n , all e ∈ R ( ˆ D ( ξ )) , and some κ > , (D2) there exists ω ∈ [0 , π su h that for all ξ ∈ R n : σ ( ˆ D ( ξ )) ⊆ S ω . In ea h L p , let D a t on its natural domain D p ( D ) := { u ∈ L p ; Du ∈ L p } . In Theorem 5.1 weprove:3.2. Theorem. Let < p < ∞ . Under the assumptions (D1) and (D2), the operator D isbise torial in L p with angle ω , and has a bounded H ∞ fun tional al ulus in L p with angle ω .3.3. Remark. (a) In (D2), the bise tor S ω an be repla ed by the se tor Σ ω where ≤ ω < π . Theoperator D is then se torial (with angle ω ) and has a bounded H ∞ fun tional al ulus (with angle ω ) in the se torial sense, i.e. f ( D ) is bounded for fun tions f ∈ H ∞ (Σ θ ) with any θ ∈ ( ω, π ) .(b) Assuming that (D1) holds for all e ∈ C N would pla e us in a more lassi al ontext, inwhi h proofs are substantially simpler. We insist on this weaker ellipti ity ondition sin e theoperators we want to handle have, in general, a non-trivial null spa e.( ) Using Bourgain's version of Mikhlin's multiplier theorem [13℄ , the above theorem extendsto fun tions with values in X N , where X is a UMD Bana h spa e.3.B. Hodge-Dira operators with onstant oe(cid:30) ients. We now turn to (cid:28)rst order opera-tors of the form Π = Γ + Γ , where Γ = − i P nj =1 ˆΓ j ∂ j , a ts on S ′ ( R n ; C N ) as a Fourier multiplier with symbol ˆΓ = ˆΓ( ξ ) = n X j =1 ˆΓ j ξ j , ˆΓ j ∈ L ( C N ) , the operator Γ is de(cid:28)ned similarly, and both operators are nilpotent in the sense that ˆΓ( ξ ) = 0 and ˆΓ( ξ ) = 0 for all ξ ∈ R n .3.4. De(cid:28)nition. We all Π = Γ+Γ a Hodge-Dira operator with onstant oe(cid:30) ients if its Fouriermultiplier symbol ˆΠ = ˆΓ + ˆΓ satis(cid:28)es the following onditions:( Π κ | ξ || e | ≤ | ˆΠ( ξ ) e | for all e ∈ R ( ˆΠ( ξ )) , all ξ ∈ R n , and some κ > , ( Π σ ( ˆΠ( ξ )) ⊆ S ω for some ω ∈ [0 , π , and all ξ ∈ R n , ( Π N ( ˆΠ( ξ )) = N (ˆΓ( ξ )) ∩ N (ˆΓ( ξ )) for all ξ ∈ R n . UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 53.5. Remark. The (cid:16)Hodge-Dira (cid:17) terminology has its origins in appli ations of this formalism toRiemannian 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 wasassumed to be the adjoint of Γ . In parti ular, our operator Π does not need to be self-adjoint in L ( R n ; C N ) .In ea h L p , we let the operators Υ ∈ { Γ , Γ , Π } a t on their natural domains D p (Υ) := { u ∈ L p : Υ u ∈ L p } , where Υ u is de(cid:28)ned in the distributional sense. Ea h Υ is a densely de(cid:28)ned, losed unboundedoperator in L p with this domain. The formal nilpoten e of Γ and Γ transfers into the operator-theoreti nilpoten e R p (Γ) ⊆ N p (Γ) , R p (Γ) ⊆ N p (Γ) . where R p (Γ) , N p (Γ) denote the range and kernel of Γ as an operator on L p .In Se tion 5 we show that the identity Π = Γ+Γ is also true in the sense of unbounded operatorsin L p . Moreover, in Theorem 5.5 we prove:3.6. Theorem. The operator Π has a bounded H ∞ fun tional al ulus in L p with angle ω , andsatis(cid:28)es the Hodge de omposition L p = N p (Π) ⊕ R p (Γ) ⊕ R p (Γ) . X N , where X is a UMD Bana h spa e.3.C. Hodge-Dira operators with variable oe(cid:30) ients. We (cid:28)nally turn to Hodge-Dira operators with variable oe(cid:30) ients. The study of su h operators is motivated by [4℄, [10℄ and [18℄.3.8. De(cid:28)nition. Let < p < ∞ and p ′ denote the dual exponent of p . Let B , B ∈ L ∞ ( R n ; L ( C N )) , and identify these fun tions with bounded multipli ation operators on L p in the natural way. Alsolet Π = Γ + Γ be a Hodge-Dira operator. Then the operator(3.9) Π B := Γ + Γ B , where Γ B := B Γ B , is alled a Hodge-Dira operator with variable oe(cid:30) ients in L p if the following hold:(B1) Γ B B Γ = 0 on S ( R n ; C N ) , (B2) k u k p . k B u k p ∀ u ∈ R p (Γ) and k v k p ′ . k B ∗ v k p ′ ∀ v ∈ R p ′ (Γ ∗ ) . Note that the operator equality (3.9), involving the impli it domain ondition D p (Π B ) := D p (Γ) ∩ D p (Γ B ) , was a proposition for Hodge-Dira operators with onstants oe(cid:30) ients, but istaken as the de(cid:28)nition for Hodge-Dira operators with variable oe(cid:30) ients.The following simple onsequen es will be frequently applied. Their proofs are left to the reader.First, the nilpoten e ondition (B1), a priori formulated for test fun tions, self-improves to Γ B B Γ = 0 on D p (Γ); hen e R p (Γ B ) ⊆ N p (Γ B ) . Se ond, the oer ivity ondition (B2) implies that R p (Γ B ) = B R p (Γ B ) = B R p (Γ) , and B : R p (Γ) → R p (Γ B ) is an isomorphism . Sometimes, we also need to assume that the related operator Π B = Γ+ B Γ B is a Hodge-Dira operator with variable oe(cid:30) ients in L p , i.e. Γ B B Γ = 0 on S ( R n ; C N ) , k u k p . k B u k p ∀ u ∈ R p (Γ) and k v k p ′ . k B ∗ v k p ′ ∀ v ∈ R p ′ (Γ ∗ ) . With the same proof as in [10, Lemma 4.1℄, one an show: HYTÖNEN, MCINTOSH, AND PORTAL3.10. Proposition. Assuming Π B = Γ + Γ B is a Hodge-Dira operator with variable oe(cid:30) ients,then the operators Γ B := B Γ B and Γ ∗ B := B ∗ Γ ∗ B ∗ are losed, densely de(cid:28)ned, nilpotent opera-tors in L p and L p ′ repe tively, and Γ ∗ B = (Γ B ) ∗ .However, the Hodge-de omposition and resolvent bounds, whi h in the ontext of [10℄ (and the(cid:28)rst-mentioned one even in [18℄) ould be established as propositions, are now properties whi hmay or may not be satis(cid:28)ed:3.11. De(cid:28)nition. We say that Π B Hodge-de omposes L p if L p = N p (Π B ) ⊕ R p (Γ) ⊕ R p (Γ B ) . Π B with the propertythat it is R-bise torial in L p and Hodge-de omposes L p . If this property holds for two exponents p ∈ { p , p } , then it holds for the intermediate values p ∈ ( p , p ) as well, and hen e the set ofexponents p , for whi h the mentioned property is satis(cid:28)ed, is an interval.The proof that R-bise toriality interpolates in these spa es an be found in [19, Corollary 3.9℄,where it is formulated for R-se torial operators. As for the Hodge-de omposition, observe (cid:28)rstthat if a Hodge-Dira operator Π B is R-bise torial in L p and Hodge-de omposes L p , then theproje tions onto the three Hodge subspa es are given by P = lim t →∞ ( I + t Π B ) − , P Γ = lim t →∞ t ΓΠ B ( I + t Π B ) − , P Γ B = lim t →∞ t Γ B Π B ( I + t Π B ) − , where the limits are taken in the strong operator topology. In parti ular, if Π B has these propertiesin two di(cid:27)erent L p spa es, then the orresponding Hodge subspa es have ommon proje tions, andone dedu es the boundedness of these proje tion operators also in the interpolation spa es.The following main result on erning the operators Π B gives a hara terization of the bound-edness of their H ∞ fun tional al ulus. It will be proven as Corollary 8.12 to Theorem 8.1.3.13. Theorem. Let ≤ p < p ≤ ∞ , and let Π B be a Hodge-Dira operator with variable oe(cid:30) ients in L p whi h Hodge-de omposes L p for all p ∈ ( p , p ) . Assume also that Π B is aHodge-Dira operator with variable oe(cid:30) ients in L p . Then Π B has a bounded H ∞ fun tional al ulus (with angle µ ) in L p ( R n ; C N ) for all p ∈ ( p , p ) if and only if it is R -bise torial (withangle µ ) in L p ( R n ; C N ) for all p ∈ ( p , p ) .This hara terization leads to perturbation results su h as the following, proven in Corol-lary 8.16, thanks to the perturbation properties of R-bise toriality.3.14. Corollary. Let ≤ p < p ≤ ∞ , and let Π A be a Hodge-Dira operator with variable oe(cid:30) ients, whi h is R-bise torial in L p and Hodge-de omposes L p for all p ∈ ( p , p ) . Then forea h p ∈ ( p , p ) , there exists δ = δ p > su h that, if Π B and Π B are Hodge-Dira operators withvariable oe(cid:30) ients, and if k B − A k ∞ + k B − A k ∞ < δ , then Π B has a bounded H ∞ fun tional al ulus in L p and Hodge-de omposes L p .3.15. Remark. The results in this paper on erning Hodge-Dira operators with variable oe(cid:30) ients an be extended to the Bana h spa e valued setting, provided the target spa e has the so- alledUMD and RMF properties, and also its dual has RMF. The UMD property, whi h passes tothe dual automati ally, is a well-known notion in the theory of Bana h spa es, f. [14℄. Weintrodu ed the RMF property in [18℄ in relation with our Radema her maximal fun tion. It holdsin ( ommutative or not) L p spa es for < p < ∞ and in spa es with type 2, and fails in L .We do not know whether it holds in every UMD spa e. This paper, espe ially in Se tion 8, usesextensively the te hniques from [18℄. We hoose not to formulate the results in a Bana h spa evalued setting to make the paper more readable, but all proofs are naturally suited to this moregeneral ontext. 