In this note we present a symbolic pseudo-differential calculus on graded
nilpotent Lie groups and, as an application, a version of the sharp Garding
inequality. As a corollary, we obtain lower bounds for positive Rockland
operators with variable coefficients as well as their Schwartz-hypoellipticity

Fischer, Veronique

Ruzhansky, Michael

Comptes Rendus Mathematique

English

arXiv

Véronique Fischer

Michael Ruzhansky

Comptes Rendus Mathématique

Lower bounds for operators on graded Lie groups

Crossref

Fischer, Véronique

Ghent University Academic Bibliography

arXiv:1301.0642v2  [math.FA]  21 Feb 2013LOWER BOUNDS FOR OPERATORS ON GRADED LIEGROUPSVE´RONIQUE FISCHER AND MICHAEL RUZHANSKYRe´sume´. In this note we present a symbolic pseudo-differential calculus onany graded (nilpotent) Lie group and, as an application, a version of thesharp G˚arding inequality. As a corollary, we obtain lower bounds for posi-tive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.1. IntroductionIn this note we present a symbolic pseudo-differential calculus on any graded Liegroup. As applications, we obtain a version of the sharp G˚arding inequality andresults on the Schwartz hypoellipticity for operators in this context.In the usual Euclidean setting, the positivity of the full symbol is required for thesharp G˚arding inequality as well as, for instance, for the Fefferman-Phong inequa-lity. This contrasts with the G˚arding or Melin-Ho¨rmander inequalities, for example,where knowing the principal (or subprincipal) symbol is sufficient. Thus the latterinequalities can be proved on manifolds using the standard Ho¨rmander theory ofpseudo-differential operators together with the usual Kohn-Nirenberg quantisationon Rn. Since the geometric control of the full symbol of an operator is impossiblewith these tools in general, the study of sharp G˚aring inequalities appears limited.However, the sharp G˚arding inequality on compact Lie groups was recently establi-shed in [13]. This approach uses the notion of a full matrix-valued symbol definedin terms of the representation theory of the group. In this note we explain howwe follow the same strategy in the case of the Heisenberg group or more generalnilpotent Lie groups.The pseudo-differential calculus that we define is different from the severalpseudo-differential calculi already developed on the Heisenberg group : see e.g.Taylor [15] and [16] for symbol classes coming from standard Ho¨rmander classesthough the exponential mapping ; Bahouri, Fermanian-Kammerer and Gallagher[1] for classes defined in terms of explicit formulae coming from the Schro¨dingerrepresentation ; or Beals and Greiner [2] or Ponge [11] for different types of analy-sis on the Heisenberg manifolds. On more general nilpotent groups, Christ, Geller,Glowacki and Polin [4] proposed an approach to pseudo-differential operators, ho-wever based on the properties of kernels and not on a symbolic calculus. FollowingRuzhansky and Turunen [14] and [12], we define symbol classes directly on thegroup. As such, our approach can be extended for general graded nilpotent LieDate: February, 2013.1991 Mathematics Subject Classification. Primary 35S05 ; Secondary 43A80.Key words and phrases. harmonic analysis, nilpotent Lie groups, pseudo-differential operators.12 V. FISCHER AND M. RUZHANSKYgroups, and by developing the symbolic calculus and the Friedrichs approximationon the group, we obtain the corresponding sharp G˚arding inequality and Schwartzhypoellipticity results.While the symbol classes in [12] are based on the spectral theory of the Laplace-Beltrami operator, here, it is not available and it becomes natural to use thesub-Laplacian on stratified groups or more general Rockland operators on gradedgroups. Moreover, surpassing [13], since a dilation structure is present, we establishthe sharp G˚arding inequality for suitable (ρ, δ) classes of operators by establishinga Caldero´n–Vaillancourt type theorem in this context. This is, in fact, the bestknown lower bound available in the (ρ, δ)-setting already on Rn.The appearing operators are Caldero´n–Zygmund in the sense of Coifman andWeiss [5, ch. III], so that Lp results follow as