A quantitative approach to weighted Carleson Condition

Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea in the 80's for the operator \[ \mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{|Q|}\int_{Q}|f(x)|dx \qquad x\in\mathbb{R}^{n}, \, t \geq0 \] are obtained. As a consequence, some sufficient conditions for the boundedness of $\mathcal{M}$ in the two weight setting in the spirit of the results obtained by C. P\'erez and E. Rela and very recently by M.T. Lacey and S. Spencer for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained.


Introduction
In 1962 L. Carleson [C] studied a variation of the Hardy-Littlewood maximal operator defined on R n+1 The interest of the study of this operator stems from the fact that it controls pointwise the Poisson integral. Indeed, if we call P (x, t) = c n t (|x| 2 + t 2 ) n+1 2 the Poisson integral is defined as P f (x, t) =ˆR n f (y)P (x − y, t)dy x ∈ R n , t ≥ 0 and it's not hard to check that (cf. [GCRdF] Chapter 2) Carleson characterized the positive Borel measures on R n+1 + such that M is of strong type (p, p) for p > 1 and of weak type (1, 1). The precise statement of his result is the following Theorem I. Let µ be a positive Borel measure on R n+1 + and let 1 < p < ∞. Then, the following statements are equivalent, (1) M : L p (R n ) → L p (R n+1 + , µ) or M : L 1 (R n ) → L 1,∞ (R n+1 + , µ) (2) There is a constant C such that for each cube Q in R n µ(Q) ≤ C|Q| whereQ = {(x, t) : x ∈ Q, 0 ≤ t ≤ l(Q)} .
In fact, it can be established that exactly the same result holds for the Poisson integral (cf. [GCRdF] Chapter 2). Condition (2) in the preceding Theorem is the so called "Carleson Condition". Later on, in the 70's [FS] Fefferman and Stein obtained the following weighted result Theorem II. Let µ be a positive Borel measure on R n+1 + and let w be a weight in R n such that Then we have both M : L p (R n , w) → L p (R n+1 , µ), and M : L 1 (R n , w) → L 1,∞ (R n+1 + , µ) This result was extended in the 80's by F. Ruiz and J. L. Torrea [R, RT]. The conditions obtained by these authors are in the spirit of the ones obtained by B. Muckenhoupt's [M] and E. Sawyer's [S] for the Hardy-Littlewood maximal function. The main results are the following.
Theorem III. Let µ be a positive Borel measure on R n+1 + and let w be a weight in R n . Let also 1 < p < ∞, then the following statements are equivalent: (1) M : Theorem IV. Let µ be a positive Borel measure on R n+1 + and let v be a weight in R n . Let also 1 < p < ∞, then the following conditions are equivalent: ( Theorems III and IV characterize qualitatively the weak and the strong type L p boundedness of M in the sense that they characterize the L p boundedness but they don't provide a quantitative relationship between the operator norm and the relevant constants associated to the couple (µ, v). Quantitative estimates for the main operators in Harmonic Analysis have been widely considered by many authors in the last years. The first result can be traced back to the work of Buckley [B] where a quantitative version of the classical Muckenhupt theorem is obtained, namely where M is the classical Hardy-Littlewood maximal function. Later on and motivated by the A 2 conjecture for the Ahlfors-Beurling transform formulated in [AIS], the sharp dependence of weighted inequalities on the A p constant has been studied thoroughly for operators such as Calderón-Zygmund operators (see for instance [PV,P1,P2,H,Le2] or the more recent works [CAR,LN,La,Le3]), rough singular integrals (c.f. [HRT]), commutators (c.f. [CPP]) or the square function (for example [HL]).
Our aim in this paper is to obtain quantitative versions of Theorems III and IV which, as a consequence, will provide two new quantitative sufficient conditions for the boundedness of M. Since M controls pointwise P , as a direct consequence of those quantitative estimates for M we will also obtain corresponding quantitative estimates for the Poisson integral.
This paper is organized as follows. In section 2 we state our main results. In section 3 we introduce some definitions needed to understand in full detail the main results. Finally in section 4 we prove our main results.
Aknowledgements. I would like to thank my advisor Carlos Pérez for turning my attention to this problem and for his guidance and support during the elaboration of this paper.

