p-Carleson Measures for a Class of Hardy-Orlicz Spaces

An alternative interpretation of a family of weighted Carleson measures is used to characterize p-Carleson measures for a class of Hardy-Orlicz spaces admitting a nice weak factorization. As an application, we provide with a characterization of symbols of bounded weighted composition operators and Cesàro-type integral operators from these Hardy-Orlicz spaces to some classical holomorphic function spaces.


Introduction
Hardy-Orlicz spaces are the generalization of the usual Hardy spaces.We raise the question of characterizing those positive measures μ defined on the unit ball n of n such that these spaces embed continuously into the Lebesgue spaces L p dμ .More precisely, let denote by dV the Lebesgue measure on n and dσ the normalized measure on the unit sphere Ë n which is the boundary of n .H n denotes the space of holomorphic functions on n .Let Φ be continuous and nondecreasing function from 0, ∞ onto itself.That is, Φ is a growth function.The Hardy-Orlicz space H Φ n is the space of function f in H n such that the functions f r , defined by f r w f rw satisfy sup r<1 inf λ > 0 : We denote the quantity on the left of the above inequality by f lux H Φ or simply f H Φ when there is no ambiguity.Let us remark that f lux H Φ sup r<1 f r lux L Φ , where f lux L Φ denotes the Luxembourg quasi -norm defined by Given two growth functions Φ 1 and Φ 2 , we consider the following question.For which positive measures μ on n , the embedding map I μ : H Φ 2 n → L Φ 1 dμ , is continuous?When Φ 1 and Φ 2 are power functions, such a question has been considered and completely answered in the unit disc and the unit ball in 1-6 .For more general convex growth functions, an attempt to solve the question appears in 7 , in the setting of the unit disc where the authors provided with a necessary condition which is not always sufficient and a sufficient condition.The unit ball version of 7 is given in 8 .To be clear at this stage, let us first introduce some usual notations.For any ξ ∈ Ë n and δ > 0, let

1.3
These are the higher dimension analogues of Carleson regions.We take as Φ 1 the power functions, that is, Φ 1 t t p for 1 ≤ p < ∞.Thus, the question is now to characterize those positive measures μ on the unit ball such that there exists a constant C > 0 such that We call such measures p-Carleson measures for H Φ n .We give a complete answer for a special class of Hardy-Orlicz spaces H Φ n with Φ t t/ log e t s , 0 < s ≤ 1.For simplicity, we denote this space by H s n .We prove the following result.Theorem 1.1.Let 0 < s ≤ 1 and 1 ≤ p < ∞.Then the following assertions are equivalent.
i There exists a constant K 1 > 0 such that for any ξ ∈ Ë n and δ > 0, To prove the above result, we combine weak-factorization results for Hardy-Orlicz spaces see 9, 10 and some equivalent characterizations of weighted Carleson measures for which we provide an alternative interpretation.We also provide with some further applications of our characterization of the measures considered here to the boundedness of weighted Cesàro-type integral operators from our Hardy-Orlicz spaces to some holomorphic function spaces in Section 3.
All over the text, C, C j and, K j , j 1, . .., will denote positive constant not necessarily the same at each occurrence.
This work can be also considered as an application of some recent results obtained by the author and his collaborators 9-11 .

λ-Hardy p-logarithmic Carleson Measures
For z z 1 , . . ., z n and w w 1 , . . ., w n in n , we let z, w Recall that when Φ is a power function, the Hardy-Orlicz space H Φ n is just the classical Hardy space.More precisely, for 0 < p < ∞, let H p n denote the Hardy space which is the space of all f ∈ H n such that We denote by H ∞ n , the space of bounded analytic functions in n .Let ρ be a continuous increasing function from 0, ∞ onto itself, and such that for some α on 0, 1 st ≤ s α t 2.2 for s > 1, with st ≤ 1.We define the space BMO ρ by where for B B δ ξ 0 , the space P N B is the space of polynomials of order ≤ N in the 2n − 1 last coordinates related to an orthonormal basis whose first element is ξ 0 and second element Áξ 0 .The integer N is taken larger than 2nα − 1.For C, the quantity appearing in the definition of BMO ρ , we note f BMO ρ : Clearly, BMOA ρ coincides with the space of holomorphic functions in H 2 n such that their boundary values lie in BMO ρ .The space BMOA 1 is the usual space of function with bounded mean oscillation BMOA while the space of function of logarithmic mean oscillation LMOA is given by 1/ρ t log 4/t.

