Approximation and entropy numbers of composition operators

Open Access. ©2020 Daniel Li et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Concr. Oper. 2020; 7:166–179 Research Article Open Access Daniel Li*, Hervé Queffélec, and Luis Rodríguez-Piazza Approximation and entropy numbers of composition operators https://doi.org/10.1515/conop-2020-0106 Received October 29, 2019; accepted September 15, 2020 Abstract: We give a survey on approximation numbers of composition operators on the Hardy space, on the disk and on the polydisk, and add corresponding new results on their entropy numbers, revealing how they are different. Keywords: approximation numbers; composition operator; entropy numbers; polydisk; pluricapacity MSC: primary: 47B33; secondary: 30H10; 32A35; 46B06 1Introduction This paper surveys results on approximation numbers of composition operator on the Hardy space, and gives new results, on their entropy numbers, in one or several dimensions. In various papers (see [3, 19–22]), pretty sharp estimates are obtained for them, either for classes of examples like the lens maps or the cusp maps, or in the general case. In particular, a few properties are investigated, related with the so-called “spectral radius type formula”, obtained, in dimension one through a result of Widom in [21], and, partially indimension N≥2 [22, 23], through a result of Nivoche [26] and Zakharyuta [33]. One of our main results (quoted in dimension one) was the implication (using Green capacity considerations): kφk∞= 1 ⇒lim n→∞[an(Cφ)]1/n= 1 ,(1.1) where an(Cφ)is the n-th approximation number of Cφ. Note that it is straightforward that if kφk∞=r<1, then limn→∞[an(Cφ)]1/n≤r<1. Another way of measuring the compactness of operators is using the entropy numbers instead of the approximation numbers. Those numbers stand a little apart in the jungle of “s-numbers”, even though they seem to be the most natural for the study of compactness, since their membership in c0characterizes compactness, even in the general framework of arbitrary Banach spaces. Given a compact operator T:H1→H2between Hilbert spaces, the relation between its entropy numbers en(T)and its approximation (if one prefers singular) numbers an(T)is theoretically known, through a general result on diagonal operators on `2, recalled in Theorem 2.2 to follow, and through the Schmidt decomposition of T. This comparison can be thought useless, since in principle we do not know better the numbers an(T) than the numbers en(T). But in our case, with T=Cφ, namely T(f) = f◦φwhere φis an analytic self-map of the polydisk DN, the situation is slightly different. Answering a question of J. Wengenroth [30] about the *Corresponding Author: Daniel Li: Univ. Artois, Laboratoire de Mathématiques de Lens (LML) UR 2462, & Fédération Mathématique des Hauts-de-France, CNRS FR 2037, Faculté Jean Perrin, Rue Jean Souvraz, S.P. 18 F-62 300 LENS, France, E-mail: [email protected] Hervé Queffélec: Univ. Lille Nord de France, USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524 & Fédération Mathématique des Hauts-de-France, CNRS FR 2037, F-59 655 VILLENEUVE D’ASCQ Cedex, France, E-mail: [email protected] Luis Rodríguez-Piazza: Universidad de Sevilla, Facultad de Matemáticas, Departamento de Análisis Matemático & IMUS, Calle Tarfia s/n 41 012 SEVILLA, Spain, E-mail: [email protected] Approximation and entropy numbers of composition operators |167 behavior of the entropy numbers of composition operators, we give in this paper estimates for these numbers, analog to that on the approximation numbers. In particular, we have: kφk∞= 1 ⇒lim n→∞[en(Cφ)]1/√n= 1 .