Invariant Distributions and Time Averages for Horocycle Flows

INVARIANT DISTRIBUTIONS AND TIME AVERAGES FOR HOROCYCLE FLOWS

LIVIO FLAMINIO AND GIOVANNI FORNI †

ABSTRACT. There are infinitely many obstructions to existence of smooth solutions of the cohomological equation Uu = f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann M surface of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows.

1. INTRODUCTION The classical horocycle flow is the flow on (compact) homogeneous spaces of the form 1 t Γ\PSL(2, ) given by right multiplication by the one-parameter subgroup . R 0 1 The ergodic properties of this flow have been an active subject of study for a long time since it has the intriguing characteristic of presenting at the same time some of properties of “orderly” ergodic systems, (zero entropy [21], minimality [23, 19, 57], unique ergodicity [14, 35, 36] etc.), and some of the properties which are more frequently associated with “chaotic” systems, (e.g. multiple strong mixing [44, 38, 37]). The works of M. Ratner in the 80-90’s [48, 47, 50] have also put in evidence the tight relation of the ergodic properties of the horocycle flow with the geometry of the underlying space Γ\PSL(2, R). Representation theory is a natural tool for the study of flows on homogeneous spaces ([16, 17, 41, 2], etc.); in particular, M. Ratner [49] and C. Moore [42] have obtained precise estimates for the mixing rate of the geodesic and horocycle flow and, more recently, M. Burger [5] has proved uniform upper bounds for the deviation of ergodic averages of sufficiently regular functions along the orbits of the horocycle flow on open complete surfaces with positive injectivity radius and on compact surfaces. Invariant distributions for geodesic flows on manifolds of constant curvatures were studied in [11]. In this article we establish precise asymptotics for the ergodic averages of sufficiently regular functions along the orbits of the horocycle flow on compact surfaces, improving on Burger’s results. For cuspidal horocycles on non-compact surfaces of finite area such asymptotics were obtained by P. Sarnak in [51], inspired by D. Zagier [58], by a method based on Eisenstein series. Our approach does not use automorphic forms and it is based on the study of the cohomological equation and of invariant distributions for the horocycle flow,

Date: June 6, 2002. 1991 Mathematics Subject Classification. 37D40, 37A20, 37A30, 22E46, 58J42. Key words and phrases. Horocycle Flow, Cohomology of flows, Ergodic averages. † Alfred P. Sloan Research Fellow. G. Forni was supported by NSF grant # DMS-9704791. 1 2 LIVIO FLAMINIO AND GIOVANNI FORNI † via representation theory. Such a method yields sharp results in the compact case while in the non-compact (finite area) case we obtain a generalisation of Sarnak’s result to arbitrary horocycle arcs. G. Margulis and D. Kleinbock [31] proved, for finite volume quotients of general semisimple Lie groups, exponential decay of the deviation from equidistribution for smooth measures supported on a compact subset of any horosphere under the action of diagonal subgroups. Our results yield precise asymptotics in the particular case of quotients of the semisimple group SL(2, R). We recall that for a flow {φt} we say that G is a coboundary if there exists a solution F (continuous , L1, L2, depending on the context) of the cohomological equation

d t (1) F ◦ φ = G. dt t=0 The cohomological equation arises in several problems in dynamics e.g. in the study of the existence of invariant measures, in conjugacy problems, in the study of repararametrisa- tions of flows etc. It is well understood in two opposite dynamical setups: • linear toral flows • Anosov flows In the first case, provided that a Diophantine condition on the rotation number is satisfied (and generically this is the case), the only obstruction to the solution of the cohomological equation is given by the unique invariant measure: given a sufficiently regular G of mean zero there exist a solution F of the equation (1) with a loss of regularity controlled by the Diophantine condition. Hence, for any sufficiently regular function G, ergodic averages converge to the mean with an error bounded by Const./T . The Denjoy-Koksma inequality [32] yields estimates for functions which are only of bounded variation. In the Anosov case the celebrated Livshitz Theorem [33, 6, 20, 7] states that a Hölder function G is a coboundary iff G integrates to zero along every periodic orbit, or, equiv- alently, if it has mean zero with respect to all invariant measures. Thus, even in this case the only obstructions to the solution of the cohomological equation are represented by measures. In contrast, the strong ergodic properties of these flows are reflected in the fact that ergodic averages converge to the mean as stochastically independent processes do, i.e. the central limit Theorem holds [46]. The horocycle flow, differently from the cases above, has invariant distributions which are not signed measures [29]. In the spirit of [12, 13] we find that these distributions control the asymptotics of ergodic averages. Furthermore the exponents of the asymptotic expansion coincide with the Sobolev order of the invariant distributions.

1.1. Statement of the results. The group PSL(2, R) acts (on the left) by isometries and holomorphically on Poincaré upper half plane 2 2 H = ({z ∈ C | =z > 0}, |dz| /(=z) ). The quotient space M = Γ\H, by a discrete subgroup of PSL(2, R) acting without fixed points, is a Riemannian manifold of constant curvature −1 and also a complex curve (a Riemann surface). The isometric action of PSL(2, R) on H induces a free transitive action on the unit tangent bundle SH of H and by fixing (i, i) as an origin in SH we identify PSL(2, R) TIME AVERAGES FOR HOROCYCLE FLOWS 3 to SH. By means of this identification elements of the Lie algebra sl(2, R) of PSL(2, R) are identified with the generators of some flows on SH which project to flows on the quotient space SM = Γ\SH ≈ Γ\PSL(2, R). The sl(2, R) matrices 0 1 1/2 0 U = ,X = 0 0 0 −1/2 U define respectively the generators of the (stable) horocycle flow {φt } and of the geodesic X flow {φt } on the unit tangent bundle SM of an hyperbolic surface M := Γ\H. For all hyperbolic surfaces of finite area, the spectrum σ(∆M ) of the Laplace-Beltrami operator ∆M on M has 0 as a simple eigenvalue and there is a “spectral gap”, i.e. the bottom of the non-zero spectrum µ0 := inf σ(∆M ) \{0} is strictly positive. If M is compact, by standard elliptic theory, its spectrum is pure point and discrete with eigenvalues of finite multiplicity and it satisfies the Weyl asymptotics. Examples of ‘pinched’ compact surfaces with arbitrarily small spectral gap µ0 > 0 were first constructed by B.Randol [45] (see [54], [9] for more recent sharper results). If M is not compact, its spectrum can be described as follows (see for instance [53] and references therein, in particular D.Hejal [24]): • Lebesgue spectrum on the interval [1/4, ∞[ with multiplicity equal to the number of cusps of M; • possibly, finitely many eigenvalues of finite multiplicity in the interval ]0, 1/4[; • possibly, embedded eigenvalues of finite multiplicity in the interval [1/4, ∞[. Examples of non-compact ‘pinched’ surfaces with an arbitrarily small spectral gap, hence with non-empty (pure point) spectrum in the open interval ]0, 1/4[, were already constructed by A. Selberg [55] (see [9] for a more recent approach). In the non-compact case, the Weyl asymptotics for the pure point spectrum fails conjecturally for a generic subgroup in any given Teichmüller space with the exception of the Teichmüller space of the once-punctured torus [8], [52]. + The picture is quite different for a relevant class of arithmetic subgroups. For all N ∈ Z , let Γ(N) := {g ∈ PSL(2, Z) | g = id mod N}. The subgroups Γ < P SL(2, R) such that + Γ(1) > Γ > Γ(N),N ∈ Z , are called congruence subgroups. For all such subgroups, Selberg’s spectral gap conjecture [55] claims that there are no eigenvalues in ]0, 1/4[, hence µ0 = 1/4. Selberg’s conjecture is known to be true for all congruence subgroups Γ > Γ(N), N ≤ 17 [27], [28]. In the case of a general congruence subgroup, the classical Selberg’s lower bound µ0 ≥ 3/16 [55] was significantly improved by W. Luo, Z. Rudnick and P. Sarnak [34] who proved that µ0 ≥ 171/784 ≈ 0.218 ... . Recently, H. Kim and P. Sarnak [30] have announced a further step in the direction of the Selberg’s conjecture: 975 µ ≥ ≈ 0.238 ... 0 4096 For all congruence subgroups, the Weyl asymptotics holds for the pure point spectrum [26], [10]. As we shall see the (pure point) spectrum within the interval ]0, 1/4[ is especially relevant to the asymptotics of ergodic averages of horocycles, both in the compact and the non- compact case. We shall denote by σpp the set of all eigenvalues of the Laplace operator ∆M 4 LIVIO FLAMINIO AND GIOVANNI FORNI † and by C the set of cusps of M. If M is compact, the Laplacian ∆M has pure point spectrum supported on the set σpp and C = ∅. Let L2(SM) be the space of all complex-valued square-integrable functions with respect s to the PSL(2, R)-invariant volume on SM. Let W (SM), s ∈ R, be the Sobolev spaces of functions f ∈ L2(SM) with ∆s/2f ∈ L2(SM), where ∆ is an elliptic second order s 0 element of the enveloping algebra of sl(2, R) (a Laplacian). The dual space (W (SM)) of linear bounded functionals on W s(SM) is isomorphic to the space W −s(SM), according to the standard theory of Sobolev spaces [1]. Let E0(SM) be the dual space of the space C∞(SM) of infinitely differentiable functions on SM. The space 0 I(SM) := D ∈ E (SM) | LU D = 0 of U-invariant distributions is completely determined by the spectrum of the Laplacian ∆M and by the genus of M, as follows. Theorem 1.1. The space I(SM) has infinite countable dimension. There is a decomposition M M M (2) I(SM) = Iµ ⊕ In ⊕ Ic + µ∈σpp n∈Z c∈C where (1) for µ = 0, the space I0 is spanned by the PSL(2, R)-invariant volume; + − ± −s (2) for 0 < µ < 1/4, there is a splitting Iµ = I ⊕ I , where I ⊂ W (SM), √ µ µ µ 1± 1−4µ iff s > 2 , and each subspace has dimension equal to the multiplicity of µ ∈ σpp; −s (3) for µ ≥ 1/4, the space Iµ ⊂ W (SM), iff s > 1/2, and it has dimension equal to twice the multiplicity of µ ∈ σpp; + −s (4) for n ∈ Z , the space In ⊂ W (SM), iff s > n and it has dimension equal to twice the rank of the space of holomorphic sections of the n-th power of the canonical line bundle over M; −s (5) for c ∈ C, the space Ic ⊂ W (SM), iff s > 1/2, and it has infinite countable dimension. The Sobolev regularity of U-invariant distributions can be summarised as follows. The Sobolev order of a distribution D ∈ E0(SM), is the extended real number + −s SD := inf s ∈ R | D ∈ W (SM) . By Theorem 1.1, we have:  √ 1±< 1−4µ if D ∈ I± , for µ > 0 ;  2 µ + (3) SD = n if D ∈ In , for n ∈ Z ;  1 2 if D ∈ Ic , for c ∈ C ;

Let µ0 > 0 be the bottom of the non-zero spectrum of ∆M . Let ν0 ∈ [0, 1[ be deﬁned as

(√ 1 1 − 4µ0, if µ0 < 4 ; (4) ν0 := 1 0, if µ0 ≥ 4 . TIME AVERAGES FOR HOROCYCLE FLOWS 5

A complete set of obstructions to the existence of smooth solutions of the cohomological equation Uf = g, for functions g ∈ W s(SM), is given by the following space of invariant distributions: s −s I (SM) := D ∈ W (SM) | LU D = 0 . , The space Is(SM) is completely determined by Theorem 1.1.

1+ν0 Theorem 1.2. For all s > 2 and all t ∈ R, there exists a constant C := C(ν0, s, t) > 0 such that, for all g ∈ W s(SM),

1+ν0 • if t < − 2 and g has zero average on SM, or • if t < s − 1 and D(g) = 0, for all D ∈ Is(SM), then the cohomological equation Uf = g has a solution f ∈ W t(SM) which satisﬁes the Sobolev estimate

k f k t ≤ C k g k s. A solution f ∈ W t(SM) of the cohomological equation Uf = g is unique up to additive 1−ν0 constants if and only if t ≥ − 2 . By deﬁnition, invariant distributions of order S > 0 are obstructions to the existence of solutions of the cohomological equation of (Sobolev) regularity S +1. However, it turns out that they obstruct also the existence of solutions of lower regularity. In fact, Theorem 1.2 is sharp, in the following sense:

s 1+ν0 Theorem 1.3. Let g ∈ W (SM), s > 2 , and let D ∈ I(SM) be an U-invariant t distribution of order SD < s. If the equation Uf = g has a solution f ∈ W (SM) for any t ≥ SD − 1, then D(g) = 0. The following remarkable result holds:

X Theorem 1.4. The action of the geodesic one-parameter group {φt } on the distributional space I(SM) has a spectral representation, if 1/4 6∈ σpp, and it has a generalized spectral representation with a ﬁnite number of 2 × 2 Jordan blocks, if 1/4 ∈ σpp. The splitting (2) of X I(SM) is {φt }-invariant, hence the subspaces M M M (5) Ipp := Iµ , Id := Ipp ⊕ In , IC := Ic . + µ∈σpp n∈Z c∈C X X are {φt }-invariant. The spectrum of {φt } on Id is discrete, while the spectrum on IC is Lebesgue with ﬁnite multiplicity equal to the number of cusps. X The discrete spectrum of {φt } on Id can be described as follows. For µ = 0, the X subspace I0 is generated by the PSL(2, R)-invariant, hence {φt }-invariant, volume form. + − For all µ ∈ σpp \{0}, there is a splitting Iµ = Iµ ⊕ Iµ into subspaces of equal dimension, which coincides for 0 < µ < 1/4 with the splitting induced by the Sobolev order (see ± Theorem 1.1, (2)), such that the following holds. For all µ 6= 1/4, the subspaces Iµ are X eigenspaces for {φt }, in fact √ 1 ± 1 − 4µ (6) φX | I± = exp t I. t µ 2 6 LIVIO FLAMINIO AND GIOVANNI FORNI †

X If 1/4 ∈ σpp, the subspace I1/4 6= {0} and {φt } | I1/4 has a Jordan normal form. In + − ± ± fact, I1/4 is generated by pairs {D , D } such that D ∈ I1/4 and

+ + X D − t 1 0 D (7) φt − = e 2 t − , D − 2 1 D

+ X For all n ∈ Z , the subspace In is an eigenspace for {φt }. In fact, X −nt (8) φt | In = e I.

