Quantum ergodicity for pseudo-Laplacians

We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.

will denote a semiclassical quantization for compactly supported symbols, see section 1. Our main result is the following quantum ergodicity statement for this operator: Theorem 1. Let X be a Riemannian surface with a finite number of constant curvature hyperbolic cusps such that the geodesic flow on S * X is ergodic. Let u j ∈ D c be an orthonormal family of eigenfuctions of ∆ c − 1 4 with eigenvalues λ 2 0 < λ 2 1 ≤ λ 2 2 ≤ · · · , covering all the eigenvalues of ∆ c − 1 4 except a finite number of non-positive ones. Let a ∈ S 0 (T * X) be compactly supported in space.
For a precise review of the geometry of the considered Riemannian manifolds, we refer to Section 1.1, while for the definition of the pseudo-Laplacian this is done in Section 1.2. There are very natural examples of such manifolds given by negatively curved surfaces with finite volume and hyperbolic cusps.
Let us make several remarks about the Theorem. First, by a standard argument (see for instance [Zw,Section 15.5]) Theorem 1 implies that Op(a)u j , u j → S * X a for a sequence of density one, when a has compact support. Moreover, since we are only interested in quantizing symbols with a compact support in the space variable, we can use a standard quantization procedure, see for instance [Zw,Section 14.2]. That means however that the estimates are not uniform far in the cusp.
In the same geometric setting, we also mention that there are other works by Dyatlov [Dy] and Bonthonneau [Bo] on the microlocal limits of non-L 2 eigenfunctions of the Laplacian but with complex eigenvalues, where one instead get a sort of "quantum unique ergodicity".
For simplicity, the proof will be presented in the case where there is one cusp, the argument being the same with several cusps. The method of proof follows the scheme from [ZeZw] and has two steps: 1) a pointwise "ellipticity bound" that states that the eigenfunctions are microlocalized on the cosphere bundle. This implies that in the limit λ → ∞, M(a, λ) 2 := 1 N(λ) λ j ≤λ Op h j (a)u j , u j − S * X a 2 is controlled by a |S * X 2 L 2 .
2) Taking a symbol with average zero, we propagate it by the geodesic flow to get a new symbol that is small on the cosphere bundle (by the L 2 ergodic theorem); we have to prove that this does not modify much M(a, λ): this is the point of the "flow invariance" theorems. We stress that working with a pseudo-Laplacian entails new difficulties, as compared to the compact setting. For the first step, since we are working with a pseudo-Laplacian, the pointwise ellipticity bound (and the subsequent microlocalization) works only outside the singular circle, and we need to prove that the needed correction is small enough. This requires a precise control of the eigenfunctions of the pseudo-Laplacian. For the second step, it is important to notice that the eigenfunctions we are interested in are not eigenfunctions of the propagator we are using for the proof. We are able to prove that M(a, λ) does not change much when replacing a by a • Φ t if Φ t is the geodesic flow, but we have to assume that the symbol a is supported quite far away from the singular circle. Since the admissible support has full measure, the L 2 control of lim sup M(a, λ) we still get at step 1 leads to the same result. Finally, [ZeZw] work with a compact manifold, and M(1, λ) thus vanishes. This is not the case for us since we only use symbols with a compact support in space. Our proof has a third step which consists in finding symbols a with average close to 1 such that lim sup M(a, λ) is arbitrarily close to zero. For that purpose, we shall prove that the modes of the eigenfunctions of the pseudo-Laplacian are microlocalized in the cusp.
1. Preliminaries 1.1. Notations. We let X be a Riemannian surface with one hyperbolic cusp, i.e. a cusp with constant curvature. This means that X can be split into two parts, X = X 0 ∪ X 1 , where X 1 is a compact Riemannian surface with boundary, and X 0 = (c 0 , ∞) r × (R/Z) θ with metric dr 2 + e −2r dθ 2 . Using the notation ξdr + ηdθ for cotangent vectors in X 0 , the Hamiltonian induced by the metric in the cusp is given by p(r, θ; ξ, η) = ξ 2 + e 2r η 2 .
In X 0 , any u ∈ L 2 loc function can be expanded into Fourier series in the θ variable: where the u n are in L 2 loc ((c 0 , ∞); e −r dr). The metric induces a natural measure µ, called Liouville measure, on the unit cotangent bundle S * X and for simplicity we shall normalize it so that it is a probability measure. The projection T * X → X on the base will be denoted by π. Finally, C > 0 will denote a generic constant that is independent of the parameters we consider (except when indicated), and that will change from line to line.