Main Results
Before we state our main results we would like to note that the precise definitions of the operators and the constants involved are summarized in section 3.
Our first of result is a quantitative characterization of the weak-type (p, p) boundedness.
Theorem 1. Let µ be a positive Borel measure on R n+1 + and let σ be a weight in R n . Let also 1 < p < ∞, then Theorem 2. Let µ, σ and p as in the previous theorem. Then Relying on the preceding result and following the proof for the Hardy-Littlewood maximal operator obtained by Pérez and Rela [PR] we can obtain the following quantitative sufficient condition for the boundedness of the operator in the weighted setting.
Theorem 3. Let µ, σ and p as before. Also, let Φ be a Young function with conjugate functionΦ. Then The definitions of the constants involved in this result can be found in section 3. These kind of conditions, often called "bump" conditions, were introduced in the 90's in [Pe1] and considered to study sharp two weighted estimates for Singular Integrals [Pe2] and have been often used in recent literature.
A very interesting recent result obtained by L. Slavíková is that this condition is not neccesary. We remit to [Sl] for details.
The following result generalizes the result due to Lacey and Spencer [LS] which is based on the very recent idea of entropy introduced in [TV].
Theorem 4. Let µ, σ and p as before. Then It's clear that if we replace M by the standard Hardy-Littlewood maximal operator and if we take µ(x, t) = w(x)δ 0 (t), where δ 0 denotes Dirac's delta, we recover all the results in their classical setting.
As a consequence of the preceding results and (1.1), we obtain also analogue estimates for the Poisson integral. We compile all those estimates in the following Corollary 1. Let σ be a weight in R n and let µ be a Borel measure on R n+1 + and 1 < p < ∞. Then the Poisson integral satisfies the following estimates

Preliminaries and definitions
3.1. Basic definitions. In this section we recall some basic facts that play a main role in this paper. We also give the precise definitions of the operators and the constants used in our results.
Definition 1. Let f a locally integrable function in R n , we define the function Mf on R n+1 We also consider dyadic versions of this operator. First we recall the definition of dyadic grid (c.f. [Le1]).
Definition 2. We say that a family of cubes D is a dyadic grid if it satisfies the following properties ( The cubes of a fixed sidelength 2 k form a partition of R n We will also use following Definition 3. Let D a dyadic grid. Given a cube Q ∈ D we call D(Q) the family of all the cubes of D that are contained in Q.
Definition 4. Let f a locally integrable function in R n and D a dyadic grid. We define the function N D f on R n+1 We shall drop the superscript D when we work with just one dyadic grid.
We denote byQ the cube built from a cube Q as follows in other words,Q is the cube in R n+1 + having Q as a face. Using the argument given in [CUMP,Lemma 5.38 pg. 111], we can obtain the following Lemma that we will use in the sequel.
Lemma 1. Given a Young function Φ for every Q and every (x, t) ∈Q we have that 3.2. Orlicz averages. We recall that Φ is a Young function if it is a continuous, nonnegative, strictly increasing and convex function defined on [0, ∞) such that Φ(0) = 0 and lim t→∞ Φ(t) = ∞. The localized Luxembourg norm of a function f with respect to a Young function Φ can be defined as follows We note that the case Φ(t) = t corresponds to the usual average. We can see these localized norms as a "different" way of taking averages. We can also define the maximal function associated to Φ as For each Young function there exists an associated complementary Young functionΦ that satisfies the following inequalities and that 3.3. A p and bump type conditions. In this section we give precise definitions of all the "A p type" conditions that appear in this paper. All them resemble in some way their classical counterparts as A p or bump type conditions. We begin with the constant involved in the weak type (p, p) inequality.
Definition 5. Let 1 < p < ∞. Given a weight σ and a Borel measure µ on R n+1 We define now the constants involved in the characterization of the L p boundedness.
Definition 6. Let 1 < p < ∞. Given a weight σ ≥ 0 and a Borel measure µ on R n+1 we define where the supremum is taken over all cubes of R n The definition of the dyadic variant of this constant is almost the same Definition 7. Let D a dyadic grid. Let 1 < p < ∞. Given a weight σ ≥ 0 and a Borel measure µ on R n+1 we define We observe that the constants we have just defined are quite natural. Indeed, if we consider µ(x, t) = w(x)δ(t) where w is a weight and δ is Dirac's delta we recover the quantitative result obtained in terms of the S p constant by Moen in [Mo].
To end this section we give precise definitions of the constants involved in the quantitative sufficient conditions provided in our main results. First we focus on the constants involved in Theorem 3.
Definition 8. Given a weight σ, a Borel measure µ on R n+1 + and a Young function Φ we define the quantity We say that µ, σ belong to the Definition 9. Given a weight σ, a Borel measure µ on R n+1 + and a Young function Φ we define the quantity which is the well known A ∞ constant that was discovered by Fujii in [F], rediscovered by Wilson in [W] and shown to be the most suitable one in [HP] (see also [HPR]).
Let us now turn our attention to the constant involved in Theorem 4.
Definition 10. Let ε be a monotonic increasing function that satisfieŝ We define the quantity , It's worth compairing the constants involved in Theorems 4 and 3 when we choose Φ(t) = t p ′ . In that case the constant in Theorem 3 is In contrast with that couple of supremums, the definition of ⌈µ, σ⌉ 1 p p,ε "includes in some way" the A ′ p constant and a "bumped" A ∞ constant. It's not clear which of those quantities is larger.
Observe that since ρ(Q) ≥ 1 which is a not unexpected fact since the strong-type (p, p) implies the weak-type (p, p). Trivially, we have also that

