Spectral estimates for Ruelle operators with two parameters and sharp large deviations

We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + \i b)\tau + (c + \i d) g}$ with two complex parameters and H\"{o}lder functions $f,\: g$ generalizing the case $\Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].


1.
Introduction. Let M be a C 2 complete Riemannian manifold, and let ϕ t : M −→ M be a C 2 weak mixing Axiom A flow ( [5], [7]). Let Λ be a basic set for ϕ t , that is, Λ is a compact locally maximal invariant subset of M and ϕ t is hyperbolic and transitive on Λ.
As in [9], we will use a symbolic coding of the flow on Λ provided by a a fixed Markov family {R i } k i=1 . More precisely, we consider a Markov family of pseudorectangles R i = [U i , S i ] = {[x, y] : x ∈ U i , y ∈ S i } (see section 2 for more details). Denote by P : R = ∪ k i=1 R i −→ R the related Poincaré map, by τ (x) > 0 the first return time function on R, and by σ : U = ∪ k i=1 U i −→ U the shift map given by σ = π (U ) • P, where π (U ) : R −→ U is the projection along stable leaves. The flow ϕ t on Λ is naturally related to the suspension flow σ τ t on the suspension space R τ (see section 2 for details). There exists a natural semi-conjugacy projection π(x, t) : R τ −→ Λ which is one-to-one on a residual set (see [2]). For x ∈ R set τ n (x) := τ (x) + τ (σ(x)) + ... + τ (σ n−1 (x)).
Given Hölder continuous functions F, G : Λ −→ R, define f, g : R −→ R by The main object of study in this paper are the Ruelle transfer operators of the form L f −sτ +zg v(x) = σy=x e f (y)−sτ (y)+zg(y) v(y) , s, z ∈ C , x ∈ U, 6392 VESSELIN PETKOV AND LUCHEZAR STOYANOV depending on two complex parameters s and z. Under certain assumptions, strong spectral estimates for such operators have been established in [9] and some significant applications to the study of zeta functions depending on two complex parameters have been made.
We denote by m H the equilibrium state corresponding to H : R τ −→ R and by µ k the equilibrium state corresponding to k : R −→ R. More precisely, Pr σ (k) = h(σ, µ k ) + where h(σ, µ) is the metric entropy of σ with respect to µ and h(σ t τ , m) is the metric entropy of the suspended flow σ t τ with respect to m. Let P = Pr σ (f ). Let h 0 denote the standard sup norm of h on U . For |b| ≥ 1, and β > 0, as in [4], define the norm on the space C β (U ) of β-Hölder functions on U . Let U be the set of those points x ∈ U such that P m (x) is not a boundary point of a rectangle for any integer m. In a similar way define R.
Our first aim in this paper is to prove the following theorem. over the basic set Λ, and let 0 < β < α. Let R = {R i } k i=1 be a Markov family for ϕ t over Λ as in section 2. Then for any real-valued functions f, g ∈ C α ( U ) and any constants > 0 and B > 0 there exist constants 0 < ρ < 1, a 0 > 0, b 0 ≥ 1 and C = C(B, ) > 0 such that if a, c ∈ R satisfy |a|, |c| ≤ a 0 then L m f −(a+ib)τ +(c+iw)g h β,b ≤ C e P m ρ m |b| h β,b (1.1) for all h ∈ C β (U ), all integers m ≥ 1 and all b, w ∈ R with |b| ≥ b 0 and |w| ≤ B |b|.
In Theorem 5.1 in [9] the above estimate has been proved in the case P = 0 assuming |w| ≤ B |b| ν for some constant ν ∈ (0, 1). The present results is significantly stronger. See also Remark 1 below.
In the proof of Theorem 1 we will use some arguments from the proof of Theorem 5.1 in [9] with necessary modifications. Notice that in Theorem 1 above we do not assume that pressure P of f is zero, unlike what has been done in previous papers. This contributes the term e P m in the right-hand-side of (1.1) which is significant especially in the case P < 0 which occurs in the applications concerning large deviations (see section 3). In previous papers the authors consider the case P = 0 and remark that the general case follows from this. However a more careful argument shows that an estimate of the form (1.1) does not follow immediately from a similar estimate with P = 0. See Remark 4 in section 4 below.
