SUMMING CARLESON EMBEDDINGS ON BERGMAN SPACES B O He

Joelle Jreis
email: joelle.jreis@univ-artois.fr

Pascal Lef Èvre
email: pascal.lefevre@univ-artois.fr

Zengjian Lou
email: zjlou@stu.edu.cn

ABSOLUTELY

Keywords: r-summing operators, Bergman spaces, Carleson embeddings, Berezin transform, Composition operators. 2010 Mathematics Subject Classification. 47B10,47B38, 30H20, 47B33

In this paper, we focus on Carleson embeddings from Bergman spaces A p into L p (µ), where µ is a positive measure on the unit disk. We describe when this injection is r-summing on A p . We complete the full characterization of such operators when p > 1, and r ≥ 1. As an immediate application, we get the characterization of absolutely summing weighted composition operators on Bergman spaces. In passing we also prove a new connection between the boundedness of the Berezin transform and the Carleson embedding on Bergman spaces.

INTRODUCTION

In this paper, we investigate particular properties of Carleson embeddings on the classical Bergman spaces A p when p > 1. In the following, the unit disk of the complex plane is denoted D = z ∈ C |z| < 1 . We denote by H (D) the class of holomorphic functions on the unit disk, and we recall that Bergman spaces are defined by

A p = f ∈ H (D) D f z p dA < ∞ with f A p = D f z p dA 1/p
, where A stands for the normalized area measure on D. When there is no ambiguity, we write simply • p instead of • A p . This definition makes sense for every p > 0. See for instance [START_REF] Duren | Bergman Spaces[END_REF] and [START_REF] Hedenmalm | Theory of Bergman Spaces[END_REF] to know more on Bergman spaces. Now, let us turn to our main subject. Given a positive Borel measure µ on the open unit disk D, we consider the formal identity J µ from the Bergman space A p into L p (D, µ) (we keep the notation J µ instead of J p,µ in the sequel for sake of lightness):

J µ : A p -→ L p (D, µ) f -→ f
A famous result of Carleson (see [START_REF] Carleson | Interpolations by bounded analytic functions and the corona problem[END_REF]), introduced and characterized measures such that the corresponding embedding on Hardy spaces are defined and bounded. On Bergman spaces, the boundedness of J µ was characterized by Hastings [START_REF] Hastings | A Carleson measure theorem for Bergman spaces[END_REF]: J µ is well defined and bounded if and only if µ is a 2-Carleson measure, i.e. Once we know that J µ is bounded, it is natural to wonder which specific operator properties are satisfied. The most natural is probably compactness and J µ is compact if and only if µ is a vanishing Carleson measure:

sup ξ∈T µ W(ξ, h) = o(h 2 ),
when h → 0.

Actually, our approach in this paper involves hyperbolic disks rather than Carleson windows (see (2.1) and Section 2 for definitions and the corresponding results about boundedness and compactness).

One standard application of Carleson embeddings is that it allows to recover some operator properties of weighted composition operators on A p . Let us recall that, given a symbol, i.e. an analytic function ϕ : D → D, the composition operator C ϕ : A p → A p is well defined and automatically bounded (see the monographs [START_REF] Cowen | Composition Operators on Spaces of Analytic Functions[END_REF] or [START_REF] Shapiro | Composition Operators and Classical Function Theory[END_REF] for example). When we are given a weight u, i.e. an analytic function u on D, then the weighted composition operator is (now formally) defined by uC ϕ ( f )(z) = u(z) • f ϕ(z) . Many operator properties of C ϕ (and of u • C ϕ ), can be expressed in terms of Carleson measures thanks to the transfer formula. We explain here how, for composition operators, the pullback measure of A associated to ϕ plays a crucial role:

A ϕ (E) = A ϕ -1 (E) = A z ∈ D| ϕ(z) ∈ E
for every Borel subsets E of D.

The transfer formula gives

f • ϕ A p = f L p (D,A ϕ ) for every f ∈ A p .
Hence many properties of the operator C ϕ are shared with the ones of the operator J A ϕ , in particular compactness, r-summingness.

The case of weighted composition operators can also be treated in the same manner.

The purpose of this paper is the characterization of (finite positive) measures µ on D such that J µ is r-summing for some r ≥ 1, and as a natural immediate application, we characterize r-absolutely summing weighted composition operators on Bergman spaces.

Let us recall that (1-)absolutely summing operators are the operators T : X → Y which do take unconditionally summable sequences {x n } in X to absolutely summable sequences {T x n } in Y. More generally, let us recall Definition 1.1. Let r ≥ 1 and T : X → Y be a linear operator between Banach spaces. We say that T is r-summing if there is a constant C ≥ 0 such that regardless of the natural number n and regardless of the choice of x 1 , x 2 , . . . , x n in X, we have where X * denotes the dual space of X, and B X * denotes the unit ball of the Banach space X * . The best constant C for which the inequality always holds is denoted by π r (T ).

The most natural example is the operator f ∈ C(K) -→ f ∈ L r (K, ν) where ν is a finite positive measure on the compact K. Pietsch's theorem expresses that it is the generic example of r-summing operators.

We shall write Π r (X, Y) for the set of all r-summing operators from X into Y. If r = 1, we say that Π 1 (X, Y) is the class of absolutely summing operators. The r-summing norm of T , denoted by π r (T ), is the least suitable constant C ≥ 0. The class of r-summing operators forms an operator ideal (for instance see [START_REF] Diestel | Absolutely Summing Operators[END_REF] for more details).

This notion was studied for the first time by Grothendieck in the 50's. One of his famous result around this topic is that every continuous linear operator from 1 to 2 is absolutely summing. However, it was only in late 60's that this property received a new attention with many fruitful applications in functional analysis (operator theory and geometry of Banach spaces): for instance, Pietsch established the basic properties of this class of operators (in particular the famous factorization theorem which plays a key role in this paper) and studied these operators on Lebesgue type spaces, whereas Lindenstrauss and Pełczyński [START_REF] Lindenstrauss | Absolutely summing operators in L p -spaces and their applications[END_REF] enlighted how these properties are helpful in Banach space geometry. There are many other deep works in the 70's and after (see the monographs [START_REF] Diestel | Absolutely Summing Operators[END_REF], [START_REF] Li | Introduction to Banach spaces: analysis and probability[END_REF] or [START_REF] Wojtaszczyk | Banach Spaces for Analysts[END_REF] for some aspects). The monograph [START_REF]Banach spaces of analytic functions and absolutely summing operators[END_REF] enlights more particularly this topic on Banach spaces of analytic functions. Let us mention that some other ideals of operators have close connections with absolutely summing operators such as order bounded operators, r-integral operators, r-nuclear operators (see [START_REF] Diestel | Absolutely Summing Operators[END_REF] or [START_REF] Wojtaszczyk | Banach Spaces for Analysts[END_REF] for examples).

Recently, in [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF], Lefèvre and Rodríguez-Piazza gave a complete characterization of rsumming Carleson embeddings on the Hardy spaces H p when p > 1 (for every r ≥ 1). We consider here the absolutely summing Carleson embedding viewed on Bergman spaces. In other words, we aim to characterize the measures µ such that the natural embedding map J µ from the Bergman spaces A p to L p (µ) is r-summing: Problem (P): let p > 1, r ≥ 1 and µ be a positive finite measure on D. Which condition on µ ensures that J µ is an r-summing operator?

We solve this problem for every p > 1 and every r ≥ 1. Let us mention that Domenig [START_REF] Domenig | Composition operators belonging to operator ideals[END_REF] (see too his PhD thesis [START_REF] Domenig | Composition operators on weighted Bergman spaces and Hardy spaces[END_REF]) characterized r-summing composition operators. Our approach is different from his point of view, which seems to rely rather strongly on the fact that he works with a pullback measure and the underlying analytic properties of the symbol. Nevertheless, a common feature of the two methods is that (in some cases only in our paper) a link is build with some specific diagonal operators on sequence spaces p . This idea is not surprising, having in mind that the Bergman space A p is isomorphic to p , but we do not use explicitly this isomorphism. Still, our approach is highly inspired by [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF], though some specific difficulties appear, so the proofs require also some novelties. For instance, in passing, we prove a new connection between the boundedness of the Berezin transform viewed from L p (D, A) to L q (D, µ), and the boundedness of the Carleson embedding from A p to L q (D, µ). This result is an ingredient required for the success of our approach. Strangely we did not find this result in the literature. On the other hand, we make a heavy use of hyperbolic disks and tools whereas they do not appear in [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF].

Concretely, let us describe the organization of the paper. The first section is an introduction to the framework of this paper. In the second section, we give basic properties of r-summing operators and remind some facts on hyperbolic geometry. We state the main results of the paper in the third section and give some direct applications to weighted composition operators. In the fourth section, we get some preliminary general results, mainly constructing some links between the Carleson embedding and some multipliers on classical sequence spaces. In Section 5, we focus on the Berezin transform and show a new link with the Carleson embedding. The case 1 < p ≤ 2 is solved in Section 6. In Sections 7, 8 and 9, we solve the case p ≥ 2: actually we split this case into three cases r ≤ p , p ≤ r ≤ p and r ≥ p, where as usual, p is the conjugate exponent of p: 1/p + 1/p = 1. In Section 10, we state the corresponding results in the more general framework of weighted Bergman spaces A p α . In the last section, we give several applications, in particular an application to some composition operators between weak (resp. strong) vector valued Bergman spaces.

DEFINITIONS, NOTATIONS AND REMINDERS.

2.1. Absolutely summing operators. We already gave the definition of absolutely summing operators in Definition 1.1. It is simple to check that Π r (X, Y) is included in B(X, Y), the space of all continuous linear operators from X into Y, and that π r defines a norm on Π r (X, Y) with

u ≤ π r (u) for all u ∈ Π r (X, Y).
It is well known that Π r (X, Y) is a Banach space under the norm π r .

The following results are well known properties about r-summing operators and will be used in this paper, see for example [START_REF] Diestel | Absolutely Summing Operators[END_REF]. We give them here for sake of completeness 1) For any Banach spaces X and Y, T : X → Y is r-summing (r ≥ 1) if and only if there is a constant C > 0 such that for any measurable space (Ω, Σ, ν) and any continuous function F : Ω → X, we have

Ω T • F r dν 1/r ≤ C sup s∈B X * Ω |s(F)| r dν 1/r .
Moreover, the best C is π r (T ) .

2) Π r (X, Y) is an operator ideal between Banach spaces: for any T ∈ Π r (X, Y), and for any two Banach spaces X 0 , Y 0 such that S ∈ B(X 0 , X) and

U ∈ B(Y, Y 0 ), we have UT S ∈ Π r (X 0 , Y 0 ) with π r (UT S ) ≤ U π r (T ) S .
3) The class of spaces Π r (X, Y) where r ≥ 1 is monotone. That is, for any 1 ≤ r ≤ s < ∞, we have Π r (X, Y) ⊂ Π s (X, Y) and the relationship π s (T ) ≤ π r (T ), for any T ∈ Π r (X, Y).

4)

Cotype property: we say that a Banach space X has cotype q ≥ 2, if there is a constant C ≥ 0 such that no matter how we select finitely many vectors x 1 , x 2 , . . . , x n from X, we have

n k=1 x k q 1/q ≤ C 1 0 n k=1 r k (t)x k 2 dt 1/2
, where (r k ) is a Rademacher sequence.

For any measure space (Ω, Σ, µ), the Lebesgue type space L q (Ω, µ) has cotype max{q, 2}. Then for any Banach spaces X and Y, we have the following properties:

1) If X has cotype 2, then Π 2 (X, Y) = Π 1 (X, Y). 2) If X has cotype 2 < q < ∞, then Π 1 (X, Y) = Π r (X, Y) for all 1 < r < q . 3) If X and Y both have cotype 2, then Π r (X, Y) = Π 1 (X, Y) for every 1 < r < ∞ .
The following lemmas about absolutely summing operators are reformulation or direct consequences of Propositions 5.5 and 5.18, and Theorem 2.21 in [START_REF] Diestel | Absolutely Summing Operators[END_REF] respectively. Lemma 2.1. Let X be a Banach space and µ be any Borel measure on D. Let T : X → L p (µ) , 1 ≤ p < ∞, be an order bounded operator. Then T is p-summing with

π p (T ) ≤ sup f ∈B X |T ( f )| .
Lemma 2.2. Let X be a Banach space and H be a Hilbert space. We assume that the operator T : X → H is such that its adjoint T * : H → X * is r-summing for some r ≥ 1. Then T is 1-summing with π 1 (T ) π r (T * ).

2.2.

Carleson embedding, Bergman metric and Luecking rectangles.

Definition 2.3. Suppose that µ is a finite positive Borel measure on the unit disk D. J µ is the identity operator from A p into L p (µ)

J µ : A p -→ L p (µ) f -→ f
which is formally defined until now, but we recall below that it is defined precisely when µ is a 2-Carleson measure.

