aa r X i v : . [ m a t h . SP ] J a n SEMICLASSICAL ANALYSIS ON COMPACT NIL-MANIFOLDS
V´ERONIQUE FISCHER
Abstract.
In this paper, we develop a semi-classical calculus on compact nil-manifolds. As appli-cations, we obtain Weyl laws for positive Rockland operators on any graded compact nil-manifoldsand results on quantum ergodicity in position for sub-Laplacians on any stratified nil-manifolds.
Contents
1. Introduction and main results 22. Preliminaries on compact nil-manifold 42.1. Definition and examples 42.2. Discrete co-compact subgroups vs lattices 52.3. Γ-periodicity and periodisation 62.4. Spaces of periodic functions 82.5. Convolution and periodicity 92.6. Operators on M and G G G and the Plancherel theorem 133.4. Positive Rockland operators on G G and on M R M E . Introduction and main results
For the last twenty years, the analysis of hypoelliptic operators in sub-Riemannian settings hasmade fundamental progress in powers and heat expansions and in index theory (see respectivelyPonge’s memoires [23] and van Erp’s work in [28, 29] and the references in both). The underlyingmethods and ideas are so comprehensive that their natural setting is not restricted to the class ofsub-Riemannian manifolds but extends more generally to filtered manifolds. In fact, a significanttool for these results has turned out to be the tangent groupoid to a filtered manifold [30, 8].Unfortunately, this dosed not seem to be suited for micro-local and semi-classical questions wherea notion of symbol is often needed. Very few results in that part of analysis have been obtained forhypoelliptic operators. For instance, quantum ergodicity in this context has been studied in specificexamples amongst contact or quasi-contact manifolds, and the list consists of [32, 3, 33, 6, 21, 24]to the author’s knowledge to-date.The main aim of this paper is to start the development of semi-classical analysis for a largeclass of hypoelliptic operators; this class includes the natural sub-Laplacians on compact nil-manifolds without any further hypothesis than a stratified structure on the underlying group.The methods and results presented here support the systematic approach of semi-classical analysisin sub-Riemannian and sub-elliptic settings developed by the author and her collaborator ClotildeFermanian-Kammerer on nilpotent Lie groups [11, 10, 12]. Indeed, the next natural context to testour approach after the group case is the one of compact nil-manifolds as they are the analogues ofthe torus T n (the quotient of R n by a lattice) in spectral Euclidean geometry: although easily com-prehended, tori provide a rich context for the usual (i.e. Euclidean and commutative) semiclassicalanalysis, see e.g. [4, 2].As applications, we obtain the following results: Theorem 1.1.
Let Γ be a discrete co-compact subgroup of a stratified nilpotent Lie group G . AHaar measure having been fixed on G , we denote by d ˙ x the corresponding natural measure on thecompact nil-manifold M := Γ \ G . Let X , . . . , X n be a basis of the first stratum g of the Lie algebraof G . Let L M be the associated self-adjoint sub-Laplacian on L ( M ) . We denote its eigenvaluesand its spectral counting function by λ < λ ≤ λ ≤ . . . and N (Λ) := |{ j ∈ N , λ j ≤ Λ }| , Weyl Law:
The spectral counting function admits the asymptotic N (Λ) ∼ c Λ − Q/ , as Λ → + ∞ , with a constant c depending on M , G and L . More precisely, c = Vol( M ) c /Q, where Vol( M ) denotes the volume of M , Q the homogeneous dimension of G and c is apositive constant depending only on L and G . Quantum Ergodicity in position:
Let ( ϕ j ) be an orthonormal basis of the Hilbert space L ( M ) consisting of eigenfunctions of L M satisfying L M ϕ j = λ j ϕ j . There exists a subse-quence ( j k ) of density one lim Λ →∞ |{ j k : λ j k ≤ Λ }| N (Λ) = 1 , uch that for any continuous function a : M → C lim k → + ∞ Z M a ( ˙ x ) | ϕ j k ( ˙ x ) | d ˙ x = Z M a ( ˙ x ) d ˙ x. Weyl laws for hypoelliptic operators in sub-Riemannian contexts have been obtained using heatexpansions, see Ponge’s memoire [23] and references therein. Note that the constant involved intheir Weyl laws is expressed in terms of a coefficient of these expansions and that this result wasgeneralised recently to operators satisfying the Rockland condition on filtered manifold in [8, p.347]. By contrast, our Weyl law above is a natural consequence of the semi-classical approachalready developed in the group context [11, 10, 12] and on nil-manifolds of Heisenberg type [13];furthermore, the constant we obtain is more explicit than the one from the heat expansion approachas our constant c is determined in Theorem 3.6 and Formula (3.6). Moreover, our result admits asemi-classical generalisation stated in Theorem 4.12, which is new. All these results are proved inthe more general setting of positive Rockland operators on any graded nil-manifold in Sections 3and 4.The second application of the semi-classical analysis on M discussed in this paper concernsquantum variance for sub-Laplacians, see Corollary 6.4 for the most general form. The result onquantum variance and standard manipulations imply the quantum ergodicity in position stated inTheorem 1.1. In the commutative case, that is, on the torus. T n , our result on quantum varianceboils down to quantum ergodicity for the 0-energy level (i.e. the analogue of [34, Theorem 9.4] for c = 0) for the standard Laplacian and for the basic semi-classical calculus; by basic semi-classicalcalculus, we mean the algebra of operators a ( ˙ x, εD ) obtained via the Kohn-Nirenberg quantisationof symbols a ∈ C ∞ ( T n ; S ( R n )) with Fourier variables dilated by ε ∈ (0 , H or product of these.Indeed, in its ‘usual Euclidean’ micro-local meaning, quantum ergodicity was obtained recently onthree dimensional contact manifolds in [6]. This approach seems possible for the Heisenberg group[3] and when the local structure is essentially H [6, 21, 24]. They imply the result for ergodicityin position in Theorem 1.1 but only for nil-manifolds which are products of the canonical quotientof H . The studies of hypoelliptic operators mentioned in the first paragraph of this introductionsuggest that this ‘Euclidean’ strategy may become unviable as the non-commutativity becomesmore involved. By contrast, our result on quantum ergodicity in position holds on any nil-manifoldwhere a natural sub-Laplacian can be defined.As mentioned earlier, the methods of the paper follow the approach developed by the authorand her collaborator Clotilde Fermanian-Kammerer on nilpotent Lie groups G , see [11, 10, 12, 13],and presented independently in Section 4. The calculus is formed by operators obtained fromsymbols and the natural quantisation based on the group Fourier transform. The group Fouriertransform of a function at a (unitary irreducible) representation is an operator on the space of therepresentation. Consequently, denoting by b G the unitary dual, i.e. the set of unitary irreduciblerepresentations of G modulo equivalence, the symbols of pseudo-differential operators introduced in
14] are measurable fields of operators on G × b G for operators on G , or M × b G for operators on thenil-manifold M . The ideas behind this symbolic ergodicity comes from computing the commutatorof L M with a semi-classical pseudo-differential operator:[ L M , Op ( ε ) ( σ )] = Op ( ε ) ( L M σ ) − ε − Op ( ε ) ( b E σ ) + ε − Op ( ε ) (cid:16) [ b L , σ ] (cid:17) , where b E := P n j =1 X M,j π ( X j ). In Section 5, we apply von Neuman’s mean ergodic theorem tothe one-parameter group of unitary operators e i t b E . As the difference between the two formalexpansions e − itε L M Op ( ε ) ( σ ) e itε L M = Op ( ε ) ( σ ) − itε [ L M , Op ( ε ) ( σ )] + . . . , Op ( ε ) ( e i t b E σ ) = Op ( ε ) ( σ ) + 2 it Op ( ε ) ( b E σ ) + . . . , is O ( ε ) if [ c L , σ ] = 0, we can use an Egorov-type argument to show the property of the quantumvariance of Section 6. The methods presented here could be easily adapted to operators of theform L M + V where V is a (real-valued and regular enough) potential. But this is left for thefuture works regarding full quantum ergodicity for hypo-elliptic operators with our semi-classicaland micro-local approaches on more general sub-Riemanian or even filtered manifolds. Notation: If V and W are topological vector spaces, we denote by L ( V, W ) the space of linearcontinuous mappings from V to W . If V = W , then we write L ( V, V ) = L ( V ). If V is atopological vector space and M a smooth manifold, we denote by C ∞ ( M : V ) the space of smoothfunctions on M valued in V .2. Preliminaries on compact nil-manifold
In this section, we set our notation for nil-manifolds and recall some elements of analysis in thissetting.2.1.
Definition and examples.
A compact nil-manifold is the quotient M = Γ \ G of a nilpotentLie group G by a discrete co-compact subgroup Γ of G . In this paper, a nilpotent Lie group isalways assumed connected and simply connected unless otherwise stated.The vocabulary varies, and a discrete co-compact subgroup may also be called uniform or latticein some literature, see e.g. [7, Section 5].A concrete example of uniform subgroup is the natural discrete subgroup of the Heisenberggroup, as described in [7, Example 5.4.1]. Abstract constructions for graded groups are discussedin Section 3.2, and will use the following statement which recalls some abstract examples andcharacterisations from [7, Section 5.1] (the definition of Malcev bases is recalled below): Theorem 2.1.
Let G be a nilpotent Lie group. (1) A subgroup Γ of G is discrete co-compact if and only if Exp Y j ∈ Γ for j = 1 , . . . , n for someweak or strong Malcev basis Y , . . . , Y n of g . (2) A subgroup Γ of G is discrete co-compact if and only if it can be written as Γ = Exp( Z Y ) . . . Exp( Z Y n ) for some weak or strong Malcev basis Y , . . . , Y n of g .Moreover, a group G admits a uniform subgroup Γ if and only if its Lie algebra g has a rationalstructure. If this is the case, then a choice of rational structure g Q is the Q -span of log Γ .Furthermore, if Y , . . . , Y n is a strong Malcev basis whose structural constants c i,j,k from [ Y i , Y j ] = P k c i,j,k Y k are all rational, then there exists a positive integer K ∈ N such that the set Exp( K Z Y ) . . . Exp( K Z Y n ) is a discrete co-compact subgroup of G . et us recall the notion of Malcev bases: an (ordered) basis Y , . . . Y n of g is a strong (resp. weak)Malcev basis when for each m = 1 , . . . , n , the subspace R Y ⊕ . . . ⊕ R Y m is a Lie sub-algebra (resp.ideal) of the Lie algebra g . We refer the reader to [7] for examples and properties of these bases,and the reader unfamiliar with this notion can just consider this as a technical property satisfiedby important bases, for instance by the basis constructed in Section 3.1.1 in the case of graded Liegroups.2.2. Discrete co-compact subgroups vs lattices.
There is a close connection between discreteco-compact subgroups and lattices in g described in [7, Section 5.4]: Theorem 2.2.
Let Γ be a discrete co-compact subgroup of a nilpotent Lie group G . Then thereexists Γ and Γ discrete co-compact subgroups of G such that • log Γ and log Γ are lattices of the vector space g ∼ R n , • the inclusions Γ ⊂ Γ ⊂ Γ hold, and • Γ / Γ and Γ / Γ are finite sets.Furthermore, having written Γ as Exp( Z Y ) . . . Exp( Z Y n ) for some strong Malcev basis Y , . . . , Y n of g , we may choose Γ = Exp( K Z Y ) . . . Exp( K Z Y n ) for some suitable integer K > , and Γ =Exp( k Z Y ) . . . Exp( k n Z Y n ) with k j = K − N n − j , j = 1 , . . . , n . Corollary 2.3.
Let Γ be a discrete co-compact subgroup of a nilpotent Lie group G . We identify G with R n via the exponential mapping and a basis of g . Then for any N > n and any norm | · | on the vector space R n ∼ G , the sum P γ ∈ Γ (1 + | γ | ) − N finite.Proof. We may assume that the basis of g is a strong Malcev basis Y , . . . , Y n such that Γ =Exp( Z Y ) . . . Exp( Z Y n ). Theorem 2.2 implies that we may assume that Λ = log Γ is a lattice of g ∼ R n . In this case, X γ ∈ Γ (1 + | γ | ) − N = X m ∈ Λ (1 + | m | ) − N . The result follows by comparisons with integrals on R n for a suitable norm on R n , as all the normson R n are equivalent. (cid:3) Structures on M . An element of M is a class˙ x := Γ x of an element x in G . If the context allows it, we may identify this class with its representative x .The quotient M is naturally equipped with the structure of a compact smooth manifold. Fur-thermore, fixing a Haar measure on the unimodular group G , M inherits a measure d ˙ x which isinvariant under the translations given by M −→ M ˙ x ˙ xg = Γ xg , g ∈ G. Recall that the Haar measure dx on G is unique up to a constant and, once it is fixed, d ˙ x is theonly G -invariant measure on M satisfying for any function f : G → C , for instance continuous withcompact support,(2.1) Z G f ( x ) dx = Z M X γ ∈ Γ f ( γx ) d ˙ x. We denote by Vol( M ) = R M d ˙ x the volume of M .Some ‘nice’ fundamental domains are described in [7, Section 5.3]: roposition 2.4. Let Γ be a discrete co-compact subgroup of a nilpotent Lie group G described as Γ = Exp( Z Y ) . . . Exp( Z Y n ) for some weak Malcev basis Y , . . . , Y n of g (see Theorem 2.1.We set R := [ − , ) × . . . × [ − , ) = [ − , ) n and for every m ∈ R n : R m := m + R and D m := { Exp( t Y ) . . . Exp( t n Y n ) : t = ( t , . . . , t n ) ∈ R m } . Then D m is a fundamental domain for M = Γ \ G . Furthermore, the map Θ : (cid:26) R n −→ Mt ΓExp( t Y ) . . . Exp( t n Y n ) , maps R m onto M and the Lebesgue measure dt restricted to R m to the G -invariant measure on M .If t, u ∈ R m and γ ∈ Γ satisfy Θ( t ) = γ Θ( u ) then t = u and γ = 0 . Furthermore, if t ∈ R and γ ∈ Γ satisfy Θ( t ) − γ Θ( t ) = 0 then γ = 0 .About the proof of Proposition 2.4. Proposition 2.4 follows from simple modifications from Theo-rem 5.3.1 in [7] and its proof. (cid:3) -periodicity and periodisation.