Remark 1. In the proof of Theorem 2 in section 3 we apply Theorem 1 with b = Cw for some constant C > 0; then |w| = 1 C |b|. The relevant part of Theorem 5.1 in [9] assumes |w| ≤ B|b| ν for some ν ∈ (0, 1) and this is clearly not sufficient for the proof of Theorem 2 below. Let F, G : Λ −→ R be Hölder continuous functions and let again f, g : R −→ R be defined by We will now assume that G > 0 everywhere on Λ. Consider a number where ξ(p) is determined by the equation Let 0 < ρ < 1 be the constant from Theorem 1, and let 0 < α 0 = − log ρ 2 . Fix an arbitrary 0 < δ ≤ α 0 and consider the sequence {δ n } n∈N , where δ n = e −δn .
Clearly the property , On the other hand, since cn ≤ τ n (x) ≤ c 1 n, ∀x ∈ R, ∀n ∈ N with some constants 0 < c ≤ c 1 for every x, the interval − δn τ n (x) , δn is exponentially shrinking to 0 as n → ∞.
Our second problem concerns the analysis of the asymptotic of µ{x : g n (x) − τ n (x)p ∈ (−δ n , δ n )}, n → ∞, and for p = R τ Gdm F we obtain a large deviation result. On the other hand, as in the previous paper [8], we examine the measure of points x ∈ R for which the difference g n (x) τ n (x) − p stays in an exponentially shrinking interval. Next, assume that the flow σ t τ is topologically weak mixing and the function G is not cohomologous to a constant function. This implies that the set has a non-empty interior and setting β(t) = Pr στ (F + tG), one has Moreover, β (ξ(p)) = p and ξ(p) is differentiable with ξ (p) = 1 β (ξ(p)) > 0. Without loss of generality, by adding a constant, we may assume that Pr στ (F ) = 0. Then 6394 VESSELIN PETKOV AND LUCHEZAR STOYANOV m F and Pr στ (F + tG) do not change and Pr σ (f − Pr στ (F )τ ) = 0 yields Pr σ (f ) = 0. Introduce the rate function Then and γ(p) ≤ 0 is a concave function with strict maximum 0 at p = R τ Gdm F . Since G is not cohomologous to a constant function, the function g p = g − pτ is not cohomologous to a function in C(R : 2πZ), and this yields From now on for simplicity of the notation we will write Pr instead of Pr σ . Consider the rate function where η(p) is the unique real number such that Notice that .
Here we have used the fact that F + η(p)G and F + η(p)G − η(p)p have the same equilibrium state in R τ . Since dPrσ τ (F +tG) dt is increasing, there exists an unique ξ(p) such that dPr στ (F + ξ(p)G) dt = p, therefore ξ(p) = η(p). Hence In section 2 we show that This implies J(p) ≤ 0 and J(p) = 0 if and only if p = R τ Gdm F and ξ(p) = 0. We prove the following large deviation theorem which is also a local limit result.
Theorem 2. Let the assumptions of Theorem 1 be satisfied. Assume that G : Λ −→ (0, ∞) is a Hölder continuous function for which there exists a Markov family for the flow ϕ t on Λ such that G is constant on the stable leaves of all "rectangular boxes" . . , k. Assume in addition that the flow σ t τ is topologically weak mixing and the function G is not cohomologous to a constant function. Let 0 < ρ < 1 be the constant in Theorem 1 and let δ n = e −δn , 0 < δ ≤ − log ρ 2 . Then A similar result for the measure of points x ∈ R for which the difference stays in a exponentially shrinking interval has been obtained in [8] under the conditions that G is a Lipschitz function on Λ and Lip G min G < µ 0 with a suitable positive constant µ 0 . In the present paper we improve the result in [8] assuming that G is only Hölder. Moreover, here we study the more natural difference This progress is essentially based on the spectral estimates for the Ruelle operator with two complex parameters established in Theorem 1. A further improvement would be the analysis of the asymptotic of as T → +∞ and this is an interesting open problem. On the other hand, the case when the interval − e −δT T , e −δT T is replaced by an interval α T , β T , α < β, has been studied in [13]. Comparing (1.5) with Theorem 1 in [13], one observes that in the case we deal with, by scaling, the variable tending to +∞ can take the form T n = n R τ dµ τ +ξ(p)(g−pτ ) . Setting one may write the leading term in (1.5) as 2δ n C(p) 2πβ (ξ(p))T n e Tnγ(p) which modulo the constant C(p) is similar to the asymptotic in [13] with T n → ∞, where the rate function is precisely γ(p).