4. Properties of the symbolsIn this se tion we onsider the symbols of the Fourier multipliers de(cid:28)ned in Subse tions 3.Aand 3.B. As a onsequen e of the assumptions made in these subse tions, we obtain the variousUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 7estimates whi h we need in the next se tions to establish an L p 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 di(cid:27)erential operator with onstant matrix oef-(cid:28) ients, satisfying (D1) and (D2). Then, denoting M = sup | ξ | =1 | ˆ D ( ξ ) | , we have that(a) σ ( ˆ D ( ξ )) ⊂ ( S ω ∩ A ( κ | ξ | k , M | ξ | k )) ∪ { } , (b) C N = N ( ˆ D ( ξ )) ⊕ R ( ˆ D ( ξ )) ,( ) ∀ µ ∈ ( ω, π ) | ( ζI − ˆ D ( ξ )) − | . | ζ | − ∀ ξ ∈ R n ∀ ζ ∈ C \ ( S µ ∩ A ( κ | ξ | k , M | ξ | k )) .Using ompa tness for | ξ | = 1 and homogeneity for | ξ | 6 = 1 , this is a onsequen e of the followinglemma.4.2. Lemma. Let T ∈ L ( C N ) , κ > , and ω ∈ [0 , π ) , and suppose that(i) κ | e | ≤ | T e | for all e ∈ R ( T ) , and(ii) σ ( T ) ⊂ S ω . Then we have that(a) σ ( T ) ⊂ ( S ω ∩ A ( κ, | T | )) ∪ { } , (b) C N = N ( T ) ⊕ R ( T ) ,( ) ∀ µ ∈ ( ω, π ) | ( ζI − T ) − | . | ζ | − ∀ ζ ∈ C \ ( S µ ∩ A ( κ, | T | )) .Proof. Let us (cid:28)rst remark that, for a non zero eigenvalue λ with eigenve tor e , we have that | λ || e | = | T e | ≥ κ | e | . This gives (a). Moreover, (i) also gives that N ( T ) = N ( T ) . Thus, writing T in Jordan anoni al form, we have the splitting C N = N ( T ) ⊕ R ( T ) . The resolvent bounds hold on N ( T ) .On R ( T ) , the fun tion ζ ζ ( ζI − T ) − is ontinuous from the losure of C \ ( S µ ∩ A ( κ, | T | )) to L ( R ( T ) , C N ) and is bounded at ∞ , and thus is bounded on C \ ( S µ ∩ A ( κ, | T | )) . (cid:3) Assuming (D1) and (D2), we thus have that for all θ ∈ (0 , π − ω ) and for all ξ ∈ R n , ∃ C > ∀ τ ∈ S θ | ( I + iτ ˆ D ( ξ )) − | L ( C N ) ≤ C. For τ ∈ S θ , we use the following notation: ˆ R τ ( ξ ) := ( I + iτ ˆ D ( ξ )) − , ˆ P τ ( ξ ) := 12 ( ˆ R τ ( ξ ) + ˆ R − τ ( ξ )) = ( I + τ ˆ D ( ξ ) ) − , ˆ Q τ ( ξ ) := i R τ ( ξ ) − ˆ R − τ ( ξ )) = τ ˆ D ( ξ ) ˆ P τ ( ξ ) . If λ / ∈ σ ( ˆ D ( ξ )) for some ξ ∈ R n , then also λ / ∈ σ ( ˆ D ( ξ ′ )) for all ξ ′ in some neighbourhood of ξ .One he ks dire tly from the de(cid:28)nition of the derivative that ∂ ξ j ( λ − ˆ D ( ξ )) − = ( λ − ˆ D ( ξ )) − ( ∂ ξ j ˆ D )( ξ )( λ − ˆ D ( ξ )) − . By indu tion it follows that ( λ − ˆ D ( ξ )) − is a tually C ∞ in a neighbourhood of ξ for λ / ∈ σ ( ˆ D ( ξ )) .In parti ular, for τ ∈ S θ , the fun tion ˆ R τ ( ξ ) is C ∞ in ξ ∈ R n , and(4.3) ∂ ξ j ˆ R τ ( ξ ) = ˆ R τ ( ξ )( − iτ ∂ ξ j ˆ D ( ξ )) ˆ R τ ( ξ ) . C N = N ( ˆ D ( ξ )) ⊕ R ( ˆ D ( ξ )) , the omplementary proje tions P N ( ˆ D ( ξ )) and P R ( ˆ D ( ξ )) = I − P N ( ˆ D ( ξ )) are in(cid:28)nitely di(cid:27)erentiable in R n \ { } and satisfy the Mikhlinmultiplier onditions | ∂ αξ P N ( ˆ D ( ξ )) | . α | ξ | −| α | , ∀ α ∈ N n . Proof. The proje tions P N ( ˆ D ( ξ )) are obtained by the Dunford(cid:21)Riesz fun tional al ulus by inte-grating the resolvent around a ontour, whi h ir ums ribes the origin and no other point of the HYTÖNEN, MCINTOSH, AND PORTALspe trum of ˆ D ( ξ ) . By Lemma 4.1, for ξ in a neighbourhood of the unit sphere (say < | ξ | < ),we may hoose P N ( ˆ D ( ξ )) = 12 πi Z ∂ D (0 , − k − κ ) ( λ − ˆ D ( ξ )) − d λ. Using the smoothness of ( λ − ˆ D ( ξ )) − dis ussed before the statement of the lemma, di(cid:27)erentiationof arbitrary order in ξ under the integral sign may be routinely justi(cid:28)ed. This shows that P N ( ˆ D ( ξ )) is C ∞ in a neighbourhood of the unit sphere.To omplete the proof, it su(cid:30) es to observe that ˆ D ( tξ ) = t k ˆ D ( ξ ) for t ∈ (0 , ∞ ) . Hen e N ( ˆ D ( ξ )) and R ( ˆ D ( ξ )) , and therefore the asso iated proje tions, are invariant under the s alings ξ tξ . Itis a general fa t that smooth fun tions, whi h are homogeneous of order zero, satisfy the Mikhlinmultiplier onditions. Indeed, for any ξ ∈ R n \ { } and t ∈ (0 , ∞ ) , we have ∂ αξ P N ( ˆ D ( ξ )) = ∂ αξ ( P N ( ˆ D ( tξ )) ) = t | α | ( ∂ α P N ( ˆ D ( · )) )( tξ ) , and setting t = | ξ | − and using the boundedness of the ontinuous fun tion ∂ αξ P N ( ˆ D ( ξ )) on the unitsphere, the Mikhlin estimate follows. (cid:3) Noti e that, by (D1), ˆ D ( ξ ) is an isomorphism of R ( ˆ D ( ξ )) onto itself for ea h ξ ∈ R n \ { } . Wedenote by ˆ D − R ( ξ ) its inverse.4.5. Lemma. The fun tion ˆ D − R ( ξ ) P R ( D ( ξ )) is smooth in R n \ { } and satis(cid:28)es the Mikhlin-typemultiplier ondition ∂ αξ (cid:0) ˆ D − R ( ξ ) P R ( ˆ D ( ξ )) (cid:1) . α | ξ | − k −| α | , ∀ α ∈ N n . Proof. By the Dunford(cid:21)Riesz fun tional al ulus, we have m ( ξ ) := ˆ D − R ( ξ ) P R ( ˆ D ( ξ )) = 12 πi Z γ ( λ − ˆ D ( ξ )) − d λλ , where γ is any path oriented ounter- lo kwise around the non-zero spe trum of ˆ D ( ξ ) . Sin e ˆ D ( rξ ) = r k ˆ D ( ξ ) , a hange of variables and Cau hy's theorem shows that m ( rξ ) = r − k m ( ξ ) , andthe smoothness of m in R n \ { } is he ked as in the previous proof by di(cid:27)erentiating under theintegral sign. The modi(cid:28)ed Mikhlin ondition follows from this in a similar way as in the previousproof. (cid:3) ν ∈ (0 , π − ω ) , the following Mikhlin onditions hold uniformly in τ ∈ S ν : | ∂ αξ ˆ R τ ( ξ ) | . α | ξ | −| α | , ∀ α ∈ N n . Similar estimates hold for ˆ P τ ( ξ ) and ˆ Q τ ( ξ ) .Proof. We use indu tion to establish the desired bounds. For α = 0 , this was proven in Lemma4.1. In order to make the indu tion step, we will need the identity(4.7) ∂ αξ ˆ R τ = X (cid:8) θ ≤ α (cid:18) αθ (cid:19) ( ∂ α − θ ˆ R τ )( − iτ ∂ θξ ˆ D ) ˆ R τ , α (cid:9) . We use the multi-index notation as follows: the binomial oe(cid:30) ients are (cid:18) αθ (cid:19) := n Y i =1 (cid:18) α i θ i (cid:19) , theorder relation θ ≤ α means that θ i ≤ α i for every i = 1 , . . . , n , whereas θ (cid:8) α means that θ ≤ α but θ = α ; (cid:28)nally, it is understood that (cid:18) αθ (cid:19) = 0 if θ α .Let us prove the identity (4.7) by indu tion. For | α | = 1 , the formula was already establishedin (4.3). Assuming (4.7) for some α (cid:9) , we prove it for α + e j , where e j is the j th standard unitUNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 9ve tor. Indeed, ∂ α + e j ξ ˆ R τ = ∂ ξ j ∂ αξ ˆ R τ = X (cid:8) θ ≤ α (cid:18) αθ (cid:19)h ( ∂ α + e j − θ ˆ R τ )( − iτ ∂ θξ ˆ D ) ˆ R τ + ( ∂ α − θ ˆ R τ )( − iτ ∂ θ + e j ξ ˆ D ) ˆ R τ + ( ∂ α − θ ˆ R τ )( − iτ ∂ θξ ˆ D ) ˆ R τ ( − iτ ∂ ξ j ˆ D ) ˆ R τ i = h X (cid:8) θ ≤ α (cid:18) αθ (cid:19) + X e j (cid:8) θ ≤ α + e j (cid:18) αθ − e j (cid:19)i ( ∂ α + e j − θ ˆ R τ )( − iτ ∂ θξ ˆ D ) ˆ R τ + ( ∂ α ˆ R τ )( − iτ ∂ e j ξ ˆ D ) ˆ R τ = X (cid:8) θ ≤ α + e j h(cid:18) αθ (cid:19) + (cid:18) αθ − e j (cid:19)i ( ∂ α + e j − θ ˆ R τ )( − iτ ∂ θξ ˆ D ) ˆ R τ , and the proof of (4.7) is ompleted by the binomial identity (cid:0) αθ (cid:1) + (cid:0) αθ − e j (cid:1) = (cid:0) α + e j θ (cid:1) . Noti e thatthe indu tion hypothesis was used twi e: (cid:28)rst to expand ∂ αξ ˆ R τ in the se ond step, and then toevaluate the summation over the last of the three terms in the third one.We then pass to the indu tive proof of the assertion of the lemma. Let α (cid:9) , and assumethe laim proven for all β (cid:8) α . We (cid:28)rst onsider ( ∂ αξ ˆ R τ ) P R ( ˆ D ) . By the indu tion assumption, weknow that the fa tors ∂ α − θξ ˆ R τ in (4.7) satisfy | ∂ α − θξ ˆ R τ ( ξ ) | . | ξ | | θ |−| α | . Furthermore, ( − iτ ∂ θξ ˆ D ) ˆ R τ P R ( ˆ D ) = ( ∂ θξ ˆ D ) ˆ R τ ( − iτ ˆ D ) ˆ D − R P R ( ˆ D ) = ( ∂ θξ ˆ D )( ˆ R τ − I ) ˆ D − R P R ( ˆ D ) , where