Proofs of the main results
4.1. Proof of Theorem 1. We first prove that holds for any 1 < p < ∞. Let us assume that M(·σ) L p (R n ,σ)→L p,∞ (R n+1 + ,µ) < ∞ since otherwise there's nothing to prove. We fix a cube Q in R n . For all (x, t) ∈Q by the definition of M it's clear that This yieldsQ If we choose f = χ Q we obtain the desired conclusion.
Now we want to prove that We may assume that f is bounded with compact support. Fix α > 0. If (x, t) then we can find a cube R containing x with l(R) ≥ t and such that 1 |R|ˆR |f (y)σ(y)|dy > α.
Let k be the only integer such that 1 2 n(k+1) < |R| ≤ 1 2 kn there is some dyadic cube Q with sidelength 2 k such that o R ∩ Q = ∅ and R∩Q |σ(y)f (y)|dy > α|R| 2 n > α|Q| 4 n then 1 |Q|ˆQ |σ(y)f (y)| > α 4 n and we have that Q ⊆ Q j ∈ C α for some j where C α 4 n denotes the family of maximal dyadic cubes P such that 1 |P |ˆP |σ(y)f (y)| > α 4 n (observe that we can consider such a family since f is integrable). Also x ∈ R ⊆ 3Q ⊆ 3Q j . This yields t ≤ l(R) ≤ l(3Q j ) and then (x, t) ∈3 Q j . Thus we have seen that