1.2. Definition of the pseudo-Laplacian. Definition 1.1. Let c > c 0 . Let us denote L 2 0,c (resp. H 1 0,c ) the space of all u ∈ L 2 (X) (resp. u ∈ H 1 (X)) such that u 0 (r) = 0 for every r ≥ c. The pseudo-Laplacian ∆ c is the unbounded non-negative self-adjoint operator on L 2 0,c defined by the quadratic form using the Friedrichs method The Riemannian measure dv g and the gradient are with respect to g.
We note that the spaces L 2 0,c and H 1 0,c are closed vector subspaces of L 2 (X), and H 1 (X). The circle r = c in X 0 will be referred to as the singular circle.
The following results are proved in [CdV3,Theorem 2].
Proposition 1.2. The operator ∆ c is an unbounded, non-negative, self-adjoint operator with compact resolvent and discrete spectrum.
We will denote (u j ) j an orthonormal family of eigenfunctions with positive eigenvalues of ∆ c − 1 4 , that is, ∆ c u j = λ 2 j + 1 4 u j , where (λ j ) j is a positive, non-decreasing sequence going to +∞. Note that the orthogonal of Span{u j , j ≥ 0} in L 2 0,c is a finite-dimensional space that possesses an orthonormal basis of eigenfuctions of ∆ c . We will denote, for each j ≥ 0, h j = λ −1 j . Note that we extend ∆ c as an unbounded self-adjoint operator from L 2 to L 2 with compact resolvent by declaring that ∆ c v = 0 whenever v ∈ L 2 (X) has support in {r ≥ c} and v does only depend on r.
1.3. Review of semiclassical analysis. We shall use the following semiclassical quantization procedure, which is similar to [Zw,Chapter 14.2]: we fix a cover by countably may open sets U i of X, i ≥ 0, with diffeomorphisms ϕ i : U i → V i , where the V i ⊂ R 2 are open sets, and take a partition of unity (χ 2 i ) i associated with it. A compactly supported symbol a ∈ S m comp (X) is a smooth function a ∈ C ∞ (T * X) whose support projects to X into a compact set and satisfying uniform bounds |∂ α x ∂ β ξ a(x, ξ)| ≤ C α,β ξ m−|β| for all multi-indices α, β. Then for any symbol a ∈ S m comp (X) with compact space support, and h > 0, we define where b w (x, hD) means the Weyl quantization. When U i ∩ π(supp(a)) = ∅ (which always happen but for a finite number of i), i does not contribute to the sum, because ((ϕ i ) * a) = 0. In any case, (ϕ i ) * a ∈ S m loc (V i ). The specific choice of the partition of unity is not important, because the difference between two different such quantizations is then an O(h) L 2 →L 2 for any S 0 comp (X) symbol. We shall thus make the following choices: and ϕ 1 is a shift in the second coordinate only.
With this procedure all the useful properties (about composition, Lie brackets, L 2 operators bounds for S 0 quantized symbols) hold: the proofs from [Zw,Chapters 14,15] still apply when the symbols are compactly supported in x, however the constants depend on the size of the supports.
2. Estimates on the singular circle 2.1. Riemannian Laplacian of the eigenfunctions. In this section, we study the the family (∆ − λ 2 j − 1 4 )u j . We will denote by δ c the Lebesgue measure with total mass 1 on the circle r = c of the cusp of X.
The following lemma is an easy application of Stokes's theorem.
Lemma 2.1. Let ϕ be a smooth function on X such that ϕ(r, θ) =φ(r) in the cusp on r > c − ε for some 0 < ε < c − c 0 , whereφ : (c 0 , ∞) → C is smooth. Assume thatφ(c) = 0, Now, we can write the Laplacian of u j as a function of its zero mode.
To estimate the ∆u j , we need an adequate description of the constants α j from Corollary 2.2.
Proposition 2.3. There exists a smooth compactly supported functionφ on X, and a sequence (I j ) j≥0 such that I j α j = u j ,φ (it is the L 2 inner product) for every j ≥ 0 and Proof. Let φ be any smooth compactly supported function on R such that: Letφ(r, θ) = e r/2 φ(r) (andφ is zero outside the cusp), such thatφ is well-defined on X, smooth, compactly supported. Now, sinceφ has no non-zero θ-Fourier mode, using its support property and the nature of the hyperbolic metric, we know that u j ,φ = α j I j , where and we are done.