For simplicity, we shall use the terminology Carleson measure instead of "2-Carleson" or "A p -Carleson" measure. It may happen that we consider J µ acting from A p 1 to L p 2 (µ) with p 1 , p 2 different from p, but keeping the same notation J µ for simplicity. Nevertheless, we shall be careful to avoid ambiguity.

It is well known that (cf. [START_REF] Zhu | Operator Theory in Function Spaces[END_REF], Theorem 7.4), J µ is bounded if and only if sup µ(D(a, r))

(1 -|a| 2 ) 2 : a ∈ D < +∞ , (2.1)
where D(a, r) (r > 0) denotes the Bergman disk defined by

D(a, r) = {z ∈ D : β(z, a) < r}.
Let us recall that the hyperbolic metric (Bergman metric) on the unit disk D is defined by

β(z, w) = 1 2 log 1 + ρ(z, w) 1 -ρ(z, w)
and ρ(z, w) denotes the pseudo-hyperbolic, defined by

ρ(z, w) = |z -w| |1 -zw| • Recall that J µ is compact if and only if µ is a vanishing Carleson measure (in the hyperbolic sense), that is, lim |a|→1 - µ(D(a, r)) (1 -|a| 2 ) 2 = 0 .
In view of these results on Carleson embeddings, it is clear that the hyperbolic disks play a crucial role in this framework. Let us go further in this direction. Definition 2.4. A sequence {a k } in D is called a t-lattice (t > 0) in the Bergman metric if the following conditions are satisfied:

1) The unit disk is covered by the Bergman metric disks D(a k , t).

2) β(a i , a j ) ≥ t/2 for all i and j with i j .

There exists a positive constant N with the following property: If 0 < t < 1 and {a k } is a t-lattice in the Bergman metric, then every point z ∈ D belongs to at most N disks D(a k , t) .

The existence of t-lattices in the Bergman metric is ensured by Lemma 4.8 in [26] : Lemma 2.5. For any t > 0 there exists a t-lattice {a k } in the Bergman metric.

Given a t-lattice, the sequence D k given by the following Lemma 2.6 plays a crucial role in the sequel. Lemma 2.6. [26, Lemma 4.10] Suppose 0 < t < 1 and {a k } is a t-lattice in the Bergman metric. For each k, there exists a measurable set D k with the following properties:

1) D(a k , t/4) ⊂ D k ⊂ D(a k , t) for all k ≥ 1 . 2) D j D i = ∅ for i j . 3) D 1 D 2 • • • = D .
Observe that, for any z ∈ D k there are several equivalence relations:

|1 -za k | ≈ 1 -|z| 2 ≈ 1 -|a k | 2 and |D k | ≈ (1 -|a k | 2 ) 2
where the underlying constants depend only on t (see consequences of Proposition 4.5 in [START_REF] Zhu | Operator Theory in Function Spaces[END_REF]). These observations will be used repeatedly throughout this paper .

We also recall the definition of Luecking rectangles: let

Γ n = z ∈ D | 1 - 1 2 n ≤ |z| < 1 - 1 2 n+1
, where n = 0, 1, 2, • • • . Then each Γ n is divided into 2 n similar pieces R n, j , 0 ≤ j < 2 n , that we call Luecking rectangles (or sometimes in the literature Luecking windows):

R n, j = z ∈ D | 1 - 1 2 n ≤ |z| < 1 - 1 2 n+1 , 2π j 2 n < arg(z) ≤ 2π( j + 1) 2 n .
So clearly the family of the R n, j with n ≥ 0 and 0 ≤ j < 2 n , forms a disjoint partition of D.

We shall use the following notations:

• D z instead of D(z, r 0 ) for some fixed 0 < r 0 < 1 and L p instead of L p (A).

• B X denotes the unit ball of a Banach space X and B + X the positive element in the unit ball of X, when X is a Banach lattice.

• C p will denote a positive constant depending only on the parameter p, but it may change along the paper. • The notation A B (and A B) for non negative quantities A and B means that there is a constant C such that A ≤ CB (and A ≥ CB, respectively). Finally, A ≈ B means that both A B and B A hold. • We are going to often use |E| to denote the area of any measurable set E in D with respect to the normalized area measure on D. • We denote the cardinal of finite E by #(E).

THE MAIN RESULTS

In this section, we state our main results, first of all the characterization of absolutely summing Carleson embeddings on classical Bergman spaces.

Main Theorem: Let 0 < t < 1/4 and µ be a Carleson measure on D. For any t-lattice {a k } and the relevant sets D k as defined in Lemma 2.6, we have the following statements:

(1) Let 1 < p ≤ 2 and r ≥ 1. Then J µ :

A p → L p (µ) is r-summing if and only if D µ(D z ) 2 p (1 -|z| 2 ) 4 p +2 dA(z) < ∞ .
Moreover, we have

π 2 (J µ ) ≈ ∞ k=1 µ(D k ) |D k | 2/p 1/2 ≈ n, j 4 n µ(R n, j ) 2/p 1/2 ≈ D µ(D z ) 2 p (1 -|z| 2 ) 4 p +2 dA(z) 1 2 ≈ D D 1 |1 -wz| p+2 dµ(w) 2 p dA(z) 1 2 . (3.1) (2) Let p ≥ 2 and 1 ≤ r ≤ p . Then J µ : A p → L p (µ) is r-summing if and only if D µ(D z ) p p (1 -|z| 2 ) 2p dA(z) < ∞.
Moreover, we have

π 1 (J µ ) ≈ π r (J µ ) ≈ ∞ k=1 µ(D k ) |D k | p /p 1/p ≈ n, j 4 n µ(R n, j ) p /p 1/p ≈ D µ(D z ) p p (1 -|z| 2 ) 2p dA(z) 1/p ≈ D D 1 |1 -wz| 2p dµ(w) p p dA(z) 1 p . (3.2) (3) Let p ≥ 2 and p ≤ r ≤ p. Then J µ : A p → L p (µ) is r-summing if and only if D µ(D z ) r p (1 -|z| 2 ) 2r p +2 dA(z) < ∞.
Moreover, we have

π r (J µ ) ≈ ∞ k=1 µ(D k ) |D k | r/p 1/r ≈ n, j 4 n µ(R n, j ) r/p 1/r ≈ D µ(D z ) r p (1 -|z| 2 ) 2r p +2 dA(z) 1/r ≈ D D 1 |1 -wz| 2+ 2p r dµ(w) r p dA(z) 1 r . (3.3) (4) Let p ≥ 2 and r ≥ p. Then J µ : A p → L p (µ) is r-summing if and only if D 1 (1 -|z| 2 ) 2 dµ(z) < ∞ . Moreover, we have π r (J µ ) ≈ π p (J µ ) ≈ ∞ k=1 µ(D k ) |D k | 1/p ≈ n 4 n µ(Γ n ) 1/p ≈ D 1 (1 -|z| 2 ) 2 dµ(z) 1/p (3.4)
Let us mention several remarks. (R1) We shall prove below (see Prop.4.1) that, for any α > 0,

∞ k=1 µ(D k ) |D k | α ≈ n≥0 0≤ j<2 n µ(R n, j ) |R n, j | α
and in Prop.4.2 the equivalence of the estimates with integrals, explaining why we obtain equivalent estimates of the π r -norm in the preceding theorem. (R2) The case p = 2 reduces to the characterization of Hilbert-Schmidt operator which is easy. More precisely, J µ is r-summing on the Hilbert space A 2 for some r ≥ 1 if and only if J µ is a Hilbert-Schmidt operator. Indeed, let {e n = ( √ n + 1)z n , n ∈ N} an orthonormal basis of A 2 , then we have

J µ HS = n≥1 J µ (e n ) 2 L 2 (µ) 1/2 = n≥1 D (n + 1)|z| 2n dµ(z) 1/2 = D dµ(z) (1 -|z| 2 ) 2 1/2 . (R3)
The characterizations in the main theorem actually do not depend on the choice of t and r 0 (as soon as 0 < t < 1/4 and 0 < r 0 < 1).

Characterization of absolutely summing weighted composition operators:

As an application of our main results, we can characterize r-summing weighted composition operators for any r ≥ 1 and p > 1. Let uC ϕ be the weighted composition operator (formally) defined by

(uC ϕ )( f ) = u • ( f • ϕ)
, where u : D → C is an analytic function and ϕ : D → D is an analytic map. When u = 1, we simply write C ϕ , which is the composition operator with symbol ϕ.

We consider the measure dµ = (|u| p dA) ϕ and we have:

(uC ϕ ) f A p = D | f • ϕ| p |u| p dA 1/p = f L p (µ) .
Therefore, uC ϕ is r-summing if and only if J µ is r-summing, hence it suffices to apply the main theorem to the measure µ. Theorem 3.1. Let ϕ : D → D be an analytic map and u : D → C be an analytic function. Let uC ϕ be the weighted composition operator viewed from A p to A p and dµ = (|u| p dA) ϕ , then uC ϕ is r-summing if and only if the corresponding π r (uC ϕ ) norm is finite, where

(1) When 1 < p ≤ 2 , for any r ≥ 1,

π r (uC ϕ ) ≈ π 2 (uC ϕ ) ≈ n, j 4 n µ(R n, j ) 2/p 1/2 ≈ D D |u(w)| p |1 -zϕ(w)| p+2 dA(w) 2 p dA(z) 1 2 . (2) When p ≥ 2 , for 1 ≤ r ≤ p , π r (uC ϕ ) ≈ π 1 (uC ϕ ) ≈ n, j 4 n µ(R n, j ) p /p 1/p ≈ D D |u(w)| p |1 -zϕ(w)| 2p dA(w) p p dA(z) 1 p . (3) When p ≥ 2 , for p ≤ r ≤ p, π r (uC ϕ ) ≈ n, j 4 n µ(R n, j ) r/p 1/r ≈ D D |u(w)| p |1 -zϕ(w)| 2+ 2p r dA(w) r p dA(z) 1 r . ( 4
) When p ≥ 2 , for p ≤ r, π r (uC ϕ ) ≈ n, j 4 n µ(R n, j ) 1/p ≈ D |u| p (1 -|ϕ| 2 ) 2 dA 1/p .
As an immediate corollary we get a characterization for composition operators, recovering the results of Domenig (which were expressed in terms of Luecking windows in [START_REF] Domenig | Composition operators belonging to operator ideals[END_REF], [START_REF] Domenig | Composition operators on weighted Bergman spaces and Hardy spaces[END_REF]).

Corollary 3.2. Let ϕ : D → D be an analytic map. On A p , the composition operator C ϕ with symbol ϕ is r-summing if and only if the corresponding π r (C ϕ ) norm is finite, where

(1) When 1 < p ≤ 2 , for any r ≥ 1, π r (C ϕ ) ≈ π 2 (C ϕ ) ≈ n, j 4 n A ϕ (R n, j ) 2/p 1/2 ≈ D D 1 |1 -zϕ(w)| p+2 dA(w) 2 p dA(z) 1 2 . (2) When p ≥ 2 , for 1 ≤ r ≤ p , π r (C ϕ ) ≈ π 1 (C ϕ ) ≈ n, j 4 n A ϕ (R n, j ) p /p 1/p ≈ D D 1 |1 -zϕ(w)| p+2 dA(w) 2 p dA(z) 1 2 (3) When p ≥ 2 , for p ≤ r ≤ p, π r (C ϕ ) ≈ n, j 4 n A ϕ (R n, j ) r/p 1/r D D 1 |1 -zϕ(w)| 2+ 2p r dA(w) r p dA(z) 1 r . ( 4
) When p ≥ 2 , for p ≤ r, π r (C ϕ ) ≈ D dA (1 -|ϕ| 2 ) 2 1/p .
Another consequence of the main theorem is the following corollary:

Corollary 3.3. Let p > 1, for any two Carleson measures µ and ν, if µ(R n, j ) ≈ ν(R n, j ) for all (n, j) , then we have π r (J µ ) ≈ π r (J ν ).
Of course, there is a similar statement in terms of the family (D k ).

TOOLS AND PRELIMINARY RESULTS

In this section, we present several ingredients employed in the solution of problem (P). We begin with some estimates useful to prove the equivalence of the different viewpoints (hyperbolic disks, Luecking windows, or integrals) in our characterizations in the main theorem.

First of all we prove the following proposition proving the equivalent formulation either with help of Luecking rectangles or the family (D k ) k .