(1.2) The proofs are not difficult, but as indicated for example by the comparison between (1.1) and (1.2), the statements feature a very different behavior of those entropy numbers, which deserves attention. This difference is actually more transparent in the polydisk DN, and the main interest of this paper is to point out how the dependence of the entropy numbers with respect to the dimension Ndiffers from that of the approximation numbers. The paper is organized as follows. Section 1 is this introduction. In Section 2 we recall the necessary background. In Section 3, we survey results on approximation numbers and give the corresponding new results on entropy numbers. We first begin with general facts, and then give specific results, with a particular interest to the examples of the lens maps and the cusp map. For any non-constant analytic map φ:D→D, we have: lim n→∞[an(Cφ)]1/n= exp −1/Cap [φ(D)],(1.3) where Cap [φ(D)] is the Green capacity of φ(D), from which it follows that limn→∞[an(Cφ)]1/n= 1 if and only if kφk∞= 1 (Theorem 3.1). We moreover prove that: lim n→∞[en(Cφ)]1/√n= exp −plog 2/Cap [φ(D)],(1.4) from which it follows that limn→∞[en(Cφ)]1/√n= 1 if and only if kφk∞= 1 (Theorem 3.2). For the lens map λθwith parameter θ, we have (the constants depending only on θ): αe−C n1/2≤an(Cλθ)≤βe−c n1/2,(1.5) whereas: α0e−C0n1/3≤en(Cλθ)≤β0e−c0n1/3.(1.6) For the cusp map χ, we have, for absolute constants: αe−Cn/log n≤an(Cχ)≤βe−cn/log n,(1.7) and α0e−C0√n/log n≤en(Cχ)≤β0e−c0√n/log n.(1.8) Section 4 is concerned by the multivariate case, again first in the general case, then on specific cases. We are in particular interested by the multi-lens map Λθ, defined by: Λθ(z1,. . . ,zN) = λθ(z1),λθ(z2),. . . ,λθ(zN), and the multi-cusp map Ξdefined by: Ξ(z1,. . . ,zN) = χ(z1),χ(z2),. . . ,χ(zN). We prove that: aexp (−C n1/(2N))≤an(CΛθ)≤bexp(−c n1/(2N))(1.9) a0exp (−C0n1/(2N+1))≤en(CΛθ)≤b0exp (−c0n1/(2N+1))(1.10) and aexp[−C n1/N/log n]≤an(CΞ)≤bexp[−c n1/N/log n](1.11) a0exp −C0n1/(N+1) (log n)−N/(N+1)≤en(CΞ) ≤b0exp −c0n1/(N+1) (log n)−N/(N+1). (1.12) Section 5 is more specifically devoted to the multidimensional case, in connection with the notion of Monge-Ampère (or Bedford-Taylor) pluricapacity, which recently turned out to play an important role in connection with composition operators [23]. 168 |Daniel Li et al. Figure 1: Lens map domain Figure 2: Cusp map domain 2Background and notation We denote by Dthe open unit disk and by T=∂Dthe unit circle; mis the normalized Lebesgue measure on T: dm(t) = dt/2π, and H2is the usual Hardy space on D. By Littlewood’s subordination principle, every analytic self-map φ:D→D(also called Schur function) defines a bounded operator Cφ:H2→H2by Cφ(f) = f◦φ, called the composition operator of symbol φ. 2.1 Two types of Schur functions Lens maps For 0<θ<1, the lens map λθwith parameter θis obtained by sending conformally the unit disk Donto the right half-plane Π={z∈C;Rez>0}; then making u7→ uθ, and coming back to D(see [29, page 27]). Namely: λθ(z) = (1 + z)θ−(1 −z)θ (1 + z)θ+ (1 −z)θ·(2.1) It is a conformal map from Donto the domain represented on Figure 1. Cusp map The cusp map is a conformal mapping χsending the unit disk Donto the domain represented on Figure 2. This map was first introduced in [14]. To obtain it, we first map Donto the half-disk D+={z∈D;Rez>0}. To do that, map Donto itself by z7→ iz; then map Donto the upper half-plane H={z∈C;Imz>0}by T(u) = i1 + u 1−u· Take the square root