Remark 2.
The condition G > 0 in Theorem 2 is not a restriction since we can replace G by G + C > 0 for some large constant C > 0. Then p = R τ Gdm F + C, and the asymptotic (1.5) is independent of the constant C. The assumption that G is constant on stable leaves of rectangular boxes B i is significant, however it seems difficult to remove when very sensitive asymptotics such as (1.5) are obtained. For "standard" large deviation results, this assumption is not necessary, since one can use Sinai's Lemma (see e.g. Proposition 1.2 in [7]) to replace an arbitrary Hölder G by a cohomologous function which is constant on stable leaves. In [13] and [10], where instead of (−e −δn , e −δn ) the authors deal with significantly larger intervals, however still smaller than (−c/n, c/n) for a constant c > 0, claims have been made that the general case of Hölder functions on two-sided shifts is easily derived from the one for one-sided shifts. However in both papers there are no proofs of these 6396 VESSELIN PETKOV AND LUCHEZAR STOYANOV claims. For sharp estimates similar to (1.5), it is tempting to believe that such claims would be difficult to justify.
The paper is organised as follows. In section 2 we collect some facts about hyperbolic flows necessary for our exposition. In section 3 we prove Theorem 2 exploiting the spectral estimates (1.1). In the proof an integral involving a Ruelle operators is used with two complex parameters tending to infinity. The Standing Assumptions are stated in section 4 where also some crucial Lemmas are established. Theorem 1 is proved in section 5. The Appendix contains the proof of a technical lemma.
2. Preliminaries. As in section 1, let ϕ t : M −→ M be a C 2 Axiom A flow on a Riemannian manifold M , and let Λ be a basic set for ϕ t . The restriction of the flow on Λ is a hyperbolic flow [7]. For any x ∈ M let W s (x), W u (x) be the local stable and unstable manifolds through x, respectively (see [2], [5], [7]). When M is compact and M itself is a basic set, ϕ t is called an Anosov flow. It follows from the hyperbolicity of Λ that if 0 > 0 is sufficiently small, there exists 1 > 0 such that if x, y ∈ Λ and d(x, y) < 1 , then W s 0 (x) and ϕ [− 0, 0] (W u 0 (y)) intersect at exactly one point [x, y] ∈ Λ (cf. [5]). That is, there exists a unique t ∈ [− 0 , 0 ] such that ϕ t ([x, y]) ∈ W u 0 (y). Setting ∆(x, y) = t, defines the so called temporal distance function.
We will use the set-up and some arguments from [11] and [9]. As in these papers, fix a (pseudo) Markov family The shift map σ : U −→ U is given by σ = π (U ) • P, where π (U ) : R −→ U is the projection along stable leaves.
It is possible to construct a Markov family R so that A is irreducible and aperiodic (see [2]). Consider the suspension space where by ∼ we identify the points (x, τ (x)) and (σx, 0). The corresponding suspension flow is defined by σ τ t (x, s) = (x, s + t) on R τ taking into account the identification ∼ . For a Hölder continuous function f on R, the topological pressure Pr(f ) with respect to σ is defined by where M σ denotes the space of all σ-invariant Borel probability measures and h(σ, m) is the entropy of σ with respect to m. We say that f and g are cohomologous and we denote this by f ∼ g if there exists a continuous function w such that The proof of (1.4) follows from the definitions of γ(p) and ξ(p) and the following computation: .

Proof of Theorem 2.
In this section we prove Theorem 2 exploiting the spectral estimates obtained in Theorem 1. We work under the assumptions of Theorem 2, in particular, G is constant on stable leaves of rectangular boxes B i for a certain Markov family R = {R i } k i=1 . Then the function g(x) depends only on x ∈ U. We may replace f by a Hölder functionf depending only on x ∈ U so that with some Hölder function z(x) we have Therefore for all t ∈ R we have Pr(f + t(g − pτ )) = Pr(f + t(g − pτ )) and µ f = µf . Below we use again the notation f assuming that f (x) depends only on x ∈ U.