X −t/2 The Lebesgue spectrum of the operator φt on IC is supported on the circle of radius e in C, for all t ∈ R. X It follows from Theorem 1.4 that the action of the geodesic flow {φt } on the infinite dimensional space I(SM) has a well-defined Lyapunov spectrum and an Oseledec’s decomposition. It turns out by comparing the formulae in Theorem 1.4 with the Sobolev orders (3) of invariant distributions, that the Lyapunov exponent of any generalized eigenvector D ∈ Id of the geodesic flow and of any D ∈ IC is equal to the negative of its Sobolev order. Theorems 1.1-1.4 are derived from abstract results on unitary representations of the Lie group PSL(2, R) (Theorems 3.2, 4.1 and 4.6). Theorem 3.2 states that the invariant distributions for the “horocycle vector field” in each irreducible representation of parameter µ 6= 1/4 are generated by eigendistributions of the “geodesic vector field”. Such eigendistributions correspond to the conical distributions for the Lie group SL(2, R) in the sense of S. Helgason [25], §2. In the exceptional case of irreducible representations with parameter µ = 1/4, the subspace of conical distributions is one-dimensional, while the subspace of horocycle-invariant distributions has dimension 2. Theorem 1.1 is based on a direct analysis of the Sobolev regularity of such (conical) distributions and Theorem 1.4 on the explicit computation of the action of the geodesic flow on invariant distributions. The abstract Theorem 4.1, on existence of smooth solutions of the cohomological equation, holds under the only condition that the Casimir operator associated with the unitary representation has a ‘spectral gap’. Theorem 1.2 on solutions of the cohomological equation on hyperbolic surfaces of finite area follows immediately, since the gap property holds. Theorem 4.6 is a converse result which proves that, in each irreducible component, invariant distributions are obstructions to the existence of solutions of smoothness lower than expected. This result shows that, in a sense, our result on existence of solutions of the cohomological equation is optimal as far as the loss of Sobolev regularity is concerned. Theorem 1.3, the corresponding statement for hyperbolic surfaces, plays a non-trivial role in obtaining L2 lower bounds for ergodic averages. X Let B ⊂ Id be a basis of (generalized) eigenvectors for {φt } on Id with the following + properties. For all µ ∈ σpp \{1/4} and all n ∈ Z , the sets Bµ := B ∩ Iµ, Bn := B ∩ In X X + are basis of eigenvectors for {φt } | Iµ and {φt } | In. If 1/4 ∈ σpp, the set B1/4 := + X + − − B ∩ I1/4 is a basis of eigenvectors for {φt } | I1/4, the set B1/4 := B ∩ I1/4 is a basis X − + − of generalized eigenvectors (of order 2) for {φt } | I1/4 and B1/4 := B1/4 ∪ B1/4 is a union + − ± ± of pairs {D , D } such that D ∈ B1/4 and formula (7) holds. TIME AVERAGES FOR HOROCYCLE FLOWS 7

1 Let I+(SM) be the complement of the line generated by the invariant volume form 1 −1 1 1 in I (SM) ⊂ W (SM). The subset B+ := B ∩ I+(SM) is a basis of (generalized) X 1 1 eigenvectors for {φt } | I+(SM). The basis B, B+ have the following decompositions:

[ [ 1 [ (9) B = Bµ ∪ Bn , B+ = Bµ . + µ∈σpp n∈Z µ∈σpp\{0} In the compact case, Theorems 1.2, 1.3 and 1.4 imply the following quantitative unique ergodicity result. U Theorem 1.5. The horocycle ﬂow {φt } on the unit tangent bundle SM of a compact hyperbolic surface M has a deviation spectrum in the following sense. For any s > 3 and for + s all (x, T ) ∈ SM × , there exist a sequence of real-valued functions {c (x, T )} 1 R D D∈B+ s s −s and distributional functions D1(x, T ) ∈ I1, R (x, T ) ∈ W (SM) such that, for all f ∈ W s(SM) and all T ≥ 0,

1 Z T Z X (10) f(φU (x))dt = f dvol + cs (x, T )D(f) T −SD + T t D 0 SM 1 + D∈B+\B1/4 X Ds(x, T )(f) log+ T + Rs(x, T )(f) + cs (x, T )D(f) T −SD log+ T + 1 . D T + D∈B1/4 s s s The functions cD, D1 and R satisfy the following uniform estimates. There exists C(s) > 0, + such that, for all (x, T ) ∈ SM × R , X s 2 |cD(x, T )| ≤ C(s); 1 D∈B+ s k D1(x, T ) k −s ≤ C(s); s k R (x, T ) k −s ≤ C(s) . 1 For every invariant distribution D ∈ B+, there exists C(D, s) > 0 such that, for sufﬁciently large T > 0, s k cD(·,T ) k 0 ≥ C(D, s) . Theorem 1.5 implies that the Central Limit Theorem does not hold for the horocycle ﬂow on compact hyperbolic surfaces. Corollary 1.6. There exist zero average functions f ∈ C∞(SM) such that any weak limit (as T → ∞) of the probability distributions of the functions

R T U 0 f(φt (·)) dt R T U k 0 f(φt (·)) dt k 0 has compact support 6= {0}. In the non-compact case, we obtain by the same methods the following generalisation of a result proved by P. Sarnak [51] for closed cuspidal horocycle arcs. 8 LIVIO FLAMINIO AND GIOVANNI FORNI †

1/2 Let I+ (SM) be the orthogonal complement of the line generated by the invariant vo- 1/2 −1/2 1/2 1/2 lume in the space I (SM) ⊂ W (SM). The subset B+ := B ∩I+ (SM) is a basis X 1/2 of eigenvectors for {φt } | I+ (SM). Let σpp(0, 1/4) := σpp∩]0, 1/4[. By Theorem 1.1, 1/2 [ (11) B+ = Bµ . µ∈σpp(0,1/4) 1/2 It follows that B+ has finite cardinality equal to the total multiplicity of the eigenvalues of the Laplacian ∆M within the interval ]0, 1/4[. In particular, it is empty for hyperbolic surfaces satisfying the Selberg’s conjecture. Let γx,T be the uniformly distributed probability measure along the unstable horocycle + X arc in SM of initial point x ∈ SM and length T > 0. For all t ∈ R , let φt (γx,T ) X be the push-forward of γx,T under the action of the geodesic flow. The measure φt (γx,T ) is equal to the uniformly distributed probability measure along the horocycle arc of initial X t point φt (x) and length Tt = e T . The following result is derived from Theorem 5.14, which contains more precise asymptotics for the push-forward of a horocycle arc depending on the rate of escape in the cusps of its endpoints. s Theorem 1.7. For any s > 3, there exist bounded coefficients {cD(x, T, t)} 1/2 and a D∈B+ bounded distributional coefficient Rs(x, T, t) ∈ W −s(SM) such that, for all f ∈ W s(SM) and all t ≥ 0, Z X X s −SD (12) φt (γx,T )(f) = f dvol + cD(x, T, t) D(f) Tt + SM 1/2 D∈B+ s −1/2 2 + R (x, T, t)(f) Tt log Tt . + + There exists a continuous function C := Cs : SM × R → R such that for all t ≥ 0, X s 2 |cD(x, T, t)| ≤ C(x, T ); 1/2 D∈B+ s k R (x, T, t) k −s ≤ C(x, T ) . 1/2 For every invariant distribution D ∈ B+ , there exists a constant C(D, s) > 0 such that, if T > 0 is sufficiently large, then for all t ≥ 0, −1 s C(D, s) ≤ k cD(·, T, t) k 0 ≤ C(D, s) .

In the particular case that γx,T is supported on a closed cuspidal horocycle, we prove by s the same methods that all the coefﬁcients cD(x, T, t) are constant functions of t ≥ 0 and that −1/2 the remainder term is O(Tt log Tt) (Proposition 5.15). Hence, we obtain an asymptotic formula with remainder purely by methods based on representation theory. We remark that 1/2 P. Sarnak [51] obtained a sharper asymptotics with remainder o(Tt ) by Eisenstein series methods (Rankin-Selberg method), taking into account the explicit spectral decomposition of cuspidal horocycles and the absolute continuity of the continuous part of the spectrum. In the particular case of the modular group, the remainder term for cuspidal horocycles is −3/4+ O(Tt ), for all > 0, iff the Riemann hypothesis holds [58], [51]. TIME AVERAGES FOR HOROCYCLE FLOWS 9

2. FOURIER ANALYSIS 2.1. Sobolev spaces. We choose as generators for sl(2, R) the elements 1/2 0 0 −1/2 0 1/2 X = ,Y = , Θ = . 0 −1/2 −1/2 0 −1/2 0 Recall the commutations rules: (13) [X,Y ] = −Θ, [Θ,X] = Y, [Θ,Y ] = −X. The basis element X ∈ sl(2, R) is the “geodesic vector field”, in the sense that it corre- sponds to the generator of the geodesic flow on SH. The elements 0 0 0 1 V = = −Y − Θ,U = = −Y + Θ 1 0 0 0 are respectively the unstable and the stable “horocycle vector fields”, in the sense that they correspond to the generators of the unstable and the stable horocycle flows on SH. In fact, the following commutation relations hold: (14) [X,V ] = −V, [X,U] = U. The Laplacian determined by the basis {X,Y, Θ} is the elliptic element of the enveloping algebra of sl(2, R) defined as (15) ∆ := −(X2 + Y 2 + Θ2) . Let H be the (Hilbert) space of a unitary representation of PSL(2, R). Any element of the Lie algebra sl(2, R) acts on H as an essentially skew-adjoint operator and the Laplacian ∆ + acts as an essentially self-adjoint operator [43]. The Sobolev space of order s ∈ R is the maximal domain W s(H) ⊂ H of the operator (I + ∆)s/2 endowed with the inner product s (16) hf, gis := h(I + ∆) f, giH . The spaces W s(H) are Hilbert spaces which coincide with the completion of the sub- ∞ space C (H) ⊂ H of infinitely differentiable vectors with respect to the norm k f k s := s/2 ∞ k (I + ∆) f k H, induced by the inner product (16). The subspace C (H) coincides with the intersection of the spaces W s(H) for all s ≥ 0. The Sobolev spaces with negative exponent W −s(H), s > 0, defined as the Hilbert space duals of the spaces W s(H), are subspaces of the space E0(H) of distributions, defined as the dual space of C∞(H). The Sobolev order of a distribution D ∈ E0(H) is the extended real number

−s (17) SD := inf{s ∈ R | D ∈ W (H)} . 2.2. Direct decompositions. The Casimir operator, which generates the centre of the enveloping algebra of sl(2, R), is the element 2 2 2 (18) := −X − Y + Θ . + 2 + The Casimir operator acts as a constant µ ∈ R ∪{−n +n | n ∈ Z } on the Hilbert space of each irreducible unitary representation and its value classiﬁes all non-trivial irreducible unitary representations according to three different types. Let Hµ be the Hilbert space of an irreducible unitary representation on which the Casimir operator takes the value µ ∈ + 2 + R ∪ {−n + n | n ∈ Z }. The representation is said to belong to the principal series 10 LIVIO FLAMINIO AND GIOVANNI FORNI † if µ ≥ 1/4, to the complementary series if 0 < µ < 1/4 and to the discrete series if µ ≤ 0 [15, 3]. Let H be the Hilbert space of any non-trivial unitary representation of PSL(2, R). Since the Casimir operator is in the centre of the enveloping algebra of sl(2, R) and acts on H as an essentially self-adjoint operator, there exists a PSL(2, R)-invariant direct integral decomposition [39, 40, 18], Z (19) H = Hµ ⊕ with respect to a positive Stieltjes measure ds(µ) over the spectrum σ(). The Casimir operator acts as the constant µ ∈ σ() on every Hilbert space Hµ. The representations induced on Hµ do not need to be irreducible. In fact, Hµ is in general the direct sum of an (at most countable) number of unitary representations equal to the spectral multiplicity of µ ∈ σ(). All the operators in the enveloping algebra are decomposable with respect to the direct integral decomposition (19). Hence there exists for all s ∈ R an induced direct decomposition of the Sobolev spaces: Z s s (20) W (H) = W (Hµ) ⊕ with respect to the measure ds(µ). The existence of the direct integral decompositions (19), (20) allows us to reduce our analysis of invariant distributions and of the cohomological equation for the horocycle vector ﬁeld to irreducible unitary representations.

2.3. Hyperbolic surfaces. If M is a hyperbolic surface of finite area, the spectrum σ(∆M ) of the Laplacian on M has a pure point discrete component of finite multiplicity and an absolutely continuous component on the interval [1/4, ∞[ with finite multiplicity equal to the number of (standard) cusps of M [24]. Hence, the Laplacian ∆M has a “spectral gap”, in the sense that σ(∆M ) \{0} has a lower bound µ0 > 0. Examples of ‘pinched’ surfaces with µ0 < 1/4 were constructed in [54, 45, 9]. We shall denote σpp the pure point spectrum of ∆M and C will denote the (finite) set of cusps of M. If M is compact, then the Laplacian has pure-point spectrum supported on the set σpp and C = ∅. 2 There is a standard unitary representation of PSL(2, R) on the Hilbert space L (SM). The corresponding Sobolev spaces W s(SM) have the following splitting as a direct integral of irreducible representations (see (19) and (20)):

s M s (21) W (SM) = W (Hµ)⊕ µ∈σpp M s s M s ⊕ (W (H+n) ⊕ W (H−n)) ⊕ W (Hc) . + n∈Z c∈C For µ = 0, the sub-representation Hµ is the trivial representation, which appears with multiplicity 1, otherwise the sub-representations Hµ, H±n and Hc are not yet (in general) irreducible. + + Let mµ ∈ Z denote the multiplicity of an eigenvalue µ ∈ σpp \{0} and mn ∈ Z denote the dimension of the space of holomorphic (anti-holomorphic) sections of the n-th TIME AVERAGES FOR HOROCYCLE FLOWS 11 power of the canonical line bundle, which can be computed by the Riemann-Roch Theorem. We have:

mµ s M s (i) W (Hµ) = W (Hµ ); i=1 mn (22) s M s (i) W (H±n) = W (H±n); i=1 Z s s W (Hc) = W (Hc(µ)) dsc(µ) . ⊕ (i) The sub-representations Hµ are irreducible with Casimir parameter µ > 0, hence they belong to either the principal series (µ ≥ 1/4) or to the complementary series (µ0 ≤ µ < (i) (i) 2 1/4); the sub-representations H+n, H−n are irreducible with Casimir parameter −n + + n, n ∈ Z , hence they belong to the holomorphic, respectively to the anti-holomorphic, discrete series; the sub-representations Hc(µ) are irreducible with Casimir parameter µ ≥ 1/4, hence they belong to the principal series, and, for all c ∈ C, the Stieltjes measures dsc(µ) are supported on [1/4, +∞[ and are absolutely continuous.

2.4. Orthogonal basis. Let Hµ be the Hilbert space of an irreducible unitary representation + 2 + of Casimir parameter µ ∈ R ∪ {−n + n | n ∈ Z }. In order to construct a convenient orthogonal basis of Hµ, we consider the following elements of the enveloping algebra

η+ = X − iY, η− = X + iY , which have the property of raising and lowering eigenvalues of −iΘ. In fact, from

[−iΘ, η±] = ±η± it follows that, if −iΘu = ku, then

−iΘ(η±u) = η±(−iΘu) + [−iΘ, η±]u = kη±u ± η±u = (k ± 1)(η±u) .

If µ ≥ 0 (i.e. if Hµ belongs to the complementary or principal series), an orthogonal basis of Hµ will be k 2 2 k . . . , η− v0, . . . , η− v0, η−v0, v0, η+v0, η+ v0, . . . , η+ v0,... where v0 is a Θ-invariant vector (Θv0 = 0). 2 If µ = −n + n and Hµ belongs to the holomorphic discrete series, an orthogonal basis of Hµ will be 2 k vn, η+vn, η+ vn, . . . , η+ vn,... where Θvn = invn. The anti-holomorphic case is similar. In fact, there is a complex anti-linear isomorphism between holomorphic and anti-holomorphic irreducible representations of the discrete series of identical Casimir parameter. It is therefore not restrictive to explicitly consider only the holomorphic case. In all cases such a basis is formed by analytic vectors, since they are by deﬁnition eigen- 2 vectors of the operator Θ, hence eigenvectors of the Laplacian ∆ = − 2Θ . 12 LIVIO FLAMINIO AND GIOVANNI FORNI †

Rather then dealing with the basis {vk}, we introduce an adapted orthogonal basis {uk} of Hµ. Let ν be a complex solution of the equation

(23) 1 − ν2 = 4µ . We remark that

• if Hµ belongs to the principal series, then µ ≥ 1/4 and therefore ν is purely imaginary; • if Hµ belongs to the complementary series, then µ ∈]0, 1/4[ and ν is real belonging to ] − 1, 1[ \{0}; 2 • if Hµ belongs to the discrete series, then µ = −n + n and ν = ±(2n − 1), n a positive integer.