to map Hin the first quadrant Q1={z∈H;Rez>0}, and go back to the half-disk {z∈D;Imz<0}by T−1:T−1(s) = 1+is is−1; finally, make a rotation by ito go onto D+. We get χ0(z) = z−i iz −11/2−i −iz−i iz −11/2+ 1 ·(2.2) One has χ0(1) = 0,χ0(−1) = 1,χ0(i) = −iand χ0(−i) = i. The half-circle {z∈T;Rez≥0}is mapped onto the segment [−i,i]and the segment [−1,1] onto the segment [0,1]. Set now, successively, χ1(z) = log χ0(z),χ2(z) = −2 πχ1(z) + 1,χ3(z) = 1 χ2(z),(2.3) Approximation and entropy numbers of composition operators |169 and finally: χ(z) = 1 −χ3(z).(2.4) Hence: 1−χ(z) = 1 1 + 2 πlog 1/|χ0(z)|−i2 πarg χ0(z)·(2.5) Note that χ2maps Donto the half-strip {z∈C;Rez>1and |Imz|<1}. One has φ(1) = 1,φ(−1) = 0, φ(i) = (1 + i)/2and φ(−i) = (1 −i)/2. The domain φ(D)is edged by three circular arcs of radii 1/2and of respective centers 1/2,1 + i/2and 1−i/2. The real interval ]−1,1[ is mapped onto the real interval ] 0,1[ and the half-circle {eiθ ;|θ|≤π/2}is sent onto the two circular arcs tangent at 1to the real axis. 2.2 Approximation and entropy numbers Given an operator T:X→Ybetween Banach spaces, recall (see [6]) that we can attach to this operator five non-increasing sequences (an),(bn),(cn),(dn),(en)of non-negative numbers (depending on T), respectively the sequences of approximation,Bernstein,Gelfand,Kolmogorov, and entropy numbers of T. We only define here the first one and the last one. The approximation numbers are defined as: an(T) = inf{kT−Rk; rank (R)<n},n≥1.(2.6) The entropy numbers are defined for n≥1as: en(T) = inf{ε>0 ; NT(BX),εBY≤2n−1},n≥1,(2.7) where BXand BYare the respective closed unit balls of Xand Y, and where, for A,B⊆Y,N(A,B)denotes the smallest number of translates of Bneeded to cover A(see [6, Chapter 1], or [28, Chapter 5]). All those sequences (an),(bn),(cn),(dn),(en), say (un), share the ideal property: un(ATB)≤kAkun(T)kBk.(2.8) For Hilbert spaces, it turns out that an=bn=cn=dn=sn,(2.9) where (sn)designates the sequence of singular numbers; but entropy numbers stay a little apart. For Banach spaces Xand Yand T:X→Y, we have, in general, for α>0: sup 1≤k≤n kαek(T)≤Cαsup 1≤k≤n kαak(T) ([5, Theorem 1], see also [28, Theorem 5.2]), and, if Xand Y*are of type 2: an(T)≤K en(T),for all n≥1 ([12, Corollary 1.6]), where K=κ[T2(X)T2(Y*)]2; in particular, if Tacts between Hilbert spaces: an(T)≤4en(T),for all n≥1 (see [28, Theorem 5.3]). 170 |Daniel Li et al. Recall (see [1, Definition 6.2.10, p. 137], [17, Définition III.3, p. 162], or [18, Definition IV.3, p. 180]) that a Banach space Xhas type 2 if there is a constant Csuch that for every finite sequence x1,x2,. . . ,xnin Xwe have: 1 2nX ε1,...,εn=±1    n X k=1 εkxk    21/2 ≤Cn X k=1 kxkk21/2 , the smallest such constant Cis denoted T2(X). Every Hilbert space has type 2, thanks to the parallelogram identity. Those inequalities indicate that entropy numbers are always bigger than singular numbers, up to a constant, and that, as far as the scale of powers nαis implied, they are dominated by approximation numbers in a weak sense. But it turns out that, individually, they can be much bigger than the latter for composition operators, as we shall see. We will rely on the following estimate ([6, Proposition 1.3.2, p. 17]), in which `2denotes the space of square-summable sequences x= (xk)k≥1of complex numbers. This estimate is given for the sequence (εn), but en=ε2n−1, by definition. Theorem 2.1. (see [6, p. 17]) There exist absolute constants 0<a<bsuch that, for any diagonal compact operator ∆:`2→`2with positive and non-increasing eigenvalues (σk)k≥1, namely ∆(xk)k= (σkxk)k, we have, for all n≥1: asup k≥12−n/2kk Y j=1 σj1/k≤en(∆)≤bsup k≥12−n/2kk Y j=1 σj1/k.