The Berezin Transform of Toeplitz Operators on the Weighted Bergman Space

In this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

Let dA denote the Lebesgue area measure on D. If u is a positive locally integrable function on D, i.e. positive u ∈ L 1 loc (dA), let L p (u) denote the space of measurable functions on D that are pth power integrable with respect to udA. That is The Bergman space A p (u) is defined to be a subspace of analytic functions in L p (u) with L p (u)-norm. We write A p = A p (1) for short. The most common reproducing kernel for the unit disc has the form K w (z) = 1 (1 −wz) 2 , for w, z ∈ D, and it corresponds to the space A 2 .
The following notations will be used throughout the paper. For a weight u and E ⊂ D, we set u(E) = E udA, A(E) = E dA. We denote by for integrable f and measure μ. If we define P by (1 −wz) 2 dA(w). The problem of characterizing the weights for which the Bergman projection P is a bounded orthogonal projection from L p (u) to A p (u) was solved by Békollé and Bonami [1,2]. They found that these weights are precisely u ∈ B p . where 1/p + 1/p = 1. Recently, the sharp estimates for the L p -continuity of the Bergman projection are investigated in [9] and [10] respectively. The inner product of the Hilbert space A 2 (u) is given by where f, g ∈ A 2 (u). The reproducing kernel of A 2 (u) will be denoted by K(z, w). It is well known that K(z, w) = K(w, z). If L z is the point evaluation at z ∈ D, that is L z f = f (z) for every f ∈ A 2 (u). It follows by the Riesz representation that K(z, z) = K(., z), K(., z) A 2 (u) = K(., z) 2 A 2 (u) = L z Let μ be a finite positive Borel measure on D that satisfies the condition Then the Toeplitz operator T μ is well-defined on A 2 (u). Recall that the pseudohyperbolic metric d : Denote by (z, r) := {w ∈ D : d(z, w) < r} the pseudohyperbolic disk centered at z with radius r. For a finite positive Borel measure μ on D and r > 0, the average function μ r is defined as It is well known that the Berezin trasform plays a role in the theory of Toeplitz operator. The Berezin transform of the Toeplitz operator T μ is given bỹ where k z (w) := K(w, z)/ K(·, z) A 2 (u) is the normalized reproducing kernel of A 2 (u). By a straightforward computation one has (1.1) It follows that the Berezin transformμ can be formulated bỹ Constantin [5] characterized the Toeplitz operator on A 2 (u) in terms of the Carleson measure. The motivation of this paper is to characterize the Toeplitz operator in terms of its Berezin transform. Now we are in the position to state our main theorems. Theorem 1.1 Let p 0 > 1 and u ∈ B p 0 . Suppose that δ ∈ (0, 1) is the one in Theorem 2.7 and 0 < r ≤ δ. The following assertion are equivalent: The following theorem charaterizes the compact Toeplitz operators on A 2 (u) with Békollé-Bonami weights Theorem 1.2 Let p 0 > 1 and u ∈ B p 0 . Suppose that δ ∈ (0, 1) is the one in Theorem 2.7 and 0 < r ≤ δ. The following assertions are equivalent: Next, we will study the Schatten class of Toeplitz operators T μ ∈ S p A 2 (u) in terms of the Berezin transform. Recall that if T is a compact operator on a Hilbert space H , then there are orthonormal sets {e n } and {σ n } in H such that s n x, e n H σ n , x ∈ H, where s n = s n (T ) is the nth singular value of T . The Schatten class S p = S p (H ) consists of those compact operators T on H for which the singular numbers sequence {s n } of T belongs to p , that is n |s n | p < ∞.
Let K be the set of all compact operators on a Banach space B. For any bounded linear operator T : B → B, the essential norm of T is defined by It is clear that T e = 0 if and only if T ∈ K. Finally, we show the conditions for Toeplitz operators to be compact, see the above theorem, in term of the essential norm estimates because essential norm estimates give us a further information. The essential norm of a bounded operator is the distance from the operator to the space of the compact operators. Throughout the paper, we use the following notations: • Q 1 Q 2 means that there is a constant C > 0 (independent of the key variable(s)) such that Q 1 ≤ CQ 2 ; • Q 1 Q 2 if both Q 1 Q 2 and Q 2 Q 1 .