Proposition 4.1. Let a, α > 0. Then, we have

∞ k=1 µ(D k ) α |D k | a ≈ n≥0 0≤ j<2 n µ(R n, j ) α |R n, j | a .
Proof. Let us mention that R n, j can be covered by a uniform finite number of D k (actually, this is stated without proof in pages 339 and 344 in [START_REF] Luecking | Embedding theorems for spaces of analytic functions via Khinchine's inequlity[END_REF] and we shall give here an explanation but we postpone it until the end of our proof): there exists N > 0 (not depending on n and j)

such that R n, j ⊂ k∈J n, j D k (4.1)
where

J n, j = {k ≥ 1; D k ∩ R n, j ∅} and #(J n, j ) ≤ N.
In the same way, D k can be covered by a (uniform) finite number of R n, j : there exists N > 0 (not depending on k) such that

D k ⊂ (n, j)∈E k R n, j (4.2)
where

E k = {(n, j) ; D k ∩ R n, j ∅} and #(E k ) ≤ N .
We claim that when k ∈ J n, j , we have |R n, j | ≈ |D k |. Indeed, by definition, for any k ∈ J n, j , there exists some z ∈ R n, j ∩ D k ∅. But since z ∈ R n, j , we have 1 -|z| ≈ 2 -n . Therefore, since z ∈ D k as well, we obtain that 2

-n ≈ 1 -|z| ≈ 1 -|a k | . It is well known that |D k | ≈ (1 -|a k |) 2 , hence we get that |R n, j | ≈ 4 -n ≈ |D k | . (4.
3) Since #(J n, j ) ≤ N for any n ≥ 0 and for any j ∈ {0, 1, ..., 2 n -1}, we have from (4.1) the following estimate:

µ(R n, j ) α = k∈J n, j µ(D k ∩ R n, j ) α ≤ k∈J n, j µ(D k ) α k∈J n, j µ(D k ) α .
We get, using also (4.3) and (4.2),

n≥0 0≤ j<2 n µ(R n, j ) α |R n, j | a n≥0 0≤ j<2 n k∈J n, j µ(D k ) α |D k | a = ∞ k=1 (n, j)∈E k µ(D k ) α |D k | a = ∞ k=1 #(E k ) µ(D k ) α |D k | a ∞ k=1 µ(D k ) α |D k | a •
By symmetry of the properties if the two families of sets (the R n, j 's and the D k 's), the reverse inequality is proved in the same way. Now let us justify (4.2) and (4.1). Of course, the relation R n, j ⊂ k∈J n, j D k is obvious and the real interesting fact is that J n, j is uniformly finite. Let us fix an integer (n, j) with 0 ≤ j < 2 n and k ∈ J n, j . there exists

w ∈ D k ∩ R n, j ∅, so 1 -a k ≈ 1 -w ≈ 2 -n .
On the other hand, since the D k 's are disjoint, we have

k∈J n, j D k = k∈J n, j D k . But k∈J n, j D k ⊂ D z n, j , λ2
-n where z n, j is the "center" of R n, j and λ > 0 does not depend on n. Indeed every z belonging to some D l with l ∈ J n, j satisfies

d z n, j , z ≤ d z n, j , w + d w, z 2 -n + 1 -a l 2 -n
where w ∈ D l ∩ R n, j . We obtain

k∈J n, j 4 -n ≈ k∈J n, j D k = k∈J n, j D k D z n, j , λ2 -n ≈ 4 -n .
Therefore J n, j is finite and there exists N (not depending on (n, j)) such that

4 -n # J n, j ≤ N4 -n
which proves (4.1). Clearly (4.2) can be proved in the same way.

We continue with some estimates comparing integrals.

Proposition 4.2. Let 0 < t < 1/4, and the set {D j } as defined in Lemma 2.6. Let µ be a positive finite measure on the unit disk D, η, γ > 0. Then we have

D D(z,2t) 1 (1 -|w| 2 ) η/γ dµ(w) γ dA(z) ≈ D µ(D(z, 2t)) γ (1 -|z| 2 ) η dA(z) ≈ ∞ j=1 µ(D j ) γ |D j | η/2-1 (4.4)
Moreover when η > max(γ, 2) then we have

D D(z,2t) 1 (1 -|w| 2 ) η/γ dµ(w) γ dA(z) ≈ D D 1 |1 -wz| η/γ dµ(w) γ dA(z) (4.5)
with constants depending only on t, η and γ.

In particular, when γ = 1 and η = 4 we have:

D 1 (1 -|z| 2 ) 2 dµ(z) ≈ ∞ j=1 µ(D j ) |D j | •
Proof. Let us begin with the particular case. Since the family (D j ) j≥1 forms a partition of D and (1

-|z| 2 ) 2 ≈ |D j | whenever z ∈ D j , we get D 1 (1 -|z| 2 ) 2 dµ(z) = j≥1 D j dµ(z) (1 -|z| 2 ) 2 ≈ j≥1 µ(D j ) |D j | .
Obviously the first quantity is equivalent to the second one because

1 -|z| 2 ≈ 1 -|w| 2 , for w ∈ D(z, 2t) . Now, again since (D j ) j≥1 forms a partition of D, D j ⊂ D(z, 2t) and (1 -|z| 2 ) 2 ≈ |D j | whenever z ∈ D j , we get D µ(D(z, 2t)) γ (1 -|z| 2 ) η dA(z) = j≥1 D j µ(D(z, 2t)) γ (1 -|z| 2 ) η dA(z) j≥1 D j µ(D j ) γ |D j | η/2 dA(z) = j≥1 µ(D j ) γ |D j | η/2 |D j | = ∞ j=1 µ(D j ) γ |D j | η/2-1 •
Hence, it remains to prove the reverse inequality: for every z ∈ D, we consider the set I z = j; D(z, 2t) ∩ D j ∅ . We claim that the cardinal

|I z | ≤ C , (4.6)
where the constant C depends only on t. Indeed, let j 1 , ..., j n distincts such that D(z, 2t)∩D j i ∅. Then there exists w i ∈ D(z, 2t) such that w i ∈ D j i ⊂ D(a j i , t), with 1 ≤ i ≤ n. Take w ∈ 1≤i≤n D(a j i , t/4), then w belongs to at least one D(a j i , t/4); and we have

β(w, z) ≤ β(w, a j i ) + β(a j i , w i ) + β(w i , z) < t/4 + t + 2t < 4t.
Hence, w ∈ D(z, 4t) and we get n i=1 D(a j i , t/4) ⊂ D(z, 4t) . Therefore, since the disks are disjoint, we have

n i=1 |D(a j i , t/4)| ≤ |D(z, 4t)| .
In other words, we have

n i=1 (1 -|a j i | 2 ) 2 ≤ (1 -|z| 2 ) 2 and since 1 -|a j i | 2 ≈ 1 -|z| 2 when z ∈ D satisfies D(z, 2t) ∩ D j i ∅,
we obtain that n ≤ C, where the constant C depends only on t, which is our claim (4.6).

Since the family (D j ) j≥1 forms a partition of D, we have

µ(D(z, 2t)) = j∈I z µ(D j ∩ D(z, 2t)). (4.7)
Hence, from (4.6) and (4.7), we get

µ(D(z, 2t)) γ j∈I z µ(D j ) γ ,
where the underlying constant depends only on t and γ. For every z ∈ D such that j ∈ I z , we have (1

-|z| 2 ) ≈ (1 -|a j | 2 ) ≈ |D j | 1/2 . We obtain D µ(D(z, 2t)) γ (1 -|z| 2 ) η dA(z) j D µ(D j ) γ (1 -|z| 2 ) η 1 { j∈I z } dA(z) j µ(D j ) γ |D j | η/2 | z ∈ D; D(z, 2t) ∩ D j ∅ |
where 1 { j∈I z } = 1 when j ∈ I z and 0 else. Observe that

| z ∈ D; D(z, 2t) ∩ D j ∅ | ≤ |D(a j , 3t)| ≈ (1 -|a j | 2 ) 2 ≈ |D j | . Finally, D µ(D(z, 2t)) γ (1 -|z| 2 ) η dA(z) ∞ j=1 µ(D j ) γ |D j | η/2-1 •
This ends the proof of (4.4). Now let us focus on (4.5). First assume that γ > 1. If µ is the Dirac at point 0 then it is easily checked, so that we may assume that µ({0}) = 0. The argument follows almost directly from a result stated in [START_REF] Peláez | Embedding Bergman spaces into tent spaces[END_REF]Th.3(iii)] (see especially (c),(d),(e); see [20, Th.2] too): for any ν positive measure on D, with ν({0}) = 0, and q > p > 1, we have for λ > 1 (we must mention that actually the hypothesis in [START_REF] Peláez | Embedding Bergman spaces into tent spaces[END_REF] is "for λ large enough" but following carefully the proof, it appears that λ > 1 is sufficient when we restrict to the case q > p > 1, and not every q > p > 0).

Ψ L p/(p-q) ≈ Φ L p/(p-q) ≈ Υ L p/(p-q) . (4.8)
where

Ψ(z) := D (1 -|w|) λ-2 |1 -wz| λ dν(w) Φ(z) := ν(D z ) (1 -|z| 2 ) 2 and Υ(z) := Σ z dν(w) (1 -|w| 2 ) 2 . and Σ z = ξ ∈ D| arg(z) -arg(ξ)| < 1 2 (1 -|ξ| |z| )
is the non-tangential approach region. In other words, we have

D D (1 -|w|) λ-2 |1 -wz| λ dν(w) p/(p-q) dA(z) ≈ D D z 1 (1 -|w|) 2 dν(w) p/(p-q) dA(z) ≈ D Σ z 1 (1 -|w|) 2 dν(w) p/(p-q) dA(z)
Now we apply this with the measure dν = (1 -|w| 2 ) 2-η γ dµ, q = 2 and p = 2γ γ -1 > 2. We also choose λ = η γ > 1 and we get the conclusion once we point out again that 1 -|w| ≈ 1 -|z|

when w ∈ D z .
Finally we assume that γ ≤ 1 and that (n, j)

2 (η-2)n µ(R n, j ) γ is finite. With the preceding
proposition and the first part of our proof, it is equivalent to

D D(z,2t) 1 (1 -|w| 2 ) η/γ dµ(w) γ dA(z) < +∞
Of course there an obvious inequality since |1 -wz| 1 -|z| ≈ 1 -|w| 2 when w ∈ D(z, 2t). On the other hand, we have

D D 1 |1 -wz| η/γ dµ(w) γ dA(z) = (m,k) R m,k (n, j) R n, j 1 |1 -wz| η/γ dµ(w) γ dA(z) which is lower than (m,k) (n, j) 4 -m µ(R n, j ) γ sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j . (4.9)
since

• 1 ≤ • γ (recall that γ ≤ 1)
. On the other hand, we observe that, for w z, 1 |1 -wz| ≤ 1 |w -z| hence when there is no contact between R m,k and R n, j , then z ∈ R m,k and w ∈ R n, j are different

and 1 |1 -wz| ≤ d(R m,k , R n, j ) -1 .
This explains why we are going to split our argument into several cases. More precisely, we consider the set V (n, j) of indexes (m, k) such that R m,k ∩ R n, j ∅. It is a (uniformly) finite set (cardinality less than 9). Moreover (n, j) belongs to V (m,k) if and only if (m, k) belongs to V (n, j) . We get an upper estimate for these terms:

(n, j) (m,k)∈V (n, j) 4 -m µ(R n, j ) γ 2 nη (n, j) 4 -n µ(R n, j ) γ 2 nη < ∞ (4.10) using the fact that m ≈ n when (m, k) ∈ V (n, j) and that #V (n, j) 1. When (n, j) does not belong to V (m,k) then d(R m,k , R n, j
) can be under estimated by 2 -n when m > n (looking the radius), but also by |k2 -m -( j+1)2 -n | or |(k+1)2 -m -j2 -n | (up to a constant) according on which side the rectangles are from each other, when we focus on the argument. Indeed, we have 2 sin(θ/2) ≈ θ, moreover the radius 1 -2 -max(n,m) may be considered as close to 1 since we focus on what happens near the boundary. Now, we focus on an upper estimate for the quantity

(m,k) (n, j) V (m,k) n≥m 4 -m µ(R n, j ) γ sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j (n, j) l≥1 n≥m 4 -m µ(R n, j ) γ l 2 m -η . Indeed d(R m,k , R n, j
) is larger (up to a constant) than the difference of angles between the borders of the corresponding Luecking rectangles and by symmetry, the value of these differences runs over {2π(d

2 -m -2 -n ) | 2 ≤ d ≤ 2 m-1
, n ≥ m} (the values are taken at most two times by symmetry)). We get Finally we focus on

(m,k) (n, j) V (m,k) n≥m 4 -m µ(R n, j ) γ sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j ≤ (n, j) 4 -n µ(R n, j ) γ 2 nη
(m,k) (n, j) V (m,k) n<m 4 -m µ(R n, j ) γ sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j which is the same than (n, j) µ(R n, j ) γ (m,k) V (n, j) m>n 4 -m sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j .
We split the set of the values of "k" such that (m, k) V (n, j) into two subsets according that R m,k touches the sector {z ∈ D| arg(z) ∈ [2π j2 -n , 2π( j + 1)2 -n ]} or not. We denote S m,n, j the set of values of k such that R m,k touches this sector: it has a cardinality 2 m-n + 2 so, taking into account the radius, the sum

4 -m k∈S m,n, j sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j is lower (up to a constant) than 4 -m k∈S m,n, j 1 2 -n -2 -m -η 4 -m 2 nη 2 m-n .
Summing over m > n, we get

m>n k∈S m,n, j 4 -m sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j 4 -n 2 nη .
Taking into account the argument (like in (4.11)) and the symmetry if the situation, the sum 1) .