The basis {uk} is defined as follows: 2 u = c (η u ), c = for k > 0, k k + k−1 k 2k − 1 + ν (24) 2 u = c (η u ), c = for k < 0, k k − k+1 k −2k − 1 + ν where the initial definition is u0 = v0 ( k u0 k = 1), in the case µ > 0 while un = vn 2 ( k un k = 1 ), for µ = −n + n. We remark that, for an irreducible representation of the discrete series with µ = −n2 +n, the basis {uk} is well defined only for ν = 2n − 1. The basis {uk} is an orthogonal, but not orthonormal, basis of eigenvectors of the opera- 2 tor Θ, hence of the Laplacian operator ∆ = − 2Θ . In fact, for all k ∈ Z (k ≥ n), 2 (25) Θuk = ikuk , ∆uk = (µ + 2k )uk . The norms of its vectors are given, for all k 6= 0 (or k 6= n) by the formula: ( 2 2 k uk−1 k , if ν ∈ iR, i.e. if µ ≥ 1/4 , (26) k uk k = 2|k|−1−ν 2 2|k|−1+ν k uk−1 k if ν ∈ R , i.e. if µ < 1/4 . ∗ In fact, since the adjoint η+ = −η− and 2 2 (27) η+η− = − − iΘ + Θ , η−η+ = − + iΘ + Θ , we obtain that for k > 0 2 2 2 k uk k = k ck k hη+uk−1, η+uk−1i = − k ck k hη−η+uk−1, uk−1i = 2k − 1 − ν = k c k 2h( − iΘ − Θ2)u , u i = k u k 2 k k−1 k−1 2k − 1 +ν ¯ k−1 and a similar computation holds for k < 0. We introduce for convenience the sequence k Y 2i − 1 − ν 1 + <(ν) (28) Π = for any integer k ≥ i = ν,k 2i − 1 + ν ν 2 i=iν +1

(empty products are set equal to 1, hence, if k = iν, then Πν,k = 1 in all cases). By (26) we have 2 (29) k uk k = Πν,|k| . TIME AVERAGES FOR HOROCYCLE FLOWS 13

The behaviour of the norms k uk k is described by the following Lemma whose proof is postponed to the Appendix.

Lemma 2.1. If ν ∈ iR, for all k ≥ iν = 0,

(30) |Πν,k| = 1.

There exists C > 0 such that, if ν ∈ ] − 1, 1[ \{0}, for all k > iν = 0, we have 1 − ν 1 − ν (31) C−1 (1 + k)−ν ≤ Π ≤ C (1 + k)−ν ; 1 + ν ν,k 1 + ν if ν = 2n − 1, for all k ≥ ` ≥ iν = n, we have k − n + 1−ν Π k − n + ν −ν (32) C−1 ≤ ν,k ≤ C . ` − n + 1 Πν,` ` − n + ν

The Sobolev norms of the vectors of the orthogonal basis {uk} are given by the identities 2 s 2 s 2 (33) k uk k s = h(I + ∆) uk, uki = (1 + µ + 2k ) k uk k P∞ s By (33)and (29), a vector f = −∞ fkuk ∈ Hµ belongs to W (Hµ) (s > 0) iff the Sobolev norm

1 ∞ ! 2 X 2 s 2 (34) k f k s := (1 + µ + 2k ) |Πν,k||fk| < ∞ . −∞ By Lemma 2.1,

2 2s−<(ν) (35) k uk k s ≈ (1 + |k|) . It follows that

1 ∞ ! 2 X 2s−<(ν) 2 (36) k f k s ≈ (1 + |k|) |fk| . −∞

We recall that if Hµ belongs to the principal series then ν is purely imaginary; if Hµ belongs to the complementary series then ν ∈ ] − 1, 1[ \{0}; and ν = 2n − 1 if Hµ belongs 2 + to the discrete series (µ = −n + n, n ∈ Z ).

3. INVARIANT DISTRIBUTIONS For any unitary representation of PSL(2, R) on a Hilbert space H, let 0 I(H) := {D ∈ E (H) | LU D = 0} be the space of all U-invariant distributions in E0(H). The analysis of I(H) and of its subspaces Is(H) of U-invariant distributions of order ≤ s can be reduced to the case of irreducible unitary representations. In fact, we have: 14 LIVIO FLAMINIO AND GIOVANNI FORNI †

Lemma 3.1. Let H be a direct integral of unitary representations Hλ of PSL(2, R) with respect to a Stieltjes measure ds. The spaces I(H), Is(H) of U-invariant distributions have direct integral decompositions: Z I(H) = I(H) ds(λ) , ⊕ Z Is(H) = Is(H) ds(λ) . ⊕ Proof. The space C∞(H), its dual space E0(H), all Sobolev spaces W s(H) are decomposable and the horocycle vector ﬁeld U, as a densely deﬁned essentially skew-adjoint operator on H, is also decomposable. Hence the Lemma holds. Invariant distributions for irreducible unitary representations are described by the following

Theorem 3.2. Let Hµ be an irreducible unitary representation of PSL(2, R) of Casimir parameter µ. Then

• if Hµ belongs to the principal or complementary series, the space I(Hµ) has dimen- ± sion 2 and it is generated by two eigenvectors Dµ of the “geodesic vector ﬁeld” X 1±ν 1±<(ν) of eigenvalues − 2 and Sobolev order 2 respectively. 2 • if Hµ belongs to the discrete series and µ = −n + n, the space I(Hµ) has dimen- + sion 1 and it is generated by an eigenvector Dµ of X of eigenvalue −n and Sobolev order n. Theorems 1.1 and 1.4 can be immediately derived from Lemma 3.1 and Theorem 3.2, by the decompositions (21), (22) of Sobolev spaces into irreducible sub-representations.

0 3.1. Formal invariant distributions. Any distribution D ∈ E (Hµ) is uniquely determined by its Fourier coefﬁcients dk = D(uk), k ∈ Z (principal and complementary series) or k ≥ n (discrete series). Since LU D = 0 iff ∞ D(Uf) = 0, for all f ∈ C (Hµ), a distribution D is U-invariant iff D(Uuk) = 0, for all k ∈ Z or all k ≥ n. We need therefore to compute the action of U on the vectors of the basis {uk}. The resulting formula (37) below, as well as the similar formula (45) in §3.3 for the action of the ‘geodesic vector ﬁeld’, can be found in the literature on SL(2, R) harmonic analysis (see for instance [4], formula (4.4)).

Lemma 3.3. Let Iν = Z if µ parametrises the principal or complementary series or Iν = 2 [n, ∞[ ⊂ Z if µ = −n + n parametrises the holomorphic discrete series. Then i(2k + 1 + ν) i(2k − 1 − ν) (37) Uu = − u + iku − u , for all k ∈ I k 4 k+1 k 4 k−1 ν (for ν = 2n − 1 and k = n the above equation must be read as Uun = −inun+1 + inun). Proof. Since i Uu = (−Y + Θ)u = (Θ − (η − η ))u k k 2 + − k TIME AVERAGES FOR HOROCYCLE FLOWS 15 for the principal ot the complementary series we have i Uu = Uu = − (η u − η u ) = iν 0 2 + 0 − 0 i u1 u−1 i(ν + 1) i(ν + 1) = − − = − u1 + u−1, 2 c1 c−1 4 4 + while for the holomorphic discrete series with parameter ν = 2n − 1, n ∈ Z , we have i i un+1 Uuiν = Uun = (Θ − η+)un = inun − = −inun+1 + inun 2 2 cn+1

Thus the Lemma is true in these particular cases. For any k > iν we have instead

i i Uu = iku − η u + η (c η u ) k k 2 + k 2 − k + k−1 i ick = ikuk − uk+1 + η−η+uk−1 2ck+1 2 i ick 2 = − uk+1 + ikuk + (− + iΘ + Θ )uk−1 . 2ck+1 2

A straightforward computation, based on the values of ck−1, ck given in (24) and on (25), yields (37) for all k > iν. A similar computation shows that, for the principal or the complementary series, the equation (37) is also valid for all k < 0. ˆ Let Lν be the linear difference operator that to a sequence d = (dk)k∈Iν assigns the sequence Lνdˆdeﬁned by

i(2k + 1 + ν) i(2k − 1 − ν) (38) (L dˆ) = − d + ikd − d , k ∈ I . ν k 4 k+1 k 4 k−1 ν We remark that, if ν = 2n − 1 (discrete series) and k = n, formula (38) should be read as (Lνdˆ)n = −indn+1 + indn . ˆ Lemma 3.3 immediately implies that the sequence d of the Fourier coefﬁcients dk = D(uk) 0 of an U-invariant distribution D ∈ E (Hµ) must satisfy the difference equation

(39) Lνdˆ= 0 . It is immediate to check that + (40) dk = 1, for all k ∈ Iν, is a solution of the equation (39) for any value of ν. In the case where ν = 2n − 1 parametrises the discrete series there are no other linearly independent solutions. In fact the identity (Lνdˆ)n = −indn+1 + indn = 0 implies dn+1 = dn, hence, by (38) and (39), dk = dn for all k ∈ Iν. If ν 6= 0 parametrises the principal or the complementary series, another linearly inde- − pendent solution can be obtained as follows. Let uk be the vectors obtained by replacing the parameter ν by −ν in the deﬁnition (24) of the adapted Fourier basis. The difference equa- − tion for the Fourier coefﬁcients of U-invariant distributions with respect to the basis {uk } 16 LIVIO FLAMINIO AND GIOVANNI FORNI †

ˆ + − becomes L−νd = 0. For ν 6= 0, by writing the solution dk = 1 for the basis {uk } in terms of the basis {uk}, we obtain as a second solution

|k| Y 2i − 1 − ν (41) d− = = Π for k 6= 0 . k 2i − 1 + ν ν,|k| i=1 For ν = 0 (µ = 1/4), a second solution can be found by directly solving the difference equation:

|k| X 1 (42) d− = 0, d− = for k 6= 0 . 0 k 2i − 1 i=1 0 We have thus shown that the space I(Hµ) ⊂ E (Hµ) of invariant distributions is at most two dimensional if Hµ belongs to the principal or the complementary series, and it is at most one-dimensional if Hµ belongs to the discrete series.

3.2. Sobolev order. The linear functional ∞ ∞ + X X Dµ : f = fkuk 7→ fk −∞ −∞ P∞ t 2 is deﬁned, for any t > 1, on the set of f ∈ Hµ for which the series −∞(1 + |k|) |fk| converges. In fact, there exists a constant Cν,t > 0 such that

∞ !1/2 + X t 2 (43) |Dµ (f)| ≤ Cν,t (1 + |k|) |fk| . −∞ + s By (36) we conclude that Dµ deﬁnes a continuous linear functional on W (Hµ) iff s > 1+<(ν) 2 . If ν 6= 0, the linear functional ∞ ∞ − X X Dµ : f = fkuk 7→ fkΠν,|k| −∞ −∞ P∞ is deﬁned, for any t > 1 − 2<(ν), on the set of f ∈ Hµ for which the series −∞(1 + t 2 |k|) |fk| converges. In fact, by Lemma 2.1, there exists a constant Cν,t > 0 such that

∞ !1/2 − X t 2 (44) |Dµ (f)| ≤ Cν,t (1 + |k|) |fk| . −∞

+ s 1−<(ν) By (36) it follows that Dµ deﬁnes a continuous linear functional on W (Hµ) iff s > 2 . If ν = 0, it is immediate to check that the linear functional

∞ ∞  |k|  X X X 1 D− : f = f u 7→ f µ k k k  2i − 1 −∞ −∞ i=1 TIME AVERAGES FOR HOROCYCLE FLOWS 17

P∞ t 2 is deﬁned, for any t > 1, on the set of f ∈ Hµ for which the series −∞(1 + |k|) |fk| − s converges. It follows that Dµ deﬁnes a continuous linear functional on W (Hµ) iff s > 1 1−<(ν) 2 = 2 .

+ − 3.3. Eigenvectors. The generators {Dµ , Dµ } of the space I(Hµ) of invariant distributions for the “horocycle vector ﬁeld” are for all µ 6= 1/4 distributional eigenvectors for the Lie derivative operator LX with respect to the “geodesic vector ﬁeld” X. In other terms, they are exactly the conical distributions for the Lie group PSL(2, R), in the sense of S. Helga- + son [25]. In the special case µ = 1/4, the distribution Dµ is still an eigenvector (a conical − distribution), but Dµ is not. In this case, the matrix of the operator LX with respect to the + − basis {D1/4, D1/4} is a 2 × 2 Jordan block.

Lemma 3.4. Let Iν = Z if µ parametrises the principal or complementary series or Iν = 2 [n, ∞[ ⊂ Z if µ = −n + n parametrises the holomorphic discrete series. Then 2k + 1 + ν 2k − 1 − ν (45) Xu = u − u for all k ∈ I k 4 k+1 4 k−1 ν

(for ν = 2n − 1, and k = n the above equation must be read as Xun = nun+1). Proof. We have [Θ,Y ] = −X and U = −Y + Θ, hence X = [Θ,U]. By Lemma 3.3, since Θuk = i k uk for all k ∈ Iν, an immediate calculation yields the above formula for Xuk. Lemma 3.5. If µ > 0 (principal or complementary series) and µ 6= 1/4, 1 ± ν (46) L D± = − D± . X µ 2 µ If µ = 1/4 (ν = 0), + + Dµ 1 1 0 Dµ (47) LX − = − − . Dµ 2 1 1 Dµ 2 + If µ = −n + n, n ∈ Z (discrete series), 1 + ν L D+ = − D+ = −nD+. X µ 2 µ µ + + Proof. The distribution Dµ is determined by Dµ (uk) = 1 for all k ∈ Iν. Hence, by Lemma 3.4, 2k + 1 + ν 2k − 1 − ν 1 + ν (48) L D+(u ) = −D+(Xu ) = − − = − D+(u ). X µ k µ k 4 4 2 µ k − − The distribution Dµ is determined, if ν 6= 0, by Dµ (uk) = Πν,|k|. Hence, by Lemma 3.4, for all k ∈ Z, 2k + 1 + ν 2k − 1 − ν L D−(u ) = −D−(Xu ) = − Π − Π = X µ k µ k 4 ν,|k+1| 4 ν,|k−1| 2k + 1 − ν 2k − 1 + ν 1 − ν = − − Π = − D−(u ). 4 4 ν,|k| 2 µ k 18 LIVIO FLAMINIO AND GIOVANNI FORNI †

− If ν = 0, the distribution Dµ is determined by |k| X 1 (49) D−(u ) = 0, D−(u ) = for k 6= 0. µ 0 µ k 2i − 1 i=1 Hence, for all k ∈ Z,

 |k+1| |k−1|  2k + 1 X 1 2k − 1 X 1 L D−(u ) = − − = X µ k  4 2i − 1 4 2i − 1 i=1 i=1 |k| 1 2k + 1 2k − 1 X 1 1 1 = − − − = − D+(u ) − D−(u ). 2 4 4 2i − 1 2 µ k 2 µ k i=1 Theorem 3.2 is therefore completely proved.

4. THECOHOMOLOGICALEQUATION Let H be a unitary representation of PSL(2, R). We prove that, if the Casimir operator on H has a ‘spectral gap’ then the only obstructions to the existence of a smooth solution f ∈ H of the equation Uf = g, for any smooth vector g ∈ H, are given by U-invariant distributions.

Theorem 4.1. If there exists µ0 > 0 such that σ() ∩ ]0, µ0[ = ∅, then the following holds. 1+ν0 Let ν0 be deﬁned as in (4). Let s > 2 and t ∈ R. Then there exists a constant C := s C(ν0, s, t) > 0 such that, for all g ∈ W (H), 1+ν0 • if either t < − 2 and g has no component on the trivial sub-representation of H, or • if t < s − 1 and D(g) = 0, for all D ∈ Is(H), then the equation Uf = g has a solution f ∈ W t(H) which satisﬁes the Sobolev estimate

k f k t ≤ C k g k s. A solution f ∈ W t(H) of the equation Uf = g is unique modulo the trivial sub-representa- 1−ν0 tion if and only if t ≥ − 2 . Theorem 4.1 immediately implies Theorem 1.2.

