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1.1. Definitions and results. Before precisely stating our main result, we intro- duce some notation. Recall that M is a compact smooth Riemannian manifold and f : M → M is of class C r with r> 1. Let f be an expanding map, i.e., there exist n n constants C > 0 and λ0 > 1 such that |Df (x)v| ≥ Cλ0 |v| for each x ∈ M and v ∈ TxM. Let us set the minimal Lyapunov exponent
1 n χmin = lim log inf inf |Df (x)v|. n→∞ n x∈M v∈TxM |v|=1
Then χmin > 0 when f is an expanding map. (For the properties of expanding maps, the reader is referred to [19, 21, 22] e.g.) Let g be a complex-valued C r˜ function on M with 0 < r˜ ≤ r. We assume a technical condition r˜ ≥ 1 or r˜ ≤ r − 1. (1.1) Note that (1.1) is satisfied for an important application g = | det Df|−1, and that if r ≥ 2, then the condition (1.1) always holds (see also Remark 6 for the condition). (n) n−1 j With the notation g (x)= j=0 g(f (x)), we set 1/n (n) n R(g) = lim Rn(g)Q , Rn(g)= kg det Df kL∞ (n ≥ 1). (1.2) n→∞ s s r˜ For each 0 ≤ s ≤ r˜, the transfer operator Lf,g : C (M) → C (M) of f with a C weight function g is defined by
s Lf,gu(x)= g(y) · u(y), u ∈ C (M). f(Xy)=x Standard references for transfer operators are [4, 8].
Remark 1. Denote by r(A|E ) and ress(A|E) the spectral radius and the essential spectral radius of a bounded operator A : E → E on a Banach space E, respectively. Due to Ruelle [23], r(Lf,g |C s(M)) is bounded by exp Ptop(log |g|), and is equal to exp Ptop(log g) when g is real-valued and strictly positive (i.e., infx∈M g > 0), where Ptop(φ) is the topological pressure of a continuous function φ : M → R (refer to [32] for the definition of topological pressure). It also follows from [8, Lemma 2.16] that R(g) ≥ exp Ptop(log |g|). Furthermore, in view of [11, Lemma A.1] regarding coincidence of eigenvalues of abstract linear operators in different Banach spaces outside of the essential spectral Bs radii, we get the following: If ress(Lf,g| pq (M)) < exp Ptop(log g) with real-valued s Bs B and strictly positive g, then r(Lf,g | pq (M)) = exp Ptop(log g), where pq(M) is the Besov space on M (see below for definition). In particular, the transfer operator Bs on pq(M) is quasi-compact (has a spectral gap), which is our goal in this paper. p Lf,g can be extended to a bounded operator on L (M) with p ∈ [1, ∞]. (Since some Besov spaces do not coincide with the completion of C s(M) with respect to its Besov norm, this extension would be necessary; see [27, §A.1].) Indeed, a change of variables shows
Lf,gu · ϕdLebM = u · ϕ ◦ f · g · | det Df|dLebM , (1.3) Z Z for any u ∈ L∞(M) and ϕ ∈ L1(M). On the other hand, the operator ϕ 7→ ′ ϕ ◦ f · g · | det Df| is bounded on Lp (M) with 1/p′ +1/p =1: notice that kϕ ◦ f · SPECTRAOFEXPANDINGMAPSONBESOVSPACES 3
′ ′ ′ g det DfkLp ≤ R1(g)kϕ ◦ fkLp and that in the case 1 ≤ p < ∞,
1/p′ ′ 1/p′ ′ p ′ kϕ ◦ fkLp = Lf,| det Df|−1 1 · |ϕ| dLebM ≤ kLf,| det Df|−1 1kL∞ kϕkLp Z by virtue of (1.3), while kϕ ◦ fkL∞ ≤kϕkL∞ . Therefore, the transfer operator has a continuous extension to Lp(M) by the duality (1.3). The extension will also be denoted by Lf,g. Below we define Besov spaces associated with a partition of unity of M. We first recall the definition of the Besov spaces on Rd, where d is the dimension of M. Let ρ : R → R be a C ∞ function such that
1 (t ≤ 1), 0 ≤ ρ ≤ 1, ρ(t)= (1.4) (0 (t ≥ 2).
C ∞ Rd R For each nonnegative integer n, define radial functions ψn ∈ 0 ( , ) by
ρ(|ξ|) (n = 0), ψn(ξ)= −n −n+1 (1.5) (ρ(2 |ξ|) − ρ(2 |ξ|) (n ≥ 1).
For a tempered distribution u (that is, u is in the dual space of the set of rapidly −1 decreasing test functions), the operator ∆n is given by ∆nu = F [ψnFu] with n ≥ 0, where F is the Fourier transform. Then we have n≥0 ∆nu = u, called the Littlewood-Paley (dyadic) decomposition. This decomposition was first employed P in context of dynamical systems theory to analyse the spectra of transfer opera- tors of Anosov diffeomorphisms by Baladi and Tsujii [10]. They also applied the decomposition to spectral analysis of transfer operators of expanding maps in the C s little Hölder space ∗ (M) in a survey [12, Subsection 3.2], see Remark 4 for the C s definition of ∗ (M). R s Rd For s ∈ , p, q ∈ [1, ∞], we define the Besov space Bpq( ) as a set of tempered distributions u on Rd whose norm
1/q sqn q n≥0 2 k∆nukLp (q< ∞), kukBs = pq sn p supPn≥0 2 k∆nukL (q = ∞) s Rd is finite. We remark that u∈ Bpq( ) if and only if there are a constant C > 0 and q a nonnegative sequence {cn}n≥0 ∈ ℓ with k{cn}n≥0kℓq ≤ 1 such that
−ns k∆nukLp ≤ C2 cn (1.6)
