Tent Carleson measures for Hardy spaces

We completely characterize those positive Borel measures $\mu$ on the unit ball $\mathbb{B}_ n$ such that the Carleson embedding from Hardy spaces $H^p$ into the tent-type spaces $T^q_ s(\mu)$ is bounded, for all possible values of $0


Introduction
The concept of a Carleson measure was introduced by L. Carleson [6,7] when studying interpolating sequences for bounded analytic functions on the unit disk, in route of solving the famous corona problem.Carleson's originally result is a characterization of those positive Borel measures on the unit disk D for which the embedding I : H p → L p (D, µ) is bounded, for 0 < p < ∞.Later on, variations and extensions of the results to the unit ball B n of C n were obtained by Hörmander, Duren and Luecking [12,16,19], obtaining a full description of the boundedness of the embedding I : H p → L q (B n , µ) for all possible choices of 0 < p, q < ∞.Moreover, generalizations of the Carleson embedding were subsequently studied by replacing the Hardy space H p by other function spaces, making Carleson type measures important tools for the study of modern function and operator theory.We recall that, for 0 < p < ∞, the Hardy space H p := H p (B n ) consists of those holomorphic functions f in B n with where dσ is the surface measure on the unit sphere S n := ∂B n normalized so that σ(S n ) = 1.We refer to the books [2], [25] and [30] for the theory of Hardy spaces in the unit ball.
Apart from replacing the Hardy space by another function space in the study of the embedding, the next more natural step for generalizing such study, is to replace L q (B n , µ) by the tent space T q s (µ), that we are going to define in a moment.We mention here that L q (B n , µ) = T q q (µ).As far as we know, tent spaces were introduced by Coifman, Meyer and Stein [11] in order to study several problems in harmonic analysis.These spaces turned to be quite useful in developing further the classical theory of Hardy spaces, closely related to tent spaces due to the Calderon's area theorem.This motivates the study of the boundedness of the embedding I : H p → T q s (µ) for all possible choices of 0 < p, q, s < ∞.Recently, natural analogues of tent spaces for Bergman spaces have been introduced [23,24], playing a similar role for the theory of weighted Bergman spaces as the original ones in the Hardy space.Other interesting results on Carleson measures related to Hardy spaces can be found in [9,13,18,26].
In order to define the tent spaces T q s (µ), we need to introduce first the Korányi admissible approach region Γ(ζ), defined for ζ ∈ S n and γ > 1 as Let 0 < q, s < ∞ and µ be a positive Borel measure on B n .The tent space T q s (µ) consists of those µ-measurable functions f : B n → C with The aperture γ > 1 of the Korányi region is suppressed from the notation, as it is well-known that any two apertures generate the same tent space with equivalent quasinorms.
A full description of the boundedness of the embedding from Hardy spaces H p into the tent spaces T q s (µ) in one dimension in the case p ≤ q was obtained by Cohn [10] and Gong-Lou-Wu [15] in terms of the area-type operator A µ,s defined as It is obvious that I : H p → T q s (µ) is bounded if and only if A µ,s : H p → L q (S n ) is bounded.Related results can be found in [3,8,22,27,28].
Our first main result is a generalization of the description of the boundedness of the area operator A µ,s : H p → L q (S n ) in the case p ≤ q to the setting of higher dimensions.We mention here, that the proof given in one dimension in [10,15] was somewhat technical, using tools of harmonic analysis such as Calderon-Zygmund decompositions among others.The proof we present here, valid for all dimensions, is much more simpler and natural, and it is based on a trick that has its roots on [21] when studying integration-type operators acting on Hardy spaces.Theorem 1.1.Let 0 < p ≤ q < ∞, and µ be a positive Borel measure on B n .Then A µ,s : . Moreover, we have We recall that, for α > 0, a finite positive Borel measure on B n is called an α-Carleson measure if µ(B δ (ξ)) δ nα for all ξ ∈ S n and δ > 0, where B δ (ξ) denotes the non-isotropic metric ball The famous Carleson measure embedding theorem was extended to several complex variables by Hörmander [16], and to the case p < q by Duren [12].A simple proof of Duren's theorem in the setting of the unit ball can be found in [21] for example.Joining all the results, the Carleson-Hörmander-Duren's theorem asserts that, for 0 < p ≤ q < ∞, the embedding I : H p → L q (µ) := L q (B n , µ) is bounded if and only if µ is a q/p-Carleson measure.Moreover, one has the estimate I H p →L q (µ) ≍ µ 1/q CM q/p , where, for α > 0, we set Concerning the description of the boundedness of A µ,s : H p → L q (S n ) when 0 < q < p < ∞, in one dimension several cases where solved in [15].It was proved that, when q > s, and 0 < q < p < ∞, then A µ,s : H p (B 1 ) → L q (S 1 ) is bounded if and only if the function µ belongs to L r/s (S 1 ), where Really, in [15], this result was proved for s = 1, but in one dimension once such a description is obtained for a fixed s, the result for general s follows directly from the strong factorization for functions in Hardy spaces.In [15], they also conjecture that the description obtained holds also for all possible values of p, q, s.Our second main result is a proof of this conjecture in all dimensions.