4 -m k S m,n, j sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j is lower (up to a constant) than 4 -m d≥2 m-n 1 d2 -m η 4 -m 2 mη 2 (m-n)(η-1) = 2 -m 2 n(η-
Summing over m > n, we get

m>n k S m,n, j 4 -m sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j 4 -n 2 nη .
Gathering the previous estimates, we obtain

(m,k) (n, j) V (m,k) n<m 4 -m µ(R n, j ) γ sup 1 |1 -wz| η z ∈ R m,k , w ∈ R n, j (n, j) µ(R n, j ) γ 4 -n 2 nη (4.12)
which is the expected result in this case.

The inequalities (4.10), (4.11) and (4.12) together gives the final conclusion. Now, we focus on preliminary results about the membership of various ideals of operators.

The following proposition belongs to the folklore. We give a proof for sake of completeness.

Proposition 4.3. Let µ be a Carleson measure on D and p ≥ 1. Then J µ :

A p → L p (µ) is order bounded if and only if D 1 (1 -|z| 2 ) 2 dµ(z) < ∞ .
Proof. J µ is order bounded if and only if there exists some h ∈ L p (µ) such that for every

f ∈ B A p , we have | f | ≤ h a.e on D. Since A p is separable, it follows that J µ is order bounded if and only if D sup f ∈A p f ≤1 | f (z)| p dµ(z) < ∞ , which is equivalent to D δ z p dµ(z) < ∞ ,
where δ z is the point evaluation at z ∈ D, viewed as a functional on A p . It is well known that δ z = 1/(1 -|z| 2 ) 2/p and this completes the proof.

For p ≥ 1, the multiplier operator M β on p is defined by

M β (e n ) = β n e n , n = 1, 2, • • • , (4.13)
where {e n } n≥1 denotes the canonical basis of p and β = (β 1 , β 2 , . . .) is a sequence of complex numbers. The following result is usually referred to as the atomic decomposition for Bergman spaces, and a proof can be found in [START_REF] Zhu | Operator Theory in Function Spaces[END_REF]. Then there exists a constant δ > 0 such that for any t-lattice {a k } in the Bergman metric, where 0 < t < δ, the space A p consists exactly of functions of the form

f (z) = ∞ k=1 c k f k (z) , (4.19)
where

f k (z) = (1 -|a k | 2 ) (pb-2)/p
(1āk z) b and {c k } ∈ p . The series in (4. [START_REF] Peláez | Embedding Bergman spaces into tent spaces[END_REF]) converges in A p -norm and there exist constants C 1 and C 2 (depending only on p and δ) such that

C 1 f p ≤ inf ∞ k=1 |c k | p : {c k } satisfies (4.19) ≤ C 2 f p .
We can see from this lemma that for any z ∈ D k ,

| f k (z)| p = (1 -|a k | 2 ) pb-2 |1 -āk z| pb ≈ 1 (1 -|a k | 2 ) 2 ≈ |D k | -1 .
We will need the following important lemma.

| f (z)| p ≤ C (1 -|z| 2 ) 2 D(z,t) | f (w)| p dA(w)
for all f ∈ H(D) and every z ∈ D.

We can get the following proposition, where the { f k } are defined as in Lemma 4.5 : Proposition 4.7. Let p ≥ 1 and µ be a Carleson measure on A p . We fix a t-lattice {a k }. Define the operator ∆ from p to L p (µ) by

∆ : c = {c k } ∈ p -→ f = ∞ k=1 c k f k 1 D k ∈ L p (µ), (4.20)
and let β be the sequence defined by:

β = µ(D 1 ) |D 1 | 1/p , µ(D 2 ) |D 2 | 1/p , . . . , µ(D k ) |D k | 1/p , . . . .
Then ∆ is an r-summing operator if and only if M β is r-summing on p . Moreover, π r (∆) ≈ π r (M β ). (4.21)

Proof. Without loss of generality, we may (and do) assume that µ(D k ) 0 for every k.

First, we consider the following composition:

p M β -→ p Ψ -→ L p (µ) ,
where Ψ is defined by: for any

c = {c k } ∈ p Ψ(c)(z) = ∞ k=1 c k |D k | µ(D k ) 1/p f k (z) 1 D k (z) .
Now we prove that Ψ is bounded from p to L p (µ):

Ψ(c) L p (µ) = D ∞ k=1 c k |D k | µ(D k ) 1/p f k (z) 1 D k (z) p dµ(z) 1/p = ∞ k=1 D k |c k | p |D k | µ(D k ) | f k (z)| p dµ(z) 1/p ≈ ∞ k=1 |c k | p 1/p = c p , (4.22)
where we used in (4.22) the fact that

| f k (z)| p ≈ |D k | -1 on D k . This implies that Ψ is a bounded operator with norm ≈ 1. It turns out that, ∆ = Ψ • M β . Furthermore, we have π r (∆) Ψ π r (M β ) ≈ π r (M β ) . (4.23) Now, let Z = span{ f k 1 D k } L p (µ)
which is a closed subspace of L p (µ). Let us consider :

Ψ : p -→ Z c -→ ∞ k=1 c k |D k | µ(D k ) 1/p f k 1 D k .
Then Ψ satisfies Ψ(c) Z ≈ c p . Hence the range of Ψ is closed, and contains the Ψ(e k )'s, a fortiori the vector space spanned by the f k 1 D k . We obtain that Im Ψ is closed and contains Z, therefore Ψ is onto. It follows that Ψ is an isomorphism.

When f = ∞ k=1 c k f k 1 D k , then Ψ -1 ( f ) = c k β k k≥1
Hence,

M β = Ψ -1 • ∆ ,
where ∆ is the map ∆ with range Z. It turns out that Proof. We assume that J µ is an r-summing operator for some r ≥ 1. Let δ = (δ 1 , δ 2 , . . . , δ k , . . .) be a sequence of Rademacher variables on a probability measurable space (Ω, Σ, P) taking their values in {±1}.

π r (M β ) π r ( ∆) = π r (∆). ( 4
We define an operator Φ ω as follows: for every ω ∈ Ω,

Φ ω : p -→ A p c -→ k c k δ k (ω) f k ,
where { f k } are defined as in Lemma 4.5. Using Lemma 4.5 again, Φ ω is bounded from p to A p with norm less than 1.

Define for every ω ∈ Ω

M ω : f ∈ L p (µ) -→ ∞ k=1 δ k (ω)1 D k f ∈ L p (µ).
Then it is easy to check that M ω is an isometry, since it is actually a multiplier by the function M ω where M ω (z) = δ k (ω) when z ∈ D k . We clearly have |M ω | = 1 everywhere since the family (D k ) k forms a partition. Now we can take the expectation E with respect to the measure P which satisfies E(δ j δ k ) = δ jk , with δ jk is the Kronecker symbol. Then for each k, we have

E M ω • J µ • Φ ω e k = Ω j δ j (ω)1 D j f k δ k (ω) dP = 1 D k f k = ∆e k ,
where ∆ is defined as in Proposition 4.7. It follows that

∆ = E M ω • J µ • Φ ω .
Then the convexity and the properties of ideal norm imply that

π r (M β ) ≈ π r (∆) ≤ π r (J µ ).
This proves the proposition.

BEREZIN TRANSFORM

The purpose of this section is to study the boundedness of the Berezin transform viewed from L p to L q (µ) for p, q > 0 and µ a positive Borel measure. Let us start with the definition of Berezin transform which is defined on D by

B f (z) = D (1 -|z| 2 ) 2 f (w) |1 -wz| 4 dA(w), for z ∈ D,
where f ∈ L 1 . We refer the reader to [START_REF] Zhu | Operator Theory in Function Spaces[END_REF] for more details about the Berezin transform properties.

In order to prove the main result in this section, we need various tools. We know that B f is not subharmonic (in general) but the following proposition provides us with a very useful estimate for our purpose.

Proposition 5.1. Let 0 < q < ∞ and r > 0. Then we have

B f q (z) 1 |D(z, r)| D(z,r) B f q (w) dA(w)
for every measurable f ≥ 0 and z ∈ D, where the underlying constant depends only on r.

Proof. We begin with the case q = 1. Using Fubini's theorem and the fact that

|D(z, r)| ≈ (1 -|z| 2 ) 2 ≈ (1 -|w| 2 ) 2 ,
whenever w ∈ D(z, r), we get:

1 |D(z, r)| D(z,r) D (1 -|w| 2 ) 2 |1 -ūw| 4 f (u) dA(u) dA(w) ≈ D f (u) 1 (1 -|z| 2 ) 2 D(z,r) (1 -|w| 2 ) 2 |1 -ūw| 4 dA(w) dA(u) .
We can see from Lemma 4.30 in [START_REF] Zhu | Operator Theory in Function Spaces[END_REF] that |1 -ūz| ≈ |1 -ūw| for every z, w, u in D with β(z, w) < r, hence

1 |D(z, r)| D(z,r) B f (w) dA(w) ≈ D f (u) 1 (1 -|z| 2 ) 2 D(z,r) (1 -|z| 2 ) 2 |1 -ūz| 4 dA(w) dA(u) ≈ D f (u) (1 -|z| 2 ) 2 |1 -ūz| 4 dA(u) = B( f )(z).
This special case together with Jensen's inequality (point out that dA/|D(z, r)| is a probability measure on D(z, r)) implies that the desired inequality holds for every q ≥ 1:

B f q (z) ≈ D(z,r) B f (w) dA(w) |D(z, r)| q D(z,r) B f q (w) dA(w) |D(z, r)| = 1 |D(z, r)| D(z,r)
B f q (w) dA(w).

To deal with the case q ∈ (0, 1): we fix z 0 ∈ D, then we have

P w (u) = 1 -|w| 2 |1 -ūw| 2 ≈ 1 -|z| 2
|1 -ūz| 2 = P z (u), for w, z ∈ D(z 0 , r) and u ∈ D, where the constants depend only on r. Indeed, from Lemma 4.30 in [START_REF] Zhu | Operator Theory in Function Spaces[END_REF], we have |1 -ūz| ≈ |1ūz 0 | for every z, z 0 , u in D with β(z, z 0 ) < r and |1 -ūw| ≈ |1ūz 0 | for every z 0 , w, u in D with β(w, z 0 ) < r where the constants depend only on r. Moreover, it is well known that 1 -|w| 2 ≈ 1 -|z 0 | 2 whenever β(z 0 , w) < r and 1 -|z| 2 ≈ 1 -|z 0 | 2 whenever β(z 0 , z) < r (cf. [26, p.69)]. Hence, we can see that there exist positive constants C 1 , C 2 which depend only on r such that

C 1 P z (u) ≤ P z 0 (u) ≤ C 2 P z (u)
and C 1 P w (u) ≤ P z 0 (u) ≤ C 2 P w (u) for all u ∈ D and w, z ∈ D(z 0 , r). Therefore, there exist C and C which depend only on r such that CP z (u) ≤ P w (u) ≤ C P z (u) for all u ∈ D and w, z ∈ D(z 0 , r). Now we can focus on our estimate. Since |D(z 0 , r)| ≈ (1 -|z 0 |) 2 , we get for every z ∈ D(z 0 , r):

1 |D(z 0 , r)| D(z 0 ,r) D (1 -|w| 2 ) 2 |1 -ūw| 4 f (u) dA(u) dA(w) ≈ 1 (1 -|z 0 |) 2 D f (u) D(z 0 ,r) (1 -|w| 2 ) 2 |1 -ūw| 4 dA(w) dA(u) ≈ 1 (1 -|z 0 |) 2 D f (u) D(z 0 ,r) (1 -|z| 2 ) 2 |1 -ūz| 4 dA(w) dA(u) ≈ |D(z 0 , r)| (1 -|z 0 | 2 ) 2 D f (u) (1 -|z| 2 ) 2 |1 -ūz| 4 dA(u) ≈ B( f )(z) . It follows that sup z∈D(z 0 ,r) B f (z) 1 (1 -|z 0 | 2 ) 2 D(z 0 ,r) B f (w) dA(w) 1 (1 -|z 0 | 2 ) 2 D(z 0 ,r) (B f ) q (w) dA(w) sup w∈D(z 0 ,r) B( f ) 1-q (w) .
Hence,

B( f ) q (z 0 ) ≤ sup z∈D(z 0 ,r) (B f ) q (z) 1 (1 -|z 0 | 2 ) 2 D(z 0 ,r) (B f ) q (w) dA(w) .
The proposition is proved for any q > 0.

We state as a lemma the following result on Bergman Carleson embeddings: Lemma 5.2. [17, Theorem 1] Let 0 < q < p < ∞, µ be a positive measure on D and K

(z) = µ(D z )/|D z |, z ∈ D. Then D | f (z)| q dµ(z) 1/q D | f (z)| p dA(z) 1/p
for every f ∈ A p if and only if K ∈ L p/(p-q) . Note that this condition is independent of the choice of r 0 (cf. Lemma 2 in [START_REF] Luecking | Multipiers of Bergman spaces into Lebesgue spaces[END_REF]).