s Rd s for all n ≥ 0. Once we know u ∈ Bpq( ), we can take C = kukBpq . I Since M is compact, there are a finite open covering {Vi}i=1 and a system of I C ∞ Rd local charts {κi}i=1 such that κi : Vi → Ui is of class with an open set Ui ⊂ I C ∞ I for each 1 ≤ i ≤ I. Let {φi}i=1 be a family of functions such that {φi ◦ κi}i=1 I is a partition of unity of M subordinate to the covering {Vi}i=1, that is, φi ◦ κi is ∞ a C function on M with values in [0, 1] ⊂ R whose support is contained in Vi for I each 1 ≤ i ≤ I and i=1 φi ◦ κi(x)=1 for all x ∈ M. In this paper, the support of a continuous function φ : M → R is defined as the closure of {x ∈ M | φ(x) 6=0}. P 4 YUSHINAKANOANDSHOTASAKAMOTO
Bs Definition 2. The Besov space pq(M) on M is the space of tempered distribu- tions u on M whose norm −1 Bs s kuk pq = kφi · u ◦ κi kBpq 1≤Xi≤I is finite. This definition does not depend on the choice of charts or the partition of unity, see [30]. Let U ⊂ Rd be a nonempty bounded open set. We write C s(U), Lp(U) and s C s Rd p Rd s Rd Bpq(U) for the subspace of ( ), L ( ) and Bpq( ), respectively, such that the s support of each element in these spaces is included in U. Then we have Bpq(U) ⊂ Lp(U) for s > 0 and p, q ∈ [1, ∞].1 Hence, from the argument following (1.3) it p Bs holds that Lf,gu is in L (M) for each u ∈ pq(M) with s > 0 and p, q ∈ [1, ∞]. Bs In particular, Lf,gu is a tempered distribution and kLf,guk pq is well-defined (but possibly takes +∞). Here we provide our main theorem for an upper bound of the essential spectral Bs radius of the transfer operator on pq (M), which concludes (due to Remark 1) a Bs spectral gap of the transfer operator on pq(M) when g is real-valued and strictly positive and s is sufficiently large: Theorem 3. Let f : M → M be a C r expanding map with r > 1, and g : M → C a C r˜ function with 0 < r˜ ≤ r satisfying (1.1). Let s ∈ (0, r˜] and p, q ∈ [1, ∞]. In addition, when q < ∞, assume that s is strictly smaller than r˜. Then Lf,g can be Bs extended to a bounded operator on pq (M), and it holds
Bs ress(Lf,g| pq (M)) ≤ exp(−sχmin) · R(g), where R(g) is defined in (1.2). Remark 4. The Besov spaces can represent many other function spaces. As an s Rd s Rd important example, B∞∞( ) is known to coincide with the Hölder space C ( ) s Rd s Rd when s is not an integer (whereas B∞∞( ) strictly includes C ( ) when s is an integer), see [27, §A.1]. Hence Theorem 3 is a generalisation of a well-known result C s by Ruelle [23, Theorem 3.2]. We note that the little Hölder space ∗ (M) given in C ∞ s [12] is defined as the completion of (M) for the norm k·kB∞∞ (which is smaller than C s(M), see [27, §A.1]), so that one can see that the proof of Theorem 3 can be translated literally to the case when the functional space where the transfer operator C s acts is ∗ (M) (this was essentially done in [12, Theorem 3.1] with r˜ = r − 1). Furthermore, for the Sobolev space W s,p(Rd) with s ∈ R and p ∈ (1, 2], it holds s Rd s,p Rd s Rd s Rd s,p′ Rd s Rd Bpp( ) ⊂ W ( ) ⊂ Bp2( ) and Bp′2( ) ⊂ W ( ) ⊂ Bp′p′ ( ), where 1/p +1/p′ = 1 (if p 6= 2, these inclusions are strict; see [26, p.161] for the cases N W s,2 Bs when s ∈ ). These inclusions imply (M)= 22(M), thus Theorem 3 recovers a part of the result by Baillif and Baladi [3].
1 Indeed, by Hölder’s inequality, for 1 ≤ q ≤∞ it holds
kuk p = ∆ u ≤ k∆ uk p L X n Lp X n L n≥0 n≥0 ′ 1/q ′ 1/q ′ ′ sqn q −sq n −sq −1/q ≤ 2 k∆nuk p 2 = 1 − 2 kuk s , X L X Bpq n≥0 n≥0 where 1/q + 1/q′ = 1 with a usual modification for q = ∞. SPECTRAOFEXPANDINGMAPSONBESOVSPACES 5
Other famous functional spaces constructed by using the Littlewood-Paley de- s Rd composition are Triebel-Lizorkin spaces Fpq( ). It seems possible to prove The- s Rd s Rd orem 3 with Bpq( ) replaced by Fpq( ) in the same manner as in the proof of s,p Rd s R Theorem 3. We note that W ( )= Fp2( ) for each 1 − sup exp −sχµ + hµ + log |g|dµ , (1.7) µ∈Erg(f) Z − where Erg(f) is the set of ergodic f-invariant probability measures, χµ is the small- est Lyapunov exponent of µ (for Df over f), and hµ is the Kolmogorov-Sinai entropy of µ (over f). We refer to [31, 32] for the definitions of Lyapunov exponents and Kolmogorov-Sinai entropy. By the variation principle for topological pressure (see [32, §9] e.g.), we have Ptop(φ) = sup hµ + φdµ , µ∈Erg(f) Z − so that, since χµ ≥ χmin for any µ ∈ Erg(f), (1.7) is bounded by exp(−sχmin + Ptop(log |g|)). Therefore, when g is real-valued and strictly positive, the transfer operator on C s(M) is quasi-compact for any 0 2. Proof of Theorem 3 2.1. Transfer operators on Rd. We shall deduce Theorem 3 from a theorem for transfer operators on Besov spaces of compactly supported functions on Rd. Let U and U ′ be nonempty bounded open subsets of Rd. Let F be a C r mapping on Rd 6 YUSHINAKANOANDSHOTASAKAMOTO with r> 1 such that U ′ ∩ F −1(U) 6= ∅ and λ = min min |DF (x)v| (2.1) ′ − x∈U ∩F 1(U) v∈Rd |v|=1 is larger than 1. Let G be a C r˜ function whose support is included in U ′ ∩ F −1(U). N(F ) Furthermore, we assume that there are finitely many compact subsets {Kj}j=1 of Rd such