4.1. Formal Green operators. Let Hµ be any irreducible unitary representation of Casimir 0 parameter µ and let f, g ∈ E (Hµ) be distributions satisfying the equation (50) Uf = g . P P Let f = k fkuk and g = k gkuk be the Fourier expansions of the distributions f, g ˆ with respect to the adapted basis (24) of Hµ. Let f = (fk)k∈Iν , gˆ = (gk)k∈Iν denote the sequences of the Fourier coefﬁcients of f and g. The equation (50) is equivalent to the difference equation: ∗ ˆ (51) Lνf =g ˆ TIME AVERAGES FOR HOROCYCLE FLOWS 19

∗ where Lν is the operator transposed of Lν. More explicitly we have, for all k ∈ Iν, 2k + 1 − ν 2k − 1 + ν (52) (L∗fˆ) = −i f + ikf − i f = g . ν k 4 k+1 k 4 k−1 k If ν = 2n − 1 (discrete series) and k = n, equation (52) should be read as i inf − f = g . n 2 n+1 n We remark that when ν ∈ i R ∪ ]0, 1[ parametrises the principal or the complementary ∗ series we have Lν = L−ν; this equality holds formally also for the discrete series. Formal solutions of the homogeneous equation Uf = 0 are (formal) vectors f ∈ Hµ representing invariant distributions with respect to the Hµ inner product. It follows that, ˆ if D ∈ I(Hµ), the sequence f = (fk)k∈Iν given by

D(uk) D(uk) fk = 2 = k uk k Πν,|k| ∗ is a solution of the homogeneous equation Lνf = 0. By (29), (40) and (41), we obtain the ∗ following formulae for a basis of the kernel of the operator Lν. For the principal or the complementary series: (1) −1 (53) fk = Πν,|k| = Π−ν,|k| and, if ν 6= 0, (2) (54) fk = 1 for all k ∈ Z. If ν = 0, a second independent solution is |k| (2) (2) X 1 (55) f = 0, f = for all k ∈ \{0}. 0 k 2i − 1 Z i=1 ∗ Such formulae can also be deduced from (40), (41) by the identity Lν = L−ν. For the discrete series: (1) −1 (56) fk = Πν,k , for all k ∈ Iν. A standard construction based on the solutions f (1), f (2) yields the Green operator ∗ for Lν. For the principal and complementary series, if ν 6= 0 we have

( 2i Πν,|`| ν Π − 1 k > ` (57) Gν(k, `) = ν,|k| 0 k ≤ ` As ν → 0 formula (57) converges to the Green operator for ν = 0: ( P|`| 1 P|k| 1 4i i=1 2i−1 − i=1 2i−1 k > ` (58) G0(`, k) = 0 k ≤ ` We remark that the Green operator is chosen so that if the sequence gˆ of the Fourier coef- + − ficients of the function g ∈ Hµ has finite support and furthermore Dµ (g) = Dµ (g) = 0, then the sequence fˆ = Gν(ˆg) has also finite support and therefore yields a bona-fide solu- ∞ tion f ∈ C (Hµ). 20 LIVIO FLAMINIO AND GIOVANNI FORNI †

The Green operator in the case of the (holomorphic) discrete series can be taken as

( 2i Πν,|`| − iν ≤ k < ` (59) G (k, `) = ν Πν,|k| ν 2i − ν iν ≤ ` ≤ k which has the property that if the sequence gˆ of the Fourier coefficients of g ∈ Hµ has + ˆ finite support and Dµ (g) = 0, then f has also finite support and therefore yields a bona-fide ∞ solution f ∈ C (Hµ). In all cases it is straightforward to check that the sequences (Gˆ)k = Gν(k, `) satisfy the equation ∗ ˆ (LνG)k = δ`k , hence the Green operators (57), (58) and (59) are well-defined.

4.2. Sobolev estimates. We prove Sobolev estimates for the Green operator Gν in each irreducible component Hµ. The dependence of the estimates on the Casimir parameter µ is studied explicitly, since it will be crucial in the construction of solutions of the equation Uf = g for general unitary representations by ‘gluing’ the solutions obtained in each irreducible ‘component’ of a direct integral decomposition. Sharp estimates for the principal or the complementary series are based on the following Lemma proven in the Appendix. Lemma 4.2. There exists C > 0 such that, if ν ∈ iR or ν ∈ ]−1, 1[\{0}, for all k ≥ ` ≥ 0, Π Π 1 + k | ν,` − 1| ≤ C|ν| max{1, | ν,` |} 1 + log . Πν,k Πν,k 1 + `

Lemma 4.3 (Principal series). For all s > 1/2, t < s − 1, there exists Cs,t > 0 such that, s for all ν ∈ iR (µ ≥ 1/4) and for all g ∈ W (Hµ) we have: (a) if t < −1/2 or + − (b) if Dµ (g) = Dµ (g) = 0, t then Gνg ∈ W (Hµ) and k Gνg k t ≤ Cs,t k g k s. Proof. In the ﬁrst case, the Lemma is equivalent to saying that, if s > 1/2 and t < −1/2, the operator ! Gν ¯ X k uk k t g¯ = (¯g`)`∈Z →7−→ f = Gν(k, `) g¯` k u` k s `∈ Z k∈Z 2 2 is a bounded operator with uniformly bounded norm from ` (Z) to ` (Z). We claim that, in fact, it is a Hilbert-Schmidt operator with Hilbert-Schmidt norm k Gν k HS bounded uniformly with respect to µ ≥ 1/4. If ν 6= 0, 2 2 t 2 X 2 k uk k t 4 X X Πν,|`| 2 (1 + µ + 2k ) k Gν k HS := |Gν(k, `)| 2 = 2 | − 1| 2 s . k u` k s |ν| Πν,|k| (1 + µ + 2` ) k,`∈Z k∈Z `

Πν,|`| (60) | − 1| ≤ C|ν| [1 + log(1 + |k|) + log(1 + |`|)] . Πν,|k| TIME AVERAGES FOR HOROCYCLE FLOWS 21

Hence, if s > 1/2, by the integral inequality, 2 X Πν,|`| log (1 + µ) | − 1|2 (1 + µ + 2`2)−s ≤ C |ν|2 [1 + log2(1 + |k|)] , Π s (1 + µ)s−1/2 ` 1/2, with Dµ (g) = Dµ (g) = 0 and let f := Gνg. If ν 6= 0, we have 2i X Πν,|`| 2i X Πν,|`| f = ( − 1) g = − ( − 1) g . k ν Π ` ν Π ` ` 0 such that log2(1 + µ) k f k 2 ≤ C0 I k g k 2 ≤ C k g k 2 . t s,t (1 + µ)s−t−1 s s,t s Hence the second case of the Lemma is proved for ν 6= 0. If ν = 0, we have X − − X − − fk = 4i (d` − dk ) g` = −4i (d` − dk ) g` , `∈Z,` 2 and g ∈ s W (Hµ) we have 22 LIVIO FLAMINIO AND GIOVANNI FORNI †

1+ν (a) if t < − 2 or 1−ν − (b) if t < − 2 and Dµ (g) = 0, or + − (c) if t < s − 1 and Dµ (g) = Dµ (g) = 0, t then Gνg ∈ W (Hµ). Furthermore there exists a constant Cs,t > 0 such that the following estimate holds. Setting  C max{1, [2s − (1 + ν)]−1/2[−2t + 1 + ν]−1/2} in case (a),  s,t Cs,t −1/2 −1/2 Cs,t,ν := ν max{1, [2s − (1 + ν)] [−2t + 1 − ν] } in case (b),  −1/2 Cs,t max{1, [2s − (1 + ν)] } in case (c); we have Cs,t,ν k Gνg k t ≤ √ k g k s, 1 − ν 1+ν 1+ν Proof. We claim that, if s > 2 and t < − 2 , the operator ! Gν ¯ X k uk k t g¯ = (¯g`)`∈Z →7−→ f = Gν(k, `) g¯` k u` k s `∈ Z k∈Z 2 2 is a Hilbert-Schmidt operator from ` (Z) to ` (Z) with Hilbert-Schmidt norm

Cs,t,ν (62) k Gν k HS ≤ √ . 1 − ν The claim immediately implies the Lemma in case (a). We have 2 2 t 2 X 2 k uk k t 4 X X Πν,|`| 2 Πν,|k| (1 + µ + 2k ) k Gν k HS := |Gν(k, `)| 2 = 2 | − 1| 2 s . k u` k s ν Πν,|k| Πν,|`| (1 + µ + 2` ) k,`∈Z k∈Z `

Since 0 < µ < 1/4, for all t ∈ R there exists a constant Ct > 0 such that, for all k ∈ Z,

−1 2t 2 t 2t (63) Ct (1 + |k|) ≤ (1 + µ + 2k ) ≤ Ct(1 + |k|) . If ν ∈ ]0, 1[, by Lemma 4.2, for all k, ` ∈ Z,

4 Πν,|`| 2 Πν,|k| Πν,|`| Πν,|k| (64) 2 | − 1| ≤ C max{ , }× ν Πν,|k| Πν,|`| Πν,|k| Πν,|`| × [1 + log(1 + |k|) + log(1 + |`|)] . Hence, by the estimate (31) in Lemma 2.1,

2t−ν Cs,t X X (1 + |k|) k G k 2 ≤ { [1 + log(1 + |`|)]+ ν HS 1 − ν (1 + |`|)2s−ν k∈Z |`|≥|k| X (1 + |k|)2t+ν + [1 + log(1 + |k|)]}. (1 + |`|)2s+ν |`|<|k| The estimate (62) then follows by the integral inequality. TIME AVERAGES FOR HOROCYCLE FLOWS 23

s 1+ν − Let g ∈ W (Hµ), s > 2 , with Dµ (g) = 0 and let f := Gνg. We have   2i X Πν,|`| 2i X Πν,|`| X f = ( − 1)g = − g + g . k ν Π ` ν  Π ` ` `

Hence   2 X Πν,|`| X |fk| ≤  |g`| + |g`| . ν Πν,|k| |`|≥|k| `

It follows by Lemma 2.1 that there exists a constant Cs,t > 0 such that   2t+ν 2t−ν Cs,t X (1 + |k|) X (1 + |k|) k f k 2 ≤ k g k 2 + . t ν2(1 − ν) s  (1 + |`|)2s+ν (1 + |`|)2s−ν  |`|≥|k| `

The integral inequality then implies that, if t < s − 1,

X (1 + |k|)2t+ν < C [2s − (1 + ν)]−1 (1 + |`|)2s+ν s |`|≥|k|

1+ν 1−ν and, if s > 2 and t < − 2 ,

X (1 + |k|)2t−ν < C [2s − (1 + ν)]−1[−2t + 1 − ν]−1. (1 + |`|)2s−ν s,t ` 2 , with Dµ (g) = Dµ (g) = 0 and let f := Gνg. We have

2i X Πν,|`| 2i X Πν,|`| f = ( − 1)g = − ( − 1)g . k ν Π ` ν Π ` `

It follows by (63), (64) and Lemma 2.1 that

2 t 2 4 2 X X Πν,|`| 2 Πν,|k| (1 + µ + 2k ) (65) k f k t ≤ 2 k g k s | − 1| 2 s ≤ ν Πν,|k| Πν,|`| (1 + µ + 2` ) k∈Z |`|≥|k| 2t−ν Cs,t X X (1 + |k|) ≤ k g k 2 [1 + log(1 + |`|)]. 1 − ν s (1 + |`|)2s−ν k∈Z |`|≥|k|

The estimate in case (c) then follows by the integral inequality. + 2 Lemma 4.5 (Discrete series). For all n ∈ Z ( ν = 2n − 1, µ = −n + n), s (a) if s > 1 − n, t < min(s − 1, n − 1) and g ∈ W (Hµ) or s + s (b) if s > n, t < s − 1, and g ∈ W (Hµ) satisfy Dµ (g) = 0 g ∈ W (Hµ), 24 LIVIO FLAMINIO AND GIOVANNI FORNI †

t then Gνg ∈ W (Hµ) and there exists a constant Cs,t > 0 such that the following estimates hold. Let  −1/2 −1/2 Cs,t max{1, (s − n) (n − 1 − t) } in case (a), if s > n;  −1/2 Cs,t max{1, (n − 1 − t) } in case (a), if s = n; Cs,t,ν := C max{1, (s + n − 1)−1/2, (n − s)−1/2} in case (a), if s < n;  s,t  −1/2 Cs,t max{1, (s − n) } in case (b); then C k G g k ≤ s,t,ν k g k . ν t νs−t s Proof. We claim that, if s > 1 − n and t < min(s − 1, n − 1), the operator   Gν ¯ X k uk k t g¯ = (¯g`)`∈Iν →7−→ f =  Gν(k, `) g¯` k u` k s `≥n k∈Iν 2 2 is a Hilbert-Schmidt operator from ` (Iν) to ` (Iν) with Hilbert-Schmidt norm

C (66) k G k ≤ s,t,ν . ν HS νs−t We have 2 t 2 t 2 4 X X Πν,k (1 + µ + 2k ) X Πν,` (1 + µ + 2k ) k Gν k HS = 2 { 2 s + 2 s } . ν Πν,` (1 + µ + 2` ) Πν,k (1 + µ + 2` ) k≥n n≤`≤k `>k 2 Since ν = 2n − 1 ≥ 1 and µ = −n + n, there exists a constant C1 > 0 such that, for + all n ∈ Z and all k ≥ n, −1 2 2 2 (67) C1 [ν + (k − n)] ≤ 1 + µ + 2k ≤ C1[ν + (k − n)] and, by the upper bound (32) in Lemma 2.1, for all k ≥ ` ≥ n, −ν Πν,k k − n + ν (68) ≤ C1 . Πν,` ` − n + ν

Hence, there exists a constant Cs,t > 0 such that X X [1 + (k/ν)]2t−ν X [1 + (k/ν)]2t+ν k G k 2 ≤ C ν2t−2s−2 { + }. ν HS s,t [1 + (`/ν)]2s−ν [1 + (`/ν)]2s+ν k≥0 0≤`≤k `>k

By the integral inequality, there exists a constants C2 > 0, C3 > 0 such that  −1 C2 max{1, (s − n) } if s > n; X 1  [1 + (`/ν)]−2s+ν ≤ C log(1 + k/ν) if s = n; ν 2 0≤`≤k  −1 −2s+ν+1 C2 max{1, (n − s) }(1 + k/ν) if s < n; hence, if t < min(s − 1, n − 1),  −1 −1 C3 max{1, (s − n) (n − 1 − t) } if s > n; X X 1 [1 + (k/ν)]2t−ν  ≤ C max{1, (n − t − 1)−1} if s = n; ν2 [1 + (`/ν)]2s−ν 3 k≥0 0≤`≤k  −1 −1 C3 max{1, (n − s) (s − 1 − t) } if s < n; TIME AVERAGES FOR HOROCYCLE FLOWS 25 and, if s > 1 − n, X 1 [1 + (`/ν)]−2s−ν ≤ C (s + n − 1)−1[1 + (k/ν)]−2s−ν+1; ν 2 `>k hence, if t < s − 1, X X 1 [1 + (k/ν)]2t+ν (69) ≤ C (s + n − 1)−1(s − 1 − t)−1. ν2 [1 + (`/ν)]2s+ν 3 k≥0 `>k

It follows that there exists a constant Cs,t > 0 such that (66) holds and such an upper bound for the Hilbert-Schmidt norm of the operator Gν implies the Lemma in case (a). s + (α) (β) Let g ∈ W (Hµ), s > n, with Dµ (g) = 0 and let f := Gνg. For all k ≥ n, let f , f be deﬁned by

