Trace estimates of Toeplitz operators on Bergman spaces and applications to composition operators
aa r X i v : . [ m a t h . C A ] J a n TRACE ESTIMATES OF TOEPLITZ OPERATORS ON BERGMANSPACES AND APPLICATIONS TO COMPOSITION OPERATORS
O. EL-FALLAH AND M. EL IBBAOUI
Abstract.
Let Ω be a subdomain of C and let µ be a positive Borel measure on Ω. In thispaper, we study the asymptotic behavior of the eigenvalues of compact Toeplitz operator T µ acting on Bergman spaces on Ω. Let ( λ n ( T µ )) be the decreasing sequence of the eigenvaluesof T µ and let ρ be an increasing function such that ρ ( n ) /n A is decreasing for some A >
0. Wegive an explicit necessary and sufficient geometric condition on µ in order to have λ n ( T µ ) ≍ /ρ ( n ). As applications, we consider composition operators C ϕ , acting on some standardanalytic spaces on the unit disc D . First, we give a general criterion ensuring that thesingular values of C ϕ satisfy s n ( C ϕ ) ≍ /ρ ( n ). Next, we focus our attention on compositionoperators with univalent symbols, where we express our general criterion in terms of theharmonic measure of ϕ ( D ). We finally study the case where ∂ϕ ( D ) meets the unit circlein one point and give several concrete examples. Our method is based on upper and lowerestimates of the trace of h ( T µ ), where h is suitable concave or convex functions. Introduction
Spectral properties of Toeplitz operators associated with positive measures play an impor-tant role in spectral theory of several operators: Hankel operators, composition operators andintegration operators. In this paper, we are interested in the behavior of the eigenvalues ofcompact Toeplitz operators acting on analytic spaces on a subdomain Ω of C with applica-tions to composition operators.Let Ω be a domain of C . We denote by H (Ω) the class of all holomorphic functions on Ω.Let ω : Ω → (0 , ∞ ) be a continuous weight on Ω. The weighted Bergman space associatedwith ω is given by A ω = { f ∈ H (Ω) : k f k ω = (cid:18)Z Ω | f ( z ) | dA ω ( z ) (cid:19) / < ∞} , where dA ω ( z ) = ω ( z ) dA ( z ) and dA is the Lebesgue measure on C .Clearly, A ω is a reproducing kernel space. The reproducing kernel of A ω will be denoted by K ( or K ω if necessary).In this paper, we call the standard Bergman spaces, denoted by A α , the Bergman spaces on D associated with ω ( z ) := α +1 π (1 − | z | ) α , where α > −
1. The standard Fock spaces F α corresponds to Ω = C and ω ( z ) = απ e − α | z | , where α > T µ , acting on A ω , induced by a positive Borel measure µ on Ω is given Mathematics Subject Classification.
Key words and phrases.
Bergman spaces, Fock spaces, Hardy space, Toeplitz operators, Composition oper-ators, Univalent functions, Harmonic measures.Research partially supported by ”Hassan II Academy of Sciences and Technology” for the first author. by T µ ( f )( z ) = Z Ω f ( ζ ) K ( z, ζ ) ω ( ζ ) dµ ( ζ ) . The boundedness, compactness and membership to Schatten classes of Toeplitz operatorswere studied in several papers (see for instance [14, 25, 24, 23, 37, 3, 31, 10]). It is proved,under some regularity conditions on ω , that T µ is bounded (resp. compact) if and only if µ ( R n ) /A ( R n ) = O (1) (resp. o (1)), where ( R n ) is a suitable lattice of Ω with respect to ω .Our goal in this paper is to study the asymptotic behavior of the eigenvalues of compactToeplitz operators on A ω . First, we fix some notations. The class of weights on Ω consideredin this paper, denoted by W (Ω), contains all standard weights. Some examples are listedin section 2. For ω ∈ W (Ω), we associate L ω , which consists of suitable lattices of Ω, withrespect to ω , the definitions of W and L ω are given in section 2.Throughout this paper we suppose that T µ is compact. The decreasing sequence of theeigenvalues of T µ will be denoted by ( λ n ( T µ )). It is proved in [10], that λ n ( T µ ) = O (1 / log γ ( n ))for some γ > c > X n exp − c (cid:18) A ( R n ) µ ( R n ) (cid:19) /γ ! < ∞ , for some ( R n ) n ∈ L ω .In this paper we are interested in compact Toeplitz operators, T µ , such that 1 /λ n ( T µ ) = O ( n A )for some A > T µ is compact, we have lim n → + ∞ µ ( R n ) /A ( R n ) = 0 . Let ( R n ( µ )) be an enu-meration of ( R n ) such that the sequence a n ( µ ) := µ ( R n ( µ )) /A ( R n ( µ )) , is decreasing. First,we will prove the following result. Theorem A.
Let ( R n ) ∈ L ω , where ω ∈ W . Let ρ : [1 , + ∞ ) → (0 , + ∞ ) be an increasingfunction such that ρ ( x ) /x A is decreasing for some A > . Let µ be a positive Borel measureon Ω such that T µ defines a compact operator on A ω . Then(1) λ n ( T µ ) = O (1 /ρ ( n )) ⇐⇒ a n ( µ ) = O (1 /ρ ( n )) .(2) λ n ( T µ ) ≍ /ρ ( n ) ⇐⇒ a n ( µ ) ≍ /ρ ( n ) . A preliminary version of this theorem, in the case of standard Bergman spaces of the unitdisc, was announced in [6]. Before going on, two remarks on Theorem A are in order.(i) The growth condition on ρ is, in some sense, necessary. Indeed, let ρ be an increasingfunction such that ρ ( x ) = o ( ρ (2 x )) when x → + ∞ . One can construct (see Subsection4.4) a Toeplitz operator T µ such that for any lattice ( R n ) n we havelim sup n →∞ λ n ( T µ ) a n ( µ ) = + ∞ . where a n ( µ ) is the decreasing rearrangement of ( µ ( R n ) /A ( R n )). RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 3 (ii) In general, the sequence ( a n ( µ )) n is not sufficient to give asymptotic estimates of( λ n ( T µ )) n . Indeed, one can construct two positive Borel measures µ and ν on the unitdisc D such that a n ( µ ) = a n ( ν ) and lim sup n →∞ λ n ( T µ ) /λ n ( T ν ) = ∞ . Next, we analyze the connection between the behavior of the eigenvalues of T µ and thebehavior of the Berezin transform of T µ . Recall that the Berezin transform of a Toeplitzoperator T µ acting on A ω is given by˜ µ ( z ) = h T µ K z , K z ik K z k , ( z ∈ Ω) . Let ( R n ) n ≥ ∈ L ω and let z n be the center of R n . It is known that T µ is compact if and onlyif lim n →∞ ˜ µ ( z n ) = 0 . As before, let ( z n ( µ )) be an enumeration of ( z n ) such that the sequence ( b n ( µ )) n , defined by b n ( µ ) := ˜ µ ( z n ( µ )) , is decreasing.First, we consider Toeplitz operators T µ such that 1 /λ n ( T µ ) = O ( n γ ) for some γ ∈ (0 , Theorem B.
Let ω ∈ W and let µ be a positive Borel measure on Ω such that T µ iscompact. Let ρ : [1 , + ∞ ) → (0 , + ∞ ) be an increasing positive function such that ρ ( x ) /x γ isdecreasing for some γ ∈ (0 , . We have λ n ( T µ ) ≍ /ρ ( n ) ⇐⇒ b n ( µ ) ≍ /ρ ( n ) . The case λ n ( T µ ) . /n A for some A >
1, is rather different. Indeed, to have a descriptionof the behavior of the eigenvalues of such Toeplitz operators in terms of ( b n ( µ )) it is necessarythat C p ( A ω , ( R n )) := sup n ≥ X j ≥ ˜ ν pn ( z j ) < ∞ , ( p ∈ (0 , , (1)where dν n = dA | R n (see Theorem 5.3).We will prove the following converse. Theorem C.
Let ω ∈ W , ( R n ) n ∈ L ω . Let µ be a positive Borel measure on Ω such that T µ is compact. Suppose that C p ( A ω , ( R n ) n ) < ∞ for all p ∈ (0 , . Let ρ : [1 , + ∞ ) → (0 , + ∞ ) be an increasing positive function satisfying ρ ( t ) /t γ is increasing for some γ > and ρ ( t ) /t β is decreasing for some large β . Then we have λ n ( T µ ) ≍ /ρ ( n ) ⇐⇒ b n ( µ ) ≍ /ρ ( n ) . The proofs of these theorems are based on upper and lower estimates of the trace of h ( T µ )for convex and concave functions h . O. EL-FALLAH AND M. EL IBBAOUI
As application, we consider composition operators on H α = { f ∈ H ( D ) : f ′ ∈ A α } , whichwas the original motivation of this work. Let ϕ be an analytic self map of D . The compositionoperator on H α induced by a symbol ϕ is defined by C ϕ ( f ) = f ◦ ϕ, ( f ∈ H α ) . Using Theorem A and a standard connection between composition operators and Toeplitzoperators, we give estimates of the singular values s n ( C ϕ , H α ) of general composition oper-ators C ϕ , when 1 /s n ( C ϕ , H α ) doesn’t increase faster than all polynomials. These estimatesare given in terms of the mean values of generalized counting function associated with ϕ .We also express these estimates in terms of the harmonic measure of ϕ ( D ), when ϕ is univa-lent and ϕ ( D ) is a Jordan domain.Next, we consider composition operators induced by univalent symbol ϕ such that ∂ϕ ( D ) ∩ ∂ D is reduced to one point. Namely, We suppose that ∂ϕ ( D ) has, in a neighborhood of +1, apolar equation 1 − r = γ ( | θ | ), where γ :]0 , π ] → ]0 ,
1] is a differential increasing function with γ (0) = 0, and satisfying the following conditions γ ( t ) t is increasing , γ ′ ( t ) = O ( γ ( t ) /t ) ( t → + ) , (2)and γ ( t ) = O (cid:16) t/ log β (1 /t ) (cid:17) for some β > / . (3)Recall that by Tsuji-Warschwski’s theorem, (see [33]), C ϕ is compact if and only if Z γ ( s ) s ds = ∞ . It is proved in [11] that the composition operator C ϕ on H α is in p -Schatten class ( p > Z e − pα Γ( t ) γ ( t ) dt < ∞ , (4)where Γ( t ) = 2 π Z t γ ( s ) s ds. We have the following result.
Theorem D.
Let α > and let Ω , γ, ϕ as before. Suppose that Z γ ( t ) t dt = ∞ . We have (1) If lim t → + γ ( t ) log(1 /t ) t = ∞ , then s n ( C ϕ , H α ) = O (1 /n A ) for all A > . (2) If γ ( t ) log(1 /t ) t = O (1) , then s n ( C ϕ , H α ) ≍ exp (cid:16) − α x n ) (cid:17) , where x n is given by Z x n dtγ ( t ) = n . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 5
As examples, we obtain
Corollary 1.1.