We will examine the sequence Remark 3. In [8] the asymptotic of the function z(n) = U χ n (g n (x) − np) dµ has been studied, where instead of the function Passing to the Fourier transform in the proof of Proposition 1 below, we must deal with the Ruelle operator L f +(ξ(p)+iu)gp instead of the operator L f +(ξ(p)+iu)(g(x)−p) involved in the analysis in [8]. In Theorem 1 the Ruelle operator L f +(ξ(p)+iu)(g−pτ ) involves two phase functions τ (x) and g(x) with parameters b = pu and w = u. Certain spectral estimates for the iterations of the Ruelle operator are necessary to cover the asymptotic when |u| → ∞. This was one of the motivation to examine Ruelle operators with two complex parameters.

Proposition 1.
Under the assumptions of Theorem 2 we have the asymptotic Proof. The Ruelle operator L f +ξ(p)gp has a simple eigenvalue and so for all sufficiently small u ∈ C the operator L f +(ξ(p)+iu)gp has a simple eigenvalue e Pr(f+(ξ(p)+iu)gp) and the rest of the spectrum of L f +(ξ(p)+iu)gp is contained in a disk of radius θλ p with some 0 < θ < 1. Note that Clearly for the Fourier transformχ n of χ n we getχ n (u) = δ nχ (δ n u). Set ω n (y) = e −ξ(p)y χ n (y). Since Pr(f ) = 0, the Ruelle operator L f has a simple eigenvalues 1 and the adjoint operator L * f satisfies L * f µ = µ, where we denote µ = µ f as in section 1.
Using this property and applying the Fourier transform, we have By Taylor expansion for small |u| one gets We choose 0 > 0 sufficiently small and changing the coordinates on , we write . The analysis of the asymptotic of this integral is given in section 4.1 in [10]. The leading term has the form Thus we deduce Next consider the integral with c 1 sufficiently large. Since g p is non-lattice, for 0 < 0 ≤ |u| ≤ c p the operator L f +(ξ(p)+iu)gp has no eigenvalues λ with |λ| = λ p and the spectral radius of L f +(ξ(p)+iu)gp is strictly less than λ p . Thus, there exists α = α(p, c), 0 < α < 1, such that for n ≥ N (p, c) we have On the other hand, with c 0 > 0 depending on the support of χ. Applying (3.5) and (3.6) with k = 0, for large n we get

Now consider
We are going to use the spectral estimates established in Theorem 1 for the Ruelle operator Then |u| ≤ 1 |p| |pu| and for sufficiently large |u| ≥ c p and for every > 0 we are in situation to apply the spectral estimates Fix 0 < ≤ 1/2 and apply the estimate (3.6) with k = 2 and (3.7) for . This gives Recall that we have the condition from which we deduce the inequality n log ρ + 2δn − log n ≤ 0.

VESSELIN PETKOV AND LUCHEZAR STOYANOV
Thus, we conclude that Consequently, and this completes the proof of Proposition 1.
To establish Theorem 2, as in [10], [8], we approximate the characteristic function 4. Ruelle operators -definitions and assumptions. Assume as in section 1 that ϕ t : M −→ M is a C 2 weak mixing Axiom A flow and Λ is a basic set for ϕ t . Here we work under the same assumptions as these in [9]. One of these is: The above condition may seem complicated at a first glance, however a careful look at it shows that it is just a rather natural non-integrability condition.
Given x ∈ Λ, T > 0 and δ ∈ (0, ] set We will say that ϕ t has a regular distortion along unstable manifolds over the basic set Λ if there exists a constant 0 > 0 with the following properties: ) for any z ∈ Λ and any T > 0.
(b) For any ∈ (0, 0 ] and any ρ ∈ (0, 1) there exists δ ∈ (0, ] such that for any z ∈ Λ and any In this paper we work under the following Standing Assumptions: (A) ϕ t has Lipschitz local holonomy maps over Λ, (B) the local non-integrability condition (LNIC) holds for ϕ t on Λ, (C) ϕ t has a regular distortion along unstable manifolds over the basic set Λ.
A rather large class of examples satisfying the conditions (A) -(C) is provided by imposing the following pinching condition: (P): There exist constants C > 0 and β ≥ α > 0 such that for every We should note that (P) holds for geodesic flows on manifolds of strictly negative sectional curvature satisfying the so called 1 4 -pinching condition. (P) always holds when dim(M ) = 3.
Simplifying Assumptions: ϕ t is a C 2 contact Anosov flow satisfying the condition (P).
By [12], the pinching condition (P) implies that ϕ t has Lipschitz local holonomy maps and regular distortion along unstable manifolds. This and Proposition 6.1 in [12] show that: the Simplifying Assumptions imply the Standing Assumptions.