Let Γ be a discrete co-compact subgroup of a nilpotentLie group G .We say that a function f : G → C is Γ-left-periodic or just Γ-periodic when we have ∀ x ∈ G, ∀ γ ∈ Γ , f ( γx ) = f ( x ) . This definition extends readily to measurable functions and to distributions.There is a natural one-to-one correspondence between the functions on G which are Γ-periodicand the functions on M . Indeed, for any map F on M , the corresponding periodic function on G is F G defined via F G ( x ) := F ( ˙ x ) , x ∈ G, while if f is a Γ-periodic function on G , it defines a function f M on M via f M ( ˙ x ) = f ( x ) , x ∈ G. Naturally, ( F G ) M = F and ( f M ) G = f .We also define, at least formally, the periodisation φ Γ of a function φ ( x ) of the variable x ∈ G by: φ Γ ( x ) = X γ ∈ Γ φ ( γx ) , x ∈ G. If E is a space of functions or of distributions on G , then we denote by E Γ the space of elementsin E which are Γ-periodic. Let us recall that G is a smooth manifold which is identified with R n via the exponential mapping and polynomial coordinate systems. This leads to a correspondingLebesgue measure on g and the Haar measure dx on the group G , hence L p ( G ) ∼ = L p ( R n ). Thisalso allows us [7, p.16] to define the spaces D ( G ) ∼ = D ( R n ) and S ( G ) ∼ = S ( R n )of test functions which are smooth and compactly supported or Schwartz, and the correspondingspaces of distributions D ′ ( G ) ∼ = D ′ ( R n ) and S ′ ( G ) ∼ = S ′ ( R n ) . Note that this identification with R n does not usually extend to the convolution: the group convo-lution, i.e. the operation between two functions on G defined formally via( f ∗ f )( x ) := Z G f ( y ) f ( y − x ) dy, s not commutative in general whereas it is a commutative operation for functions on the abeliangroup R n .We also define the set of functions C ∞ b ( G ) := (cid:26) f ∈ C ∞ ( G ) : sup G | Y α f | < ∞ for every α ∈ N n (cid:27) , for some basis Y , . . . , Y n of g identified with left-invariant vector fields and Y α = Y α Y α · · · Y α n n , α ∈ N n . We check readily that the vector space C ∞ b ( G ) and its natural topology are independent of a choiceof basis Y , . . . , Y n and that C ∞ ( G ) Γ = C ∞ b ( G ) Γ . Furthermore, we have:
Lemma 2.5.
The periodisation of a Schwartz function φ ∈ S ( G ) is a well-defined function φ Γ in C ∞ b ( G ) Γ . Furthermore, the map φ φ Γ yields a surjective morphism of topological vector spacesfrom S ( G ) onto C ∞ b ( G ) Γ and from D ( G ) onto C ∞ b ( G ) Γ .Proof of Lemma 2.5. We first need to set some notation. By Theorem 2.1, we may assume thatΓ = Exp( Z Y ) . . . Exp( Z Y n ) for some strong Malcev basis Y , . . . , Y n of g . As strong Malcev basesyields polynomial coordinates, we may identify G with R n via the exponential mapping: y =( y , . . . , y n ) Exp( y Y + . . . + y n Y n ). We fix a Euclidean norm | · | on R n ∼ G . Note that | y − | = | − y | = | y | and that the Baker-Campbell-Hausdorff formula implies the following modifiedtriangle inequality:(2.2) ∃ C > ∀ a, b ∈ G | ab | ≤ C (1 + | a | ) s (1 + | b | ) s , where s is the step of G .We first show that the periodisation of a Schwartz function φ ∈ D ( G ) makes sense as a functionon G . As φ is Schwartz, for all N ∈ N there exists C = C φ,N such that ∀ x ∈ G | φ ( x ) | ≤ C (1 + | x | ) − N , so by (2.2), ∀ x ∈ G, γ ∈ Γ | φ ( γx ) | ≤ C (1 + | γx | ) − N ≤ CC N (1 + | x | ) N (1 + | γ | ) − N/n . The sum P γ ∈ Γ (1+ | γ | ) − N/s is finite by Corollary 2.3 for N large enough (it suffices to have N > ns ).Hence the function φ Γ is well defined on G . Furthermore, it is now a routine exercise to check that φ Γ ∈ C ∞ b ( G ) Γ and that φ φ Γ is a morphism of topological vector spaces between S ( G ) to C ∞ b ( G ) Γ , and also from D ( G ) to C ∞ b ( G ) Γ .It remains to show the surjectivity of D ( G ) ∋ φ φ Γ ∈ C ∞ b ( G ) Γ . We observe(2.3) ∀ φ ∈ C ∞ b ( G ) Γ , ∀ ψ ∈ D ( G ) φ ψ ∈ D ( G ) and ( φψ ) Γ = φ ψ Γ . We fix ψ ∈ D ( G ) valued in [0 ,
1] and such that ψ = 1 on a fundamental domain of M , for instancethe fundamental domain D from Proposition 2.4 to fix the ideas. We observe that ψ Γ0 ∈ C ∞ b ( G ) Γ is bounded above but also below by 1, and furthermore that 1 /ψ Γ0 is also in C ∞ b ( G ) Γ . Given any φ ∈ C ∞ b ( G ) Γ , applying (2.3) to φ and ψ = ψ /ψ Γ0 shows the surjectivity of D ( G ) ∋ φ φ Γ ∈ C ∞ b ( G ) Γ and concludes the proof of Lemma 2.5. (cid:3) .4. Spaces of periodic functions.
We now examine E Γ for some spaces of functions E on thenilpotent Lie group G , where Γ is a discrete co-compact subgroup of a nilpotent Lie group G .Although D ( G ) Γ = { } = S ( G ) Γ , many other E Γ are isomorphic to important spaces of functions (or distributions) on M as thefollowing lemmata suggest.The definition of F G and f M extend to measurable functions and we have: Lemma 2.6.
For every p ∈ [1 , ∞ ] , the map F F G is an isomorphism of the topological vectorspaces (in fact Banach spaces) from L p ( M ) onto L ploc ( G ) Γ with inverse f f M . The proof follows readily from the description of fundamental domains above, see Proposition2.4. It is left to the reader.We also check readily:
Lemma 2.7.
The mapping F F G is an isomorphsim of topological vector spaces from D ( M ) onto C ∞ b ( G ) Γ with inverse f f M . Consequently, D ′ ( M ) is isomorphic to the dual of C ∞ b ( G ) Γ . This allows for a first distributionalmeaning to F F G with F G in the continuous dual of ( C ∞ b ( G )) Γ and extended to C ∞ b ( G ) by Hahn-Banach’s theorem. However, we prefer to extend the definition of F G to the case of distributionsin the following way: if F ∈ D ′ ( M ), then F G given by ∀ φ ∈ S ( G ) h F G , φ i = h F, ( φ Γ ) M i . is a tempered distribution by Lemmata 2.7 and 2.5. One checks easily that it is periodic and thatit coincides with any other definition given above, for instance on ∪ p ∈ [1 , ∞ ) L p ( M ). Furthermore, wehave: Lemma 2.8.
We have D ′ ( G ) Γ = S ′ ( G ) Γ , and the map F F G yields an isomorphism of topologicalvector spaces from D ′ ( M ) onto S ′ ( G ) Γ .Proof. Lemmata 2.5 and 2.7 imply easily that F F G is a morphism of topological vector spacesfrom D ′ ( M ) to S ′ ( G ) Γ . We can construct its inverse using the function ψ ∈ D ( G ) from the proofof Lemma 2.5. If f ∈ D ′ ( G ) Γ then we define ∀ ψ ∈ D ( M ) h f M , ψ i := (cid:28) f, ψ G ψ ψ Γ0 (cid:29) . Lemma 2.7 implies easily that f M is a distribution on M and that f f M is a morphism oftopological vector spaces from D ′ ( G ) Γ to D ′ ( M ). Furthermore, it gives the inverse of F F G since we have first from the definitions of these two mappings: ∀ φ ∈ D ( G ) h ( f M ) G , φ i = (cid:10) f M , ( φ Γ ) M (cid:11) = (cid:28) f, φ Γ ψ ψ Γ0 (cid:29) = h f, φ i , by periodicity of f . Hence f = ( f M ) G ∈ S ′ ( G ) Γ for any f ∈ D ′ ( G ) Γ . The statement follows. (cid:3) One checks easily that the inverse f f M of F F G constructed in the proof above coincideswith any other definition of f f M given above, for instance on ∪ p ∈ [1 , ∞ ) L ploc ( G ). Moreover, forevery p ∈ [1 , ∞ ), since L p ( M ) ⊂ D ′ ( M ) , we have L ploc ( G ) Γ ⊂ S ′ ( G ) Γ by Lemma 2.8. .5. Convolution and periodicity.
We already know that the convolution of a tempered distri-bution with a Schwartz function is smooth on a nilpotent Lie group G . When the distribution isperiodic under the discrete co-compact subgroup of G , we also have the following properties, inparticular a type of Young’s convolution inequality: Lemma 2.9.
Let F ∈ S ′ ( G ) Γ and κ ∈ S ( G ) . Then F ∗ κ ∈ C ∞ b ( G ) Γ . Viewed as a function on M , ( F ∗ κ ) M ( ˙ x ) = Z M F M ( ˙ y ) ( κ ( · − x ) Γ ) M ( ˙ y ) d ˙ y = Z M F M ( ˙ y ) X γ ∈ Γ κ ( y − γx ) d ˙ y. If f ∈ L p ( M ) for p ∈ [1 , + ∞ ] , then f G ∈ S ′ ( G ) Γ and we have k ( f G ∗ κ ) M k L p ( M ) ≤ k f k L p ( M ) k κ k L ( G ) Proof.
We check readily for x ∈ G and γ ∈ Γ F ∗ κ ( γx ) = Z G F ( y ) κ ( y − γx ) = Z G F ( γz ) κ ( z − x ) dz = Z G F ( z ) κ ( z − x ) dz = F ∗ κ ( x ) . The formula on M follows from (2.1).Let f ∈ L p ( ˙ M ) for p ∈ [1 , + ∞ ]. As a consequence of Lemmata 2.6 and 2.8, f G ∈ S ′ ( G ) Γ ∩ L ploc ( G ) Γ . By Lemmata 2.5 and 2.7, for each fixed ˙ x ∈ M , we can set d ˙ x ( ˙ y ) := κ ( · − x ) Γ ( y ) = X γ ∈ Γ κ ( y − γx ) , and this defines a smooth function d ˙ x on M . Furthermore, ˙ x d ˙ x is continuous from M to D ( M ).This function allows us to write the more concise formula( f G ∗ κ ) M ( ˙ x ) = Z M f ( ˙ y ) d ˙ x ( ˙ y ) d ˙ y. The decomposition of the Haar measure in (2.1) and its invariance under translation imply k d ˙ x k L ( M ) ≤ Z M X γ ∈ Γ | κ ( y − γx ) | d ˙ y = Z G | κ ( y − x ) | dy = k κ k L ( G ) , ˙ x ∈ M (fixed) , (2.4) Z M | d ˙ x ( ˙ y ) | d ˙ x ≤ Z M X γ ∈ Γ | κ ( y − γx ) | d ˙ x = Z G | κ ( y − x ) | dx = k κ k L ( G ) , ˙ y ∈ M (fixed) . (2.5)The case p = + ∞ follows from (2.4) since we have | ( f G ∗ κ ) M ( ˙ x ) | ≤ k f k L ∞ ( M ) k d ˙ x k L ( M ) = k f k L ∞ ( M ) k κ k L ( G ) . Let p ∈ [1 , + ∞ ). By Jensen’s inequality, we have for any fixed ˙ x ∈ M | ( f G ∗ κ ) M ( ˙ x ) | p = (cid:12)(cid:12)(cid:12)(cid:12)Z M f ( ˙ y ) d ˙ x ( ˙ y ) d ˙ y (cid:12)(cid:12)(cid:12)(cid:12) p ≤ k d ˙ x k pL ( M ) Z M | f ( ˙ y ) | p | d ˙ x ( ˙ y ) k d ˙ x k L ( M ) d ˙ y so that, now integrating against ˙ x ∈ M and using (2.5), k ( f G ∗ κ ) M k pL p ( M ) ≤ k κ k p − L ( G ) Z M | f ( ˙ y ) | p Z M | d ˙ x ( ˙ y ) | d ˙ x d ˙ y = k κ k pL ( G ) Z M | f ( ˙ y ) | p d ˙ y. The statement follows. (cid:3)
The proofs of Lemmata 2.9 and 2.5 can be modified in a classical manner to obtain: orollary 2.10. Let κ ∈ L ( G ) . We assume that there is an N > large enough ( N = ns + 1 issufficient, where n and s are the dimension and the step of G ) such that ∃ C > ≤ κ ( x ) ≤ C (1 + | x | ) − N for almost every x ∈ G. Then K ( ˙ x, ˙ y ) := X γ ∈ Γ κ ( y − γx ) defines a non-negative integrable function ( ˙ x, ˙ y ) K ( ˙ x, ˙ y ) on M × M satisfying Z M K ( ˙ x, ˙ y ) d ˙ y = k κ k L ( G ) for a.e. ˙ x ∈ M, and Z M K ( ˙ x, ˙ y ) d ˙ x = k κ k L ( G ) for a.e. ˙ y ∈ M. Furthermore, if f ∈ L p ( M ) for p ∈ [1 , + ∞ ) , then Z M (cid:18)Z M | f ( ˙ y ) | K ( ˙ x, ˙ y ) d ˙ y (cid:19) p d ˙ x ≤ k κ k pL ( G ) k f k pL p ( M ) . Operators on M and G . Consider a linear continuous mapping T : S ( G ) → S ′ ( G ) which isinvariant under Γ in the sense that ∀ F ∈ S ( G ) , ∀ γ ∈ Γ , T ( F ( g · )) = ( T F )( g · ) . Then it naturally induces an operator T M on M via T M f = ( T f G ) M . Furthermore, T M : D ( M ) → D ′ ( M ) is a linear continuous mapping by Lemmata 2.7 and 2.8. Notethat if T is invariant under G , then it is invariant under Γ.Invariant differential operators keep many of their feature from G to M : Lemma 2.11.
Let T be a smooth differential operator on G which is invariant under Γ . Then T M is a smooth differential operator on M .If T is hypo-elliptic, then so is T M .If T is symmetric and positive on S ( G ) in the sense that ∀ F , F ∈ S ( G ) Z G T F ( x ) F ( x ) dx = Z G F ( x ) T F ( x ) dx and Z G T F ( x ) F ( x ) dx ≥ , then T M is also symmetric and positive on D ( M ) . Moreover, T is essentially self-adjoint on L ( G ) and T M is essentially self-adjoint on L ( M ) . These arguments are very classical and we only sketch their proofs.