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Let μ denote a positive Borel measure on n .The measure μ is called an s-Carleson measure, if there is a finite constant C > 0 such that for any ξ ∈ Ë n and any 0 < δ < 1, 2.5 When s 1, μ is just called Carleson measure.The infinimum of all these constants C will be denoted μ s .We use the notation μ for μ 1 .In this section, we are interested in Carleson measure with weights involving the logarithmic function.Let μ be a positive Borel measure on n and 0 < s < ∞.For ρ, a positive function defined on 0, 1 , we say μ is a ρ, s -Carleson measure if there is a constant C > 0 such that for any ξ ∈ Ë n and 0 < δ < 1, We will restrict here to the case ρ t log 4/t p loglog e 4 /t q , 0 < p, q < ∞ studied by the author in 11 see also 12 for a special case in one dimension .But here we go beyond the interpretation provided in 11 .

λ-Hardy ρ-Carleson Measures
In this section, we recall some results of 11 and the notion of λ-Hardy Carleson measures.
We then provide with an alternative interpretation of the results of 11 that will be useful to our characterization.From now on, the notation K 1 ≈ K 2 , where K 1 and K 2 are two positive constants, will mean there exists an absolute positive constant M such that and in this case, we say K 1 and K 2 are comparable or equivalent.The notation

2.8
We first recall the following higher dimension version of the theorem of Carleson 1 and its reproducing kernel formulation.ii There is a constant iii There is a constant

2.10
We note that the constants K 1 , K 2 in Theorem 2.1 are both comparable to μ .The proof of this theorem can be found in 13 .
We now recall some basic facts about λ-Hardy measures.
Definition 2.2.Let 0 < p, q < ∞ and λ q/p.We say a positive measure μ on n is a λ-Hardy Carleson measure if there exists a constant C > 0 such that for all f ∈ H p n ,

2.11
The following high dimension Peter Duren's characterization of λ-Hardy Carleson measures is useful for our purpose.Proposition 2.3.Let 0 < p, q < ∞ and λ q/p > 1.Let μ be a positive measure on n .Then the following assertions are equivalent.
i There exists a constant K 1 > 0 such that for any ξ ∈ Ë n and any 0 < δ < 1, ii There exists a constant K 2 > 0 such that iii There exists a constant

2.14
The constants K 1 , K 2 , and K 3 in the above proposition are equivalent.That i ⇔ ii can be found in 11 .The equivalence i ⇔ iii can be found in 14 for example.We have the following elementary consequence.
Corollary 2.4.Let 0 ≤ p, q < ∞, p / 0 and let μ be a positive measure on n .Then the following assertion are equivalent.
i There exists a constant K 1 > 0 such that for any ξ ∈ Ë n and any 0 < δ < 1,

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ii There exists a constant K 2 > 0 such that iii There exists a constant K 3 > 0 such that for all f ∈ H p n , sup iv There exists a constant K 4 > 0 such that for all f ∈ H p n and any g ∈ H r n , Proof.The equivalence i ⇔ ii is a special case of Proposition 2.3.Note that iii is equivalent in saying that for any f ∈ H p n , the measure |f z | q dμ z / f q H p is a Carleson measure which is equivalent to iv .The implication iv ⇒ i follows from the usual arguments.Thus, it only remains to prove that ii ⇒ iii .First by Proposition 2.3, ii is equivalent in saying that there exists a constant K 2 > 0 such that for any f ∈ H p n , n f z p q dμ z ≤ K 2 f p q H p .