Moreover,

J µ A p →L q (µ) ≈ D D z 1 (1 -|w| 2 ) 2 dµ(w) p/(p-q) dA(z) (p-q)/pq .
The following crucial lemma is certainly well known from the specialists . Lemma 5.3. Let 0 < p ≤ q < ∞ and 0 < r < 1. Let µ be a positive Borel measure on D. Then the following assertions are equivalent :

1) The Bergman space A p is boundedly contained in L q (µ).

2) µ(D(z, r))/|D(z, r)| q/p is bounded on D.

3) µ(D(a k , r))/|D(a k , r)| q/p is bounded on D for the sequence r-lattice {a k } as described in Section 2.

Proof. The equivalence of (1) and ( 2) is proved in [START_REF] Luecking | Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives[END_REF]. (2) implies (3) is obvious. For the proof of (3) implies (1) see Theorem 3.1 in [START_REF] Tong | Carleson measures on the weighted Bergman spaces with Békollé weights[END_REF]. Now we can state the main result of this section.

Theorem 5.4. Let p > 1 and q > 0. Let µ be a positive Borel measure on D. Then the following assertions are equivalent:

1) The Berezin transform B viewed from L p to L q (µ) is bounded;

2) The embedding J µ :

A p → L q (µ) is bounded. Moreover, B ≈ J µ Proof.
For the proof we need to split cases: (a) The case 0 < q < p < ∞: First, we assume that B is bounded from L p to L q (µ). Let us consider this composition

A p i p -→ L p B -→ L q (µ)
where i p is the natural embedding map. From the fact that (cf. [26, Proposition 6.13]) Bg = g for every g ∈ A p , it follows that J µ = B • i p . Since i p is a contraction, we get

J µ ≤ B .
Now assume that J µ is bounded from A p to L q (µ). From now on, we may assume that f ≥ 0 since |B f | ≤ B(| f |), and f L p = | f | L p . Using Theorem 5.1 with r > 0 , we have

B f q (w) 1 (1 -|w| 2 ) 2 D(w,r) B f q (z) dA(z) .
The sufficiency will be clarified by the following computation:

D B f (w) q dµ(w) D 1 (1 -|w| 2 ) 2 D(w,r) B f (z) q dA(z) dµ(w) = D 1 (1 -|w| 2 ) 2 D 1 D(w,r) (z) B f (z) q dA(z) dµ(w) = D B f (z) q D 1 D(w,r) (z) (1 -|w| 2 ) 2 dµ(w) dA(z), to Fubini's theorem. We get, D B f (w) q dµ(w) D B f (z) q D 1 D(z,r) (w) (1 -|z| 2 ) 2 dµ(w) dA(z), since 1 D(w,r) (z) = 1 D(z,r) (w) and 1 -|z| 2 ≈ 1 -|w| 2 whenever β(z, w) < r . We obtain, D B f (w) q dµ(w) D B f (z) q µ(D(z, r)) (1 -|z| 2 ) 2 dA(z) ≈ D B f (z) q K(z) dA(z),
where

K(z) = µ(D(z, r)) (1 -|z| 2 )
2 is introduced in Lemma 5.2. Since J µ is bounded from A p to L p (µ), Lemma 5.2 implies that K ∈ L p/p-q and is a multiplier from L p to L q , hence

D B f (w) q dµ(w) D B f (z) p dA(z) q/p K L p/(p-q) .
Finally, we conclude that

D B f (w) q dµ(w) f q L p . K L p/(p-q) ≈ f q L p . J µ q A p →L q (µ) , since K L p/(p-q) ≈ J µ q A p →L q (µ) (recall Lemma 5.
2) and B is bounded from L p to L p . We get, B J µ .

(b) The case 1 < p ≤ q < ∞ : Once again, we only need to prove that (2) implies (1). Using Theorem 5.1 with r > 0, the fact that D(w, r) ⊂ D(a k , 2r) and |D(w, r)| ≈ |D(a k , r)| whenever w ∈ D(a k , r), we have

D B f (w) q dµ(w) k≥1 D(a k ,r) 1 |D(w, r)| D(w,r) B f (z) q dA(z) dµ(w) k≥1 D(a k ,r) 1 |D(a k , r)| D(a k ,2r) B f (z) q dA(z) dµ(w) = k≥1 µ(D(a k , r)) |D(a k , r)| D(a k ,2r) B f (z) q dA(z) .
We claim that

B f (z) f A p (1 -|z| 2 ) 2/p .
Indeed, it is clear for p = 1. When p > 1, using Hölder's inequality and Lemma 3.10 in [START_REF] Zhu | Operator Theory in Function Spaces[END_REF], we have |D(a k , r)| (q-p)/p B f (z)

B f (z) = D (1 -|z| 2 ) 2 |1 -zw| 4 f (w) dA(w) ≤ f A p D (1 -|z| 2 ) 2 |1 -zw| 4 p dA(w) 1/p f A p 1 (1 -|z| 2 ) 2/p • Now,
q-p B f (z) p dA(z) f q-p A p k≥1 D(a k ,2r) B f (z) p dA(z) f q-p A p B f p L p f q
A p , where we used the fact that B is bounded from L p to L p .

THE CASE 1 < p ≤ 2

Let us recall that, since 1 < p ≤ 2, A p and L p (µ) have both cotype 2, we know that π r (J µ ) ≈ π 2 (J µ ) for any r ≥ 1. Therefore, in this section we focus on 2-summing operators. Theorem 6.1. Let 1 < p ≤ 2 and µ be a positive measure on D. The following assertions are equivalent:

1) J µ :

A p → L 2 (µ) is 2-summing.
2)

D D dµ(z) |1 -zw| 4 p /2 dA(w) < ∞ .
3) The Bergman projection viewed from L p to L 2 (µ)

T : f ∈ L p -→ T f ∈ L 2 (µ) with T ( f )(z) = D f (w) (1 -zw) 2 dA(w) is 1-summing.
Moreover, we have

π 2 J µ : A p → L 2 (µ) ≈ D D dµ(z) |1 -zw| 4 p /2 dA(w) 1/p . (6.1)
Proof. We assume that p < 2 since the case p = 2 is clear (cf. Remarks in Section 3). 1)⇒2). For r ∈ (0, 1) and for w ∈ D, we consider the function

k w (z) = 1 (1 -rzw) 2 , z ∈ D,
then k w ∈ A p for every w ∈ D . For any ξ in the unit ball of the dual of A p , there exists a function g 1 in the unit ball of L p , such that

ξ(k w ) = D k w (z)g 1 (z)dA(z) = g(rw) ,
where Pg 1 = g and P is the Bergman projection from L p onto A p which is bounded. Then Then, since p ≥ 2, thanks to Fubini's Theorem and the definition of p -summing, we have

π 2 (J µ ) ≥ π p (J µ ) ≥ 1 P D k w p L 2 (µ) dA(w) 1/p = 1 P D D dµ(z) |1 -rzw| 4 p /2 dA(w) 1/p .
Let r → 1 -and thanks to the Fatou's Lemma, we obtain

D D dµ(z) |1 -zw| 4 p /2 dA(w) 1/p ≤ P π 2 (J µ ) . 2)⇒3). Consider, R : f ∈ L 2 (µ) -→ R( f ) ∈ L p formally defined by R( f )(z) = D f (w) (1 -zw) 2 dµ(w),
which is an analytic function on D.

We are going to justify that R actually defines an order bounded operator. Indeed, for every z ∈ D, we have sup

f L 2 (µ) ≤1 D f (w) (1 -zw) 2 dµ(w) = D dµ(w) |1 -zw| 4 1/2 .
Under the hypothesis conditions, we have

D sup f ∈B L 2 (µ) |R f | p dA(z) = D D dµ(w) |1 -zw| 4 p /2 dA(z) < ∞.
This shows that R is order bounded. From Lemma 2.1, we get that R is p -summing. Actually its adjoint operator R * : L p → L 2 (µ) is the operator T :

L p -→ L 2 (µ) f -→ T f with T f (z) = D f (w) (1 -zw) 2 dA(w).
Hence, R * is the Bergman projection acting from L p to L 2 (µ). Moreover, we have 3)⇒1). For any f ∈ A p , we have T f = f . Considering the maps

T * = R * * = R. Therefore, T = R *
A p i p -→ L p T -→ L 2 (µ),
where i p : A p → L p is the natural embedding with a norm equal to 1. Then

J µ = T • i p .
J µ is a 2-summing operator since it is 1-summing with norm

π 2 (J µ ) ≤ π 1 (J µ ) ≤ π 1 (T ) i p ≤ π 1 (T ) . Proposition 6.2. Let 1 < p ≤ 2, 1 ≤ q ≤ 2 and assume that J µ : f ∈ A p → f ∈ L q (µ)
is an r-summing operator for some r ≥ 1, then

D 1 (1 -|z| 2 ) q dµ(z) < ∞ .
In other words, dν = dµ/(1 -|z| 2 ) q is a finite positive measure on D.

Proof. Since A p and L q (µ) have both cotype 2 when 1 ≤ p, q ≤ 2, then

Π 1 A p , L q (µ) = Π r A p , L q (µ)
for any r ≥ 1. In particular, J µ is a q-summing operator. We shall work with a probability space (Ω, Σ, P). We consider a (finite) sequence of independent Bernoulli variables (r n ) 0≤n≤N on this space taking values in {±1} (i.e. a Rademacher sequence), and the function F ω defined by:

F ω (z) = N n=0 r n (ω)(n + 1) 1/2 n , where ω ∈ Ω.
Applying property 1 (cf. Section 2) to the A p -valued random variable ω ∈ Ω → F ω , we get

Ω J µ • F ω q L q (µ) dP(ω) ≤ π q (J µ ) q sup α∈B (A p ) * Ω |α • F ω | q dP(ω). (6.2)
On one hand, using Fubini's theorem and Khinchin's inequality, we have

Ω D N n=0 r n (ω)(n + 1) 1/2 z n q dµ(z) dP(ω) = D Ω N n=0 r n (ω)(n + 1) 1/2 z n q dP(ω) dµ(z) ≈ D N n=0 (n + 1)|z| 2n
q/2 dµ(z) .

(6.3)
On the other hand, for every α in B (A p ) * , we can write

α(F ω ) = g, F ω = D g(z)F ω (z) dA(z) ,
where g ∈ A p with a norm bounded by some c p (depending only on p). Then, by Khinchin's inequality

Ω α(F ω ) q dP(ω) = Ω N k=0 ĝ(k)r k (ω) (k + 1) 1/2 q dP(ω) ≈ N k=0 |ĝ(k)| 2 k + 1 q/2 ≤ g q A 2 ≤ g q A p ≤ c p 1 . (6.4)
Combining (6.3) with (6.4) and the definition of q-summing operator, we get for arbitrary large

N D N k=0 (k + 1)|z| 2k q/2 dµ(z) π q (J µ ) q . Note that lim N→∞ N k=0 (k + 1)|z| 2k q/2 = 1 (1 -|z| 2 ) q •
Finally, we obtain:

D 1 (1 -|z| 2 ) q dµ(z) < ∞ ,
for every z ∈ D.

We shall need the following tools for the proof of our main theorem in this section.

Lemma 6.3. [13, Lemma 7.5] Let δ > 0, (Ω, Σ, µ) be a measure space and h : Ω → [0, +∞) be a measurable function. Then

inf Ω h F dµ : F ∈ L δ (µ), F ≥ 0, Ω F δ dµ ≤ 1 = Ω h δ/(δ+1) dµ (δ+1)/δ
. (6.5) Lemma 6.4. [13, Lemma 7.6] Let 1 ≤ q < 2, X be a Banach space and T : X → L q (µ) be a continuous linear operator. Then T is a 2-summing operator if and only if there exists F ∈ L 2q/(2-q) (µ) with F ≥ 0 (µ-a.e.), such that

T : X → L 2 (µ/F 2 )
is well defined and 2-summing operator. Moreover π 2 T : X → L q (µ) ≈ inf π 2 T : X → L 2 µ/F 2 ) : F ≥ 0, F L 2q/(2-q) (µ) ≤ 1 . (6.6) Lemma 6.5. (Ky Fan s Lemma) (See [START_REF] Diestel | Absolutely Summing Operators[END_REF], page 190) Let E be a Hausdorff topological vector space, and F be a compact convex subset of E. Let M be a set of functions on F with values in (-∞, +∞] having the following properties: 1) Each f ∈ M is convex and lower semi-continuous.

2) If g ∈ conv(M), the convex closure of M, there is a f ∈ M with g(x) ≤ f (x), for all x ∈ F.

3) There is an r ∈ R such that for each f ∈ M has a value less and equal to r. Then, there exists at least one x 0 ∈ F, such that f (x 0 ) ≤ r for all f ∈ M.

