summary:Given  $\alpha $, $0<\alpha <n$, and $b\in {\mathrm BMO}$, we give sufficient conditions on weights for the commutator of the fractional integral operator, $[b,I_\alpha ]$, to satisfy weighted endpoint inequalities on  $\mathbb{R}^n$ and on bounded domains. These results extend our earlier work [3], where we considered unweighted inequalities on  $\mathbb{R}^n$

Cruz-Uribe, D., SFO

Fiorenza, A.

English

Institute of Mathematics AS CR, v. v. i.

Czechoslovak Mathematical JournalDavid Cruz-Uribe; Alberto FiorenzaWeighted endpoint estimates for commutators of fractional integralsCzechoslovak Mathematical Journal, Vol. 57 (2007), No. 1, 153–160Persistent URL: http://dml.cz/dmlcz/128162Terms of use:© Institute of Mathematics AS CR, 2007Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 57 (132) (2007), 153–160WEIGHTED ENDPOINT ESTIMATES FOR COMMUTATORSOF FRACTIONAL INTEGRALSD. Cruz-Uribe, SFO, Hartford, A. Fiorenza, Napoli(Received November 9, 2004)Abstract. Given α, 0 < α < n, and b ∈ BMO, we give sufficient conditions on weightsfor the commutator of the fractional integral operator, [b, Iα], to satisfy weighted endpointinequalities on  n and on bounded domains. These results extend our earlier work [3],where we considered unweighted inequalities on  n .Keywords: fractional integrals, commutators, BMO, weights, Orlicz spaces, maximalfunctionsMSC 2000 : 42B20, 42B251. IntroductionGiven α, 0 < α < n, define the fractional integral operator Iα byIαf(x) =∫nf(y)|x− y|n−α dy;for b ∈ BMO, define the commutator [b, Iα] by[b, Iα]f(x) = b(x)Iαf(x)− Iα(bf)(x) =∫n(b(x)− b(y)) f(y)|x− y|n−α dy.Commutators were first introduced by Chanillo [1] who proved Lp estimates, 1 <p < ∞. In [3] we proved the following endpoint estimate:|{x ∈  n : |[b, Iα]f(x)| > t}| 6 CΨ(∫nB(‖b‖BMO|f(x)|t)dx),where B(t) = t log(e + t) and Ψ(t) = [t log(e + tα/n)]n/(n−α).153We initially conjectured that the corresponding weighted inequality wasw({x ∈  n : |[b, Iα]f(x)| > t}) 6 CΨ(∫nB(‖b‖BMO|f(x)|t)w(x)1/q dx),where w ∈ A1 and q = n/(n− α). However, this conjecture proved to be false, andwe gave a counterexample [3, Example 1.8].At the time, we were not able to make a new conjecture. One difficulty we hadwas that locally, our original conjecture appeared to be true; the counterexampleworks because it exploits the decay of the weight at infinity. Careful considerationof this behavior yielded two results. First, we found the correct weighted endpointinequality for the commutator on  n . Second, we showed that if we restrict ourselvesto a bounded domain, then there is a sharper endpoint inequality which is evensimpler than our original conjecture.Theorem 1.1. Given α, 0 < α < n, and a function b ∈ BMO, let B(t) =t log(e + t), Ψ(t) = [t log(e + tα/n)]n/(n−α), Θ(t) = t1−α/n log(e + t−α/n), and q =n/(n− α). Then for each weight w ∈ A1, there exists a constant C such that(1.1) w({x ∈  n : |[b, Iα]f(x)| > t}) 6 CΨ(∫nB(‖b‖BMO|f(x)|t)Θ(w(x)) dx).But, given any bounded domain Ω ⊂  n ,(1.2) w({x ∈ Ω: |[b, Iα]f(x)| > t}) 6 C(∫ΩB(‖b‖BMO|f(x)|t)w(x)1/q dx)q.Remark 1.2. If we replace f by fχΩ, Ω unbounded, (1.1) yields a nominallymore general result for unbounded domains. However, in both (1.1) and (1.2), theA1 weights are still defined on all of  n .Theorem 1.1 is best understood by comparing inequalities (1.1) and (1.2) to theweighted