Throughout we work under the Standing Assumptions. In what follows we will use arguments similar to those in section 4 in [9], however technically they will be more complicated, since the numbers of parameters involved will increase. E.g. where we had functions f at , h at , etc., depending on two parameters, now we have to deal with functions f atc , h atc , etc., depending on three parameters. While some of the arguments we use here are almost the same as corresponding arguments in [9] (and we omit them), there are others that require more significant modification and we do them in some detail. Let . Then (2.1) hold for some constants c 0 ∈ (0, 1] and γ 1 > γ 0 > 1. Fix a number α > 0 and two real-valued functions f and g in C α ( U ). Let P = P f be the unique real number so that Pr(f − P τ ) = 0, where Pr is the topological pressure with respect to σ. For any t ∈ R with t ≥ 1, let f t be the average of f over balls in U of radius 1/t obtained as follows: fix an arbitrary extension f ∈ C α (V ) (with the same Hölder constant), where V is an open neighbourhood of U in M , and then take the averages in question. Then f t ∈ C ∞ (V ) and: ( Remark 4. As we remarked immediately after the statement of Theorem 1, in this theorem we do not assume that the pressure P of f is zero, and thus we get the term e P m in the right-hand-side of (1.1). Assuming that the latter holds with P = 0 implies a weaker estimate than (1.1) in the general case. Indeed, assume we have proved (1.1) in the case P = 0, and then deal with the general case using the standard approach. Given a, b as in the theorem and h ∈ C β ( U ), we have We can now apply (1.1) in the case P = 0 replacing h by e P τ m (v) h. Since 0 < c ≤ τ (u) ≤ c 1 for some constants c and c 1 , assuming e.g. P < 0 (the other case is similar), we get e P τ m h ∞ ≤ e mP c h ∞ , and |e P τ m h| β ≤ e mP c |h| β + |e P τ m | β h ∞ .

VESSELIN PETKOV AND LUCHEZAR STOYANOV
Given u, v ∈ U i , assuming e.g. e P τ m (u) > e P τ m (v) and using (2.1), we deduce Replacing c by an appropriate c 0 < c, we get |e P τ m | β ≤ Const e mP c0 . This implies Combining the latter with ( * ) gives As one can see this estimate is a bit worse than (1.1), since c > c 0 > 0 can be rather small constants.
Let G : Λ −→ R be a fixed α-Hölder function which is constant on the stable leaves of all "rectangular boxes" Given a large parameter t > 0, define G t as above, so that G t is again constant on the stable leaves of all rectangular boxes B i and In particular, for some constant C 0 > 0 we have Lip(G t ) ≤ C 0 t.
Then define g t : R −→ R by Clearly g t is α-Hölder and constant on stable leaves, so we can regard g t as a function on U . Thus, g t ∈ C α (U ). Let λ 0 > 0 be the largest eigenvalue of L f , i.e. λ 0 = e P , and letν 0 be the (unique) probability measure on U with L * fν 0 = λ 0ν0 . Fix a corresponding (positive) eigenfunction h 0 ∈ C α (U ) such that U h 0 dν 0 = 1. Then dν 0 = h 0 dν 0 defines a σ-invariant probability measure ν 0 on U . Setting Given real numbers a, c and t (with |a| + 1 |t| small and c ∈ I), denote by λ atc the largest eigenvalue of L ft−aτ +cgt on C Lip (U ) and by h atc the corresponding (positive) eigenfunction such that U h atc dν atc = 1, whereν atc is the unique probability measure on U with L * ft−a τ +cgtν atc = λ atcνatc . Setting dν atc = h atc dν atc defines a σ-invariant probability measure ν atc on U .
Given θ ∈ (0, 1), consider the metric d θ on U defined by d θ (x, x) = 0 and d θ (x, y) = θ m , where m is the largest integer such that x = y belong to the same cylinder of length m. Taking θ ∈ (0, 1) sufficiently close to 1 and β ∈ (0, α) sufficiently close to 0 we have (d(x, y)) α ≤ Const d θ (x, y) and d θ (x, y) ≤ Const (d(x, y)) β for all x, y ∈ U . In what follows we assume that θ and β satisfy these assumptions.
By the properties of the approximations f t and g t stated above, there exists a constant C 0 > 0, depending on f and α but independent of β, such that for all |a|, |c| ≤ 1 and t ≥ 1.