well. In Section 2 we fix the notationconcerning Lie groups that we are working on. In Section 3 we formulate the results.2. PreliminariesLet us first briefly recall the necessary notions and set some notation. In general,we will be concerned with graded Lie groups G which means that G is a connectedand simply connected Lie group (of step s > 1) with the gradation of its Lie algebrag given by g =∞⊕ℓ=1gℓ with [gℓ, gℓ′ ] ⊂ gℓ+ℓ′ for any ℓ, ℓ′ ∈ N, where the gℓ, ℓ = 1, 2, . . .are vector subspaces of g, almost all equal to {0}. This implies that the group G isnilpotent. If the whole g is generated by g1 in this way, the group G is said to bestratified.Let {X1, . . .Xn1} be a basis of g1 (this basis is possibly reduced to {0}), let{Xn1+1, . . . , Xn1+n2} a basis of g2 and so on, so that we obtain a basis X1, . . . , Xnof g adapted to the gradation. Via the exponential mapping expG : g → G, weidentify the points (x1, . . . , xn) ∈ Rn with the points x = expG(x1X1+ · · ·+xnXn)in G. This leads to a corresponding Lebesgue measure on g and the Haar measuredx on the group G. We define the spaces of Schwartz functions S(G) and tempereddistributions S ′(G) of the group G as those on Rn. The coordinate function x =(x1, . . . , xn) ∈ G 7→ xj ∈ R is denoted by xj . More generally we define for everymulti-index α ∈ Nn0 , xα := xα11 xα22 . . . xαnn , as a function on G. Similarly we setXα = Xα11 Xα22 · · ·Xαnn in the universal enveloping Lie algebra of g.For any r > 0, we define the linear mapping Dr : g → g by DrX = rℓX forevery X ∈ gℓ, ℓ ∈ N. Then the Lie algebra g is endowed with the family of dilations{Dr, r > 0} and becomes a homogeneous Lie algebra in the sense of [8]. The weightsof the dilations are the integers υ1, . . . , υn given by DrXj = rυjXj , j = 1, . . . , n.The associated group dilations are defined byr · x := (rυ1x1, rυ2x2, . . . , rυnxn), x = (x1, . . . , xn) ∈ G, r > 0.In a canonical way this leads to the notions of homogeneity for functions and opera-tors. For instance the degree of homogeneity of xα and Xα, viewed respectively asa function and a differential operator on G, is [α] =∑j υjαj . Indeed, let us recallthat a vector of g defines a left-invariant vector field on G and more generally thatthe universal enveloping Lie algebra of g is isomorphic with the left-invariant diffe-rential operators ; we keep the same notation for the vectors and the correspondingoperators.The dimension of G is n =∑ℓ nℓ while its homogeneous dimension is Q =∑ℓ ℓnℓ = υ1 + υ2 + . . .+ υn.LOWER BOUNDS FOR OPERATORS ON GRADED LIE GROUPS 3We denote by Ĝ the set of equivalence classes of (continuous) irreducible unitaryrepresentations of G. We will often identify a representation of G with its equiva-lence class. We will also keep the same notation for the corresponding infinitesimalrepresentation. For π ∈ Ĝ, we denote by Hπ the representation space of π and byH∞π its subspace of smooth vectors. For f ∈ L1(G), we define its Fourier transformat π ∈ Ĝ by f̂(π) =∫Gf(g)π(g)∗dg, with the integral understood in the Bochnersense. Denoting by µ the Plancherel measure on Ĝ, the inverse Fourier formulaholds :f(g) =∫Ĝtrace(π(g)f̂(π))dµ(π) when∫Ĝtrace∣∣∣f̂(π)∣∣∣ dµ(π) <∞ .Let R be a positive (left) Rockland operator on G ; this means that R is a left-invariant differential operator, homogeneous of degree ν necessarily even, positivein the operator sense, and such that for every non-trivial π ∈ Ĝ the operator π(R)is injective on H∞π . The operator R admits an essentially self-adjoint extension onC∞0 (G) (see [8]), and we will still denote this extension by R. Examples of suchoperators are given in the stratified case by R = −L where L =∑1≤j≤n1X2i is aKohn-sub-Laplacian, and in the graded case by the operators∑1≤j≤nj(−1)νoυjX2 νoυjj and∑1≤j≤njX4 νoυjj ,where νo denotes some common multiple of υ1, . . . , υn. In fact our class of operatorsdo not depend on the choice of such an operator R.3. ResultsWe aim at defining the symbol classes in terms of the operators R as above.Definition 3.1. A symbol is a family of operators σ = {σ(x, π) : x ∈ G, [π] ∈ Ĝ},such that1. for each x ∈ G, the family {σ(x, π), π ∈ Gˆ} is a µ-measurable field of operatorsH∞π → Hπ ;2. there