endpoint inequality for the fractional integral operators due to Mucken-houpt and Wheeden [7]. They showed that if w ∈ A1, then(1.3) w({x ∈  n : |Iαf(x)| > t}) 6 C(1t∫n|f(x)|w(x)1/q dx)q.Intuitively, Theorem 1.1 shows that the commutator [b, Iα] is more singular than Iαitself, and that its singularity is worse at infinity.154Theorem 1.1 should also be compared to the analogous result for commutatorsof singular integral operators (formally corresponding to the case α = 0) due toPérez [9]. If T is a singular integral operator, b ∈ BMO and w ∈ A1, thenw({x ∈  n : |[b, T ]f(x)| > t}) 6 C∫nB(‖b‖BMO|f(x)|t)w(x) dx.The proof of Theorem 1.1 follows the same outline as the proof of the unweightedresult on  n [3, Theorem 1.1]. The first step is to prove that the associated Orliczfractional maximal operator,Mα,Bf(x) = supQ3x|Q|α/n‖f‖B,Q,satisfies inequalities analogous to (1.1) and (1.2).Theorem 1.3. With the same notation and hypotheses as in Theorem 1.1, wehave that(1.4) w({x ∈  n : Mα,Bf(x) > t}) 6 CΨ(∫nB( |f(x)|t)Θ(w(x)) dx)and(1.5) w({x ∈ Ω: Mα,Bf(x) > t}) 6 C(∫ΩB( |f(x)|t)w(x)1/q dx)q.We prove Theorem 1.3 in Section 2 below; since we can do so with essentially nomore work, we prove a generalization that holds for a large class of Young functions.The proof of Theorem 1.1 now proceeds as in the unweighted case. Here we sketchthe main steps, and we refer the reader to [3] for details. On the left-hand sideof (1.1) (or (1.2)) we replace |[b, Iα]f(x)| byMd([b, Iα]f)(x), whereMd is the dyadicmaximal operator. We then use the good-λ inequality relating M d and the sharpmaximal operator M# (Lemma 6.1 in [3], which remains true in the weighted casesince w ∈ A1) to replace this with M#([b, Iα]f)(x). Next, we apply the inequalityM#([b, Iα]f)(x) 6 C‖b‖BMO[Iαf(x) + Mα,Bf(x)](Theorem 1.3 in [3]), which then reduces the estimate to endpoint inequalities for Iαand Mα,B . To complete the proof we apply inequality (1.3) and Theorem 1.3, anduse the fact that w1/q 6 Θ(w).1552. Endpoint results for Orlicz fractional maximal operatorsIn this section we state and prove two endpoint inequalities for the Orlicz fractionalmaximal operator. Theorem 1.3 will be an immediate consequence of these results.Hereafter we will assume that the reader is familiar with the basic facts about Orliczspaces, maximal operators and Muckenhoupt Ap weights, and we refer the readerto [4], [5], [6], [10] for further information. Also, we will draw heavily on our work [3],and we urge the reader to consult that paper.Remark 2.1. The conclusions of our main results in this section, Theorems 2.3and 2.5, remain true if α = 0. (See Pérez [8].) However, to avoid technical difficulties,we restrict ourselves to α > 0.To state our results, we need one definition.Definition 2.2. Given an increasing function ϕ, define a function hϕ byhϕ(s) = supt>0ϕ(st)ϕ(t), 0 6 s < ∞.The function hϕ could be infinite if s > 1, but if ϕ is doubling, then it is finite forall 0 < s < ∞. (See Maligranda [6, Theorem 11.7].) If ϕ is submultiplicative, thenhϕ ≈ ϕ. Also, for all s, t > 0, ϕ(st) 6 hϕ(s)ϕ(t).Theorem 2.3. Given α, 0 < α < n, let B be a Young function such thatB(t)/tn/α is decreasing for all t > 0, and let w ∈ A1. Then there exists a constant Cdepending only on B and the A1 constant of w such that for all t > 0, Mα,B satisfiesthe modular weak-type inequality(2.1) Φ(w({x ∈  n : Mα,Bf(x) > t})) 6 C∫nB(f(x)t)hΦ(w(x)) dxfor all non-negative f ∈ LB(  n ), where(2.2) Φ(s) =0 if s = 0,shB(sα/n)if s > 0.Inequality (1.4) follows easily from Theorem 2.3. Since B is submultiplicative,hB ≈ B, soΦ(t) ≈ t1−α/nlog(e + tα/n).156The function Φ is invertible withΦ−1(t) ≈ Ψ(t) = [t log(e + tα/n)]n/(n−α).Thus, inequality (2.1) yields (1.4), provided thathΦ(t) 6 CΘ(t) = Ct1−α/n log(e + t−α/n).However, this follows from the definition: since hB is submultiplicative and hB ≈ B,hΦ(s) = supt>0Φ(st)Φ(t)= s supt>0hB(tα/n)hB((st)α/n)6 shB(s−α/n) 6 CsB(s−α/n) = CΘ(s).Remark 2.4. Note that if we let t = s−1 and use the fact that hB(t) > cB(t), weget hΦ(s) > cΘ(s). Hence, Θ is the best possible function we can get by this means.Theorem 2.5. Given a bounded domain Ω ⊂  n and given α, 0 < α < n, let B bea Young function such that B(t)/tn/α is decreasing for all t > 0. Suppose furtherthat there exists r, 1 6 r 6 n/α, such that hB(t) 6 Ctr for all t 6 6 diam(Ω)α. Thenfor each w ∈ A1 there exists a constant C, depending on B, diam(Ω) and the A1constant of w, such that for all non-negative f ∈ LB(Ω),(2.3) w({x ∈ Ω: Mα,Bf(x) > t})1−rα/n 6 C∫ΩB(f(x)t)w(x)1−rα/n dx.Inequality (1.5) is an immediate consequence of Theorem 2.5. Since B(t) =t log(e + t), hB ≈ B, so if t 6 6 diam(Ω), then hB(t) 6 C log(e + 6 diam(Ω))t.Thus inequality (2.3) holds with r = 1, and this yields (1.5).	of Theorem 2.3. Our proof is very similar to the proof of Theorem 3.3in [3], and we refer the reader there for many lemmas and technical details.Fix a non-negative function f and t > 0. Define Et = {x ∈  n : Mα,Bf(x) > t}.For each x ∈ Et there exists a cube Qx 3 x such that|Qx|α/n‖f‖B,Qx > t.The collection {Qx}x∈Et covers Et. Thus, by Lemma 3.14 in [3], there exists β > 0and a collection of disjoint dyadic cubes {Pj} such that Et ⊂⋃j3Pj and|Pj |α/n‖f‖B,Pj > βt.157By the properties of the Luxemburg norm on Orlicz spaces and by Definition 2.2,1 <1|Pj |∫PjB( |Pj |α/nf(x)βt)dx(2.4)6 ChB(|3Pj |α/n)|3Pj |∫PjB(f(x)t)dx=CΦ(|3Pj |)∫PjB(f(x)t)dx.The growth conditions assumed on B imply (see Lemma 3.12 in [3]) thatΦ(w(Et)) 6 Φ(∑jw(3Pj))6∑jΦ(w(3Pj)).Hence, if we combine the two inequalities above and apply Definition 2.2, we getΦ(w(Et)) 6 C∑jΦ(w(3Pj))Φ(|3Pj |)∫PjB(f(x)t)dx6 C∑jhΦ(w(3Pj)|3Pj |) ∫PjB(f(x)t)dx;since w ∈ A1 and the Pj ’s are disjoint,Φ(w(Et)) 6 C∑j∫PjB(f(x)t)hΦ(w(x)) dx6 C∫nB(f(x)t)hΦ(w(x)) dx.This completes the proof. The proof of Theorem 2.5 requires two lemmas.Lemma 2.6. Given α, 0 < α < n, let B be a Young function such that B(t)/tn/αis decreasing for all t > 0. Then hB(s) 6 sn/α for all s > 1.	. Fix s > 1. Then for all t > 0,B(st)(st)n/α6 B(t)tn/α, or equivalently,B(st)B(t)6 sn/α.Taking the supremum over all t we get the desired inequality. 158Lemma 2.7. Given α, 0 < α < n, let B be a Young function such that B(t)/tn/αis decreasing for all t > 0. If Q and Q̄ are cubes and f is a function such thatsupp(f) ⊂ Q ⊂ Q̄, then|Q̄|α/n‖f‖B,Q̄ 6 |Q|α/n‖f‖B,Q.	