The following lemma is a variant of the claim in the proof of Theorem 7.3 in [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF]. Lemma 6.6. Let 1 < p, q < 2, set t = q/(2q), k = p/(2p). Define

I 1 = inf f ∈B + L t (µ) sup g∈B + L k D D g(w) |1 -wz| 4 f (z)
dµ(z) dA(w) ,

I 2 = sup g∈B + L k inf f ∈B + L t (µ) D D g(w) |1 -wz| 4 f (z)
dµ(z) dA(w) .

Then I 1 = I 2 .

Proof. The proof is nothing but a copy and paste of the one in [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF] up to a small change of the kernel, but we include it for sake of completeness. The fact that I 1 ≥ I 2 is obvious by the definitions of I 1 and I 2 . In the sequel, we assume that I 2 is finite else there is nothing to do and the inequality is trivial.

Let

Φ g ( f ) = D D g(w) |1 -wz| 4 f (z) dµ(z) dA(w) , for f ∈ B + L t (µ)
and

M = Φ g | g ∈ B + L k .
We point out that B + L t (µ) is a compact set under its weak topology by the Alaoglu's theorem once t > 1. Then M appears as a set of functions on B + L t (µ) and since the maps g → Φ g are linear and B + L t (µ) is convex, M is also a convex set. We have: (a) For any Φ g ∈ M, Φ g is convex (thanks to the convexity of 1/x) and lower semicontinuous. Indeed, fix δ > 0 , g > 0 and take

K g = { f ∈ B + L t (µ) | Φ g ( f ) ≤ δ} . Let { f n } in K g , converging to some f ∈ L t (µ)
. Or, we can assume that it is a pointwise convergence a.e up to an extraction, then by Fatou's lemma, we have

Φ g ( f ) ≤ lim inf n→∞ Φ g ( f n ) ≤ δ .
It implies that K g is a closed set of L t (µ) and so Φ g is lower semi-continuous on B + L t (µ) . (b) For fixed ε > 0, by the definition of I 2 , for any

g ∈ B + L k inf f ∈B + L t (dµ) Φ g ( f ) < I 2 + ε = r , so, there is a f 0 ∈ B + L t (µ) such that Φ g ( f 0 ) ≤ r .
Now, Ky Fan's theorem shows that there exists at least one

f 1 ∈ B + L t (µ) such that Φ g ( f 1 ) ≤ r = I 2 + ε , for any g ∈ B +
L k . Note that the second condition in Ky Fan's theorem is obvious since M is convex. This implies that I 1 ≤ I 2 + ε . Now we can state and prove the main theorem of this section. Point out that the measure ν in the following statement is finite as soon as J µ is r-summing for some r ≥ 1, thanks to Proposition 6.2. On the other hand, as soon as J ν is bounded, it is clear that ν is a finite measure (just testing the constant function 1). Theorem 6.7. Let 1 < p < 2, 1 ≤ q ≤ 2 . Let µ be a positive Borel measure and dν = dµ/(1 -|z| 2 ) q . Then the following statements are equivalent:

1) J µ : A p -→ L q (µ) is 2-summing.
2) J µ : A p -→ L q (µ) is r-summing for some r ≥ 1.

3) J

ν : A p 2-p -→ L q 2 ν is bounded. 4) κ(z) = µ(D z ) |D z |(1 -|z| 2 ) q ∈ L s , where s = 2p/(2p -2q + pq).
Moreover, we have

π 2 (J µ ) ≈ D |κ(z)| s dA(z) 1/sq
. Therefore (3.1) follows from the following immediate corollary. Corollary 6.8. Let r ≥ 1, 1 < p ≤ 2 and µ be a Carleson measure on A p . Then J µ : A p → L p (µ) is an r-summing operator if and only if

D µ(D z ) (1 -|z| 2 ) p+2 2/p dA(z) < ∞ . Moreover π r (J µ ) ≈ π 2 (J µ ) ≈ D µ(D z ) (1 -|z| 2 ) p+2 2/p dA(z) 1/2 .
Proof of Theorem 6.7. Since A p and L q (µ) have both cotype 2 when 1 ≤ p, q ≤ 2, then J µ : A p -→ L q (µ) is r-summing for some r ≥ 1 if and only if J µ is 2-summing.

Since p/(2p) > 1 ≥ q/2 > 0 then Lemma 5.2 shows that κ ∈ L s if and only if

J ν : A p/(2-p) -→ L q/2 ν
is bounded. So we only need to prove that 1) is equivalent to 3).

We already mentioned before the statement of the theorem why we know that ν is a finite positive Borel measure.

Set k = p/(2p) and t = q/(2q). From Lemma 6.4 and Theorem 6.1, we know that:

π 2 J µ : A p → L q (µ) 2 ≈ inf F∈B + L 2t (µ) π 2 J µ : A p → L 2 (µ/F 2 ) 2 ≈ inf f ∈B + L t (µ) D D 1 |1 -wz| 4 f (z) dµ(z) p /2 dA(w) 2/p . (6.7)
Hence, by duality between L k and L p /2 , we get

π 2 J µ : A p → L q (µ) 2 ≈ inf f ∈B + L t (µ) sup g∈B + L k D D g(w) |1 -wz| 4 f (z)
dµ(z) dA(w) .

Using Lemma 6.6, Lemma 6.3 and then Lemma 5.4, it follows

π 2 J µ : A p → L q (µ) 2 ≈ sup g∈B + L k inf f ∈B + L t (µ) D Bg(z) (1 -|z| 2 ) 2 f (z) dµ(z) = sup g∈B + L k Bg (1 -|z| 2 ) 2 L q/2 (µ) = sup g∈B + L k Bg L q/2 (ν) ≈ sup g∈B L k Bg L q/2 (ν) ≈ J ν A k →L q/2 ν (6.8)
where the (positive) operator B is the Berezin transformation viewed from L k = L k (A) to L q/2 ν . In other words, we obtain that ( 1) is equivalent to (3).

THE CASE 2 ≤ p ≤ r

In this section, we focus on the case 2 ≤ p ≤ r. Actually, we shall also obtain some results useful to treat the other cases. In particular, some general necessary conditions are obtained below.

Let us begin with the simple case r = p.

Proposition 7.1. Let p ≥ 2 and µ be a Carleson measure. Then J µ :

A p → L p (µ) is p- summing if and only if dµ(z)/(1 -|z| 2 ) 2 is a finite (positive) measure, that is D 1 (1 -|z| 2 ) 2 dµ(z) < ∞ . (7.1)
Moreover

π p (J µ ) ≈ D 1 (1 -|z| 2 ) 2 dµ(z) 1/p .
This result is the counterpart in the Bergman space A p of the corresponding result in [START_REF] Shapiro | Compact, nuclear, and Hilbert-Schmidt composition cperators on H 2[END_REF] which characterizes p-summing composition operators on H p (their proof extends obviously to general Carleson measures on H p ).

Proof. First we assume that the measure dµ(z)/(1 -|z| 2 ) 2 is finite, then by Proposition 4.3, this is equivalent to the fact that J µ is an order bounded operator, therefore J µ is p-summing as well. Hence,

π p (J µ ) D dµ(z) (1 -|z| 2 ) 2 1/p .

Now we define the following (normalized) sequence in

A p . e n (z) = pn + 2 2 1/p z n , n ≥ 0. Let g ∈ A p with a Taylor expansion g(z) = ∞ k=0 ĝ(k)z k . It is well known (cf. [11, Corollary 8.2.10]) that, since p ≥ 2, ∞ k=0 |ĝ(k)| p (k + 1) p-1 1/p g A p .
Then, we have for every g in the unit ball of A p :

∞ n=0 | e n , g | p ∞ n=0 |ĝ(n)| p (n + 1) p-1 1 .
The definition of the p-summing operator yields

∞ n=0 e n p L p (µ) ≤ π p (J µ ) p sup g∈B A p ∞ n=0 | e n , g | p π p (J µ ) p . But ∞ n=0 e n p L p (µ) = D ∞ n=0 pn 2 + 1 |z| pn dµ(z) ≥ D ∞ n=0 (n + 1)|z| pn dµ(z) = D 1 (1 -|z| p ) 2 dµ(z) . We obtain D dµ(z) (1 -|z| 2 ) 2 1/p π p J µ .
We recall that the multiplier operator M β on p is defined by M β (e n ) = β n e n , where {e n } denotes the canonical basis of p and β

= (β 1 , β 2 , • • • ) is a sequence. Proposition 7.2. Let p ≥ 2 and µ be a Carleson measure on A p . Assume that J µ is r- summing. Then 1) For 1 ≤ r ≤ p , ∞ k=1 µ(D k ) |D k | p /p 1/p π r (J µ ) . (7.2) 2) For p ≤ r ≤ p , ∞ k=1 µ(D k ) |D k | r/p 1/r π r (J µ ) . (7.3) 3) For p ≤ r , D 1 (1 -|z| 2 ) 2 dµ(z) 1/p ≈ ∞ k=1 µ(D k ) |D k | 1/p π r (J µ ) . (7.4)
Proof. Assume that J µ is r-summing. From Proposition 4.8 , we get that M β is r-summing as well, where

β = µ(D k ) |D k | 1/p k≥1 .
1 (1 -|z| 2 ) 2 dµ(z) < ∞ . (7.5) Furthermore π r (J µ ) ≈ D 1 (1 -|z| 2 ) 2 dµ(z) 1/p ≈ k≥1 µ(D k ) |D k | 1/p . (7.6)
Proof. Let us assume that (7.5) is satisfied. By Proposition 7.1, J µ is a p-summing operator, hence it is r-summing since p ≤ r and we get

π r (J µ ) π p (J µ ) D 1 (1 -|z| 2 ) 2 dµ(z) 1/p .
The reverse inequality follows immediately from Proposition 7.2 .

THE CASE p ≤ r ≤ p

In this section, we focus on the case p ≤ r ≤ p with p ≥ 2. We shall denote

A p (D k , A) = H(D) L p (D k , A), with f p A(D k ,A) = D k | f | p dA 1 p
.

Given a sequence {X k } of Banach spaces, the space p X k is equipped by the norm

{x k } p X k = k≥1 x k p X k 1 p
, where x k ∈ X k for every k ≥ 1.

For any f ∈ L p (µ), since the family (D k ) k≥1 forms a partition of D,

D | f | p dµ = ∞ k=1 D k | f | p dµ = ∞ k=1 f p L p (D k , µ) .
In other words,

f ∈ L p (µ) -→ f | D k k ∈ ⊕ p L p (D k , µ) , (8.1)
is an isomorphic isometry. This can be rewritten as

L p (µ) = ⊕ p L p (D k , µ) .
Point out that D = k D(a k , 4t) , with 0 < t < 1/4, and recall that there exists a positive integer N such that each point of D belongs to at most N of the sets D(a k , 4t) (cf. Section 2). For p ≥ 1, the following map is continous:

f ∈ A p -→ f | D(a k ,4t) k ∈ ⊕ p A p (D(a k , 4t), A) . (8.2)
Indeed, for any f ∈ A p we have

f | D(a k ,4t) k p A p (D(a k ,4t), A) = k≥1 D(a k ,4t) | f (z)| p dA(z) 1/p ≤ N 1/p D | f (z)| p dA(z) 1/p = N 1/p f A p .
For any t-lattice {a k } (0 < t < 1/2), we shall use the notation D k instead of D(a k , 2t).

Lemma 8.1. Let 1 ≤ r ≤ p < ∞, 0 < t < 1/2, {a k } be a t-lattice on D and {D k } be the relevant subsets of D defined in Lemma 2.6. Then for a positive Borel measure µ on D k (k = 1, 2, . . .), the natural embedding map

T r,p : A r (D k , A) → L p (D k , µ) is bounded if and only if µ(D k ) 1/p |D k | 1/r < ∞ . Furthermore, T r,p ≈ µ(D k ) 1/p |D k | 1/r • Proof. Using Lemma 4.6, the fact that (1 -|z| 2 ) 2 ≈ (1 -|a k | 2 ) 2 ≈ |D k | whenever z ∈ D k (cf.
Section 2), and the fact that D(z, t) ⊂ D(a k , 2t) = D k for every z ∈ D k , we get, for any

f ∈ A r (D k , A), D k | f (z)| p dµ(z) ≤ sup z∈D k | f (z)| p µ(D k ) sup z∈D k 1 (1 -|z| 2 ) 2p/r D(z,t) | f (w)| r dA(w) p/r µ(D k ) 1 (1 -|a k | 2 ) 2p/r D k | f (w)| r dA(w) p/r µ(D k ) f p r µ(D k ) |D k | p/r • (8.3)
This gives that

T r,p µ(D k ) 1/p |D k | 1/r • Now take f (z) = (1 -|a k | 2 ) 2 (1 -āk z) 4 1/r , for z ∈ D . Since (1 -|a k | 2 ) 2 ≈ |1 -āk z| 2 ≈ |D k | ≈ |D k | whenever z ∈ D k (so z ∈ D k ) with constants independent of k, we get f A r (D k , A) ≈ 1 ,
and

D k | f (z)| p dµ(z) ≈ µ(D k ) |D k | p/r • Therefore T r,p µ(D k ) 1/p |D k | 1/r •
As an immediate corollary, we have by renormalization:

Corollary 8.2. Let 1 ≤ r ≤ p < ∞, 0 < t < 1/2
, {a k } be a t-lattice on D and {D k } be the relevant subsets of D defined in Lemma 2.6. Then the operator T (k) r,p defined by

T (k) r,p : f ∈ A r D k , A |D k | -→ 1 |D k | 1/p f | D k ∈ L p (D k , µ) is bounded if and only if µ(D k ) |D k | < ∞ .
Moreover its norm is

T (k) r,p ≈ µ(D k ) |D k |
(D k ) |D k | < ∞ . Then, S k : f ∈ A p D(a k , 4t), A -→ f | D k ∈ L p (D k , µ) is 1-summing with π 1 (S k ) µ(D k ) |D k | 1/p .
Proof. Let H ∞ (D k ) be the class of the bounded analytic functions on D k with the natural supremum norm. We consider the commutative diagram

A p D(a k , 4t), A S k / / P 1 L p k , µ) H ∞ (D k ) j 1 / / A 1 D k , ν k P 2 O O
where ν k is the normalized area measure on D k : 1 |D k | dA, and P 2 := T (k) 1,p and with

P 1 f (z) = |D k | 1/p f (z),
for f ∈ A p D(a k , 4t), A , we know from Lemma 8.2 that P 2 is bounded with norm µ(D k )/|D k | 1/p .

