Two-weight, weak-type norm inequalities for fractional integrals, Calderón-Zygmund operators and commutators

Two-weigh , Weak- ype No m Inequali ies o F ac ional In eg als, Calde ´ on-Zygmund Ope a o s and Commu a o s D. CRUZ-URIBE,SFO,C.P´ EREZ ABSTRACT.Wegi eAp- ype condi ions which a e sufficien o he wo-weigh , weak- ype (p, p) inequali ies o ac ional in eg al ope a o s, Calde ´ on-Zygmund ope a o s and commu a- o s. Fo ac ionalin eg alope a o s, hissol esap oblemposed by Sawye and Wheeden [28]. A he hea o all o ou p oo s is an inequali y ela ing he Ha dy-Li lewood maximal unc ion and he sha p maximal unc ion which is s ongly eminiscen o he good-λinequali y o Feffe man and S ein [13]. .1INTRODUCTION Le Mbe he Ha dy–Li lewood maximal ope a o . Gi en a pai o weigh s (u, ) and p,1<p<∞, i is well known ha he weak- ype inequali y u({x∈Rn:M (x) > })≤C pRn| |p dx(1.1) holds i and only i (u, ) ∈Ap: he e exis s a posi i e cons an Ksuch ha o all cubes Q, 1 |Q|Qudx1 |Q|Q −p/p dxp/p ≤K.(1.2) 697 Indiana Uni e si y Ma hema ics Jou nal c , Vol. 49, No. 2 (2000) 698 D. CRUZ-URIBE,SFO&C. P´ EREZ Fo o he classical ope a o s, howe e , he Apcondi ion is no sufficien o he weak (p, p) inequali y. In ac , o he ope a o s we a e in e es ed in, a necessa y andsufficien condi ion o heweak(p, p) inequali yisknownonly o ac ional in eg alope a o s. (SeeSawye [27].) This esul isin e es ingandimpo an , bu i has he d awback ha he condi ion in ol es he ac ional in eg al ope a o . Sufficien , Ap- ype condi ions can also be go en om sufficien condi ions o he s ong (p, p) inequali y. Neugebaue [18] showed ha 1 |Q|Qu dx1/ p 1 |Q|Q − p/p dx1/ p ≤C, > 1,(1.3) issufficien o hes ong(p, p) inequali y o hemaximalope a o , o Calde ´ on- Zygmundope a o sandcommu a o s. Sawye andWheeden[28]showed ha o 0<α<n, |Q|α/n 1 |Q|Qu dx1/ p 1 |Q|Q − p/p dx1/ p ≤C, > 1,(1.4) is sufficien o he s ong- ype (p, p) inequali y o ac ional in eg al ope a o s. (Addi ionalsufficien condi ionsa e oundin[20],[21],and[24]. Wegi ep ecise de ini ions o hese ope a o s in Sec ion 2 below.) In gene al, sufficien condi ions o he weak (p, p) inequali y which a e de- i ed om s ong (p, p) condi ions a e no sha p. The pu pose o his pape is o show ha o he ope a o s we conside , he e a e condi ions ha a e weake han (1.3) and (1.4), which a e sufficien o he weak- ype inequali y. Roughly, i suffices o s eng hen he Apcondi ion (1.2) by in oducing a “powe bump” on he le -hand e m alone, a he han on bo h e ms as in (1.3) and (1.4). Ou i s esul is o ac ional in eg al ope a o s. I sol es a p oblem posed by Sawye and Wheeden [28]. Theo em 1.1. Gi en a pai o weigh s (u, ),p,1<p<∞, and α,0<α< n, suppose ha o some >1and o all cubes Q, |Q|α/n 1 |Q|Qu dx1/ p 1 |Q|Q −p/p dx1/p ≤C<∞.