Off-Diagonal Two Weight Bumps for Fractional Sparse Operators
aa r X i v : . [ m a t h . C A ] J a n OFF–DIAGONAL TWO WEIGHT BUMPS FOR FRACTIONAL SPARSEOPERATORS
ROB RAHM
Abstract.
In this paper, we continue some recent work on two weight boundedness of sparseoperators to the "off–diagonal" setting. We use the new "entropy bumps" introduced in byTreil–Volberg ([21]) and improved by Lacey–Spencer ([9]) and the "direct comparison bumps"introduced by Rahm–Spencer ([19]) and improved by Lerner ([10]). Our results are "sharp" inthe sense that they are sharp in various particular cases. A feature is that given the currentmachinery, the proofs are almost trivial. Introduction
The topic of this article is two–weight bump conditions for sparse operators in the “off–diagonal” setting (i.e q > p ). We continue the line of investigation concerning entropy bumpsthat began with Treil–Volberg in [21] and was continued in [9, 18, 19] and the line of investigationconcerning "direct comparison bumps" introduced in [19] and continued and improved in [10].We will be concerned with sparse operators of the form ( ≤ α < d ): T α, S f := X Q ∈S ( | Q | αd h | f | i Q ) Q , where h f i Q := | Q | Z Q f. A more general class of these operators are studied in [5]. Unlike that paper, we make noassumption about the weights being in A ∞ and we only concentrate on the off–diagonal case.The literature is saturated with results of this type. The main contributions of this paper are(1) the results are new; (2) the proofs are succinct and highlight the development of this areaand (3) indicate the challenges in proving various "separated bump theorems".The first deals with entropy bumps. It was proven in the p = q = case in [21] and in thegeneral p = q case in [9]; both with α = . Theorem 1.1.
Let T α, S be a sparse operator and ∞ then: k T α, S σ · : L p ( w ) → L p ( σ ) k . E + E ∗ , where: E = sup Q w ( Q ) σ ( Q ) ′ | Q | − αd ρ ( Q ; σ ) ε ( ρ ( Q ; σ )) Mathematics Subject Classification.
Key words and phrases. sparse operators, separated bumps, entropy bumps, direct comparison bumps. and E ∗ = sup Q w ( Q ) σ ( Q ) ′ | Q | − αd ρ ( Q ; σ ) ′ ε ( ρ ( Q ; σ )) ′ and ρ ( Q ; σ ) is the "local A ∞ characteristic": ρ ( Q ; σ ) := ( Q ) Z P M ( σ11 P ) and ρ ( Q ; w ) := ( Q ) Z P M ( w11 P ) , and ε is a monotonic increasing function that satisfies P ∞ r = ε ( r ) − < ∞ . The next theorem was introduced in [19] and improved in [10] (for p = q and α = ): Theorem 1.2.
Let T α, S be a sparse operator and ∞ and p > 1 then: k Tσ · : L p ( w ) → L p ( σ ) k . D + D ∗ , where: D = sup w ( Q ) σ ( Q ) ′ | Q | − αd ε ( h σ i Q ) and D ∗ = sup w ( Q ) σ ( Q ) ′ | Q | − αd ε ( h w i Q ) ′ and ε is a function that is decreasing on (
0, 1 ) increasing on ( ∞ ) and satisfies P ∞ r =− ∞ ε ( r ) − . Background and Discussion
It was noted in [6, 12, 13] that the standard Muckenhoupt condition was necessary but notsufficient for the two–weight boundedness of operators of interest. The purpose of the various"bumps" are to replace the standard A p (or A p,q in our case) condition with a slightly biggercondition that is sufficient to give two–weight boundedness. This is a well–developed area, see[1–4, 8, 11, 14, 15, 17] and the references therein for more information.The "entropy bumps" in Theorem 1.1 were introduced by Treil–Volberg in [21] in the p = q = case. This was extended to the general p = q case by Lacey–Spencer in [9]. These operatorswere also studied in the off–diagonal p ≤ q , α > 0 setting in [16, 18]. Our theorem here is"sharp" in the sense that when σ and w are A ∞ , we recover the sharp results of, for example[2, 3, 7]. The "direct comparison bumps" were introduced in [19] and improved in [10].The off–diagonal setting allows us to replace the normal norm in the testing inequalities with L norms (this is Proposition 3.1 below; see also [2, Theorem1.1]). This is why off–diagonalresults are sharper than on–diagonal results. Based on comparisons with Orlicz conjectures, thefollowing conjecture is made (Lerner almost proves this in [10]): Conjecture 2.1. k Tσ · : L p ( σ ) → L p ( w ) k . D p,p + D ∗ p,p . FF–DIAGONAL BUMPS 3
The Orlicz, entropy, and direct comparison bumps are not strictly comparable. The Orliczbumps are the most established but require the most in terms of local integrability. The entropybumps are guaranteed to be bounded in the one weight setting (and record important A ∞ information about the operator norms). The direct comparison bumps require the least in termsof local integrability but are not as popular. They are also easier to verify than Orlicz bumps orentropy bumps. The weights σ ( x ) = ( − log x ) , w ( x ) = x . satisfy the two weight conditions in [10] or even in [19]. Yet σ is not in L log L and so neither theOrlicz nor the entropy bumps will detect the boundedness of sparse operators with these weights.3. Preliminaries and Notation
A collection of cubes, S , is called λ –Sparse ( ) if for every Q ∈ S , there holds: X Q ∈S : Q ⊂ Q ,Q is maximal | Q | ≤ λ | Q | . If S is sparse, the sets E Q = Q \ ∪ Q ′ ∈S : Q ′ ⊂ Q Q ′ are pairwise disjoint and | E Q | ≃ | Q | . If Q ∈ S then using the pairwise disjointness of the { E Q } we have the following well–know estimate: X Q ∈Q : Q ⊂ Q σ ( Q ) ≃ Z Q X Q ∈Q : Q ⊂ Q h σ i Q E Q ≤ Z Q M ( σ11 Q ) = ρ ( Q ; σ ) σ ( Q ) . (3.1)A consequence of [2, Theorem 1.1] is (note the L norms): Proposition 3.1.
