Some Entropy Bump Conditions for Fractional Maximal and Integral Operators

We investigate weighted inequalities for fractional maximal operators and fractional integral operators. We work within the innovative framework of"entropy bounds"introduced by Treil--Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.


Introduction
We are concerned with two-weight inequalities for the fractional maximal and fractional integral operators. The goal is to find simple, A p −like conditions for a pair of weights (non-negative, locally integrable functions) σ, w to ensure where T denotes a fractional maximal or fractional integral operator, and T σ (f ) := T (σf ). One popular approach, initiated by Neugebauer in [9] and developed by Pérez in [10,11], has been to slightly strengthen the A p characteristic by introducing new factors. These new factors, known as bumps, have come in different forms. For example, Neugebauer requires that the weights σ 1+ǫ and w 1+ǫ belong to A p , while Pérez requires that the two weights have finite Orlicz norm. The Orlicz approach is also taken by Cruz-Uribe and Moen in [3]. See the recent paper of Cruz-Uribe [1] and the references therein for more information.
In the context of Calderón-Zygmund operators, Treil-Volberg have recently introduced the notion of entropy bounds and are able to deduce stronger results than have been obtained using the Orlicz approach [14]. In [8], Lacey and the second author combine the entropy bound approach with the theory of sparse operators, introduced by Lerner [6], to efficiently deduce stronger results than in [14]. We use these techniques to prove similar results for fractional integral and fractional maximal operators. In particular, we require that our weights satisfy certain "bump" or "separated bump" conditions (to be defined below.) Before stating the main theorems, we give some definitions. For 0 < α < n, the fractional maximal operator for functions defined on R n is and the fractional integral operator is One reasonable generalisation of the Muckenhoupt A p condition to the present setting is to set [σ, w] := sup Q a cube σ(Q) 1/p ′ w(Q) 1/q |Q| α/n−1 . Ideally, we would like for (1.1) to hold when [σ, w] is finite. This condition is insufficient (see [2] for a counter example in the case of the fractional maximal operator). This condition is enough, however, to deduce weak-type bounds (we couldn't find a proof of this well-known result, so we include one in Section 1.1); in particular, there holds: w] defined as above, T the fractional maximal operator or fractional integral operator, and 1 ≤ p ≤ q ≤ ∞, there holds: Since the finiteness of [σ, w] is not enough to deduce strong bounds, we use two types of bumped conditions to deduce the strong estimates. The first set of conditions on the weights that we consider require a single bump (compare with the separated bumps to be discussed later). Set ρ σ (Q) := 1 σ(Q) Q M(σ½ Q ), and define ρ w similarly, where M is the Hardy-Littlewood maximal operator. We deal first with the fractional maximal operator. Theorem 1.2. Let σ and w be two weights, 1 < p ≤ q < ∞, and M α be the fractional maximal operator. Let ǫ q be a monotonic increasing function on (1, ∞) that satisfies and set ⌈σ, w⌉ := sup Q∈D β(Q). Then The corresponding theorem for the fractional integral operator is: Theorem 1.3. Let 1 ≤ p ≤ ∞ and σ and w be two weights and let I α be the fractional integral operator. Let ǫ p be a monotonic increasing function on (1, ∞) that satisfies ∞ 1 dt tǫ p p (t) = 1 and similarly for ǫ q ′ . Define: and set ⌊σ, w⌋ := sup Q∈Q β(Q). Then The C α,n constant in the above and below theorems arise below in (2.6). The condition in the next theorem is called a "separated bump" for obvious reasons. We use a bump defined in terms of the fractional maximal operator, namely or simply ̺ σ or ̺ when clear.
Theorem 1.4. Let σ and w be weights with densities, 1 < p ≤ q < ∞, and ε q , ε p ′ : R + → R be nonincreasing on (0, 1) and nondecreasing on (1, ∞) such that There holds: In Section 2, we give some preliminary information and lemmas that will be used below. In Section 3, we give a proof of the weak estimates. Sections 4 and 5 contain the proofs of the one-bump theorems for the fractional maximal and fractional integral operators. The proofs in these sections use the theory of sparse operators, discussed below, but avoid the explicit use of testing inequalities. Finally, Section 6 contains the proof of the separated bump theorem for the fractional integral operator. The proof uses both sparse operators and testing inequalities but is still elementary.