the di(cid:27)erent fa tors are bounded by | ∂ θξ ˆ D | . | ξ | k −| θ | , | ˆ R τ − I | . , | ˆ D − R ( ξ ) P R ( ˆ D ( ξ )) | . | ξ | − k . Multiplying all these estimates, we get | ( ∂ αξ ˆ R τ ) P R ( ˆ D ) | . | ξ | −| α | , as required.It remains to estimate ( ∂ αξ ˆ R τ ) P N ( ˆ D ) . We have ( ∂ αξ ˆ R τ ) P N ( ˆ D ) = ∂ αξ ( ˆ R τ P N ( ˆ D ) ) − X ≤ β (cid:8) α ( ∂ βξ ˆ R τ ) ∂ α − β P N ( ˆ D ) = ∂ αξ P N ( ˆ D ) − X ≤ β (cid:8) α O ( | ξ | −| β | ) O ( | ξ | −| α − β | ) = O ( | ξ | −| α | ) , where the O -bounds follow from the indu tion assumption and the Mikhlin bounds for P N ( ˆ D ( ξ )) established in Proposition 4.4. The proof is omplete. (cid:3) µ ∈ ( ω, π ) , and let f ∈ H ∞ ( S µ ) . Then f ( D ) is a Mikhlin multiplier, andmore pre isely its symbol [ f ( D )( ξ ) = f ( ˆ D ( ξ )) satis(cid:28)es, for ξ = 0 , | ∂ αξ f ( ˆ D ( ξ )) | . µ | ξ | −| α | sup (cid:8) | f ( λ ) | : λ ∈ S µ , κ | ξ | k ≤ | λ | ≤ M | ξ | k (cid:9) , where M = sup | ξ | =1 | ˆ D ( ξ ) | .Proof. Pi k θ ∈ ( ω, µ ) . By the de(cid:28)nition of the fun tional al ulus, we have(4.9) ∂ αξ [ f ( D ) = 12 πi Z ∂S θ f ( ζ ) ∂ αξ ( ζ − ˆ D ) − d ζ = ∂ αξ f ( ˆ D ) . From (4.7) one sees that ∂ αξ ( ζ − ˆ D ) − = ζ − ∂ αξ ˆ R i/ζ has poles at ζ ∈ σ ( ˆ D ) . By Lemma 4.1, thenon-zero spe trum satis(cid:28)es σ ( ˆ D ( ξ )) \ { } ⊂ { ζ ∈ S ω : κ | ξ | k ≤ | ζ | ≤ M | ξ | k } . ξ ∈ R n , we may deform the integration path in (4.9) to ∂ [ S θ ∩ D (0 , M | ξ | k ) \ D (0 , κ | ξ | k )] ∪ ∂ [ S θ ∩ D (0 , ε )] =: γ ∪ γ . On γ , there holds | ζ | h | ξ | k , while the length of the path is also ℓ ( γ ) h | ξ | k . Hen e (cid:12)(cid:12)(cid:12) Z γ f ( ζ ) ∂ αξ ( ζ − ˆ D ( ξ )) − d ζ (cid:12)(cid:12)(cid:12) ≤ sup {| f ( ζ ) | : ζ ∈ S µ ∩ D (0 , M | ξ | k ) \ D (0 , κ | ξ | k ) } Z γ | ∂ αξ ˆ R i/ζ ( ξ ) | | d ζ || ζ | . sup (cid:8) | f ( λ ) | : λ ∈ S µ , κ | ξ | k ≤ | λ | ≤ M | ξ | k (cid:9) · | ξ | −| α | , whi h has a bound of the desired form. On the other hand, the integral over γ vanishes in thelimit as ε ↓ . (cid:3) We shall make use of operators with the following parti ular symbols:4.10. Corollary. For t ∈ R and θ ∈ N k with | θ | = k , the symbols σ t ( ξ ) := t ξ θ ˆ D ( ξ )( I + t ˆ D ( ξ ) ) − are C ∞ away from the origin and satisfy the Mikhlin multiplier estimates | ∂ αξ σ t ( ξ ) | . α | ξ | −| α | ∀ α ∈ N n . Proof. The symbols are σ t ( ξ ) = ξ θ ˆ D − R ( ξ ) t ˆ D ( ξ )( I + t ˆ D ( ξ ) ) − = (cid:0) ξ θ ˆ D − R ( ξ ) P R ( ˆ D ( ξ )) (cid:1)(cid:0) I − ˆ P t ( ξ ) (cid:1) , where both fa tors are smooth in R n \ { } , and the (cid:28)rst satis(cid:28)es the Mikhlin multiplier onditionby Lemma 4.5 and the se ond by Proposition 4.6. (cid:3) L p theory for operators with onstant oeffi ientsIn this se tion, we onsider the Fourier multipliers in L p , whi h orrespond to the symbolsstudied in Se tion 4. We denote by R t , P t , Q t the multiplier operators with the symbols ˆ R t , ˆ P t , ˆ Q t .We start with the operators D from Subse tion 3.A. The estimates obtained in the pre edingse tion give Theorem 3.2, restated here for onvenien e:5.1. Theorem. Let < p < ∞ . Under the assumptions (D1) and (D2), the operator D isbise torial in L p with angle ω , and has a bounded H ∞ fun tional al ulus in L p with angle ω .Proof. By the Mikhlin multiplier theorem, the bise toriality follows from Proposition 4.6, whilethe boundedness of the H ∞ fun tional al ulus follows from Proposition 4.8. (cid:3) The oer ivity ondition (D1) for the symbol has the following rein arnation on the level ofoperators:5.2. Proposition. For all u ∈ R p ( D ) ∩ D p ( D ) , there holds u ∈ D p ( ∇ k ) , and k∇ k u k p . k Du k p . Proof. For u ∈ D p ( D ) ∩ R p ( D ) , we have (for real t ) u t := t DP t ( Du ) = t D P t u = ( I − P t ) u → P R p ( D ) u = u, t → ∞ . The operators t ∂ θ DP t , | θ | = k , are bounded on L p by Corollary 4.10. It follows that u t ∈ D p ( ∂ θ ) ,and k ∂ θ u t k p = k t ∂ θ DP t ( Du ) k p . k Du k p . Let w be a test fun tion in the dual spa e. Then |h ∂ θ u, w i| = |h u, ∂ θ w i| = lim t →∞ |h u t , ∂ θ w i| = lim t →∞ |h ∂ θ u t , w i|≤ lim inf t →∞ k ∂ θ u t k p k w k p ′ . k Du k p k w k p ′ . Thus u ∈ D p ( ∂ θ ) and k ∂ θ u k p . k Du k p for all | θ | = k . (cid:3) UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 11We turn to the Hodge-Dira operators
Π = Γ + Γ whi h satisfy the hypotheses at the start ofSubse tion 3.B, and note that they then satisfy the onditions on D with k = 1 . In parti ular,Theorem 5.1 and Proposition 5.2 hold for D = Π and k = 1 . Moreover there is an operator versionof the symbol ondition ( Π N p (Π) = N p (Γ) ∩ N p (Γ) .Proof. The in lusion ⊇ is lear. Let u ∈ N p (Π) . Then ˆΠˆ u = 0 in the sense of distributions.It follows from Lemma 4.5 that the fun tion Υ( ξ ) := ˆΓ( ξ ) ˆΠ − R ( ξ ) P R (ˆΠ( ξ )) is C ∞ away from theorigin. Hen e, if ψ ∈ C ∞ c ( R n \ { } ) , then also ψ Υ is in the same lass, and the produ t ψ Υ · ˆΠˆ u is well-de(cid:28)ned and vanishes as a distribution. But Υ( ξ ) ˆΠ( ξ ) = ˆΓ( ξ ) ˆΠ − R ( ξ ) P R (ˆΠ( ξ )) ˆΠ( ξ ) = ˆΓ( ξ ) P R (ˆΠ( ξ )) = ˆΓ( ξ ) by ( Π ψ ˆΓˆ u = ψ Υ ˆΠˆ u = 0 for every ψ ∈ C ∞ c ( R n \ { } ) . This meansthat the distribution ˆΓˆ u is at most supported at the origin, and hen e Γ u = P , a polynomial. Butalso Γ u = − i P nj =1 ˆΓ j ∂ j u = P nj =1 ∂ j u j , where u j = − i ˆΓ j u ∈ L p . Let ˆ φ ∈ S ( R n ) be identi allyone in a neighbourhood of the origin. Then ˆ P = ˆ P ˆ φ and hen e P = P ∗ φ . But P ∗ φ ( y ) = h P, φ ( y − · ) i = n X j =1 h u j , ( ∂ j φ )( y − · ) i → as | y | → ∞ (using just the fa t that u j ∈ L p and ∂ j φ ∈ L p ′ ), and a polynomial with this propertymust vanish identi ally. This shows that Γ u = P = 0 , and then also Γ u = Π u − Γ u = 0 . (cid:3) This then implies:5.4. Proposition. The operator identity
Π = Γ + Γ holds in L p , in the sense that D p (Π) = D p (Γ) ∩ D p (Γ) and Π u = Γ u + Γ u for all u ∈ D p (Π) .Proof. It is lear that D p (Γ) ∩ D p (Γ) ⊆ D p (Π) . Sin e Π is bise torial in L p , there is the topologi alde omposition L p = N p (Π) ⊕ R p (Π) . Write u ∈ D p (Π) as u = u + u in this de omposition.Then u ∈ N p (Π) = N p (Γ) ∩ N p (Γ) , and u = u − u ∈ D p (Π) ∩ R p (Π) . By Proposition 5.2, u ∈ D p ( ∇ ) ⊆ D p (Γ) ∩ D p (Γ) . Thus also D p (Π) ⊆ D p (Γ) ∩ D p (Γ) . The oin idende of Π u and Γ u + Γ u is lear from the distributional de(cid:28)nition. (cid:3) We are ready to prove Theorem 3.6, restated here:5.5. Theorem. Let Π be a Hodge-Dira operator with onstant oe(cid:30) ients, and let < p < ∞ .Then the operator Π has a bounded H ∞ fun tional al ulus in L p with angle ω , and satis(cid:28)es thefollowing Hodge de omposition L p = N p (Π) ⊕ R p (Γ) ⊕ R p (Γ) , where R p (Π) = R p (Γ) ⊕ R p (Γ) .Proof. The fa t that Π is a bise torial operator with a bounded H ∞ fun tional al ulus is aparti ular ase of Theorem 3.2. The bise toriality already implies the de omposition L p = N p (Π) ⊕ R p (Π) , whi h we now want to re(cid:28)ne.We (cid:28)rst he k that R p (Π) ⊆ R p (Γ) + R p (Γ) . If u ∈ R p (Π) , then u = lim j →∞ Π y j for some y j ∈ D p (Π) ∩ R p (Π) ⊆ D p ( ∇ ) . Then k Γ( y j − y k ) k p . k∇ ( y j − y k ) k p . k Π( y j − y k ) k p → (usingProposition 5.2), and hen e Γ y j onverges to some v ∈ R p (Γ) with k v k p . k u k p . Similarly, Γ y j onverges to w ∈ R p (Γ) , and u = v + w ∈ R p (Γ) + R p (Γ) .Next we show that R p (Γ) ⊆ R p (Π) . Indeed, R p (Γ) = Γ( D p (Γ)) = Γ( D p (Γ) ∩ R p (Π)) (by thede omposition in the (cid:28)rst paragraph and Lemma 5.3) ⊆ Γ( D p (Γ) ∩ ( R p (Γ) + R p (Γ))) (by theprevious paragraph) = Γ( D p (Γ) ∩ R p (Γ)) (be ause Γ is nilpotent) = Π( D p (Π) ∩ R p (Γ)) (be ause Γ is nilpotent) ⊆ R p (Π) . Therefore R p (Γ) ⊆ R p (Π) . In a similar way, we see that R p (Γ) ⊆ R p (Π) .2 HYTÖNEN, MCINTOSH, AND PORTALOn ombining these two results with that in the pre eding paragraph, we obtain R p (Γ)+ R p (Γ) = R p (Π) . To show that the sum is dire t, observe that R p (Γ) ∩ R p (Γ) ⊆ R p (Π) ∩ N p (Γ) ∩ N p (Γ) = R p (Π) ∩ N p (Π) = { } by nilpoten e, Lemma 5.3 and the de omposition