(2.10) A useful corollary of Theorem 2.1 is the following. Theorem 2.2. Let T:H1→H2be a compact operator between the complex Hilbert spaces H1and H2, and let (an)n≥1be its sequence of approximation numbers. Then, for all n≥1: αsup k≥12−n/2kk Y j=1 aj1/k≤en(T)≤βsup k≥12−n/2kk Y j=1 aj1/k,(2.11) where αand βare positive numerical constants. Proof. Let Tx =P∞ n=1 sn(x|un)vnthe Schmidt decomposition of T, where (un)nand (vn)nare orthonormal sequences of H1and H2, respectively, and (sn)nis the sequence of singular numbers of T. Let ∆:`2→`2the diagonal operator with diagonal values sn,n≥1. Then T=V1∆ U1and ∆=V2TU2, with U1x=(x|un)n, V1(tn)n=Pntnvn,U2(tn)n=Pntnunand V2x=(x|vn)n. We have kU1k,kV1k,kU2k,kV2k≤1; hence the result follows from Theorem 2.1 and the ideal property (2.8). This theorem might be thought useless, because we do not know better the numbers anthan the numbers en! In our situation, this is not the case, since we made a more or less systematic study of the approximation numbers anfor composition operators in [3, 19–21] for example. 3The 1-dimensional case 3.1 General results In [21], we coined the parameter β1(T) = lim n→∞an(T)1/n(3.1) and its versions β+ 1(T),β− 1(T)with an upper limit and a lower limit respectively. The following result, proved in [21, Theorem 3.1] for kφk∞<1and in [21, Theorem 3.14] for kφk∞= 1, shows in particular that no lower or Approximation and entropy numbers of composition operators |171 upper limit is needed for β=β1, and provides a simpler proof of the second item in Theorem 3.1 below than in our initial proof of [19, Theorem 3.4]. For the definition of the Green capacity Cap (A)of a Borel subset Aof D,0≤Cap (A)≤ ∞, we refer to [21, Section 2.3]. Theorem 3.1. Let Ω=φ(D), with φ:D→Da non-constant analytic map. Then: 1) One always has β− 1(Cφ) = β+ 1(Cφ) =: β1(Cφ)and β1(Cφ) = exp[−1/Cap (Ω)] >0.(3.2) 2) In particular, one has the equivalence β1(Cφ) = 1 ⇔ kφk∞= 1 .(3.3) The first item says in particular that we always have: an(Cφ)&rn(3.4) for some positive constant r<1[19, Theorem 3.1]. This was actually first pointed out by Parfenov [27]. The second item says that the behavior an(Cφ)≈rnis only obtained when kφk∞<1([19, Theorem 3.4] or [21, Theorem 3.14]). For entropy numbers, another parameter emerges: γ1(T) = lim n→∞en(T)1/√n(3.5) and its γ+ 1(T)and γ− 1(T)versions. Theorem 3.2. Let φ:D→Dbe a non-constant symbol and Ω=φ(D). Then 1) γ− 1(Cφ) = γ+ 1(Cφ) =: γ1(Cφ)and γ1(Cφ) = exp −plog 2/Cap (Ω)>0.(3.6) 2) In particular, one has the equivalence γ1(Cφ) = 1 ⇔ kφk∞= 1 .(3.7) Proof. Set ρ= 1/Cap (Ω)for simplicity of notations. Let ε>0, and Cεa positive constant which depends only on εand can vary from a formula to another. Theorem 3.1 implies ak≤Cεeεke−kρ, whence (a1· · · ak)1/k≤Cεeεk/2e−ρk/2. Theorem 2.2 gives en(Cφ)≤Cεsup k≥1 [ eεk/2e−(n/k)(log 2/2)+(ρk/2)]. This supremum is essentially attained for k=p(nlog 2)/ρwhere d.estands for the integer part, and gives en(Cφ)≤Cεeε 2√nlog 2/ρe−√n ρ log 2 . This implies γ+ 1(Cφ)≤eε 2√log 2/ρe−√ρlog 2, and finally γ+ 1(Cφ)≤e−√ρlog 2 . The lower bound γ− 1(Cφ)≥e−√ρlog 2 is proved similarly. This clearly ends the proof, since we know that Cap (Ω) = ∞if and only if kφk∞= 1 (see [21, Theorem 3.13]). 