For there holds k T α, S ( σ · ) : L p ( σ ) → L q ( w ) k . T + T ∗ where: T := sup S is sparse sup R ∈S σ ( R ) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈S : R ⊂ Q ( | Q | q αd h σ11 R i Q ) q E Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( w ) T ∗ = sup S is sparse sup R ∈S w ( R ) − ′ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈S : R ⊂ Q | Q | ( αd − ) p ′ w ( Q ) p ′ E Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ′ L ( σ ) . Indeed, [2, Theorem 1.1] says that if p < q then: k T α, S σ · : L p ( σ ) → L q ( w ) k ≃ k M α σ · : L p ( σ ) → L q ( w ) k + (cid:13)(cid:13)(cid:13) M α σ · : L q ′ ( w ) → L p ′ ( σ ) (cid:13)(cid:13)(cid:13) , where M α f ( x ) := sup Q | Q | αd h | f | i Q Q ( x ) is the fractional maximal operator. The well–known([20]) testing conditions for this operator reduce to the ones in Proposition 3.1. This is becausethe linearization of M α is the function inside the norm in this proposition and the pairwisedisjointness of the { E Q } allows us to pass the exponent q (and p ′ ) under the sum. ROB RAHM Proof of Theorem 1.1
We will estimate T from Proposition 3.1 (the estimate for T ∗ is dual).Let S a be those cubes with ρ ( Q ; σ ) ≃ a (observe that a ≥ ) and let S ∗ a be the maximalcubes in S a . The integral in the definition of T can be organized as follows: X Q ∈S : Q ⊂ R | Q | q αd h σ i qQ w ( Q ) = X a ≥ X Q ∗ ∈S ∗ a X Q ∈S a : Q ⊂ Q ∗ ( w ( Q ) σ ( Q ) qp ′ | Q | q − q αd ) σ ( Q ) qp . (4.1)Concerning the inner sum, we bound this by: E q a ε ( a ) X Q ∈S a : Q ⊂ Q ∗ σ ( Q ) qp E q a ε ( a ) σ ( Q ∗ ) qp − X Q ∈S a : Q ⊂ Q ∗ σ ( Q ) ≤ E q σ ( Q ∗ ) qp ε ( a ) . where we multiplied and divided by ( ρ ( Q ; σ ) ε ( ρ ( Q ; σ )) and used the fact that ρ ( Q ; σ ) ≃ a .The " ≤ " uses (3.1). Now we insert this estimate into (4.1) and using again that qp ≥ , themaximality of the cubes Q ∗ and the summability of ε we get: X a ≥ X Q ∗ ∈S ∗ a X Q ∈S a : Q ⊂ Q ∗ | Q | q αd h σ i qQ w ( Q ) . E q X a ≥ ( a ) X Q ∗ ∈S ∗ a σ ( Q ∗ ) qp ≤ E q σ ( R ) qp . Taking q th roots we conclude that T . E .5. Proof of Theorem 1.2
We will estimate T from Proposition 3.1 (the estimate for T ∗ is dual).Let S a be those cubes with h σ i Q ≃ a . (observe that − ∞ < r < ∞ ) and let S ∗ a be themaximal cubes in S a . The integral in the definition of T can be organized as follows: X Q ∈S : Q ⊂ R | Q | q αd h σ i qQ w ( Q ) = X a ∈ Z X Q ∗ ∈S ∗ a X Q ∈S a : Q ⊂ Q ∗ | Q | q αd h σ i qQ w ( Q ) . Concerning the inner sum, this is (similar to the above): X Q ∈S a : Q ⊂ Q ∗ ( w ( Q ) σ ( Q ) qp ′ | Q | q − q αd ) σ ( Q ) qp . D q ( a ) σ ( Q ∗ ) qp − X Q ∈S r : Q ⊂ Q ∗ σ ( Q ) ≤ D q σ ( Q ∗ ) qp ε ( a ) . where we multiplied and divided by ε ( h σ i Q ) and used the fact that h σ i Q ≃ a . The “ ≤ ” isobtained using sparseness combined with σ ( Q ) ≃ a | Q | for the cubes in question.Similar to the above proof, the summability condition on ε implies that T . D . References [1] David Cruz-Uribe,
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