Corollary 2.4. We have: Proof. Since h j ∼ −I j as j goes to infinity, and since Q j = F α j (h j ) −1 for some constant F , we find that |Q j | 2 h 4 j is positive and is equivalent to |F | 2 α 2 j I 2 j = |F | 2 u j ,φ 2 (with the above notations). Now, since the (u j ) j forms an orthonormal family in L 2 , which proves the claim.
2.2. Pseudo-differential operators acting on δ c . Following up on the previous subsection, we have: where · S −2 is some S −2 seminorm (in every estimate of that kind in the following, the seminorm will have to be universal).
Lemma 2.6. Let a ∈ S −2 comp (R 2 ), with π(supp(a)) ⊂ (c − ε, c + ε)×(0, 1). Let χ : R → [0, 1] be smooth and zero outside (0, 1). Let ν, ϕ = 1 0 χ(θ)ϕ(c, θ)dθ. Then, for some universal constant C > 0, Proof. We may assume that a is compactly supported, if we find out that C does not depend on the support of a. A computation gives . ϕ ξ,θ is a smooth function from R 6 to R, and its gradient is zero at the only point r Besides, at that point, the Hessian matrix of ϕ ξ,θ is so it has full rank and we see from the stationary phase method (say, [Zw,Theorem 3.16]), that for some constants F, C, The conclusion is easily drawn from this. Now let us prove proposition 2.5: Proof. Let ψ be a smooth function on X such that ψ = 1 on {2|r − c| < ε}, and ψ = 0 on {|r − c| > ε}. Then write a = a(1 − ψ) + aψ. The support π(supp(a(1 − ψ))) is at distance at most ε and at least ε for some universal constant C > 0. Now, apply lemma 2.6 to the explicit quantization (as explained in section 1.2) of a(1 − ψ) (where the only non-vanishing terms are for the charts 0 and 1).

Ellipticity and variance bound
In this section, we complete what we have called in the introduction the first step. We use the results of the former section, as well as an ellipticity estimate similar to the one from [ZeZw], to prove that the microlocalization of the eigenfunctions on the energy surface still holds, albeit on average only.
Definition 3.1. We define, for any symbol a ∈ S 0 comp (X) and for any h > 0, λ > 0, Remark. The bound M(a + b, t) ≤ M(a, t) + M(b, t) holds, and similarly for Y .
Let us mention the following very important result: Proof. Actually, N(λ) is, up to some additive constant, the number of eigenvalues of ∆ c that are not greater than λ 2 + 1 4 . The result is then proved in [CdV3, Theorem 6]. 3.1. Ellipticity in the mean. Lemma 3.3. Let a ∈ S 0 comp (X) be a symbol and assume that a |S * X = 0. Then where the constant in the O(h 2 j ) depends only on some S 0 seminorms of a and on π(supp(a)), C is universal and Q j is the constant of Corollary 2.2.
the O referring to L 2 → L 2 operator norm, and the constant satisfies the relevant dependencies.
. Now, since the phase space support of b does not meet the wave front set of δ c (which is {r = c, η = 0}), Op h j (bψ)δ c is a smooth O(h ∞ j ) function (with the required dependencies for the constants). Besides, proposition 2.5 gives us the upper bound for Op h (b(1 − ψ))δ c 2 L 2 . Proposition 3.4 (Ellipticity in the mean). Let a j ∈ S 0 comp for every j ≥ 1 and assume that ∪ j (π(supp(a j )) ⊂ K for some fixed compact set K ⊂ X and that the family is bounded in S 0 . Assume that for each j, (a j ) |S * X = 0. Then where · S 0 is some S 0 seminorm, and C is universal.
Proof. Let I be the supremum over j ≥ 0 of the where C is the constant in (1) and K is the constant in the O(h 2 j ) of (1). Then, using Weyl's law and corollary 2.4: and we conclude by considering that for some suitable · S 0 , I ≤ sup{ a j 2 S 0 }.

3.2.
Bound on the variance. In this section, we shall prove some bounds on the variance M(a, λ) defined in Definition 3.1.