Preliminaries and Basic Properties
The pseudohyperbolic metric obeys the following so-called strong triangle inequality: Furthermore, if 0 < r < 1, then whenever z, w ∈ D with ρ(z, w) < r, and for all ζ ∈ D 1 −ζ z 1 −ζ w 1 where the constants involved depend only on r. We will denote by (z, r) := {w ∈ D : ρ(z, w) < r} the pseudohyperbolic disk centered at z with radius r. We will also use the following class of weights which is denoted by C p . For p > 1, a positive locally integrable weights u belongs to C p , or say u satisfies C p condition if where 1/p + 1/p = 1. Condition C p seems to depend on a choice of r < 1, but it is known that the same class of weights is obtained for any r ∈ (0, 1) and B p ⊂ C p . To see this, we note that for a given r, there is a a ∈ D such that (a, r) ⊂ S a with comparable volumes, for more details see [7].
It is not hard to see that S(a) is "equivalent" to the set S(ζ, h) for ζ = a/|a| ∈ ∂D and See more details in [7]. The point evaluations on A p (u) are bounded linear functionals for p > 0. To be precise, we have the following estimate.
where the constant C > 0 depends on r, p and the In the Békollé setting, Bergman metric balls have comparable weighted areas when their centers are close.

Lemma 2.2 (Lemma 2.2 in [5])
Suppose u ∈ C p for some p > 1. Let t, s ∈ (0, 1), and z, w ∈ D with ρ(z, w) < r for some r > 0. Then we have where the constant is independent of z and w.
Similarly, if u ∈ B p 0 , it is worthy to be noted that whenever (a, r) ⊂ S a with comparable volumes. To interested readers we can refer [7] and Lemma 5.23 in [13] for more details. For s > 0 and 0 < r < 1, we denote by Test functions play a crucial role in our proofs. Constantin [4] gives an estimate of the the norm of G s w in terms of the weighted area of Euclidean disks inside D. We can adopt an alternative method to estimate the norm of G s w in terms of the weighted area of S(a) (or S(ζ, h) equivalently). Our method relies on a popular decomposition of D which is used repeatedly in many papers. See Theorem 1 in [12] for instance. The first two authors obtain the same estimate on the unit ball by an analogue method, see [11]. For the sake of clarity, we reprove it here.

Lemma 2.3
Let p > 0, p 0 > 1 and the weight u ∈ B p 0 . We have where the constant involved is independent of w ∈ D.
Rearranging this inequality, we have 1 − |w| ≥ |1 − zw|/3, and it follows immediately that To prove the rest conclusions of the lemma, we firstly consider the case when s > 2p 0 /p. We denote by Then we can obtain the following estimate under this decomposition of D.
Since u ∈ B p 0 , for every positive integer k, we have Noting that s > 2p 0 /p, we can estimate the norm G s w L p (u) as follows, Now we have proved that (2.3) holds for s > 2p 0 /p. The case s = 2p 0 /p follows from Lemma 3.1 in [4] and also Lemma 2.1 in [5]. So we have That completes the proof.
The following covering lemma will play a role.
Lemma 2.4 (Theorem 2.23 in [13]) There exists a positive N such that for any 0 < r ≤ 1 we can find a sequence {a k } in D with the following properties.
Any sequence satisfying the conditions in Lemma 2.4 will be called an r-lattice. Note that |a k | → 1 − as k → ∞. In what follows, the sequence {a k } will always refer to the sequence chosen in Lemma 2.4.