Theorem 1.2.Let 0 < q < p < ∞ and s > 0. Let µ be a positive Borel measure on B n .Then the operator A µ,s : H p → L q (S n ) is bounded if and only if µ ∈ L r/s (S n ) with r = pq/(p − q).Moreover, one has The proof of the sufficiency of the condition µ ∈ L r/s (S n ) is somewhat easy, so that the interest of Theorem 1.2 is its necessity.The case r > s is proved using Luecking description [19] of those positive Borel measures µ on B n for which the embedding I : H p → L q (µ) is bounded for 0 < q < p < ∞.That is, Luecking's theorem states that for 0 < q < p < ∞, the Carleson embedding I : H p → L q (µ) is bounded if and only if µ ∈ L p p−q (S n ), and moreover one has the estimate The necessity in the case 0 < r ≤ s seems to be much more challenging, and our proof uses: Khintchine's type inequalities, that has been widely used in recent years in order to get the necessity part when studying the boundedness with loss of certain operators; the theory of tent sequence spaces, with a crucial use of the factorization of such tent spaces; and several new techniques whose ideas came to us when studying Luecking's results [19] on embedding derivatives of Hardy spaces into Lebesgue spaces.We hope that these new techniques are going to be useful in the future in order to deal with related problems.
Throughout the paper, constants are often given without computing their exact values, and the value of a constant C may change from one occurrence to the next.We also use the notation a b to indicate that there is a constant C > 0 with a ≤ Cb.Also, the notation a ≍ b means that the two quantities are comparable.
The paper is organized as follows: in Section 2 we recall some well known results that will be used in the proofs.Theorems 1.1 is proved in Section 3, and Theorem 1.2 is proved in Section 4, section that is divided in several subsections according to what case is proved: sufficiency, and necessity for r > s, r = s and r < s, as each of these cases requires different techniques.

Some preliminaries
In this section, we are going to collect some results and estimates needed for the proofs of the main theorems of the paper.
2.1.Tent sequence spaces.A sequence of points {z j } in B n is said to be separated if there exists δ > 0 such that β(z i , z j ) ≥ δ for all i and j with i = j, where β(z, w) denotes the Bergman metric on B n .We use the notation D(a, r) = {z ∈ B n : β(z, a) < r} for the Bergman metric ball of radius r > 0 centered at a point a ∈ B n .
By [30,Theorem 2.23], there exists a positive integer N such that for any 0 < r < 1 one can find a sequence {a k } in B n with the properties: Any sequence {a k } satisfying the above conditions is said to be an r-lattice (in the Bergman metric).It is clear that any r-lattice is a separated sequence.For 0 < p, q < ∞ and a fixed separated sequence Z = {z j } ⊂ B n , let T p q (Z) consist of those sequences λ = {λ j } of complex numbers with The following result can be thought as the holomorphic analogue of Lemma 3 in Luecking's paper [19].The current version can be found in [4,17,21].
is bounded.
We will also need the following duality result for the tent spaces of sequences [4,17,19].
A crucial step for the proof of Theorem 1.2 will be an appropriate use of the following result concerning factorization of tent sequence spaces, which can be found in [20].
λ T p q (Z) .2.2.Some estimates and the admissible maximal function.
where ϕ is any positive measurable function and ν is a finite positive measure.
We recall that, for a continuous function f : B n → C, the admissible (non-tangential) maximal function f * is defined by An important well known result [30,Theorem 4.24] is the L p -boundedness of the admissible maximal function:

Area function description of Hardy spaces.
Another function we need is the admissible area function Af defined on S n by , where Rf denotes the radial derivative of f .The following result [1,14,21] describing the functions in the Hardy space in terms of the admissible area function, is the version for the unit ball of C n of the famous Calderón area theorem [5].