(1.5) Then he ac ional in eg al ope a o Iαsa is ies he weak (p, p) inequali y u({x∈Rn:|Iα (x)|> })≤C pRn| |p dx.(1.6) Ou second esul is o Calde ´ on-Zygmund ope a o s. Two-weigh , Weak- ype No m Inequali ies 699 Theo em 1.2. Le Tbe a Calde ´ on-Zygmund ope a o . Gi en a pai o weigh s (u, ) and p,1<p<∞, suppose ha o some >1and o all cubes Q, 1 |Q|Qu dx1/ p 1 |Q|Q −p/p dx1/p ≤C<∞.(1.7) Then Tsa is ies he weak (p, p) inequali y u({x∈Rn:|T (x)|> })≤C pRn| |p dx.(1.8) Rema k 1.3. Though o cla i y we ha e s a ed Theo em 1.2 o Calde ´ on- Zygmund ope a o s, i is ue o amuchla ge classo ope a o s. Tobep ecise: i he eexis ssomeδ,0<δ<1,andacons an Cδsuch ha o e e y ∈C∞ 0(Rn), M#(|T |δ)(x)1/δ ≤CδM (x),(1.9) hen (1.7) implies (1.8). Al a ez and P´ e ez [3] showed ha inequali y (1.9) holds o Calde ´ on- Zygmund ope a o s. In his case i can be hough o as ex ending he classical es ima e M#(T )(x) ≤C M(| | )(x)1/ ,(1.10) whe e Tis a egula singula in eg al ope a o and >1, (see Ga c´ ıa-Cue a and Rubio de F ancia [14, p. 204]). In some sense, (1.9) con ains mo e in o ma ion han (1.10) since he la e does no suffice o p o e Theo em 1.2. Al a ez and P´ e ez also showed ha inequali y (1.9), and so Theo em 1.2, hold o he ollowing ope a o s: weakly s ongly singula in eg al ope a o s (see C. Feffe man [12]), some pseudo-diffe en ial ope a o s in he H¨ o mande class (see H¨ o mande [15]), and a class o oscilla o y in eg al ope a o s ela ed o hose in oduced by Phong and S ein [25]. They used (1.9) o gene alize Coi man’s heo em [7] ela ing he Lpno m o singula in eg al ope a o s and he maximal unc ion. Rema k 1.4. Fo Calde ´ on-Zygmund ope a o s we ha e been able o p o e s onge esul s;see[11]. Bydiffe en me hodsweshowed ha wemay eplace he “powe bump” in (1.7) by a “bump” in he scale o O licz spaces. Mo e p ecisely, we eplace he L no m by he L(logL)p−1+δno m wi h δ>0. Howe e we a e unable o ex end hese esul s o he b oade class o ope a o s discussed in he p e ious ema k. 700 D. CRUZ-URIBE,SFO&C. P´ EREZ Rema k 1.5. Condi ions(1.5)and(1.7)a esufficien o he ac ionalmax- imal ope a o and he Ha dy-Li lewood maximal ope a o o be bounded om Lp(u−p/p) o Lp( −p/p). (See [21], [22].) We conjec u e ha he bounded- nesso heco espondingmaximalope a o isi sel sufficien o inequali ies(1.6) and (1.8) o hold. In pa icula we belie e ha he O licz space condi ions gi en in [21] and [22] a e sufficien . Ou las esul is abou (linea ) commu a o s. These ope a o s a e de ined by Ck b (x)=(b(x) −b(y))kK(x,y) (y)dy, whe e Kis a ke nel sa is ying he s anda d es ima es and bis a locally in eg able unc ion. (See Sec ion 2 o a p ecise de ini ion.) Sincecommu a o sha eag ea e deg eeo “singula i