Sketch of the proof.
The manifold M have natural charts given by the descriptions of the funda-mental domains in Proposition 2.4. They imply that T M is a smooth differential operator on M andalso that if T is hypo-elliptic (resp. symmetric and positive), then so is T M . The self-adjointnessfollows from the fact that the positivity imply the density of the ranges of the operators T + i I and T − i I, and T M + i I and T M − i I. (cid:3) . Preliminaries on graded groups and Rockland operators
Graded nilpotent Lie group.
In the rest of the paper, we will be concerned with gradedLie groups. References on this subject includes [16] and [14].A graded Lie group G is a connected and simply connected Lie group whose Lie algebra g admitsan N -gradation g = ⊕ ∞ ℓ =1 g ℓ where the g ℓ , ℓ = 1 , , . . . , are vector subspaces of g , almost all equalto { } , and satisfying [ g ℓ , g ℓ ′ ] ⊂ g ℓ + ℓ ′ for any ℓ, ℓ ′ ∈ N . This implies that the group G is nilpotent.Examples of such groups are the Heisenberg group and, more generally, all stratified groups (whichby definition correspond to the case g generating the full Lie algebra g ).3.1.1. Dilations and homogeneity.
For any r >
0, we define the linear mapping D r : g → g by D r X = r ℓ X for every X ∈ g ℓ , ℓ ∈ N . Then the Lie algebra g is endowed with the family ofdilations { D r , r > } and becomes a homogeneous Lie algebra in the sense of [16]. We re-write theset of integers ℓ ∈ N such that g ℓ = { } into the increasing sequence of positive integers υ , . . . , υ n counted with multiplicity, the multiplicity of g ℓ being its dimension. In this way, the integers υ , . . . , υ n become the weights of the dilations.We construct a basis X , . . . , X n of g adapted to the gradation, by choosing a basis { X , . . . X n } of g (this basis is possibly reduced to ∅ ), then { X n +1 , . . . , X n + n } a basis of g (possibly { } aswell as the others) We have D r X j = r υ j X j , j = 1 , . . . , n .The associated group dilations are defined by D r ( x ) = rx := ( r υ x , r υ x , . . . , r υ n x n ) , x = ( x , . . . , x n ) ∈ G, r > . In a canonical way, this leads to the notions of homogeneity for functions, distributions and oper-ators and we now give a few important examples.The Haar measure is Q -homogeneous where Q := X ℓ ∈ N ℓ dim g ℓ = υ + . . . + υ n , is called the homogeneous dimension of G .Identifying the element of g with left invariant vector fields, each X j is a υ j -homogeneous differ-ential operator of degree. More generally, the differential operator X α = X α X α · · · X α n n , α ∈ N n is homogeneous with degree [ α ] := α υ + · · · + α n υ n . Approximate identity. On G , we can easily construct approximation of the identity operatorin the following way: if κ ∈ S ( G ), then for each s ∈ (0 , κ s ∈ S ( G ) via κ s ( y ) = s − Q κ ( s − y ). Setting c κ = R G κ ( y ) dy , we have the convergence [14, Section 3.1.10] f ∗ κ s −→ s → c κ f, in S ( G ) for any f ∈ S ( G ) and in S ′ ( G ) for any f ∈ S ′ ( G ). The convergence also takes place in L p ( G ), p ∈ (0 , ∞ ), but we will not use this here. .1.3. Homogeneous quasi-norms.
An important class of homogeneous map are the homogeneneousquasi-norms, that is, a 1-homogeneous non-negative map G ∋ x
7→ k x k which is symmetric anddefinite in the sense that k x − k = k x k and k x k = 0 ⇐⇒ x = 0. In fact, all the homogeneousquasi-norms are equivalent in the sense that if k · k and k · k are two of them, then ∃ C > ∀ x ∈ G C − k x k ≤ k x k ≤ C k x k . Examples may be constructed easily, such as k x k = ( n X j =1 | x j | N/υ j ) − N for any N ∈ N , with the convention above.An important property of homogeneous quasi-norms is that they satisfy the triangle inequalityup to a constant: ∃ C > ∀ x, y ∈ G k xy k ≤ C ( k x k + k y k ) . However, it is possible to construct a homogeneous quasi-norm on G which yields a distance on G ∼ R n in the sense that the triangle inequality above is satisfied with constant 1 [14, Theorem3.1.39].Using polar coordinates [14, Proposition 3.1.42], it is easily seen that for any homogeneousquasi-norm k · k on G , we have for any s > Q (3.1) Z G (1 + k x k ) − s dx < ∞ and Z k y k >ε | y | − s dy ≤ C s,G ε − s + Q , for some finite constant C s,G > Discrete co-compact subgroups of G . If the structural constants c i,j,k from [ X i , X j ] = P k c i,j,k X k are all rational, then then there exists a positive integer K ∈ N such that the setΓ := Exp( K Z X n ) . . . Exp( K Z X ) = Exp( K Z X ) . . . Exp( K Z X n )is a discrete co-compact subgroup of G as a consequence of Theorem 2.1 and the following statement: Lemma 3.1.
Let G be a graded Lie group. The basis constructed in Section 3.1.1 but ordered as X n , . . . , X is a strong Malcev basis of g .Proof of Lemma 3.1. The properties of the dilations implies that the Lie bracket of an element X ∈ g with X m is in the linear span of X j ’s with weights > m , hence in R X m +1 ⊕ . . . ⊕ R X n . (cid:3) Remark . • In what follows, we do not require that Γ is constructed out of a basis con-structed in Section 3.1.1. • Although we will not need this property, we observe that the same proof yields the fact that X n , . . . , X is a strong Malcev basis of g through the sequence of ideals g (1) ⊂ g (2) ⊂ . . . ⊂ g ( k ) = g defined as follows. Re-labeling the weights as a sequence of strictly decreasingintegers { ≤ υ ≤ . . . ≤ υ n } = { υ n = ω > . . . > ω k = υ } , g ( j ) denotes the vector space spanned by the elements in g with weights ≥ ω j for each j = 1 , . . . , k .We will need the following property which is similar to Corollary 2.3 but for homogeneous quasi-norms: emma 3.3. Let Γ be a discrete co-compact subgroup of a graded group G . Then for any N > υ n n and any homogeneous quasi-norm k · k on G , the sum P γ ∈ Γ \{ } k γ k − N finite.Proof. As all the homogeneous quasi-norms are equivalent, it suffices to prove the result for theone given by k x k = max j =1 ,...,n | x j | /υ j , where we have written x = Exp P nj =1 x j X j , for the basis X , . . . , X n adapted to the gradation and constructed in Section 3.1.1. We observe that it sufficesto show that the sum over γ ∈ Γ with k γ k ≥ x ∈ G k x k ≥ ⇒ k x k ≥ ( max j =1 ,...,n | x j | ) /υ n . Now x max j =1 ,...,n | x j | is a norm on R n , and we can conclude with Corollary 2.3. (cid:3) A consequence of the analysis in Section 2 is the existence of approximate identity:
Corollary 3.4.
Let Γ be a discrete co-compact subgroup of a graded group G . Let κ ∈ S ( G ) , thenfor each s ∈ (0 , , we define κ s ∈ S ( G ) via κ s ( y ) = s − Q κ ( s − y ) and set c κ = R G κ ( y ) dy . Then forany f ∈ D ′ ( M ) , we have the convergence in D ′ ( M ) : ( f G ∗ κ s ) M −→ s → c κ f, Proof of Corollary 3.4.
This is a consequence of the convergence of the approximate identity in S ′ ( G ), see Section 3.1.2, and Lemma 2.8. (cid:3) The dual of G and the Plancherel theorem. In this paper, we always assume that therepresentations of the group G are strongly continuous and acting on separable Hilbert spaces.Unless otherwise stated, the representations of G will also be assumed unitary. For a representation π of G , we keep the same notation for the corresponding infinitesimal representation which actson the universal enveloping algebra U ( g ) of the Lie algebra of the group. It is characterised by itsaction on g :(3.2) π ( X ) = ∂ t =0 π ( e tX ) , X ∈ g . The infinitesimal action acts on the space H ∞ π of smooth vectors, that is, the space of vectors v ∈ H π such that the mapping G ∋ x π ( x ) v ∈ H π is smooth.We will use the following equivalent notations for the group Fourier transform of a function f ∈ L ( G ) at π π ( f ) ≡ b f ( π ) ≡ F G ( f )( π ) = Z G f ( x ) π ( x ) ∗ dx. We denote by b G the unitary dual of G , that is, the unitary irreducible representations of G modulo equivalence and identify a unitary irreducible representation with its class in b G . The set b G is naturally equipped with a structure of standard Borel space. The Plancherel measure is theunique positive Borel measure µ on b G such that for any f ∈ C c ( G ), we have:(3.3) Z G | f ( x ) | dx = Z b G kF G ( f )( π ) k HS ( H π ) dµ ( π ) . Here k · k HS ( H π ) denotes the Hilbert-Schmidt norm on H π . This implies that the group Fouriertransform extends unitarily from L ( G ) ∩ L ( G ) to L ( G ) onto L ( b G ) := Z b G H π ⊗ H ∗ π dµ ( π ) , which we identify with the space of µ -square integrable fields on b G . Consequently (3.3) holds forany f ∈ L ( G ); this formula is called the Plancherel formula. It is possible to give an expression or the Plancherel measure µ , see [7, Section 4.3], although we will not need this in this paper. Wededuce the inversion formula: for any κ ∈ S ( G ), we have [14, Proposition 5.1.15](3.4) Z b G Tr |F κ ( π ) | dµ ( π ) < ∞ , ∀ x ∈ G Z b G Tr( π ( x ) F G κ ( π )) dµ ( π ) = κ ( x ) . We also recall that b G inherits a dilation from the one on G , see [11, Section 2.2]. We denote by r · π the element of b G obtained from π through dilatation by r , that is, r · π ( x ) = π ( rx ), r > x ∈ G .3.4. Positive Rockland operators on G . Let us briefly review the definition and main propertiesof positive Rockland operators. References on this subject includes [16] and [14].3.4.1.
Definitions. A Rockland operator R on G is a left-invariant differential operator which ishomogeneous of positive degree and satisfies the Rockland condition, that is, for each unitaryirreducible representation π on G , except for the trivial representation, the operator π ( R ) is injectiveon the space H ∞ π of smooth vectors of the infinitesimal representation.Recall [19] that Rockland operators are hypoelliptic. In fact, they may equivalently. be charac-terised as the left-invariant differential operators which are hypoelliptic. If this is the case, then R + P [ α ] <ν c α X α where c α ∈ C and ν is the homogeneous degree of ν is hypoelliptic.A Rockland operator is positive when ∀ f ∈ S ( G ) , Z G R f ( x ) f ( x ) dx ≥ . Any sub-Laplacian with the sign convention − ( Z + . . . + Z n ′ ) of a stratified Lie group is a positiveRockland operator; here Z , . . . , Z n ′ form a basis of the first stratum g . The reader familiar withthe Carnot group setting may view positive Rockland operators as generalisations of the naturalsub-Laplacians. Positive Rockland operators are easily constructed on any graded Lie group [14,Section 4.2].A positive Rockland operator is essentially self-adjoint on L ( G ) and we keep the same notationfor their self-adjoint extension. Its spectrum is included in [0 , + ∞ ) and the point 0 may be neglectedin its spectrum, see [10, Lemma 2.13].For each unitary irreducible representation π of G , the operator π ( R ) is essentially self-adjointon H ∞ π and we keep the same notation for this self-adjoint extension. Its spectrum is a discretesubset of (0 , ∞ ) if π = 1 b G is not trivial while π ( R ) = 0 if π = 1 R . We denote by E π its spectralmeasure of π ( R ) = R R λdE π ( λ ).3.4.2. Spectral multipliers in R and in b R . If ψ : R + → C is a measurable function, the spectralmultiplier ψ ( R ) is well defined as a possibly unbounded operator on L ( G ). Its domain is thespace of function ψ ∈ L ( G ) such that R + ∞ | ψ ( λ ) | d ( E λ ψ, ψ ) is finite where E denotes the spectralmeasure of R = R + ∞ λdE λ . For instance, if ψ is bounded, then ψ ( R ) is bounded on L ( G ).However, we will be concerned with multipliers ψ that may not be in L ∞ ( R ). If the domain of ψ ( R ) contains S ( G ) and defines a continuous map S ( G ) → S ′ ( G ), then it is invariant under right-translation and, by the Schwartz kernel theorem, admits a right-convolution kernel ψ ( R ) δ ∈ S ′ ( G )which satisfies the following homogeneity property:(3.5) ψ ( r ν R ) δ ( x ) = r − Q ψ ( R ) δ ( r − x ) , x ∈ G. urthermore, for each unitary irreducible representation π of G , the domain of the operator ψ ( π ( R )) = R R ψ ( λ ) dE π ( λ ) contains H ∞ π and we have F G { ψ ( R ) ϕ } ( π ) = ψ ( π ( R )) b ϕ ( π ) , φ ∈ S ( G ) . The following statement is the famous result due to Hulanicki [20]:
Theorem 3.5 (Hulanicki’s theorem) . Let R be a positive Rockland operator on G . If ψ ∈ S ( R ) then ψ ( R ) δ ∈ S ( G ) . For instance, the heat kernels p t := e − t R δ , t > , are Schwartz - although this property is in fact used in the proof of Hulanicki’s Theorem.The following result was mainly obtained by Christ for sub-Laplacians on stratified groups [5,Proposition 3]. As explained below, the proof extends to positive Rockland operators: Theorem 3.6.