Since the map f → Pr(f ) is analytic (see e.g. Proposition 4.7 in [7]) and the related eigenfunction projection is also analytic in f (cf. e.g. Ch. 4 in [7]), it follows that the maximal eigenvalue λ atc = e Pr(ft−aτ +cgt) of the Ruelle operator L ft−aτ +cgt depends analytically on the parameters. Thus, there exists a constant a 0 > 0 such that, taking C 0 > 0 sufficiently large, we have 3) for |a|, |c| ≤ a 0 and 1/t ≤ a 0 . We take C 0 > 0 and a 0 > 0 so that Given real numbers a, c and t with |a|, |c|, 1/t ≤ a 0 consider the functions f atc = f t − aτ + cg t + ln h atc − ln(h atc • σ) − ln λ atc and the operators M atc = L fatc : C(U ) −→ C(U ). One checks that M atc 1 = 1.
Taking the constant C 0 > 0 sufficiently large, we may assume that The proof of the following lemma is given in [9] when c = 0. In the case with three parameters the proof is almost the same, so we omit it. Lemma 1. Taking the constant a 0 > 0 sufficiently small, there exists a constant T > 0 such that for all a, t, c ∈ R with |a|, |c| ≤ a 0 and t ≥ 1/a 0 we have h atc ∈ C Lip ( U ) and Lip(h atc ) ≤ T t.
We are going to study Ruelle operators of the form L f −sτ +zg , where s = a+ib and z = c + iw, a, b, c, w ∈ R, and |a|, |c| ≤ a 0 for some constant a 0 > 0, approximating them by Ruelle operators of the form Since f atc − ibτ + zg t is Lipschitz, the operators L abtz preserve each of the spaces C α ( U ) for 0 < α ≤ 1 including the space C Lip ( U ) of Lipschitz functions h : U −→ C. For such h we will denote by Lip(h) the Lipschitz constant of h. For |b| ≥ 1, define the norm .
The main step in proving Theorem 1 is the following.
Theorem 3. Under the assumptions in Theorem 1 there exist constants 0 < ρ < 1, Throughout we work under the Standing Assumptions made above and with fixed real-valued functions f, g ∈ C α ( U ) as in section 1, where α > 0 is a fixed number. Another fixed number β ∈ (0, α) will be used later.
Assuming that all rectangles R i are sufficiently small we have diam(U i ) < 1 for all i. Recall the metric D on U defined in [11]: D(x, y) = min{diam(C) : x, y ∈ C , C is a cylinder contained in U i } if x, y ∈ U i for some i = 1, . . . , k, and D(x, y) = 1 otherwise. As shown in [11], d(x, y) ≤ D(x, y) if x, y ∈ U i for some i, and for any cylinder C in U the characteristic function χ C of C on U is Lipschitz with respect to D and Lip D (χ C ) ≤ 1/diam(C). for all u, u ∈ U that belong to the same U i for some i = 1, . . . , k.
Fix an arbitrary constantγ with 1 <γ < γ 0 . The following lemma is similar to Lemma 5.2 in [9], hoverer some technical details are different, so we sketch its proof in the Appendix.
Lemma 2. Assuming a 0 > 0 is chosen sufficiently small, there exists a constant A 0 > 0 such that for all a, c, t ∈ R with |a|, |c| ≤ a 0 and t ≥ 1 the following hold: for any v, v ∈ U i , i = 1, . . . , k, then for any integer m ≥ 1 and any b, w ∈ R with |b|, |w| ≥ 1, for z = c + iw we have whenever u, u ∈ U i for some i = 1, . . . , k. In particular, if e A0t t ≤ |b| and |w| ≤ B|b| for some constant B > 0, then for some constant A 1 > 0, depending on B.
From now on we will assume that a 0 and A 0 are fixed with the properties in Lemma 2 above and a, b, c, w, t ∈ R are such that |a|, |c| ≤ a 0 , |b|, t, |w| ≥ 1 and |w| ≤ B|b|. As before, set z = c + id.
As in [9], we need the entire set-up and notation from section 4 in [11], so we will now recall some of it.
Definitions ( [11]). (a) For a cylinder C ⊂ U 0 and a unit vector ξ ∈ E u (z 0 ) we will say that a separation by a ξ-plane Let S ξ be the family of all cylinders C contained in U 0 such that a separation by an ξ-plane occurs in C.