L p = N p (Π) ⊕ R p (Π) . (cid:3) We on lude this se tion with an analogue, in our matrix-valued ontext, of Bourgain's [13,Lemma 10℄. It is an important property of Hodge-Dira operators with onstant oe(cid:30) ients whi hwe use to study Hodge-Dira operators with variable oe(cid:30) ients in Se tion 8.5.6. Proposition. For all z ∈ R n , there holds E (cid:13)(cid:13)(cid:13) X k ε k τ k z Q k u (cid:13)(cid:13)(cid:13) p . (1 + log + | z | ) k u k p . where τ z denotes the translation operator τ z u ( x ) := u ( x − z ) .Proof. Let us (cid:28)x a test fun tion ϕ ∈ D ( R n ) su h that B (0 , − ) ≤ ϕ ≤ B (0 , , and write ψ ( ξ ) := ϕ ( ξ ) − ϕ (2 ξ ) , ψ m ( ξ ) := ψ (2 − m ξ ) , and ϕ m ( ξ ) := ϕ (2 − m ξ ) for m ∈ Z + . Let Φ m and Ψ m , m ∈ Z ,be the orresponding Fourier multiplier operators with symbols ϕ m and ψ m . Then we havethe partition of unity ϕ k + P ∞ m =1 ψ k + m ≡ , and the orresponding operator identity Φ k + P ∞ m =1 Ψ m + k = I for all k ∈ Z .Sin e the support of the Fourier transform of Q k Ψ m − k is ontained in B (0 , m − k ) , by Bourgain's[13, Lemma 10℄, there holds(5.7) E (cid:13)(cid:13)(cid:13) X k ε k τ k z Q k Ψ m − k u (cid:13)(cid:13)(cid:13) p . (1 + log + (2 m | z | )) E (cid:13)(cid:13)(cid:13) X k ε k Q k Ψ m − k u (cid:13)(cid:13)(cid:13) p . The same reasoning applies with Φ in pla e of Ψ .We now estimate the right side of (5.7) as a Fourier multiplier transformation. The symbol ofthe operator a ting on u is given by σ ( ξ ) = X k ε k f (2 k ˆΠ( ξ )) ψ (2 k − m ξ ) , f ( τ ) = τ (1 + τ ) − . For every α ∈ { , } n , a omputation shows that ∂ α σ ( ξ ) = X k ε k X β ≤ α ∂ β f (2 k ˆΠ( ξ ))( ∂ α − β ψ )(2 k − m ξ )2 ( k − m ) | α − β | . By the support property of ψ , the series in k redu es to at most two non-vanishing terms for whi h k − m | ξ | h . By Proposition 4.8, there moreover holds(5.8) | ∂ β f (2 k ˆΠ( ξ )) | . k | ξ | k | ξ | ) | ξ | −| β | , whi h shows that, for m ∈ Z + , | ∂ α σ ( ξ ) | . − m | ξ | −| α | . Hen e the asso iated Fourier multiplier is bounded with norm . − m .A similar omputation an be made with Φ − k in pla e of Ψ m − k , but the estimation of thesymbol then involves an in(cid:28)nite series of terms: | ∂ α σ ( ξ ) | . X β ≤ α X k | ∂ β f (2 k ˆΠ( ξ )) || ( ∂ α − β ϕ )(2 k ξ ) | k | α − β | . X β ≤ α X k : | k ξ |≤ k (1+ | α − β | ) | ξ | −| β | . | ξ | −| α | , where (5.8) was used again in the se ond estimate. We on lude that the operator a ting on u in(5.7), with Φ − k in pla e of Ψ m − k , is also bounded.UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 13Colle ting all the estimates, we have shown that E (cid:13)(cid:13)(cid:13) X k ε k τ k z Q k u (cid:13)(cid:13)(cid:13) p . (1 + log + | z | ) (cid:13)(cid:13)(cid:13) X k ε k Q k Φ − k u (cid:13)(cid:13)(cid:13) p + ∞ X m =1 ( m + log + | z | ) (cid:13)(cid:13)(cid:13) X k ε k Q k Ψ m − k u (cid:13)(cid:13)(cid:13) p . ∞ X m =0 (max { , m } + log + | z | )2 − m k u k p . (1 + log + | z | ) k u k p , whi h is the asserted bound. (cid:3)
6. Properties of Hodge de ompositionsIn this se tion, we olle t various results on erning the Hodge de omposition. These in ludeduality results, a relation of the Hodge de ompositions of the operator Π B and its variant Π B ,and (cid:28)nally some stability properties of the Hodge de omposition under small perturbations of the oe(cid:30) ient matri es B and B . These will be needed in proving the stability of the fun tional al ulus of the Hodge-Dira operators under small perturbations later on.6.1. Lemma. Let < p < ∞ and let Π B be a Hodge-Dira operator with variable oe(cid:30) ients in L p . The following assertions are equivalent:(1) Π B Hodge-de omposes L p .(2) ( L p = N p (Γ) ⊕ R p (Γ B ) ,L p = N p (Γ B ) ⊕ R p (Γ) . Proof. (1) ⇒ (2). We (cid:28)rst show that N p (Π B ) = N p (Γ) ∩ N p (Γ B ) . If u ∈ N p (Π B ) then Γ u = − Γ B u ∈ R p (Γ) ∩ R p (Γ B ) = { } . It follows that N p (Π B ) ⊕ R p (Γ) ⊆ N p (Γ) . Also N p (Γ) ∩ R p (Γ B ) ⊆ N p (Π B ) ∩ R p (Γ B ) = { } . Hen e N p (Π B ) ⊕ R p (Γ) = N p (Γ) , and thus L p = N p (Γ) ⊕ R p (Γ B ) . These ond part of (2) is similarly proven.(2) ⇒ (1). By (2), u ∈ L p an be de omposed as u = v + v + u where v + v ∈ N p (Γ) , v ∈ N p (Γ B ) , v ∈ R p (Γ) , and u ∈ R p (Γ B ) . Then Π B v = Γ( v + v − v ) = 0 , and k v k p + k v k p . k v + v k p . k u k p . Moreover N p (Π B ) ∩ R p (Γ) ⊆ N p (Γ B ) ∩ R p (Γ) = { } , N p (Π B ) ∩ R p (Γ B ) ⊆ N p (Γ) ∩ R p (Γ B ) = { } , R p (Γ B ) ∩ R p (Γ) ⊆ N p (Γ) ∩ R p (Γ B ) = { } . The proof is omplete. (cid:3) D and D be losed, densely de(cid:28)ned operators in L p . Then L p = N p ( D ) ⊕ R p ( D ) if and only if L p ′ = N p ′ ( D ∗ ) ⊕ R p ′ ( D ∗ ) . Proof. Assuming the (cid:28)rst de omposition, we have that L p ′ = R p ( D ) ⊥ ⊕ N p ( D ) ⊥ = N p ′ ( D ∗ ) ⊕ R p ′ ( D ∗ ) . The other impli ation follows by symmetry. (cid:3) Π B be a Hodge-Dira operator with variable oe(cid:30) ients in L p whi h Hodgede omposes L p . Then Π ∗ B is a Hodge-Dira operator with variable oe(cid:30) ients in L p ′ whi h Hodgede omposes L p ′ .Proof. We have to show that B ∗ = ( B ∗ , B ∗ ) satisfy (B2) in L p ′ . Let us remark that B is anisomorphism from R p (Γ) onto R p (Γ B ) . By Lemma 6.1, this means that B is an isomorphism from R p (Γ) onto L p / N p (Γ) , and thus B ∗ is an isomorphism from R p ′ (Γ ∗ ) onto L p ′ / N p ′ (Γ ∗ ) . This givesthe part of the result on erning B ∗ thanks to Lemma 6.2. The ase of B ∗ is handled in the same4 HYTÖNEN, MCINTOSH, AND PORTALway. Condition (B1) is obtained by duality, and the proof is on luded by applying Lemma 6.1and Lemma 6.2. (cid:3) Re all that Π B := Γ + B Γ B .6.4. Lemma. Let < p < ∞ and suppose that Π B and Π B are both Hodge-Dira operator withvariable oe(cid:30) ients in L p , and that Π B Hodge-de omposes L p . Then Π B also Hodge-de omposes L p . If, moreover, Π B is an R-bise torial operator in L p , then so is Π B .Proof. By Lemmas 6.1 and 6.2, the assumption that Π B Hodge-de omposes L p is equivalent to(6.5) L p = N p (Γ) ⊕ R p (Γ B ) , L p ′ = N p ′ (Γ ∗ ) ⊕ R p ′ (Γ ∗ B ) , whereas the laim that Π B Hodge-de omposes L p is equivalent to(6.6) L p = N p ( B Γ B ) ⊕ R p (Γ) , L p ′ = N p ′ ( B ∗ Γ ∗ B ∗ ) ⊕ R p ′ (Γ ∗ ) . We show that the (cid:28)rst de omposition in (6.5) implies the (cid:28)rst one in (6.6). Let u ∈ L p andwrite B u = v + w where v ∈ N p (Γ) and w ∈ R p (Γ B ) . Let w = B x for x ∈ R p (Γ) . Then u − x ∈ N p ( B Γ B ) and k x k p . k B x k p = k w k p . k B u k p . k u k p . The dedu tion of the se ond de omposition in (6.6) from the se ond one in (6.5) is analogous.We now turn to the se ond statement. Let us denote by P the proje tion on R p (Γ) , by P theproje tion on R p ( B Γ B ) , by P the proje tion on R p (Γ) , and by P the proje tion on R p ( B Γ B ) .Let ( t k ) k ∈ N ⊂ R and ( u k ) k ∈ N ⊂ L p , and remark (cid:28)rst that ( I + it k Π B ) − = I − ( t k Π B ) ( I + ( t k Π B ) ) − − it k Π B ( I + ( t k Π B ) ) − . The R-bise toriality of Π B will thus follow on e we have proven that(6.7) E (cid:13)(cid:13)(cid:13) X k ε k it k Π B ( I + ( t k Π B ) ) − u k (cid:13)(cid:13)(cid:13) p . E (cid:13)(cid:13)(cid:13) X k ε k u k (cid:13)(cid:13)(cid:13) p and(6.8) E (cid:13)(cid:13)(cid:13) X k ε k ( t k Π B ) ( I + ( t k Π B ) ) − u k (cid:13)(cid:13)(cid:13) p . E (cid:13)(cid:13)(cid:13) X k ε k u k (cid:13)(cid:13)(cid:13) p . To do so we note that, sin e Π B and Π B are Hodge-Dira operators with variable oe(cid:30) ients,we have: Γ B ( I + ( t k Π B ) ) − P = Γ( I + ( t k Π B ) ) − B P , Γ( I + ( t k Π B ) ) − B P = Γ B ( I + ( t k Π B ) ) − P . We an now pro eed with the estimates, using the R-bise toriality of Π B . E (cid:13)(cid:13)(cid:13) X k ε k it k Π B ( I + ( t k Π B ) ) − P u k (cid:13)(cid:13)(cid:13) p = E (cid:13)(cid:13)(cid:13) X k ε k it k B Γ B ( I + ( t k Π B ) ) − P u k (cid:13)(cid:13)(cid:13) p , = E (cid:13)(cid:13)(cid:13) X k ε k it k B Γ( I + ( t k Π B ) ) − B P u k (cid:13)(cid:13)(cid:13) p , = E (cid:13)(cid:13)(cid:13) X k ε k it k B Π B ( I + ( t k Π B ) ) − B P u k (cid:13)(cid:13)(cid:13) p , . E (cid:13)(cid:13)(cid:13) X k ε k B P u k (cid:13)(cid:13)(cid:13) p . E (cid:13)(cid:13)(cid:13) X k ε k u k (cid:13)(cid:13)(cid:13) p . UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 15Introdu ing v k su h that B P v k = P u k , we also get: E (cid:13)(cid:13)(cid:13) X k ε k it k Π B ( I + ( t k Π B ) ) − P u k (cid:13)(cid:13)(cid:13) p = E (cid:13)(cid:13)(cid:13) X k ε k it k Γ( I + ( t k Π B ) ) − B P v k (cid:13)(cid:13)(cid:13) p , = E (cid:13)(cid:13)(cid:13) X k ε k it k Γ B ( I + ( t k Π B ) ) − P v k (cid:13)(cid:13)(cid:13) p , . E (cid:13)(cid:13)(cid:13) X k ε k it k Π B ( I + ( t k Π B ) ) − P v k (cid:13)(cid:13)(cid:13) p , . E (cid:13)(cid:13)(cid:13) X k ε k P v k (cid:13)(cid:13)(cid:13) p . E (cid:13)(cid:13)(cid:13) X k ε k u k (cid:13)(cid:13)(cid:13) p . The estimate (6.8) is proven in the same way. (cid:3) X = X ⊕ X = P X ⊕ P X , then there is δ > su hthat for all T ∈ L ( X , X ) with k T k < δ , it also splits as X = X ⊕ ( I − T ) X = ˜ P X ⊕ ˜ P X ,with k ˜ P − P k + k ˜ P − P k . δ. Proof. For δ := k P k , I − T P is invertible. Let U := ( I − T P ) − , and observe the identity U = I + U T P . De(cid:28)ne the operators ˜ P := P U, ˜ P := ( I − T P ) P U. Then R (˜ P ) = X , R (˜ P ) = ( I − T ) X , and ˜ P + ˜ P = ( I − P + ( I − T P ) P ) U = ( I − T P ) U = I. It remains to show that ˜ P (and then also ˜ P ) is a proje tion. This follows from ˜ P ˜ P = P U P U = P ( I + U T P ) P U = P U = ˜ P , where we used P P = 0 and P = I .Sin e ˜ P − P = P U T P , ˜ P − P = P U T P − T P U , and k U k ≤ , we also have that k ˜ P − P k + k ˜ P − P k . δ. (cid:3) p ∈ (1 , ∞ ) , and Π A be a Hodge-Dira operator with variable oe(cid:30) ientsin L p whi h Hodge-de omposes L p . There exists δ > su h that, if Π B is another Hodge-Dira operator with variable oe(cid:30) ients in L p with k B − A k ∞ + k B − A k ∞ < δ , then Π B Hodge-de omposes L p . Moreover, the asso iated Hodge-proje tions satisfy k P A − P B k + k P A Γ − P B Γ k + k P A Γ A − P B Γ B k . δ. Proof. Consider the ondition (6.5) equivalent to the Hodge-de omposition. Let us de(cid:28)ne T ∈ L ( R p (Γ A ) , L p ) by T A Γ u := ( A − B )Γ u whi h, by (B2), gives a well-de(cid:28)ned operator of norm k T k . δ , and we have B Γ = ( I − T ) A Γ .On the dual side, we de(cid:28)ne ˜ T ∈ L ( R p ′ (Γ ∗ A ) , L p ′ ) by ˜ T A ∗ Γ ∗ v = ( A ∗ − B ∗ )Γ v , whi h issimilarly well-de(cid:28)ned and satis(cid:28)es k ˜ T k . δ . Let then T := ( ˜ T P Γ ∗ A ) ∗ ∈ L ( L p ) , where P Γ ∗ A is the proje tion in L p ′ asso iated to the de omposition in (6.5). By duality, it follows that R p ( B − A ( I − T )) ⊆ N p (Γ) , whi h means that Γ B = Γ A ( I − T ) . Sin e the operators I − T and I − T P Γ A are invertible for δ small enough, we then have that R (Γ B ) = R (( I − T )Γ A ( I − T )) = ( I − T ) R (Γ A ) . Similarly, with B ∗ Γ ∗ = ( I − T ) A ∗ Γ ∗ and Γ ∗ B ∗ = Γ ∗ A ∗ ( I − T ) , there holds R (Γ ∗ B ) = ( I − T ) R (Γ ∗ A ) . Hen e the laim follows from two appli ations of Lemma 6.9 with ( X , X , T ) =( N p (Γ) , R (Γ A ) , T ) and ( X , X , T ) = ( N p ′ (Γ ∗ ) , R (Γ ∗ A ) , T ) (cid:3) By (B2), the restri tion A : R p (Γ) → R p (Γ A ) is an isomorphism, and we denote by A − itsinverse. Thus the operator A − P Γ A is well-de(cid:28)ned. We shall also need to perturb su h operators:6 HYTÖNEN, MCINTOSH, AND PORTAL6.11. Corollary. Under the assumptions of Proposition 6.10, there also holds k A − P A Γ A − B − P B Γ B k . δ. Proof. We use the same notation as in the proof of Proposition 6.10 and observe from the identity B Γ = ( I − T ) A Γ that I − T : R p (Γ A ) → B R p (Γ) , and(6.12) B − ( I − T ) P A Γ A = A − P A Γ A . We further re all from the proof of Lemma 6.9 that the proje tion P B Γ B related to the splitting L p = N p (Γ) ⊕ R p (Γ B ) , where R p (Γ B ) = ( I − T ) R p (Γ A ) , is given by P B Γ B = ( I − T ) P A Γ A ( I − T P A Γ A ) − . Combining this with (6.12) shows that B − P B Γ B = A − P A Γ A ( I − T P A Γ A ) − = A − P A Γ A − ( A − P A Γ A ) T P A Γ A ( I − T P A Γ A ) − , whi h proves the laim, sin e the se ond term ontains the fa tor T of norm k T k . δ . (cid:3)
7. Fun tional al ulusIn this se tion we olle t some general fa ts about the fun tional al ulus of bise torial operatorsin re(cid:29)exive Bana h spa es. We provide, in the ontext of the dis rete randomized quadrati estimates required in [18℄, versions of results originally obtained in [15℄. Lemma 7.1 an be seen asa dis rete Calderón reprodu ing formula, and Lemma 7.2 as a S hur estimate, while Propositions7.3 and 7.5 express the fundamental link between fun tional al ulus and square fun tion estimates.Su h results are not new, and have been developed from [15℄ by various authors, most notablyKalton and Weis ( f. [20℄). Here we hope, however, to provide simpler versions of both thestatements and the proofs of these fa ts.Let A denote a bise torial operator in a re(cid:29)exive Bana h spa e, with angle ω , and let θ ∈ ( ω, π ) .We use the following notations. r ( A ) = ( I + i A ) − , p ( A ) = r ( A ) r ( −A ) = ( I + A ) − , q ( A ) = i r ( A ) − r ( −A )) = A I + A . X k ∈ Z q (2 k A ) q (2 k +1 A ) = P R ( A ) . Proof. Observe (cid:28)rst that p ( t A ) → P N ( A ) as t → ∞ and p ( t A ) → I as t → . Hen e P R ( A ) = I − P N ( A ) = X k ( p (2 k A ) − p (2 k +1 A ))= X k p (2 k A )[( I + 2 k +1) A ) − ( I + 2 k A )] p (2 k +1 A )= 32 X k p (2 k A )(2 k A )(2 k +1 A ) p (2 k +1 A ) = 32 X k q (2 k A ) q (2 k +1 A ) , as we wanted to show. (cid:3) A be R -bise torial and let η ( x ) := min { x, /x } (1 + log max { x, /x } ) . Then theset { η ( s/t ) − q ( t A ) f ( A ) q ( s A ) : t, s > f ∈ H ∞ ( S θ ) , k f k ∞ ≤ } is R-bounded.UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 17Proof. Denoting q t ( λ ) := q ( tλ ) , observe that q ( t A ) f ( A ) q ( s A ) = ( q t · f · q s )( A ) = 12 πi Z γ ( q t f q s )( λ )( I − λ A ) − d λλ , where γ denotes ∂S θ ′ , for some θ ′ ∈ ( θ, π ) , parameterized by ar length and dire ted anti- lo kwise around S θ , and the resolvents ( I − λ − A ) − belong to an R -bounded set. The operators q ( t A ) f ( A ) q ( s A ) are hen e in a dilation of the absolute onvex hull of this R -bounded set (see [21℄for information on R-boundedness te hniques). To evaluate the dilation fa tor, observe that | q t f q s ( λ ) | . k f k ∞ s | λ | s | λ | ) t | λ | t | λ | ) , and splitting the integral into three regions (depending on the position of | λ | with respe t to min( t , s ) and max( t , s ) ) it follows that Z γ | q t f q s ( λ ) | | d λ || λ | . k f k ∞ η ( s/t ) , whi h implies the asserted R -bound. (cid:3) A be R -bise torial (with angle µ ) and satisfy the two-sided quadrati esti-mate(7.4) k u k h E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) u (cid:13)(cid:13)(cid:13) , u ∈ R ( A ) . Then A has a bounded H ∞ fun tional al ulus (with angle µ ).Proof. Suppose (7.4) holds. Let u ∈ R ( A ) , θ ∈ ( µ, π ) , and f ∈ H ∞ ( S θ ) . Then k f ( A ) u k . E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) f ( A ) u (cid:13)(cid:13)(cid:13) h E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) f ( A ) X j q (2 j A ) q (2 j +1 A ) u (cid:13)(cid:13)(cid:13) . X m E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) f ( A ) q (2 k + m A ) q (2 k + m +1 A ) u (cid:13)(cid:13)(cid:13) . X m η (2 m ) k f k ∞ (cid:13)(cid:13)(cid:13) X k ε k q (2 k + m +1 A ) u (cid:13)(cid:13)(cid:13) . X m (1 + | m | )2 −| m | k f k ∞ k u k . k f k ∞ k u k , where we used . from (7.4), Lemma 7.1, the triangle inequality after relabelling j = k + m ,Lemma 7.2, and & from (7.4). (cid:3) We also use the following variant.7.5. Proposition. Let A be R -bise torial (with angle µ ) and satisfy the two quadrati estimates E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) u (cid:13)(cid:13)(cid:13) X . k u k X , u ∈ X, E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ∗ ) v (cid:13)(cid:13)(cid:13) X ∗ . k v k X ∗ , v ∈ X ∗ , (7.6)Then A has a bounded H ∞ fun tional al ulus (with angle µ ).Proof. Let