With the same notations as above, we have(1) If γ ( t ) = κt/ log( e/t ) with κ > , then s n ( C ϕ , H α ) ≍ n ακ/ π . (2) If γ ( t ) = κt/ log( e/t ) log log( e /t ) with κ > , then s n ( C ϕ , H α ) ≍ n ) ακ/π . The article is organized as follows: In section 2, we recall some classical results on compactoperators and introduce the weighted Bergman spaces considered throughout this paper. InSection 3, we show how to obtain estimates of the eigenvalues of a compact operator fromtrace estimates. Section 4 is devoted to proving the estimates of tr ( h ( T µ )) where h satisfiessome concave/convex conditions. It is important to note that the proof presented in thispaper, in particular in the concave case, is different from luecking’s proof [24] and does notrequire off-diagonal kernel estimates. This section contains the proof of Theorem A. In Sec-tion 5, we study the behavior of the eigenvalues of T µ in terms of its Berezin transform. Insection 6, we consider composition operators C ϕ with general symbol ϕ and give estimates ofthe singular values of C ϕ in terms of the generalized Nevanlinna function associated with ϕ .Section 7, is devoted to composition operators with univalent symbols. We express the as-ymptotic behavior of the singular values of C ϕ in terms of the harmonic measure of ϕ ( D ) andwe give explicit examples. In the last section we consider examples of composition operatorsacting on the Hardy space and on the classical Dirichlet space. Notations:
Throughout this paper, we will use the following notations • x . y if there exists a constant C > x ≤ Cy . • x ≍ y if x . y and y . x . • C ( x , .., x n ) is a constant which depends on x , .., x n .2. Preliminaries
Compact operators.
Let H be a complex Hilbert space and let T be a bounded oper-ator on H . The class of compact operators on H will be denoted S ∞ or ( S ∞ ( H ) if necessary).Let T ∈ S ∞ . The sequence ( s n ( T )) n ≥ (or ( s n ( T, H )) n ) denotes the non increasing sequenceof eigenvalues of ( T ∗ T ) / . If T is positive ( s n ( T )) n ≥ is the sequence of eigenvalues of T andwe write in this case s n ( T ) = λ n ( T ).By the spectral decomposition of compact operators, every compact operator T on H canbe written as follows T f = X n s n ( T ) h f, f n i g n , ( f ∈ H ) , where ( f n ) and ( g n ) n ≥ are orthonormal systems of H .So, it is easy to see that s n ( T ) = inf {k T − R k , dim R ( H ) < n } . O. EL-FALLAH AND M. EL IBBAOUI
In particular, if T and S are two compact operators such that T = XS , where X is acontraction, then s n ( T ) ≤ s n ( S ) , (for all n ≥ . Recall that a compact operator T on H belongs to the p − Schatten class S p (for p >
0) if k T k p := X n ≥ s n ( T ) p /p < ∞ . The following result is known as the monotonicity Weyl’s Lemma.
Lemma 2.1.
Let
T, S be two positive bounded operators on a complex Hilbert space H suchthat T ≤ S . If S is compact, then T is compact and λ n ( T ) ≤ λ n ( S ) for all n ≥ . We will also need the following general result [29].
Lemma 2.2.
Let ( T n ) n ≥ be a sequence of positive compact operators on a Hilbert space H and let T = X n ≥ T n (with norm-operator convergence). Let h : [0 , + ∞ ) → [0 , + ∞ ) be anincreasing function such that h (0) = 0 . Then (1) If h is convex, then Tr ( h ( T )) ≥ X n Tr ( h ( T n )) . (2) If h is concave, then Tr ( h ( T )) ≤ X n Tr ( h ( T n )) . The following classical result will be used in section 4.
Lemma 2.3.
Let p ≥ and let ( a n ) n ≥ , ( b n ) n ≥ be two positive decreasing sequences. Supposethat n X k =1 a /pk ≤ n X k =1 b /pk , ( for all n ≥ . Then, for every increasing positive function h such that h ( t p ) is convex, we have n X k =1 h ( a n ) ≤ n X k =1 h ( b n ) . (5) Proof.
This is a direct consequence of Corollary 3.3 of Chapter IV in [13]. (cid:3)
Weighted Bergman spaces.
In this subsection we recall briefly the definition of theclass of weights W introduced in [10] . Let Ω be a domain (bounded or not) of C and let ∂ Ωdenotes the boundary of Ω. Let ∂ ∞ Ω = ∂ Ω if Ω is bounded and ∂ ∞ Ω = ∂ Ω ∪ {∞} if Ω is notbounded. Let ω be a positive continuous weight on Ω. In what follows, we suppose that thereproducing kernel K of A ω satisfies the following two conditionslim z → ∂ ∞ Ω k K z k = ∞ . (6)And for every ζ ∈ Ω | K ( ζ, z ) | = o ( k K z k ) ( z → ∂ ∞ Ω) . (7)Let τ ( z )(= τ ω ( z )) := 1 ω ( z ) k K z k , ( z ∈ Ω) . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 7
We say that ω ∈ W (or W (Ω)) if there exists constants a, C > z, ζ ∈ Ωsatisfying | z − ζ | ≤ aτ ( z ), we have k K z kk K ζ k ≤ C | K ( ζ, z ) | , C τ ( ζ ) ≤ τ ( z ) ≤ Cτ ( ζ ) , (8)and τ ( z ) = O (min(1 , dist ( z, ∂ ∞ Ω))) . (9)Now, we give some examples • Standard Bergman spaces on the unit disc D . Let α > − A α := (cid:26) f ∈ H ( D ) : k f k α = Z D | f ( z ) | dA α ( z ) < ∞ (cid:27) , . The reproducing kernel is given by K αz ( w ) = 1(1 − zw ) α and τ α ( z ) := τ ( z ) = (1 + α )(1 − | z | ) . • Weighted Bergman spaces on D . Let D be the class of Oleinik-Perel’man weights on D (see [25, 3, 10]). It is easy to see from [23, 3], that if ω ∈ D , then ω ∈ W , k K ωz k ≍ ω − ( z )∆(log(1 /ω ( z )) and τ ω ( z ) ≍ /ω ( z ))) . For more general situation see [16]. • Standard Fock spaces. Let α > F α := F α ( C ) = { f ∈ H ( C ) : k f k = απ Z C | f ( z ) | e − α | z | dA ( z ) < ∞} . (10)Then the reproducing kernel is given by K ( z, w ) = e αz ¯ w and τ ( z ) ≍ • Weighted Fock spaces. In this case A ω will be denoted by F ω .Let ω ( z ) = e − Ψ( | z | ) be a positive weight on C . We say that ω ∈ R if Ψ : [0 , + ∞ ) → (0 , + ∞ ) ∈ C and satisfies the following conditionsΨ ′ > , Ψ ′′ ≥ , Ψ ′′′ ≥ , (11)and Φ ′′ ( x ) = O ( x − / (cid:0) Φ ′ ( x ) (cid:1) η ) for some η < / , (12)where Φ( x ) = x Ψ ′ ( x ). This class of spaces was considered by K. Seip and E. H.Youssfi in [31]. One can see that since polynomials are dense in F ω , then conditions(6) and (7) are satisfied. It is proved in [31] that τ ω ( z ) − =: K ( z, z ) ω ( z ) ≍ Φ ′ ( | z | ) . Using Lemma 3.2 of [31], it is not hard to prove that ω ∈ W .It is proved in [10], that if ω ∈ W then there exist B ω > , δ ω ∈ (0 , a/ B ω ) such that for all δ ∈ (0 , δ ω ) there exists ( z n ) ∈ Ω such that • Ω = ∪ n ≥ D ( z n , δτ ω ( z n )) = ∪ n ≥ D ( z n , B ω δτ ω ( z n )). O. EL-FALLAH AND M. EL IBBAOUI • D ( z n , δB ω τ ω ( z n )) ∩ D ( z m , δB ω τ ω ( z m )) = ∅ , for n = m . • z ∈ D ( z n , δτ ω ( z n )) implies that D ( z, δτ ω ( z )) ⊂ D ( z n , B ω δτ ω ( z n )). • There exists an integer N such that every D ( z n , B ω δτ ω ( z n )) cuts at most N sets ofthe family ( D ( z m , B ω δτ ω ( z m ))) m ). We say that ( D ( z n , B ω δτ ω ( z n ))) n is of finite mul-tiplicity.We say that ( R n ) n ∈ L ω if ( R n ) n = ( D ( z n , δτ ω ( z n ))) n and satisfies the above conditions.In the following, we consider ω ∈ W and suppose that T µ is compact. That is µ ( R n ) /A ( R n ) = o (1). As mentioned before ( R n ( µ )) will denote an enumeration of ( R n ) n such that a n ( µ ) := µ ( R n ( µ )) /A ( R n ( µ )) , is decreasing. Lemma 2.4.
Let ω ∈ W and let ( R n ) ∈ L ω . Let µ be a positive Borel measure on Ω suchthat T µ is compact on A ω . Denote by b = B ω and by µ n the restriction of µ to ∪ k ≤ n R k ( µ ) .Let ν n , ν be the following measures dν n = n X j =1 a j ( µ ) dA | bR j ( µ ) and dν = X j ≥ a j ( µ ) dA | bR j ( µ ) . Then, there exists a constant C = C ( ω, ( R n )) > such that(1) T µ n ≤ CT ν n and T µ ≤ CT ν .(2) k T µ k ≤ Ca ( µ ) .(3) λ k ( T µ n ) ≤ Cλ k ( T ν n ) , ( k ≥ ).(4) λ k ( T µ n ) ≤ λ k ( T µ ) ≤ λ k ( T µ n ) + Ca n +1 ( µ ) , ( k ≥ ).Proof. By the subharmonicity inequality applied to the function z → | f ( z ) /K ( z, ζ ) | , thereexists a constant C >
0, which depends on ω and ( R n ) n , such that | f ( ζ ) | ω ( ζ ) ≤ C A ( R n ) Z bR n | f ( z ) | ω ( z ) dA ( z ) , ( ζ ∈ R n ) . (13)It gives that Z R n | f ( ζ ) | ω ( ζ ) dµ ( ζ ) ≤ C a n ( µ ) Z bR n | f ( z ) | ω ( z ) dA ( z ) . (14) RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 9
This implies h T µ n f, f i = Z Ω | f ( ζ ) | ω ( ζ ) dµ n ( ζ ) ≤ n X j =1 Z R j ( µ ) | f ( ζ ) | ω ( ζ ) dµ ( ζ ) ≤ C n X j =1 a j ( µ ) Z bR j ( µ ) | f ( z ) | ω ( z ) dA ( z )= C Z Ω | f ( z ) | ω ( z ) dν n ( z )= h C T ν n f, f i . This means that T µ n ≤ C T ν n , which proves the part (1) of the lemma.Let N be the multiplicity of ( D ( z n , B ω δτ ω ( z n ))) n . From part (1) we have0 ≤ T µ ≤ C T ν ≤ N C a ( µ ) Id A ω . Then k T µ k ≤ N C a ( µ ).Clearly, part (3) is a consequence of part 1 and Lemma 2.1.Since µ n ≤ µ , we have T µ n ≤ T µ . Then by Lemma 2.1, we get λ k ( T µ n ) ≤ λ k ( T µ ). For thesecond inequality note that λ k ( T µ ) ≤ λ k ( T µ n ) + k T µ − T µ n k . Using the part (2), applied to µ − µ n , we obtain k T µ − T µ n k = k T µ − µ n k ≤ Ca ( µ − µ n ) ≤ Ca n +1 ( µ ) . Combining the two last inequalities, we obtain λ k ( T µ ) ≤ λ k ( T µ n ) + Ca n +1 ( µ ) . (cid:3) A general argument
Let β > δ >
0. The function h β,δ defined on [0 , ∞ ) is given by h β,δ ( t ) = ( t β − δ ) + := max( t β − δ, . The functions h β,δ , will play an important role in our study. First, note that h β,δ is convexfor β ≥ β ∈ (0 , h β,δ ( t p ) and h pβ,δ are convex if and only if p ≥ /β .The following two lemmas will be used in the sequel to obtain estimates of eigenvalues ofpositive compact operator T from upper and lower estimates of the trace of h ( T ) for somesuitable functions h . Lemma 3.1.