3. Proof of Theorem 1.1 ). Assume first that 0 < q ≤ s.As, for α = p(s−q) q , we have Hence, by Hölder's inequality with exponents s/(s − q) and s/q, we get Therefore, A µ,s : Next, we consider the case q > s.As A µ,s f s L q (Sn) = (A µ,s f ) s L q/s (Sn) , by duality we must show that, for any function ϕ ∈ L q/(q−s) (S n ) with ϕ ≥ 0, we have where As q > s, then t > 1 with conjugate exponent Hence, applying Hölder's inequality, we obtain On the other hand, we have and by the well known version for Poisson integrals of the Carleson-Hörmander-Duren's theorem, we also have All together, we have . By standard approximation arguments, it is enough to prove the estimate assuming that µ is already a β-Carleson measure.It is well known [29,Theorem 45] that, for any t > 0, we have For each a ∈ B n , consider the function f a defined as Then, by (2.1), Choose 0 < ε < 1 so that εs < q.Then, by Hölder's inequality with exponent .
By the boundedness of A µ,s , we have Set It is easy to check that 1 + s( 1 p 1 − 1 q 1 ) = β.Therefore, by the sufficiency already proved in the previous subsection, we have Putting these estimates together, we obtain Bearing in mind (3.2), we get Finally, by the typical integral estimates (see [30, Theorem 1.12]) we have that gives the desired result.
4. Proof of Theorem 1.2 4.1.Sufficiency.For f ∈ H p , we have Then, applying Hölder's inequality with exponent p q (that has conjugate exponent p p−q ), and using the L p -boundedness of the admissible maximal function, we obtain Hence A µ,s : 4.2.Necessity: the case r > s.Suppose A µ,s : H p → L q (S n ) is bounded.From Luecking's theorem, we have where the number t is chosen so that p p−t = r s , that is, Observe that t > 0 as r > s, and also p > t.From (2.1), we have If t = s, then q = s and we get directly the result.In case that t > s, then q/s > 1 having conjugate exponent q/(q − s) = p/(t − s).Then Hölder's inequality together with the L p boundedness of the admissible maximal function f * gives It remains to deal with the case t < s.By standard approximation arguments, it suffices to show the estimate µ L r/s (Sn) A µ,s s H p →L q (Sn) assuming that the function µ is already in L r/s (S n ).By Hölder's inequality with exponent s/t > 1, we get It is also easy to check that t < q since q < s.Then, using by Hölder's inequality again, now with exponent q/t (that has conjugate exponent q/(q − t) = r/(s − t)), gives From this, it follows that and we obtain the desired result.

4.3.
Necessity: the case r = s.For the proof of this case, we use an averaging function related to the measure µ.For t > 0, let If spt(µ) denotes the support of the measure µ, it is then clear that µ t (z) = 0 for z ∈ spt(µ).
The following conditions are equivalent: Proof.(i) implies (ii).Observe first that where Γ(ζ) is another approach region Γ γ ′ (ζ) with a bigger aperture γ ′ so that (4.2) In order to prove (4.1), we first use the subharmonic result in [30, Lemma 2.24] to obtain This gives For w ∈ D(z, t), we have D(w, t) ⊂ D(z, 2t), and µ(D(w, t)) > 0 if z ∈ spt(µ).Therefore, if A denotes the set of points w ∈ B n with µ(D(w, t)) > 0, then proving (4.1).Now, using Cauchy-Schwarz and the inequality in (4.1), we see that the quantity in (ii) is less than constant times Finally, applying Hölder's inequality with exponents (2+σ)/2 and (2+σ)/σ, together with the area description of functions in Hardy spaces, we see that the quantity in (ii) is less than constant times Now assume that (ii) holds.We test the inequality with the function Using the argument with Kahane and Khintchine's inequalities as in [20], we obtain Since, for z ∈ D(a k , t), we have µ t (z) µ 2t (a k ), we get (4.4) For proving that, we claim that it is enough to show that where one takes the indexes k with µ t (a k ) = 0.That is, it is enough to prove Indeed, we have Then, by the Cauchy-Schwarz inequality To prove (4.5), by the duality of tent sequence spaces in Theorem B, we must prove that Using the factorization of tent sequence spaces in Theorem C, we can write . Then Finally, an application of Hölder's inequality with exponent 4σ/(2 + σ) that has conjugate exponent 4σ/(3σ − 2), together with our condition (4.4) gives C µ , finishing the proof of the Proposition.