u ∈ X , v ∈ X ∗ , θ ∈ ( µ, π ) , and f ∈ H ∞ ( S θ ) . Then |h f ( A ) u, v i| ≤ X k |h q (2 k A ) f ( A ) u, q (2 k +1 A ∗ ) v i| . E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) f ( A ) u (cid:13)(cid:13)(cid:13) X E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ∗ ) v (cid:13)(cid:13)(cid:13) X ∗ . E (cid:13)(cid:13)(cid:13) X k ε k q (2 k A ) f ( A ) u (cid:13)(cid:13)(cid:13) X k v k X ∗ . (cid:3) L p theory for operators with variable oeffi ientsIn this se tion, we give the proofs of the results stated in Subse tion 3.C, and some variations.The ore result, Theorem 8.1, is a generalization of [18, Theorem 3.1℄. The key ingredients of theproof are ontained in [18℄. Here we indi ate where [18℄ needs to be modi(cid:28)ed, using the resultsfrom the pre eding se tions. Let us re all that Π denotes an Hodge-Dira operator with onstant oe(cid:30) ients as de(cid:28)ned in De(cid:28)nition 3.4, and that Π B denotes a Hodge-Dira operator with variable oe(cid:30) ients as de(cid:28)ned in De(cid:28)nition 3.8. We also use the following notation. R Bt := ( I + it Π B ) − = r ( t Π B ) ,P Bt := ( I + t Π B ) − = p ( t Π B ) ,Q Bt := t Π B P Bt = q ( t Π B ) , and denote by R t , P t , Q t the orresponding fun tions of Π .8.1. Theorem. Let ≤ p < p ≤ ∞ , and let Π B be an R-bise torial Hodge-Dira operator withvariable oe(cid:30) ients in L p whi h Hodge-de omposes L p for all p ∈ ( p , p ) . Then(8.2) E (cid:13)(cid:13)(cid:13) X k ε k Q B k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) , and(8.3) E (cid:13)(cid:13)(cid:13) X k ε k ( Q B k ) ∗ v (cid:13)(cid:13)(cid:13) p ′ . k v k p ′ , v ∈ R p ′ (Γ ∗ ) . Proof. We (cid:28)rst prove (8.2). Let us re all the following notation from [18℄. Let △ := [ k ∈ Z △ k , △ k := (cid:8) k ([0 , n + m ) : m ∈ Z n (cid:9) . denote a system of dyadi ubes, and A k u ( x ) := h u i Q := 1 | Q | Z Q u ( y ) d y, x ∈ Q ∈ △ k . be the orresponding onditional expe tation proje tions. Let(8.4) γ k ( x ) w := Q B k ( w )( x ) := X Q ∈△ k Q B k ( w Q )( x ) , x ∈ R n , w ∈ C N . denote the prin ipal part of Q B k , whi h we also identify with the orresponding pointwise multi-pli ation operator.The proof of (8.2) now divides into the following four estimates:(8.5) E (cid:13)(cid:13)(cid:13) X k ε k Q B k ( I − P k ) u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . (8.6) E (cid:13)(cid:13)(cid:13) X k ε k ( Q B k − γ k A k ) P k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . (8.7) E (cid:13)(cid:13)(cid:13) X k ε k γ k A k ( I − P k ) u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . (8.8) E (cid:13)(cid:13)(cid:13) X k ε k γ k A k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . Inequality (8.5) follows from the fa t that Π B Hodge-de omposes L p , and from the R-bise torial-ity of Π B : as in [18, Lemma 6.3℄, denoting by P Γ B the proje tion onto R p (Γ B ) in the Hodge-de omposition, we have Q B k ( I − P k ) u = ( I − P B k ) P Γ B Q k u, u ∈ R p (Γ) , UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 19and(8.9) E (cid:13)(cid:13)(cid:13) X k ε k ( I − P B k ) P Γ B Q k u (cid:13)(cid:13)(cid:13) p . E (cid:13)(cid:13)(cid:13) X k ε k Q k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ L p , where the last inequality follows from Theorem 3.6.To prove (8.6) and (8.7) we (cid:28)rst note that ommutators of the form [ ηI, Γ] are multipli ationoperators by an L ∞ fun tion bounded by k∇ η k ∞ . Then the R-bise toriality of Π B and [18,Proposition 6.4℄ give the following o(cid:27)-diagonal R-bounds: for every M ∈ N , every Borel subsets E k , F k ⊂ R n , every u k ∈ L p , and every ( t k ) k ∈ Z ⊆ { j } j ∈ Z su h that dist( E k , F k ) /t k > ̺ for some ̺ > and all k ∈ Z , there holds E (cid:13)(cid:13)(cid:13) X k ε k E k Q Bt k F k u k (cid:13)(cid:13)(cid:13) p . (1 + ̺ ) − M E (cid:13)(cid:13)(cid:13) X k ε k F k u k (cid:13)(cid:13)(cid:13) p . (8.10)This, in turn, gives that the family ( γ k A k ) k ∈ Z is R-bounded exa tly as in [18, Lemma 6.5℄.Now let us prove (8.6). This is similar to [18, Lemma 6.6℄ but, sin e a few modi(cid:28) ations needto be made, we in lude the proof. Letting v k = P k u , and using the o(cid:27)-diagonal R-bounds, wehave: E (cid:13)(cid:13)(cid:13) X k ε k ( Q B k − γ k A k ) v k (cid:13)(cid:13)(cid:13) p = E (cid:13)(cid:13)(cid:13) X k ε k X Q ∈△ k Q Q B k (cid:0) v k − h v k i Q ) (cid:13)(cid:13)(cid:13) p ≤ X m ∈ Z n E (cid:13)(cid:13)(cid:13) X k ε k X Q ∈△ k Q Q B k (cid:0) Q − k m ( v k − h v k i Q ) (cid:1)(cid:13)(cid:13)(cid:13) p . X m ∈ Z n (1 + | m | ) − M E (cid:13)(cid:13)(cid:13) X k ε k X Q ∈△ k Q ( v k − h v k i Q +2 k m ) (cid:13)(cid:13)(cid:13) p . (8.11)A version of Poin aré's inequality [18, Proposition 4.1℄ allows to majorize the last fa tor by Z [ − , n Z E (cid:13)(cid:13)(cid:13) X k ε k k ( m + z ) · ∇ τ t k ( m + z ) P k u (cid:13)(cid:13)(cid:13) p d t d z. We then use Propositions 5.2 and 5.6 to estimate this term by (1 + | m | )(1 + log + | m | ) k u k p , and the proof is thus ompleted by pi king M large enough.To prove (8.7), we (cid:28)rst use the R-boundedness of ( γ k A k ) and the idempoten e of A k . Wethus have to show that E (cid:13)(cid:13)(cid:13) X k ε k A k ( I − P k ) u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . This is essentially like [18, Proposition 5.5℄. We indi ate the beginning of the argument, wherethe operator-theoreti Lemma 7.1 repla es a Fourier multiplier tri k used in [18℄. Indeed, with u ∈ R p (Γ) ⊆ R p (Π) , we have E (cid:13)(cid:13)(cid:13) X k ε k A k ( I − P k ) u (cid:13)(cid:13)(cid:13) p h E (cid:13)(cid:13)(cid:13) X k ε k A k ( I − P k ) X j Q j Q j +1 u (cid:13)(cid:13)(cid:13) p ≤ X m E (cid:13)(cid:13)(cid:13) X k ε k (cid:0) A k ( I − P k ) Q k + m (cid:1) Q k + m +1 u (cid:13)(cid:13)(cid:13) p . Thanks to the se ond estimate in (8.9), it su(cid:30) es to show that the operator family (cid:8) A k ( I − P k ) Q k + m : k ∈ Z (cid:9) ⊂ L ( L p ) is R-bounded with R-bound C − δ | m | for some δ > . This is done by repeating the argument of[18, Proposition 5.5℄.0 HYTÖNEN, MCINTOSH, AND PORTALFinally, the proof of (8.8) is done exa tly as in [18, Theorem 8.2 and Proposition 9.1℄. Noti ethat this last part is the only pla e where we need the assumptions for p in an open interval ( p , p ) ; all the other estimates work for a (cid:28)xed value of p .This ompletes the proof of (8.2).Let us turn our attention to (8.3). By Lemma 6.3, we have that Π B ∗ = Γ ∗ + B ∗ Γ ∗ B ∗ isa Hodge-Dira operator with variable oe(cid:30) ients in L p ′ whi h Hodge-de omposes L p ′ . It is alsoR-bise torial, as this property is preserved under duality (see [20, Lemma 3.1℄). So the proof of(8.2) adapts to give the quadrati estimate (8.3) involving ( Q Bt ) ∗ = t Π B ∗ ( I + ( t Π B ∗ ) ) − . (cid:3) We now prove Theorem 3.13 as a orollary.8.12. Corollary. Let ≤ p < p ≤ ∞ , µ ∈ ( ω, π/ , and let Π B be a Hodge-Dira operator withvariable oe(cid:30) ients in L p whi h Hodge-de omposes L p for all p ∈ ( p , p ) . Assume also that Π B is a Hodge-Dira operator with variable oe(cid:30) ients in L p . Then Π B has a bounded H ∞ fun tional al ulus (with angle µ ) in L p ( R n ; C N ) for all p ∈ ( p , p ) if and only if it is R -bise torial (withangle µ ) in L p ( R n ; C N ) for all p ∈ ( p , p ) .Proof. The fa t that a bounded H ∞ fun tional al ulus implies R-bise toriality is a generalproperty (see Remark 2.5). To prove the other dire tion, assume that Π B is R-bise torial on L p ( R n ; C N ) for all p ∈ ( p , p ) . By Theorem 8.1, we have that(8.13) E (cid:13)(cid:13)(cid:13) X k ε k Q B k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . Moreover, sin e Π B also satis(cid:28)es the assumptions of Theorem 8.1 (using Lemma 6.4), we havethat(8.14) E (cid:13)(cid:13)(cid:13) X k ε k k Π B ( I + (2 k Π B ) ) − u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ) . For u ∈ R p (Γ) , there holds k Π B ( I + (2 k Π B ) ) − u = 2 k B Γ B ( I + (2 k Π B ) ) − u = 2 k B Γ( I + (2 k Π B ) ) − B u = 2 k B Π B ( I + (2 k Π B ) ) − B u. Thus by (B2), the estimate (8.14) implies(8.15) E (cid:13)(cid:13)(cid:13) X k ε k Q B k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ R p (Γ B ) . Combining (8.13) and (8.15) with the Hodge-de omposition and the obvious fa t that Q B k anni-hilates N p (Π B ) , we arrive at E (cid:13)(cid:13)(cid:13) X k ε k Q B k u (cid:13)(cid:13)(cid:13) p . k u k p , u ∈ L p . In the same way, one gets the dual estimate E (cid:13)(cid:13)(cid:13) X k ε k ( Q B k ) ∗ u (cid:13)(cid:13)(cid:13) p ′ . k u k p ′ , u ∈ L p ′ , where p ′ denotes the onjugate exponent of p . The fun tional al ulus then follows from Proposi-tion 7.5. (cid:3) ≤ p < p ≤ ∞ , and let Π A be a Hodge-Dira operator with variable oe(cid:30) ients, whi h is R-bise torial in L p and Hodge-de omposes L p for all p ∈ ( p , p ) . Then forea h p ∈ ( p , p ) , there exists δ = δ p > su h that, if Π B and Π B are Hodge-Dira operators withvariable oe(cid:30) ients su h that k B − A k ∞ + k B − A k ∞ < δ , then Π B has an H ∞ fun tional al ulus in L p and Hodge-de omposes L p .UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 21Proof. Let p ∈ ( p , p ) . By Proposition 6.10, we have that, for δ small enough, Π B Hodge-de omposes L p . We need to show that Π B is R-bise torial in L p provided δ is su(cid:30) iently small.As in the proof of Proposition 6.10, let T ∈ L ( R p (Γ A ) , L p ) and T ∈ L ( L p ) be operators of norm k T i k . δ su h that B Γ = ( I − T ) A Γ , Γ B = Γ A ( I − T ) . Then Π B = Γ + B Γ B = Γ + ( I − T ) A Γ A ( I − T ) = ( I − T P Γ A )Π A ( I − P Γ T ) , where P Γ and P Γ A are the Hodge-proje tions asso iated to Π A , onto R p (Γ) and R p (Γ A ) , respe -tively. Hen e I + it Π B = ( I − T P Γ A )( I + it Π A )( I − P Γ T ) + ( T P Γ A + P Γ T )= ( I − T P Γ A )( I + it Π A )( I − P Γ T ) × (cid:2) I + ( I − P Γ T ) − ( I + it Π A ) − ( I − T P Γ A ) − ( T P Γ A + P Γ T )] , where the inverses involving T i exist for δ small enough. Hen e ( I + it Π B ) − an be expressedas a Neumann series involving powers of the operators ( I + it Π A ) − , whi h are R-bounded byassumption, times powers of (cid:28)xed bounded operators, in luding T P Γ A + P Γ T whi h has norm atmost Cδ . For δ small enough, the R-boundedness of ( I + it Π B ) − follows from this representation.Given ε ∈ ( p − p , p − p ) we thus have that there exists δ p,ε su h that Π B is R-bise torial in L p − ε and in L p + ε , and Hodge de omposes L p − ε and L p + ε . By interpolation ( f. Remark 3.12), Π B is R-bise torial in L ˜ p and Hodge de omposes L ˜ p for all ˜ p ∈ ( p − ε, p + ε ) .Now the onditions of Corollary 8.12 are veri(cid:28)ed for the operators Π B and Π B , so the mentionedresult implies that Π B has a bounded H ∞ fun tional al ulus in L p , as laimed. (cid:3) With potential appli ations to boundary value problems in mind (see [4℄), we on lude thisse tion with the following spe ial ase. This proof is essentially the same as in the L ase [10,Theorem 3.1℄.8.17. Corollary. Let ≤ p < p ≤ ∞ . Let D = − i P nj =1 ˆ D j ∂ j be a (cid:28)rst order di(cid:27)erentialoperator with ˆ D j ∈ L ( C N ) , and A ∈ L ∞ ( R n ; L ( C N )) be su h that | ξ || e | . | ˆ D ( ξ ) e | for all e ∈ R ( ˆ D ( ξ )) , and σ ( ˆ D ( ξ )) ⊆ S ω for some ω ∈ (0 , π and all ξ ∈ R n , (H1)(H2) k u k p . k Au k p for all u ∈ R p ( D ) , (H3) k u k p ′ . k A ∗ u k p ′ for all u ∈ R p ′ ( D ∗ ) , for all p ∈ ( p , p ) . Then we have the following:(1) The operator DA has an H ∞ fun tional al ulus (with angle µ ) in L p for all p ∈ ( p , p ) if and only if it is R-bise torial (with angle µ ) in L p for all p ∈ ( p , p ) .(2) If the equivalent onditions of (1) hold, then for ea h p ∈ ( p , p ) , there exists δ = δ p > su h that, if another ˜ A ∈ L ∞ ( R n ; L ( C N )) satis(cid:28)es k A − ˜ A k ∞ < δ , then D ˜ A also has an H ∞ fun tional al ulus in L p .Proof. (1) The philosophy of the proof is to redu e the onsideration of an operator of the form DA to the Hodge-Dira operator with variable oe(cid:30) ients Π B , whi h we already understand fromthe previous results. On C N ⊕ C N , onsider the matri es ˆΓ j := (cid:18) D j (cid:19) , ˆΓ j := (cid:18) D j (cid:19) , B := (cid:18) A
00 0 (cid:19) , B := (cid:18) A (cid:19) . We de(cid:28)ne the asso iated di(cid:27)erential operators Γ , Γ and Π a ting in L p := L p ( R n ; C N ⊕ C N ) asin Subse tions 3.B, and the operators Π B = (cid:18) ADAD (cid:19) , Π B = (cid:18) DADA (cid:19) , Π B and Π B are then Hodge-Dira operators with variable oe(cid:30) ients in L p whi h Hodge-de ompose L p for all p ∈ ( p , p ) . Indeed, the hypotheses (H1), (H2) and (H3)guarantee the onditions ( Π Π
2) and (B2). The remaining requirements ( Π
3) and (B1), aswell as the Hodge de omposition (see [18, Lemma 3.5℄), are satis(cid:28)ed be ause of the spe ial formof Π B .A omputation shows that ( I + it Π B ) − = (cid:18) I − itADA I (cid:19) (cid:18) I
00 ( I + t ( DA ) ) − (cid:19) (cid:18) I − itD I (cid:19) . Assuming that DA is R-bise torial, we he k that so is Π B . This amounts to verifying the R-boundedness of the families of operators ( I + t ( DA ) ) − , − itADA ( I + t ( DA ) ) − , − it ( I + t ( DA ) ) − D, where t ∈ R . For the (cid:28)rst two, this is immediate from the R-bise toriality of DA and theboundedness of A . For the third one, we need the stability of R-boundedness in the L p spa esunder adjoints, and hypothesis (H3) whi h allows to redu e the adjoint − itD ∗ ( I + t ( A ∗ D ∗ ) ) − to a fun tion of ( DA ) ∗ = A ∗ D ∗ by omposing with A ∗ from the left.Thus, by Corollary 8.12, Π B has an H ∞ fun tional al ulus on L p for all p ∈ ( p , p ) . But theresolvent formula above also gives f (Π B ) (cid:18) Auu (cid:19) = (cid:18) Af ( DA ) uf ( DA ) u (cid:19) , for f ∈ H ∞ ( S θ ) and u ∈ L p , and hen e we (cid:28)nd that DA has an H ∞ fun tional al ulus, too.This ompletes the proof that the R-bise toriality of DA implies fun tional al ulus. The onversedire tion is a general property of the fun tional al ulus in L p (see Remark 2.5).(2) We turn to the se ond part and assume that DA is R-bise torial. We (cid:28)rst note that ˜ A alsosatis(cid:28)es the hypotheses (H2) and (H3) when δ is small enough. Indeed, for u ∈ R p ( D ) , we have k ˜ Au k p ≥ k Au k p − k A − ˜ A k ∞ k u k p ≥ ( c − δ ) k u k p , and (H3) is proven similarly. Moreover, theR-bise toriality of DA implies the same property for D ˜ A by a Neumann series argument, as I + itD ˜ A = I + itDA − itD ( A − ˜ A ) = ( I + itDA )[ I − ( I + itDA ) − itD ( A − ˜ A )] , where the family { ( I + itDA ) − itD : t ∈ R } is R-bounded (by duality, (H3), and the R-bise toriality of DA ), and the fa tor k A − ˜ A k ∞ < δ ensures onvergen e for δ small enough.The H ∞ al ulus then follows from part (1) applied to ˜ A in pla e of A . (cid:3)
9. Lips hitz estimatesIn this (cid:28)nal se tion, we prove Lips hitz estimates of the form k f (Π B ) u − f (Π A ) u k p . max i =1 , k A i − B i k ∞ k f k ∞ k u k p , for small perturbations of the oe(cid:30) ient matri es involved in the Hodge-Dira operators. Su hestimates are obtained via holomorphi dependen e results for perturbations B z depending on a omplex parameter z . This te hnique an be seen as one of the original motivations for studyingthe Kato problem for operators with omplex oe(cid:30) ients. We start with the operators studied inCorollary 8.17, and then dedu e similar estimates for general Hodge-Dira operators with variable oe(cid:30) ients, as in [3, Se tion 10.1℄.Let D be a (cid:28)rst order di(cid:27)erential operator as in Corollary 8.17. Let U be an open set of C and ( A z ) z ∈ U a family of multipli ation operators su h that the map z A z ∈ L ∞ ( R n ; C N ) isholomorphi . Let ≤ p < p ≤ ∞ , z ∈ U , and assume that DA z has a bounded H ∞ fun tional al ulus in L p for all p ∈ ( p , p ) . By Corollary 8.17, for ea h p ∈ ( p , p ) , there then exists a δ = δ p > su h that DA z has a bounded H ∞ fun tional al ulus in L p for all z ∈ B ( z , δ ) .Moreover, we have the following.9.1. Proposition. For θ ∈ ( µ, π ) and f ∈ H ∞ ( S θ ) , the fun tion z f ( DA z ) is holomorphi on D ( z , δ ) .UNCTIONAL CALCULUS OF HODGE-DIRAC OPERATORS 23Proof. This is entirely similar to [10, Theorem 6.1, Theorem 6.4℄. Letting τ ∈ C \ S θ , we have ddz ( I + τ DA z ) − = − ( I + τ DA z ) − τ DA ′ z ( I + τ DA z ) − . From (H1), (H3) and the bise toriality of DA z , these operators are uniformly bounded for z ∈ U ,and thus the fun tions z ( I + τ DA z ) − are holomorphi . The result is then obtained by passingto uniform limits in the strong operator toplogy. (cid:3) The Lips hitz estimates now follow.9.2. Corollary. Let ≤ p < p ≤ ∞ , let D be a (cid:28)rst order di(cid:27)erential operator, and A ∈ L ∞ ( R n ; C N ) a multipli ation operator whi h satisfy the hypotheses (H1), (H2) and (H3) of Corol-lary 8.17. Let moreover DA be R-bise torial. Then, for ea h p ∈ ( p , p ) , there exists δ = δ p > su h that, if ˜ A ∈ L ∞ ( R n ; C N ) satis(cid:28)es k A − ˜ A k ∞ < δ , then D ˜ A has a bounded H ∞ fun tional al ulus in L p with some angle ω ∈ (0 , π ) , and for θ ∈ ( ω, π ) , f ∈ H ∞ ( S θ ) , and u ∈ L p we have k f ( DA ) u − f ( D ˜ A ) u k p . k A − ˜ A k ∞ k f k ∞ k u k p . Proof. Let A z := A + z ( ˜ A − A ) / k ˜ A − A k ∞ . Then A = A , A z = ˜ A for z = k ˜ A − A k ∞ , and z A z is holomorphi . For z ∈ D (0 , δ ) , where δ is small enough, DA z has a bounded H ∞ fun tional al ulus in L p by Corollary 8.17, and z f ( DA z ) is holomorphi for f ∈ H ∞ ( S θ ) byProposition 9.1. By the S hwarz Lemma, k f ( DA ) u − f ( DA z ) u k p . | z | k f k ∞ k u k p , whi h gives the assertion. (cid:3) We (cid:28)nally turn to the Lips hitz estimates for Hodge-Dira operators with variable oe(cid:30) ients,using the same approa h as in [3, Se tion 10.1℄.9.3. Corollary. Let ≤ p < p ≤ ∞ , and let Π A and Π A be Hodge-Dira operators with variable oe(cid:30) ients, where Π A is R-bise torial in L p and Hodge-de omposes L p for all p ∈ ( p , p ) . Then,for ea h p ∈ ( p , p ) , there exists δ = δ p > su h that, if Π B and Π B are also Hodge-Dira operators with variable oe(cid:30) ients with k A − B k ∞ + k A − B k ∞ < δ , then both Π A and Π B havea bounded H ∞ fun tional al ulus with some angle ω ∈ (0 , π ) , and for θ ∈ ( ω, π ) , f ∈ H ∞ ( S θ ) ,and u ∈ L p there holds k f (Π A ) u − f (Π B ) u k p . max i =1 , k A i − B i k ∞ k f k ∞ k u k p . Proof. The philosophy of the proof is analogous to that of Corollary 8.17 but goes in the oppositedire tion: we now dedu e results for operators of the form Π A from what we already know for theoperators DA . To this end, onsider the spa e L p ⊕ L p ⊕ L p and the operators D := , A := A
00 0 A . Let us write P A , P A Γ and P A Γ A for the Hodge-proje tions asso iated to Π A . By (B2), the re-stri tion A : R p (Γ) → R p (Γ A ) is an isomorphism, and we write A − for its inverse. Then a omputation shows that ( I + itDA ) − = I A − P A Γ A ( I + t Π A ) − A − it Γ A ( I + t Π A ) − − it Γ( I + t Π A ) − A P A Γ ( I + t Π A ) − , and one an he k that the R-bise toriality and the Hodge-de omposition of Π A imply the R-bise toriality of DA . Indeed, for the diagonal elements above it is immediate, and for the non-diagonal elements it follows after writing Γ A = A − Γ A = A − P A Γ A Π A and Γ = P A Γ Π A .We next de(cid:28)ne operators S A : L p → L p ⊕ L p ⊕ L p and T A : L p ⊕ L p ⊕ L p → L p by S A u := ( P A u, A − P A Γ A u, P A Γ u ) ,T A ( u, v, w ) := P A u + P A Γ w + P A Γ A A v. T A S A = I and S A Π A = ( DA ) S A . Hen e S A ( λ − Π A ) − = ( λ − DA ) − S A , andthen by the de(cid:28)nition of the fun tional al ulus, f (Π A ) u = T A f ( DA ) S A u, f ∈ H ∞ ( S θ ) , u ∈ L p . We repeat the above de(cid:28)nitions and observations with B in pla e of A , and then f (Π A ) u − f (Π B ) u = [ T A − T B ] f ( DA ) S A u + T B [ f ( DA ) − f ( DB )] S A u + T B f ( DB )[ S A − S B ] u. The asserted Lips hitz estimate then follows by using Corollary 9.2 for the middle term, andProposition 6.10 and Corollary 6.11 for the other two terms. (cid:3)
Referen es[1℄ D. Albre ht, X. Duong, A. M Intosh, Operator theory and harmoni analysis. In Instru tional Workshopon Analysis and Geometry, Part III (Canberra 1995), Pro . Centre Math. Appl. Austral. Nat. Univ., 34 (1996),77(cid:21)136.[2℄ P. Aus her, On ne essary and su(cid:30) ient onditions for L p estimates of Riesz transforms asso iated to ellipti operators on R n and related estimates. Mem. Amer. Math. So . 871 (2007).[3℄ P. Aus her, A. Axelsson, A. M Intosh, On a quadrati estimate related to the Kato onje ture andboundary value problems. In Pro eedings of the 8th International Conferen e on Harmoni Analysis and PDE's(El Es orial 2008), Contemp. Math., Amer. Math. So ., Providen e, RI, to appear (math.CA/0810.3071).[4℄ P. Aus her, A. Axelsson, A. M Intosh, Solvability of ellipti systems with square integrable boundarydata, Ark. Mat., to appear (math.AP/0809.4968).[5℄ P. Aus her, S. Hofmann, M. La ey, A. M Intosh, Ph. T hamit hian, The solution of the Kato squareroot problem for se ond order ellipti operators on R n . Ann. of Math. (2) 156 (2002), no. 2, 633(cid:21)654.[6℄ P. Aus her, J. M. Martell, Weighted norm inequalities, o(cid:27)-diagonal estimates and ellipti operators. PartI: General operator theory and weights, Adv. Math. 212 (2007), 225(cid:21)276.[7℄ P. Aus her, J. M. Martell, Weighted norm inequalities, o(cid:27)-diagonal estimates and ellipti operators. PartII: O(cid:27)-diagonal estimates on spa es of homogeneous type, J. Evol. Equ. 7 (2007), 265(cid:21)316.[8℄ P. Aus her, J. M. Martell, Weighted norm inequalities, o(cid:27)-diagonal estimates and ellipti operators. PartIII: Harmoni Analysis of ellipti operators, J. Fun t. Anal. 241 (2006), 703(cid:21)746.[9℄ P. Aus her, A. M Intosh, E. Russ, Hardy spa es of di(cid:27)erential forms and Riesz transforms on Riemannianmanifolds. J. Geom. Anal., 18 (2008), 192(cid:21)248.[10℄ A. Axelsson, S. Keith, A. M Intosh, Quadrati estimates and fun tional al uli of perturbed Dira operators. Invent. Math. 163 (2006), no. 3, 455(cid:21)497.[11℄ A. Axelsson, S. Keith, A. M Intosh, The Kato square root problem for mixed boundary value problems,J. London Math. So . 74 (2006), 113(cid:21)130.[12℄ S. Blun k, P. C. Kunstmann, Calderón-Zygmund theory for non-integral operators and the H ∞ -fun tional al ulus, Rev. Mat. Iberoameri ana 19 (2003), no. 3, 919(cid:21)942.[13℄ J. Bourgain, Ve tor-valued singular integrals and the H -BMO duality. In Probability theory and harmoni analysis (Cleveland, Ohio, 1983). Monogr. Textbooks Pure Appl. Math., 98, Dekker, New York (1986), 1(cid:21)19.[14℄ D. L. Burkholder, Martingales and singular integrals in Bana h spa es. In Handbook of the geometry ofBana h spa es, Vol. I, 233(cid:21)269, North-Holland, Amsterdam, 2001.[15℄ M. Cowling, I. Doust, A. M Intosh, A. Yagi, Bana h spa e operators with a bounded H ∞ fun tional al ulus, J. Austral. Math. So . Ser. A 60 (1996), 51(cid:21)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 fun tional al ulus for se torial operators, Operator Theory: Advan es and Appli ations 169,Birkhäuser Verlag, Basel (2006)[18℄ T. Hytönen, A. M Intosh, P. Portal, Kato's square root problem in Bana h spa es. J. Fun t. Anal. 254(2008), no. 3, 675(cid:21)726.[19℄ N. J. Kalton, P. C. Kunstmann, L. Weis, Perturbation and interpolation theorems for the H ∞ - al uluswith appli ations to di(cid:27)erential operators, Math. Ann, 336 (2006) no. 4, 747(cid:21)801.[20℄ N. J. Kalton, L. Weis, The H ∞ - al ulus and sums of losed operators, Math. Ann. 321 (2001), no. 2,319(cid:21)345.[21℄ P. C. Kunstmann, L. Weis, Maximal L p regularity for paraboli problems, Fourier multiplier theoremsand H ∞ -fun tional al ulus, In Fun tional Analyti Methods for Evolution Equations (Editors: M. Iannelli,R. Nagel, S. Piazzera). Le t. Notes in Math. 1855, Springer-Verlag (2004).[22℄ A. M Intosh, Operators whi h have an H ∞ fun tional al ulus. In Mini onferen e on operator theory andpartial di(cid:27)erential equations (North Ryde, 1986). Pro . Centre Math. Appl. Austral. Nat. Univ., 14 (1986),210(cid:21)231.[23℄ A. M Intosh, A. Yagi, Operators of type ω without a bounded H ∞∞