Let R be a positive Rockland operator of homogeneous degree ν on G . If themeasurable function ψ : R + → C is in L ( R + , λ Q/ν dλ/λ ) , then ψ ( R ) defines a continuous map S ( G ) → S ′ ( G ) whose convolution kernel ψ ( R ) δ is in L ( G ) . Moreover, we have k ψ ( R ) δ k L ( G ) = c Z ∞ | ψ ( λ ) | λ Qν dλλ , where c = c ( R ) is a positive constant of R and G . In other words, the map ψ ψ ( R ) δ is an isometry from L ((0 , ∞ ) , c λ Q/ dλ/λ ) onto its imagewhich is a closed subspace of L ( G ). Remark . • By plugging the function ψ ( λ ) = e − λ/ in the formula of Theorem 3.6 andnoticing k p / k L = p (0), we obtain the following expression for the constant in the state-ment in terms of the heat kernel p t of R :(3.6) c = c ( R ) = p (0)Γ( Q/ν ) , where Γ( Q/ν ) = Z ∞ e − λ λ Q/ν dλλ . • Using the Plancherel formula (3.3), we have an expression for k ψ ( R ) δ k L ( G ) = Z b G k ψ ( b R ( π )) k HS dµ ( π ) , in terms of the Plancherel measure. Taking for instance ψ = 1 [0 , easily leads c = Qν Z b G Tr (cid:16) [0 , ( b R ( π )) (cid:17) HS dµ ( π ) . More generally, taking ψ = 1 [ a,b ] where 0 ≤ a < b , we have(3.7) c νQ ( b Qν − a Qν ) = Z b G Tr (cid:16) [ a,b ] ( b L ( π )) (cid:17) d ˙ xdµ ( π ) . • We can check Theorem 3.6 in the familiar setting of the canonical Laplacian ∆ R n = − P j ∂ j on the abelian group G = R n . Using the Euclidean Fourier transform anda change in polar coordinates, we readily obtain c (∆ R n ) = (Γ( n/ n π n/ ) − which isindeed equal to p (0) / Γ( n/
2) since the heat kernels in this setting are the Gaussians p t ( x ) = (4 πt ) − n/ e −| x | / t . We can also check Theorem 3.6 and the formula in (3.6) in the case of the Heisenberg group.Realising the Heisenberg group H n as R n +1 = R n × R n × R with the group law( x, y, t )( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ + 12 ( x · y ′ − y · x ′ )) , the canonical sub-Laplacian L H n is well understood, see e.g. [26] or [14]. In particular, therelations between the Fourier transform of H n and the functional calculus of L H n yield k ψ ( L H n ) δ k L ( H n ) = (2 π ) − (3 n +1) Z + ∞−∞ ∞ X a =0 ( n + a − n − a ! | ψ ( | λ | (2 a + n )) | | λ | n dλ = c ( L H n ) Z ∞ | ψ ( λ ) | λ Q/ dλλ , with Q = 2 n + 2, so c ( L H n ) := (2 π ) − (3 n +1) ∞ X a =0 ( n + a − n − a ! (2 a + n ) − n − . • If R is a positive Rockland operator, then any positive powers R ℓ is also a positive Rocklandoperator, and a simple change of variable on (0 , + ∞ ) implies(3.8) c ( R ℓ ) = 1 ℓ c ( R ) for any ℓ ∈ N . Sketch of the proof of Theorem 3.6.
Since the heat kernels are Schwartz, we can argue as in theproof of [5, Proposition 3]: for any b >
0, the kernel φ := 1 (0 ,b ] ( R ) δ is in L ( G ) , and, for every ψ ∈ L ∞ (0 , b ], the kernel ψ ( R ) δ is in L ( G ) with k ψ ( R ) δ k L ( G ) = Z ∞ | ψ ( λ ) | d ( E λ φ , φ ) . This implies the existence and uniqueness of a sigma-finite Borel measure m on R + satisfying k ψ ( R ) δ k L ( G ) = Z ∞ | ψ ( λ ) | dm ( λ ) , for every ψ ∈ L ∞ ( R + ) with compact support. From the uniqueness and the homogeneity propertyin (3.5), it follows that the measure m is homogeneous of degree Q/ν on R + . This means that λ − Q/ν dm ( λ ) is a Haar measure for the multiplicative group R + , and is therefore a constant multipleof dλ/λ . This shows the theorem for any ψ ∈ L ∞ ( R + ) with compact support.Let us now prove the theorem for ψ in L ( R + , λ Q/ν dλ/λ ). The first problem is to ensure that ψ ( R ) δ makes sense. For this, let ( ψ j ) j ∈ N be a sequence of bounded compactly supported functionsthat converges to ψ in L ( R + , λ Q/ν dλ/λ ) as j → ∞ . As the statement is shown for the ψ j ’s,the kernels ψ j ( R ) δ , j ∈ N , form a Cauchy sequence in the Hilbert space L ( G ); we denote by κ ∈ L ( G ) its limit. We observe that for each φ ∈ S ( G ), Fatou’s inequality yields Z ∞ | ψ ( λ ) | d ( E λ φ, φ ) ≤ lim inf j →∞ Z ∞ | ψ j ( λ ) | d ( E λ φ, φ ) ≤ c − k φ k L ( G ) k κ k L ( G ) , since we have c Z ∞ | ψ j ( λ ) | d ( E λ φ, φ ) = k ψ j ( R ) φ k L ( G ) = k φ ∗ ψ j ( R ) δ k L ( G ) ≤ k φ k L ( G ) k ψ j ( R ) δ k L ( G ) , y the Young convolution inequality. This shows that ψ ( R ) defines a continuous map S ( G ) → S ′ ( G )thus admits a right-convolution kernel ψ ( R ) δ ∈ S ′ ( G ) in the sense of distributions. It remains toshow κ = ψ ( R ) δ . For this we observe that we have for any χ ∈ D ( R ) and any φ ∈ S ( G ): k ( ψ ( R ) − ψ j ( R )) χ ( R ) φ k L ( G ) ≤ k ( ψ − ψ j ) χ ( R ) δ k L ( G ) k φ k L ( G ) , and, since the statement is proved for functions with compact supports, k ( ψ − ψ j ) χ ( R ) δ k L ( G ) = c Z ∞ | ( ψ − ψ j ) χ ( λ ) | λ Qν dλλ ≤ c k χ k L ∞ ( R ) k ψ − ψ j k L ( R + ,λ Q/ν dλ/λ ) . The last expression converges to 0 as j → ∞ by hypothesis. Hencelim j →∞ k ( ψ ( R ) − ψ j ( R )) χ ( R ) φ k L ( G ) = 0 = k ( χ ( R ) φ ) ∗ ( ψ ( R ) δ − κ ) k L ( G ) . This implies that h χ ( R ) φ, ψ ( R ) δ − κ i = 0 and thus ψ ( R ) δ − κ = 0 as tempered distributionsbecause χ ( r R ) δ is an approximate identity (see Section 3.1.2 and Hulanicki’s theorem in Theorem3.5 with (3.5)). This concludes the sketch of the proof of Theorem 3.6. (cid:3) Semiclassical calculus on graded compact nil-manifold
Semiclassical pseudodifferential operators.
The semiclassical pseudodifferential calculusin the context of groups of Heisenberg type was presented in [11, 12], but in fact extends readilyto any graded group G . Here, we show how to define it on the quotient manifold M .We consider the class of symbols A of fields of operators defined on M × b Gσ ( ˙ x, π ) ∈ L ( H π ) , ( ˙ x, π ) ∈ M × b G, that are of the form σ ( ˙ x, π ) = F G κ ˙ x ( π ) , where ˙ x κ ˙ x is a smooth map from M to S ( G ). Equivalently, the map ˙ x κ ˙ x in C ∞ ( M : S ( G ))may also be viewed as the x -periodic map x κ ˙ x in C ∞ ( G : S ( G )) Γ , and the symbol σ = { σ ( ˙ x, π ) : ( ˙ x, π ) ∈ M × b G } as a field of operators on G × b G which is Γ-periodicin x ∈ G . The group Fourier transform yields a bijection ( ˙ x κ ˙ x ) ( ˙ x σ ( ˙ x, · ) = F ( κ ˙ x ))from C ∞ ( M : S ( G )) onto A . We equip A of the Fr´echet topology so that this mapping is anisomorphism of topological vector spaces.We observe that A is an algebra for the usual composition of symbol. Furthermore, it is alsoequip with the involution σ σ ∗ , where σ ∗ = { σ ( ˙ x, π ) ∗ , ( ˙ x, π ) ∈ M × b G } . We say that a symbol σ ∈ A is self-adjoint if σ ∗ = σ . If σ ∈ A , we then define the symbols ℜ σ := 12 ( σ + σ ∗ ) and ℑ σ := 12 i ( σ − σ ∗ ) , so that σ = ℜ σ + i ℑ σ , and both ℜ σ and ℑ σ are self-adjoint and in A .Note that by the Fourier inversion formula (3.4), we have κ ˙ x ( z ) = Z b G Tr( π ( z ) σ ( ˙ x, π )) dµ ( π ) = Z b G Tr( π ( z ) σ G ( x, π )) dµ ( π ) . We then define the operator Op G ( σ ) at F ∈ S ′ ( G ) viaOp G ( σ ) F ( x ) := F ∗ κ ˙ x ( x ) , x ∈ G. his makes sense since, for each x ∈ G , the convolution of the tempered distribution F with theSchwartz function κ ˙ x yields a smooth function F ∗ κ ˙ x on G . Because of the Fourier inversion formula(3.4), it may be written formally asOp G ( σ ) F ( x ) = Z G × b G Tr( π ( y − x ) σ G ( x, ε · π )) F ( y ) dydµ ( π ) . If F is periodic, then Op G ( σ ) F is also periodic with Op G ( σ ) F ∈ C ∞ ( G ) Γ and we can view F and Op G ( σ ) F as functions on M , see Section 2.4. In other words, we set for any f ∈ D ′ ( M ) and˙ x ∈ M : Op( σ ) f ( ˙ x ) := Op G ( σ ) f G ( x ) = ( f G ∗ κ ˙ x ) M ( ˙ x ) = Z M f ( ˙ y ) X γ ∈ Γ κ ˙ x ( y − γx ) d ˙ y, and this defines the function Op( σ ) f ∈ D ( M ). We say that κ ˙ x is the kernel associated with thesymbol σ or Op( σ ).The results in Section 2.4, especially Lemma 2.9, yield: Lemma 4.1.
Let σ ∈ A and let κ ˙ x be its associated kernel. Then Op( σ ) maps D ′ ( M ) to D ( M ) continuously, and its Schwartz integral is the smooth function K on M × M given by K ( ˙ x, ˙ y ) = X γ ∈ Γ κ ˙ x ( y − γx ) . Consequently, the operator
Op( σ ) is Hilbert-Schmidt on L ( M ) with Hilbert-Schmidt norm k Op( σ ) k HS = k K k L ( M × M ) . Let ε ∈ (0 ,
1] be a small parameter. For every symbol σ ∈ A , we consider the symbol(4.1) σ ( ε ) := { σ ( ˙ x, επ ) : ( ˙ x, π ) ∈ M × b G } , whose associated kernel is then(4.2) κ ( ε )˙ x ( z ) := ε − Q κ ˙ x ( ε − · z ) , z ∈ G, if κ ˙ x = κ (1)˙ x is the kernel associated with the symbol σ = σ (1) . The semi-classical pseudo-differentialcalculus is then defined viaOp ε ( σ ) := Op( σ ( ε ) ) and Op εG ( σ ) := Op G ( σ ( ε ) ) . Properties of the semiclassical calculus.
The first two results in this section will justifythe introduction of the following semi-norm on A : k σ k A := Z G sup ˙ x ∈ M | κ ˙ x ( y ) | dy, where κ ˙ x is the kernel associated with σ ∈ A .4.2.1. Boundedness in L ( M ) . Proposition 4.2.
For every ε ∈ (0 , and σ ∈ A , k Op( σ ) k L ( L ( M )) ≤ k σ k A and k σ k A = k σ ( ε ) k A where σ ( ε ) is given in (4.1) . Consequently, we have: k Op ε ( σ ) k L ( L ( M )) ≤ k σ k A . roof. We observe that for f ∈ L ( M ), | Op( σ ) f ( ˙ x ) | = (cid:12)(cid:12) Z M f ( ˙ y ) X γ ∈ Γ κ ˙ x ( y − γx ) d ˙ y (cid:12)(cid:12) ≤ Z M | f ( ˙ y ) | X γ ∈ Γ sup ˙ x ∈ M | κ ˙ x ( y − γx ) | d ˙ y. Furthermore, we can apply Corollary 2.10 to the function defined on G by sup ˙ x ∈ M | κ ˙ x | since˙ x κ ˙ x is a smooth function from G to S ( G ). We obtain k Op( σ ) f k L ( M ) ≤ k f k L ( M ) k sup ˙ x ∈ M | κ ˙ x |k L ( G ) . This implies the first inequality. For the equality, a simple change of variable y = ε − · z yields k σ ( ε ) k A = k sup ˙ x ∈ M | κ ( ε )˙ x |k L ( G ) = Z G sup ˙ x ∈ M | κ ˙ x | ( ε − · z ) ε − Q dz = Z G sup ˙ x ∈ M | κ ˙ x ( y ) | dy = k σ k A . The rest follows. (cid:3)
Singularity of the operators as ε → . The following lemma is similar to Proposition 3.4 in[10] and shows that the singularities of the integral kernels of the operators Op ( ε ) ( σ ) concentrateon the diagonal as ε →
0. It may also justify for many semi-classical properties that the kernelassociated with a symbol may be assumed to be compactly supported in the variable of the group:
Lemma 4.3.
Let χ ∈ D ( G ) be identically equal to close to . Let σ ∈ A and let κ ˙ x ( z ) denoteits associated kernel. For every ε > , the symbol σ ε defined via σ ε ( ˙ x, π ) = F G ( κ ˙ x χ ( ε · )) , that is, the symbol with associated kernel κ ˙ x ( z ) χ ( ε · z ) , is in A . Furthermore, for all N ∈ N , thereexists a constant C = C N,σ,χ > such that ∀ ε ∈ (0 , k σ ε − σ k A ≤ Cε N . Proof.
As the function χ is identically 1 close to z = 0, for all N ∈ N , there exists a boundedcontinuous function θ , identically 0 near 0, such that ∀ y ∈ G, − χ ( y ) = θ ( y ) k y k N , where k · k is a fixed homogeneous quasi-norm on G (see Section 3.1.1). This notation implies κ ˙ x ( z ) − κ ˙ x ( z ) χ ( ε · z ) = κ ˙ x ( z ) θ ( ε · z ) k ε · z k N . As k ε · z k = ε k z k , we obtain k σ ε − σ k A = Z G sup ˙ x | κ ˙ x ( z ) − κ ˙ x ( z ) χ ( ε · z ) | dz ≤ ε N k θ k L ∞ Z G sup ˙ x ∈ M | κ ˙ x ( z ) |k z k N dz. This last integral is finite and this concludes the proof. (cid:3)
Symbolic calculus.