y” han heco espond- ingCalde ´ on-Zygmundope a o s,weneedasligh lys onge condi ion. Roughly, weneed o “bump” he igh -hand e mas well, bu i suffices odosoin hescale o O licz spaces. Recall ha i Bis an inc easing Young unc ion and i Qis any cube, we de ine he mean Luxembu g no m o a measu able unc ion wi h espec o Bby  B,Q =in λ>0: 1 |Q|QB| | λdx ≤1. (Fo mo e in o ma ion on O licz spaces, see Sec ion 2 below.) Theo em 1.6. Le Tbe a Calde ´ on-Zygmund ope a o and ba unc ionin BMO. Gi en a pai o weigh s (u, ),p,1<p<∞, and k≥0, suppose ha o some >1and o all cubes Q, 1 |Q|Qu dx1/ p  −1/pCk,Q ≤C<∞,(1.11) whe e Ck( ) = plog(e + )kp. Then he commu a o Ck bsa is ies he weak (p, p) inequali y u({x∈Rn:|Ck b (x)|> })≤C pRn| |p dx.(1.12) When k=0, C0 b=T, and so in his case Theo em 1.6 educes o Theo em 1.2. Two-weigh , Weak- ype No m Inequali ies 701 Rema k 1.7. As a co olla y o Theo em 1.6 we ge a new p oo o he one- weigh , s ong (p, p) no m inequali y o commu a o s, which was i s p o ed in a mo e gene al o m by Al a ez, Bagby, Ku z, and P´ e ez [2] and Sego ia and To ea [29]. I w∈Ap, henwand w−p/p bo h sa is y he e e se H¨ olde inequali y and so inequali y (1.11) holds o some >1and o p±ε.The s ong- ype inequali y ollows by in e pola ion. The p oo s o Theo ems 1.1, 1.2 and 1.6 all ollow he same ou line. Each elies on ou so-called p incipal lemma, Theo em 3.4 below, which ela es he Ha dy-Li lewood maximal ope a o and he Feffe man-S ein sha p maximal op- e a o ia an inequali y s ongly eminiscen o a good-λinequali y. To apply Theo em 3.4 we use h ee esul s which ela e he gi en ope a o , he sha p max- imal ope a o and he maximal ope a o . Fo Calde ´ on-Zygmund ope a o s his is inequali y (1.9). Simila inequali ies hold o ac ional in eg al ope a o s and commu a o s: see Lemmas 4.4 and 6.1. The emainde o his pape is o ganized as ollows: in Sec ion 2 we gi e a numbe o de ini ions and lemmas needed in la e sec ions. The hea o he pape is Sec ion 3, whe e we p o e Theo em 3.4. Finally, in Sec ions 4, 5, and 6 we p o e Theo ems 1.1, 1.2, and 1.6. Th oughou his pape all no a ion is s anda d o will be de ined as needed. All cubes a e assumed o ha e hei sides pa allel o he coo dina e axes. Gi en acubeQ,#(Q) will deno e he leng h o i s sides and o any >0, Q will deno e he cube wi h he same cen e as Qand such ha #( Q) = #(Q).We will deno e he collec ion o all dyadic cubes by ∆and by ∆(Q) he collec ion o all dyadic subcubes ela i e o he (no necessa ily dyadic) cube Q.Byweigh swe will always mean non-nega i e, locally in eg able unc ions which a e posi i e on a se o posi i e measu e. Gi en a Lebesgue measu able se Eand a weigh w,|E| will deno e he Lebesgue measu e o Eand w(E) =Ewdx.Gi en1<p<∞, p=p/(p −1)will deno e he conjuga e exponen o p. Finally, Cwill deno e a posi i e cons an whose alue may change a each appea ance. .2PRELIMINARY IDEAS In his sec ion we gi e a numbe o de ini ions and lemmas needed in la e sec ions. Themainope a o s. Fi s wede ine heope a o sinTheo ems1.1,1.2,and 1.6. 