4.2.
Proof of Theorem 2. The proof of Theorem 2 is based on the corresponding dyadic version, namely, Theorem 5. Let µ be a positive Borel measure on R n+1 + and let σ be a weight in R n . Let D a dyadic grid and let 1 < p < ∞. Then The proof of Theorem 5 consists essentially in tracking the constants in the proof of this result in [RT].
Proof of Theorem 5. The proof is the same for every choice of dyadic grid D so for simplicity we denote N D by N and [µ It suffices to choose f = χ Q . We focus now on the converse. As always we may assume that f is bounded with compact support. We need to introduce the following truncations of the dyadic maximal operator: We observe that N R f (x, t) = 0 for t > R and that Let We will also need a Calderón-Zygmund decomposition suited to that truncations. The following Lemma contains that decomposition.
Lemma 2 ( [RT]). Let g ∈ L 1 . For each k ∈ Z we can choose a family {Q k j } j∈J k ⊆ D such that (2) The interiors ofQ k j are disjoint.
The proof of that lemma is the same as the one contained [RT] with minor modifications. Applying Lemma with g = σf , we can consider the sets The sets E k j have disjoint interiors and Following ideas from Sawyer and Jawerth we introduce the following notation Using this notation we have that If we call Q i the maximal cubes of the family {Q k j : (k, j) ∈ Γ(λ)} we can rewrite the preceding sum as follows Now we use 2 and we obtain Now from the definition of Γ(λ) and the g ′ jk s it's clear that We have obtained that for every R > 0 R n+1 Since N R f (x, t) ↑ N f (x, t) by monotone convergence theorem we're done.
Theorem 2 will be proved reducing it to the dyadic case. To do that we will show next that M is controlled pointwise by a sum of 2 n N D j operators.
Lemma 3. We can build 2 n dyadic grids such that Proof. To prove this lemma we need to use the fact that we can find a finite family of dyadic grids such that every cube is "well controlled" by a cube in one these families Lemma ([Le1, Prop. 5.1]). There are 2 n dyadic grids D j such that for any cube Q, there exists a cube Q j ∈ D j such that Q ⊆ Q j and l(Q j ) ≤ 6l(Q).
We note that this result appears implicitly in [GCRdF,. Armed with that lemma we can establish our result. Let us take (x, t) ∈ R n+1 + and consider If we choose any of the averages involved in that supremum we can take Q j ∈ D j such that Q ⊆ Q j and l(Q) ≤ l(Q j ) ≤ 6l(Q). This yields To end this section we give the proof of Theorem 2.
is exactly the same of the dyadic version Theorem 5. It's simply testing with cubes. Now suppose that [µ, σ] S ′ p ,D j < ∞, since otherwise there's nothing to prove. It's clear that for every dyadic grid D j Consequently Theorem 5 applies for every dyadic grid, that is, . Now we apply Lemma 3 and use (4.1) to obtain as we wanted to prove.

Proof of Theorems 3 and 4.
To proof both Theorems we need a Calderón-Zygmund type decomposition suited to our purposes. We obtain that decomposition in the following Lemma 4. Let D be a dyadic grid. Let f be a locally integrable function such that supp f ⊆ Q 0 ∈ D. Let k 0 ∈ Z be the smallest integer such that (2) The interiors ofQ k j with j ∈ J k are pairwise disjoint.
Furthermore, the family of cubes {Q k j } j∈∪J k is sparse, that is, for each Q k j we can take a measurable subset E k j such that 1 2 |Q k j | ≤ |E k j | and the sets E k j are pairwise disjoint.
Proof. First of all a dyadic version of Lemma 1 allows us to say that for each (x, t) ∈Q 0 we have that Then we observe that for each (x, t) ∈ Ω k we can find a cube Q (x,t) ∈ D such that x ∈ Q (x,t) Q (x,t) ⊆ Q 0 and l(Q (x,t) ) ≥ t. Since all that cubes are contained in Q 0 we can choose among them the maximal ones and call them {Q k j } j∈J k . It's clear that this collection of cubes satisfies conditions (1), (2) and (3). To end the proof we have to prove that the family {Q k j } ∪ k≥k 0 J k is sparse.
Let us call for each k ≥ k 0 m ⊆ Q k j by maximality. Taking that into account and using (1) and (2) and we obtain the desired conclusion taking E k j = Q k j \ H k+1 . To give the proof of Theorem 3 we will adapt the argument in Pérez-Rela [PR].
Proof of Theorem 3. As in the proof of Theorem 2 it is enough to consider N D j , Hence, it is enough to prove that Let us fix a cube Q. Consider k 0 ∈ Z such that
Now we see thatQ dividing by σ(Q) and raising to the power 1 To end this section we give the proof of Theorem 4. We will adapt the proof of [LS].
Proof. As in the proof of Theorem 2 it is enough to consider N D j , Hence, it is enough to prove that [µ, σ] Sp,D j ⌈µ, σ⌉ p,ε .
To simplify the notation we fix one of these j and denote D j and N D j by D and N Let us fix Q ∈ D. Arguing as we did in the proof of Theorem 3, we can writêQ First we observe that To end the proof we have to control II. To simplify we denote by S the family of cubes {Q k i } and everything is left is to understand where σ S = σ(S) |S| . We divide now the collection S into subcollections S a,r as follows. S ∈ S a,r for some a ∈ Z and r ∈ {0, 1, 2, . . . } if and only if 2 a−1 ≤ σ p−1 S µ(S) |S| ρ σ,ε (S) ≤ 2 a and 2 r ≤ ρ(S) ≤ 2 r+1 .
This ends the proof.