(α) 2i X f := − g ; k ν ` n≤`≤k ∞ (β) 2i X Πν,` fk := g` ν Πν,k `>k Since f = f (α) +f (β), estimates on the Sobolev norms of f (α), f (β) imply corresponding estimates for f by the triangular inequality. We have that 2 t (β) 2 4 2 X X Πν,` (1 + µ + 2k ) k f k t ≤ 2 k g k s 2 s , ν Πν,k (1 + µ + 2` ) k≥n `>k hence, if s > n and t < s − 1, by the estimates (67), 68 and (69), there exists a constant Cs,t > 0 such that C2 k f (β) k 2 ≤ s,t,ν k g k 2 . t ν2s−2t s + Since Dµ (g) = 0, (α) 2i X 2i X f = − g = g . k ν ` ν ` n≤`≤k `>k It follows that (α) 4 X |f |2 ≤ k g k 2 Π−1(1 + µ + 2`2)−s; k ν2 s ν,` n≤`≤k (α) 4 X |f |2 ≤ k g k 2 Π−1(1 + µ + 2`2)−s. k ν2 s ν,` `>k Let n+ν k 2 t (1) 4 X X Πν,k (1 + µ + 2k ) Σs,t,ν := 2 2 s ; ν Πν,` (1 + µ + 2` ) k=n `=n ∞ ∞ 2 t (2) 4 X X Πν,k (1 + µ + 2k ) Σs,t,ν := 2 2 s . ν Πν,` (1 + µ + 2` ) k=n+ν+1 `=k+1 26 LIVIO FLAMINIO AND GIOVANNI FORNI †

We then have (α) 2 (1) (2) 2 k f k t ≤ Σs,t,ν + Σs,t,ν k g k s. (1) The function Σs,t,ν can be estimated as above. In fact, there exists a constant Cs,t > 0 such that ν k 2t−ν (1) X X [1 + (k/ν)] Σ ≤ C ν2t−2s−2 . s,t,ν s,t [1 + (`/ν)]2s−ν k=0 `=0 Since, by the integral inequality ,if s > n, k X 1 [1 + (`/ν)]−2s+ν ≤ C max{1, (s − n)−1}; ν 1 `=0 and ν X 1 [1 + (k/ν)]2t−ν ≤ 2t , ν k=0 there exists a constant Cs,t > 0 such that (1) 2 −1 2t−2s Σs,t,ν ≤ Cs,t max{1, (s − n) } ν . (2) The estimate of the function Σs,t,ν, can be carried out as follows. By the lower bound (32) in Lemma 2.1, there exists a constant C4 > 0 such that, if ` > k, ν Πν,k ` − n + 1 ≤ C4 . Πν,` k − n + 1

It follows that there exists a constant Cs,t > 0 such that ∞ ∞ 2t ν (2) X X [1 + (k/ν)] 1 + ` Σ ≤ C ν2t−2s−2 . s,t,ν s,t [1 + (`/ν)]2s 1 + k k=ν+1 `=k+1 Since, for k ≥ ν, we have [1 + (k/ν)]2t ≤ 22t(k/ν)2t and since 1 + ` 1 + `/ν ≤ , 1 + k k/ν it follows that ∞ ∞ 2t−ν (2) X X k Σ ≤ C0 ν2t−2s−2 [1 + (`/ν)]−2s+ν . s,t,ν s,t ν k=ν+1 `=k+1 By the integral inequality, if s > n (hence −2s + ν + 1 < 0), ∞ −2s+ν+1 X 1 k [1 + (`/ν)]−2s+ν ≤ (s − n)−1 , ν ν `=k+1 and, if t < s − 1, ∞ 2t−2s+1 X 1 k ≤ (s − 1 − t)−1. ν ν k=ν+1 Hence, there exists a constant Cs,t > 0 such that (2) 2 −1 2t−2s Σs,t,ν ≤ Cs,t(s − n) ν . TIME AVERAGES FOR HOROCYCLE FLOWS 27

It follows that (α) 2 2 −1 2t−2s k f k t ≤ Cs,t max{1, (s − n) } ν .

Proof of Theorem 4.1. Let H be a direct integral of non-trivial unitary representations Hλ s of PSL(2, R) with respect to a Stieltjes measure ds. Every vector g ∈ W (H) has a decomposition Z g = gλ ds(λ)

s with gλ ∈ W (Hλ). We claim that: s (a) D(g) = 0 for all D ∈ I (H) iff, for ds-almost all λ ∈ R, Dλ(g) = 0 for all Dλ ∈ s I (Hλ); (b) if there exists a constant C(s, t) > 0 such that, for ds-almost all λ ∈ R, the equa- t tion Ufλ = gλ has a solution fλ ∈ W (Hλ) satisfying a uniform Sobolev estimate

(70) k fλ k t ≤ C(s, t) k gλ k s , then the equation Uf = g has a solution f ∈ W t(H) with the same Sobolev bound:

k f k t ≤ C(s, t) k g k s . The claim (a) follows immediately from Lemma 3.1. The claim (b) can be proved as follows. Let Z f := fλ ds(λ) .

Since the operator U is decomposable and Z 2 2 k f k t = k fλ k t ds(λ) ≤ Z 2 2 2 2 ≤ C(s, t) k gλ k s ds(λ) = C(s, t) k g k s < +∞ , the vector f ∈ W t(SM) yields a well-defined solution of the equation Uf = g which satisfies the required Sobolev bound. The claim is therefore completely proved. The existence part of the Theorem then follows from Lemmata 4.3, 4.4, 4.5 and the direct integral decompositions (19), (20). In fact, if the Casimir operator has a ‘spectral gap’, the uniform Sobolev estimates (70) holds for all irreducible unitary sub-representations. The uniqueness part is proved as follows. A solution f ∈ W t(H) of the equation Uf = g is not unique modulo the trivial sub-representation iff there exists a non-trivial invariant t 1−ν0 distribution D ∈ W (H). By Lemma 3.1 and Theorem 3.2, that is the case iff t ≥ − 2 . 4.3. Converse results. Let H be the Hilbert space of a unitary representation of the group s PSL(2, R). It is immediate by the definition of an U-invariant distribution D ∈ I (H) that, if g ∈ H is a vector such that the equation Uf = g has a solution f ∈ W t(H) with t ≥ s+1, then D(g) = 0. We prove that D(g) = 0 under weaker regularity assumptions on the solution f. In fact, it turns out that, if g is not in the kernel of U-invariant distributions, then Theorem 4.1 gives the optimal Sobolev regularity for the solution. Such converse result will 28 LIVIO FLAMINIO AND GIOVANNI FORNI † also be a crucial tool in the proof of lower bounds on the deviations of ergodic averages of the horocycle flow.

s 1+ν0 0 Theorem 4.6. Let g ∈ W (H), s > 2 , and let D ∈ E (H) be a U-invariant distribution t of Sobolev order SD < s. If the equation Uf = g has a solution f ∈ W (H) with t ≥ SD − 1, then D(g) = 0. Theorem 4.6 implies Theorem 1.3, for all ﬁnite area hyperbolic surfaces. The proof of Theorem 4.6 can be reduced, by direct decomposition into irreducible sub- representations, to the following Lemmata.

s Lemma 4.7 (Principal series). Let ν ∈ iR ( µ ≥ 1/4). Let g ∈ W (Hµ), s > 1/2, be t any vector such that the cohomological equation Uf = g has a solution f ∈ W (Hµ). + − If t ≥ −1/2, then Dµ (g) = Dµ (g) = 0. s Lemma 4.8 (Complementary series). Let ν ∈ ]0, 1[ ( 0 < µ < 1/4). Let g ∈ W (Hµ), 1+ν s > 2 , be any vector such that the cohomological equation Uf = g has a solution f ∈ t W (Hµ). 1+ν − (1) If t ≥ − 2 , then Dµ (g) = 0 ; 1−ν + − (2) if t ≥ − 2 , then Dµ (g) = Dµ (g) = 0. 2 + s Lemma 4.9 (Discrete Series). Let ν = 2n − 1 ( µ = −n + n), n ∈ Z . Let g ∈ W (Hµ), s > n, be any vector such that the cohomological equation Uf = g has a solution f ∈ t + W (Hµ). If t ≥ n − 1, then Dµ (g) = 0. We remark that the proof of Lemmata 4.7, 4.8 and 4.9 can be reduced to the case that the Fourier series gˆ of g ∈ Hµ with respect to the basis {uk} has finite support. In fact, for s 1+<(ν) ˆ any g ∈ W (Hµ), s > 2 , there exists a vector h ∈ Hµ, with Fourier series h of finite s support, such that D(h) = D(g), for all D ∈ I (Hµ). By Theorem 3.2, the sequence hˆ can be determined as a solution of a linear system of two equations, in the case of the principal or the complementary series, or of a single equation, in the case of the discrete t series. By Theorem 4.1, the equation Uf = h − g has a solution which belongs to W (Hµ) for all t < s − 1. Hence, for any t < s − 1, the equation Uf = h has a solution which t belongs to W (Hµ) iff the equation Uf = g does. It is therefore sufficient to consider the case of a vector g ∈ Hµ with Fourier series of finite support. Proof of Lemma 4.7. Let ν 6= 0. If f is a distributional solution of the equation Uf = g, the sequence fˆ of its Fourier coefficients satisfies the linear second order difference equation (52). The general solution can be written as

2i X Πν,|`| (1) (2) (71) f = ( − 1)g + c f + c f , k ν Π ` 1 k 2 k `

(1) −1 (2) where fk = Πν,|k|, fk = 1, for all k ∈ Z, and c1, c2 ∈ C. If gˆ has ﬁnite support, there − − − exists k ∈ Z such that, for all k < k < 0, (1) (2) fk = c1fk + c2fk . TIME AVERAGES FOR HOROCYCLE FLOWS 29

t It follows that, if f ∈ W (Hµ), t ≥ −1/2, then c1 = c2 = 0. In fact, if |c1|= 6 |c2|, − iθ since |Πν,|k|| = 1, for all k < k , |fk| ≥ ||c1| − |c2|| > 0. If |c1| = |c2|, let c1 = e c2 iθk i(θ−θk) and Πν,|k| = e . Then |fk| ≥ |c2||e + 1|. By the definition of Πν,|k|, if ν = is 6= 0, we have |k| X s θ = −2 tan−1( ) . k 2i − 1 i=1 1 − Let I ⊂ S be a closed subinterval such that θ − π 6∈ I. The set {k < k |θk ∈ I} − + − contains infinitely many intervals [kj , kj ] ⊂ Z such that there exists a constant c > 1 − + with |kj /kj | ≥ c for all j ∈ N. Since I is closed, there exists a constant d > 0 such 0 0 − + i(θ−θk) that |θ − π − θ | ≥ d for all θ ∈ I. Hence, if k ∈ [kj , kj ], since θk ∈ I, |e + 1| ≥ d > 0, for all j ∈ N. In both cases it follows that, if c1 6= 0 or c2 6= 0, then 2 X |fk| = +∞ , |k| k k > 0, 2i f = D−(g)Π−1 + D+(g) . k ν µ ν,|k| µ t + − The above argument implies that, if f ∈ W (Hµ) with t ≥ −1/2, then Dµ (g) = Dµ (g) = 0. Let ν = 0. The general solution of the difference equation (52) is X − − (1) (2) fk = 4i (d` − dk )g` + c1fk + c2fk , k<` (1) where fk = 1, for all k ∈ Z, c1, c2 ∈ C and |k| (2) X 1 d− = f = . k k 2i − 1 i=1 − − − If gˆ has finite support, there exists k ∈ Z such that, for all k < k < 0, (1) (2) fk = c1fk + c2fk . (2) t Since fk is unbounded, it is immediate that, if f ∈ W (Hµ) with t ≥ −1/2, then c1 = + + + c2 = 0. Hence there exists k ∈ Z such that, for all k > k , − + − fk = 4i Dµ (g) − Dµ (g)dk − (2) t + and again, since dk = fk is unbounded, if f ∈ W (Hµ) with t ≥ −1/2, then Dµ (g) = − Dµ (g) = 0. Proof of Lemma 4.8. The sequence fˆ of the Fourier coefficients can be written as in (71). − − − If gˆ has finite support, there exists k ∈ Z such that, for all k < k < 0, (1) (2) −1 fk = c1fk + c2fk = c1Πν,|k| + c2. 30 LIVIO FLAMINIO AND GIOVANNI FORNI †

1+ν By the estimates (31) in Lemma 2.1 and by (36), it follows that (1) if t ≥ − 2 , then c1 = 0; 1−ν (2) if t ≥ − 2 , then c1 = c2 = 0. + + + If gˆ has finite support, there exists k ∈ Z such that, for all k > k > 0, 2i f = D−(g)Π−1 + D+(g) + c f (1) + c f (2). k ν µ ν,|k| µ 1 k 2 k 1+ν − Again by Lemma 2.1, it follows that (1) if t ≥ − 2 , since c1 = 0, then Dµ (g) = 0; (2) 1−ν + − if t ≥ − 2 , since c1 = c2 = 0, then Dµ (g) = Dµ (g) = 0. Proof of Lemma 4.9. The sequence fˆ of the Fourier coefficients of the solution f satisfies the linear difference equation (52) on the set Iν = [n, +∞] ⊂ Z. Since the space of solutions of the corresponding homogeneous equation is one-dimensional, the general solution can be written as k ∞ ! 2i X X Πν,|`| (1) f = − g + g + c f , k ν ` ` Π 1 k `=n `=k+1 ν,|k| (1) −1 where c1 ∈ C and fk = Πν,k, for all k ≥ n. If gˆ has finite support, there exists k+ > n such that, for all k > k+, 2i f = − D+(g) + c f (1) . k ν µ 1 k t Let f ∈ W (Hµ). By the estimates (32) in Lemma 2.1, it follows that, if t ≥ −n, then c1 = + 0 and, if t ≥ n − 1, then c1 = 0 and Dµ (g) = 0.