In order to write down the symbolic properties of the semi-classical calcu-lus, we need to introduce the notions of difference operators. They aim at replacing the derivativeswith respect to the Fourier variable in the Euclidean case.If q is a smooth function on G with polynomial growth, we define ∆ q via∆ q b f ( π ) = F G ( qf )( π ) , π ∈ b G, or any function f ∈ S ( G ) and even any tempered distribution f ∈ S ′ ( G ). In particular, consideringthe basis ( X j ) constructed in Section 3.1.1, we consider [14, Proposition 5.2.3] the polynomials q α such that X β q α = δ α,β for all α, β ∈ N n . We then define∆ α := ∆ q α ( · − ) . We can now describe the symbolic calculus. The quantitative results on the symbolic calculusfor the pseudo-differential calculus on G in [14, Section 5.5] imply the following statement: Proposition 4.4. (1) If σ , σ ∈ A , then for every N ∈ N , we have Op ( ε ) ( σ )Op ( ε ) ( σ ) = X [ α ] Lemma 4.6. Let σ ∈ A with associated kernel κ ˙ x ( z ) . For every ε ∈ (0 , , Op ( ε ) ( σ ) is a Hilbert-Schmidt and trace-class operator on M , with Hilbert-Schmidt norm k Op ( ε ) ( σ ) k HS satisfying forevery N ∈ N k Op ( ε ) ( σ ) k HS = ε − Q Z Z G × M | κ ˙ x ( z ) | dzd ˙ x + O ( ε ) N , and trace satisfying for every N ∈ N Tr (cid:16) Op ( ε ) ( σ ) (cid:17) = ε − Q Z M κ ˙ x (0) d ˙ x + O ( ε ) N . This means that there exists a constant C = C N,σ,G, Γ > ε ∈ (0 , (cid:12)(cid:12)(cid:12)(cid:12) k Op ( ε ) ( σ ) k HS − ε − Q Z Z G × M | κ ˙ x ( z ) | dzd ˙ x (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε N and (cid:12)(cid:12)(cid:12)(cid:12) Tr (cid:16) Op ( ε ) ( σ ) (cid:17) − ε − Q Z M κ ˙ x (0) d ˙ x (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε N . roof of Lemma 4.6. Lemma 4.1 implies that Op ( ε ) ( σ ) is Hilbert-Schmidt and trace-class, withHilbert-Schmidt norm and trace that can be expressed in terms of their smooth kernels.First, let us consider the Hilbert-Schmidt norm for ε = 1. By Lemma 4.1, we have k Op( σ ) k HS = Z Z M × M (cid:12)(cid:12) X γ ∈ Γ κ ˙ x ( y − γx ) (cid:12)(cid:12) d ˙ yd ˙ x = Z Z M × M X γ ,γ ∈ Γ κ ˙ x (( γ y − ) x ) κ ˙ x (( γ y − ) γ x ) d ˙ yd ˙ x = Z Z G × M X γ ∈ Γ κ ˙ x ( y − x ) κ ˙ x ( y − γ x ) dyd ˙ x, by (2.1). Using the change of variable z = y − x , we have obtained k Op( σ ) k HS = Z Z G × M X γ ∈ Γ κ ˙ x ( z ) κ ˙ x ( zx − γ x ) dzd ˙ x = X γ ∈ Γ Z Z G × R κ ΓΘ( t ) ( z ) κ ΓΘ( t ) ( z Θ( t ) − γ Θ( t )) dzdt having used the fundamental domain R described in Proposition 2.4. Note that the term corre-sponding to γ = 0 in this last sum is equal to RR G × M | κ ˙ x ( z ) | dzd ˙ x .For every ε ∈ (0 , k Op ( ε ) ( σ ) k HS = X γ ∈ Γ ρ ε,γ , where ρ ε,γ := Z Z G × R κ ( ε )ΓΘ( t ) ( z ) κ ( ε )ΓΘ( t ) ( z Θ( t ) − γ Θ( t )) dzdt = ε − Q Z Z G × R κ ΓΘ( t ) ( z ) κ ΓΘ( t ) ( z ε − · (Θ( t ) − γ Θ( t ))) dzdt, after the change of variable z ε · z . For γ = 0, we see ρ ε, = Z Z G × M | κ ( ε )˙ x ( z ) | dzd ˙ x = ε − Q Z Z G × M | κ ˙ x ( z ) | dzd ˙ x. We now consider γ ∈ Γ \ { } . We fix a homogenous quasi-norm k · k on G (see Section 3.1.1).In order to avoid introducing unnecessary constants, we may assume that it yields a distance on G ∼ R n . By assumption on the kernel κ ˙ x ( z ) associated with σ , we have ∀ N ∈ N ∃ C N ∀ ˙ x ∈ M, z ∈ G | κ ˙ x ( z ) | ≤ C N (1 + k z k ) − N . These estimates imply for any N , N ∈ N : | ρ ε,γ | ≤ C N C N Z Z G × R (1 + k z k ) − N (1 + k z ε − · (Θ( t ) − γ Θ( t )) k ) − N dzdt ≤ C N C N Z G (1 + k z k ) − N + N dz Z R (1 + ε − k Θ( t ) − γ Θ( t ) k ) − N dt. Let N ∈ N be fixed. We choose N = N + Q + 1 so that the last integral over G is finite, see (3.1).For the integral over R , we observe that the function t 7→ k Θ( t ) − γ Θ( t ) k is continuous from R n o [0 , + ∞ ) and never vanishes by Proposition 2.4 and the properties of the quasi-norms; let c γ > Z R (1 + ε − k Θ( t ) − γ Θ( t ) k ) − N dt ≤ (1 + c γ ε − ) − N ≤ c − Nγ ε N . We will use this for the finite number of γ ∈ Γ \ { } such that k γ k ≤ t ∈ ¯ R k Θ( t ) k . For theothers, the triangle inequality and k γ k > t ∈ ¯ R k Θ( t ) k imply that k Θ( t ) − γ Θ( t ) k ≥ k γ k / 2, so Z R (1 + ε − k Θ( t ) − γ Θ( t ) k ) − N dt ≤ (1 + ε − k γ k / − N ≤ N ε N k γ k − N Summing over γ ∈ Γ \ { } , we obtain the estimate X γ ∈ Γ \{ } | ρ ε,γ | ≤ ǫ N X < k γ k≤ t ∈ ¯ R k Θ( t ) k c − Nγ + 2 N X k γ k > k γ k − N . By Lemma 3.3, the very last sum is finite for N large, N > nν n being sufficient. Hence theright-hand side above is . ε N , and this shows the estimate for the Hilbert-Schmidt norm in thestatement.The proof for the trace is similar. Indeed, we haveTr(Op ( ε ) ( σ )) = Z M X γ ∈ Γ κ ( ε )˙ x ( x − γx ) d ˙ x = Z M κ ( ε )˙ x (0) d ˙ x + ρ ε , where | ρ ε | ≤ X γ ∈ Γ \{ } Z R (cid:12)(cid:12)(cid:12) κ ( ε )ΓΘ( t ) ( z Θ( t ) − γ Θ( t )) (cid:12)(cid:12)(cid:12) dt. Proceeding as above, the right-hand side is O ( ε N ) for any N ∈ N . This concludes the proof ofLemma 4.6. (cid:3) The Hilbert space L ( M × b G ) . We open a brief parenthesis devoted to the tensor product ofthe Hilbert spaces L ( M ) and L ( b G ) (see Section 3.3 for the latter): L ( M × b G ) := L ( M ) ⊗ L ( b G ) . We may identify L ( M × b G ) with the space of measurable fields of Hilbert-Schmidt operators σ = { σ ( ˙ x, π ) : ( ˙ x, π ) ∈ M × b G } such that k σ k L ( M × b G ) := Z Z M × b G k σ ( ˙ x, π ) k HS ( H π ) d ˙ xdµ ( π ) < ∞ . Here µ is the Plancherel measure on b G , see Section 3.3. The group Fourier transform yields anisomorphism between the Hilbert spaces L ( M × b G ) and L ( M × G ), and F − G σ ( ˙ x, · ) = κ ˙ x will stillbe called the associated kernel of σ . Naturally A ⊂ L ( M × b G ).Note that the first formula in Lemma 4.6 may be written as ∀ σ ∈ A k Op ( ε ) ( σ ) k HS = ε − Q k σ k L ( G × M ) + O ( ε ) N , while by the Fourier inversion formula (3.4), the second formula is ∀ σ ∈ A Tr (cid:16) Op ( ε ) ( σ ) (cid:17) = ε − Q Z Z M × b G Tr ( σ ( ˙ x, π )) d ˙ xdµ ( π ) + O ( ε ) N . .3. Properties of positive Rockland operators on G and on M . This section is devotedto the general properties of positive Rockland operators. Many of these properties, for instanceregarding self-adjointness and heat kernels, are well-known for general sub-Laplacians on smoothmanifolds [25, p. 261-262], and some of them are known for Rockland operators on compactmanifolds from the recent paper [8].Let R be a positive Rockland operator on G . The operator R M it induces on M is a smoothdifferential operator which is positive and essentially self-adjoint on L ( M ), see Section 2.6. We willkeep the same notation for R M and for its self-adjoint extension. The properties of the functionalcalculus for R imply: Lemma 4.7. Let R be a positive Rockland operator on G . Let ψ ∈ S ( R ) . (1) The operator ψ ( R M ) defined as a bounded spectral multiplier on L ( M ) coincides with theoperator Op( σ ) on C ∞ ( M ) , with symbol σ ( π ) := ψ ( b R ( π )) in A independent of ˙ x ∈ M .The associated kernel is κ := ψ ( R ( π )) δ ∈ S ( G ) . (2) For every ε ∈ (0 , , we have ψ ( ε ν R M ) = Op ( ε ) ( σ ) on C ∞ ( M ) , where ν is the degree ofhomogeneity of R . (3) The integral kernel of ψ ( R M ) is a smooth function on M × M given by K ( ˙ x, ˙ y ) = X γ ∈ Γ κ ( y − γx ) . The operator ψ ( R M ) is Hilbert-Schmidt on L ( M ) .Proof. We have ψ ( R ) = Op G ( σ ) with symbol σ ( π ) = ψ ( b R ( π )) independent of ˙ x ∈ M , see [11, 10,14]. Its associated kernel is κ := ψ ( R ( π )) δ is Schwartz by Hulanicki’s theorem (Theorem 3.5).Hence σ ∈ A . This shows Part (1). The homogeneity properties of R and its functional calculusshow Part (2). Part (3) follows from Lemma 4.1. (cid:3) By Lemma 4.7 applied to ψ ( λ ) = e − tλ , the heat operators e − t R M , t > 0, admits the followingsmooth kernels K t on M × M : K t ( ˙ x, ˙ y ) = X γ ∈ Γ p t ( y − γx ) , ˙ x, ˙ y ∈ M, where p t = e − t R δ , t > 0, are the heat kernels for R .The following statement is classical for sub-Laplacians on compact manifolds. We provide herea self-contained proof for R M : Proposition 4.8. (1) The spectrum Sp( R M ) of R M is a discrete and unbounded subset of [0 , + ∞ ) . Each eigenspace of R M has finite dimension. (2) The constant functions on M form the 0-eigenspace of R M . Note that a consequence of Part (1) is that the resolvent operators ( R − z ) − , z ∈ C \ Sp( R M ),are compact on L ( M ). Proof of Proposition 4.8 (1). The heat operator e − t R M is positive and Hilbert-Schmidt on L ( M ),so its spectrum is a discrete subset of [0 , ∞ ) and the only possible accumulation point is zero.Furthermore, the eigenspaces of e − t R for positive eigenvalues have finite dimensions. Let us showthat the kernel of each e − t R M is trivial. If e − t R M f = 0 for some f ∈ L ( M ) then e − t ′ R M f = 0 for t ′ = t, t/ , t/ , . . . since k e − ( t ′ / R M f k L ( M ) = ( e − t ′ R M f, f ) = 0. By Section 2.4, f G is a periodictempered distribution in L loc ( G ) satisfying e − t ′ R f G = 0 for t ′ = t, t/ , t/ , . . . , but this implies G = 0 since e − s R converges to the identity on S ′ ( G ) as s → f = 0. We have obtained that the kernel of each e − t R M is trivial and thus that their spectrum isincluded in (0 , + ∞ ).The spectrum of R M is the discrete set Sp( R M ) = − ln Sp( e −R ) ⊂ R . It is unbounded since R M is a (non-constant) differential operator. It is included in [0 , + ∞ ) as R M is positive. Theeigenspaces for R M and for its heat kernels are in one-to-one correspondence, and therefore havefinite dimensions. (cid:3) Proof of Proposition 4.8 (2). If a function is constant on M , then it is a 0-eigenfunction. Let usprove the converse. Let f be a 0-eigenfunction, i.e. f ∈ L ( M ) and R M f = 0. By Section 2.4, f G is a periodic tempered distribution in L loc ( G ) satisfying R f G = 0. By the Liouville theorem forhomogeneous groups [14, Theorem 3.2.45] due to Geller [17], f G is a polynomial on G ∼ R n . As itis periodic, it must be a constant. Hence f is a constant. (cid:3) A consequence of Sections 2.6 and 3.4.1 is that the operator R M is hypoelliptic on the manifold M . The same argument shows that the operator R M − E I is hypoelliptic for every constant E ∈ C ,and this implies: Lemma 4.9. The eigenfunctions of R M are smooth on M . Weyl laws for R M . In this section, we consider a positive Rockland operator R on a gradedLie group G and its corresponding operator R M on the nil-manifold M = Γ \ G where Γ is a discreteco-compact subgroup of G . By Proposition 4.8, we may order the eigenvalues of R M (counted withmultiplicity) into the sequence0 = λ < λ ≤ λ ≤ . . . ≤ λ j −→ + ∞ as j → + ∞ . We denote its spectral counting function by N (Λ) := |{ j ∈ N , λ j ≤ Λ }| . This section is devoted to Weyl laws for R M , starting with the following statement: Theorem 4.10 (Weyl law) . We have lim Λ → + ∞ Λ − Q/ν N (Λ) = Vol( M ) c ν/Q where Q