2.19
It follows from the hypotheses, the latter, and H ölder's inequality that

2.20
Thus ii ⇒ iii .The proof is complete.
Next, we recall the following result proved in 11 .
Theorem 2.5.Let 0 ≤ p, q < ∞, s ≥ 1, and let μ be a positive Borel measure on n .Then the following conditions are equivalent.
i There is K 1 > 0 such that for any ξ ∈ Ë n and 0 < δ < 1, ii There is K 2 > 0 such that iii There is K 3 > 0 such that for any f ∈ BMOA, iv There is K 4 > 0 such that for any g ∈ LMOA, 2.24 v There is K 5 > 0 such that for any f ∈ BMOA and any g ∈ LMOA,

2.25
Definition 2.6.Let 0 < p, q < ∞ and λ q/p.Let ρ be a positive function defined on 0, ∞ .We say a positive measure μ on n is a λ-Hardy ρ-Carleson measure if for any f ∈ H p n , the measure is a ρ-Carleson measure.
We have the following characterization of λ-Hardy ρ-Carleson measure which is in fact an alternative interpretation of Theorem 2.5.Theorem 2.7.Let 0 ≤ p, q, r, s < ∞, s / 0, and let μ be a positive Borel measure on n .Then the following conditions are equivalent.
i There is K 1 > 0 such that for any ξ ∈ Ë n and 0 < δ < 1, p loglog e 4 /δ q . 2.27 ii There is K 2 > 0 such that for any f ∈ BMOA, and any h ∈ H s n , sup

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iii There is K 3 > 0 such that for any g ∈ LMOA, and any h ∈ H s n , 2.29 iv There is K 4 > 0 such that for any f ∈ BMO, any g ∈ LMOA, and any h ∈ H s n , sup 2.30 v There is K 5 > 0 such that for any f ∈ BMOA, g ∈ LMOA, and any h ∈ H s n and l ∈ H m n ,

2.31
Proof.i ⇔ iv : we first observe with Theorem 2.5 that i is equivalent in saying that there is a constant C 1 such that for any f ∈ BMOA and any g ∈ LMOA, sup 2.32 By Corollary 2.4, the latter is equivalent to iv .
ii ⇔ iii ⇔ iv : by rewriting ii as sup where d μ z |h z | r / h r H s dμ z , it follows directly from Theorem 2.5 that ii ⇔ iii ⇔ iv .
That iv ⇔ v is a consequence of Theorem 2.1.The proof is complete.

p-Carleson Measures for Hardy-Orlicz Spaces
In this section, we characterize p-Carleson measures of some special Hardy-Orlicz spaces.For this, we will need a weak factorization result of functions in these spaces which follows from the one in 10 .

2.34
ii Any function f in the unit ball of H s n admits the following representation (weak factorization):

2.36
Let us remark that the space H 1 n is the predual of LMOA.The following theorem gives a characterization of p-Carleson measures of the Hardy-Orlicz spaces considered here.
Theorem 2.9.Let 0 < s ≤ 1, 1 ≤ p < ∞.Let H s n be the Hardy-Orlicz space H Φ n corresponding to the function Φ t t/ log e t s .Then, for μ a positive measure on n , the following assertions are equivalent.
i There exists a constant K 1 > 0 such that for any ξ ∈ Ë n and any 0 < δ < 1, 39 and ∞ j 0 f j H s g j BMOA º f H s .It follows using the equivalent assertion iv of Theorem 2.7 that International Journal of Mathematics and Mathematical Sciences

2.40
Now we prove that ii ⇒ i .That ii holds implies in particular that for any f ∈ H s n and any g ∈ BMOA,

2.41
We observe with Corollary 2.4 that 2.41 is equivalent in saying that for any g ∈ BMOA, the measure is a p/s -Carleson measure or equivalently, sup a∈ n n K a z p/s g z p dμ z ≤ K 3 g p BMOA .