5. DEVIATION OF ERGODIC AVERAGES 5.1. Spectral decomposition of horocycle orbits. Since Is(SM) ⊂ W −s(SM) is closed, there is an orthogonal splitting (72) W −s(SM) = Is(SM) ⊕⊥ Is(SM)⊥ . s X Although the space I (SM) is {φt }-invariant, the action of the geodesic one-parameter X s X group {φt } on W (SM) is not unitary and the orthogonal splitting (72) is not {φt }- invariant. X According to Theorems 1.1 and 1.4, the one-parameter group {φt } has a (generalized) s X spectral representation on the space I (SM). In fact, for all s > 0, there is a {φt }-invariant orthogonal splitting s s ⊥ s (73) I (SM) = Id ⊕ IC X s s and the spectrum of φt is discrete on the subspace Id := Id ∩ I (SM) and Lebesgue of −t/2 s ﬁnite multiplicity with spectral radius equal to e on IC, for all t ∈ R. X Let B ⊂ Id be a basis of (generalized) eigenvectors for {φt } | Id such that B∩ Id I1/4 X is a basis of eigenvectors for {φt } | Id I1/4 and, if 1/4 ∈ σpp, the set B1/4 := B ∩ I1/4 X is a basis which brings {φt } | I1/4 into its Jordan normal form. For any D ∈ B \ B1/4 of Sobolev order SD > 0, there exists λD ∈ C with <(λD) = −SD < 0 such that, for all t ∈ R,

X λDt (74) φt (D) = e D ; TIME AVERAGES FOR HOROCYCLE FLOWS 31

+ − if 1/4 ∈ σpp, the subset B1/4 ⊂ B is the union of a ﬁnite number of pairs {D , D } such ± ± ± that the distributions D ∈ B1/4 = B ∩ I1/4 have the same Sobolev order equal to 1/2 and formula (7) holds: + + X D −t/2 1 0 D (75) φt − = e t − . D − 2 1 D s s X The set B := B ∩ Id is a basis of (generalized) eigenvectors for the action of {φt } s on Id. By Theorem 1.1 and (9), for all s > 1, there is a decomposition s [ [ (76) B = Bµ ∪ Bn . µ∈σpp 1≤n

~~Lemma 5.1. There exists a constant C1 := C1(s) > 0 such that, for all t ∈ R, X s −t/2 (77) k φt IC k −s ≤ C1 (1 + |t|) e .~~

Proof. By §§3.1 and 3.3, if Hµ is the Hilbert space of an irreducible representation of the s principal series, then I (Hµ) has dimension 2 and it is generated by two complex conjugate ± ± invariant distributions Dµ . By Lemma 3.5, the distribution Dµ is an eigenvector of the op- X −(1±ν)t/2 erator φt with eigenvalue the complex number e , where ν is the purely imaginary + − parameter defined by (23). The distortion of the basis {Dµ , Dµ } is uniformly bounded as the parameter µ varies over subsets of the real line bounded away from 1/4 (which corre- ± sponds to the imaginary parameter ν being bounded away from 0). Let dµ be the sequences ± of the Fourier coefficients of the distributions Dµ , with respect to the basis (24) of Hµ. By ± − the definition of dµ in §3.1, an explicit calculation of the derivative of the coefficients dµ of − the distribution Dµ yields d+ − d− 1 d µ µ − + lim = − dν = d0 . ν→0 2ν 2 dν ν=0 Since the Fourier basis (24) depends continuously on the parameter ν ∈ i R ∪ ] − 1, 1[, the + − s basis { 0 such that, for all µ ≥ 1/4, the X s −t/2 s norm of the operator φt on I (Hµ) is bounded by C1(1 + |t|)e . Since the space IC s decomposes as a direct integral of the spaces I (Hµ) over the interval [1/4, +∞[ and the X operator φt is also decomposable, the desired estimates follows. According to (72) and (73), every γ ∈ W −s(SM) can be written as

X (78) γ = cD(γ)D + C(γ) + R(γ) D∈Bs s s ⊥ with C(γ) ∈ IC and R(γ) ∈ I (SM) . The real number cD(γ) will be called the D- component of γ along D ∈ Bs and the distribution C(γ) the (U-invariant) continuous component of γ. We recall that the continuous component vanishes for all γ ∈ W −s(SM) if 32 LIVIO FLAMINIO AND GIOVANNI FORNI †

~~M is compact. The following Lemma tells us that bounds on the norms of distributions in W −s(SM) are equivalents to bounds on their coefﬁcients.~~

~~Lemma 5.2. There exists a constant C2 := C2(s) > 0 such that~~

−2 2 X 2 2 2 2 2 (79) C2 k γ k −s ≤ |cD(γ)| + k C(γ) k −s + k R(γ) k −s ≤ C2 k γ k −s . D∈Bs Proof. The splittings (72) and (73) are orthogonal with respect to the Hilbert structure of W −s(SM). The basis Bs is not orthogonal, however we claim that its distortion is uniformly bounded. In fact, vectors of the basis supported on different irreducible representations are + − s orthogonal; if Dµ , Dµ ∈ B are normalised eigenvectors supported on the same irreducible + representation of Casimir parameter µ ∈ R (principal or complementary series), a cal- + − culation shows that the function hDµ , Dµ i−s is continuous on the interval ]1/4, +∞[, it s converges to 0 as µ → +∞ and to 1 as µ → 1/4. Since Id is contained in the pure point + component of the the spectral representation of the Casimir operator, the angle between Dµ − and Dµ has a strictly positive uniform lower bound as µ ∈ σpp.

+ 5.1.1. Horocycle orbits. For x ∈ SM and T ∈ R , let γx,T be the probability measure uniformly distributed on the horocycle orbit of length T starting at x. More precisely, for any continuous function f on SM, we deﬁne Z T 1 U γx,T (f) = f(φt (x)) dt T 0

By the Sobolev embedding Theorem (see [1]), for s > 3/2, the measures γx,T are continuous functionals on W s(SM) (which depend weakly-continuously on x ∈ SM + and T ∈ R ). Thus the splitting (78) can be applied to horocycle orbits. We set

cD(x, T ) := cD(γx,T ), C(x, T ) := C(γx,T ) , R(x, T ) := R(γx,T ) . so that X (80) γx,T = cD(x, T )D + C(x, T ) + R(x, T ) . D∈Bs The proofs of Theorems 1.5 and 1.7 will reduce to estimates on the norms of the three parts of this splitting. We start by showing in next section that since the parts of this splitting in- P variant by the horocycle flow, namely D∈Bs cD(x, T )D and C(x, T ), vanish on coboundaries, the remainder part R(x, T ) must be of the order of 1/T ; furthermore the individual coefficients cD(x, T ) and the continuous component C(x, T ) cannot be too small. The uniform norm of functions on a compact manifold can be bounded in terms of a Sobolev norm by the Sobolev embedding theorem. In the case of a non-compact hyperbolic surfaces M of finite area, since the injectivity radius is not bounded away from zero, the Sobolev embedding theorem holds only locally. We therefore prove a version of the Sobolev embedding theorem on compact subsets of the unit tangent bundle SM, with an explicit bound on the constant. We fix once for all a point x0 ∈ SM and let d0 : SM → R be the distance function from x0. Constants in the following statements will implicitly depend (in a inessential way) on this choice. TIME AVERAGES FOR HOROCYCLE FLOWS 33

~~Lemma 5.3. There exists a constant C3 := C3(x0,M) such that for any function F ∈ W 2(SM), we have that F is continuous and~~

d0(x)/2 |F (x)| ≤ C3 e k F k 2 . Proof. Recall that if G is a locally W 2 function on Poincaré’s plane H then G is continuous and there exists C() > 0 such that Z |G(x)|2 < C() (|G|2 + |dG|2 + |∆G|2) dy B(x,) for any x ∈ H ([22] page 63 ). For x in SM denote by ρ(x) the radius of injectivity of SM at x. Let < π and set A0 the open set of points x ∈ SM where ρ(x) > . By choosing c sufﬁciently small we can assume that complement A0 consists of k connected components 1 + Vi each contained in SAi, the tangent unit bundle of disjoint open cusps Ai ≈ S × R whose boundary horocycle has length 2. By the Sobolev embedding theorem mentioned above there exists C() > 0 such that for any x ∈ A0 we have Z |F (x)|2 < C() (|F |2 + |dF |2 + |∆F |2) dy. B(x,) −d For x ∈ Vi let d be the distance of x from ∂Ai. It’s easy to see that 2e ≤ ρ(x) ≤ 4e−d. Let F˜ denote the lift of F to Poincare’s half-plane H and let x˜ be a point SH projecting to x. Then, by the same embedding theorem, Z |F˜(x)|2 < C() (|F˜|2 + |dF˜|2 + |∆F˜|2) dy B(˜x,) and, since, the ball B(˜x, ) ⊂ SH covers the ball B(x, ) ⊂ SM at most [ed/2] + 1 times, we get Z |F (x)|2 < edC() (|F |2 + |dF |2 + |∆F |2) dy . B(x,) 0 The proof is ﬁnished by observing that d0 is bounded on A0 and d0(x) < d + C (). Lemma 5.3 allows us to derive the following upper bound for the uniform norm of components and remainder terms of horocycle arcs.

~~Corollary 5.4. For all s ≥ 2, there exists a constant C4 := C4(s, x0) > 0 such that, if the horocycle arc γx,T ⊂ B(x0, d), then~~

X 2 2 2 2 d (81) |cD(x, T )| + k C(x, T ) k −s + k R(x, T ) k −s ≤ C4 e . D∈Bs Proof. Let s ≥ 2. By Lemma 5.3, for any function f ∈ W s(SM), we have: d/2 |γx,T (f)| ≤ max{|f(x)| : x ∈ B(x0, d)} ≤ C3e k f k s . The estimate (81) then follows immediately from Lemma 5.2. 34 LIVIO FLAMINIO AND GIOVANNI FORNI †

5.2. Coboundaries. Let {φt} be a measure preserving ergodic flow on a probability space. We recall that a function g is a coboundary for {φt} if it is a derivative of a function f along this flow. The Gottschalk-Hedlund Theorem, or rather its proof, yields upper bounds for the uniform or the L2 norm of ergodic averages of a coboundary g in terms of the uniform, or respectively, the L2 norm of its primitive f. A key consequence is that the uniform bound for the remainder term R(x, T ) proved in Corollary 5.4 can be significantly improved.

Lemma 5.5. Let x0 ∈ SM. For every s > 3, there exists a constant C5 := C5(s, x0) such that if the end-points of γx,T are in B(x0, d), then C (82) k R(x, T ) k ≤ 5 ed/2 . −s T Proof. Let I := Is(SM). The orthogonal splitting (72) induces a dual orthogonal splitting (83) W s(SM) = Ann(I) ⊕ Ann(I⊥) . s Hence, any function g ∈ W (SM) has a unique (orthogonal) decomposition g = g1 + g2, ⊥ ⊥ ⊥ where g1 ∈ Ann(I) and g2 ∈ Ann(I ). Since R(x, T ) ∈ I , the function g2 ∈ Ann(I ) and g1 ∈ Ann(I), we have:

~~(84) R(x, T )(g) = R(x, T )(g1 + g2) = R(x, T )(g1) = γx,T (g1) .~~

The function g1 is a coboundary for the horocycle ﬂow. In fact, it belongs to the kernel of all U-invariant distributions of order ≤ s; hence, by Theorem 1.2, there exists a func- t tion f1 ∈ W (SM), with 2 < t < s − 1, such that Uf1 = g1 and k f1 k t ≤ C k g1 k s. U Let d > 0 be such that x, φT (x) ∈ B(x0, d). By the Sobolev embedding Theorem, the function f1 is continuous and by Lemma 5.3 d/2 0 d/2 (85) max{|f1(x)| : x ∈ B(x0, d)} ≤ C3 e k f k t ≤ C3 e k g1 k s . By the Gottschalk-Hedlund argument and the inequality (85), 1 2C0 (86) |γ (g )| = |f ◦ φU (x) − f (x)| ≤ 3 ed/2 k g k . x,T 1 T 1 T 1 T 1 s Since the dual splitting (83) is orthogonal, by the estimates (84) and (86), we get ed/2 2C0 |R(x, T )(g)| ≤ 2C0 k g k ≤ ed/2 k g k . 3 T 1 s T s The lemma is therefore proved. A similar argument proves the following L2 bound.

Lemma 5.6. For every s > 1, there exists a constant C6 := C6(s), such that, for all functions g ∈ W s(SM), C (87) k R(·,T )(g) k ≤ 6 k g k . 0 T s X Proof. According to (84), since g1 = Uf1 is a coboundary and φt is volume preserving, by Theorem 1.3 and the orthogonality of the splitting (83), we have 2 C C k R(·,T )(g) k = k γ (g ) k ≤ k f k ≤ 6 k g k ≤ 6 k g k . 0 x,T 1 0 T 1 0 T 1 s T s TIME AVERAGES FOR HOROCYCLE FLOWS 35

The following lemma is a widely known L2 version for ergodic measurable flows. It will be a crucial tools in the proof of the L2 lower bounds of Theorems 1.5 and 1.7. Lemma 5.7 (Gottschalk-Hedlund Lemma). If an L2-function F is a solution of the equation dF ◦ φ (88) t = G, dt t=0 then the one-parameter family of functions {GT }T ∈R defined by Z T GT (·) := G (φt(·)) dt 0 2 is equibounded in L by 2 k F k . Conversely if the family {GT }T ≥0, is equibounded, then the equation (88) has an L2 solution. Proof. If there exists a solution F of the equation (88), then Z T G (φt(x)) dt = F (φT (x)) − F (x); 0 hence Z T k G (φt(·)) dt k 0 ≤ 2 k F k 0 . 0 2 Conversely, if the family of functions {GT }T ≥0 is equibounded in L , then the family of functions {FT }T ≥0 defined by Z T Z t 1 s FT (·) := − G (φ (·)) ds dt T 0 0 2 2 is equibounded in L . Since {φt} is ergodic, by Von Neumann L ergodic Theorem, the 2 2 function G has zero average and any weak limit F ∈ L of {FT } is a (weak) L solution of the equation (88). We have the following L2 bounds for the components of horocycle arcs:

~~s Lemma 5.8. For every D ∈ B there exists a constant CD > 0 such that, for all T > 0, the 2 D-component cD(x, T ) of γx,T satisﬁes the L upper bound~~

~~(89) k cD(·,T ) k 0 ≤ CD . 1 2 If D ∈ I (SM), the D-component cD(x, T ) satisfy the L lower bound~~

~~(90) sup T k cD(·,T ) k 0 = +∞ . + T ∈R If the continuous spectrum is not empty, we also have~~

~~(91) sup T k C(·,T ) k −s,0 = +∞ + T ∈R for the continuous component C(x, T ) of γx,T . 36 LIVIO FLAMINIO AND GIOVANNI FORNI †~~

Proof. Let D ∈ Bs. There exists a unique function g ∈ Ann(Is(SM)⊥) ⊂ W s(SM) 0 0 s 0 0 s such that D(g) = 1, D (g) = 0, for all D ∈ B with D 6= D and all D ∈ IC. By the + splitting (80), for all (x, T ) ∈ SM × R ,

γx,T (g) = cD(x, T ) . It follows that, since the horocycle flow is volume preserving, Z T 1 U k c(·,T ) k 0 ≤ k g φt (x) dt k 0 ≤ k g k 0 . T 0 If the distribution D ∈ Bs has Sobolev order S ≤ 1, since D(g) 6= 0, by Theorem 1.3 the equation Uf = g has no solution f ∈ L2(SM). By the Gottschalk-Hedlund Lemma 5.7, it follows that the family of functions Z T U T cD(x, T ) = T γx,T (g) = g φt (x) dt 0 is not equibounded in L2(SM). Hence (90) is proved. The proof of (91) is completely analogous. Suppose that the continuous spectrum is not s s ⊥ s empty and let g ∈ W (SM) be a function belonging to Ann I (SM) ∩ Ann (Id) and s s such that D(g) 6= 0 for some D ∈ IC. Since any D ∈ IC has Sobolev order equal to 1/2, by Theorem 1.3, the equation Uf = g has no solution f ∈ L2(SM). Then, as before, by the Gottschalk-Hedlund Lemma 5.7, the family of functions {T C(·,T )(g)}T ∈R is not equibounded in L2(SM). In fact, Z T U T C(x, T )(g) = T γx,T (g) = g φt (x) dt . 0 Since k C(·,T )(g) k 0 ≤ k C(·,T ) k −s,0 k g k s , 2 the family of functions {T k C(·,T ) k −s,0}T ∈R is not equibounded in L (SM), proving (91). X 5.3. Iterative estimates. By the commutation relations (14), the geodesic flow {φt } ex- V t pands the orbits of unstable horocycle flow {φs } by a factor e and it contracts the orbits of U −t stable horocycle flow {φs } by a factor e : X U U X X V V X (92) φt ◦ φs = φse−t ◦ φt , φt ◦ φs = φset ◦ φt . It follows that, in the distributional sense, X (93) φ (γx,T ) = γ X t . t φ−t(x),e T Were the splitting (72), hence the splitting (80), invariant under the action geodesic flow, Theorems 1.5 and 1.7 would follow immediately. In fact, by the eigenvalue identities (74) s X s on Id and by the bound on the norm of φt on IC given by Lemma 5.1, the invariance of the s splitting (80) would entail that, for all D ∈ B \B1/4 and for the continuous component,

~~X t λDt cD(φ−t(x), e ) = cD(x, 1) e , X t −t/2 k C(φ−t(x), e ) k −s ≤ C1(1 + |t|) e k C(x, 1) k −s TIME AVERAGES FOR HOROCYCLE FLOWS 37~~

~~+ − and, if 1/4 ∈ σpp, by identity (75) for all pairs {D , D } ⊂ B1/4,~~

X t t −t/2 cD+ (φ−t(x), e ) = [cD+ (x, 1) − cD− (x, 1)] e , (94) 2 X t −t/2 cD− (φ−t(x), e ) = cD− (x, 1) e , allowing us to conclude. The aim of this section is to prove that, in spite of the lack of invariance of the split- s ting (80), the projections onto IC and D ∈ B of the geodesic push-forwards of the remainder term R(x, T ) are negligible. Let x ∈ SM, T > 0. It will be convenient to discretise the geodesic ﬂow time t ≥ 1 and X to consider the push-forwards of the arc γx,T by φ`h, where h ∈ [1, 2] and ` ∈ N. Then the X distribution (measure) φ`h(γx,T ) has a splitting

X X (95) φ`h(γx,T ) = cD(x, T, `)D + C(x, T, `) + R(x, T, `) . D∈Bs 2 We will prove pointwise and L bounds on the sequences of functions cD(·, T, `), C(·, T, `) and R(·, T, `). By the identity (93) and the deﬁnition (80), we have:

~~ X `h cD(x, T, `) = cD φ−`h(x), e T ,~~

~~ X `h (96) C(x, T, `) = C φ−`h(x), e T ,~~

~~ X `h R(x, T, `) = R φ−`h(x), e T .~~

2 Uniform bounds and L bounds on the functions cD(·, T, `), C(·, T, `) and R(·, T, `) are 2 `h `h `h clearly equivalent to uniform and L bounds on cD(·, e T ), C(·, e T ) and R(·, e T ) respectively. Let X rD(x, T, `) := cD φh R(x, T, `) ∈ R , (97) X s RC(x, T, `) := C φh R(x, T, `) ∈ IC .