that N(F ) int(Ki) ∩ int(Kj )= ∅ when i 6= j, and supp G = ∪j=1 Kj , (2.2) and for each 1 ≤ j ≤ N(F ) one can find a small neighbourhood Vj of Kj satisfying that r F : Vj → F (Vj ) is a C diffeomorphism. (2.3) s d s Define a bounded operator LF,G : C (R ) → C (U) by LF,Gu(x)= G(y) · u(y). F (Xy)=x Then, in a manner similar to one below (1.3), for each p ∈ [1, ∞] we can extend p d p LF,G to a bounded operator from L (R ) to L (U). The only essential difference from the argument below (1.3) is that in the case 1 ′ ′ kϕ ◦ F · G det DF kLp = k1U ′∩F −1(U) · ϕ ◦ F · G det DF kLp (2.4) ′ ≤kG det DF kL∞ k1U ′∩F −1(U) · ϕ ◦ F kLp ′ ′ −1 with 1/p +1/p =1, where 1U ′∩F −1(U) is the indicator function of U ∩F (U). Fur- p d thermore, it follows from this argument that the operator norm of LF,G : L (R ) → Lp(U) is bounded by a nonnegative number α ≡ α(F,G,p), given by 1/p′ 1 sup kG det DF kL∞ (1 C1 = sup k∆nukLp (2.6) kukLp =1 (such C1 is bounded and independent of n: see [2, Remark 2.11]). Notice also that ∞ if we let ϕ0 ∈ C (U) such that ϕ0 ◦F ≡ 1 on the support of G, then M : u 7→ u·ϕ0 s Rd s M is a bounded operator from Bpq( ) to Bpq(U). Denote the operator norm of by C˜ = C˜(s,p,q,ϕ0) > 0. For s> 0, let 5s −s −1 γ˜s = C12 (1 − 2 ) and γs = C˜γ˜s. (2.7) Theorem 3 will follow from the next Lasota-Yorke type inequality. Theorem 7. Assume that (s,p,q) satisfies the condition in Theorem 3, and let F L s ′ and G be as above. Then, F,G can be extended to a bounded operator from Bpq(U ) s ˆ to Bpq(U). Furthermore, there are a constant CF,G > 0 and σ ∈ (0,s) such that −s L s s ˆ σ k F,GukBpq ≤ γsαλ kukBpq + CF,GkukBpq s ′ for all u ∈ Bpq(U ), where the constants are defined in (2.1), (2.5), and (2.7). SPECTRAOFEXPANDINGMAPSONBESOVSPACES 7 Remark 8. The proof of Theorem 7 works even when λ ≤ 1, although it may give no spectral information for the original dynamical system f (in our setting (1.1) about n r˜): the role of F will played by f in local chart and λ ≤ 1 corresponds to χmin ≤ 0, Bs so that Theorem 3 only means ress(Lf,g| pq(M)) is bounded by exp(−sχmin) · R(g) with exp(−sχmin) ≥ 1 and R(g) ≥ exp Ptop(log |g|) (see Remark 1). However yet, Theorem 7 with F = id (λ = 1) seems to be actually helpful to demonstrate a spectral gap with r˜ in full generality (that is, with r˜ being not in the range of (1.1)), see [8, Theorem 2.15] for the Sobolev case. s ′ Proof. We follow [12] and [8]. Let u ∈ Bpq(U ). It follows from (1.6) that both p Rd L ℓ:ℓ֒→n ∆ℓu and ℓ:ℓ6֒→n ∆ℓu are in L ( ) for any n ≥ 0. Thus, if we define 0,nu and L1,nu with n ≥ 0 by P P L′ L L M L′ 0,nu = ∆n F,G∆ℓu, 0,nu = ◦ 0,nu, ℓ:Xℓ֒→n L′ L L M L′ 1,nu = ∆n F,G∆ℓu, 1,nu = ◦ 1,nu, ℓ:Xℓ6֒→n p then both L0,nu and L1,nu are in L (U) (due to (2.5) and (2.6)), where the sum- mations are taken over nonnegative integers ℓ and we write ℓ ֒→ n if 2n ≤ λ−12ℓ+4 n −1 ℓ+4 and ℓ 6֒→ n if 2 > λ 2 . It is obvious from the fact ℓ≥0 ∆ℓu = u that L L L ∆n F,Gu = 0,nu + 1,nu, P so we have by Minkowski’s inequality that sqn q 1/q sqn q 1/q L s L p L p k F,GukBpq ≤ 2 k 0,nukL + 2 k 1,nukL n≥0 n≥0 X X in the case q< ∞ (a corresponding inequality also holds for q = ∞). Therefore, it suffices to show the following two inequalities for each q < ∞ (and corresponding two inequalities for q = ∞): 1/q sqn ′ q −s L p s 2 k 0,nukL ≤ γ˜sαλ kukBpq , (2.8) nX≥0 and 1/q sqn ′ q L p ¯ σ 2 k 1,nukL ≤ CF,GkukBpq (2.9) nX≥0 with some constant C¯F,G > 0 and σ ∈ (0,s). First we show (2.8). We focus on the case when q is finite, but the other case is analogous. Using (1.6), (2.5) and (2.6), we can find a nonnegative ℓq sequence {cℓ}ℓ≥0 satisfying k{cℓ}ℓ≥0kℓq ≤ 1 such that 1/q q 1/q sqn ′ q nsq −sℓ L p s 2 k 0,nukL ≤ C1αkukBpq 2 2 cℓ nX≥0 nX≥0 ℓ:Xℓ֒→n q 1/q s(n−ℓ) s (C1αkukBpq 2 1{0֒→(·)}(n − ℓ)cℓ , (2.10 ≥ nX≥0 Xℓ≥0 .{where 1{0֒→(·)} is the indicator function of {0 ֒→ (·)} = {ℓ ∈ Z | 0 ֒→ ℓ To estimate it, we recall Young’s inequality for the locally compact group Z. We q define ℓ (Z) as the set of (two-sided) nonnegative sequences a(·) ≡ {aℓ}ℓ∈Z such 8 YUSHINAKANOANDSHOTASAKAMOTO q 1/q that ka(·)kℓq(Z) := ℓ∈Z aℓ is finite. Then, it follows from [2, Lemma 1.4] that 1 q ka ∗ b k q Z ≤ ka k 1 Z kb k q Z for all a ∈ ℓ (Z) and b ∈ ℓ (Z), where (·) (·) ℓ ( ) P(·) ℓ ( ) (·) ℓ ( ) (·) (·) a(·) ∗ b(·) is the convolution of a(·) and b(·) given by a(·) ∗ b(·)(n)= ℓ∈Z an−ℓbℓ for n ∈ Z. Thus, if we let {c˜ℓ}ℓ∈Z be as c˜ℓ = cℓ for ℓ ≥ 0 and = 0 for ℓ < 0, then we have P q s(n−ℓ) n − ℓ)cℓ){(·)→0֒}1 2 n≥0 ℓ≥0 X X q q s(n−ℓ) s(·) (·)˜n − ℓ)˜cℓ = 2 · 1{0֒→(·)} ∗ c){(·)→0֒}1 2 ≥ q Z Z ℓ ( ) n∈ ℓ∈Z X X q q s(·) q sℓ (c˜(·) q Z ≤ 2 . (2.11 {(·)→0֒}1 · 2 ≥ ℓ1(Z) ℓ ( ) {(·)→ℓ∈{0֒ X On the other hand, since Z (ℓ ∈ | ℓ ≤ 4 − log2 λ}, (2.12} = {(·) →֒ 0} we have that 25s 2sℓ ≤ 2s(5−log2 λ) 2−sℓ = λ−s. (2.13) 1 − 2−s ℓ∈{X0֒→(·)} Xℓ≥0 By (2.10), (2.11) and (2.13), we obtain (2.8). d Next, we shall show (2.9). Let {ψ˜ℓ}ℓ≥1 be a family of smooth functions on R given by −1 ˜ ρ(2 |ξ|) (n = 0), ψn(ξ)= −n−1 −n+2 (ρ(2 |ξ|) − ρ(2 |ξ|) (n ≥ 1) d for ξ ∈ R , where ρ is defined in (1.4). Notice that ψ˜n ≡ 1 on the support of ψn for −1 each n ≥ 0. For a tempered distribution u, we define ∆˜ nu by ∆˜ nu = F [ψ˜nFu] for each n ≥ 0. Then it is straightforward to see that L′ L ˜ p ′ 1,nu = ∆n F,G∆ℓ∆ℓu for u ∈ L (U ). ℓ:Xℓ6֒→n We borrow the following crucial lemma by Baladi [8] (from the proof of Lemma 2.21 of [8]), which has been proven in the special case r˜ = r − 1 and p = ∞ in the survey [12] by Baladi and Tsujii (inequality (15) and a comment following inequality (19) of [12]). Since it is a key estimate in the proof of Theorem 7, we give a full proof in Appendix A. Lemma 9. There exists a constant CF,G > 0 such that for any p ∈ [1, ∞], u ∈ Lp(Rd), n ≥ 0 and ℓ 6֒→ n, the following holds: If r˜ ≤ r − 1, then −r˜max{n,ℓ} k∆nLF,G∆˜ ℓukLp ≤ CF,G2 kukLp . If r˜ ≥ 1, then min{n,ℓ}−r˜ max{n,ℓ} k∆nLF,G∆˜ ℓukLp ≤ CF,G2 kukLp . SPECTRAOFEXPANDINGMAPSONBESOVSPACES 9 Fix σ ∈ (0,s) satisfying that σ>s − r˜ +1+ δ when r>˜ 1 (2.14) min{n,ℓ} with some δ ∈ (0, r˜ − 1). Let γn,ℓ = 1 when r˜ ≤ 1 and γn,ℓ = 2 when r˜ > 1. Then, applying Lemma 9 together with (1.6), we can find a nonnegative ℓq sequence {cℓ}ℓ≥0 satisfying k{cℓ}ℓ≥0kℓq ≤ 1 such that L′ −r˜ max{n,ℓ} k 1,nukLp ≤ CF,G γn,ℓ2 k∆ℓukLp ℓ:Xℓ6֒→n −σℓ−r˜ max{n,ℓ} σ ≤ CF,G γn,ℓ2 cℓkukBpq . ℓ:Xℓ6֒→n Therefore, in the case q = ∞, sn ′ sn−σℓ−r˜ max{n,ℓ} L p σ sup 2 k 1,nukL ≤ CF,GkukBpq sup γn,ℓ2 . n≥0 n≥0 ℓ:Xℓ6֒→n Since −r˜max{n,ℓ}≤−rn˜ , and we assumed r˜ ≥ s and σ> 0, 2sn−σℓ−r˜ max{n,ℓ} ≤ (1 − 2−σ)−1 < ∞. ℓ:Xℓ6֒→n Furthermore, (2.14) implies that when r>˜ 1, we have −(˜r − s)(n − ℓ) − δℓ (ℓ 1/q sqn ′ q −τℓ L p σ 2 k 1,nukL ≤ CF,GkukBpq 2 (2.16) Z nX≥0 ℓ∈X:06֒→ℓ −τ(3−log λ) CF,G2 2 ≤ kuk σ . 1 − 2−τ Bpq This completes the proof of (2.9). 10 YUSHINAKANOANDSHOTASAKAMOTO Remark 10. We cannot show (2.9) for s =r ˜ and q < ∞ in the same manner as the case q = ∞. Indeed, in a manner similar to obtaining (2.16), we have rn˜ −σℓ−r˜ max{n,ℓ} −σℓ .ℓ:ℓ6֒→n 2 ≥ ℓ:ℓ6֒→n,ℓ≤n 2 ≥ 1 for any sufficiently large n ≥ 0 rn˜ −σℓ−r˜ max{n,ℓ} q .(∞ > PIt follows that ℓ:ℓ6֒→n 2 P is not in ℓ (q n≥0 n o 2.2. CompletionP of the proof. Now we can complete the proof of Theorem 3 by n reducing the transfer operator Lf n,g(n) = Lf,g to a family of transfer operators on the local charts and applying Theorem 7 for each n ≥ 1. Proof of Theorem 3. For the time being, we shall fix n ≥ 1 and suppress it from Bs I s the notation. Let ι be an isometric embedding from pq(M) to i=1 Bpq(Ui) I I (equipped with the norm k(ui) k I s = kuikBs ) given by i=1 Li=1 Bpq (Ui) i=1 pq L −1 I PBs ι(u)= φi · u ◦ κi i=1 , u ∈ pq(M). n −1 For 1 ≤ i, j ≤ I, we let Uij ≡ Un,ij = κj (Vj ∩ (f ) (Vi)) and let Fij ≡ Fn,ij be a C r mapping on Rd such that n −1 Fij (x)= κi ◦ f ◦ κj (x) for x ∈ Uij . r˜ d Furthermore, let Gij ≡ Gn,ij be a compactly supported C function on R such that (n) −1 Rd Gij (x)= φi ◦ Fij (x) · φj (x) · g ◦ κj (x) for x ∈ . By the existence of Markov partitions for expanding maps [21], one can find finitely N(f n) N(f n) many compact sets {Kℓ}ℓ=1 and their neighbourhoods {Vℓ}ℓ=1 satisfying (2.2) L L I s I s and (2.3). Finally we define ≡ n : i=1 Bpq(Ui) → i=1 Bpq(Ui) by I L L I L (u )I = L u I for (u )I ∈ Bs (U ), (2.17) i i=1 ij j i=1 i i=1 pq i j=1 i=1 X M L L s s where ij ≡ n,ij : Bpq(Uj ) → Bpq(Ui) for 1 ≤ i, j ≤ I is given by d Lij u(x)= Gij (y) · u(y) for x ∈ R . (2.18) FijX(y)=x L I C s (More precisely, we define as a bounded operator on i=1 (Ui) by (2.17) and (2.18), and it can be extended to a bounded operator on I Bs (U ) by virtue L i=1 pq i of Theorem 7.) Then it is straightforward to see that the following commutative diagram holds: L Ln Bs f,g Bs pq(M) → pq(M) ↓ ι ↓ ι (2.19) L I s I s i=1 Bpq(Ui) → i=1 Bpq(Ui). n L Bs In particular, kL ukBLs = k ◦ ι(u)k I sL for each u ∈ (M). f,g pq Li=1 Bpq(Ui) pq It follows from Theorem 7 that one can find a constant Cˆn > 0 and σ ∈ (0,s) s such that for each 1 ≤ i, j ≤ I satisfying Uij 6= ∅ and each u ∈ Bpq(Uj ), we get the SPECTRAOFEXPANDINGMAPSONBESOVSPACES 11 L s bound of k ij ukBpq by 1/p′ 1 γs sup kGij det DFij kL∞ x∈Fij (Uij ) | det DFij (y)| Fij (y)=x X −s s ˆ σ × min min |DFij (x)v| kukBpq + CnkukBpq , (2.20) x∈Uij v∈Rd |v|=1 where the second factor of the first term is replaced with 1 when p =1. On the other hand, it is straightforward to see that there is a constant C2 > 0 independently of n,i,j such that the first term of (2.20) is bounded by −s ′ n 1/p n −1 ∞ s C2γskLf,| det Df| 1M kL Rn(g) min min |Df (x)v| kukBpq . x∈M v∈TxM |v|=1 n ∞ Moreover, it follows from the estimate (4.5) in [9] that kLf,| det Df|−1 1M kL is bounded by a constant C3 > 1 which is independent of n. n Bs