702 D. CRUZ-URIBE,SFO&C. P´ EREZ F ac ional in eg al ope a o s. Gi en α,0<α<n, de ine he ac ional in eg al ope a o o o de αby Iα (x)=Rn (y) |x−y|n−αdy. Fo mo e in o ma ion, see S ein [31, pp. 117-120]. Calde on-Zygmund ope a o s. Gi en a ke nel Kon Rn×Rn—i.e. alocally in eg able, complex- alued unc ion de ined off hediagonal—wesay ha i sa - is ies he s anda d es ima es i he e exis δ,0<δ≤1, and C ini e such ha o all dis inc poin s xand yin Rn,andallzsuch ha |x−z|<1 2|x−y|: (1) |K(x,y)|≤C|x−y|−n; (2) |K(x,y) −K(z,y)|≤C|x−z|δ|x−y|n+δ; (3) |K(y,x) −K(y,z)|≤C|x−z|δ|x−y|n+δ. A bounded linea ope a o T:C∞ 0(Rn)→D (Rn)(he e Dis he space o dis ibu ions) is said o be associa ed wi h a ke nel Ki T ,g=RnRnK(x,y)g(x) (y)dx dy o all and gin C∞ 0(Rn)wi h supp( ) ∩supp(g) =∅.Tis said o be a Calde ´ on-Zygmundope a o i i sassocia edke nelsa is ies hes anda des ima es and i ex ends o a bounded linea ope a o on L2. Fo mo e in o ma ion, see Coi man and Meye [8] and Ch is [6]. Impo an examples o such ope a o s a e he Calde ´ on-Zygmund singula in eg al ope a o s: T (x)=p. .Rnk(x −y) (y)dy, whe ek∈L1 loc(Rn {0})andK(x,y) =k(x−y)sa is ies hes anda des ima es. Fo mo e in o ma ion see Ga c´ ıa-Cue a and Rubio de F ancia [14, p. 192]. Commu a o s. Gi en a Calde ´ on-Zygmund ope a o Tand a unc ion bin BMO, le Mbdeno e mul iplica ion by b. We de ine he linea ope a o s Ck bby C0 b=T,C1 b=[Mb,T] =MbT−MbT,and o k>1, Ck b=[Mb,Ck−1 b].I ∈C∞ 0(Rn), hen Ck b (x)=(b(x) −b(y))kK(x,y) (y)dy, x ∈ supp( ). Two-weigh , Weak- ype No m Inequali ies 703 Commu a o swe ein oducedbyCoi man,Rochbe gandWeiss[9],whoshowed hey a e bounded on Lp,1<p<∞. Maximal ope a o s. Key o he p oo s o ou esul s a e a numbe o maxi- mal ope a o s. Fo comple eness we gi e hei de ini ions he e. The maximal ope a o . Gi en a locally in eg able unc ion and α,0≤ α<n ,de ine Mα (x)=sup Qx 1 |Q|1−α/n Q| |dy. I α=0 his is he Ha dy-Li lewood maximal ope a o and we w i e M o M0 ;i 0<α<n his is he ac ional maximal ope a o o o de α.Weuse he Ha dy-Li lewoodmaximalope a o ocon olCalde ´ on-Zygmundope a o sand commu a o s, and he ac ional maximal ope a o o con ol ac ional in eg al ope a o s. (See inequali y (1.9) and Lemmas 4.4 and 6.1.) We de ine he dyadic maximal and ac ional maximal ope a o s Mdand Md α simila lyexcep hesup emumsa e es ic ed odyadiccubescon ainingx.Gi en δ>0wede ine heδ-maximal ope a o by Mδ (x) =M(| |δ)(x)1/δ.We de ine Md δsimila