exist two constants γ1, γ2 ∈ R such that for every x ∈ G, the operatorπ(I +R)γ1σ(x, π)π(I +R)γ2 is bounded on Hπ uniformly in π ∈ Gˆ ;3. for any π ∈ Ĝ and any u, v ∈ Hπ, the scalar function x 7→ (σ(x, π)u, v)Hpi issmooth on G.Here, the powers π(I +R)γ are defined by the spectral theorem for the positiveoperator π(R). The existence of γ1, γ2 in the second condition is used to guaranteethat the following formula makes sense :Tf(x) =∫Ĝtrace(π(x)σ(x, π)f̂ (π))dµ(π) , f ∈ S(G), x ∈ G ;indeed such operator T = Op(σ) is well-defined and continuous S(G)→ S ′(G). Wenote that if the operator T is left-invariant, then its symbol is independent of x.Definition 3.2. The difference operators ∆α, α ∈ Nn0 , are densely defined on theC∗-algebra of the group via(∆αf̂)(π) := (̂xαf)(π) , f ∈ S(G) .Let now m ∈ R and 0 ≤ δ ≤ ρ ≤ 1 with δ 6= 1.4 V. FISCHER AND M. RUZHANSKYDefinition 3.3. The symbol class Smρ,δ is defined as the set of symbols σ satisfyingfor all α, β ∈ Nn0 and every γ ∈ R :(3.1) supx∈G,π∈Ĝ‖π(I +R)ρ[α]−m−δ[β]+γν Xβx∆ασ(x, π)π(I +R)−γν ‖op <∞.(The supremum over π is in fact the essential supremum over the Plancherel measureµ.)It is easy to see that if m1 ≤ m2, δ1 ≤ δ2 and ρ1 ≥ ρ2, then Sm1ρ1,δ1⊂ Sm2ρ2,δ2 .Furthermore, the expressions in (3.1) define a Fre´chet topology on the linear spaceSmρ,δ.In the abelian case, that is, Rn endowed with the addition law and R = −L, Lbeing the Laplace operator, Smρ,δ boils down easily to the usual Ho¨rmander class.However our initial motivation did not come from the abelian case : we wantedto define the difference operators and the symbol classes in analogy with the onesdefined in [14] on compact Lie groups. In this case, a definition similar to (3.1)would formally give the same classes of symbols defined in [14] since, R = −L, Lbeing the Laplace-Beltrami operator, the operator π(I +R) is scalar. In our case,i.e. G being a graded non-abelian Lie group, the operator R is not even central andthe introduction of γ in (3.1) assures that⋃m∈R Smρ,δ is an algebra.We have the following properties for the operators classes Ψmρ,δ := Op(Smρ,δ)defined using the quantisation procedure σ 7→ Op(σ) described above.Theorem 3.4. Let 0 ≤ δ ≤ ρ ≤ 1. We have the following properties :(1) The symbol classes is an algebra of operators⋃m∈R Smρ,δ stable by taking theadjoint. Each vector space Smρ,δ does not depend on the choice of the positiveRockland operator R.(2) For ρ 6= 0, the operator class⋃m∈RΨmρ,δ is an algebra stable by taking theadjoint.(3) For any α ∈ Nn0 , we have Xα ∈ Ψ[α]1,0.(4) For any positive Rockland operator of homogeneous degree ν, we have (I +R)mν ∈ Ψm1,0.(5) If ρ ∈ [0, 1) then the operators in Ψ0ρ,ρ are continuous on L2(G).(6) Let ρ 6= 0. The integral kernel K(x, y) of an operator T ∈ Ψmρ,δ is smoothon (G × G)\{(x, y) : x = y}. It is of Caldero´n-Zygmund type in the sense ofCoifman and Weiss [5, ch.III]. It decreases rapidly as |xy−1| → ∞ (here wehave fixed a homogeneous norm | · | on G, i.e. a continuous function, homoge-neous of degree one and vanishing only at 0) : i.e. for any M > 0 there existsCM > 0 such that|xy−1| ≥ 1 =⇒ |K(x, y)| ≤ CM |xy−1|−M .At the diagonal it satisfies|xy−1| ≤ 1 =⇒ |K(x, y)| ≤ C|xy−1|−Q+mρ .By [5, ch.III the´ore`me 2.4] Property (6) implies that the operators in Ψ0ρ,δ, 1 ≥ρ ≥ δ ≥ 0, ρ 6= 0, δ 6= 1, are continuous on Lp(G), 1 < p <∞.By Properties (2) and (4), any operator in Ψmρ,δ, 1 ≥ ρ ≥ δ ≥ 0, ρ 6= 0, δ 6= 1,is continuous on the natural Sobolev spaces (denoted by L2a(G)) associated withthe dilations and the loss of derivatives is controlled by the order m. The SobolevLOWER BOUNDS FOR OPERATORS ON GRADED LIE GROUPS 5space L2a(G) is defined as the set of tempered distribution f ∈ S′(G) such that(I +R)aν f ∈ L2(G) but does not depend on the choice of R. These Sobolev spacesenjoy properties similar to the stratified case proved by Folland [7], in particularfor interpolation (see [6]).We now give the sharp G˚arding inequality.Theorem 3.5. Let 0 ≤ δ ≤ ρ ≤ 1, ρ 6= 0, δ 6= 1, and let T ∈ Ψmρ,δ with symbolσ = {σ(x, π)}. Assume that each {σ(x, π)} of T is non-negative on Hπ (in theoperator sense). Assume also that there exists a Rockland operator R such thateach