Therefore, by (2.19), we obtain the bound of kLf,guk pq by −s ′ 1/p n Bs ˆ Bσ IC2C3 γs min min |Df (x)v| Rn(g)kuk pq + ICnkuk pq x∈M v∈TxM |v|=1 Bs Bσ ,for any n ≥ 1. Since the embedding pq(M) ֒→ pq(M) is compact (see [2 Corollary 2.96]), we can apply Hennion’s theorem [18] (see also [8, Appendix A]): we finally have −s 1/n ′ 1/p n Bs ress(Lf,g| pq(M)) ≤ lim inf IC2C3 γs min min |Df (x)v| Rn(g) . n→∞ x∈M v∈TxM |v|=1 This completes the proof of Theorem 3. Appendix A. Proof of Lemma 9 N(F ) N(F ) We follow [8, 12]. Recall {Vj }j=1 in (2.3). Let {θj}j=1 be a partition of unity N(F ) subordinate to {Vj }j=1 . Given 1 ≤ j ≤ N(F ), let Ij be the set of integers i in d d {1,...,N(F )} such that F (Vi) ∩ Vj 6= ∅. For 1 ≤ j ≤ N(F ), let Tj : R → R be a C r mapping such that −1 Tj (x) = (F |Vj ) (x) for x ∈ F (Vj ), d that Tj (x) 6∈ Vj if x 6∈ F (Vj ), and that |Tj (x) − Tj(y)| >cj |x − y| for any x, y ∈ R with some constant cj > 0. Then we have N(F ) LF,Gu = θj · (θi · G · u) ◦ Ti. (A.1) j=1 X iX∈Ij 12 YUSHINAKANOANDSHOTASAKAMOTO Indeed, we have N(F ) θj (x)(θi · G · u) ◦ Ti(x)= θj (x) (θi · G · u) ◦ Ti(x). j=1 X iX∈Ij j:Xx∈Vj iX∈Ij If x 6∈ F (Vi), then Ti(x) 6∈ Vi and it yields (θi · G · u) ◦ Ti(x)=0, thus −1 θj (x) (θi · G · u) ◦ Ti(x)= θj (x) (θi · G · u) ◦ (F |Vi ) (x). j:Xx∈Vj iX∈Ij j:Xx∈Vj i∈Ij :Xx∈F (Vi) On the other hand, for j such that x ∈ Vj , x ∈ F (Vi) implies i ∈ Ij , so −1 −1 (θi · G · u) ◦ (F |Vi ) (x)= (θi · G · u) ◦ (F |Vi ) (x). i∈Ij :Xx∈F (Vi) i:x∈XF (Vi) −1 Furthermore, for i such that x ∈ F (Vi), setting y = (F |Vi ) (x) gives F (y) = x and y ∈ Vi, therefore −1 (θi · G · u) ◦ (F |Vi ) (x)= (θi · G · u)(y)= (G · u)(y). i:x∈XF (Vi) F (Xy)=x i:Xy∈Vi F (Xy)=x Hence, we get (A.1), and it follows from Minkowski’s inequality that N(F ) k∆nLF,G∆ℓukLp ≤ k∆nLij ∆ℓukLp , j=1 X iX∈Ij r˜ r where Lij u = [θj · (θi · G) ◦ Ti] · u ◦ Ti with a C function θj · (θi · G) ◦ Ti and a C mapping Ti. Therefore, it suffices to show the following local version of Lemma 9: Let T : Rd → Rd be a C r mapping such that |T (x)−T (y)| > c˜|x−y| for any x, y ∈ Rd with some constant c>˜ 0, and that T : V → T (V ) is a C r diffeomorphism on a bounded open set V . Let G˜ be a C r˜ function whose support is included in V . Define (with L p Rd p Rd slight abuse of notation) T,G˜ : L ( ) → L ( ) by L ˜ p Rd T,G˜ u = G · u ◦ T, u ∈ L ( ). Also, we use the notation ℓ 6֒→ n when it holds 2n > Λ2ℓ+4, with Λ= max max |DT tr(w)v|, w∈supp(G˜) v∈Rd |v|=1 where DT tr(w) is the transpose matrix of DT (w). Lemma 11. There exists a constant CT,G˜ > 0 such that for any p ∈ [1, ∞], u ∈ Lp(Rd), n ≥ 0 and ℓ 6֒→ n, it holds L ˜ −r˜ max{n,ℓ} k∆n T,G˜ ∆ℓukLp ≤ CT,G˜ 2 kukLp (A.2) when r˜ ≤ r − 1 and L ˜ min{n,ℓ}−r˜ max{n,ℓ} k∆n T,G˜ ∆ℓukLp ≤ CT,G˜ 2 kukLp (A.3) when r˜ ≥ 1. SPECTRAOFEXPANDINGMAPSONBESOVSPACES 13 L ˜ p Rd p Rd Proof. We start the proof by noticing that ∆n T,G˜ ∆ℓ : L ( ) → L ( ) is the ℓ M M p Rd composition Hn ◦ 1 of a bounded operator 1 :Φ 7→ Φ ◦ T · | det DT | on L ( ), whose operator norm is bounded by a constant CT > 0 (see (2.4)), and an integral ℓ ℓ p Rd operator Hn :Φ 7→ Rd Vn (·,y)Φ(y)dy on L ( ) with kernel ℓ R i(x−w)ξ+i(T (w)−T (y))η ˜ ˜ Vn (x, y)= e G(w)ψn(ξ)ψℓ(η)dwdξdη. (A.4) R3d Z ℓ We will give a bound of Vn by a convolution kernel together with the factor 2−r˜max{n,ℓ}: Let b : Rd → (0, 1] be the integrable function given by 1 (|x|≤ 1), b(x)= −d−1 (A.5) (|x| (|x| > 1), and we set d dm m bm : R → R, bm(x)=2 · b(2 x) for m> 0. (A.6) ˜ ,Then, we will show that there is a constant CT,G˜ > 0 such that for all ℓ 6֒→ n ℓ ˜ −r˜max{n,ℓ} |Vn (x, y)|≤ CT,G˜ 2 bmin{n,ℓ}(x − y) ifr ˜ ≤ r − 1, (A.7) and ℓ ˜ min{n,ℓ}−r˜max{n,ℓ} |Vn (x, y)|≤ CT,G˜ 2 bmin{n,ℓ}(x − y) ifr> ˜ 1. (A.8) It follows from (A.7), by Young’s inequality, that ℓ ˜ −r˜max{n,ℓ} kHnΦkLp ≤ CT,G˜ 2 kbmin{n,ℓ} ∗ ΦkLp ˜ −r˜max{n,ℓ} ≤ CT,G˜ 2 kbmin{n,ℓ}kL1 kΦkLp , ˜ −r˜max{n,ℓ} which is bounded by CT,G˜ kbkL1 2 kΦkLp since kbmkL1 = kbkL1 for any ˜ m ≥ 1, and we get (A.2) with CT,G˜ = CT CT,G˜ kbkL1 . Similarly, (A.3) follows from (A.8). In the rest of the proof, we are dedicated to showing (A.7) and (A.8). Step 1: We first prove (A.7). We further divide the proof of (A.7) according to whether r˜ is an integer. Case (1-a): r˜ is an integer. We recall integrations by parts in w: for each C 2 function φ : Rd → R and a compactly supported C 1 function h : Rd → R with d 2 j=1(∂j φ) 6=0 on the support of h, we set P i∂ φ(w) · h(w) hφ(w)= k k =1, . . . , d, (A.9) k d 2 j=1(∂j φ(w)) 1 Rd belonging to C0 ( ). Then weP get d d iφ(w) iφ(w) φ iφ(w) φ e h(w)dw = − i∂kφ(w)e · hk (w)dw = e · ∂khk (w)dw, Z k=1 Z Z k=1 X X (A.10) d Rd where w = (wk)k=1 ∈ and ∂k denotes partial differentiation with respect to wk. We call this operation integration by parts for h over φ. Define d φ Tφh = ∂khk , φ(ξ,η,x,y)(w) = (x − w)ξ + (T (w) − T (y))η. (A.11) Xk=1 14 YUSHINAKANOANDSHOTASAKAMOTO We simply write Tξ,η for Tφ(ξ,η,x,y) (note that Tφ(ξ,η,x,y) h does not depend on x and y). Then, we have ℓ iφ(ξ,η,x,y)(w) r˜ ˜ ˜ Vn (x, y)= e Tξ,ηG(w)ψn(ξ)ψℓ(η)dwdξdη, (A.12) Z ˜ by integrating (A.4) by parts for G over φ(ξ,η,x,y) r˜ times (that is possible by definition of r˜). Put ˜ n ℓ r˜1 ˜ ˜ Gn,ℓ(ξ,η,w)= Gn,ℓ(2 ξ, 2 η, w) with Gn,ℓ(ξ,η,w)= Tξ,ηG(w)ψn(ξ)ψℓ(η). Then, by scaling we have −1 dn+dℓ −1 n ℓ F Gn,ℓ(u,v,w)=2 F G˜n,ℓ(2 u, 2 v, w), where F−1 is the inverse Fourier transform with respect to the variable (ξ, η). There- fore, by (A.11) and (A.12), we have ℓ F−1 Vn (x, y)= Gn,ℓ(x − w,T (w) − T (y), w)dw (A.13) Z dn+dℓ −1 n ℓ =2 F G˜n,ℓ(2 (x − w), 2 (T (w) − T (y)), w)dw. Z ′ Next we will see that for any integers k, k ≥ 0, there is a constant Ck,k′ > 0 such that −1 −k −k′ −r˜ max{n,ℓ} |F G˜n,ℓ(u,v,w)| ≤ |u| |v| Ck,k′ 2 (A.14) for (u,v,w) ∈ R3d. This immediately follows if one can see that for any multi- indices α, β, there is a constant Cα,β > 0 such that α β ˜ −r˜max{n,ℓ} |∂ξ ∂η Gn,ℓ(ξ,η,w)|≤ Cα,β2 (A.15) for (ξ,η,w) ∈ R3d, where ∂α = ∂α1 ∂α2 ··· ∂αd for α = (α ,...,α ) ∈ {0, 1,...}d. ξ ξ1 ξ2 ξd 1 d (Just consider integration by parts with respect to (ξ, η) for −1 −2d i(uξ+vη) F G˜n,ℓ(u,v,w)=(2π) e G˜n,ℓ(ξ,η,w)dξdη, Z 2 with noticing that the support of G˜n,ℓ(·, ·, w) is included in [−2, 2] .) Hence, we will show (A.15). We first note that, by induction with respect to r˜, Θ (ξ,η,w) T r˜ G˜(w)= 3˜r , ξ,η |ξ − DT tr(w)η|4˜r with some function Θ3˜r ≡ Θ3˜r,T,G˜ , which is a polynomial function of degree 3˜r in ξ and η whose coefficients are C 0 in w. Therefore, we have (by induction again) that Θ(1) (ξ,η,w) ∂α∂βT r˜ G˜(w)= 3˜r+|α|+|β| ξ η ξ,η |ξ − DT tr(w)η|4˜r+2|α|+2|β| (1) with some function Θ3˜r+|α|+|β|, which is a polynomial function of degree 3˜r+|α|+|β| C 0 in ξ and η whose coefficients are in w, where |α| = j αj for each multi-index d α = (α1,...,αd) ∈ {0, 1,...} . Notice also that by definition, there is an integer P d N1(T ) > 0 such that for each ξ ∈ supp(ψn), η ∈ supp(ψ˜ℓ) and w ∈ R , .ξ − DT tr(w)η|≥ 2max{n,ℓ}−N1(T ) if ℓ 6֒→ n| SPECTRAOFEXPANDINGMAPSONBESOVSPACES 15 3d Hence, for each multi-indices α, β and (ξ,η,w) ∈ R satisfying ψn(ξ) · ψ˜ℓ(η) 6= 0 with ℓ 6֒→ n, we have α β r˜ ˜ ˜ −(˜r+|α|+|β|)max{n,ℓ} ˜ −n|α|−ℓ|β|−r˜max{n,ℓ} |∂ξ ∂η Tξ,ηG(w)|≤ Cα,β2 ≤ Cα,β2 , (A.16) ˜ α ∞ −n|α| for some constant Cα,β > 0. Since k∂ξ ψnkL ≤ Cα(ρ)2 with some constant Cα(ρ) > 0 (recall (1.5)), it follows that for each multi-indices α, β and (ξ,η,w) ∈ R3d, α β −n|α|−ℓ|β|−r˜max{n,ℓ} |∂ξ ∂η Gn,ℓ(w)|≤ Cα,β2 , for some constant Cα,β > 0. Thus, we immediately obtain (A.15). ℓ Combining (A.13) and (A.14), we get that |Vn (x, y)| is bounded by C′2dn+dℓ−r˜max{n,ℓ} b(2n(x − w))b(2ℓ(T (w) − T (y)))dw, (A.17) ZK with some constant C′ > 0 and K = supp G˜. If ℓ ≤ n, we have 2dn b(2n(x − w))b(2ℓ(T (w) − T (y)))dw ≤ 2dn b(2n(x − w))dw = b(u)du ZK Z Z n ′ ′ by setting u =2 (x − w). Hence, with C1 = C kbkL1 , we get ℓ ′ dℓ−r˜max{n,ℓ} ′ d min{n,ℓ}−r˜max{n,ℓ} |Vn (x, y)|≤ C12 ≤ C12 . In the other case (when ℓ>n), by setting u˜ =2ℓ(T (w)−T (y)), in a similar manner ℓ ′ dn−r˜max{n,ℓ} ′ d min{n,ℓ}−r˜ max{n,ℓ} one can see that |Vn (x, y)|≤ C22 ≤ C22 with some ′ ′ constant C2 ≡ C2,T > 0. Recalling (A.5), the above estimates imply (A.7) for |x − y| ≤ 2− min{n,ℓ}. So, we now assume that |x − y| > 2− min{n,ℓ}. We consider the case ℓ ≤ n (the other case can be dealt similarly, as above). Let q0 = ⌊− log2 |x − y|⌋, so that either |x − w| ≥ 2−q0−1 or |w − y| ≥ 2−q0−1 holds for any w ∈ Rd. Hence with the −q0−1 notation K1 = {w ∈ K | |x − w| ≥ 2 } and K2 = {w ∈ K | |w − y| ≥ −q0−1 −q0−1 2 }, noting that w ∈ K2 implies |T (w) − T (y)|≥ c˜2 , we have the bound n ℓ of K b(2 (x − w))b(2 (T (w) − T (y)))dw by R |2n(x − w)|−d−1b(2ℓ(T (w) − T (y)))dw ZK1 + b(2n(x − w))|c˜2ℓ(w − y)|−d−1dw ZK2 ′ −dℓ−(n−q0−1)(d+1) −dn−(ℓ−q0−1)(d+1) ′ −dn −ℓ ≤C3(2 +2 ) ≤ C42 b(2 (x − y)) ′ ′ ′ ′ with some constants C3 ≡ C3,T,d > 0 and C4 ≡ C4,T,d > 0, which completes the proof of (A.7) due to (A.17). Case (1-b): r˜ is not an integer. We recall regularised integration by parts in w: for each C 1+δ function φ : Rd → R and a compactly supported C δ function Rd R d 2 φ h : → , for δ ∈ (0, 1), with j=1(∂j φ) 6=0 on the support of h, each hk given δ Rd φ φ in (A.9) belongs to C0 ( ) forPk = 1,...,d. Let hk,ǫ := hk ∗ vǫ for ǫ > 0, where −d ∞ d vǫ(x)= ǫ v(x/ǫ) with a C function v : R → R+ supported in the unit ball and satisfying v(x)dx = 1. There is a constant Cv > 0, independent of φ, such that for each C δ function u : Rd → R and each small ǫ> 0, R δ−1 δ k∂k(u ∗ vǫ)kC0 ≤ Cvǫ kukCδ , ku − u ∗ vǫkC0 ≤ Cvǫ kukCδ . (A.18) 16 YUSHINAKANOANDSHOTASAKAMOTO Finally, for every real number L ≥ 1, it holds d iLφ(w) iLφ(w) φ e h(w)dw = − i∂kφ(w)e · hk (w)dw Z Xk=1 Z d d eiLφ(w) = · ∂ hφ dw − i∂ φ(w)eiLφ(w) · (hφ(w) − hφ (w))dw. L k k,ǫ k k k,ǫ Z kX=1 Xk=1 Z (Compare with (A.10).) We call this operation regularised integration by parts for h over φ. Define d d (0,L) −1 φ (1,L) φ φ Tφ h = L ∂khk,L−1 , Tφ h = −i ∂kφ · (hk − hk,L−1 ), (A.19) Xk=1 Xk=1 and denote T (0,L) and T (1,L) by T (0,L) and T (1,L), respectively. Then, by φ(ξ,η,x,y) φ(ξ,η,x,y) ξ,η ξ,η ˜ integration of (A.4) by parts for G over φ(ξ,η,x,y) [˜r] times and regularised inte- −1 gration by parts over φ(ξ,η,x,y) one time with δ =r ˜ − [˜r] and ǫ = L (note that φ(ξ,η,x,y)(w)= Lφ(ǫξ,ǫη,x,y)(w)), we get (0,L) [˜r] ℓ iφ(ξ,η,x,y)(w) ˜ ˜ Vn (x, y)= e Tǫξ,ǫη Tξ,η G(w)ψn(ξ)ψℓ(η)dwdξdη Z (1,L) [˜r] iφ(ξ,η,x,y)(w) ˜ ˜ + e Tǫξ,ǫη Tξ,η G(w)ψn(ξ)ψℓ(η)dwdξdη, Z (0,L) (1,L) which we denote by Vn,ℓ (x, y)+ Vn,ℓ (x, y). It follows from (A.18) and (A.19) that (with L−1 = ǫ) d α β (0,L) α β φ(ǫξ,ǫη,x,y) |∂ξ ∂η Tǫξ,ǫη h(w)|≤ ǫ ∂k (∂ξ ∂η hk ) ∗ vǫ (w) (A.20) Xk=1 d δ α β φ(ǫξ,ǫη,x,y) ≤ Cvǫ k∂ξ ∂η hk kCδ . Xk=1 α β (1,L) In a similar manner, we get the bound of |∂ξ ∂η Tǫξ,ǫη h(w)| by d δ α(1) β(1) φ(ǫξ,ǫη,x,y) α(2) β(2) Cvǫ k∂ξ ∂η hk kCδ k∂ξ ∂η (∂kφ(ǫξ,ǫη,x,y))kC0 , (1) (1) (2) (2) Xk=1 (α ,β X,α ,β ) where the second summation is taken over multi-indices (α(1),β(1), α(2),β(2)) ∈ {0, 1,...}4d such that α(1) + α(2) = α and β(1) + β(2) = β. On the other hand, −1 φ iǫ Θˆ (w,ξ,η)h(w) ∂ φ = ǫΘˆ (w,ξ,η), h (ǫξ,ǫη,x,y) (w)= 1 (A.21) k (ǫξ,ǫη,x,y) 1 k |ξ − DT tr(w)η|2 tr where Θˆ 1(w,ξ,η) is the k-th element of −ξ + DT (w)η, which is a polynomial function of degree 1 in ξ and η whose coefficients are C r−1 (at least C r˜) in w. [˜r] ˜ Therefore, applying (A.20) and (A.21) for h = Tξ,η G, together with (A.16), we get α β (j,L) [˜r] ˜ ˜ δ −n|α|−ℓ|β|−[˜r] max{n,ℓ} |∂ξ ∂η Tǫξ,ǫη Tξ,η G(w)|≤ CvCα,βǫ 2 (j =0, 1) SPECTRAOFEXPANDINGMAPSONBESOVSPACES 17 − max{n,ℓ} for any ξ ∈ supp ψn and η ∈ supp ψ˜ℓ with ℓ 6֒→ n, implying that, with ǫ =2 and δ =r ˜ − [˜r], α β (j,L) [˜r] ˜ ˜ −n|α|−ℓ|β|−r˜max{n,ℓ} |∂ξ ∂η Tǫξ,ǫη Tξ,η G(w)|≤ CvCα,β2 (j =0, 1). ℓ (0,L) Therefore, in a manner similar to one in the case (1-a) (replacing Vn with Vn,ℓ (1,L) and Vn,ℓ ), one can get (A.7). Step 2: Next we show (A.8), according to whether r˜ is an integer. ℓ Case (2-a): r˜ is an integer. Recall Vn from (A.4) and bm from (A.6). For each (ξ,η,x,y), define ˜ i(φ1−φ2) φ(ξ,η,x,y)(w) = (x − w)ξ + DT (y)(w − y)η, Rφ1,φ2 h = e h, Rd R −1 for continuous functions φ1, φ2,h : → . We simply write R0 and R0 for ˜ Rφ1,φ2 and Rφ2,φ1 , respectively, with φ1 = φ(ξ,η,x,y) and φ2 = φ(ξ,η,x,y). Denote (0,L) (1,L) ˜ ˜ (0,L) ˜ (1,L) Tφ˜ , T ˜ and T ˜ by Tξ,η,y, Tξ,η,y and Tξ,η,y , respectively. Then (ξ,η,x,y) φ(ξ,η,x,y) φ(ξ,η,x,y) (A.4) can be rewritten as ˜ ℓ iφ(ξ,η,x,y) (w) ˜ ˜ Vn (x, y)= e R0G(w)ψn(ξ)ψℓ(η)dwdξdη. (A.22) Z ˜ d ∞ ˜ r˜ Note that φ(ξ,η,x,y) : R → R is of class C and R0G is C because r˜ < r, so that d r˜−1 T˜ξ,η,yR0G˜ : R → R is well-defined and of class C . ˜ ˜ Integrating (A.22) by parts once on w (for R0G over φ(ξ,η,x,y)), we obtain ˜ ℓ iφ(ξ,η,x,y) (w) ˜ ˜ ˜ Vn (x, y)= e Tξ,η,yR0G(w)ψn(ξ)ψℓ(η)dwdξdη (A.23) Z iφ(ξ,η,x,y) (w) −1 ˜ ˜ ˜ = e R0 Tξ,η,yR0G(w)ψn(ξ)ψℓ(η)dwdξdη. Z Integrating (A.23) by parts r˜ − 1 times on w, we obtain ℓ iφ(ξ,η,x,y) (w) r˜−1 −1 ˜ ˜ ˜ Vn (x, y)= e Tξ,η R0 Tξ,η,yR0G(w)ψn(ξ)ψℓ(η)dwdξdη. Z In a manner similar to the case (1-a), it can be shown that α β r˜−1 −1 ˜ ˜ ˜ ℓ −n|α|−ℓ|β|−r˜max{n,ℓ} ∂ξ ∂η Tξ,η R0 Tξ,η,yR0G ≤ Cα,β2 2 , (A.24) L∞ because one can have, by induction, that Θ(2) (ξ,η,w)+Θ(3) (ξ,η,w)η T r˜−1R−1T˜ R G˜ = 3(˜r−1)+1 3(˜r−1)+1 , (A.25) ξ,η 0 ξ,η,y 0 |ξ − DT tr(w)η|4(˜r−1)|ξ − DT tr(y)η|2 (2) (3) with some functions Θ3(˜r−1)+1 and Θ3(˜r−1)+1, which are polynomial functions of degree 3(˜r − 1)+1 in ξ and η whose coefficients are C 0. r˜−1 −1 ˜ r˜ r˜ We considered Tξ,η R0 Tξ,η,yR0 instead of Tξ,η since Tξ,η is not well-defined when r˜ > r − 1. The price we have to pay for this change is the factor η in (A.25), {resulting in 2ℓ in (A.24). By definition of ℓ 6֒→ n, 2ℓ is dominated by 2min{n,ℓ multiplied by a constant. Consequently, as in the case (1-a), one can see that (A.24) implies (A.8) . 18 YUSHINAKANOANDSHOTASAKAMOTO Case (2-b): r˜ is not an integer. We perform single integration by parts of ˜ ˜ (A.4) for G over φ(ξ,η,x,y), [˜r]−1 integration by parts over φ, and a single regularised ˜ integration by parts over φ(ξ,η,x,y) with δ =r ˜ − [˜r], resulting in ˜ ℓ iφξ,η,x,y(w) ˜ (0,L) [˜r]−1 −1 ˜ ˜ ˜ Vn (x, y)= e Tξ/L,η/L,yR0Tξ,η R0 Tξ,η,yR0G(w)ψn(ξ)ψℓ(η)dwdξdη Z ˜ iφξ,η,x,y (w) ˜ (1,L) [˜r]−1 −1 ˜ ˜ ˜ + e Tξ/L,η/L,yR0Tξ,η R0 Tξ,η,yR0G(w)ψn(ξ)ψℓ(η)dwdξdη. Z Then in view of the cases (1-b) and (2-a), we obtain (A.8). Acknowledgements This work was partially supported by JSPS KAKENHI Grant Numbers 16J03963 and 17K05283. The second author shows his deep gratitude to the members of Faculty of Engineering in Kitami Institute of Technology for their warm hospitality when he visited there in August 2016 and October 2017. The authors also would like to express their gratitude to Masato Tsujii for many fruitful discussions. Finally, the authors are deeply grateful to an anonymous reviewer for many important suggestions, all of which substantially improved the paper. References [1] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps, Stochastics and Dynamics 1 (2001), no. 02, 193–237. [2] H. Bahouri, J.