ly. F om he con ex he e should be no con usion be ween he ac ional maximal ope a o and he δ-maximal ope a o . The sha p maximal ope a o . Gi en a locally in eg able unc ion and a cube Q,le Qdeno e he a e age o o e Q: Q=1 |Q|Q dx. De ine he sha p maximal unc ion o by M# (x)=sup Qx 1 |Q|Q| (y)− Q|dy. The sha p maximal unc ion was in oduced by Feffe man and S ein [13]. Again, de ine he dyadic sha p maximal unc ion M#,d by es ic ing he sup emum o dyadic cubes. Gi en δ>0, de ine he sha p δ-maximal unc ion by M# δ (x)=M#(| |δ)(x)1/δ, and de ine M#,d δsimila ly. 704 D. CRUZ-URIBE,SFO&C. P´ EREZ O licz spaces. In Sec ion 6 we will need he ollowing ac s abou O licz spaces. (Fo u he in o ma ion see Benne and Sha pley [4] o Rao and Ren [26].) A unc ion B:[0,∞)→[0,∞)is a Young unc ion i i is con ex and inc easing, and i B(0)=0andB( ) →∞as →∞. Gi en a Young unc ion B, de ine he mean Luxembu g no m o on a cube Qby  B,Q =in λ>0: 1 |Q|QB| | λdy ≤1. When B( ) = p,1≤p<∞,  B,Q =1 |Q|Q| |pdx1/p ; ha is, he Luxembu g no m coincides wi h he (no malized) Lpno m. The e is ano he cha ac e iza ion o he Luxembu g no m, due o K asnosel’ski˘ ıand Ru icki˘ ı [17, p. 92] (also see Rao and Ren [26, p. 69]) which we will need:  B,Q ≤in s>0s+s |Q|QB| | sdx≤2 B,Q.(2.1) Gi en h ee Young unc ions A,B,andCsuch ha o all >0, A−1( )C−1( ) ≤B−1( ),(2.2) henweha e he ollowinggene alizedH¨ olde ’sinequali ydue oO’Neil[19]: o any cube Qand all unc ions and g,  gB,Q ≤2 A,Q gC,Q.(2.3) De ine he maximal ope a o MBby MB (x)=sup Qx  B,Q. The dyadic maximal ope a o Md Bis de ined in simila ly, excep he sup emum is es ic ed o dyadic cubes con aining x. I ollows om an inequali y due o S ein [30] ha o k≥1, i Bk( ) = log(e + )k−1, henMk ≈MBk ,whe e Mk=M·M···Mis he k- h i e a e o he maximal unc ion. (See Ca ozza and Passa elli di Napoli [5] and he e e ences gi en he e.) Two-weigh , Weak- ype No m Inequali ies 705 The Calde ´ on-Zygmund decomposi ion. Ou p oo s depend hea ily on he Calde ´ on-Zygmunddecomposi ionandagene aliza iono i oO liczspaceno ms. To be p ecise and o es ablish no a ion, we s a e he esul he e. Fo a p oo see [22]; his is an adap a ion o he classical p oo gi en in Ga c´ ıa-Cue aandRubio de F ancia [14, p. 137]. Lemma 2.1. Gi en a Young unc ion B, suppose is a non-nega i e unc ion such ha  B,Q ends oze oas#(Q) ends o in ini y. Then o each >0 he e exis s a disjoin collec ion o dyadic cubes {C i}such ha o each i, < B,C i≤ 2n , {x∈Rn:Md B (x)> }=  i C i, {x∈Rn:MB (x)> 4n }⊂ i3C i. Mo eo e , hecubesa emaximal: i Qisadyadiccubesuch ha Q⊂{Md B (x)> }, hen Q⊂C i o some i. To ecap u e he classical lemma, le B( ) = and no e ha i ∈Lq o some q,1≤q<∞, hen  B,Q =1 |Q|Q dx→0as|Q|→∞. Mo e gene ally, o apply Lemma 2.1 i suffices o assume ha is bounded and has compac suppo . .3THE PRINCIPAL LEMMA In hissec ion we p o e ou p incipal lemma: an inequali y linking he sha p maximal unc ion and he Ha dy-Li lewood maximal unc ion. In spi i , hough no inde ail i esembles he good-λinequali y o Feffe manand S ein[13]. (Also see Ga c´ ıa-Cue a and Rubio de F ancia [14, pp. 161-3] and Jou n´ e [16, p. 41].) To s a e he p incipal lemma we i s need a de ini ion and a lemma. De ini ion 3.1. Gi en >1andaweigh u, de ine he se unc ion A uon measu able se s E⊂Rnby A u(E) =|E|1/ Eu dx1/ =|E|1 |E|Eu dx1/ . (The second equali y holds p o ided |E|>0.) 712 D. CRUZ-URIBE,SFO&C. P´ EREZ and (by he Lebesgue diffe en ia ion heo em) sup >0 pu({x∈Rn:|Id α (x)|> })≤sup >0 pu({x∈Rn:Md(Iα )(x) > }) ≤Csup >0 p j A u(Q j). Fix ; hen by Lemma 4.4, o each j, Q j⊂{x∈Rn:Md α (x)>εD−1 α }. By an a gumen analogous o ha o he dyadic maximal ope a o (c . Lemma 2.1), we can w i e he igh -hand side as he union o disjoin dyadic cubes {P k} such ha o each k, |P k|α/n−1P k dx>εD −1 α . Fu he , he P k’s a e maximal wi h his p ope y; in pa icula , o each j he e exis s ksuch ha Q j⊂P k. The e o e, byLemma 3.2, Condi ion (3), p j A u(Q j) = p k Q j⊂P k A u(Q j)≤ p k A u(P k) ≤(ε−1Dα)p k |P k|1 |P k|P k u dx1/ |P k|α/n−1P k dx p . By H¨ olde ’s inequali y and inequali y (1.5), ≤C k |P k|αp/n 1 |P k|P k u dx1/ 1 |P k|P k −p/p dxp/pP k p dx ≤C kP k p dx≤CRn p dx. The cons an is independen o ,soi we ake hesup emumo e all >0 we ge inequali y (1.6). ❐ Rema k 4.5. A he cos o a mo e complex a gumen simila o ha o Calde ´ on-Zygmund ope a o s (c . Lemma 5.1 below) we could dispense wi h he dyadic ac ional in eg al ope a o and p o e Theo em 1.1 di ec ly o Iα. The key inequali y is he non-dyadic analogue o Lemma 4.4 due o Adams [1]: M#(Iα )(x) ≤CMα (x). Two-weigh , Weak- ype No m Inequali ies 713 .5CALDER´ ON-ZYGMUND OPERATORS In hissec ionwep o eTheo em1.2. Thep oo issimila o ha o Theo em 1.1, bu is complica ed by he ac ha we canno pass o an equi alen dyadic ope a o . To compensa e we need he ollowing lemma which is also needed in he p oo o Theo em 1.6. Lemma 5.1. Le BbeaYoung unc ion. Suppose ha o some unc ion ∈Lq, 1≤q<∞, and o some >0 he e exis a cons an µ,0<µ≤1, and a collec ion o dyadic cubes {Qj}such ha o each j, |Qj∩{x∈Rn:MB (x)> }| ≥ µ|Qj|. Then he e exis s a cons an ν>0, depending on nand µ, and a subcollec ion {Pk} o he Calde ´ on-Zygmund decomposi ion wi h espec o Bo a heigh ν ,{Cν i}, such ha o each j,Qj⊂3Pk o some k. I we eplace MBby Md Bin he hypo hesis, hen we can s eng hen he conclusion by inding Pk’s such ha Qj⊂Pkand by le ing µ=ν. P oo . We i s conside he non-dyadic case. By Lemma 2.1, E ={x∈Rn:MB (x)> }⊂ i3Cγ i, whe e γ=4−n.I wehadQj⊂3Cγ i o some iwe would be done, bu his need no be he case, e en i µ=1. Howe e , o each j he e is a collec ion o indices Ajsuch ha Qj∩E ⊂ i∈Aj 3Cγ iand 3Cγ i∩Qj≠∅,i∈Aj. The e a e wo possibili ies: i s , he e exis s i∈Ajsuch ha #(Qj)≤#(3Cγ i). Then Qj⊂9Cγ iand by inequali y (2.1), 2 B,9Cγ i≥in s>0  s+s |9Cγ i|9Cγ i B| | sdx   ≥9−nin s>0  s+s |Cγ i|Cγ i B| | sdx   =9−n B,Cγ i>9−nγ . 