Let β ∈ (0 , and let ( a n ) n ≥ be a decreasing sequence. Let ρ : [1 , + ∞ ) → (0 , + ∞ ) be an increasing positive function such that ρ ( x ) /x γ is decreasing for some γ ∈ (0 , /β ) . Suppose that there exists B > such that for every δ ∈ (0 , , we have X n ≥ h β,δ (cid:18) Bρ ( n ) (cid:19) ≤ X n ≥ h β,δ ( a n ) ≤ X n ≥ h β,δ (cid:18) Bρ ( n ) (cid:19) . (15) Then a n ≍ /ρ ( n ) . Proof.
Without loss of generality, we suppose that ρ is strictly increasing and β = 1. Let δ > h δ ( t ) = ( t − δ ) + . By (15) we have X a n ≥ δ a n ≤ X a n ≥ δ ( a n − δ ) ≤ X n h δ ( a n ) ≤ X n h δ ( B/ρ ( n )) ≤ X ρ ( n ) ≤ B/δ Bρ ( n ) . and X ρ ( n ) ≤ Bδ Bρ ( n ) ≤ X n ( 1 Bρ ( n ) − δ ) + ≤ X n ( a n − δ ) + ≤ X a n ≥ δ a n . These inequalities can be written as follows X ρ ( n ) ≤ / Bδ ρ ( n ) . X a n ≥ δ a n . X ρ ( n ) ≤ B/δ ρ ( n ) . (16)We have X ρ ( n ) ≤ x ρ ( n ) ≍ ρ − ( x ) x Indeed, obviously we have ρ − ( x ) x . X ρ ( n ) ≤ x ρ ( n ) . Conversely, using the fact that ρ ( x ) /x γ is decreasing, we have X ρ ( n ) ≤ x ρ ( n ) = X ρ ( n ) ≤ x n γ ρ ( n ) 1 n γ . (cid:0) ρ − ( x ) (cid:1) γ x X ρ ( n ) ≤ x n γ ≍ ρ − ( x ) x . Then X a n ≥ δ a n . δB ρ − (cid:18) Bδ (cid:19) . Let N ( δ ) := Card { n : a n ≥ δ } . Since δN ( δ ) ≤ P a n ≥ δ a n , we obtain N ( δ ) . B ρ − (cid:18) Bδ (cid:19) (17)Let A >
1. Since x /γ /ρ − ( x ) is decreasing, we have ρ − ( x/A ) ≤ A − γ ρ − ( x ). Then X a n ≥ Aδ a n . AδB ρ − (cid:18) BAδ (cid:19) . (cid:18) A (cid:19) − γγ δB ρ − (cid:18) Bδ (cid:19) Then, for sufficiently large A , we have X a n ≥ δ a n ≍ X δ ≤ a n ≤ Aδ a n . Using the left inequality of (16), we get δρ − (1 / Bδ ) . X ρ ( n ) ≤ / Bδ ρ ( n ) . X δ ≤ a n ≤ Aδ a n . AδN ( δ ) . (18)Combining (17) and (18) we obtain a n ≍ /ρ ( n ). (cid:3) The following lemma will be used in section 5.
RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 11
Lemma 3.2.
Let ( a n ) n ≥ be a positive decreasing sequence. Let ρ : [1 , + ∞ ) → (0 , + ∞ ) bean increasing positive function. Suppose that there exist β > and γ > such that ρ ( t ) /t γ isincreasing and ρ ( t ) /t β is decreasing.Let p ∈ (0 , /β ) and suppose that there exists B > such that for every increasing concavefunction satisfying h ( t ) /t p is increasing, we have B X n ≥ h (cid:18) ρ ( n ) (cid:19) ≤ X n ≥ h ( a n ) ≤ B X n ≥ h (cid:18) ρ ( n ) (cid:19) . (19) Then a n ≍ /ρ ( n ) .Proof. Let δ > h be the concave function given by h ( t ) = t t ∈ (0 , δ ) δ − p t p t ≥ δ. Clearly h ( t ) /t p is increasing. Then (19) implies1 B X ρ ( n ) > /δ ρ ( n ) + δ − p X ρ ( n ) ≤ /δ ρ ( n ) p ≤ X a n >δ a n + δ − p X a n ≥ δ a pn ≤ B X ρ ( n ) > /δ ρ ( n ) + δ − p X ρ ( n ) ≤ /δ ρ ( n ) p (20)Let N ( δ ) = Card { n : a n ≥ δ } . It is clear that δN ( δ ) ≤ δ − p X a n ≥ δ a pn . Then δN ( δ ) . X ρ ( n ) > /δ ρ ( n ) + δ − p X ρ ( n ) ≤ /δ ρ ( n ) p . Using the fact that ρ ( n ) /n γ is increasing and ρ ( n ) p /n pβ is decreasing with γ > pβ ∈ (0 , X ρ ( n ) ≥ /δ ρ ( n ) ≍ δρ − (1 /δ ) and δ − p X ρ ( n ) ≤ /δ ρ ( n ) p ≍ δρ − (1 /δ ) . Then δN ( δ ) . δρ − (1 /δ ), which implies that a n . /ρ ( n ).For the reverse inequality we repeat the argument used in the proof of Lemma 3.1. Indeed,one can verify that X a n <δ/K a n + δ − p X a n ≥ Kδ a pn ≤ C ( K ) δρ − (1 /δ ) , with lim K →∞ C ( K ) = 0 . So, from (20) we obtain δ − p /B X ρ ( n ) ≤ /δ ρ ( n ) p ≤ X a n <δ a n + δ − p X a n ≥ δ a pn ≤ X a n <δ/K a n + δ − p X a n ≥ Kδ a pn + X δ/K ≤ a n <δ a n + δ − p X δ ≤ a n The following result is implicitly proved in [10]. Here we give a directand short proof. Theorem 4.1. Let ω ∈ W and let ( R n ) ∈ L ω . Let µ be a positive Borel measure on Ω suchthat T µ is compact on A ω . Let h be a convex increasing function such that h (0) = 0 . We have X n h (cid:18) B a n ( µ ) (cid:19) ≤ X n h ( λ n ( T µ )) ≤ X n h ( Ba n ( µ )) . Where B is a positive constant which depends on ω and ( R n ) .Proof. We will use here the same notations as in Lemma 2.4. Since lim n →∞ a n ( ν ) = 0, T ν is acompact operator. Let ( f n ) n ≥ be an orthonormal basis of eigenfunctions of T ν . We have X n ≥ h ( λ n ( T µ )) ≤ X n ≥ h ( Cλ n ( T ν ))= X n ≥ h ( C h T ν f n , f n i )= X n ≥ h (cid:18) C Z Ω | f n ( z ) | ω ( z ) dν ( z ) (cid:19) ≤ X n ≥ h X k N Ca k ( µ ) Z R k ( µ ) | f n ( z ) | dA ω ( z ) ! , where N is the multiplicity of ( bR n ) n .Since h is convex, by Jensen’s inequality, we get X n ≥ h ( λ n ( T µ )) ≤ X n,k h ( N Ca k ( µ )) 1 N Z R k ( µ ) | f n ( z ) | dA ω ( z )= X k h ( N Ca k ( µ )) 1 N Z R k ( µ ) X n | f n ( z ) | dA ω ( z ) . X k h ( N Ca k ( µ )) Z R k ( µ ) k K z k dA ω ( z ) . X k h ( N Ca k ( µ )) . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 13 Conversely, let ¯ µ j = µ | R j ( µ ) and put ¯ µ = X j ≥ ¯ µ j . We have T ¯ µ ≤ N T µ . So, by Lemma 2.4 andLemma 2.2, we have Tr( h ( T µ )) ≥ Tr( h ( N T ¯ µ )) ≥ X j ≥ Tr (cid:18) h ( 1 N T ¯ µ j ) (cid:19) ≥ X j ≥ h ( 1 N λ ( T ¯ µ j )) = X j ≥ h ( 1 N k T ¯ µ j k ) ≥ X j ≥ h (cid:18) N h T ¯ µ j K z j k K z j k , K z j k K z j k i (cid:19) where z j is the center of R j ( µ ).Now, since h T ¯ µ j K z j k K z j k , K z j k K z j k i = Z R j ( µ ) (cid:12)(cid:12)(cid:12)(cid:12) K z j ( ζ ) k K z j k (cid:12)(cid:12)(cid:12)(cid:12) ω ( z ) dµ ( z ) ≍ a j ( µ ) , we obtain the second inequality. This ends the proof. (cid:3) Concave case. Theorem A will be obtained from the following result. Theorem 4.2. Let ω ∈ W and let ( R n ) ∈ L ω . Let µ be a positive Borel measure on Ω suchthat T µ is compact on A ω . Let h be a concave increasing function such that h (0) = 0 . Wehave B X n ≥ h ( a n ( µ )) ≤ X n ≥ h ( λ n ( T µ )) ≤ B X n ≥ X k ≥ h (cid:16) a n ( µ ) e − γk (cid:17) . In addition, if h ( t ) /t p is increasing for some p ∈ (0 , , then X n ≥ h ( λ n ( T µ )) ≤ Bp X n ≥ h ( a n ( µ )) , where B, γ > are constants which depend only on ω and ( R n ) . We will need the following lemma in the proof of Theorem 4.2. Lemma 4.3. Let ω ∈ W and let ( R n ) ∈ L ω . Let ν n = dA | R n , then T ν n is compact on A ω and λ k ( T ν n ) ≤ Be − γk , where B, γ > depend on ω and ( R n ) .Proof. Following the same proof of Theorem 3.8 of [23], there exist B > δ ∈ (0 , 1) suchthat k T ν n k pp ≤ B (1 − δ p ) − ≤ C ( δ, B ) p , ( p ∈ (0 , / . It implies that kλ pk ( T ν n ) ≤ Cp , where C = C ( δ, B ). Then, for p = keC , we obtain the desiredresult, with γ = eC . (cid:3) Proof of Theorem 4.2. We use the same notations as in Lemma 2.4. Let ˜ µ be the positiveBorel measure given by d ˜ µ = X k N Ka k ( µ ) µ | R k ( µ ) , where K > NKa k ( µ ) µ | R k ( µ ) = 0 if a k ( µ ) = 0. By Lemma 2.4, T ˜ µ is bounded and k T ˜ µ k ≤ C sup n ˜ µ ( R n ) A ( R n ) . We have ˜ µ ( R n ) A ( R n ) = 1 N K X k : R k ∩ R n = ∅ µ ( R n ∩ R k ) a k ( µ ) A ( R n ) ≤ N K X k : R k ∩ R n = ∅ µ ( R k ) a k ( µ ) A ( R n ) ≍ N K X k : R k ∩ R n = ∅ µ ( R k ) a k ( µ ) A ( R k ) . K . Then for a large K we have k T ˜ µ k ≤ f n ) n ≥ be an orthonormal basis of A ω of eigenfunctions of T µ . We have X n ≥ h ( λ n ( T µ )) = X n ≥ h ( h T µ f n , f n i )= X n ≥ h (cid:18)Z Ω | f n ( z ) | ω ( z ) dµ ( z ) (cid:19) ≥ X n ≥ h N X k Z R k ( µ ) | f n ( z ) | ω ( z ) dµ ( z ) ! = X n ≥ h X k CKa k ( µ ) Z R k ( µ ) | f n ( z ) | ω ( z ) d ˜ µ ( z ) ! ≥ X n ≥ X k h ( CKa k ( µ )) Z