Here HS means the Hilbert-Schmidt norm. We finally apply the trace formula in A.1 (in the appendix).
Proposition 3.6. Let a ∈ S 0 comp (X). There is a universal constant C such that for all h > 0 small, is real valued such that χ = 1 on [2/5, 5/2]. From Proposition 3.4 (with the symbols a Now, since Op hτ (f ) = Op h (f τ ) (because of the quantization procedure), let us denote Since T * X |b| 2 ≤ C S * X |a| 2 , by the previous lemma, it is enough to prove that )) support, the results follows from Proposition 3.4.
Proposition 3.7 (Variance bound). If a ∈ S 0 comp (X), then for some universal constant C, Proof. Let h = 1/(2λ). Let I := S * X |a| 2 , where C is as above. Then using Cauchy Schwartz for the a term and Proposition 3.6, we get where the sum is over the k ≥ 0 such that, for some integer j ≥ 0, 2 2k−1 h ≤ h j ≤ 2 2k+1 h: in particular, if 2 2k−1 h > h 0 (ie 4 k ≤ 4h 0 λ), the corresponding term does not contribute. We conclude using again Proposition 3.2.

Egorov theorem
This section deals with step 2: similarly to [ZeZw], we want to prove that propagating some symbol a through the geodesic flow does not change M(a, λ) too much. The main difference here is the fact that the operator we study (ie the pseudo-Laplacian) is not the generator of the propagator we use. From a geometric point of view, we solve this by requiring that our symbols have a support far from the singular circle.
The main result of this section is proposition 4.4, which gives a precise statement about the idea above.

Flow invariance of the eigenfunctions.
Lemma 4.2. Let T > 0, let a ∈ C ∞ c (T * X) with supp(a) ⊂ Σ T . Then there exists some constant C > 0, depending only on a and T , such that for every j ≥ 0, and every 0 ≤ t ≤ T , Proof. Let P := (h 2 j ∆ − 1), then we have P u j = h 2 j Q j δ c + h 2 j 4 u j , and δ c is H −1 (see the beginning of section 2.1 for the definition) and (h 2 j Q j ) j is in ℓ 2 (N) (Proposition 2.4) hence bounded. Let s j (t) := Op h j (a • Φ t )u j , u j ; every s j is smooth, and In this computation, for every 0 Op h j (a)u j , u j .

Let us notice that WF
is bounded by some B > 0. Thus for some constant B depending only on a and T , |V | ≤ Bh j . Finally, The same argument as for the bound on V can be re-used to get |ih j U| ≤ B ′ h 2 j , ie |U| ≤ B ′ h j . So we have |∂ t s j (t)| ≤ (C 1 + B + B ′ )h j = Ah j uniformly in 0 ≤ t ≤ T , where clearly C 1 , B, B ′ depend only on a and T . Now, this yields |s j (t) − s j (0)| ≤ AT h j when integrating.
A direct consequence is the following: An easy argument then yields (taking into account the fact that a − a T has average zero):

Analysis far in the cusp
If we joined the main results of sections 3 and 4, we would be able to prove the main theorem for symbols with average zero. This will be done in section 6.
But if we want to prove the main theorem for general symbols in S 0 comp , we have to find some symbols with non-zero average for which the result holds. A direct proof turns out to be difficult: so we will exhibit symbols s with average arbitrarily close to some non-zero constant and such that lim sup λ→∞ M(s, λ) is arbitrarily close to 0.
Before, we need to introduce some cutoff functions: let us set some R > e c 0 , and χ R be a smooth nondecreasing function such that (2) χ R (r) = 1 on [R + 1, ∞), χ R (r) = 0 on (−∞, R]. Let φ R : X → [0; 1] be a smooth function that is zero outside the cusp and such that if r > c 0 , θ ∈ R/Z, φ R (r, θ) = χ R (r). Note that 1 − φ 4 R ∈ C ∞ c (X). We will show the following: Proposition 5.1. There exists a universal constant C > 0 such that for any R > e c , Our first step is to understand where the mass of the u j is localized. Let us write, for every j ≥ 0, r > c 0 , θ ∈ R/Z, This is similar to the expansion of u j as a Fourier series in θ, but the coefficients were renormalized, so that θ∈R/Z |u j | 2 dv g = k∈Z |v j,k | 2 (r) dr.