714 D. CRUZ-URIBE,SFO&C. P´ EREZ Al e na i ely, #(Qj)>#(3Cγ i) o all i∈Aj.Bu hen o eachi∈Aj,3Cγ i⊂ 3Qj,andso 2|3Qj| B,3Qj≥in s>0s|3Qj|+s3Qj B| | sdx ≥ i∈Aj in s>0s|Cγ i|+sCγ i B| | sdx = i∈Aj |Cγ i| B,Cγ i>3−nγ  i∈Aj |3Cγ i| ≥3−nγ |Qj∩E |≥9−nµγ |3Qj|. So in ei he case, o each j he e exis s a cube ¯ Qjcon aining Qjsuch ha  B, ¯ Qj>µγ 2·9n. Now by he same a gumen ha is used o p o e he Calde ´ on-Zygmund decomposi ion, Lemma 2.1, we can show ha he e exis s a subcollec ion {Pk}o {Cν i},ν=1 2µγ36−n=1 2µ144−n,such ha o eachj,Qj⊂¯ Qj⊂3Pk o some k. This comple es he p oo o MB. The p oo in he dyadic case is e y simila , bu is simpli ied conside ably by he ac ha i wo dyadic cubes in e sec hen one is con ained in he o he . ❐ P oo . [P oo o Theo em 1.2] By a s anda d a gumen , we may assume ha ∈C∞(Rn)and has compac suppo . Fix p,1<p<∞; henT ∈Lq,whe e q>1issuch ha p≥q/ . Hence, we may apply Theo em 3.4 o i . Fix δ<1. Then he e exis s ε>0such ha o each >0 he e exis s a sequence o disjoin dyadic cubes {Q j}such ha  1 |Q j|Q j |T |δ−(|T |δ)Q j dx  1/δ >ε 1/δ and sup >0 pu({x∈Rn:|T (x)|> })≤sup >0 pu({x∈Rn:Md δ(T )(x) > }) ≤Csup >0 p j A u(Q j). Two-weigh , Weak- ype No m Inequali ies 715 As we no ed in he In oduc ion, Tsa is ies inequali y (1.9). The e o e, o each j, Q j⊂{x∈Rn:M# δ(T )(x) > ε1/δ }⊂{x∈Rn:M (x) > β }, whe e β=C−1 δε1/δ. By Lemma 5.1 (wi h µ=1), o each >0 he eexis s asequence o disjoin dyadic cubes {P k}such ha o each j,Q j⊂3P k o some k,andsuch ha 1 |P k|P k | |dx > ρ , whe e ρ>0 depends only on βand n. Then by Lemma 3.2, Condi ion (3), o each >0, p j A u(Q j) = p k Q j⊂3P k A u(Q j) ≤ p k A u(3P k) ≤ρ−p k |3P k|1 |3P k|3P k u dx1/ 1 |P k|P k | |dxp . By H¨ olde ’s inequali y and inequali y (1.7), ≤C k1 |3P k|3P k u dx1/ 1 |3P k|3P k −p/p dxp/pP k | |p dx ≤C kP k | |p dx ≤CRn| |p dx. The cons an is independen o ,soi we ake hesup emumo e all >0 we ge inequali y (1.8). This comple es ou p oo . ❐ 716 D. CRUZ-URIBE,SFO&C. P´ EREZ .6COMMUTATORS In his sec ion we p o e Theo em 1.6. The p oo depends on Theo em 1.2 and he ollowing analogue o inequali y(1.9) o commu a o s. Lemma 6.1. Gi enaCalde ´ on-Zygmund ope a o T, a unc ion bin BMO, cons an s δ0and δ1,0<δ 0<δ 1<1, and k≥1, he e exis s a cons an K, depending on he BMO no m o b, such ha o e e y unc ion ∈C∞ 0(Rn)and any x∈Rn, M#,d