-Y. Chemin, and R. Danchin, Fourier analysis and nonlinear partial differential equations, Vol. 343, Springer Science & Business Media, 2011. [3] M. Baillif and V. Baladi, Kneading determinants and spectra of transfer operators in higher dimensions: the isotropic case, Ergodic Theory Dyn. Syst. 25 (2005), 1437–1470. [4] V. Baladi, Positive transfer operators and decay of correlations, World Scientific, 2000. [5] , The quest for the ultimate anisotropic Banach space, Journal of Statistical Physics 166 (2017), no. 3-4, 525–557. [6] , Characteristic functions as bounded multipliers on anisotropic spaces, Proceedings of the American Mathematical Society 146 (2018), no. 10, 4405–4420. [7] , Correction to: The Quest for the Ultimate Anisotropic Banach Space, Journal of Statistical Physics 170 (2018), no. 6, 1242–1247. [8] , Dynamical zeta functions and dynamical determinants for hyperbolic maps, Ergeb- nisse der Mathematik und ihrer Grenzgebiete, Springer, 2018. [9] V. Baladi, A. Kondah, and B. Schmitt, Random correlations for small perturbations of ex- panding maps, Random Comput. Dynam. 4 (1996), 179–204. [10] V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomor- phisms, Ann. Inst. Fourier 57 (2007), 127–154. [11] , Dynamical determinants and spectrum for hyperbolic diffeomorphisms (K. Burns, D. Dolgopyat, and Y. Pesin, eds.), Contemp. Math., vol. 469, American Mathematical Society, 2008. [12] , Spectra of differential hyperbolic maps (S. Albeverio, M. Marcolli, S. Paycha, J. Plazas, and K. Diederich, eds.), Aspects Math., vol. E38, Friedr. Vieweg, 2008. [13] V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys. 156 (1993), no. 2, 355–385. [14] , Erratum: “On the spectra of randomly perturbed expanding maps”, Comm. Math. Phys. 166 (1994), no. 1, 219–220. [15] F. Faure, Semiclassical origin of the spectral gap for transfer operators of a partially expand- ing map, Nonlinearity 24 (2011), 1473–1498. [16] S. Gouëzel, Limit theorems in dynamical systems using the spectral method (D. Dolgopyat, Y. Pesin, M. Pollicott, and L. Stoyanov, eds.), Proceedings of Symposia in Pure Mathematics, vol. 89, American Mathematical Society, 2015. SPECTRAOFEXPANDINGMAPSONBESOVSPACES 19 [17] V. M. Gundlach and Y. Latushkin, A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces, Ergodic Theory and Dynamical Systems 23 (2003), no. 1, 175–191. [18] H. Hennion, Sur un théoreme spectral et son application aux noyaux lipchitziens, Proceedings of the American Mathematical Society 118 (1993), no. 2, 627–634. [19] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cam- bridge University Press, 1995. [20] R. Mané, Ergodic theory and differentiable dynamics, Vol. 8, Springer Science & Business Media, 2012. [21] F. Przytycki and M. Urbański, Conformal fractals: ergodic theory methods, Vol. 371, Cam- bridge University Press, 2010. [22] D. Ruelle, Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics, Addison-Wesley, 1978. [23] , The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), 239–262. [24] , Une extension de la théorie de Fredholm, Comptes rendus de l’Académie des sciences. Série 1, Mathématique 309 (1989), no. 6, 309–310. [25] , An extension of the theory of Fredholm determinants, Publications Mathématiques de l’Institut des Hautes Études Scientifiques 72 (1990), no. 1, 175–193. [26] E. M Stein, Singular integrals and differentiability properties of functions, Vol. 2, Princeton university press, 1970. [27] M. E Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, vol. 100, Birkhäuser Boston, Inc., Boston, MA, 1991. [28] D. Thomine, A spectral gap for transfer operators of piecewise expanding maps, Discrete & Continuous Dynamical Systems-A 30 (2011), no. 3, 917–944. [29] H. Triebel, Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation, Arkiv för Matematik 11 (1973), no. 1-2, 13–64. [30] , Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Arkiv för Matematik 24 (1986), no. 1, 299–337. [31] M. Viana, Lectures on Lyapunov exponents, Vol. 145, Cambridge University Press, 2014. [32] P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982. (Yushi Nakano) Faculty of Engineering, Kitami Institute of Technology, Hokkaido, 090-8507, JAPAN E-mail address: [email protected] (Shota Sakamoto) Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan E-mail address: [email protected]