The operator j 1 is obviously bounded and j 1 ≈ 1. Using Lemma 4.6 and the fact that D(z, 2t) ⊂ D(a k , 4t) for every z ∈ D k , we get for any

f ∈ A p D(a k , 4t), A sup z∈D k | f (z)| ≤ sup z∈D k 1 (1 -|z| 2 ) 2/p D(z,2t) | f (w)| p dA(w) 1/p sup z∈D(a k ,4t) 1 (1 -|z| 2 ) 2/p f A p (D(a k ,4t), A) 1 (1 -|a k | 2 ) 2/p f A p (D(a k ,4t), A) ≈ 1 |D k | 1/p f A p (D(a k ,4t), A) . So P 1 is bounded with norm 1. Obviously, S k = T (k) 1,p • j 1 • P 1 .
Now let us observe that j 1 is 1-summing: indeed, it suffices to see that the restriction to

H ∞ (D k ) of the map id : L ∞ (D k ) → L 1 (D k , ν k ) is absolutely summing. Thus, π 1 ( j 1 ) ≤ |D k | |D k | ≈ 1 .
Therefore, the ideal property for 1-summing operators leads to

π 1 (S k ) P 1 π 1 ( j 1 ) T (k) 1,p µ(D k ) |D k | 1/p .
We shall need the following lemma.

T : ⊕ p X n -→ ⊕ p Y n (x n ) n -→ T n (x n ) n≥1 .
Then T is r-summing if and only if each T n is r-summing and n≥1 π r (T n ) r < +∞.

Moreover, we have

π r (T ) ≈ n≥1 π r (T n ) r 1/
µ(D k ) |D k | r/p < ∞ . Moreover π r (J µ ) ≈ ∞ k=1 µ(D k ) |D k | r/p 1/r . (8.4)
Proof. We consider the commutative diagram:

A p J µ / / M L p (µ) ⊕ p A p (D(a k , 4t), A) S / / ⊕ p L p (D k , µ) M O O
where M is the map defined as in (8.2), which is continuous, and M is bounded since the map as defined in (8.1) is an isomorphic isometry. Finally, S in the natural block diagonal map whose entries are the operator S k . Theorem 8.3 implies that

S k : f ∈ A p D(a k , 4t), A -→ f ∈ L p (D k , µ) is 1-summing with π 1 (S k ) µ(D k ) |D k | 1/p .
It is obvious that J µ = M • S • M and it follows from Lemma 8.4 that

π r (J µ ) π r (S ) ≈ k≥1 π r (S k ) r 1/r ≤ k≥1 π 1 (S k ) r 1/r ∞ k=1 µ(D k ) |D k | r/p 1/r
.

We get the reverse inequality from Proposition 7.2 :

∞ k=1 µ(D k ) |D k | r/p 1/r
π r (J µ ).

THE CASE 1 ≤ r ≤ p

We focus now on the last remaining case 1 ≤ r ≤ p .

Proposition 9.1. Let X n and Y n be two Banach spaces. Let T n : X n → Y n be a 1-summing operator with π 1 (T n ) 1. Then the operator

T : x = (x n ) n ∈ ⊕ 1 X n -→ T n (x n ) n≥1 ∈ ⊕ 2 Y n is 1-summing and π 1 (T ) 1.
We are indebted to L. Rodríguez-Piazza for the following argument which seems to us clearer than an alternative previous one.

Proof. Let n ≥ 1. Since T n is 1-summing with π 1 (T n ) 1, by the Pietsch Theorem (see for instance [START_REF] Diestel | Absolutely Summing Operators[END_REF], page 44), there exist a constant C > 0 depending only on n and a probability measure µ n on the unit ball K n of X * n (K n is a compact set endowed with the weak-star topology) such that

∀u ∈ X n , T n (u) ≤ C K n | χ(u)| dµ n ( χ) (9.1)
for each n.

Let (Ω, P) be a probability space and (r n ) n be a Rademacher sequence (i.e. a sequence of independent random equidistributed variables taking their values in {±1}; i.e. random choice of signs). Actually we can assume that Ω is a compact set (take {±1} N , the Cantor group and r n is the n th coordinate map). Then, writing K = n K n , we define K = Ω × K which is a compact set and the measure ν = P ⊗ µ is a probability measure on K, where µ

= ⊗ n µ n . Let x * n (ξ n ) = ξ n (x n ) where ξ = (ξ n ) ∈ K. The map: E : x = (x n ) n ∈ ⊕ 1 X n -→ n r n ⊗ x * n ∈ C( K)
is an isometry. Indeed,

E(x) = sup t∈Ω ξ n ∈K n n r n (t)x * n (ξ n ) = sup ξ n ∈K n n |ξ n (x n )| = n x n .
On the other hand, let Z 1 be the closure in L 1 ( K, ν) of the vector space V 1 spanned by

{r n ⊗ x * n ; x n ∈ X n }. The formal identity i from C( K) to Z 1 ⊂ L 1 ( K, ν) is 1-summing with π 1 (i) = ν( K) = 1 (ν is a probability measure).
Finally the map

F : f = n r n ⊗ x * n ∈ V 1 -→ T n (x n ) n ∈ ⊕ 2 Y n
is well defined, bounded and extends to a bounded operator on Z 1 . Indeed, since (9.1) holds then

n T n (x n ) 2 1/2 ≤ C n x * n 2 L 1 (K) 1/2
which is nothing but

K |x * n |dµ n 2
and is then lower than

K n |x * n | 2 1/2 dµ.
Then the Khinchin's inequality implies that

n T n (x n ) 2 1/2 K Ω n r n (t)x * n d P dµ = n r n ⊗ x * n .
Hence, π 1 (T ) ≤ F π 1 (i) E 1. The conclusion follows.

Corollary 9.2. Let X n and Y n be Banach spaces. Let T n : X n -→ Y n be a 1-summing operator. We assume that n≥1 π 1 (T n ) p < ∞.

Then the operator

T : x = (x n ) n ∈ ⊕ p X n -→ T n (x n ) n≥1 ∈ ⊕ p Y n is 1-summing with π 1 (T ) π 1 (T n ) p .
Proof. Write λ n = π 1 (T n ). We have (λ n ) n ∈ p by hypothesis. The operator S n = λ -1 n T n satisfies π 1 (S n ) = 1 by definition. From the previous proposition, we know that the operator

S : x = (x n ) n ∈ ⊕ 1 X n -→ (S n (x n )) n ∈ ⊕ 2 Y n is 1-summing with π 1 (S ) 1 .
On the other hand, we have the factorization: T = J • S • D where

D : x = (x n ) n ∈ ⊕ p X n -→ (λ n x n ) n ∈ ⊕ 1 X n is bounded since (λ n ) n ∈ p with D λ n p , and
J : y = (y n ) n ∈ ⊕ 2 Y n -→ (y n ) n ∈ ⊕ p Y n is bounded since p ≥ 2 with norm J 1 . Therefore, π 1 (T ) ≤ J π 1 (S ) D λ n p = π 1 (T n ) p .
Now we can state the main theorem of this section.

Theorem 9.3. Let p ≥ 2 and 1 ≤ r ≤ p . Let µ be a Carleson measure on A p , a t-lattice {a k } and {D k } as defined as in Lemma 2.6. Then the following statements are equivalent:

1) J µ : A p → L p (µ) is 1-summing. 2) J µ : A p → L p (µ) is r-summing. 3) k≥1 µ(D k ) |D k | p /p < ∞.
Moreover,

π 1 (J µ ) ≈ π r (J µ ) ≈ k≥1 µ(D k ) |D k | p /p 1/p ≈ D D z 1 (1 -|w| 2 ) 2(1+p/p ) dµ(w) p /p dA(z) 1/p .
Proof. 1) ⇒ 2). The proof is obvious.

2) ⇒ 3). It follows from Theorem 8.5 and the fact that r ≤ p ,

k≥1 µ(D k ) |D k | p /p 1/p π p (J µ ) ≤ π r (J µ ).
3) ⇒ 1). By hypothesis, we have in particular:

sup k µ(D k ) |D k | < ∞ . Theorem 8.3 implies that S k : f ∈ A p D(a k , 4t), A -→ f ∈ L p (D k , µ) is 1-summing with π 1 (S k ) µ(D k ) |D k | 1/p .
Following the same proof of Theorem 8.5 and using Corollary 9.2, we get J µ is 1-summing with

π 1 (J µ ) k≥1 µ(D k ) |D k | p /p 1/p .
This completes the proof.

THE RESULTS ON WEIGHTED BERGMAN SPACES

In this section, we give similar characterizations on weighted Bergman spaces A p α . For any α > -1, let dA α = (α + 1)(1 -|z| 2 ) α dA(z) and

A p α = H(D) L p (D, A α ).
The following theorem extends the main theorem to the framework of weighted Bergman spaces. We can prove it in a similar way, and so we omit the details here.

Theorem 10.1. Let 0 < t < 1/4 and α > -1. Let µ be a Carleson measure on A p α . Then

(1) Let 1 < p ≤ 2 and r ≥ 1 . Then J µ : A p α → L p (µ) is r-summing if and only if D µ(D z ) 2 p (1 -|z| 2 ) (α+2)( 2 p +1) dA α (z) < ∞ .
Moreover, we have

π 2 (J µ ) ≈ ∞ n, j 4 n(1+α/2) µ(R n, j ) 2/p 1/2 ≈ D µ(D z ) 2 p (1 -|z| 2 ) (α+2)( 2 p +1) dA α (z) 1/2 . ( 10.1)
(2) Let p ≥ 2 and 1 ≤ r ≤ p . Then J µ :

A p α → L p (µ) is r-summing if and only if ∞ n, j 4 n(1+α/2) µ(R n, j ) p /p < ∞.
Moreover, we have

π 1 (J µ ) ≈ π r (J µ ) ≈ ∞ n, j 4 n(1+α/2) µ(R n, j ) p /p 1/p ≈ D µ(D z ) p p (1 -|z| 2 ) (α+2)( p p +1) dA α (z) 1/p . (10.2)
(3) Let p ≥ 2 and p ≤ r ≤ p . Then J µ :

A p α → L p (µ) is r-summing if and only if ∞ n, j 4 n(1+α/2) µ(R n, j ) r/p < ∞ .
Moreover, we have

π r (J µ ) ≈ ∞ n, j 4 n(1+α/2) µ(R n, j ) r/p 1/r ≈ D µ(D z ) r p (1 -|z| 2 ) (α+2)( r p +1) dA α (z) 1/r . (10.3) (4) Let p ≥ 2 and r ≥ p . Then J µ : A p α → L p (µ) is r-summing if and only if D 1 (1 -|z| 2 ) 2+α dµ(z) < ∞ . Moreover, we have π r (J µ ) ≈ ∞ n, j 4 n(1+α/2) µ(R n, j ) 1/p ≈ D 1 (1 -|z| 2 ) 2+α dµ(z) 1/p . ( 10.4)
In the same way, we get for free the characterizations for weighted composition operators.

Theorem 10.2. Let ϕ : D → D be an analytic map and u : D → C be an analytic function.

Let uC ϕ be the composition operator viewed from A p α to A p α where α, α > -1, and dµ = (|u| p dA α ) ϕ . The operator uC ϕ is r-summing if and only if the following π r norm is finite.

(1) When 1 < p ≤ 2 , for any r ≥ 1 ,

π r (uC ϕ ) ≈ π 2 (uC ϕ ) ≈ D ϕ -1 (D z ) |u| p (1 -|ϕ| 2 ) (2+α)(1+p/2) dA α (w) 2/p dA α (z) 1/2 . (2) When p ≥ 2 , for 1 ≤ r ≤ p , π r (uC ϕ ) ≈ π 1 (uC ϕ ) ≈ D ϕ -1 (D z )
|u| p (1 -|ϕ| 2 ) (2+α)(1+p/p ) dA α (w) p /p dA α (z)

SOME OTHER APPLICATIONS

We mention in this section some direct applications of our results.