2.43
By Theorem 2.5, the latter is equivalent to which is equivalent to i .The proof is complete.

Some Applications
We provide in this section with some applications of p-Carleson measures of the above Hardy-Orlicz spaces to the boundedness of multiplication operators, composition operators, and Cesàro integral-type operators.Let us first introduce the generalized Bergman spaces in the unit ball.We recall that for f ∈ H n , its radial derivative Rf is the holomorphic function defined by These spaces have been studied in 15 .When α > −1, the space A p α n corresponds to the usual weighted Bergman space which consists of holomorphic function f in n such that For α −1 and p 2, the corresponding space is just the Hardy space H 2 n .Let u be a holomorphic function in n .We denote by M u the multiplication operator by u defined on H n by We recall that if ϕ is a holomorphic self map of n , then the composition operator C ϕ is defined on H n by For u a holomorphic function in n , the weighted composition operator uC ϕ is the composition operator followed by the multiplication by u.That is, For b a holomorphic function in n , the Cesàro-type integral operator T b is defined by Combining this operator with the weighted composition operator, we obtain a more general operator 3.9 When ϕ z z for all z ∈ n , we write T u,ϕ,b T u,b .The multiplication operator, the composition operator, the Cesàro-type integral, and their products have been intensively studied by many authors on various holomorphic function spaces.We refer to the following and the references therein 11, 12, 16-30 .As an application of the characterization of p-Carleson measures for the Hardy-Orlicz spaces of the previous section, we consider boundedness criteria of the above operators from Hardy-Orlicz spaces to generalized weighted Bergman spaces and weighted BMOA spaces in the unit ball.We have the following result.

3.10
Proof.Clearly, that uC ϕ is bounded from

3.13
Changing the variables back, we finally obtain that uC ϕ is bounded from dV z < ∞.

3.14
The proof is complete.
Remarking that one has we prove in the same way the following result.

3.22
The proof is complete.
The methods used in this text are quite specific to the case considered here, that is, the embedding I μ : H s n → L p n .We remark that even in the case 0 < s ≤ p < 1, the condition i of Theorem 2.9 is still necessary.The proof given here does not allow to say if it is sufficient.In general, the characterization of those positive measures μ on n such that the embedding map I μ : H Φ 1 n → H Φ 2 n Φ 1 / Φ 2 if Φ 1 and Φ 2 are convex growth functions is bounded, is still open.

Theorem 2 . 1 .
For a positive Borel measure μ on n , and 0 < p < ∞, the following are equivalent i The measure μ is a Carleson measure.

Theorem 3 . 1 .
Let 0 < s ≤ 1, 1 ≤ p < ∞ and, α > −1.Then uC ϕ is bounded from H s n to A p α nif and only if then an easy change of variables gives that 3.11 is equivalent to We recall that a function h is in BMOA ρ if and only if the measure |Rh z| 2 1 − |z| 2 dV z is a 1/ρ 2 -Carleson measure see 31 .That is |1 − z, a | 2n |Rh z | 2 1 − |z| 2 dV z < ∞. 3.18 It follows that T u,b is bounded from H s n to BMOA ρ if and only if for any f ∈ H s n , International Journal of Mathematics and Mathematical Sciences dμ z |u z | 2 |Rb z | 2 1 − |z| 2 dV z .By the equivalent definition in Theorem 2.7, this is equivalent in saying that for any f 1 ∈ BMOA, f 2 ∈ LMOA, and any g ∈ H m n , |u z | 2 |Rb z | 2 1 − |z| 2 dV z 3.21 is a 2-Carleson measure for H s n .It follows from the equivalent definitions of Theorems 2.7 and 2.9 that the latter is equivalent to .16 where dμ z |u z | p |Rb z | p 1 − |z| 2 α p dV z .Let us consider now the operator T u,b .We have the following:Theorem 3.3.Let 0 < s ≤ 1, 0 ≤ p, q < ∞,and α > −1.Let 1/ρ t log 4/t p loglog e 4 /t q .Then T u,b is bounded from H s n to BMOA ρ , if and only if 2 |Rb z | 2 1 − |z| 2 dV z .Proof.