Carleson Measures
Let 0 < p ≤ q < ∞. A positive Borel measure μ on D is called to be a q-Carleson measure for A p (u) if the embedding I : A p (u) → L q (dμ) is bounded. We have the following Carleson embedding theorem. Lemma 2.5 Suppose q ≥ p > 0, p 0 > 1 and 0 < r < 1. Let u ∈ B p 0 be a weight and μ is a positive Borel measure on D. Then the following conditions are equivalent.
(a) The embedding I : for every analytic function f on D.  > 0 such that μ( (a, r)) u( (a, r)) q/p for every a ∈ D. (d) There is an r > 0 such that μ ( (a k , r)) u( (a k , r)) q/p for the sequence {a k } described in Lemma 2.4. Proof The equivalence (a) and (c) was proved by Constantin in [5]. We are going to prove To prove (b)⇒(c), we let r be sufficiently small and fixed. It will be done to prove μ( (a, r)) u( (a, r)) q/p for each |a| ≥ tanh(2r). As we state before Lemma 2.1, there is a a ∈ D such that (a, r) ⊂ S a with comparable areas. By Eq. 2.2, we have μ( (a, r)) ≤ μ S a u S a q/p u( (a, r)) q/p .

The proof of (c) ⇒ (d) is obvious. We next prove (d) ⇒ (a). If f is holomorphic in D, then by Lemma 2.1 we have
where the last inequality is deduced by Lemma 2.4. Now we prove (a)⇒(e). Assume that the identity I : A p (u) → L q (dμ) is bounded. By Lemma 2.3, we have To see (e)⇒(c), we assume that (2.4) holds. Then Considering that 1−|w| 2 |1−zw| 1 when ρ(z, w) < r, we find that the left hand side is equivalent to μ( (w, r)) u( (w, r)) q/p . That completes the proof.

Reproducing Kernels
The key point to prove the main theorems is to estimate the normalized reproducing kernel functions k z (w) from below. We start our discussion by the following lemma which estimate the reproducing kernel functions on the diagonal. Lemma 2.6 (Lemma 4.1 in [5]) Suppose p 0 > 1 and u ∈ B p 0 . Let K(z, w) be the Bergman kernel in A 2 (u) and r ∈ (0, 1). Then we have the following estimate where the constant involved is independent of z ∈ D.
Now we can estimate the normalized reproducing kernel |k w (z)| when z and w are close enough. Our strategy is to update the method of Lemma 3.6 in [8] to our setting. Theorem 2.7 Suppose p 0 > 1 and u ∈ B p 0 . There is a sufficient small δ ∈ (0, 1), such that Proof For any fixed w 0 ∈ D, consider the subspace A 2 (u, w 0 ) of A 2 (u), which is defined by We have the decomposition where L w 0 is the one-dimensional subspace spanned by the function k w 0 (z). If we denote by K w 0 (·, ·) the reproducing kernel of A 2 (u, w 0 ), it is easy to see that Hence we have |k w 0 (z)| 2 ≤ K(z, z). To prove the reverse inequality, we only need to show that there exist constants 0 < C < 1 and 0 < δ < 1, such that whenever z ∈ (w 0 , δ). Let us consider the operator We claim that S w 0 is a bounded mapping from A 2 (u, w 0 ) into A 2 (u). Let us see the proof. For every f ∈ A 2 (u, w 0 ), we have f (z) = (z − w 0 )f (z) when z ∈ D(w 0 , ) for some > 0 small enough and holomorphicf on D(w 0 , ). It is clear that S w 0 f (z) =f (z) whenever z ∈ D(w 0 , ) and hence it is bounded on D(w 0 , ). Then we have According to Lemma 2.6, we fix a r ∈ (0, 1) so that there is a constant C independent on the choice of z ∈ D with K(z, z) ≤ Cu( (z, r)) −1 . Hence we have where k is an integer. By the reproducing property we have It follows that By Lemma 2.6, we obtain ( (w 0 , r)) , which converges to 0 as k goes to infinity. Combining this fact with g 2 A 2 (u) = I k + II k , now we can choose a k large enough such that It then follows that where C is independent on w 0 , k is an integer and r ∈ (0, 1) is fixed. Hence Since U z w 0 and L z are point evaluations on A 2 (u, w 0 ) and A 2 (u) respectively, by the Riesz representation we have U z w 0 2 A 2 (u,w 0 )→C = K w 0 (z, z) and L z 2 A 2 (u)→C = K(z, z). We let ρ(z, w 0 ) < δ where δ ∈ (0, r) will be specified later. We obtain that where C is independent on w 0 and z by Eq. 2.1. We now choose δ > 0 such that C k/r · δ < 1, and this completes the proof of (2.5) and of the theorem.
The following proposition is proved by Chacón in [3] which is going to be employed in the proof of the compactness.