The bound (4) is a direct consequence of the fact that u j 2 L 2 = 1. As for (5), note that in the cusp, (∂ r u j )(r, θ) = e r/2 k∈Z 1 2 v j,k + v ′ j,k e 2ikπθ , so and the proof is complete.
Let now h > 0 be very small, and λ = h −1 . Let ω 2 j = (hλ j ) 2 , for every j such that h/2 ≤ h j ≤ 2h (we say that j is in the range): then, if h is small enough, ω j is between 1/2 and 2.
Lemma 5.3. In the cusp region r > c 0 , (−h 2 ∂ 2 r − ω 2 j + (2khπ) 2 e 2r )v j,k = 0. We can therefore extend v j,k as a smooth function of the whole real line with the same properties.
This means that microlocally, the mass of v j,k is concentrated near the curve ξ 2 + (2khπ) 2 e 2r = ω 2 j : Lemma 5.4. There exists a constant C R depending only on R (and not on h) such that for every j in the range, for every 1 ≤ |k| ≤ 3λ, Proof. Let us denote here Now, using [Zw,Theorem 4.23], (remember that inside the symbol, when (2|k|hπ)e r > C R , the χ 1 term destroys everything, so all relevant S 0 seminorms are bounded uniformly in j, k, h, as long as h ≤ |k|h ≤ 3 and j is in the range) we may write, for some constant K R > 0 depending only on R: 1 L 2 ). Now, when h is small, 1 ≤ |k| ≤ 3λ, j is in the range, the symbols a * := a − a + are bounded by a constant depending only on R in respectively the class of symbols S( ξ −2 ), S 0 ( ξ 2 ), S(1) (using the notation of [Zw,Section 4.4.1]), thus by [Zw,Theorems 4.18,4.23] we have Op h (a − ) Op h (a + ) − Op h (a * ) L 2 →L 2 ≤ C R h, for some constant C R > 0 depending only on R. Therefore, we get that for some constant C R > 0 depending only on R, L 2 (c 0 ,+∞) ).
Corollary 5.5. There exists a constant C R depending only on R such that when j is in the range Proof. It is a consequence of the previous lemma and of corollary 5.2, more precisely estimates (4) and (5).
The result hereafter is the main property of localization we were aiming at: it tells us that each φ 4 R u j is localized along his lowest modes in the cusp, and each of these modes is microlocalized in a compact zone that depends very little on j: that is, v j,k is microlocalized in the zone |ξ| ≤ 4, R ≤ r ≤ R/2 − ln 2h|k|π.
Let us first define the operator A h,k := Op h (χ 2 (ξ)χ 2 R (r)χ h,k (r) 2 ). Proposition 5.6. Let j ≥ 0 be in the range. For some universal constant C > 0, and some constant C R > 0 depending only on R, We split the proof in several steps.
Lemma 5.7. Let j ≥ 0 be in the range. Then Proof. Using (3), we obtain the sequence of inequalities which proves the claim.
Lemma 5.8. Let j ≥ 0 be in the range. Then the following holds true: Proof. Using again (3), which proves the claim. Now, we can prove Proposition 5.6: Proof of Proposition 5.6. One easily sees that: Now, we split the sum between the |k| > 3λ, the sum of which is not greater than e −2R (Lemma 5.7), and the 1 ≤ |k| ≤ 3λ. Moreover, if 1 ≤ |k| ≤ 3λ, L 2 . From Lemma 5.8 we see that the first term contributes only as 6h 2 + 80e −R , now we have to assess the second term. Now, We saw from Corollary 5.5 that the second term contributes only as C R h 2 , where C R > 0 depends only on R, and this ends the proof. Now, we turn the pointwise localization estimate we have on the φ 4 R u j into an average estimate on j. Thanks to Hilbert-Schmidt norm estimates (operators will always be considered as from the relevant L 2 spaces into themselves) we obtain significantly better results.