R k ( µ ) | f n ( z ) | ω ( z ) d ˜ µ ( z )= X k h ( CKa k ( µ )) X n ≥ Z R k ( µ ) | f n ( z ) | ω ( z ) d ˜ µ ( z )= X k h ( CKa k ( µ )) Z R k ( µ ) k K z k ω ( z ) d ˜ µ ( z ) ≍ X k h ( a k ( µ )) . Which gives the first inequality. RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 15 Let ¯ µ j = µ | R j ( µ ) . Since µ ≤ ∞ X j =1 ¯ µ j , by Lemma 2.4 and Lemma 2.2 we haveTr( h ( T µ )) ≤ Tr( h ( ∞ X j =1 T ¯ µ j )) ≤ ∞ X j =1 Tr( h ( T ¯ µ j )) . By Lemma 2.4, T ¯ µ j ≤ CT ν j , where dν j = a j ( µ ) dA | bR j ( µ ) . Then by Lemma 14, we haveTr( h ( T ¯ µ j )) ≤ Tr (cid:0) h ( CT ν j ) (cid:1) . ∞ X k =1 h (cid:16) Ca j ( µ ) e − γk (cid:17) . ∞ X k =1 h (cid:16) a j ( µ ) e − γk (cid:17) Then we obtain the second inequality of Theorem 4.2.Now we prove the last inequality of Theorem 4.2 . Since h ( t ) /t p is increasing, we have h ( a j ( µ ) e − γk ) ≤ h ( a j ( µ )) e − pγk . It implies that Tr( h ( T µ )) . ∞ X j =1 ∞ X k =1 h (cid:16) a j µ ) e − γk (cid:17) . ∞ X j =1 ∞ X k =1 h ( a j ( µ )) e − pγk ≍ p ∞ X j =1 h ( a j ( µ )) . And the proof is complete. (cid:3) Remarks. (1) It is proved by P. Lin and R. Rochberg in [23] that if ω ∈ D then T µ ∈ S p if and onlyif ( a n ( µ )) n ∈ ℓ p for all p ≥ 1. They also proved, for p ∈ (0 , a n ( µ )) n ∈ ℓ p then T µ ∈ S p . Since D ⊂ W it is clear from Theorem 4.2 that the converse is alsotrue. (see [2] for radial weights).(2) For ω ∈ R , the class of weights introduced by K. Seip and E. H. Youssfi in [31], Theo-rem 4.2 completes the characterization of membership to Schatten classes given in [31].(3) The factor 1 /p in Theorem 4.2 can not be replaced by 1 /p − ε . Indeed, let Ω = D andlet ω = 1. Suppose that X n ≥ λ pn ( T µ ) ≤ Bp − ε X n ≥ a pn ( µ ) , ( ∀ p > , for every positive Borel measure µ on D . Then if µ is of compact support, we have nλ pn ( T µ ) ≤ X j ≥ λ pj ( T µ ) ≤ Cp − ε , ∀ p ∈ (0 , , for some constant C > 0. This implies that λ n ( T µ ) ≤ e − Kn − ε . Now, for dµ = dA | D (0 ,δ ) where δ ∈ (0 , λ n ( T µ ) = 2 Z δ r n +1 dr ≍ n + 1 δ n +2 , which gives a contradiction.The following corollary is somewhat more general than Theorem A. Corollary 4.4. Let ω ∈ W and let ( R n ) ∈ L ω . Let µ be a positive Borel measure on Ω suchthat T µ is compact on A ω . For p ∈ (0 , , we have p ε C n X j =1 a pj ( µ ) ≤ n X j =1 λ pj ( T µ ) ≤ C p n X j =1 a pj ( µ ) , where C is a positive constant which depends on ε, ω, ( R n ) and C is a positive constantwhich depends on ω and ( R n ) .Proof. Applying Theorem 4.2 to T µ n and taking into account the multiplicity of ( R n ), thereexists B > ω and ( R n ) such that n X j =1 λ pj ( T µ n ) ≤ B p n X j =1 a pj ( µ ) . By Lemma 2.4, λ j ( T µ ) ≤ λ j ( T µ n ) + Ca n +1 ( µ ) . We obtain n X j =1 λ pj ( T µ ) ≤ B p n X j =1 ( a pj ( µ )) + C p a pn +1 ( µ )) ≤ B (1 + C ) p n X j =1 a pj ( µ ) . Conversely, let q ∈ (0 , T µ n , we have ∞ X j =1 λ pqj ( T µ n ) ≤ B pq n X j =1 a pqj ( µ ) . Then λ pj ( T µ n ) ≤ B jpq n X k =1 a pqk ( µ ) ! /q ≤ (cid:18) B q (cid:19) /q n q − p /q j /q n X k =1 a pk ( µ )Then, for A > X j ≥ An +1 λ pj ( T µ n )) ≤ C ( B , q ) p /q A q − n X k =1 a pk ( µ ) RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 17 Once time again, by Theorem 4.2, we have1 /B n X j =1 a pj ( µ ) ≤ ∞ X j =1 λ pj ( T µ n ) ≤ An X j =1 λ pj ( T µ n ) + ∞ X j = An +1 λ pj ( T µ n ) ≤ A n X j =1 λ pj ( T µ n ) + C ( B , q ) p /q A q − n X j =1 a pj ( µ ) ≤ A n X j =1 λ pj ( T µ ) + C ( B , q ) p /q A q − n X j =1 a pj ( µ )For q = ε and for A big enough we obtain the result. (cid:3) Now we can state the following important consequence of Corollary 4.4. Theorem 4.5. Let ω ∈ W and let ( R n ) ∈ L ω . Let µ be a positive Borel measure on Ω suchthat T µ is compact on A ω . Let h be an increasing function on [0 , + ∞ ) such that h (0) = 0 and h ( t p ) is convex for some p > . We have n X j =1 h ( 1 B a j ( µ ))) ≤ n X j =1 h ( λ j ( T µ )) ≤ n X j =1 h ( Ba j ( µ )) , ( n ≥ , where B > is a positive constant which depends on ω , ( R n ) and p .Proof. This is a consequence of Lemma 2.3 and Corollary 4.4. (cid:3) Proof of Theorem A. We prove (1). Suppose that a n ( µ ) = O (1 /ρ ( n )). Let p ∈ (0 , 1) suchthat pA < 1. We have n X k =1 ρ p ( k ) ≍ n/ρ p ( n ) . By Corollary 4.4, since a n ( µ ) = O (1 /ρ ( n )), we obtain nλ pn ( T µ ) ≤ n X k =1 λ pk ( T µ ) . n X k =1 a pk ( µ ) . n/ρ p ( n ) . This implies that λ n ( T µ ) = O (1 /ρ ( n )). The reverse implication is obtained in the same way.The second assertion comes from Theorem 4.5 and Lemma 3.1. (cid:3) Remarks on Theorem A. In this section we provide two examples. The first one,shows that the condition ρ ( x ) /x A is decreasing for some A > 0, is necessary and sharp. Andin the second example we show that the sequence ( a n ( µ )) is not sufficient, in general, todescribe the asymptotic behavior of the eigenvalues of T µ .(1) The conclusion of Theorem A is not valid if ρ increases faster than all polynomials.Namely, suppose that lim x → + ∞ ρ (2 x ) ρ ( x ) = + ∞ . (21) Let dµ ( re it ) = 1 ρ (1 / (1 − r )) rdrdt. (22)The Toeplitz operator T µ defined on the unweighted Bergman space A ( D ) is compact.Since µ is radial, it is easy to see that f n = ( n + 1) / z n is an eigenfunction of T µ andfor all M > λ n ( T µ ) = 2 π Z r n +1 ρ (1 / (1 − r )) dr ≥ π Z − M/n r n +1 ρ (1 / (1 − r )) dr ≥ C ( M ) ρ ( n/M ) . Let p be an integer and let ( R n,j ( p )) denotes the p − adic decomposition of D , that is R n,j ( p ) = n z ∈ D ; 1 − p n ≤ | z | < − p n +1 and 2 jπp n +1 ≤ arg z < j + 1) πp n +1 o , ≤ j < p n +1 . Let ( R n ) n be a lattice of A ( D ). It is clear that for p big enough, then for all n thereexists ( k, j ) such that R k,j ( p ) ⊂ R n . Note also that we have A ( R k,j ( p )) ≍ A ( R n ).Then we obtain a n ( µ ) . a ′ n ( µ ) , n ≥ , where ( a n ( µ )) n (resp. a ′ n ( µ )) is the decreasing rearrangement of (cid:16) µ ( R n ) A ( R n ) (cid:17) n (resp. (cid:16) µ ( R n,j ( p )) A ( R n,j ( p )) (cid:17) n ). We have µ ( R n,j ( p )) A ( R n,j ( p )) . ρ ( p n ) . Then we have a n ( µ ) . a ′ n ( µ ) . ρ ( n/ p ) , n ≥ . Using (21), we obtain lim n →∞ λ ( T µ ) a n ( µ ) = ∞ . This proves our assertion.(2) Now, we construct two positive Borel measures µ and ν on D such that a n ( µ ) = a n ( ν ) and lim sup n →∞ λ n ( T µ ) /λ n ( T ν ) = ∞ . To this end, let µ be the measure given in (22) and let ( R n ) n be the dyadic ( p = 2)decomposition of D . Let ν be the measure given by ν = X n ≥ c n δ w n , where ( w n ) n is an interpolating separated sequence of A = A [30]. The sequence( w n ) n satisfies (cid:13)(cid:13)(cid:13)(cid:13)X c n K w n k K w n k (cid:13)(cid:13)(cid:13)(cid:13) ≍ X | c n | . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 19 Let R ( w n ) be the unique disc R k such that w n ∈ R k and put c n = a n ( µ ) A ( R ( w n )).Let ν n = X k ≥ n c k δ w k , we have λ n ( T ν ) ≤ k T ν n k . a n ( µ ) . This implies that lim inf n →∞ λ n ( T ν ) /λ n ( T µ ) = 0, while a n ( µ ) = a n ( ν ).5. The Berezin Transform Preliminaries. The Berezin transform of a bounded operator T acting on A ω is definedby ˜ T ( z ) = h T K z , K z ik K z k , ( z ∈ Ω) . If T is positive and compact then Tr( T ) = Z Ω ˜ T ( z ) dA ( z ) τ ( z ) . In particular T ∈ S if and only if ˜ T ∈ L (Ω , dAτ ).The following general result is standard and is well known (at least for h ( t ) = t p ) ([36]). Proposition 5.1. Let T be a positive compact operator on A ω . Let h be an increasingfunction such that h (0) = 0 . We have(1) If h is convex then X n h ( λ n ( T )) ≥ Z Ω h (cid:16) ˜ T ( z ) (cid:17) dA ( z ) τ ( z ) . (2) If h is concave then X n h ( λ n ( T )) ≤ Z Ω h (cid:16) ˜ T ( z ) (cid:17) dA ( z ) τ ( z ) . Proof. Let ( f n ) n ≥ be an orthonormal basis of A ω containing a maximal orthonormal systemof eigenfunctions of T . Set λ n = λ n ( T ) and write h T K z , K z i = X n λ n |h f n , K z i| = X n λ n | f n ( z ) | . If h is convex, then Z Ω h ( ˜ T ( z )) dA ( z ) τ ω ( z ) = Z Ω h X n λ n | f n ( z ) | k K z k ! dA ( z ) τ ω ( z ) ≤ Z Ω X n h ( λ n ) | f n ( z ) | k K z k dA ( z ) τ ω ( z )= X n h ( λ n ) Z Ω | f n ( z ) | ω ( z ) dA ( z )= X n h ( λ n ) . The concave case is obtained in the same way. (cid:3) Trace estimates and consequences. Let ( R n ) ∈ L ω . In the sequel z n will denote thecenter of R n . For the Toeplitz operator T µ acting on A ω , the Berezin transform of T µ will bedenoted by ˜ µ . In this section we use the following notationˆ µ ( z n ) = µ ( R n ) /A ( R n ) . Our goal in this section is to estimate the eigenvalues of T µ in terms of ˜ µ ( z n ). To this end, byLemma 3.1 and Lemma 3.2, it suffices to estimate Tr h ( T µ ) in terms of ( h (˜ µ ( z n ))) n ≥ . Forthe convex case we have the following result Theorem 5.2. Let ω ∈ W . Let µ be a positive Borel measure on Ω and h be a convexincreasing function such that h (0) = 0 . Then X n ≥ h ( 1 B ˜ µ ( z n )) ≤ Tr ( h ( T µ )) ≤ X n ≥ h ( B ˜ µ ( z n )) . Where B > doesn’t depend on either µ or h .Proof. By Theorem 4.1, we have X n ≥ h ( 1 B ˆ µ ( z n )) ≤ Tr ( h ( T µ )) ≤ X n ≥ h ( B ˆ µ ( z n )) . Since ˆ µ ( z n ) . ˜ µ ( z n ) ( [10]), we deduce that Tr ( h ( T µ )) ≤ X n ≥ h ( B ˜ µ ( z n )).On the other hand, by (13) we have | K ( z n , ζ ) | ω ( z n ) . A ( R n ) Z R n | K ( z, ζ ) | ω ( z ) dA ( z ) , ζ ∈ D . It implies that | K ( z n , ζ ) | k K z n k . Z R n | K ( z, ζ ) | k K z k dA ( z ) τ ω ( z ) , ζ ∈ D , and ˜ µ ( z n ) . Z R n ˜ µ ( z ) dA ( z ) τ ω ( z ) . Since h is convex, by Theorem 5.1 we obtain, for some c > X n h ( c ˜ µ ( z n )) . Z Ω h (˜ µ ( z )) dA ( z ) τ ω ( z ) ≤ Tr ( h ( T µ )) . The proof is complete. (cid:3) Proof of Theorem B. By Theorem 5.2 we have X n ≥ h ( 1 B b n ( µ )) ≤ X n ≥ h ( λ n ( T µ )) ≤ X n ≥ h ( Bb n ( µ )) . So, by Lemma 3.1 we have b n ( µ ) ≍ /ρ ( n ) if and only if λ n ( T µ ) ≍ /ρ ( n ) . (cid:3) Now, we turn to the concave case. Let p > dν n = dA | R n . Recall that C p ( A ω , ( R n )) = sup n ≥ X j ≥ ˜ ν pn ( z j ) ∈ [0 , + ∞ ) . For the concave case we have the following result RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 21 Theorem 5.3. Let ω ∈ W , ( R n ) n ∈ L ω and p ∈ (0 , . The following assertions are equiva-lent(1) C p ( A ω , ( R n ) n ) < ∞ .(2) There exists B > such that for every Borel positive measure µ on Ω and everyincreasing concave function h such that h ( t ) /t p is increasing, we have B X n ≥ h (˜ µ ( z n )) ≤ Tr ( h ( T µ )) ≤ B X n ≥ h (˜ µ ( z n )) . Proof. The same argument as before proves that we have Tr ( h ( T µ )) ≤ C X n ≥ h (˜ µ ( z n )).By Lemma 14, there exists B > T pν j ≤ B/p . So, it is obvious that thecondition C X n ≥ h (˜ µ ( z n )) ≤ Tr ( h ( T µ )), applied with µ = ν n and h ( t ) = t p , gives that C p ( A ω , ( R n ) n ) < ∞ .Conversely, suppose that C p ( A ω , ( R n ) n ) < ∞ . A standard computation gives˜ µ ( z ) . X j ≥ ˆ µ ( z j ) Z R j | K ( z, ζ ) | k K z k ω ( ζ ) dA ( ζ ) ! . . X j ≥ ˆ µ ( z j )˜ ν j ( z ) . Since h is concave and h ( t ) /t p is increasing, we have h (˜ µ ( z )) . X j ≥ h (ˆ µ ( z j ))˜ ν j ( z ) p . Consequently X n ≥ h (˜ µ ( z n )) . X j ≥ h (ˆ µ ( z j )) X n ≥ ˜ ν j ( z n ) p . . C p ( A ω , ( R n ) n ) X j ≥ h (ˆ µ ( z j )) . C p ( A ω , ( R n ) n )Tr( h ( T µ )) . The proof is complete. (cid:3) Theorem C is a direct consequence of the following result Theorem 5.4. Let ω ∈ W and ( R n ) n . Let µ be a positive Borel measure on Ω such that T µ is compact. Let p ∈ (0 , such that C p ( A ω , ( R n ) n )) < ∞ . Let ρ : [1 , + ∞ ) → (0 , + ∞ [ be anincreasing positive function. Suppose that there exist β ∈ (0 , /p ) and γ > such that ρ ( t ) /t γ is increasing and ρ ( t ) /t β is decreasing. Then λ n ( T µ ) ≍ /ρ ( n ) ⇐⇒ b n ( µ ) ≍ /ρ ( n ) . Proof. By Theorem 5.3 we have1 B X n ≥ h ( b n ( µ )) ≤ X n ≥ h ( λ n ( T µ ) ≤ B X n ≥ h ( b n ( µ )) . So, by lemma 3.2 we have b n ( µ ) ≍ /ρ ( n ) if and only if λ n ( T µ ) ≍ /ρ ( n ) . (cid:3) Examples. Now, we give some examples. • Standard Fock spaces. Let α > 0. Let F α be the standard Fock space given by(10). First, recall that the Berezin transform of T µ is given by˜ µ ( z ) = Z C e − α | z − ζ | dµ ( ζ ) ( z ∈ C ) . For more informations on Fock spaces see [38].We have C p ( F α , ( R n ) n ) < ∞ . for all p ∈ (0 , X j ≥ ˜ ν pn ( z j ) ≍ X n (cid:18)Z R n e − α | z j − ζ | dA ( ζ ) (cid:19) p ≍ X n (cid:18)Z R n e − α | ζ | dA ( ζ ) (cid:19) p = O ( p ) . • Weighted analytic spaces. Let Ω be a subdomain of C and let ω ∈ W . Let M > ω ∈ W M if the reproducing kernel of A ω satisfies | K ( z, ζ ) | ≤ C ( M ) k K z kk K ζ k (cid:18) min( τ ω ( z ) , τ ω ( ζ )) | z − ζ | (cid:19) M . (23)We will denote W ∞ = ∩ M> W M . Examples of such weights can be found in [3, 31, 16]. Proposition 5.5. Let M > and let ω ∈ W M , For every ( R n ) ∈ L ω , we have C p ( A ω , ( R n ) n )) < ∞ , ( for all p > /M ) . In particular, if ω ∈ W ∞ then C p ( A ω , ( R n ) n )) < ∞ , for all p > .Proof. Let p > /M and let z n be the center of R n . We have X j ≥ ˜ ν pn ( z j ) = X j ≥ Z R n | K z j ( ζ ) | k K z j k dA ω ( ζ ) ! p . Since ( B ω R j ) is of finite multiplicity, Λ n := { j : B ω R j ∩ B ω R n = ∅} is finite. Then X j ∈ Λ n Z R n | K z j ( ζ ) | k K z j k dA ω ( ζ ) ! p . X j ∈ Λ n (cid:18)Z R n k K ζ k dA ω ( ζ ) (cid:19) p . X j ∈ Λ n (cid:18)Z R n τ ω ( ζ ) dA ( ζ ) (cid:19) p = O (1) . On the other hand, since M p > 1, let a > M − a ) p = 1. We have RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 23 X j / ∈ Λ n Z R n | K z j ( ζ ) | k K z j k dA ω ( ζ ) ! p . X j / ∈ Λ n Z R n k K ζ k (cid:18) min( τ ω ( z j ) , τ ω ( ζ )) | z j − ζ | (cid:19) M dA ω ( ζ ) ! p ≍ X j / ∈ Λ n τ ω ( z j ) (2 M − a ) p τ apω ( z n ) | z j − z n | Mp . Z Ω \ R n τ ω ( ζ ) (2 M − a ) p − τ apω ( z n ) | ζ − z n | Mp dA ( ζ ) . Z Ω \ R n τ apω ( z n ) | ζ − z n | ap +2 dA ( ζ )= O (1) . Which implies that C p ( A ω , ( R n ) n ) < ∞ , whenever p > /M . (cid:3) • Standard Bergman spaces on D . Let α > − 1, let ω α ( z ) = (1 + α )(1 − | z | ) α andlet A α be the associated standard Bergman spaces. Recall that the kernel of A α isgiven by K αz ( w ) = 1(1 − zw ) α . We have the following proposition Proposition 5.6. Let ( R n ) n ∈ L ω α and let p ∈ (0 , . We have C p ( A α , ( R n ) n )) < ∞ if and only if p > α .Proof. Let ( R n ) ∈ L ω . We have˜ ν n ( z ) = Z R n (1 − | z | ) α | − ¯ zζ | α dA α ( ζ ) . Then X j ≥ ˜ ν pn ( z j ) = X j ≥ (cid:18)Z R n (1 − | z j | ) α | − ¯ z j ζ | α dA α ( ζ ) (cid:19) p ≍ X j ≥ (cid:18) (1 − | z j | ) α (1 − | z n | ) α | − ¯ z j z n | α (cid:19) p ≍ Z D (cid:18) (1 − | w | ) α (1 − | z n | ) α | − ¯ wz n | α (cid:19) p dA ( w )(1 − | w | ) ≍ Z D (1 − | z n | ) (2+ α ) p | − ¯ wz n | (4+2 α ) p dA ( w )(1 − | w | ) − (2+ α ) p . ¡¡ Then, the last integral is uniformly finite if and only if p > α ( [[5] Lemma 2 page32]. (cid:3) Proposition 5.6 implies that the Berezin transform is not sufficient to describe thebehavior of the eigenvalues of Toeplitz operators. In what follows, we consider a mod-ified Berezin transform which is more appropriate to our problem in this case (see forinstance [34] and [26]). Let T be a bounded operator on A α and let s > − 1. The modified Berezin trans-form, B α,s ( T ), of T is given by B α,s ( T )( z ) = h T K sz , K sz ik K sz k α Let τ ( z ) = (1 − | z | ). We have the following general result Proposition 5.7. Let α and let s such that s > α − . Let T be a positive compactoperator on A α . We have(1) Tr ( T ) ≍ Z D B α,s ( T )( z ) dA ( z ) τ ( z ) . (2) Let h be a concave function such that h (0) = 0 . ThenTr ( h ( T )) . Z D h ( B α,s ( T )( z )) dA ( z ) τ ( z ) . (3) Let h be a convex function such that h (0) = 0 . Then Z D h ( B α,s ( T )( z )) dA ( z ) τ ( z ) . Tr ( h ( T )) . All the implied constants depend on α and s .Proof. Let f = X n ≥ a n z n ∈ A α . Write K sz ( ζ ) = X n ≥ c n ( s ) z n ζ n . It is known that c n ( s ) ≍ (1 + n ) s . This implies that k K sre it k α = k K sr k α ≍ − r ) s − α . Then we have Z D |h f, K sz i| k K sz k α dA ( z ) τ ( z ) = Z (cid:18)Z π |h f, K sre it i| dt π (cid:19) rdr k K sr k α τ ( r ) ≍ Z X n ≥ | a n | r n c n ( s )(1 + n ) α (1 − r ) s − α rdr ≍ X n ≥ | a n | (1 + n ) α − s Z r n +1 (1 − r ) s − α dr ≍ k f k α . Let ( f n ) n ≥ be an orthonormal basis of A α containing a maximal orthonormal systemof eigenfunctions of T . Write h T K sz , K sz i = X n λ n |h f n K sz i| , ( λ n ( T ) = λ n ) . Then, Z D B α,s ( T )( z ) dA ( z ) τ ( z ) = X n λ n Z D |h f n , K sz i| k K sz k α dA ( z ) τ ( z ) ≍ X n λ n = Tr( T ) . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 25 To prove (2), Suppose that h is concave Z D h ( B α,s ( T )( z )) dA ( z ) τ ( z ) = Z D h X n λ n |h f n , K sz i| k K sz k α ! dA ( z ) τ ( z ) > ∼ Z D X n h ( λ n ) |h f n , K sz i| k K sz k α dA ( z ) τ ( z )= X n h ( λ n ) Z D |h f n , K sz i| k K sz k α dA ( z ) τ ( z ) . ≍ X n h ( λ n )The convex case is obtained in the same way. (cid:3) Lemma 5.8. Let α > − , and let ( R n ) n ∈ L ω α . Let h be a concave function suchthat h ( t ) /t p is increasing for some p ∈ (0 , . Let s > pα − p p and let µ be a positiveBorel measure on D , thenTr h ( T µ ) ≍ X n ≥ h ( B α,s ( T µ )( z n )) , where the implied constants depend on α, s, p and ( R n ) n .Proof. By Proposition 5.7 we haveTr( h ( T µ )) . Z D h ( B α,s ( T µ )( z )) dA ( z ) τ ( z ) ≍ X n ≥ h ( B α,s ( T µ )( z n )) . Conversely, by Theorem 4.2 we haveTr h ( T µ ) ≍ X n h ( a n ( µ )) . So, it suffices to verify that Z D h ( B α,s ( z )) dA ( z ) τ ( z ) . X n h ( a n ( µ )) . Let ν = X n a n ( µ ) dA α | bR n . By Lemma 2.4 T µ . T ν . Then h T µ K sz , K sz i . h T ν K sz , K sz i . X n a n ( µ ) Z R n | K sz ( w ) | dA α ( w ) . Using the concavity of h , we get Z D h ( B α,s ( T µ )( z )) dA ( z ) τ ( z ) = Z D h (cid:18) h T µ K sz , K sz ik K sz k α (cid:19) dA ( z ) τ ( z ) . Z D h X n a n ( µ ) Z R n | K sz ( w ) | k K sz k dA α ( w ) ! dA ( z ) τ ( z ) . Z D X n h (cid:18) a n ( µ ) Z R n | K sz ( w ) | k K sz k dA α ( w ) (cid:19) dA ( z ) τ ( z ) . On the other hand, we have Z R n | K sz ( ζ ) | k K sz k α dA α ( ζ ) ≍ Z R n (1 − | z | ) s − α | − zζ | s dA α ( ζ ) ≍ (1 − | z | ) s − α | − zz n | s (1 − | z n | ) α . Using the assumption h ( t ) /t p is increasing , we get h (cid:18) a n ( µ ) Z R n | K sz ( ζ ) | k K sz k α dA α ( w ) (cid:19) . h ( a n ( µ )) (cid:18)Z R n | K sz ( ζ ) | k K sz k α dA α ( ζ ) (cid:19) p . h ( a n ( µ )) (cid:18) (1 − | z | ) s − α (1 − | z n | ) α | − zz n | s (cid:19) p Combining all these inequalities, and using the fact that s > − p + αp p , we obtain Z D h ( B α,s ( T µ )( z )) dAz ) τ ( z ) . X n h ( a n ( µ )) Z D (cid:18) (1 − | z | ) s − α (1 − | z n | ) α | − zz n | s (cid:19) p dA ( z ) . X n h ( a n ( µ )) . The proof is complete. (cid:3) Let ( b α,sn ( µ )) n be the decreasing enumeration of ( B α,s ( T µ )( z n )) n ≥ . Theorem C is a directconsequence of the following result. Theorem 5.9. Let ω ∈ W and ( R n ) n ∈ L ω . Let µ be a positive Borel measure on D such that T µ is compact on A α . Let ρ : [1 , + ∞ ) → (0 , + ∞ [ be an increasing positive function. Supposethat there exist β > and γ > such that ρ ( t ) /t γ is increasing and ρ ( t ) /t β is decreasing.Then, for s > β + α − , we have λ n ( T µ ) ≍ /ρ ( n ) ⇐⇒ b α,sn ( µ ) ≍ /ρ ( n ) . Proof. It is a consequence of Theorem 4.2 and Lemma 5.8. (cid:3) Composition operators We consider composition operators on weighted analytic spaces on the unit disc D . Let ω ∈ W , H ω will denote the space of analytic functions f ∈ H ( D ) such that f ′ ∈ A ω .The space H ω becomes a Hilbert space if endowed with the norm k . k H ω , given by k f k H ω := | f (0) | + Z D | f ′ ( z ) | dA ω ( z ) . For ω = ω α , the space H ω α will be denote by H α .By the classical Littlewood–Paley identity, we have H = H is the Hardy space. Note alsothat for α ∈ [0 , H α := D α are the weighted Dirichlet spaces and for α > H α = A α − arethe weighted standard Bergman spaces. For more informations on these spaces see [12, 15, 8]. RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 27 Let ϕ be a holomorphic self map of D . The composition operator C ϕ with symbol ϕ actingon H ω is defined by C ϕ f = f ◦ ϕ, f ∈ H ω . Several papers gave some general criterions for boudeddness, compactness and membershipto Schatten classes of composition operators (see for instance, [32, 24, 35, 34, 19, 9, 17]).The Nevanlinna counting function, N ϕ,ω , of ϕ associated with H ω is defined by N ϕ,ω ( w ) = X z ∈ ϕ − ( w ) ω ( z ) ∈ (0 , ∞ ] if w ∈ ϕ ( D ) , if w / ∈ ϕ ( D ) . In what follows, µ ϕ,ω will denote the measure given by dµ ϕ,ω ( w ) = N ϕ,ω ( w ) ω ( w ) dA ( w ) , ( w ∈ D ) . The change of variable formula [1], can be written as follows Z D | ( f ◦ ϕ ) ′ ( z ) | dA ω ( z ) = Z D | f ′ ( z ) | ω ( z ) dµ ϕ,ω ( z ) . Using this identity, it is clear that the composition operator C ϕ on H ω is closely related tothe Toeplitz operator T µ ϕ,ω on A ω . Indeed, if we suppose that ϕ (0) = 0. Then the subspace H ω := { f ∈ H ω : f (0) = 0 } is reduced by C ϕ . If T : H ω → H ω , denotes the restriction of C ϕ to H ω , then T ∗ T is unitarily equivalent to T µ ϕ,ω on A ω . Namely, T ∗ T = V ∗ T µ ϕ,ω V, where V f = f ′ is the derivation operator which defines a unitary operator from H ω onto A ω .As consequence, we have Proposition 6.1. Let ϕ be an analytic self map of D such that ϕ (0) = 0 . Then C ϕ is compacton H ω if and only if T µ ϕ,ω is compact on A ω . In this case, we have s n ( C ϕ , H ω ) = λ n ( T µ ϕ,ω , A ω ) . As a direct consequence of Proposition 6.1 and trace estimates for Toeplitz operators, weobtain the following results. Theorem 6.2. Let ( R n ) ∈ L ω . Let p ≥ and h : [0 , + ∞ ) → [0 , + ∞ ) be an increasingfunction such that h ( t p ) is convex and h (0) = 0 . Let ϕ be an analytic self map of D satisfying ϕ (0) = 0 . We have X n h (cid:18) B (cid:18) µ ϕ,ω ( R n ) A ( R n ) (cid:19)(cid:19) ≤ X n h (cid:0) s n ( C ϕ , H ω ) (cid:1) ≤ X n h (cid:18) B (cid:18) µ ϕ,ω ( R n ) A ( R n ) (cid:19)(cid:19) , where B > depends on ω and p . Corollary 6.3. Let ω ∈ W and let ( R n ) ∈ L ω . Let ρ : [1 , + ∞ ) → (0 , + ∞ ) be an increasingfunction such that ρ ( x ) /x A is decreasing for some A > . Let ϕ be an analytic self map of D such that ϕ (0) = 0 and C ϕ is compact on H ω . Then(1) s n ( C ϕ ) = O (1 /ρ ( n )) ⇐⇒ a n ( µ ϕ,ω ) ≍ O (cid:0) /ρ ( n ) (cid:1) . (2) s n ( C ϕ ) ≍ /ρ ( n ) ⇐⇒ a n ( µ ϕ,ω ) ≍ /ρ ( n ) . Composition operators with univalent symbol on H α The goal of this section is to provide some concrete examples. We will focus our attentionon composition operators C ϕ acting on H α such that ϕ is univalent. We will give estimatesof the singular values of C ϕ in terms of the pull-back measure induced by ϕ .7.1. Composition operators with univalent symbol. Let ϕ be an analytic self map of D . The pull-back measure associated with ϕ is the positive borelian measure on D defined by m ϕ ( B ) = m ( { ζ ∈ T : ϕ ( ζ ) ∈ B } ) , where m is the normalized Lebesgue measure of T .Let Ω be a simply connected subdomain of D which contains 0. Let ϕ be a conformal mapof D onto Ω. Let σ be an automorphism of D . Since C σ is an invertible operator on H α , wehave s n ( C ϕ , H α ) ≍ s n ( C ϕ ◦ σ , H α ) ( n → ∞ ). So, without loss of generality we suppose, in thesequel, that ϕ (0) = 0.Let n, j be integers such that n ≥ j ∈ { , , .., n − } . The dyadic square R n,j isgiven by R n,j = n z ∈ D ; 1 − − n ≤ | z | < − n +1 and 2 jπ n ≤ arg z < j + 1) π n o . By following the same proofs, in all the previous results, one can see that we can replace( R n ) n ∈ L ω α by ( R n,j ) n,j . For our purpose, it is more convenient to consider the Carlesonboxes W n,j which are given by W n,j = n z ∈ D ; 1 − − n ≤ | z | and 2 jπ n ≤ arg z < j + 1) π n o . The main result of this section is the following theorem. Theorem 7.1. Let ϕ be a univalent analytic self map of D . Let h : [0 , + ∞ ) → [0 , + ∞ ) bean increasing function such that h (0) = 0 . Suppose that there exists p ≥ such that h ( t p ) isconvex. Let α > , we have X n,j h (cid:18) B (2 n m ϕ ( W n,j )) α (cid:19) ≤ X n h (cid:0) s n ( C ϕ , H α ) (cid:1) ≤ X n,j h ( B (2 n m ϕ ( W n,j )) α ) , where B > depends on α and p . Let ( m n ( ϕ )) n ≥ be the decreasing enumeration of (2 n m ϕ ( W n,j )) n,j . As a consequence ofTheorem 7.1, Lemma 3.1 and Lemma 3.2, we obtain the following result. Corollary 7.2. Let α > . Let ϕ be a univalent analytic self map of D . Let ρ : [1 , + ∞ ) → (0 , + ∞ ) be an increasing function such that ρ ( x ) /x A is decreasing for some A > . Then thefollowing are equivalent.(1) s n ( C ϕ , H α ) ≍ /ρ ( n ) .(2) m n ( ϕ ) ≍ /ρ /α ( n ) . To prove Theorem 7.1, we need some intermediate results. We begin by the elementaryfollowing lemma. RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 29 Lemma 7.3. Let p ≥ and let h : [0 , + ∞ ) → [0 , + ∞ ) be an increasing function such that h (0) = 0 and h ( t p ) is convex. We have X n ≥ n − X j =0 h (cid:18) C µ ( R n,j ) A ( R n,j ) (cid:19) ≤ X n ≥ n − X j =0 h (cid:18) C µ ( W n,j ) A ( W n,j ) (cid:19) ≤ B X n ≥ n − X j =0 h (cid:18) C µ ( R n,j ) A ( R n,j ) (cid:19) , where B > depends only on p .Proof. The first inequality comes from the facts that h is increasing, R n,j ⊂ W n,j and A ( W n,j ) = 2 A ( R n,j ). For the reverse inequality. We follow the argument given in [18].We have W n,j = [ l ≥ n [ k ∈ H l,n,j R l,k , where H l,n,j = (cid:26) k ∈ { , , ....., l − } ; j n ≤ k l < j + 12 n (cid:27) . From the above decomposition and the convexity of h ( t p ), we get ∞ X n =1 2 n − X j =0 h (cid:18) C µ ( W n,j ) A ( W n,j ) (cid:19) = ∞ X n =1 2 n − X j =0 h X l ≥ n X k ∈ H l,n,j n − l − C µ ( R l,k ) A ( R l,k ) . ∞ X n =1 2 n − X j =0 h X l ≥ n X k ∈ H l,n,j n − lp (cid:18) C µ ( R l,k ) A ( R l,k ) (cid:19) /p p . ∞ X n =1 2 n − X j =0 X l ≥ n X k ∈ H l,n,j n − lp h (cid:18) C µ ( R l,k ) A ( R l,k ) (cid:19) ≤ ∞ X l =1 2 l − X k =0 X l ≥ n X k ∈ H l,n,j n − l +1 p h (cid:18) C µ ( R l,k ) A ( R l,k ) (cid:19) ≤ B ∞ X l =1 2 l − X k =0 h (cid:18) C µ ( R l,k ) A ( R l,k ) (cid:19) . This ends the proof. (cid:3) In [20], P. Lefevre, D. Li, H. Queff´elec and L. Rodr´ıguez-Piazza give an explicit relationbetween the Nevanlinna counting function of an analytic self map ϕ of D and it’s pull-backmeasure. Namely, Theorem 7.4. There exist absolute positive constants c , c , C and C such that for everyanlytic self map ϕ of D , ζ ∈ T and every δ ∈ (0 , −| ϕ (0) | ) one has(1) N ϕ ( w ) ≤ C m ϕ ( W ( ζ, c δ )) , for every w ∈ W ( ζ, δ ) .(2) m ϕ ( W ( ζ, δ )) ≤ C δ Z W ( ζ,c δ ) N ϕ ( w ) dA ( w ) . In particular we have the following inequalities1 C m ϕ ( W ( ζ, δ/c )) ≤ sup z ∈ W ( ζ,δ ) N ϕ ( z ) ≤ C m ϕ ( W ( ζ, c δ )) , (24)For a simple proof of these results see [7].We also need a consequence of the well known Hardy-Littlewood inequality. Lemma 7.5. Let ϕ be an analytic self map of D , let α > and let ζ ∈ T . There exists anabsolute constant c > such that m ϕ ( W ( ζ, δ )) α ≤ C ( α ) δ Z W ( ζ,κδ ) ∩ D N αϕ ( z ) dA ( z ) , (0 < δ < c (1 − | ϕ (0) | )) , where κ is an absolute constant and C ( α ) depends only on α .Proof. Let R ∈ (1 , 2) and let ψ = ϕ/R . By Hardy-Littlewood inequality [27], for every z ∈ D such that 1 − | z | < −| ψ (0) | and every δ ∈ (0 , − | z | ) we have N ψ ( z ) α ≤ Cδ Z D ( z,δ ) N αψ ( w ) dA ( w ) . (25)Let z ∈ D and let δ > − | z | , δ ) < (1 − | ϕ (0) | ). Then, for R = 1 + −| ϕ (0) | ,we have δ < − | z | /R < −| ψ (0) | . By (25), we get N αϕ ( z ) = N αψ ( z/R ) ≤ Cδ Z D ( z/R,δ ) N αψ ( w ) dA ( w ) ≤ Cδ Z D ( z, δ ) N αϕ ( w ) dA ( w ) . Now let ζ ∈ T and let δ < c (1 − | ϕ (0) | ), where c = c c ) and c is the constant appearingin (24).For z ∈ W ( ζ, δ/c ), we have D ( z, δ ) ⊂ W ( ζ, (2 + 1 /c ) δ ). Then N αϕ ( z ) . δ Z W ( ζ,κδ ) N αϕ ( w ) dA ( w ) , κ = 2 + 1 /c . And the result comes from (24). (cid:3) Let c > W = W ( ζ, δ ) be a Carleson box. We will denote W c = W ( ζ, cδ ). Theorem7.1 is a direct consequence of Theorem 6.2, Lemma 7.3 and the following inequalities Lemma 7.6. Let α > . Let h : [0 , + ∞ ) → [0 , + ∞ ) be an increasing positive function suchthat h ( t p ) is convex for some p ≥ . Let ϕ be a univalent analytic self map of D and let C > , we have X n ≥ n − X j =0 h (cid:18) C µ ϕ,α ( W n,j ) A ( W n,j ) (cid:19) . X n ≥ n − X j =0 h ( C (2 n m ϕ ( W n,j )) α ) . X n ≥ n − X j =0 h (cid:18) C µ ϕ,α ( W n,j ) A ( W n,j ) (cid:19) , where the implied constants don’t depend on h . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 31 Proof. Since ϕ is univalent, N ϕ,α = N αϕ . Then, by equation (24) we have µ ϕ,α ( R n,j ) A ( R n,j ) = 1 A ( R n,j ) Z R n,j N αϕ ( z )(1 − | z | ) α dA ( z ) . αn sup z ∈ W n,j N αϕ ( z ) . (cid:16) n m ϕ ( W c n,j ) (cid:17) α . Then X n ≥ n − X j =0 h (cid:18) µ ϕ,α ( R n,j ) A ( R n,j ) (cid:19) . X n ≥ n − X j =0 h (cid:16) C ′ (cid:16) n m ϕ (cid:16) W c n,j (cid:17)(cid:17) α (cid:17) . X n ≥ n − X j =0 h ( C (2 n m ϕ ( W n,j )) α ) . Then the left inequality of Lemma 7.6 is obtained from Lemma 7.3.Conversely, by Lemma 7.5, we have(2 n m ϕ ( W n,j )) α . µ ϕ,α ( W κn,j ) A ( W κn,j ) . Which gives the remaining inequality in order to finish the proof. (cid:3) Examples. Let Ω be a subdomain of D such that 0 ∈ Ω, ∂ Ω ∩ ∂ D = { } and ∂ Ω has, ina neighborhood of +1, a polar equation 1 − r = γ ( | θ | ), where γ : [0 , π ] → [0 , 1] is a differentialcontinuous increasing function such that γ (0) = 0 and satisfying γ ′ ( t ) = O ( γ ( t ) /t ) ( t → + ).Let ϕ be a univalent map from D onto Ω with ϕ (0) = 0 and ϕ (1) = 1. By definition, theharmonic measure ̟ ( ., E, Ω) is the harmonic extension of χ E on Ω, where E is closed subsetof ∂ Ω. By conformal invariance of the harmonic measure we have ̟ (0 , E, Ω) = ̟ (0 , ϕ − ( E ) , D ) = m ( ϕ − ( E )) = m ϕ ( E ) . So to use Theorem 7.1, we have to estimate the harmonic measure of our domains. To thisend we use Ahlfors-Warschawski type estimates. The following lemma, is proved in [11]. Inthe sequel of this subsection, we suppose that γ satisfies conditions (2) and (3). Lemma 7.7. Let γ, Ω and ϕ as above. Then • ̟ (0 , W n,j ∩ ∂ Ω , Ω) . C n exp h − Γ (cid:16) π ( j +1)2 n (cid:17)i , (0 ≤ j < j n := n π γ − (2 π/ n )) . • There exists η > such that for k satisfying k +1 ≤ j n , we haveCard (cid:26) j ∈ { k , .., k +1 − } : η n exp (cid:20) − Γ (cid:18) π ( j + 1)2 n (cid:19)(cid:21) . ̟ (0 , W n,j ∩ ∂ Ω , Ω) (cid:27) ≍ k , Now, we are able to prove the following estimates Theorem 7.8. Let γ, Ω and ϕ as above. Let h : [0 , + ∞ ) → [0 , + ∞ ) be an increasing functionsuch that h (0) = 0 . Suppose that there exists p ≥ such that h ( t p ) is convex. Let α > , wehave B Z h ( be − α Γ( s ) ) γ ( s ) ds ≤ X n h (cid:0) s n ( C ϕ , H α ) (cid:1) ≤ A Z h ( ae − α Γ( s ) ) γ ( s ) ds, where A, B, a, b > depend on α and p . Proof. By Theorem 7.1 and Lemma 7.7, it suffices to prove that Z h (cid:0) CA e − α Γ( s ) (cid:1) γ ( s ) ds . ∞ X n =1 j n X j =0 h (cid:18) C exp (cid:20) − α Γ (cid:18) π ( j + 1)2 n (cid:19)(cid:21)(cid:19) . Z h (cid:0) ACe − α Γ( s ) (cid:1) γ ( s ) ds. First, remark that Z π ( j +1) / n πj/ n γ ( s ) s ds . Z π ( j +1) / n πj/ n s ds = O (1) . (26)So, there exists A > A e − α Γ(2 πj/ n ) ≤ e − α Γ( s ) ≤ Ae − α Γ(2 πj/ n ) , s ∈ (cid:18) πj n , π ( j + 1)2 n (cid:19) . Then h (cid:16) Ce − α Γ( πj n ) (cid:17) . n Z j +1) π n jπ n h (cid:16) Ce − α Γ( s ) (cid:17) ds . h (cid:16) ACe − α Γ( π ( j +1)2 n ) (cid:17) . ∞ X j n X j =1 h (cid:16) C e − α Γ(( πj n ) (cid:17) ≥ ∞ X j n X j =1 n Z πj n π ( j − n h (cid:18) C A e − α Γ( s ) (cid:19) ds ≍ ∞ X n Z γ − (2 − n )0 h (cid:18) C A e − α Γ( s ) (cid:19) ds ≍ C ∞ X n =0 n ∞ X k = n Z γ − (2 − k ) γ − (2 − k − ) h (cid:18) C A e − α Γ( s ) (cid:19) ds ≍ ∞ X k =0 k Z γ − (2 − k ) γ − (2 − k − ) h (cid:18) C A e − α Γ( s ) (cid:19) ds ≍ Z h (cid:0) C A e − α Γ( s ) (cid:1) γ ( s ) ds. This proves the first inequality.The second inequality can be obtained using similar computations. (cid:3) Proof of Theorem D. The first assertion is a direct consequence of the characterization ofmembership to p − Schatten classe given in [11]. Indeed, suppose that lim t → + γ ( t ) log(1 /t ) t = ∞ .It is easy to verify that Z e − pα Γ( t ) γ ( t ) dt < ∞ , ( ∀ p > . Then C ϕ ∈ ∩ p> S p ( H α ). This is equivalent to s n ( C ϕ , H α ) = O (1 /n A ) for all A > ρ ( x ) = exp (cid:8) α Γ(Λ − ( x )) (cid:9) , where Λ( t ) = Z t dsγ ( s ) . RACE ESTIMATES OF TOEPLITZ OPERATORS AND APPLICATIONS 33 First, we prove that ρ ( x ) /x A is decreasing, where A is such that γ ( t ) ≤ πA α t log(1 /t ) . Since γ ( t ) /t is increasing, we haveΛ( t ) = Z t dtγ ( t ) ≤ tγ ( t ) log(1 /t ) ≤ πA α t γ ( t ) . This implies that t → Λ( t ) exp( − αA Γ( t )) is decreasing and then ρ ( x ) /x A is decreasing.Note also that if h an increasing positive function, then Z h ( Ce − α Γ( s ) ) γ ( t ) dt ≍ X n ≥ h (cid:16) Ce − α Γ( x n ) (cid:17) Z x n x n +1 dtγ ( t ) = X n ≥ h (cid:16) Ce − α Γ( x n ) (cid:17) . Then, by Theorem 7.8 and Lemma 3.1 we obtain the result. (cid:3) Concluding Remarks Composition operators on the Hardy space. The Hardy space H is equal to H .The problem of estimating the singular values of composition operators on H was consideredin several papers ([18, 22, 21, 28]). Using the same arguments as those given in section 7, onecan remove the condition that ϕ is univalent in Corollary 7.2. We have the following result. Theorem 8.1. Let ϕ be an analytic self map of D Let ρ : [1 , + ∞ ) → (0 , + ∞ [ be an increasingfunction such that ρ ( x ) /x A is decreasing for some A > . Then s n ( C ϕ , H ) ≍ /ρ ( n ) ⇐⇒ m n ( ϕ ) ≍ /ρ ( n ) . Note that our method can also be applied to composition operators with outer symbol.Such composition operators was considered in [18, 4, 28]. Namely, let ϕ be the outer functiongiven by ϕ ( z ) = exp (cid:18) − Z T e it − ze it + z U ( | t | ) dt π (cid:19) , (27)where U : [0 , π ] → [0 , ∞ ) is an incresing integrable function such that U (0) = 0. It is provedin [4, 28], under some regularity conditions on U , that C ϕ is compact if and only if Z U ( s ) s ds = + ∞ . It is also proved in [4] that C ϕ ∈ S p ( H ) if and only if Z dtU ( t ) q p − U ( t ) < ∞ , where q U ( t ) = Z t U ( s ) s ds .One can extends this result. Namely, in accordance with [4], we say that U is admissible if U is concave or convex and if U ( t ) ≍ U (2 t ) ≍ tU ′ ( t ). We have Theorem 8.2. Let U be an admissible function such that t = o ( U ( t )) and U ( t ) = (cid:18) t Z πt U ( s ) s ds (cid:19) ( t → + ) . Let h be an increasing function such that h (0) = 0 . Suppose that there exists p ≥ suchthat h ( t p ) and h p are convex. We have B Z h (cid:18) bq U ( t ) (cid:19) q U ( t ) U ( t ) dt ≤ X n h (cid:0) s n ( C ϕ , H ) (cid:1) ≤ A Z h (cid:18) aq U ( t ) (cid:19) q U ( t ) U ( t ) dt, where ϕ is given by (27) and A, B, a, b > depend on p . In [28], H. Queffelec and K. Seip give some estimates of the singular values of such com-position operators. They proved that if U is sufficiently regular and q U ( t ) = O (log γ log(1 /t ))for some γ > 0, then s n ( C ϕ , H ) ≍ q q U ( e −√ n ) . The extremal decreasing case corresponds to q U ( t ) = log γ log(1 /t ), they obtained that s n ( C ϕ , H ) ≍ γ/ n . Using Theorem 8.2 and Lemma 3.1, we extend this result as follows Theorem 8.3. Under the same hypothesis of Theorem 8.2. Suppose that Z U ( s ) s ds = + ∞ . .We have(1) If lim t → + log q U ( t )log log 1 /t = ∞ , then s n ( C ϕ , H ) = O (1 /n A ) ( for all A > . (2) If log q U ( t )log log 1 /t = O (1) , then s n ( C ϕ , H ) ≍ p q U ( x n ) , where x n is given by Z πx n q U ( t ) U ( t ) dt = n. Composition operators on the Dirichlet space. The Dirichlet space, denoted by D , is given by D (:= H ) = { f ∈ H ( D ) : f ′ ∈ L ( D , dA ) } . The Nevanlinna counting function N ϕ, induced by ϕ and associated with D is the countingfunction n ϕ . That is N ϕ, ( z ) = n ϕ ( z ) = Card { ϕ − ( z ) } , ( z ∈ D ) . In particular, if ϕ is univalent then n ϕ = χ Ω and dµ ϕ, = χ ϕ (Ω) dA, (Ω = ϕ ( D )) . Let Ω , γ and ϕ as before. The compactness and membership to schatten classes of C ϕ isstudied in [11]. Recall that C ϕ is compact on D if and only iflim t → + γ ( t ) t = ∞ . 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