σ(x, π) commutes with the spectral measure of π(R) for every x ∈ G andalmost every π ∈ Ĝ. Then there exists C > 0 such that for every f ∈ S(G) we haveRe(Tf, f)L2(G) ≥ −C‖f‖L2m−(ρ−δ)2(G).The class includes– the variable coefficient Kohn-sub-Laplacians and Rockland operators of theform a(x)R, with a(x) ≥ 0 satisfying Xαa ∈ L∞(G) for all α ∈ Nn0 ,– the multipliers φ(R) for a smooth function φ : [0,∞) 7→ [0,∞) satisfying(3.2) ∀a ∈ N0 ∃C = Ca > 0 ∀λ ≥ 0 |∂αλφ(λ)| ≤ C(1 + λ)mν−a,– more generally the operators with symbols given by σ(x, π) = φx(π(R)) with(x, λ) 7→ φx(λ) being non-negative and smooth on G×[0,∞), and φx satisfying(3.2) at each x with a constant C independent of x.The condition on the commutation with the spectral measure of π(R) seems tobe reasonable : in the corresponding version of the sharp G˚arding inequality oncompact Lie groups in [13] or in the abelian case, this condition is automaticallysatisfied there because if R = −L and L is the Laplacian then π(R) is central.In [9], Helffer and Nourigat proved that R and, equivalently, I +R are hypoel-liptic. We finally give the result stating that the Schwartz version of such hypoel-lipticity is also true.Theorem 3.6. The operator I+R is Schwartz-hypoelliptic, i.e. for f ∈ S ′(G), thecondition (I +R)f ∈ S(G) implies f ∈ S(G).In fact, our symbolic calculus allows the construction of a parametrix for theoperator I+R from which both the hypoellipticity and the Schwartz hypoellipticityfollow.AcknowledgementsThe first author acknowledges the support of the London Mathematical Societyvia the Grace Chisholm Fellowship. It was during this fellowship held at King’s Col-lege London in 2011 that the work was initiated. The second author was supportedin part by the EPSRC Leadership Fellowship EP/G007233/1.Re´fe´rences[1] Bahouri H., C. Fermanian-Kammerer and I. Gallagher, Phase-space analysis and pseudodif-ferential calculus on the Heisenberg group, Aste´risque 342 (2012).[2] R. Beals and P. Greiner, Calculus on Heisenberg manifolds. Princeton University Press, 1988.[3] M. Christ, On the regularity of inverses of singular integral operators, Duke Math. J. 57(1988), 459–484.6 V. FISCHER AND M. RUZHANSKY[4] M. Christ, D. Geller, P. Glowacki and L. Polin, Pseudodifferential operators on groups withdilations, Duke Math. J. 68 (1992), 31–65.[5] R. Coifman and G. Weiss Analyse harmonique non-commutative sur certains espaces ho-moge`nes, Springer, 1971.[6] V. Fischer and M. Ruzhansky, Quantization on nilpotent Lie groups, monograph in prepara-tion.[7] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat.,13 (1975), 161–207.[8] G. B. Folland and E. Stein, Hardy spaces on homogeneous groups, Princeton University Press,1982.[9] B. Helffer and J. Nourrigat, Caracterisation des ope´rateurs hypoelliptiques homoge`nes inva-riants a` gauche sur un groupe de Lie nilpotent gradue´, Comm. Partial Differential Equations,4 (1979), 899–958.[10] A. Hulanicki, A functional calculus for Rockland operators on nilpotent Lie groups, StudiaMath., 78 (1984), 253–266.[11] R. Ponge, Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenbergmanifolds, Mem. Amer. Math. Soc. 194 (2008).[12] M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries : Backgroundanalysis and advanced topics, Birkha¨user, Basel, 2010.[13] M. Ruzhansky and V. Turunen, Sharp G˚arding inequality on compact Lie groups, J. Funct.Anal., 260 (2011), 2881–2901.[14] M. Ruzhansky and V. Turunen, Global quantization of pseudo-differential operators on com-pact Lie groups, SU(2) and 3-sphere, Int Math Res Notices IMRN (2012), 58 pages, doi :10.1093/imrn/rns122.[15] M. E. Taylor, Noncommutative microlocal analysis. I. Mem. Amer. Math. Soc. 52 (1984).[16] M. E. Taylor, Noncommutative harmonic analysis, American Mathematical Society, 1986.Universita degli studi di Padova, DMMMSA, Via Trieste 63, 35121 Padova, ItalyE-mail address: fischer@dmsa.unipd.it180 Queen’s Gate, Department of Mathematics, Imperial College London, London,SW7 2AZ, United KingdomE-mail address: m.ruzhansky@imperial.ac.uk

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Lower bounds for operators on graded Lie groups

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