the homogeneous dimension of G and c the constant from Theorem 3.6, see also (3.6) . Let us comment on Theorem 4.10 before presenting its proof: • In the particular case of the canonical Laplacian ∆ T n = − P j ∂ j on the torus T n = R n / Z n ,we recover the well known result since ν = 2, Q = n , Vol( M ) = 1 and c (∆ R n ) =(Γ( n/ n π n/ ) − (see Remark 3.7). • The Weyl law for Rockland operators has been recently obtained in greater generality byDave and Haller [8, Corollary 3]. Their proof is different from the one presented here: theydevelop heat kernel expansions on filtered manifolds, and the constant in their Weyl law ischaracterised as a coefficient of these expansions. • Let us consider the case of the canonical Heisenberg nil-manifold, that is, the quotient M = Γ \ H n of the Heisenberg group H n by the canonical lattice Γ = Z n × Z n × Z .The spectrum of the canonical sub-Laplacian L M is known [9, 15, 27]: it consists of thesingle eigenvalue 4 π | m | where m runs over Z n , and of the eigenvalue 4(2 a + n ) π | k | withmultiplicity (2 | k | ) n ( n + a − n − a ! where a and k run over N and Z \ { } respectively. ince Vol( M ) = 1 / 2, the Weyl law for L M gives as Λ → + ∞ c Λ n +1 ∼ X m ∈ Z n :4 π | m | ≤ Λ X k ∈ Z \{ } , a ∈ N a + n ) π | k | < Λ (2 | k | ) n ( n + a − n − a !where c = c ( L H n ) / (2 n + 2), and the constant c ( L H n ) was explicitly given in Remark 3.7.The Weyl law for the torus implies that the first sum is ∼ c ′ Λ n . Hence we have obtained: c Λ n +1 ∼ X k ∈ Z \{ } , a ∈ N a + n ) π | k | < Λ (2 | k | ) n ( n + a − n − a ! . • If R is a positive Rockland operator, then any positive powers of R is also a positiveRockland operator and we can check using the property (3.8) of the constant c that theWeyl law above for R implies the Weyl law for R ℓ for any ℓ ∈ N .Theorem 4.10 follows from taking ε ν = Λ − and a convenient choice of functions ψ ∈ D ( R )approximating the indicatrix of [0 , 1] in the following statement: Lemma 4.11. For any ψ ∈ S ( R ) and N ∈ N , there exists a constant C = C N,ψ,G, Γ , R > suchthat we have for every ε ∈ (0 , : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X j =0 | ψ | ( ε ν λ j ) − ε − Q Vol( M ) c Z ∞ | ψ ( λ ) | λ Qν dλλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε N . Proof of Lemma 4.11. By Lemmata 4.7 and 4.6 (see also Section 4.2.5), we have k ψ ( ε ν R M ) k HS ( M ) = ε − Q k κ k L ( M × G ) + O ( ε ) N . We can compute the L -norm with Theorem 3.6: k κ k L ( M × G ) = Vol( M ) k κ k L ( G ) = Vol( M ) c Z ∞ | ψ ( λ ) | λ Qν dλλ . Now by functional calculus, we also have: k ψ ( ε ν R M ) k HS ( M ) = ∞ X j =0 | ψ | ( ε ν λ j ) , The statement follows. (cid:3) We finish this section with the following generalised Weyl law: Theorem 4.12 (Generalised Weyl law) . Let ≤ a < b . Denoting the semi-classical countingfunction for [ a, b ] by N [ a,b ] (Λ) := { j ∈ N : Λ a ≤ λ j ≤ Λ b } , we have as Λ → + ∞ N [ a,b ] (Λ) ∼ c Λ Q/ , where the (positive, finite) constant c is c = Vol( M ) c νQ ( b Qν − a Qν ) = Vol( M ) Z b G Tr (cid:16) [ a,b ] ( b L ( π )) (cid:17) d ˙ xdµ ( π ) . urthermore, if we consider an orthonormal basis ( ϕ j ) j ∈ N of the Hilbert space L ( M ) consistingof eigenfunctions of R M : R M ϕ j = λ j ϕ j , j = 0 , , , . . . with increasing eigenvalues λ j ≤ λ j +1 , then we have for any σ ∈ A as ε → N [ a,b ] ( ε − ν ) X j : λ j ∈ ε − ν [ a,b ] D Op ( ε ) ( σ ) ϕ j , ϕ j E L ( M ) −→ Z Z M × b G Tr (cid:16) σ ( ˙ x, π )1 [ a,b ] ( b R ( π )) (cid:17) d ˙ xdµ ( π ) . This statement should be compared with its ‘commutative’ counterpart, for instance in [34,Theorem 9.3]. Proof. Taking smooth approximations of 1 [ a,b ] from below and above in Lemma 4.11 easily leadsto the estimate for N [ a,b ] ( ε − ν ), see also (3.7). Let us prove the rest of the statement. First let uscheck that we can define the following linear functional on A : ℓ ( σ ) := Z Z M × b G Tr (cid:16) σ ( ˙ x, π )1 [ a,b ] ( b L ( π )) (cid:17) d ˙ xdµ ( π ) . As 1 [ a,b ] = 1 a,b ] ∈ L ( λ Q/ν dλ/λ ), we have Z b G Tr | [ a,b ] ( b R ) | dµ = Z b G k [ a,b ] ( b R ) k HS dµ = c − k [ a,b ] ( R ) δ ) k L ( G ) < ∞ , by the Plancherel formula (3.3) and Theorem 3.6. Thus the following quantity is finite Z Z M × b G Tr (cid:12)(cid:12)(cid:12) σ ( ˙ x, π )1 [ a,b ] ( b R ( π )) (cid:12)(cid:12)(cid:12) d ˙ xdµ ( π ) ≤ sup ( ˙ x,π ) ∈ M × b G k σ ( ˙ x, π ) k L ( H π ) Z b G Tr | [ a,b ] ( b R ) | dµ < ∞ since we have (with κ ˙ x ( y ) denoting the kernel associated to σ ) k σ k L ∞ ( M × b G ) := sup ( ˙ x,π ) ∈ M × b G k σ ( ˙ x, π ) k L ( H π ) × b G ≤ sup ˙ x Z G | κ ˙ x ( y ) | dy < ∞ . We observe that if ψ ∈ D ( G ) satisfies ψ [ a,b ] = 1 [ a,b ] , then ℓ ( ψ ( ε b R )) = Vol( M ) Z Z M × b G Tr (cid:16) σ ( ˙ x, π )1 [ a,b ] ( b R ( π )) (cid:17) d ˙ xdµ ( π ) . Hence, replacing σ with σ − Vol( M ) − ℓ ( σ ) ψ ( ε b R ), we may assume that σ ∈ A with ℓ ( σ ) = 0.We are left with showing the convergence for any σ ∈ A satisfying ℓ ( σ ) = 0 S ε ( σ ) N [ a,b ] ( ε − ν ) −→ ε → , where S ε ( σ ) := X λ j ∈ ε − ν [ a,b ] D Op ( ε ) ( σ ) ϕ j , ϕ j E ;in this proof, we use the short hand h ., . i for the scalar product on L ( M ). By linearity and since σ = ℜ σ + i ℑ σ , we may assume that σ = σ ∗ . We observe that, by Remark 4.5, S ε ( σ ) = S ε ( σψ ( b R ))where ψ ∈ D ( G ) is a function satisfying ψ [ a,b ] = 1 [ a,b ] , and that for any ψ ∈ D ( G ) supported in[ a, b ] S ε ( ψ ( ε b R ) σ ) = Tr (cid:16) Op ( ε ) ( σψ ( b R )) (cid:17) = ε − Q Z Z M × b G Tr (cid:16) σψ ( b R ) (cid:17) d ˙ xdµ + O ( ε ) N , by Lemma 4.6, see also Section 4.2.5. By linearity, we have | S ε ( σ ) | ≤ | S ε ( σψ ( b R )) | + | S ε ( σ ( ψ − ψ )( b R )) | . or the second term, we observe that S ε ( σ ( ψ − ψ )( b R )) = X λ j ∈ ε − ν [ a,b ] D (Op ( ε ) ( σ )) ∗ ϕ j , ( ψ − ψ )( ε R ) ϕ j E = X λ j ∈ ε − ν [ a,b ] D Op ( ε ) ( σ ) ϕ j , ( ψ − ψ )( ε R ) ϕ j E + N ( ε − ) O ( ε ) , by the properties of the semiclassical calculus (see Proposition 4.4 (2)). Proceeding as above, wesee | S ε ( σ ( ψ − ψ )( b R )) | ≤ | S ε, | + | S ε, | + N ( ε − ) O ( ε ) , where for ψ ∈ D ( R ) S ε, := X λ j ∈ ε − ν [ a,b ] D Op ( ε ) ( σψ ( b R )) ϕ j , ( ψ − ψ )( ε R ) ϕ j E ,S ε, := X λ j ∈ ε − ν [ a,b ] D Op ( ε ) ( σ ( ψ − ψ )( b R )) ϕ j , ( ψ − ψ )( ε R ) ϕ j E . We assume that ψ = 0 on the support of ψ − ψ . This implies S ε, = 0. For the second term, wesee | S ε, | ≤ X j k Op ( ε ) ( σ ) ( ψ − ψ )( ε R ) ϕ j k L ( M ) k ( ψ − ψ )( ε R ) ϕ j k L ( M ) ≤ k Op ( ε ) ( σ ) k L ( L ( M ) X j k ( ψ − ψ )( ε R ) ϕ j k L ( M ) k ( ψ − ψ )( ε R ) ϕ j k L ( M ) ≤ k σ k A (cid:16) k ( ψ − ψ )( ε R ) k HS ( L ( M )) + k ( ψ − ψ )( ε R ) k HS ( L ( M )) (cid:17) , by Proposition 4.2. By Lemma 4.6, see also Section 4.2.5, we have k ( ψ − ψ )( ε R ) k HS ( L ( M )) = ε − Q k ( ψ − ψ )( b R ) k L ( b G )) + O ( ε ) N while the Plancherel formula (3.3) and Theorem 3.6 yield: k ( ψ − ψ )( b R ) k L ( b G )) = c k ( ψ − ψ ) k L ((0 , ∞ ) ,λ Q/νdλ/λ ) ) . We have a similar result for k ( ψ − ψ )( ε R ) k HS ( L ( M )) .Collecting all the estimates above yields:lim sup ε → | S ε ( σ ) | N − a,b ]( ε − ν ) ≤ c (cid:12)(cid:12)(cid:12)(cid:12)Z Z M × b G Tr (cid:16) σψ ( b R ) (cid:17) d ˙ xdµ (cid:12)(cid:12)(cid:12)(cid:12) + c k ψ − ψ k L ((0 , ∞ ) ,λ Q/ν dλ/λ )) + c k ψ − ψ k L ((0 , ∞ ) ,λ Q/ν dλ/λ )) , for any ψ , ψ, ψ ∈ D ( G ), with ψ [ a,b ] = 1 [ a,b ] , supp ψ ⊂ [ a, b ] and ψ = 0 on supp ( ψ − ψ ). Forthe first term on the right-hand side, since ℓ ( σ ) = 0, we have (cid:12)(cid:12)(cid:12)(cid:12)Z Z M × b G Tr (cid:16) σψ ( b R ) (cid:17) d ˙ xdµ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z M × b G Tr (cid:16) σ ( ψ − [ a,b ] )( b R ) (cid:17) d ˙ xdµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ k σ k L ∞ ( M × b G ) Vol( M ) Z b G Tr (cid:12)(cid:12)(cid:12) ( ψ − [ a,b ] )( b R ) (cid:12)(cid:12)(cid:12) dµ. urthermore, since b R ∗ = b R , the Plancherel formula (3.3) and Theorem 3.6 yield Z b G Tr (cid:12)(cid:12)(cid:12) ( ψ − [ a,b ] )( b R ) (cid:12)(cid:12)(cid:12) dµ = Z b G (cid:13)(cid:13)(cid:13)q | ψ − [ a,b ] | ( b R ) (cid:13)(cid:13)(cid:13) HS dµ = c k q | ψ − [ a,b ] |k L ((0 , ∞ ) ,λ Q/ν dλ/λ )) . It is not difficult to construct suitable sequences of functions ψ , ψ and ψ converging to 1 [ a,b ] in L ((0 , ∞ ) , λ Q/νdλ/λ ) and such that k q | ψ − [ a,b ] |k L ((0 , ∞ ) ,λ Q/ν dλ/λ )) converges to 0. This showsthat lim sup ε → N [ a,b ] − ( ε − ν ) | S ε ( σ ) | = 0 and concludes the proof of Theorem 4.12. (cid:3) Ergodicity for the symbols In this section, we discuss the ergodicity of the symbols in Section 5.2. The setting we nowconsider till the end of the paper, is the one of Theorem 1.1, that is, a discrete co-compact subgroupΓ of a stratified nilpotent Lie group G , together with a basis X , . . . , X n of the first stratum g ofits Lie algebra g . We denote the associated sub-Laplacians by L = − X − . . . − X n on G and by L M on M = Γ \ G and keep the same notation for their self-adjoint extensions.5.1. The differential operator E . We denote by E the operator on M × G E = n X j =1 X M,j X j . Here X M,j denote the differential operator on M obtained by periodisation of the vector field X j on G , see Section 2.6 for the periodisation of left-invariant operator on G . Clearly, E is a realsmooth differential operator on M × G . As the vector fields X j ’s are left invariant, E is symmetricin L ( M × G ). Furthermore, the ranges of E + i I and E − i I are dense, so E is essentially self-adjointon L ( M × G ).We are interested in its symbol b E = n X j =1 X M,j π ( X j ) , that is, in the operator b E acting on A via b E σ = F G ( E κ ˙ x ) , for any σ ∈ A with associated kernel ( ˙ x, y ) κ ˙ x ( y ) ∈ C ∞ ( M : S ( G )).In Section 6, E and its symbol will play a pivotal role, and this comes from the followingcomputational remark mentioned in the introduction: Lemma 5.1. If σ ∈ A , then [ L M , Op ( ε ) ( σ )] = Op ( ε ) ( L M σ ) − ε − Op ( ε ) ( b E σ ) + ε − Op ( ε ) (cid:16) [ b L , σ ] (cid:17) , where L M σ ( ˙ x, π ) = L M, ˙ x σ ( ˙ x, π ) . The symbols L M σ , b E σ and [ b L , σ ] are in A . We could deduce this computation from an extension of the symbolic calculus stated for symbolsin A in Proposition 4.2.3, since ε L M = Op ( ε ) ( b L ) and X [ α ] ≤ ε [ α ] (∆ α b L ) X α = n X j =1 − π ( X j ) − ε n X j =1 π ( X j ) X j + ε n X j =1 − X j . But we choose to give a direct proof. roof of Lemma 5.1. Let κ x be the kernel associated with σ . We have for any f ∈ D ( M )( L M Op( σ ) f ) G ( x ) = L x ( f G ∗ κ x ( x ))= f G ∗ L x = x κ x ( x ) + L x = x ( f G ∗ κ x ( x )) − n X j =1 X j,x = x X j,x = x f G ∗ κ x ( x ) . The result follows from L x = x ( f G ∗ κ x ( x )) = f G ∗ ( L κ x )( x ). (cid:3) In the next section, we will need the following technical properties regarding the commutatorsad( E ) B = [ E , B ] = E B − B E , for operators B given by differential operators X αM or X α or by the multiplication operator f ( ˙ x, y ) q ( y ) f ( ˙ x, y ). Lemma 5.2. We denote by s the step of the stratified Lie group G ; this is the smallest integer suchthat ad( Y ) . . . ad( Y s +1 ) = 0 for any Y , . . . , Y s +1 in g . We have the same property with any s + 1 G -invariant vector fields on M .For any α ∈ N n , if k > s | α | then ad k ( E ) X αM = 0 and ad k ( E )( X ) α = 0 , whereas ad k ( E ) B = 0 for a multiplication operator B with any homogeneous polynomial q of homogeneous degree a < k .Proof of Lemma 5.2. For B = X αM , we seead k ( E ) X αM = X ≤ j ,...,j k ≤ n ad( X M,j ) . . . ad( X M,j k )( X α ) X j . . . X M,j k . If X α = X aM, for instance, thenad( X M, )( X aM, ) = a X b =1 X b − M, ad( X M, )( X M, ) X a − bM, . More generally, ad( X M,j ) . . . ad( X M,j k )( X αM ) will be given by a similar linear combination of prod-ucts of | α | vector fields, with k -instances of ad applied in certain ways. A pigeon hole argumentshows that if k > s | α | , then ad has been applied at least s + 1 times on at least one vector withina term of the linear combination of ad( X M,j ) . . . ad( X M,j k )( X αM ) which thus vanishes.We proceed similarly for B = X α :ad k ( E )( X α ) = X ≤ j ,...,j k ≤ n X M,j . . . X M,j k ad( X j ) . . . ad( X j k )( X α ) . The argument above shows that k > s | α | this expression vanishes.Let us now consider the case of B q multiplication by a a -homogeneous polynomial q on G :ad k ( E ) B q = X ≤ j ,...,j k ≤ n X M,j . . . X M,j k ad( X j ) . . . ad( X j k ) B q . One checks easily that ad( X ) B q = B Xq for any X ∈ g . We obtain recursively that ad( X j ) . . . ad( X j k ) B q is a multiplication by a homogeneous polynomial of homogeneous of degree a − k , with the conven-tion that a polynomial of negative order is identically zero. Hence, ad k ( E ) a = 0 for k > a .This concludes the proof of Lemma 5.2. (cid:3) .2. Mean ergodicity of symbols. By ergodicity of the symbol, we mean understanding theergodic behaviour in the sense of von Neuman’s mean ergodic theorem of the one-parameter groupassociated with i b E . Here b E = P n j =1 X M,j π ( X j ) is the symbol of the differential operator E definedin Section 5.1. As E is essentially self-adjoint on L ( M × G ) with domain C ∞ ( M : S ( G )), so is b E on the Hilbert space L ( M × b G ) with domain A ; the latter Hilbert space is defined in Section4.2.5. We will keep the same notation for their self-adjoint extensions.It is instructive to look at the commutative case, that is, G = R n is abelian so M ∼ T n is atorus. An element of A is then a function a ∈ C ∞ ( T n ); S ( R n )), i.e. a function a ( ˙ x, ξ ) Schwartz in ξ and depending smoothly in ˙ x ∈ T n , and we have b E = n X j =1 ∂ x j iπ ∂ ξ j and e it b E a ( x, ξ ) = a ( x + tξ, ξ ) . Let us now go back to the general case of G stratified Lie group. This section is devoted toshowing the following properties for e it b E : Theorem 5.3. The operators e it b E , t ∈ R , form a strongly continuous one-parameter group ofunitary operators on L ( M × b G ) . (1) If σ ∈ A then e it b E σ ∈ A for every t ∈ R . Furthermore, the mapping t e it b E is continuousfrom R to L ( A ) . (2) For any σ ∈ L ( M × b G ) , we have the convergence as T → + ∞ (cid:13)(cid:13)(cid:13)(cid:13) T Z T e it b E σdt − P σ (cid:13)(cid:13)(cid:13)(cid:13) L ( M × b G ) −→ , where P is the orthogonal projection of L ( M × b G ) given by P σ ( ˙ x, π ) = Z M σ ( ˙ x ′ , π ) d ˙ x ′ for almost every ( ˙ x, π ) ∈ M × b G. The proof of the first part of Theorem 5.3 relies on the following observations of a computationalnature. We start with recalling that, at least formally, if B is a linear operator on the space ofsymbols, then we have(5.1) e − it b E Be it b E = e ad( − it b E ) B = ∞ X k =0 ( − it ) k k ! ad k ( b E ) B, where ad is short for the commutator bracket, that is, ad( A ) B = [ A, B ]. This makes sense for theoperators B of the form X αM , ∆ α and π ( X ) α because the sum over k is finite by Lemma 5.2 Proof of Theorem 5.3 (1). Let σ ∈ A and t ∈ R . At least formally, we may write for any α ∈ N n , β, γ ∈ N n , ∆ α π ( X β ) X γM e it b E σ = e it b E τ α,β,γ where τ α,β,γ := e − it b E ∆ α π ( X β ) X γM e it b E σ. By (5.1) and Lemma 5.2, every symbol τ α,β,γ = e − it b E ∆ α e it b E e − it b E π ( X β ) e it b E e − it b E X γM e it b E σ, is well defined in A and the formula above holds. Consequently, e it b E τ α,β,γ ∈ L ( M × b G ).We observe that ∆ α e it b E σ ∈ L ( M × b G ) for every α ∈ N n and the Plancherel theorem imply that(1 + k · k ) N F − G (cid:16) e it b E σ ( ˙ x, · ) (cid:17) is in L ( G ) for any N ∈ N and a well chosen homogeneous quasi-norm n G . Similarly, π ( X ) β e it b E σ ∈ L ( M × b G ) for every β ∈ N n and the Sobolev embeddings on G [14, Theorem 4.4.25] imply that the map ˙ x 7→ F − G (cid:16) e it b E σ ( ˙ x, · ) (cid:17) is in L ( M ; L ( G ) ∩ C ∞ b ( G )). Here L ( M ; B ) denotes the topological vector space of the square integrable function on M valued in aFrechet space B . Finally, X γM e it b E σ ∈ L ( M × b G ) for every γ ∈ N n and the (Euclidean) Sobolevembeddings on M imply that the map ˙ x 7→ F − G (cid:16) e it b E σ ( ˙ x, · ) (cid:17) is smooth from M to L ( G ).Performing all the three operations simultaneously shows that the map ˙ x 7→ F − G (cid:16) e it b E σ ( ˙ x, · ) (cid:17) issmooth from M to S ( G ). In other words, e it b E σ ∈ A . Furthermore, since the proof above involvesonly finite sums, the Plancherel formula implies the continuity in t of e it b E on L ( A ). (cid:3) As already mentioned, the core in the proof of the second part of Theorem 5.3 is an instance ofthe famous Mean Ergodic Theorem due to von Neumann. Variations of this theorem can be foundin many textbooks on functional analysis, and a very clear proof of the version below may be foundin [18, p.71]: Theorem 5.4 (Mean Ergodic Theorem) . Let ( U t ) t ∈ R be a strongly continuous one-parameter groupof unitary operators on a Hilbert space H . Let P denote the orthogonal projection on the space ofinvariant vectors under the group. Then for every vector v ∈ H , we have the convergence (cid:13)(cid:13)(cid:13)(cid:13) T Z T e it b E v dt − P v (cid:13)(cid:13)(cid:13)(cid:13) H −→ as T → + ∞ . Another ingredient in the proof of the second part will be the decomposition of the regularrepresentation of G on L ( M ) given by π R ( g ) f ( ˙ x ) = f ( ˙ xg ). It decomposes as a discrete direct sum π R = ⊕ ⊥ m ( ρ ) ρ, over a countable family of irreducible representations ρ of G with finite multiplicity m ( ρ ), see e.g.[31, Section 2.7] or [22, p.146]; we denote this set of ρ by d Γ \ G . The trivial representation 1 b G hasmultiplicity 1, and its corresponding space H ρ is the space of constant functions on M .The proof of the third part of Theorem 5.3 then boils dow to the following lemma: Lemma 5.5. Let ρ ∈ d Γ \ G be non-trivial, i.e. ρ = 1 b G . The operator ˜ E ρ := P n j =1 ρ ( X j ) X j isessentially self-adjoint on the Hilbert space H ρ ⊗ L ( G ) . The only vector in H ρ ⊗ L ( G ) invariantunder the one-parameter group of unitary operators e it ˜ E ρ , t ∈ R , is .Proof of Lemma 5.5. Since the vector fields X j are G -invariant, the self-adjointness of ˜ E ρ followsfrom the one of E together with the composition of the projection of L ( M ) on the ρ -componenttensored with the Fourier transform F G .Let ˜ κ ρ ∈ H ρ ⊗ L ( G ) be invariant under e it ˜ E ρ , t ∈ R . This is equivalent to ˜ κ ρ being in the domainof ˜ E ρ and ˜ E ρ ˜ κ ρ = 0. Since the vector fields X j are left invariant, we can convolve a scalar function φ ∈ D ( G ) on the left with an any element ˜ κ ′ ρ in H ρ ⊗ L ( G ), and obtain φ ∗ ˜ E ρ ˜ κ ′ ρ = ˜ E ρ ( φ ∗ ˜ κ ′ ρ ). Thisimplies that φ ∗ ˜ κ ρ is also invariant and that it suffices to prove the result for an invariant ˜ κ ρ ( z )smooth in z ∈ G . We assume so.Let φ, ψ ∈ D ( G ) be scalar valued functions such that ψ is supported where φ is constant. Thisimplies ψ ˜ E ρ (˜ κ ρ φ ) = ψφ ˜ E ρ ˜ κ ρ + n X j =1 ρ ( X j )˜ κ ρ ψX j φ = 0 , nd more generally ∀ ℓ ∈ N ψ ˜ E ℓρ (˜ κ ρ φ ) = 0 , thus ψe it ˜ E ρ (˜ κ ρ φ ) = + ∞ X ℓ =0 ( it ) ℓ ℓ ! ψ ˜ E ℓρ (˜ κ ρ φ ) = ψφ ˜ κ ρ . Taking a suitable limit in L ( G ) of functions φ , we obtain(5.2) e it ˜ E ρ (˜ κ ρ φ ∞ ) = ˜ κ ρ φ ∞ , where φ ∞ is the indicatrix 1 B of a ball B . As this is so for any B , by linearity, (5.2) holds for anysimple function φ ∞ , and by L -continuity of e it ˜ E ρ , for any φ ∞ ∈ D ( G ). Hence, we have obtained0 = ˜ E ρ (˜ κ ρ φ ) for any φ ∈ D ( G ), and thus n X j =1 ρ ( X j )˜ κ ρ ( z ) X j φ ( z ) = 0 for any point z ∈ G. As we can choose the scalars X j φ ( z ) arbitrarily, we have ρ ( X j )˜ κ ρ ( z ) = 0 in H ρ for each j = 1 , . . . , n and each z ∈ G . As the X j , j = 1 , . . . , n , generate the Lie algebra g , the vector ˜ κ ρ ( z ) in H ρ isannihilated by the infinitesimal representation ρ of g , which is irreducible and non-trivial. Thisimplies that ˜ κ ρ ( z ) = 0 for each z ∈ G , and concludes the proof of Lemma 5.5. (cid:3) Proof of Theorem 5.3 (2). By the mean ergodic theorem, it suffices to identify the space of symbols σ ∈ L ( M × b G ) invariant under e it b E , t ∈ R . Such a symbol σ may be decomposed as σ = X ρ ∈ d Γ \ G σ ρ , with σ ρ ∈ H ρ ⊗ L ( b G )satisfying F − G σ ρ ∈ H ρ ⊗ L ( G ) being invariant under e it ˜ E ρ , t ∈ R . By Lemma 5.5, for ρ = 1 b G , F − G σ ρ = 0 so σ ρ = 0, while for ρ = 1 b G , we have ˜ E ρ = 0 so e it ˜ E ρ is the identity operator on H ρ ⊗ L ( G ) ∼ L ( G ). This implies that σ = σ ρ with ρ = 1 b G . In other words, the orthogonalprojection P on the invariants is the projection onto the ρ -component with ρ = 1 b G which is givenby integration on M . This concludes the proof of Theorem 5.3. (cid:3) Quantum Variance In this section, we keep the same setting as in Section 5. We also consider an orthonormal basis( ϕ j ) j ∈ N of the Hilbert space L ( M ) consisting of eigenfunctions of L M : L M ϕ j = λ j ϕ j , j = 0 , , , . . . with increasing eigenvalues λ j ≤ λ j +1 . We denote the counting spectral function by N (Λ) := { j ∈ N : λ ≤ Λ } . This section is devoted to finishing the proof of Theorem 1.1. The Weyl law N ( ε − ) ∼ c ε − Q hasalready been obtained in Section 4.4 while the quantum ergodicity will follow from the followingstatement. Proposition 6.1 (Quantum Variance) . For any σ ∈ A satisfying R M σ ( ˙ x, π ) d ˙ x = 0 for almostevery π ∈ b G , we have the convergence N ( ε − ) X j : λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12)D Op ( ε ) ( σ ) ϕ j , ϕ j E L ( M ) (cid:12)(cid:12)(cid:12)(cid:12) −→ as ε → . e start with showing how Proposition 6.1 implies the last part of Theorem 1.1. The nextsubsection gives important observations for the proof of