Holomorphic functional calculus of Hodge-Dirac operators in Lp

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  • J. Evol. Equ. 11 (2011), 71105 2010 Springer Basel AG1424-3199/11/010071-35, published onlineSeptember 24, 2010DOI 10.1007/s00028-010-0082-y

    Journal of EvolutionEquations

    Holomorphic functional calculus of Hodge-Dirac operators in L p

    Tuomas Hytonen, Alan McIntosh and Pierre Portal

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

    1. Introduction

    A variety of problems in PDEs can be solved by establishing the boundedness,and stability under small perturbations, of the H functional calculus of certain dif-ferential operators. In particular, Axelsson et al. [10] have recovered and extendedthe solution of the Kato square root problem [5] by showing that Hodge-Dirac oper-ators with variable coefficients of the form B = + B1B2 have a boundedH functional calculus in L2(Rn; CN ), when is a homogeneous first-order differ-ential operator with constant coefficients, and B1, B2 L(Rn;L (CN )) are strictlyaccretive multiplication operators. Recently, Auscher et al. [4] have used related per-turbation results to show the openness of some sets of well-posedness for boundaryvalue problems with L2 boundary data.

    In this paper, we first consider homogeneous differential operators with constant(matrix-valued) coefficients. For such operators, the boundedness of the H func-tional calculus is established using Mikhlins multiplier theorem. However, the esti-mates on the symbols may be difficult to check in practice, especially when the nullspaces of the symbols are non-trivial. Here we provide a simple condition (invertibilityof the symbols on their ranges and inclusion of their eigenvalues in a bisector) thatgives such estimates. We then turn to operators with coefficients in L(Rn; C) ofthe form B = + B1 B2, where and are nilpotent homogeneous first-orderoperators with constant (matrix-valued) coefficients, and B1, B2 L(Rn;L (CN ))are multiplication operators satisfying some L p coercivity condition. For such oper-ators, we aim at perturbation results which give, in particular, the boundedness of the

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

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

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

    This presents two main difficulties. First of all, even in L2, the H functional cal-culus of a (bi)sectorial operator is in general not stable under small perturbations in thesense that there exist a self-adjoint operator D and bounded operators A with arbitrarysmall norm such that D(I + A) does not have a bounded H functional calculus(see [24]). Subtle functional analytic perturbation results exist (see [16] and [20]), butdo not give the estimates needed in [4] or [10]. To obtain such estimates, one needsto take advantage of the specific structure of differential operators using harmonicanalytic methods. Then, the problem of moving from the L2 theory to an L p theoryis substantial. Indeed, the operators under consideration fall outside the Caldern-Zygmund class and cannot be handled by familiar methods based on interpolation.A known substitute, pioneered by Blunck and Kunstmann in [12], and developed byAuscher and Martell [2,68], consists in establishing an extrapolation method adaptedto the operator, which allows to extend results from L2 to L p for p in a certain range(p1, p2) containing 2. In [19], we started another approach, which combines probabi-listic tools from functional analysis with the aforementioned L2 methods, and allowsL p results which do not rely on some L2 counterparts.

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

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

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

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

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

    2. Preliminaries

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

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

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

    (A) S := (), where := {z C\{0} ; | arg(z)| } {0},

    and outside the bisector it verifies the following resolvent bounds:

    (,

    2

    )C > 0 C\S (I A)1L (Y ) C. (2.2)

    We often omit the angle and say that A is bisectorial if it is bisectorial with someangle [0, 2

    ).

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

    H0 (S) :={ H(S) : , C (0,) z S |(z)| C

    z

    1 + z2}

    .

    For a bisectorial operator A with angle < < < /2, and H0 (S), wedefine

    (A)u := 12i

    S()( A)1u d,

    where S is directed anti-clockwise around S.

    DEFINITION 2.3. A bisectorial operator A, with angle , is said to admit a boundedH functional calculus with angle [, 2 ) if, for each

    (, 2

    ),

    C < H0 (S) (A)yY CyY .

  • 74 T. Hytonen et al. J. Evol. Equ.

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

    f (A)u := f (0)P0u + limnn(A)u,

    where P0 denotes the projection on the null space of A corresponding to the decom-position Y = N(A) R(A), which exists for R-bisectorial operators, and (n)nN H0 (S) is a bounded sequence which converges locally uniformly to f . See [1,15,17,22,23] for details.

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

    E

    M

    k=1k Tkuk

    Y

    E

    Mk=1

    kuk

    Y,

    where E is the expectation which is taken with respect to a sequence of independentRademacher variables k , i.e., random signs with P(k = +1) = P(k = 1) = 12 .

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

    {(I A)1 : C\S }is R-bounded for all (, /2). The infimum of such angles is called the angleof R-bisectoriality of A.

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

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

    3. Main results

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

  • Vol. 11 (2011) Functional calculus of Hodge-Dirac operators 75

    We work in the Lebesgue spaces L p := L p(Rn; CN ) with p (1,) and denoteby S (Rn; CN ) the Schwartz class of rapidly decreasing functions with values in CN ,and by S (Rn; CN ) the corresponding class of tempered distributions.3.1. General constant-coefficient operators

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

    D = (i)k

    Nn :| |=kD

    acting on S (Rn; CN ) as a Fourier multiplier with symbol D() = | |=k D ,where D L (CN ).

    ASSUMPTION 3.1. The Fourier multiplier symbol D() satisfies

    | |k |e| |D()e| for all Rn, all e R(D()), and some > 0,(D1)

    there exists [0,

    2

    )such that for all Rn : (D()) S.

    (D2)

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

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

    REMARK 3.3. (a) In (D2), the bisector S can be replaced by the sector where 0 < . The operator D is then sectorial (with angle ) and has abounded H functional calculus (with angle ) in the sectorial sense, i.e. f (D)is bounded for functions f H( ) with any (, ).

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

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

    3.2. Hodge-Dirac operators with constant coefficients

    We now turn to first-order operators of the form

    = + ,

  • 76 T. Hytonen et al. J. Evol. Equ.

    where = i nj=1 j j , acts on S (Rn; CN ) as a Fourier multiplier with symbol

    = () =n

    j=1 j j , j L (CN ),

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

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

    | ||e| |()e| for all e R(()), all Rn, and some > 0,(1)

    (()) S for some [0,

    2

    ), and all Rn, (2)

    N(()) = N(()) N( ()) for all Rn . (3)

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

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

    Dp() := {u L p : u L p},

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

    Rp() Np(), Rp( ) Np( ).

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

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

    THEOREM 3.6. The operator has a bounded H functional calculus in L pwith angle and satisfies the Hodge decomposition

    L p = Np() Rp() Rp( ).

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

  • Vol. 11 (2011) Functional calculus of Hodge-Dirac operators 77

    3.3. Hodge-Dirac operators with variable coefficients

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

    DEFINITION 3.8. Let 1 < p < and p denote the dual exponent of p. LetB1, B2 L(Rn;L (CN )),

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

    B := + B, where B := B1 B2, (3.9)is called a Hodge-Dirac operator with variable coefficients in L p if the followinghold:

    v, B2 B1 u := v, B2 B1 u = 0 for u, v S (Rn; CN ), (B1)up B1up u Rp( ) and vp B2vp v Rp( ). (B2)

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

    The foll...

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