Let us define the operator A ′ h,k := A h,k χ R , then let Aw(r, θ) := χ R (r)e r/2 1≤|k|≤3λ (A h,k (χ R w k ))(r)e 2ikπθ for every r > c 0 , θ ∈ R/Z, where w(r, θ) = e r/2 k∈Z w k (r)e 2ikπθ . Proposition 5.9. There exist constants C > 0 universal, and C R > 0 depending on R only such that: Proof. From Proposition 5.6, we know that for any j in range, φ 4 R u j 2 ≤ 200e −R + C R h 2 + 4 Au j 2 L 2 . So, using Weyl's law, and the fact the the (u j ) are orthonormal, Now, let (f p ) be an orthonormal basis of L 2 (R, ∞), let (g q ) be an orthonormal basis of {f ∈ L 2 (X), ½ r>R f = 0}. Then the family of all e r/2 f p (r)e 2ikπθ and the g q , is an orthonormal basis of L 2 (X). Therefore, when f is an element of this orthonormal basis, we realize that only when f = e r/2 f p (r)e 2ikπθ , 1 ≤ |k| ≤ 3λ, Af 2 does not vanish. Besides, it will always be lower or equal than ( ) . From this, it follows that which completes the proof. Now, it is easy to give an upper bound on the Hilbert-Schmidt norm of the operators, and to turn it into a complete estimate: Proposition 5.10. The following bound holds true for |k| ≤ 3λ: Let ψ ∈ C ∞ c (R 2 ),ψ be its Fourier transform with respect to its second variable. Let T = Op h (ψ)χ R . For any f ∈ L 2 (R), Now, we obtain This completes the proof.
Corollary 5.11. The following estimate holds true: where C > 0 is universal and C R depends only on R.
Proof. Using Stirling's formula, assuming that 2πhe R/2 ≤ 1 (else anyway the sum is just zero and the bound holds), for some universal constant C, Corollary 5.12. For some universal constant C > 0 and some constant C R depending only on R, one has Y (φ 4 R , h) 2 ≤ Ce −R/2 + C R h.
Proof. It is a consequence of all that precedes.
Proof of Proposition 5.1. We write The second term is lower than some Ce −2R for some constant C independent of R, because dv g (r, θ) = e −r drdθ. The first term is not greater than which by Corollary 5.12 is not greater than (using again Weyl's law) C λ 2 1≤4 k ≤4h 0 λ Ce −R/2 2 2−4k λ 2 + C R 2 1−2k λ ≤ Ce −R/2 + C R 1 λ .
This concludes the proof.

Proof of the main theorem
Let a ∈ S 0 comp and assume first that S * X a dµ = 0. We let T > 0 and ε > 0. We may write a = a 1 + a 2 , where a 1 is S −∞ satisfies supp(a 1 ) ⊂ Σ T , and S * X |a 2 | 2 ≤ ε 2 . Then, as λ → ∞, M(a, λ) ≤ M(a 1 , λ) + Cε ≤ M( a 1 T , λ) + Cε ≤ M( a T , λ) + 2Cε ≤ C( a T L 2 (S * X) + 2ε) where we used Proposition 3.7 for the last inequality. So we let ε go to zero first, and then T go to ∞ and the L 2 ergodic theorem proves the result.
In the general case, let a ∈ S 0 comp (X) and let α := S * X a. Let us denote, for every R > e c 0 , I R := S * X φ 8 R . Then I R = O(e −R ), thus, if R is large enough, a R = a− α 1−I R (1−φ 8 R ) belongs to S 0 comp with S * X a R = 0. Letting R → ∞ yields M(a, λ) → 0 and the proof of Theorem 1 is complete.
Proof. We may assume that a is real-valued. Let u ∈ C ∞ c (X). Let x ∈ X, let i ≥ 0 be such that x ∈ U i . Using a change of variables, the following holds: where m i (x, y) = ϕ −1 i ϕ i (x) + ϕ i (y) 2 , (ϕ i ) * ,y is the linear mapping from T * y X into R 2 such that the thus enhanced ϕ i is a symplectomorphism, (ϕ i ) y→m i (x,y) = (ϕ i ) −1 * ,m i (x,y) • (ϕ i ) * ,y .
f i (x, y) = χ i (x)χ i (y) T * y X a(m i (x, y), (ϕ i ) y→m i (x,y) (ξ)) exp i h (ϕ i (x) − ϕ i (y)) · (ϕ i ) * ,y (ξ) dξ, the measure dξ being the Liouville measure. The integral is convergent because a has compact support. Let F be the set of all indices i such that the space support of a meets U i ; F is finite. For every i / ∈ F , f i = 0; for every i ∈ F , f i : (supp χ i ) 2 → C is continuous, hence bounded; thus, F = i f i is in L ∞ (X 2 ).