172 |Daniel Li et al. 3.2 Estimates for approximation numbers Estimates of approximation numbers of composition operators can be obtained by using the boundary behavior of the symbol and Blaschke products for the upper estimates, and the radial behavior of this symbol, reproducing kernels, and interpolating sequences, for the lower estimates. In order to treat simultaneously the two cases of the lens maps and of the cusp map, we will put ourselves in a more general situation. We say that a continuous function ω: [0,1] →R+is a modulus of continuity if it is increasing, subadditive, i.e. ω(s+t)≤ω(s) + ω(t), and vanishes at 0. Upper estimates Let φbe a symbol in the disk algebra, i.e. φ:D→Dis continuous on Dand analytic in D, and such that φ(∂D)∩∂D={ξ1,. . . ,ξp}. We say that this symbol is boundary regular if, writing ξj= eitj, we have: 1) for some positive constant C, we have |φ(eit)−φ(eitj)|≤C1−|φ(eit)|, for tin a neighborhood of tj and for j= 1,. . . ,p; 2) for some modulus of continuity ωand for some positive constant c, we have c ω(|t−tj|)≤|φ(eit)− φ(eitj)|, for tin a neighborhood of tjand for j= 1,. . . ,p. The following result is proved in [20, Theorem 2.3]. Theorem 3.3. Let φbe a symbol in the disk algebra whose image touches ∂Dat the points ξ1,. . . ,ξp, and nowhere else, and such that φis boundary regular. Then, there are constants κ,K,L>0, depending only on φ, such that, for every q≥1: aq(Cφ)≤Krω−1(κ2−Nq) κ2−Nq,(3.8) where Nqis the largest integer such that p NdN<q, with dNthe integer part of σlog κ2−N ω−1(κ2−N)+ 1. This theorem allows to give an upper estimate for all approximation numbers an(Cφ),n≥1when we can interpolate between the integers NdNand (N+ 1) dN+1, but this is not the case in general. Nevertheless, this is the case in the examples below. Theorem 3.4. 1) For the lens map λθwith parameter θ, we have, for some positive constants αand β, depending on θ: an(Cλθ)≤αe−β√n.(3.9) 2) For the cusp map χ, we have, for some positive constants αand β: an(Cχ)≤αe−βn/log n.(3.10) Proof. 1) The map λθsatisfies the conditions of Theorem 3.3 with ω(h) = hθ, with p= 2. We have ω−1(h) = h1/θ. Hence dN≈N,Nq≈√q, and we then get from (3.8) that aq(Cλθ)≤α2−δN for q&N2, with δ>0, which gives (3.9). 2) The map χsatisfies the conditions of Theorem 3.3 with p= 1 and ω(h) = 1/(log 1/h). Then, ω−1(h) = e−1/hand dN≈2N, so that Nq≈log qand 2Nq≈q/log q, and (3.10) follows. The proof of Theorem 3.3 is based on the following Lemma 3.5, with a suitable choice of the Blaschke product of length pNd (da positive integer to be specified): B(z) = p Y j=1 N Y k=1 z−pj,k 1−pj,kzd , where pj,k= (1 −2−k)ξj, for j= 1,. . . ,pand k= 1,2,. . .. We do not give the details here and refer to [20, Proof of Theorem 2.3]. Approximation and entropy numbers of composition operators |173 Lemma 3.5 ([15, Lemma 2.4]).For every Blaschke product Bwith less than Nzeros (each of them being counted with its multiplicity), one has: aN(Cφ)2.sup 0<h<1,|ξ|=1 1 hZ S(ξ,h) |B(z)|2dmφ(z),(3.11) where S(ξ,h) = {z∈D;|z−ξ|≤h}and mφis the pull-back measure by φof the normalized Lebesgue measure mon T. The proof of Lemma 3.5 comes from the estimate of the Carleson norm in the Carleson’s theorem and from the fact that the subspace BH2is of codimension ≤N−1, leading to a majorization of the Gelfand number cN(Cφ), and then using (2.9). For the convenience of the reader, we reproduce the proof. Proof of Lemma 3.5. The subspace BH2is of codimension ≤N−1. Therefore, aN=cN(Cφ)≤ Cφ|BH2 , where cNis the n-th Gelfand number and where we used the equality aN=cNoccurring in the Hilbertian case, as recalled in the introduction. Now, since kBfkH2=kfkH2for any f∈H2, we have:  Cφ|BH2  2= sup kfkH2≤1Z T |B◦φ|2|f◦φ|2dm = sup kfkH2≤1Z D |B|2|f|2dmφ=kRµk2, where µ=|B|2mφand where