Next, assume that B > 1, β ∈ (0, α) and E ≥ max 4A 0 , BC 1 , 2A0 T γ−1 are fixed constants, where A 0 ≥ 1 is the constant from Lemma 2 and C 1 is the constant from the proof of Lemma 2 in the Appendix. Fix an integer N ≥ N 0 such thatγ We will also assume now that the parameter t = t(a 0 , N ) > 1 is fixed with (4.7) (Part of this condition will be needed for the proof of Theorem 1.) Clearly the above requires to assume that a 0 = a 0 (N ) satisfies a 1/(α−β) 0 ≤ t. Some other conditions on the small parameter a 0 = a 0 (N ) > 0 will be imposed later. We will also need to choose b 0 ≥ te A0t .
Let the parameters b, w ∈ R be so that |w| ≤ B |b| and |b|, |w| ≥ b 0 .
Let J be a subset of the set Define the function ω = ω J : U −→ [0, 1] by

VESSELIN PETKOV AND LUCHEZAR STOYANOV
Clearly ω ∈ C Lip D ( U ) and 1 − µ 0 ≤ ω(u) ≤ 1 for any u ∈ U . Moreover, Next, define the contraction operator N = N J (a, b, t, c) : . Using Lemma 2 above, the proof of the following lemma is very similar to that of Lemma 5.6 in [11] and we omit it. Lemma 4. Under the above conditions for N and µ the following hold : (a) N h ∈ K E|b| ( U ) for any h ∈ K E|b| ( U ); for any v, v ∈ U j , j = 1, . . . , k and |h| ≤ H on U , then for any i = 1, . . . , k and any u, u ∈ U i we have Definition. A subset J of Ξ will be called dense if for any m = 1, . . . , p there exists (i, j, ) ∈ J such that D j ⊂ C m .
Denote by J = J(a, b, z) the set of all dense subsets J of Ξ. Although the operator N here is different, the proof of the following lemma is very similar to that of Lemma 5.8 in [11] and we omit it.
Until the end of this section we will assume that h, H ∈ C Lip D ( U ) are fixed functions such that and by , , and set for all u ∈ U .
Definitions ( [11]). We will say that the cylinders D j and D j are adjacent if they are subcylinders of the same C m for some m. If D j and D j are contained in C m for some m and for some = 1, . . . , 0 there exist u ∈ D j and v ∈ D j such that d(u, v) ≥ 1 2 diam(C m ) and we will say that D j and D j are η -separable in C m .
As a consequence of Lemma 3(b) one gets the following whose proof is almost the same as that of Lemma 5.9 in [11], so we omit it. Lemma 6. Let j, j ∈ {1, 2, . . . , q} be such that D j and D j are contained in C m for some m = 1, . . . , p and are η -separable in C m for some = 1, . . . , 0 . Then The following lemma is the analogue of Lemma 5.10 in [11] and represents the main step in proving Theorem 1. To prove this we need the following lemma which is the analogue of Lemma 14 in [4] and its proof is very similar, so we omit it.
Fix , j and j with the above properties, and set Z = Z j ∪ Z j ∪ Z j . If there exist t ∈ {j, j , j } and i = 1, 2 such that the first alternative in Lemma 8(b) holds for Z t , and i, then µ ≤ 1/4 implies χ (i) (u) ≤ 1 for any u ∈ Z t .
Assume that for every t ∈ {j, j , j } and every i = 1, 2 the second alternative in Lemma 8(b) holds for Z t , and i, i.e. |h(v . Now the estimate (6.2) in the Appendix below implies Assume for example that c 0 (e −2a0T γ 0 ) N < π 12 assuming a 0 > 0 is chosen sufficiently small and N sufficiently large. So, the angle between the complex numbers e zg N t (v ) (regarded as vectors in R 2 ) is < π/6. In particular, for each i = 1, 2 we can choose a real continuous function θ i (u), u ∈ Z, with values in [0, π/6] and a constant λ i such that Fix an arbitrary u 0 ∈ Z and set λ = γ (u 0 ). Replacing e.g λ 2 by λ 2 + 2mπ for some integer m, we may assume that |λ 2 − λ 1 + λ| ≤ π. Using the above, θ ≤ 2 sin θ for θ ∈ [0, π/6], and some elementary geometry yields The difference between the arguments of the complex numbers Given u ∈ Z j and u ∈ Z j , since D j and D j are contained in C m and are η -separable in C m , it follows from Lemma 6 and the above that Thus, |Γ ( ) (u ) − Γ ( ) (u )| ≥ c2 2 1 for all u ∈ Z j and u ∈ Z j . Hence either |Γ ( ) (u )| ≥ c2 4 1 for all u ∈ Z j or |Γ ( ) (u )| ≥ c2 4 1 for all u ∈ Z j . Assume for example that |Γ ( ) (u)| ≥ c2 4 1 for all u ∈ Z j . Since Z ⊂ σ n1 (C m ), as in [11], for any u ∈ Z we get |Γ (u)| < 3π 2 . Thus, c2 4 1 ≤ |Γ ( ) (u)| < 3π 2 for all u ∈ Z j . Now as in [4] (see also [11]) one shows that χ (1) (u) ≤ 1 and Parts (a) and (b) of the following lemma can be proved in the same way as the corresponding parts of Lemma 5.3 in [11], while part (c) follows from Lemma 4(b).