The first one concerns a monotony property relatively to the index of the Bergman space.

Proposition 11.1. Let µ be a positive measure on D and 1 < p 2 < p 1 . If J µ is r-summing on A p 1 , then J µ is r-summing on A p 2 .

We can read the previous result in another way. Assume that µ is γ-Carleson. If γ ∈ (5/2, 3] then J µ is absolutely summing on A p for any p ∈ (1, 2(γ -2)) ⊂ (1, 2).

If γ > 3, then J µ is r-summing on A p for some suitable (r, p) with p > 2. In particular, it is absolutely summing on A p for any p ∈ (1, γ -1).

Proof. It suffices to check (easily) that the conditions in the main theorem ar satisfied.

For instance, for p ∈ (1, 2), since

µ(D z ) 2 p (1 -|z| 2 ) 4 p +2 (1 -|z| 2 ) 2(γ-2)
p -2 a sufficient condition to ensure the convergence of the integral is

2(γ -2) p -2 > -1 .
The other cases are treated in the same way.

Another application concerns composition operators on vector valued (weak and strong) Bergman spaces. A partial result is given by Laitila-Tylli-Wang in [START_REF] Laitila | Composition operators from weak to strong spaces of vector-valued analytic functions[END_REF]. They also treat partially the case of vector valued Hardy spaces and the result is completed in [START_REF] Blasco | Composition operators from weak to strong vector-valued Hardy spaces[END_REF] by Blasco for L q valued functions. We give below the corresponding result for Bergman spaces.

Before stating precisely the results, let us recall the framework. For any Banach space X, we use p weak (X) to show the weak p-summable sequence (x n ) in X. It is equipped with the norm defined by (x n ) p weak (X) := sup

x * ∈B X *        k | x * , x k | p        1 p
where (x n ) ∈ p weak (X). For any Banach space X, let p ≥ 1, the norm on A p (D, X) (resp. A p weak (D, X)) is defined by respectively. For any symbol ϕ : D → D (analytic), the composition operator C ϕ is well defined (and bounded) from A p into A p . For any Banach space X, we are interested in the operator C X ϕ (until now formally) defined by

f A p (D,X) := D f (z) p X dA(z)
C X ϕ : f ∈ A p weak (D, X) → f • ϕ ∈ A p (D, X).
That is, to any f (z) = ∞ n=0 x n z n ∈ A p weak (D, X) where x n ∈ X for n ≥ 0, we associate the X-valued analytic function

C X ϕ ( f )(z) = ∞ n=0
x n ϕ(z) n .

In [START_REF] Laitila | Composition operators from weak to strong spaces of vector-valued analytic functions[END_REF]Th 3.2], the authors prove the following theorem:

Theorem A. Let 2 ≤ p < ∞, ϕ be a symbol and X be any complex infinite-dimensional Banach space. Then They also prove a Hardy space version of the previous theorem, which is completed for other values of p and when X = L q (µ) by Blasco in [1, Th 1.1]: Theorem B. Let ϕ be a symbol and 1 < p ≤ 2. Let X = L q (µ) for 1 ≤ q ≤ p and infinite dimensional. Then (11.

2)

The following result is the analogue of Theorem B for vector valued Bergman spaces, and completes Theorem A in this framework.

Theorem 11.3. Let ϕ : D → D be an analytic function, 1 < p ≤ 2 and 1 ≤ q ≤ p. Let (Σ, Ω, µ) be a probability space and X := L q (µ). Then C X ϕ : A p weak (D, L q (µ)) → A p (D, L q (µ)) is bounded if and only if Proof. Suppose (11.3) is true. Then by Corollary 3.2, we get C ϕ ∈ Π q (A p , A p ). Let F(z) = ∞ n=0 f n z n ∈ A p weak (D, L q (µ)), where f n ∈ L q (µ) for every n ≥ 0. Hence, for each h ∈ L q (µ) * , we have that | G(ω), g(.) | q dµ(ω)

z ∈ D -→ h, F(z) =
1/q
, where G(ω) = ∞ n=0 f n (ω)z n ∈ A p . The fact that C ϕ ∈ Π q (A p , A p ) shows that

Ω C ϕ (G) q A p dµ(ω) 1/q ≤ π q (C ϕ ) sup g∈B A p Ω
| G(ω), g(.) | q dµ(ω)

1/q .
So the Minkowski's inequality gives that

D Ω
|C ϕ (F)| q dµ(ω) p/q dA(z)

1/p ≤ Ω D
|C ϕ (F)| p dA(z) q/p dµ(ω)

1/q ≤ π q (C ϕ ) F A p weak D,L q (µ) , and moreover For each x * ∈ L q (µ)

C X
x * , F n (z) = Therefore, for every x * in the unit ball of L q (µ), we have

x * , F n A p = n k=1 T * n (x * ), e k f k A p = sup g∈B A p k T * n (x * ), e k . f k , g ≤        k | T * n (x * ), e k | 2        1/2
. sup

g∈B A p        k | f k , g | 2        1/2 ≤ T * n .
x * ≤ 1 since T n = x * = 1. This implies that F n A p weak (D,L q (µ)) ≤ 1. Since C X ϕ is bounded from A p weak (D, L q (µ)) into A p (D, L q (µ)), using Dvoretzky theorem, we have dA(z)

≥ (1 + ε) -p D        n k=1 | f k (ϕ(z))| 2        p/2
dA(z)

≥ (1 + ε) -p        n k=1 D | f k (ϕ(z))| p dA(z) 2/p        p/2
, where the last inequality holds since 2/p ≥ 1 and using the Minkowski's inequality. Since ε is arbitrary, we get

       n k=1 C ϕ ( f k ) 2 p        1/2 ≤ C X ϕ p A p weak →A p .
By the definition of 2-summing, we have

π 2 (C ϕ ) ≤ C X ϕ A p weak →A p .

  sup ξ∈T µ W(ξ, h) = O(h 2 ), when h → 0,where W(ξ, h) is the Carleson windowW(ξ, h) = {z ∈ D | 1h ≤ |z| and | arg(z ξ)| ≤ h}.

l η converges and m≤n 4 -m 2

 42 mη 4 -n 2 nη (keep in mind η > 2).

Lemma 4 . 4 . 3 )

 443 [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF] Proposition 4.1] Define M β as in (4.13). Then with constants depending only on p and r, we have:1) For 1 ≤ p ≤ 2 and every r ≥ 1,π r (M β ) ≈ β 2 (4.14)2) For p ≥ 2 and r ≤ p , π r (M β ) ≈ β p (4.15) For p ≥ 2 and p ≤ r ≤ p, π r (M β ) = β r (4.16) 4) For p ≥ 2 and r ≥ p, π r (M β ) ≈ β p . (4.17)

Lemma 4 . 5 .

 45 [START_REF] Zhu | Operator Theory in Function Spaces[END_REF] Theorem 4.33] Suppose p > 0 and b > max 1

Lemma 4 . 6 .

 46 [START_REF] Zhu | Operator Theory in Function Spaces[END_REF] Proposition 4.13 ] Suppose p > 0 and t > 0. Then there exists a positive constant C such that

. 24 )Proposition 4 . 8 .

 2448 With (4.23) and (4.24), we get the conclusion. Let p ≥ 1 and µ be a Carleson measure on A p . If J µ : A p → L p (µ) is an r-summing operator for some r ≥ 1, then M β is an r-summing operator on p .

  since (b) is satisfied, it follows from Lemma 5.3 and the fact that (1-|z| 2 ) 2 ≈ |D(a k , r)| whenever z ∈ D(a k , r), that D B f (w) q dµ(w) k≥1 D(a k ,2r)

D

  |ξ(k w )| p dA(w) = D |g(rw)| p dA(w) So sup ξ∈B (A p ) * D |ξ(k w )| p dA(w) ≤ P p .

  is 1-summing by Lemma 2.2 and we have π 1 (T ) π p (T * )

Lemma 8 . 4 .

 84 [START_REF] Lefèvre | Absolutely summing Carleson embeddings on Hardy spaces[END_REF] Corollary 4.8] Let p ≥ 2, p ≤ r ≤ p. Assume that for every integer n ≥ 1, we have bounded operators T n : X n → Y n for Banach spaces X n and Y n . Consider the operator

.( 3 ).( 4 )

 34 When p ≥ 2 , for p ≤ r ≤ p ,π r (uC ϕ ) ≈ D ϕ -1 (D z ) |u| p (1 -|ϕ| 2 ) (2+α)(1+p/r) dA α (w) When p ≥ 2 , for p ≤ r , π r (uC ϕ ) ≈ D |u| p (1 -|ϕ| 2 ) 2+α dA α 1/p .Remark: Of course, we could add some estimates involving the family (D k ) k or the Luecking rectangles, using the fact that, for every s > 0,∞ k=1 µ(D k )/|D k | 1+α/2 s ≈ n≥0 0≤ j<2 n µ(R n, j )/|R n, j | 1+α/2 s .

  weak (D,X) := supx * ∈B X * D | x * , f (z) | p dA(z) 1 p

C 1 ( 1

 11 ϕ A p weak (D,X)→A p (D,X) ≈ D -|ϕ(z)| 2 ) 2 dA(z)

D D 1

 1 |1zϕ(w)| p+2 dA(w)

  ) f n (ω)dµ(ω) z n . belongs to A p .So we haveF A p weak (D,L q (µ)) = sup h∈B L q (µ) D | h, F(z) | p dA(z) ) f n (ω)dµ(ω) ān ; g ∈ B A p with g =

  T n (e k ) f k (z)

  e k ) f k (ϕ(z)) p L q (µ)

  Theorem 7.3. Let 2 ≤ p ≤ r and µ be a Carleson measure on A p . Then J µ : A p → L p (µ) is an r-summing operator if and only if

	Now the comparison (1); (2) and (3) of π r (J µ ) with the norm of β follows directly from
	Lemma 4.4 : more precisely , (1) follows from (4.15); (2) follows from (4.16) and (3) follows
	from (4.17).
	D

Now we can prove the main result for 2 ≤ p ≤ r.

  Let p ≥ 1, 0 < t < 1/4, a t-lattice {a k } and {D k } is defined as in Lemma 2.6. For any k, assume that µ

	Theorem 8.3.
	1/p
	.

  Now we can get easily the main theorem of this section . Let p ≥ 2, p ≤ r ≤ p and 0 < t < 1/4. Let µ be a Carleson measure on A p . For any t-lattice {a k } and relevant set {D k } as defined in Lemma 2.6, J µ : A p → L p (µ) is r-summing if and only if k

	Theorem 8.5.

r .

The research of the first and fourth authors was partly supported by the NNSF of China (Nos. 12071272, 11871293) and NSF of Guangdong Province (No. 2122A1515012117). The research of the second and third authors is partly supported by the funding ANR-17-CE40-0021 of the French National Research Agency ANR (project Front).

Proof. Fix r ≥ 1, p 1 > p 2 > 1 and let ξ := µ(D k ) |D k | k≥1 .

We have to separate the cases since our characterizations depend on the values of (p 1 , p 2 , r).

Assume that J µ : A p 1 → L p 1 (µ) is absolutely summing, then we have ξ ∈ 2/p 1 by our characterization in Section 3. It follows that ξ ∈ 2/p 2 , hence J µ : A p 2 → L p 2 (µ) is absolutely summing thanks to our characterization (3.1).

• If 2 ≤ p 2 < p 1 . Let r ≥ 1. we split this case in sub-cases:

-If 1 ≤ r ≤ p 1 , we have that ξ ∈ p 1 /p 1 . This implies that ξ ∈ p 2 /p 2 since p 1 /p 1 < p 2 /p 2 . Then, again from our main theorem in Section 3, it follows that

we get that ξ ∈ r/p 2 , since ξ ∈ r/p 1 and r/p 1 < r/p 2 , hence J µ is r-summing on A p 2 thanks to (3.3). -If p 2 ≤ r ≤ p 1 , we get that ξ ∈ 1 , since ξ ∈ r/p 1 and r < p 1 , hence J µ is r-summing on A p 2 thanks to (3.4). -If p 1 ≤ r, then we have that ξ ∈ 1 . It turns out that J µ is r-summing on A p 2 thanks to (3.4). • The case p 2 ≤ 2 < p 1 follows immediately from the previous cases: if J µ is rsumming on A p 1 then it is r-summing on A 2 by the previous case, hence it is rsumming on A p 2 by the first case.

We give another direct application of our main theorem: we have a sufficient condition to ensure that J µ is absolutely summing in terms of Carleson's condition. Proposition 11.2. Let µ be a positive measure on D. Let γ > 5/2.

Assume that µ is a positive measure on D which is γ-

Recall that the boundedness of J µ on A p is ensured exactly as soon as µ is 2-Carleson.