Proposition 2.8
Let p 0 > 1. If u ∈ B p 0 , then the normalized kernel function k w converges to zero weakly in A 2 (u).
The next Proposition is a classical result, its proof is similar to that one given by K. Zhu in [14,Theorem1.14].

Proposition 2.9
Let p 0 > 1 and u ∈ B p 0 . A linear operator T on A 2 (u) is compact if and only if Tf n A 2 (u) −→ 0 whenever f n → 0 weakly in A 2 (u).
Since ( which gives the desired result.

Proof of Theorem 1.3
Let T be a compact operator and h : R + → R + a continuous increasing function. The authors of [6] introduce To study the Schatten class of the Toeplitz operators on the Bergman spaces with Békollé-Bonami weights, we follows the strategy of [6]. The following Lemma is the generalized version of Theorem 6.2 in [6].

Lemma 5.1
Let p 0 > 1, u ∈ B p 0 and h : R + → R + an increasing convex function. Let μ be a positive Borel measure on D such that the Toeplitz operator T μ is compact on Proof Assume that T μ ∈ S h A 2 (u) . That is n h(Cs n ) < ∞ for some positive constant C. Let {e n } be an orthonormal set in A 2 (u) and T μ = ∞ n=1 s n ·, e n e n the canonical decomposition of the positive operator T μ where s n are also the eigenvalues of T μ . Note that k z is the normalized reproducing kernel functions in A 2 (u). We have n | k z , e n | 2 = 1 by the Parseval formula. Then it follows by the convexity of h and Jensen's inequality that Cs n | k z , e n A 2 (u) | 2 dλ u (z) Conversely, we assume that D h(Cμ(z))dλ u (z) < ∞ for some C > 0. By Lemmas 2.1, 2.4, 2.6 and Theorem 2.7, we obtain T μ e n , e n = D |e n (z)| 2 dμ(z) By Fubini's theorem, Lemma 2.2 and Theorem 2.7, we get T μ e n , e n = D (ξ,δ) It then follows by Jensen's formula that Therefore T μ ∈ S h (A 2 (u)).
As a direct consequence of Lemma 5.1, we give the proof of Theorem 1.3 which is the Schatten class of the Toeplitz operators on A 2 (u).
Proof of Theorem 1.3 Let h(t) = t p where p > 1 and use Lemma 5.1.
Another application of Lemma 5.1 is on the decay of the eigenvalue of T μ which is regarded as a generalization of Theorem 6.4 in [6].

Proof of Theorem 1.4
In the proof of Theorem 1.1 we obtain μ r (z) ≤μ(z), for z ∈ D and r ∈ (0, δ). Therefore, it is enough to prove lim sup Firstly, we start with the lower estimate. We take an arbitrary compact operator K on A 2 (u). By Proposition 2.8, the normalized reproducing kernel {k z } converges to 0 weakly in A 2 (u). Then, Kk z A 2 (u) → 0 as |z| → 1 − , by Proposition 2.9. Therefore, On the other hand, since T μ is bounded, we havẽ μ(z) = | T μ k z , k z A 2 (u) | ≤ T μ k z A 2 (u) .
Combining this with (6.2), we get the lower estimate.
Now we prove the upper estimate for the essential norm of Toeplitz operators T μ . Suppose {e n } is a complete orthonormal system of A 2 (u). For n ∈ N, we define an operator Q n by f, e j A 2 (u) e j , for any f ∈ A 2 (u).
The operators Q n is compact on A 2 (u). Let R n = I − Q n . It is easy to see that R * n = R n and R 2 n = R n . Furthermore, we have lim n→+∞ R n f A 2 (u) = 0, for any f ∈ A 2 (u).
For ρ > 0, let D ρ = D \ D(0, ρ) and dμ ρ (z) = χ D ρ (z)dμ(z), where χ D ρ is the characteristic function on D ρ . By the definition of the average function of μ ρ , we can see μ ρ r (z) = 1 u( (z, r)) (z,r)∩D ρ dμ(ξ ) = 1 u( (z, r)) (z,r)∩D ρ dμ(ξ ). and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.