Rµ:H2→L2(µ)is the restriction map. Of course, µis a Carleson measure for H2since µ≤mφ. Now, Carleson’s embedding theorem says us that kRµk2≤κ2sup0<h<1,ξ∈T µ[S(ξ,h)] h(see [9], Remark after the proof of Theorem 9.3, at the top of page 163; actually, in that book, Carleson’s windows W(ξ,h)are used instead of pseudo-Carleson’s windows S(ξ,h), but that does not matter, since W(ξ,h)⊆ S(ξ,2h): if r≥1−hand |t−t0|≤h, then |reit −eit0|≤|reit −eit|+|eit −eit0|≤2h). Lower estimates We consider symbols φtaking real values in the real axis (i.e. its Taylor series has real coefficients) and such that limr→1−φ(r) = 1, with a given speed. We say that φis radially regular if it takes real values on ]−1,1[ and there exists a modulus of continuity ω: [0,1] →[0,2] such that 1−φ(r)≤ω(1 −r)for 0≤r<1. Then we have the following result ([20, Theorem 3.2]). Theorem 3.6. Let φbe a radially regular symbol. Then, for the approximation numbers an(Cφ)of the composition operator Cφof symbol φ, one has the following lower bound. an(Cφ)≥csup 0<σ<1rω−1(a σn) a σnexp −20 1−σ,(3.12) where a= 1 −φ(0) >0and cis another constant depending only on φ. For our examples, we get: Theorem 3.7. 1) For the lens map λθof parameter θ, we have, for some positive constants cand C, depending on θ: an(Cλθ)≥cexp −C√n.(3.13) 2) For the cusp map χ, we have, for some positive constants cand C: an(Cφ)≥cexp(−C n/log n).(3.14) 174 |Daniel Li et al. Proof. 1) The lens map λθsatisfies the conditions of Theorem 3.6 with ω−1(h)≈h1/θ. We get the result by adjusting σ= 1 −1/√n. 2) The cusp map χsatisfies the conditions of Theorem 3.6 with ω−1(h)≈e−C0/h, and by taking σ= exp(−log n/2n), we get the result. The tools for proving Theorem 3.6 are the following. Recall (see [13, pages 194–195], or [25, pages 302–303]) that if (zj)is a Blaschke sequence, its Carleson constant δis defined as δ= inf j≥1(1 −|zj|2)|B0(zj)|, where Bis the Blaschke product whose zeros are the points zj. Recall also (see [7], [13, pages 194–195], or [25, pages 302–303]) that an interpolation sequence (zn)with (best) interpolation constant Cis a sequence (zn) (necessarily Blaschke, i.e. P∞ n=1(1 −|zn|)<∞) in the unit disk such that, for any bounded sequence (wn)of scalars, there exists a bounded analytic function f(i.e. f∈H∞) such that: f(zn) = wn,∀n≥1,and kfk∞≤Csupn≥1|wn|. Now (see [11, Chapter VII, Theorem 1.1]), every H∞-interpolation sequence (zj)is a Blaschke sequence and its Carleson constant δis connected to its interpolation constant Cby the inequalities 1/δ≤C≤κ/δ2(3.15) where κis an absolute constant (actually C≤κ1(1/δ)(1+log 1/δ)). Now, if (zj)is a H∞-interpolation sequence with constant C, the sequence of the normalized reproducing kernels fj=Kzj/kKzjk, viz fj(z) = 1−|zj|2 1−zjz, satisfies C−1X j|λj|21/2 ≤   X j λjfj   H2 ≤CX j|λj|21/2 (3.16) (see [19, Lemma 2.2]). We then use the following lemma [20, Lemma 3.3], with uj∈[0,1) defined inductively by u0= 0 and the relation: 1−φ(uj+1) = σ[1 −φ(uj)] with 1>uj+1 >uj, using the intermediate value theorem, where 0<σ<1is a fixed positive number. Note that, when setting vj=φ(uj), we have −1<vj<1, 1−vj+1 1−vj =σ,(3.17) and 1−vn=a σn,(3.18) with a= 1 −φ(0). Lemma 3.8. Let φ:D→Dbe an analytic self-map. Let u= (u1,. . . ,un)be a finite sequence in Dand set vj=φ(uj),v= (v1,. . . ,vn). Denote by δvthe Carleson constant of the finite sequence vand set µ2 n= inf 1≤j≤n 1−|uj|2 1−|φ(uj)|2· Then, for some constant c0>0, we have the lower bound: an(Cφ)≥c0δ4 vµn.(3.19) For the details of the proof, we refer to [20, Proof of Theorem 3.2, pages 556-557].