Proposition 6.1. The last subsection statesand proves a generalisation of Proposition 6.1.6.1. End of the proof of Theorem 1.1 from Proposition 6.1. The last part of Theorem 1.1follows classically ([34, Proof of Theorem 9.4 (i)] or [1, p.111]) from the separability of the spaceof continuous functions on M together with the following statement which is a consequence ofProposition 6.1: Corollary 6.2. For any continuous function a : M → C , we have the convergence N ( ε − ) X j : λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12)Z M a ( ˙ x ) | ϕ j ( x ) | d ˙ x − Z M a ( ˙ x ) d ˙ x (cid:12)(cid:12)(cid:12)(cid:12) −→ as ε → . Proof of Corollary 6.4. An argument of density shows that we may assume a ∈ D ( M ). We mayalso assume R M a ( ˙ x ) d ˙ x = 0.Let ψ ∈ D ( R ) be such that ψ = 1 [0 , ψ . Then σ ( ˙ x, π ) = a ( ˙ x ) ψ ( b L ) defines a symbol σ in A byLemma 4.7 and we have X j : λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12)Z M a ( ˙ x ) | ϕ j ( x ) | d ˙ x (cid:12)(cid:12)(cid:12)(cid:12) ≤ X j : λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12)D Op ( ε ) ( σ ) ϕ j , ϕ j E L ( M ) (cid:12)(cid:12)(cid:12)(cid:12) . We conclude with Proposition 6.1. (cid:3) Some observations. In this section, we present some remarks which will guide us in somesteps of the proof of Proposition 6.1.6.2.1. A weak result of Egorov type. The observations explained at the end of the introduction leadus to the following first result of Egorov type. Although we will not use this result, the proof shedslight on an important problem related to the non-commutativiy of our symbols: Lemma 6.3 (Weak Egorov) . Let σ ∈ A be such that σ t := e it b E σ commute with b L for every t ∈ R .For any T > , there exists a constant C = C G, Γ , ( X j ) ,σ,T > such that we have ∀ ε ∈ (0 , ∀ t ∈ [0 , T ] k e − itε L M Op ( ε ) ( σ ) e itε L M − Op ( ε ) ( σ t ) k L ( L ( M )) ≤ Cε. Proof of Lemma 6.3. We write e − itε L M Op ( ε ) ( σ ) e itε L M − Op ( ε ) ( σ t ) = Z ∂ s n e − itsε L M Op ( ε ) ( σ t (1 − s ) ) e itsε L M o ds = Z e − itsε L M (cid:16) [ − itε L M , Op ( ε ) ( σ t (1 − s ) )] − i Op ( ε ) ( b E σ t (1 − s ) ) (cid:17) e itsε L M ds = − it Z e − itsε L M Op ( ε ) (cid:16) ε − [ b L , σ t (1 − s ) ] + ε L M σ t (1 − s ) (cid:17) e itsε L M ds, by Lemma 5.1. The statement follows from the hypothesis on σ t and the properties of the semi-classical calculus (see Section 4) and of e i t b E (see Theorem 5.3 (1)). (cid:3) .2.2. Some non-commutative but real considerations. We are therefore led to study the termOp ( ε ) ([ b L , σ t (1 − s ) ]). In fact, it will appear in a scalar product, and to tackle this appearance, we willuse the following observations which require first to set some vocabulary.If σ ∈ A , we also define the symbol ( σ )˘ := ˘ σ ∈ A via˘ σ ( ˙ x, π ) := σ ( ˙ x, ¯ π ) . We observe that this operation commutes with taking the adjoint:(6.1) ( σ )˘ ∗ = ( σ ) ∗ ˘ thus ( ℜ σ )˘ = ℜ ˘ σ. If κ ˙ x ( y ) is the kernel associated with σ , then the kernels associated with σ ∗ and ˘ σ are ¯ κ ˙ x ( y − )and ¯ κ ˙ x ( y ) respectively. The relation between Op ε ( σ ) and Op ε ( σ ∗ ) is given in Proposition 4.4. For˘ σ , we see(6.2) Op ( ε ) (˘ σ ) φ = Op ( ε ) ( σ ) ¯ φ, so h Op ( ε ) ( σ ) φ, φ i L ( M ) = h Op ( ε ) (˘ σ ) ¯ φ, ¯ φ i L ( M ) , which implies easily(6.3) σ = ˘ σ = ⇒ ℜ D Op ( ε ) ( σ ) ϕ, ϕ E = ℜ D Op ( ε ) ( σ ) ℜ ϕ, ℜ ϕ E − ℜ D Op ( ε ) ( σ ) ℑ ϕ, ℑ ϕ E . We see that for τ , τ ∈ A or τ ∈ A and τ ∈ F G U ( g ) we have(6.4) [ τ , τ ] ∗ = − [ τ ∗ , τ ∗ ] and ([ τ , τ ])˘ = [˘ τ , ˘ τ ] . We will also need the following observations describing the relations of ˘ and b E :(6.5) ( b E σ )˘ = b E ˘ σ so ( e it b E σ )˘ = e − it b E ˘ σ. Proof of Proposition 6.1. As we can write σ = ℜ σ + i ℑ σ , it suffices to prove the result for σ ∈ A self-adjoint. Similarly, it suffices to prove the result for σ ∈ A such that ˘ σ = σ . Hence wemay assume σ = σ ∗ = ˘ σ . Furthermore, the properties of the calculus, especially Proposition 4.4(2), imply X λ j ≤ ε − (cid:12)(cid:12)(cid:12) ℑ D Op ( ε ) ( σ ) ϕ j , ϕ j E(cid:12)(cid:12)(cid:12) = O ( ε ) − Q ;in this section, we use the short hand h ., . i for the scalar product on L ( M ). Therefore, it sufficesto show that(6.6) lim ε → S ε N ( ε − ) = 0 where S ǫ := X λ j ≤ ε − (cid:12)(cid:12)(cid:12) ℜ D Op ( ε ) ( σ ) ϕ j , ϕ j E(cid:12)(cid:12)(cid:12) . Using (6.3), we may now assume that the ϕ j ’s are real-valued.With all these assumptions, we now start the analysis with S ǫ = X λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:28) T Z T − T e − itε L M Op ( ε ) ( σ ) e − itε L M dt ϕ j , ϕ j (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ≤ S ,ε + S ,ε ) , for T > σ t := e it b E σ , S ,ε := X λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:28) T Z T − T (cid:16) e − itε L M Op ( ε ) ( σ ) e − itε L M − Op ( ε ) ( σ t ) (cid:17) dt ϕ j , ϕ j (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) ,S ,ε := k Op ( ε ) (cid:18) T Z T − T σ t dt (cid:19) k HS = ε − Q k T Z T − T σ t dt k L ( M × b G ) + O ( ε ) N , y Lemma 4.6, see also Section 4.2.5. The properties of the calculus, especially Proposition 4.4 (2),imply S ,ε = S ′ ,ε + O ( ε − Q ) where S ′ ,ε := X λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12) ℜ (cid:28) T Z T − T (cid:16) e − itε L M Op ( ε ) ( σ ) e − itε L M − Op ( ε ) ( ℜ σ t ) (cid:17) dt ϕ j , ϕ j (cid:29)(cid:12)(cid:12)(cid:12)(cid:12) . We now proceed as in the proof of Lemma 6.3, even if having introduced the real part of σ t willproduce some additional terms: e − itε L M Op ( ε ) ( σ ) e itε L M − Op ( ε ) ( ℜ σ t ) = Z ∂ s n e − itsε L M Op ( ε ) ( ℜ σ t (1 − s ) ) e itsε L M o ds = Z e − itsε L M (cid:16) [ − itε L M , Op ( ε ) ( ℜ σ t (1 − s ) )] − ( ε ) ( ℜ i b E σ t (1 − s ) ) (cid:17) e itsε L M ds = Z e − itsε L M Op ( ε ) (cid:16) − itε − [ b L , ℜ σ t (1 − s ) ] + 2 i b E ℜ σ t (1 − s ) − ℜ ( i b E σ t (1 − s ) ) − itε L M ℜ σ t (1 − s ) (cid:1) e itsε L M ds. This together with the properties of the calculus, especially Proposition 4.4 (2), imply S ′ ,ε = S ′′ ,ε + O ( ε ) − Q where S ′′ ,ε := X λ j ≤ ε − (cid:12)(cid:12)(cid:12) ℜ D Op ( ε ) (cid:16) − iε − [ b L , τ ] + ℜ ( iτ ) (cid:17) ϕ j , ϕ j E(cid:12)(cid:12)(cid:12) , where τ and τ are the symbols given by: τ := 12 T Z T − T Z ℜ σ t (1 − s ) dstdtτ := 1 T Z T − T Z b E ℜ σ t (1 − s ) − b E σ t (1 − s ) dstdt. For the term in τ , we observe that τ is in A and satisfies˘ τ = 12 T Z T − T Z ℜ σ − t (1 − s ) dstdt = τ , because of (6.5) and (6.1). Thus, using (6.2), (6.4), ˘ L = L and ϕ j = ℜ ϕ j , we see that D Op ( ε ) ([ b L , τ ]) ϕ j , ϕ j E = D Op ( ε ) ([ b L , τ ]) ϕ j , ϕ j E , so ℜ D Op ( ε ) ( i [ b L , τ ]) ϕ j , ϕ j E = 0 . For the term in τ , we observe similarly that τ is in A and satisfies ˘ τ = τ so ℜ D Op ( ε ) ( iτ ) ϕ j , ϕ j E = 0 . We have the same observation with τ replaced with τ ∗ because of (6.1). This implies ℜ D Op ( ε ) ( ℜ ( iτ )) ϕ j , ϕ j E = 0 , and therefore S ′′ ,ε = 0.Collecting all the equalities and inequalities above, we see S ε ≤ ε − Q k T Z T − T σ t dt k L ( M × b G ) + Cε − Q , here the positive constant C may depend on T . This implieslim sup ε → S ε N ( ε − ) ≤ c − k T Z T − T σ t dt k L ( M × b G ) , where c is the constant from the Weyl law N ( ε − ) ∼ c ε − Q . By our Ergodic Theorem (see Theorem5.3 (2)), the right-hand side tends to 0 as T → + ∞ . This shows the convergence in (6.6).This concludes the proofs of Proposition 6.1 and of our main result (Theorem 1.1).6.4. A generalisation of Proposition 6.1. In this section, we prove the following generalisationof Proposition 6.1: Corollary 6.4. For any σ ∈ A , we have the convergence as ε → N ( ε − ) X λ j ≤ ε − (cid:12)(cid:12)(cid:12)(cid:12)D Op ( ε ) ( σ ) ϕ j , ϕ j E L ( M ) − Z M σ ( ˙ x, b G ) d ˙ x (cid:12)(cid:12)(cid:12)(cid:12) −→ . As explained in the introduction, a commutative counterpart on the torus would be the analogueof [34, Theorem 9.4] for the 0-energy level of the standard laplacian on the torus for the basic semi-classical calculus.Before starting the proof, we observe that the integral in Corollary 6.4 is indeed a scalar, andthat it can be computed as σ ( ˙ x, b G ) = Z G κ ˙ x ( y ) dy, so Z M σ ( ˙ x, b G ) d ˙ x = Z Z M × G κ ˙ x ( y ) d ˙ xdy, where κ ˙ x ( y ) is the kernel associated with σ . Proof of Corollary 6.4. We want to show that for any σ ∈ A we havelim ε → S ε ( σψ ) N ( ε − ) = 0 where S ε ( σ ) := X λ j ≤ ε − | u j,ε,σ | , and u j,ε,σ := D Op ( ε ) ( σ ) ϕ j , ϕ j E L ( M ) − Z M σ ( ˙ x, b G ) d ˙ x. Writing τ for the symbol in A independent of ˙ x given by τ ( π ) = R M σ ( ˙ x, π ) d ˙ x , we observe that S ε ( σ ) ≤ S ε ( σ − τ ) + S ε ( τ )) , and that the average over M of the symbol σ − τ vanishes, so we have lim ε → S ε ( σ − τ ) /N ( ε − ) = 0by Proposition 6.1. This shows that we may assume that σ is independent of ˙ x .Note that b L (1 b G ) = 0, so ψ ( b L ( π )) = ψ (0) at π = 1 b G for any ψ ∈ D ( G ). Consequently, considering ψ ∈ D ( G ) satisfying ψ [0 , = 1 [0 , , we see that σ − σ (1 b G ) ψ ( b L ) is in A with S ε ( σ ) = S ε ( σ − σ (1 b G ) ψ ( b L )) and ( σ − σ (1 b G ) ψ ( b L ))( π ) = 0 at π = 1 b G . This shows that we may assume in addtion σ (1 b G ) = 0.As explained above, we may assume σ = b κ with κ ∈ S ( G ) and σ (1 b G ) = R G κ ( y ) dy = 0. We have u j,ε,σ = D Op ( ε ) ( σ ) ϕ j , ϕ j E L ( M ) , so | u j,ε,σ | ≤ k Op ( ε ) ( σ ) k L ( L ( M ) k ϕ j k L ( M ) ≤ k σ k A , by Proposition 4.2 and since k ϕ j k L ( M ) = 1; this implies ∀ ε ∈ (0 , 1] 0 ≤ N ( ε − ) − S ε ( σ ) ≤ k σ k A . urthermore, we have u j,ε,σ = D ( ϕ j,G ∗ κ ( ε ) ) M , ϕ j E L ( M ) −→ ε → , by Corollary 3.4 since R G κ ( y ) dy = 0. The convergence is for j ∈ N fixed, but it shows that forevery j ∈ N , there exists ε ( j ) ∈ (0 , 1] such that | u j,ε,σ | ≤ / ( j + 1) holds for every ε ∈ (0 , ε ( j ) ).We may assume that the sequence ( ε ( j ) ) j ∈ N is decreasing. Consequently, if ( ε ℓ ) ℓ ∈ N is a sequencein (0 , 1] converging to 0, then we may extract a subsequence ( ε ℓ k ) k ∈ N such that ε ℓ k < ε ( k ) for all k ∈ N and we have S ε ℓk ( σ ) ≤ X ≤ j ≤ N ( ε − ℓk ) j + 1 ∼ ln N ( ε − ℓ k ) , so lim k →∞ S ε ℓk ( σ ) N ( ε − ℓ k ) = 0 . This implies lim ε → S ε ( σ ) /N ( ε − ) = 0 and concludes the proof of Corollary 6.4. (cid:3) References [1] N. Anantharaman, F. Faure and C. Fermanian-Kammerer Chaos en m´ecanique quantique , Editor: Harinck,Pascale and Plagne, Alain and Sabbah, Claude, Lectures from the Mathematical Days (X-UPS 2014) held atl’´Ecole Polytechnique, Palaiseau, April 28–29, 2014, ´Editions de l’´Ecole Polytechnique, Palaiseau, 2014.[2] N. Anantharaman and. F. 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Fischer) University of Bath, Department of Mathematical Sciences, Bath, BA2 7AY, UK Email address : [email protected]@bath.ac.uk