As a consequence of the previous proposition, we get the following result that will be the key for the proof of the necessity of Theorem 1.2 when r = s.Corollary 4.2.Let µ be a positive Borel measure on B n .We have that µ ∈ L 1 (S n ) if and only if, for 0 < p < ∞, we have Moreover, one has and the result is a consequence of Proposition 4.1 with σ = 4.
Now we are ready for the proof of the necessity in Theorem 1.2 when r = s.
Theorem 4.3.Let µ be a positive Borel measure on B n , and A µ,s s H p →L q (Sn) .Proof.We are going to use Corollary 4.2.By Cauchy-Schwarz inequality, the estimate in (4.1), Hölder's inequality with exponents 4 and 4/3, and the area functions description of Hardy spaces, we have .
Hence, we need to estimate If p = 2s, as r = s, we see that q = 2s/3, so that, in this case, we have , and the result follows by the boundedness of A µ,s : H p → L q (S n ) and Corollary 4.2.
Since pq/(p−q) = s, we see that ps = q(p+s).Then, by the L p -boundedness of the admissible maximal function, we get and we get the result from Corollary 4.2.
If p < 2s, then we apply Hölder's inequality with exponent 2s/p > 1 to obtain Hence, assuming that µ is already in L 1 (S n ), by Corollary 4.2 we have that implies the estimate µ L 1 (Sn) A µ,s s H p →L q (Sn) .The general case follows from an standard approximation argument.4.4.Necessity: the case r < s.We need first the following lemma.Lemma 4.4.Let 0 < p, q < ∞, s > 0. Let µ be a positive Borel measure on B n .If A µ,s : ≤ A µ,s q H p →L q (Sn) f m q H p = A µ,s q H p →L q (Sn) f mq H mp , which proves the desired estimate.
The next result needed can be considered as the analogue of Proposition 4.1 and Corollary 4.2 for the case r < s.Proposition 4.5.Suppose 0 < q < p < ∞, s > 0, r/s < 1 with r = pq/(p − q).For any positive integer m with Let µ be a positive Borel measure on B n .The following conditions are equivalent: Moreover, we have where the supremum is taken over all functions f, g ∈ H 2mp with f Proof.First, suppose µ ∈ L r/s (S n ), and let f, g ∈ H 2mp .Proceeding in the same way as in the proof of the estimate (4.1), we have Observe that, as r/s < 1, then s(p − q) − pq > 0. This, together with (4.6) tells us that σ > 0. Also, as mp > 2, it follows that σ < 2. Now, an application of the estimate (2.1) together with Hölder's inequality with exponent 2/σ > 1 gives As 0 < σ < 2, the condition r < s implies that 2r s(2−σ) > 1 with conjugate exponent given by 2r 2r − s(2 − σ) = mp σ .
By Cauchy-Schwarz inequality, the L p -boundedness of the admissible maximal function and the area description of functions in Hardy spaces, we have where the supremum is taken over all functions f, g For the converse, we need to show (4.8) µ Then, by (ii) we have That is, we have Now we proceed as in the proof of Proposition 4.1.We only sketch the proof without giving all the details.We test this inequality with the function F s given by (4.3), where λ . Using the argument with Kahane and Khintchine's inequalities, we obtain (4.9) In order to prove that µ ∈ L r/s (S n ), it suffices to show that { µ t (a k )} ∈ T r/s 1 (Z), and this is equivalent to .Then Finally, an application of Hölder's inequality with exponent 4mp/(2mp + σ) that has conjugate exponent 4mp/(2mp − σ), together with our condition (4.9) shows that µ is in L r/s (S n ).An examination of the estimates as done in the proof of Proposition 4.1 gives (4.8).
Now we are ready for the proof of the necessity for the case r < s in Theorem 1.2.
Proof.Take a positive integer m big enough so that (4.6) holds.From Lemma 4.4, we know that A µ,2ms : H 2mp → L 2qm (S n ) is bounded with A µ,2ms H 2mp →L 2mq (Sn) ≤ A µ,s
Putting this in the estimate (4.12), by the L p -boundedness of the admissible maximal function, we have .
(i) B n = k D(a k , r); (ii) The sets D(a k , r/4) are mutually disjoint; (iii) Each point z ∈ B n belongs to at most N of the sets D(a k , 4r).
by the L p -boundedness of the admissible maximal function and the Carleson-Hörmander-Duren's theorem, we obtain