Let a ∈ R and b, w ∈ R be such that |a| ≤ a 0 and |w| ≤ B|b|, |b|, |w| ≥ b 0 , and let J ∈ J(a, b). Then (a) follows from Lemma 4(a), while (b) follows from Lemma 5.
To check (c), assume that h, H ∈ C Lip D ( U ) satisfy (4.9) and (4.10). Now define the subset J of J(a, b) in the following way. First, include in J all (1, j, ) ∈ Ξ such that χ (1) (u) ≤ 1 for all u ∈ Z j . Then for any j = 1, . . . , q and = 1, . . . , 0 include (2, j, ) in J if and only if (1, j, ) has not been included in J (that is, for some u ∈ Z j ) and χ (2) (u) ≤ 1 for all u ∈ Z j . It follows from Lemma 7 that J is dense.
5. Proofs of theorems. Proof of Theorem 3. We use an argument from [4]. Let B > 0 be a constant. Let N ,ρ, a 0 , w 0 and E be as in Lemma 9. Given a, b, c, w, t ∈ R with |a| ≤ a 0 , |b| ≥ b 0 , |w| ≤ B|b|, let {N J } J∈J be a finite family of operators having the properties (a), (b) and (c) in Lemma 9.
Let h ∈ C Lip (U ) be such that h Lip,b ≤ 1. Then |h(u)| ≤ 1 for all u ∈ U and Lip(h) ≤ |b|. Thus, for any u, v ∈ U i . i = 1, . . . , k, we have As in [4] and [11] we need the following lemma whose proof is the same.
Lemma 10. Let β ∈ (0, α). There exists a constants A 1 > 0 such that for all a, b, c, t, w ∈ R with |a|, |c|, 1/|b|, 1/t ≤ a 0 such that |w| ≤ B|b|, and all positive integers m and all h ∈ C β (U ) we have for all u, u ∈ U i .
We will derive Theorem 1 from Theorem 3 and Lemma 10 above.

VESSELIN PETKOV AND LUCHEZAR STOYANOV
We will now approximate L f −sτ +zg by L ft−sτ +cgt in two steps. First, the above implies L n f −sτ +cg+iwgt h β,w = L n ft−sτ +zgt e (f n −f n t )+c(g n −g n t ) h β,b ≤ C λ n 0 ρ n 4 |b| e (f n −f n replacing C by a larger constant where necessary. Combining this with the previous estimate gives e (f n −f n t )+c(g n −g n t ) h β,b ≤ C n e C na0 h β , so L n f −sτ +cg+iwgt h β,b ≤ C λ n 0 ρ n 4 |b| n e C na0 h β,b . Taking a 0 > 0 sufficiently small, we may assume that ρ 4 e C a0 < 1. Now take an arbitrary ρ 5 with ρ 4 e C a0 < ρ 5 < 1. Then we can take the constant C 7 > 0 so large that n ρ n 4 e Cna0 ≤ C 7 ρ n 5 for all integers n ≥ 1. This gives L n f −sτ +cg+iwgt h β,b ≤ C 7 λ n 0 ρ n 5 |b| h β,b for all n ≥ 0. Using the latter we can write L n f −sτ +zg h β,b = L n f −sτ +cg+iwgt e iw(g n −g n t ) h β,b ≤ C 7 λ n 0 ρ n 5 |b| e iw(g n −g n t ) h β,b .