Preliminaries
In this section, we list several well-known results; we include some proofs because we could not find them in the literature. We start with some familiar definitions. For a measure µ, will write f µ Q for 1 µ(Q) Q f and f Q when µ is Lebesgue measure. Definition 2.1. A collection, D of cubes is said to be a dyadic grid if: (i) The side length of every Q ∈ D equals 2 k for some k ∈ Z.
(ii) If Q, R ∈ D and Q ∩ R is not empty then either Q ⊂ R or R ⊂ Q.
(iii) If D k = {Q ∈ D : the side length of Q equals 2 k }, then R n = ∪ Q∈D k Q.
Definition 2.2. A subset S of a dyadic grid is said to be sparse if for every P ∈ S there holds: Definition 2.3. Given a measure µ on R n and a dyadic grid, D, a sequence of positive numbers, {a Q } Q∈D , is called a p, q-Carleson Sequence if for every P ∈ D, where the implied constant depends on p, q and the best constant in (2.1).
Proof. We will treat D as a discrete measure space with measure ν where ν(Q) = a Q . We show that the operator T with rule The objective then is to show that for every λ > 0, there holds: where M is the dyadic maximal function. The lemma follows from (2.2) since the dyadic maximal function is bounded for p > 1: We now turn to proving (2.2). Fix λ > 0, and let D λ be the maximal elements Q ∈ D such that f µ Q > λ (such maximal cubes exist since f ∈ L p (µ)). Using the Carleson property of the sequence {a Q } Q∈D , there holds: The last inequality follows by the disjointness of the P ∈ D λ and the fact that q/p ≥ 1.
For the "continuous" version of this theorem, see [4]. We are certain that Lemma 2.4 is contained in a paper, but we have not been able to find a reference.
For a given dyadic grid, D, define the dyadic fractional maximal operator: and the dyadic fractional integral operator: The following lemma is well-known (for the proof of the fractional integral estimate see [3]; the proof of the estimate for the fractional maximal operator is obvious given the fact that for every cube, Q, there is a cube, P Q in a dyadic grid such that Q ⊂ P Q and |P Q | ≤ 3 n |Q|): Lemma 2.5. Let M α be the fractional maximal operator and I α be the fractional integral operator. There is a collection of 3 n dyadic grids such that the following point-wise equivalences hold for all non-negative f : Remark 2.6. When proving the estimates below for the dyadic fractional maximal operator, it is more convenient to deal with the following truncated version: We then prove estimates that are independent of Q 0 . Assuming that f is finite almost everywhere, we can further simplify matters. We start by building a stopping collection, S. Initialise {Q 0 } → S, and in the recursive stage, if P ∈ S is minimal, add to S all maximal children Q of P such that |Q| α/n f Q > 4 |P | α/n f P . For a cube Q ⊂ Q 0 , let Q S denote the S-parent of Q. Similarly, let ch(S) denote the maximal S-descendants of S. Finally, let E Q = Q \ ch(Q). A simple computation shows that for every S ∈ S, That is, the stopping collection S is sparse. Additionally, the E Q are pairwise disjoint and for almost every x ∈ Q 0 there is some Q with x ∈ E Q (this follows from the fact that f = ∞ on a set of measure zero). Thus, we may further reduce (2.3) to: We also note that if {E Q } Q∈D is any collection of pairwise disjoint sets such that E Q ⊂ Q, There is a similar reduction for the dyadic fractional integral operator. Again, we may reduce matters to: We now create the stopping family by initialising {Q 0 } → S and in the recursive stage, if P ∈ S is minimal, add to S all maximal children Q of P such that f Q > 4 f P . Note that we are stopping on averages, not fractional averages. Again, simple computations show that S is sparse. For fixed x ∈ Q 0 , and fixed S ∈ S, the sequence {|Q| α/n ½ Q (x)} Q S =S is geometric and so (2.6) Therefore, the sum in (2.5) can be estimated as: Therefore, in all estimates below, for fixed f , we can replace the operator of interest with one from the right hand side of (2.4) or (2.7); our estimates will be independent of sparse collection S and root Q 0 .
Lemma 2.7. Let 1 < p ≤ q < ∞, let D be a dyadic grid and let S ⊂ D be sparse. Let T be the operator given by T f = Q∈S |Q| α/n f Q ½ Q . Define: Then: 3. Proof of Theorem 1.1 By Lemma 2.5 and the fact that the dyadic fractional maximal operator is pointwise dominated by the dyadic fractional integral operator, Theorem 1.1 follows from the following lemma.
Lemma 3.1. Let 1 ≤ p ≤ q < ∞ and σ and w be two weights. Let D be a dyadic grid, and let I α the dyadic fractional integral operator. Define: Proof. By Remark 2.6, it suffices to prove (3.1) with I α replaced by Let D λ be the maximal elements of D contained in Q 0 such that |Q| α/n f σ Q > λ. Since f σ Q = f σ Q σ Q , there holds: Given the disjointness of the sets Q ∈ D λ , (3.1) is immediate for p = 1. For p > 1, notice the sequence {σ(Q) q/p } Q∈D λ is p, q-Carleson with respect to the measure σ.