X X X s By the identity φ(`+1)h = φh ◦φ`h, since the distributions D ∈ B \B1/4 are eigenvectors X s X of the geodesic ﬂow {φt } (see (74)) and the space IC is {φt }-invariant, we obtain by projecting on D-components and on the continuous component:

~~λDh cD(x, T, ` + 1) = cD(x, T, `) e + rD(x, T, `); (98) X C(x, T, ` + 1) = φh C(x, T, `) + RC(x, T, `) . + − If 1/4 ∈ σpp, for all pairs {D , D } ⊂ B1/4 we obtain by (75):~~

h −h/2 cD+ (x, T, ` + 1) = [cD+ (x, T, `) − cD− (x, T, `)] e + rD+ (x, T, `); (99) 2 −h/2 cD− (x, T, ` + 1) = cD− (x, T, `) e + rD− (x, T, `) . Bounds on the solutions of the difference equations (98) and (99) can be derived from the following trivial lemma. 38 LIVIO FLAMINIO AND GIOVANNI FORNI †

~~Lemma 5.9. Let Φ ∈ L(E) be a bounded linear operator on a normed space E. Let {R`}, ` ∈ N, be a sequence of elements of E. The solution {x`} of the following difference equation in E,~~

(100) x`+1 = Φ(x`) + R` , ` ∈ N , has the form `−1 ` X `−j−1 (101) x` = Φ (x0) + Φ Rj . j=0 By Lemma 5.9, the proof of Theorems 1.5 and 1.7 is essentially reduced to estimates on the ‘remainder terms’ rD(x, T, `) and RC(x, T, `). Such estimates can be derived from Lemma 5.5. Let x0 ∈ SM be a reference point as in Lemma 5.5 and let d0 : SM → R be + the distance function from x0. For each (x, T ) ∈ SM × R and each ` ∈ N, let d(x, T, `) X be the the maximum distance of the endpoints of the horocycle arc φ`h(γx,T ) from x0: X X U (102) d(x, T, `) := max{d0 φ−`h(x) , d0 φ−`h ◦ φT (x) } .

+ Lemma 5.10. There exists a constant C7 := C7(s) such that, for all (x, T ) ∈ SM × R and all ` ∈ N, 2 X C7 (103) |r (x, T, `)|2 + k R (x, T, `) k 2 ≤ exp{d(x, T, `) − 2`h} . D C −s T D∈Bs X Proof. Let CX (s) := maxh∈[1,2] k φh k −s. Then, by the definition (97) and Lemma 5.2, we have X 2 2 2 2 2 |rD(x, T, `)| + k RC(x, T, `) k −s ≤ C2 CX k R(x, T, `) k −s . D∈Bs But R(x, T, `) is the “R” component of an arc of horocycle of length e`hT whose endpoints are at a distance d(x, T, `) from the reference point x0 (cf. (96), (93), (102)). Thus by d(x,T,`)/2 `h Lemma 5.5 k R(x, T, `) k −s < C5e /e T and the lemma follows. The difference equations (98) and (99) also yield estimates for the L2 norms of the functions cD(·, T, `) and k C(·, T, `) k −s on SM. However, in the case of the continuous component, such L2 bounds are not effective because of the lack of L2 estimates on the remainder term k RC(·, T, `) k −s. For the D-components we can prove the following: s Lemma 5.11. For each D ∈ B , there exists a constant C8 := C8(D, s) > 0 such that, for all T > 0 and all ` ∈ N, C (104) k r (·, T, `) k ≤ 8 e−`h . D 0 T s 0 Proof. Let g := gD ∈ W (SM) be the unique function such that D(g) = 1, D (g) = 0 0 s 0 s s ⊥ X for all D ∈ B , D 6= D, and g ∈ Ann (IC) ∩ Ann(I (SM) ). Let gh := φ−h(g). By definition of the remainder term we have rD(x, T, `) = R(x, T, `)(gh). Again by definition X `h (see (96)) we have R(x, T, `) = R φ−`h(x), e T and we obtain, using Lemma 5.6: C C k r (·, T, `) k = k R(·, T, `)(g ) k ≤ 6 k g k ≤ 8 e−`h D 0 h 0 T e`h h −s T TIME AVERAGES FOR HOROCYCLE FLOWS 39 with C8 := C6 maxh∈[1,2] k gh k s. 5.4. Bounds on the components: the cuspidal case. In the cuspidal case, the precision of our asymptotics of geodesic push-forwards of a horocycle arc depends on the rate of escape + into the cusps of its endpoints. Let d0 : SM → R be the distance function from a fixed point x0 ∈ SM. For any given σ ∈ [0, 1] and A ≥ 0 let X VA,σ := {x ∈ SM | d0 φt (x) ≤ A + σ |t| , for all t ≤ 0} . [ Vσ := VA,σ A≥0 The sets Vσ do not depend on the choice of the point x0 ∈ SM and they are measurable as they can be written as countable unions of closed sets (hence, they are Fσ sets). Since the geodesic flow has unit speed V1 = SM. On the other hand, by the logarithmic law of geodesics, Vσ has full measure for any σ > 0 [56]. s Lemma 5.12. Let s > 3. For every D ∈ B of order SD > 0, there exists a uniformly + {K (x, T, `)} + (x, T ) ∈ V × bounded sequence of positive bounded functions D `∈Z , A,σ R , such that the following estimates hold. For every horocycle arc γx,T having endpoints x, T + s + φU (x) ∈ VA,σ and for all ` ∈ Z we have, if D ∈ B \B1/4,  K (x, T, `) e−SD`h , if S < 1 − σ ,  D D 2 −SD`h σ (105) |cD(x, T, `)| ≤ KD(x, T, `) ` e , if SD = 1 − 2 , −(1− σ )`h σ  2 KD(x, T, `) e , if SD > 1 − 2 . + If 1/4 ∈ σpp and D ∈ B1/4, we have ( −`h/2 KD(x, T, `) ` e , if σ < 1 , (106) |cD(x, T, `)| ≤ 2 −`h/2 KD(x, T, `) ` e , if σ = 1 .

There exists a positive constant KC := KC(A, σ, s, T ), such that the following estimates + hold. For every horocycle arc γx,T as above and for all ` ∈ Z , we have ( −`h/2 KC ` e , if σ < 1 , (107) k C(x, T, `) k −s ≤ 2 −`h/2 KC ` e , if σ = 1 .

~~In addition, there exists a positive constant K := K(A, σ, T, s) such that, for all γx,T with + endpoints belonging to the set VA,σ and for all ` ∈ Z ,~~

~~X 2 2 2 (108) KD(x, T, `) + KC ≤ K . D∈Bs~~

Proof. For all D 6∈ B1/4, by the ﬁrst difference equation in formula (98) and by Lemma 5.9 λ h with E := C and Φ the multiplication operator by e D ∈ C, we obtain −SD`h (109) |cD(x, T, `)| ≤ |cD(x, T, 0)| e + ΣD(x, T, `) , with `−1 X −SDh(`−j−1) ΣD(x, T, `) := |rD(x, T, j)|e . j=0 40 LIVIO FLAMINIO AND GIOVANNI FORNI †

S `h We must therefore bound the terms |cD(x, T, 0)| and e D ΣD(x, T, `) by KD(x, T, `), σ (SD−1+ )`h KD(x, T, `)` or KD(x, T, `)e 2 , according to the different values of SD, for some + {K (x, T, `)} + V × uniformly bounded sequence of functions D `∈Z on A,σ R . It follows from Corollary 5.4, taking into account the fact that the endpoints of γx,T + + belong to the set VA,σ ⊂ B(x0,A), that there exists a continuous function Cs : R × R → + R (which can be chosen constant if M has a positive injectivity radius) such that

X 2 2 (110) |cD(x, T, 0)| ≤ Cs(A, T ) ; D∈Bs thus the term |cD(x, T, 0)| in formula (109) satisﬁes estimates ﬁner than (105) and (108). Using again the fact that the endpoints of γx,T belong to the set VA,σ and the estimate (103), a calculation based on the Cauchy-Schwartz inequality yields the following bounds on the remainder terms ΣD(x, T, `) for D 6∈ B1/4. For all S > 0, A ≥ 0, σ ∈ [0, 1], there exists a constant C0 := C0(A, σ, S, s) > 0 such that

X C0 σ Σ2 (x, T, `) e2SD`h ≤ , if S < 1 − ; D T 2 2 D:SD≤S X C0 `2 σ Σ2 (x, T, `) e2SD`h ≤ , if S = 1 − ; (111) D T 2 2 D:SD=S 0 X 2 C −2(1− σ )`h σ Σ (x, T, `) ≤ e 2 if S > 1 − . D T 2 2 D:SD≥S It follows from (110) and (111) that, for each D ∈ Bs, the sequence of positive functions (112)  S `h σ |cD(x, T, 0)| + ΣD(x, T, `)e D if SD < 1 − ,  2  SD`h −1 σ KD(x, T, `) := |cD(x, T, 0)| + ΣD(x, T, `)e ` if SD = 1 − 2 ,  σ  SD`h −(SD−1+ 2 )`h σ  |cD(x, T, 0)| + ΣD(x, T, `)e e , if SD > 1 − 2 . is uniformly bounded for all γx,T with endpoints belonging to the set VA,σ. In view of the bound (109), this proves the estimate (105) for each D 6∈ B1/4. In addition, since the set s of real numbers {SD | D ∈ B } is ﬁnite, the inequalities (110) and (111) imply that, for all + γx,T with endpoints belonging to the set VA,σ and for all ` ∈ Z ,

X 2 00 (113) KD(x, T, `) ≤ C , s D∈B \B1/4 for some constant C00 := C00(A, σ, T, s) > 0, thereby proving the upper bound (108) over s all D-components with D ∈ B \B1/4. + − The proofs of the upper bounds for all pairs {D , D } ⊂ B1/4 (if 1/4 ∈ σpp) and for the continuous component are similar. In the ﬁrst case, by formula (99) we can apply 2 Lemma 5.9 with E = R and 1 −h/2 (114) Φ := e−h/2 . 0 1 TIME AVERAGES FOR HOROCYCLE FLOWS 41

~~By formula (101), we obtain~~

~~Since the endpoints of the horocycle arc γx,T belong to the set VA,σ, by the estimates (110) and (103), the sequence of positive functions~~

(117) ( `h `h/2 −1 |cD+ − 2 cD− |(x, T, 0) + ΣD+ (x, T, `) e ` , if σ < 1 , KD+ (x, T, `) := `h `h/2 −2 |cD+ − 2 cD− |(x, T, 0) + ΣD+ (x, T, `) e ` , if σ = 1 . + + is uniformly bounded by a constant KD := KD (A, σ, s, T ) > 0. − − If D = D ∈ B1/4, it follows from the second lines in (115) and (116) that the sequence σ of positive functions KD(x, T, `), defined as in (112) with SD = 1/2 ≤ 1 − 2 , is uniformly bounded by a constant KD := KD(A, σ, s, T ) > 0. (3) Since B1/4 is always a finite set, by the extimate (113) there exists a constant C := (3) C (A, s, T ) > 0 such that, for all γx,T with endpoints belonging to the set VA,σ and for all + ` ∈ Z , X 2 (3) (118) KD(x, T, `) ≤ C . D∈Bs s X For the continuous component, we apply Lemma 5.9, with E = IC and Φ = φh , to the second difference equation in formula (98). We obtain ` (119) k C(x, T, `) k −s ≤ k Φ k −s k C(x, T, 0) k −s + ΣC(x, T, `) with `−1 X `−j−1 ΣC(x, T, `) := k Φ RC(x, T, j) k −s . j=0 X s −t/2 By Lemma 5.1 the norm of the operator φt on IC is bounded by C1(1 + |t|)e . Tak- ing into account the fact that the endpoints of γx,T belong to the set VA,σ we find: (1) by Lemmata 5.2 and 5.3 we obtain that there exists a constant C(4) := C(4)(A, s, T ) such that (4) k C(x, T, 0) k −s ≤ C ; (2) using the estimate (103) for k RC(x, T, j) k −s we find that the sequence of positive functions defined by:

 `h/2 −1  k C(x, T, 0) k −s + ΣC(x, T, `) e ` , if σ < 1 , (120) KC(x, T, `) := `h/2 −2  k C(x, T, 0) k −s + ΣC(x, T, `) e ` , if σ = 1 . 42 LIVIO FLAMINIO AND GIOVANNI FORNI † is uniformly bounded by a constant KC := KC(A, σ, s, T ) > 0.

Remark 1. In the case σ = 1, the set Vσ = SM and the results of Lemma 5.12 hold for all + (x, T ) ∈ SM × R . In addition, for every bounded subset F ⊂ SM, there exists a constant A := AF > 0 such that F ⊂ VA,1. In fact, one can take AF := max{dist(x, x0) | x ∈ F }, for a ﬁxed x0 ∈ SM.