. Let s = |Q̄|/|Q| > 1. Then by the definition of the Luxemburg normand by Lemma 2.6,|Q̄|α/n‖f‖B,Q̄ = |Q|α/n‖sα/nf‖B,Q̄= |Q|α/n inf{λ > 0:1|Q̄|∫Q̄B(sα/n|f(x)|λ)dx 6 1}6 |Q|α/n inf{λ > 0:1|Q̄|∫QhB(sα/n)B( |f(x)|λ)dx 6 1}6 |Q|α/n inf{λ > 0:1|Q|∫QB( |f(x)|λ)dx 6 1}= |Q|α/n‖f‖B,Q.	of Theorem 2.5. The proof of this result is nearly the same as the proof ofTheorem 2.3. The major difference is that we must show that we can restrict the sizeof the cubes used to compute Mα,B . Fix x ∈ Ω, and let Q be any cube containing x.If `(Q) > diam(Ω), then we can maximize the quantity |Q|α/n‖f‖B,Q by taking Qto be such that Ω ⊂ Q. Further, by Lemma 2.7 we can increase this quantity bychoosing Q such that `(Q) 6 diam(Ω). Consequently, in computing Mα,B we canrestrict ourselves to cubes whose sidelength is at most the diameter of Ω.Thus, in the proof of Theorem 2.3, when we cover the set Et by cubes Qx, wemay assume that each of these cubes has sidelength bounded by diam(Ω). We thenuse Lemma 3.14 in [3] to replace these cubes by a collection of dyadic cubes Pj . Butfrom the proof of this lemma (see [2], [8]) we see that the sidelength of each Pj isbounded by twice the largest sidelength of the Qx’s; therefore `(Pj) 6 2 diam(Ω) forall j.By assumption, hB(t) 6 Ctr for t 6 6 diam(Ω)α. Therefore, hB(|3Pj |α/n) 6|3Pj |rα/n for all j. Hence, we can modify the argument that yielded (2.4) to get1 <C|3Pj |1−rα/n∫PjB(f(x)t)dx.If we now let Φ(t) = t1−rα/n, then hΦ(t) = Φ(t) and we can repeat the remainderof the argument in the proof of Theorem 2.3 to get (2.3). 159References[1] S. Chanillo: A note on commutators. Indiana Math. J. 31 (1982), 7–16. Zbl 0523.42015[2] D. Cruz-Uribe, SFO: New proofs of two-weight norm inequalities for the maximal op-erator. Georgian Math. J. 7 (2000), 33–42. Zbl 0987.42019[3] D. Cruz-Uribe, SFO, A. Fiorenza: Endpoint estimates and weighted norm inequalitiesfor commutators of fractional integrals. Publ. Mat. 47 (2003), 103–131. Zbl 1035.42015[4] J. Duoandikoetxea: Fourier Analysis. Grad. Studies Math. Vol. 29. Am. Math. Soc.,Providence, 2000. Zbl 0969.42001[5] J. García-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and RelatedTopics. Math. Studies Vol. 116. North Holland, Amsterdam, 1985. Zbl 0578.46046[6] L. Maligranda: Orlicz spaces and interpolation. Seminars in Mathematics 5, IMECC.Universidad Estadual de Campinas, Campinas, 1989. Zbl 0874.46022[7] B. Muckenhoupt, R. Wheeden: Weighted norm inequalities for fractional integrals.Trans. Am. Math. Soc. 192 (1974), 261–274. Zbl 0289.26010[8] C. Pérez: On sufficient conditions for the boundedness of the Hardy-Littlewood maximaloperator between weighted Lp-spaces with different weights. Proc. London Math. Soc.71 (1995), 135–157. Zbl 0829.42019[9] C. Pérez: Endpoint estimates for commutators of singular integral operators. J. Funct.Anal. 128 (1995), 163–185. Zbl 0831.42010[10] M.M. Rao, Z.D. Ren: Theory of Orlicz Spaces. Marcel Dekker, New York, 1991.Zbl 0724.46032Authors’ addresses:            , Dept. of Mathematics, Trinity College,Hartford, CT 06106-3100, USA, e-mail: david.cruzuribe@mail.trincoll.edu;     ! " $# , Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università diNapoli, Via Monteoliveto, 3, I-80134 Napoli, Italy, and Istituto per le Applicazioni del Cal-colo “Mauro Picone”—sezione di Napoli, Consiglio Nazionale delle Ricerche, via PietroCastellino, 111, I-80131 Napoli, Italy, e-mail: fiorenza@unina.it.160

Weighted endpoint estimates for commutators of fractional integrals

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Weighted endpoint estimates for commutators of fractional integrals

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Institute of Mathematics AS CR, v. v. i.