Proof of Theorem 1.2
By Lemma 2.5, Theorem 1.2 follows from the following lemma. We remark that while the following proof does not make explicit use of the Sawyer Maximal testing inequalities in [13], the proof does use some of the same ideas.
Lemma 4.1. Let 1 < p ≤ q < ∞, and let σ and w be two weights. Given a dyadic grid D, let M α be the dyadic fractional maximal operator. Let ǫ q be a monotonic increasing function on (1, ∞) that satisfies Set ⌈σ, w⌉ := sup Q∈Q β(Q), then Proof. Let S be any sparse subset of D. By Remark 2.6 we need to verify Taking q th roots and summing over k will imply (4.1).
Using the identity f σ Q = σ Q f σ Q and the pairwise disjointness of the sets E Q , (4.2) will follow from: Thus, by the Carleson Embedding Theorem (Lemma 2.4), it is enough to verify: for all P ∈ Q k . Using the fact that β(Q) ≃ 2 −k ⌈σ, w⌉ for Q ∈ Q k we estimate: We want to show that the sum above is dominated by σ(P ) q/p . To this end, set S r = {Q ∈ Q k , Q ⊂ P : 2 r−1 ≤ ρ σ (Q) ≤ 2 r }. Thus, the sum above is dominated by Appealing to the summability condition on ǫ q , it suffices to show that Q∈Sr σ(Q) q/p ≤ 2 qr/p σ(P ) q/p . Let S * r be the maximal elements in S r . Observe that for fixed S * ∈ S * r , and for any P ⊂ S * , there holds: Since the sets E Q are pairwise disjoint, |Q| ≃ |E Q |, and S * sup P ∈D ½ S * σ P ≤ σ(S * )ρ σ (S * ) ≃ 2 r σ(S * ) for S * ∈ S * r , we estimate Using the disjointness of the sets S * ∈ S * r , the sum in the last line above is dominated by σ(P ) q/p , completing the proof.

Proof of Theorem 1.3
By Lemma 2.5, Theorem 1.3 follows from the following lemma.
Lemma 5.1. Let 1 < p ≤ q < ∞, and let σ and w be two weights. Given a dyadic grid D, let I D α be the dyadic fractional integral operator. Let ǫ p be a monotone increasing function on (1, ∞) such that ∞ 1 dt tǫ p p (t) = 1, and similarly for ǫ q ′ . Define Set ⌊σ, w⌋ := sup Q∈Q β(Q), then Proof. We proceed by duality. Let f ∈ L p (σ) and g ∈ L q ′ (w). Below we use the identity: Using the definition of ⌊σ, w⌋, there holds: .

By Hölder's inequality, it suffices to show that
are dominated by f L p (σ) and g L q ′ (w) , respectively. Since p ≤ q, it follows that q ′ ≤ p ′ , so by the the Carleson Embedding Theorem (Lemma 2.4), it suffices to show: for all P ∈ S. But the proof of each of these estimates is similar to those in Lemma 4.1 and we omit the details 6. Proof of Theorem 1.4 From Remark 2.6 and Lemma 2.7, it is enough to show for any sparse collection Q and Q 0 ∈ Q (the dual testing condition follows identically). For the remainder, fix a root Q 0 and let Q be a sparse collection of cubes contained in Q 0 . Fix α, p, q in the respective appropriate range; we'll ignore these fixed indices where there is no confusion. It remains to show For integers a and r, set Q a,r := {Q ∈ Q : β(Q) ≃ 2 a , ̺(Q) ≃ 2 r }; notice Q a,r is empty for a large enough. Construct a stopping family S for the σ fractional averages: let S be the minimal subset of Q a,r containing the maximal cubes in Q a,r such that whenever S ∈ S, the maximal cubes Q ⊂ S, Q ∈ Q a,r with |Q| α/n σ Q > 4|S| α/n σ S are also in S. Denote by Q S the S-parent of Q. Partition Q a,r into Q a,r k , those cubes in Q a,r such that |Q| α/n σ Q ≃ 2 −k |Q S | α/n σ Q S . We temporarily denote Q a,r k by Q ′ . We will show where summing over k ≥ −2 gives Q∈Q a,r Define for each S ∈ S Since S∈S Φ S,ℓ is geometric for fixed ℓ ∈ Z + , Hölder's inequality yields 3) It is apparent that we need the following distributional estimate.
Proof. The inequality is immediate in the case w is Lebesgue measure from sparseness of Q. Notice that we have for Q ∈ Q ′ with Q S = S, w Q ≃ 2 a 2 r ε q (2 r ) (2 −k σ S |S| α/n ) −q/p ′ =: τ S , where the equivalence is independent of S. Denote by Q * the maximal cubes in Q ′ . Since the Φ S > λ2 −k |S| α/n σ S is the union of the maximal cubes P ∈ Q ′ with P S = S and inf x∈P Φ S (x) > λ2 −k |S| α/n σ S , hence a disjoint union, it follows that w Φ S > λ2 −k |S| α/n σ S ≃ τ S Φ S > λ2 −k |S| α/n σ S τ S 2 −(λ−1) The collection Q * is disjoint, so the proof is complete.