~~For the D-components we also have the following L2 upper and lower bounds:~~

~~s 0 0 Lemma 5.13. Let s > 3. For each D ∈ B , there exists a constant KD = K (D, s), such + that for all ` ∈ Z :~~

~~ 0 −S `h + K e D , if D 6∈ B & SD < 1 ;  D 1/4 0 −SD`h + (121) k cD(·, T, `) k 0 ≤ KD ` e , if D ∈ B1/4 or SD = 1 ;  0 −`h s KD e , if D ∈ B & SD > 1 .~~

~~00 00 If SD < 1, there exists KD := K (D, s) > 0, TD := T (D, s) > 0 and `D := `(D, s) ∈ + + Z (`D = 1 if D 6∈ B1/4) such that, for all T ≥ TD and all ` ≥ `D:~~

~~( 00 −SD`h + KD e if D 6∈ B1/4 ; (122) k cD(·, T, `) k 0 ≥ 00 −SD`h + KD ` e if D ∈ B1/4 .~~

Proof. The proof of L2 upper bounds is similar to that of the pointwise upper bounds in Lemma 5.12. In case D 6∈ B1/4, it is based on the difference equation for D-components in formula (98), on the upper bounds (89) of Lemma 5.8, on Lemma 5.11 and on Lemma 5.9 for the normed space E := L2(SM) and the operator Φ ∈ L(E) of multiplication by λDh e ∈ C. In case D ∈ B1/4, it is based on the vector-valued difference equation (99) and on Lemma 5.9 for the normed space E := L2(SM) × L2(SM) and the operator Φ ∈ L(E) given by the natural action of the Jordan matrix (114) on E. The proof of the lower bound (122) is as follows. In case D 6∈ B1/4, by the difference equation for D-components in formula (98) and by Lemma 5.9, we have:

~~−SD`h k cD(·, T, `) k 0 ≥ k cD(·,T, 0) k 0 e − ΣD(T, `) with `−1 X −SD(`−j−1)h ΣD(T, `) := k rD(·, T, j) k 0 e . j=0~~

~~+ − If {D , D } ⊂ B1/4, by the vector-valued difference equation (99) and by Lemma 5.9, we have~~

`h −`h/2 k cD+ (·, T, `) k 0 ≥ k cD− (·,T, 0) − cD+ (·,T, 0) k 0 e − ΣD+ (T, `) , (123) 2 −`h/2 k cD− (·, T, `) k 0 ≥ k cD− (·,T, 0) k 0 e − ΣD− (T, `) , TIME AVERAGES FOR HOROCYCLE FLOWS 43 with `−1 X (` − j − 1)h −h(`−j−1)/2 Σ + (T, `) := k r + (·, T, j) − r − (·, T, j) k e , D D 2 D 0 j=0 (124) `−1 X −h(`−j−1)/2 ΣD− (T, `) := k rD− (·, T, j) k 0 e . j=0

~~From the upper bound (104) for k rD(·, T, j) k 0 of Lemma 5.11, we obtain that, for each s + D ∈ B with SD < 1, there exists a constant C := C(D, s) > 0 such that, for all ` ∈ Z ,~~

~~( C −SD`h + T e if D 6∈ B1/4 ; (125) ΣD(T, `) ≤ C `h −`h/2 + T 2 e if D ∈ B1/4 .~~

The condition SD < 1 implies also, by Lemma 5.8, that there exists TD := T (D, s) > 1 + such that, for all T ≥ TD and all ` ∈ Z , C k c (·,T, 0) k > 2 . D 0 T 2 + + The L lower bounds hence follow from the lower bounds (123) and for D ∈ B1/4 from 2 + the L upper bound for the D -component proved in Lemma 5.8.

X For all t ≥ 0, the push-forward probability measure φt (γx,T ) is the uniformly distributed t probability measure on a stable horocycle arc of length Tt := e T . The following quanti- s s tative equidistribution result holds. Let I+(SM) ⊂ I (SM) be the subspace of invariant distributions orthogonal to the volume form.

Theorem 5.14. Let s > 3. Then there exists a constant C9 := C9(A, σ, s, T ) such that for any horocycle arc γx,T with endpoints belonging to the set VA,σ, for any t ≥ 1 and for all f ∈ W s(SM), we have Z X X s −SD (126) φt (γx,T )(f) = f dvol + cD(x, T, t) D(f) Tt + SM σ 1− 2 D∈B+ 1 σ s − 2 ασ s 2 −1 βσ + C (x, T, t)(f)Tt log Tt + R (x, T, t)(f) Tt log Tt . s s s s −s with cD(x, T, t) ∈ C, C (x, T, t) ∈ IC and R (x, T, t) ∈ W (SM) satisfying the following upper bounds:

~~X s 2 |cD(x, T, t)| ≤ C9 , σ 1− 2 D∈B+ s k C (x, T, t) k −s ≤ C9 , s k R (x, T, t) k −s ≤ C9 .~~

In the above asymptotics, the exponent ασ is 1 if σ < 1 and equals 2 if σ = 1; the expo- s σ nent βσ is 0 if every D ∈ B has Sobolev order SD 6= 1 − 2 and equals 1 otherwise. 44 LIVIO FLAMINIO AND GIOVANNI FORNI †

~~σ 1− 2 In addition, for every invariant distribution D ∈ B+ , there exists a constant CD := C(D, s) > 0 such that, if T > 0 is sufﬁciently large, then for all t ≥ 0,~~

−1 CD ≤ k cD(·, T, t) k 0 ≤ CD . + Proof. Let t ≥ 1. There exist h ∈ [1, 2] and ` ∈ Z such that t = `h. The distribu- X −s tion φt (γx,T ) ∈ W (SM) can be split as in (95), hence the expansion (126) follows. The pointwise upper bounds on the coefficients can be derived from Lemma 5.12 for the D-components and the C-component and, by its definition in (96), from Lemma 5.5 for the remainder term R(x, T, `) of the splitting (95). We remark that the term with coefficient s 1− σ R (x, T, t)(f) in (126) includes the contributions of all D-components with D 6∈ B 2 as well as the contribution of the remainder term R(x, T, `) of the splitting (95). Finally, the L2 bounds for the D-components follow from Lemma 5.13. In fact, all such estimates are uniform with respect to h ∈ [1, 2]. Theorem 1.7 follows from Theorem 5.14 (for σ = 1) and from Remark 1. In the particular case that γx,T is supported on a closed cuspidal horocycle, our methods yield a simple proof of a weaker version of a result of P. Sarnak [51]. Sarnak’s proof is based on Eisenstein series (Rankin-Selberg method).

Proposition 5.15. Let s > 2 and let γx,T be a closed cuspidal horocycle. Then there exist s 1/2 s coefﬁcients cD ∈ C for D ∈ B+ and a distribution-valued function C (t) ∈ IC, uniformly bounded with respect to t ≥ 0, such that the following asymptotics holds. For all t ≥ 0, Z X X s −SD (127) φt (γx,T )(f) = f dvol + cD D(f) Tt + SM 1/2 D∈B+ 1 s − 2 + C (t)(f)Tt log Tt . Proof. A uniformly distributed probability measure supported on a closed cuspidal horocycle is a U-invariant distribution. In fact by the Sobolev embedding theorem, if s > 2, s then γx,T ∈ I (SM). Hence, in the splitting of γx,T according to (80), the remainder term R(x, T ) = 0. The result then follows immediately from the description of the spec- X s tral representation of the one-parameter group {φt } on the space I (SM) given by Theo- rem 1.4 and Lemma 5.1.

5.5. Bounds on the components: the compact case. In the compact case, the spectrum of the Laplacian is purely discrete and therefore the splitting (80) of a horocycle arc γx,T becomes: X γx,T = cD(x, T )D + R(x, T ) . D∈Bs Lemma 5.12 can be strengthened as follows:

Lemma 5.16. Let s > 3 and let T0 ≥ 1. There exists a sequence of bounded positive + functions {KD(x, T )}D∈Bs , (x, T ) ∈ SM × R , such that, for any horocycle arc γx,T with T > T0 we have TIME AVERAGES FOR HOROCYCLE FLOWS 45

 −S + KD(x, T ) T D , if D 6∈ B & SD < 1 ;  1/4 K (x, T ) T −SD log T if D ∈ B+ (⇒ S = 1/2) ; (128) |c (x, T )| ≤ D 1/4 D D −1 s KD(x, T ) T log T, if D ∈ B & SD = 1 ;  −1 s KD(x, T ) T , if D ∈ B & SD ≥ 2 . + In addition, there exists K := K(s, T0) > 0 such that, for all (x, T ) ∈ SM × R , T ≥ T0, X 2 2 (129) KD(x, T ) ≤ K . D∈Bs 1 s 0 0 0 On the other hand, for each D ∈ B ⊂ B , there exist KD := K (D, s) > 0 and TD := 0 0 T (D, s) > 0 such that, for all T ≥ TD,

( 0 −SD + KD T , if D 6∈ B1/4 ; (130) k cD(·,T ) k 0 ≥ 0 −SD + KD T log T, if D ∈ B1/4 . + `h Proof. Let T0 > 0. For all T ≥ eT0, there exist h ∈ [1, 2] and ` ∈ Z such that T = e T0. By the ﬁrst identity in (96), it follows that X cD(x, T ) = cD φ (x),T0, ` , (131) `h k cD(·,T ) k 0 = k cD (·,T0, `) k 0 . Since M is compact, the unit tangent bundle SM is compact and has ﬁnite diameter. Hence, by Lemma 5.5 and by the argument of Lemma 5.12, there is a constant K > 0 and, for each s + D ∈ B , there exists a positive function KD(x, T ), (x, T ) ∈ SM × R , such that the estimates in (105), (106) and (108) hold with σ = 0. The desired uniform upper bounds 2 then follow. The L lower bounds follow immediately from Lemma 5.13. Theorem 1.5 follows immediately from lemma 5.5 and lemma 5.16. Lemma 5.16 implies the following pointwise lower bound on the deviation of ergodic averages. 1 s 00 00 Corollary 5.17. If D ∈ B ⊂ B , there exist constants KD := K (D, s) > 0, αD := 00 00 00 α(D, s) > 0 and TD := T (D, s) > 0 such that the following holds. For all T ≥ TD there exists a set AT ⊂ SM of measure at least αD > 0 such that, for all x ∈ AT ,

( 00 −SD + KD T , if D 6∈ B1/4 ; (132) |cD(x, T )| ≥ 00 −SD + KD T log T, if D ∈ B1/4 . Hence, the lower bound (132) holds for almost all x ∈ SM and for inﬁnitely many T > 0. 1 0 Proof. Let D ∈ B . Since SD < 1, by Lemma 5.16, there exist constants KD ≥ KD > 0 0 such that, for all x ∈ SM and all T > TD, we have KD |cD(x, T )| ≤ 0 k cD(·,T ) k 0 . KD Fix any constant K ∈ ]0, 1[ and let

~~AT := {x ∈ SM | |cD(x, T )| > K k cD(·,T ) k 0} . 46 LIVIO FLAMINIO AND GIOVANNI FORNI †~~

0 Then, for T > TD, we have 2 ! Z 2 KD 2 2 (133) K + 0 |AT | k cD(·,T ) k 0 ≥ |cD(x, T )| dvol + KD SM\AT Z 2 2 + |cD(x, T )| dvol = k cD(·,T ) k 0 . AT 0 It follows that the measure of the set AT is uniformly bounded below for all T > TD by a positive constant. In fact, we have 0 2 2 KD |AT | ≥ (1 − K ) . KD 2 The desired pointwise lower bounds hold on AT by its definition and by the L lower bounds in Lemma 5.16. By a standard Borel-Cantelli argument, the set A of x ∈ SM such that x ∈ AT for infinitely many T ≥ T0, has positive measure. By the ergodicity of the horocycle flow, the set A has full measure. We conclude by proving that the Central Limit Theorem does not hold for horocycle flows on compact hyperbolic surfaces.

1 s Proof of Corollary 1.6. Let D ∈ B ⊂ B . Let SD ∈ ]0, 1[ be its Sobolev order. Let f ∈ W s(SM), s > 3, be any zero average function such that D(f) 6= 0 and f belongs to the s kernel of all U-invariant distributions of the basis B of Sobolev order S ≤ SD. By Theorem 1.5, there exists K > 0 such that, for T > 0 sufﬁciently large, the functions Z T U FT := f(φt (·)) dt 0 are uniformly bounded above by K k FT k 0. It follows that the probability distribution ρT of FT / k FT k 0 is supported within the interval [−K,K]. By Theorem 1.3 and Corollary 5.17, there exists an interval around zero whose complement has ρT -measure larger than αD > 0, for all sufﬁciently large T > 0.

~~6. APPENDIX Proof of Lemma 2.1. The case of the principal series (ν ∈ iR) is immediate. In the case of the complementary series, if ν ∈ ]0, 1] we write, for k ≥ 1,~~

~~k 1 + ν Y 2i − 1 + ν Π−1 = Π = ν,k −ν,k 1 − ν 2i − 1 − ν i=2 (empty products are set equal to 1). We have~~

~~k k Y 2i − 1 + ν X 2ν log = log 1 + . 2i − 1 − ν 2i − 1 − ν i=2 i=2 TIME AVERAGES FOR HOROCYCLE FLOWS 47~~

We then estimate the logarithms by the inequalities x ≤ log(1 + x) ≤ x, for x ∈ +. 1 + x R Finally we estimate from above and below the resulting series by the integral inequality. In the case of the discrete series (ν = 2n − 1), we remark that, for all k ≥ n, k k Y 2i − 1 − ν Y i − n (k − n)!(2n − 1)! Π = = = . ν,k 2i − 1 + ν i + n − 1 (k + n − 1)! i=n+1 i=n+1 By the Stirling’s formula, 1 (k − n + 1)! (k − n + 1)k−n+1 k − n + 1 2 ≈ e−2 . (k + n − 1)! (k + n − 1)k+n−1 k + n − 1 where ≈ means that, for some constant C > 1 independent of n and k, the ratio of the left side to the right side is bounded above by C and below by C−1 for all n ≥ 1 and k ≥ n. Hence, k−n+1 −2 k − n + 1 −ν+ 1 − 1 (134) Π ≈ e ν! (k + n − 1) 2 (k − n + 1) 2 = ν,k k + n − 1 k+n−1 −2 k − n + 1 − 1 −ν+ 1 = e ν! (k + n − 1) 2 (k − n + 1) 2 . k + n − 1

~~1 Since k ≥ ` and the function (1 + x) x is decreasing on the interval x > −1,~~

`−n+1 ν−1 ` + n − 1 ν − 1 `−n+1 (135) = (1 + ) ν−1 ≤ ` − n + 1 ` − n + 1 ν−1 k−n+1 ν − 1 k−n+1 k + n − 1 ≤ (1 + ) ν−1 = k − n + 1 k − n + 1 and `+n−1 ν−1 ` + n − 1 ν − 1 − `+n−1 (136) = (1 − ) ν−1 ≥ ` − n + 1 ` + n − 1 ν−1 k+n−1 ν − 1 − k+n−1 k + n − 1 ≥ (1 − ) ν−1 = . k + n − 1 k − n + 1 Since k ≥ `, the function k − n + x ` − n + x + is non-increasing on R . Hence k + n − 1 k − n + ν k − n + 1 = ≤ . ` + n − 1 ` − n + ν ` − n + 1 It follows that −ν+ 1 − 1 −ν Π k − n + ν 2 k − n + 1 2 k − n + ν ν,k ≤ C ≤ C Πν,` ` − n + ν ` − n + 1 ` − n + ν 48 LIVIO FLAMINIO AND GIOVANNI FORNI † and −ν+ 1 − 1 −ν Π k − n + 1 2 k − n + ν 2 k − n + 1 ν,k ≥ C−1 ≥ C−1 . Πν,` ` − n + 1 ` − n + ν ` − n + 1

Proof of Lemma 4.2. Since |Πν,`| ≥ |Πν,k| for all k ≥ ` ≥ 0, if |ν| ≥ 1/2 the estimate is immediate by the triangular inequality. If ν ∈ iR or ν ∈] − 1/2, 1/2[, we have k k d Πν,` d Y 2i − 1 + ν X 2(2i − 1) log = log = 2 2 > 0. dν Πν,k dν 2i − 1 − ν (2i − 1) − ν i=`+1 i=`+1 By the integral inequality, d Π Π d Π Π 1 + k | ν,` | = | ν,` || log ν,` | ≤ C| ν,` | log . dν Πν,k Πν,k dν Πν,k Πν,k 1 + ` Hence, if |ν| < 1/2 the estimate follows from the intermediate value theorem.

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UNITÉ MIXTEDE RECHERCHE CNRS 8524, UNITÉDE FORMATION ET RECHERCHEDE MATHÉMA- TIQUES,UNIVERSITÉDE SCIENCESET TECHNOLOGIESDE LILLE, F59655 VILLENEUVED’ASQ CEDEX, FRANCE E-mail address: [email protected]

~~DEPARTMENT OF MATHEMATICS,NORTHWESTERN UNIVERSITY,EVANSTON, IL 60208, U.S.A. E-mail address: [email protected]~~