δ0(Ck b )(x) ≤K k−1  i=0 Md δ1(Ci b )(x) +KMk+1 (x). This esul is ound in [23, 24]. As gi en he e, he non-dyadic maximal ope a o appea s in he i s e m on he igh -hand side, bu i is immedia e om he p oo ha i is s ill ue wi h he dyadic maximal ope a o he e. P oo . [P oo o Theo em 1.6] When k=0, Theo em 1.6 educes o The- o em 1.2, so we may ix k≥1. By a s anda d a gumen we may assume ha ∈C∞(Rn)andhas compac suppo . Fixp,1<p<∞; henCi b ∈Lq,whe e 0≤i≤kand q>1issuch ha p≥q/ . Hence, we may apply Theo em 3.4 o Ck b .Fixδ0and δ1,0<δ 0<δ 1<1. Then he e exis s ε>0such ha o each >0 he e exis s a sequence o disjoin dyadic cubes {Q j}such ha  1 |Q j|Q j |Ck b |δ0−(|Ck b |δ0)Q j dx  1/δ0 >ε 1/δ0 and sup >0 pu({x∈Rn:|Ck b (x)|> })≤sup >0 pu({x∈Rn:Md δ0(Ck b )(x) > }) ≤Csup >0 p j A u(Q j). By Lemma 6.1, o each jand , Two-weigh , Weak- ype No m Inequali ies 717 Q j⊂ k−1  i=1 {x∈Rn:Md δ1(Ci b )(x) > β } ∪{x∈Rn:Md δ1(T )(x) > β } ∪{x∈Rn:Mk+1 (x)>β } ≡k−1  i=1 Fβ i∪Fβ 0∪Fβ k, whe eβ=ε1/δ0K−1(k +1)−1.Fo eachjand we canno ha e ha |Q j∩Fβ i|< (k +1)−1|Q j| o all i.Hence, o somei,|Q j∩Fβ i|≥(k +1)−1|Q j|;i hisis he case, we w i e Q j∈F β i.Thus, sup >0 p j A u(Q j)≤ k  i=0sup >0 p Q j∈Fβ i A u(Q j). To comple e he p oo we will show ha each e m o he ou e sum on he igh -hand side is domina ed by CRn| |p dx. The e a e h ee cases. 9Case 1: Cubes in Fβ k.As we no ed in Sec ion 2, he e exis s a cons an β>0such ha {x∈Rn:Mk+1 (x)>β }⊂{x∈Rn:MB (x)>β }, whe e B( ) = log(e + )k. The e o e, by Lemma 5.1 (wi h µ=(k +1)−1), he e exis s a cons an ν>0such ha , o each >0 he e exis s a collec ion o disjoin dyadic cubes {P #}such ha o each j,Q j⊂3P # o some #and such ha  B,P #>ν . We now p oceed exac ly as we did a he end o he p oo o Theo em 1.2. Since Ck( ) = plog(e + )kp,C−1 k( ) ≈ 1/plog(e + )−k, 718 D. CRUZ-URIBE,SFO&C. P´ EREZ and so 1/pC−1 k( ) ≤B−1( ). Then, by Lemma 3.2, Condi ions (2) and (3), he gene alized H¨ olde ’s inequali y (2.3) and inequali y (1.11), sup >0 p Q j∈Fβ k A u(Q j) ≤sup >0 p # A u(3P #) ≤Csup >0 # |3P #|1 |3P #|3P # u dx1/    p B,P # ≤Csup >0 #1 |3P #|3P # u dx1/   −1/p p Ck,P #P # | |p dx ≤Csup >0 #1 |3P #|3P # u dx1/   −1/p p Ck,3P #P # | |p dx ≤Csup >0 #P # | |p dx ≤CRn| |p dx. 9Case 2: Cubes in Fβ 0.Gi en >0, le s=(β )δ1. Again by Lemma 5.1 ( he dyadic case), i Q j∈F β 0, hen o somei,Q j⊂Cs i,whe e{Cs i}is he Calde ´ on-Zygmund decomposi ion o |T |δ1a heigh s. Hence, by Lemma 3.2, Condi ions (2) and (3), sup >0 p Q j∈Fβ 0 A u(Q j)≤sup >0 p i A u(Cs i). By Co olla y 3.5, he e exis ε>0 and a subcollec ion {¯ Q j}o {Cs i}such ha i x∈¯ Q j, henM#,d δ1(T )(x) > β ,whe eβ =ε1/δ1β,andsuch ha sup >0 p i A u(Cs i)≤Csup >0 p j A u(¯ Q j). We can now a gue exac ly aswe did in he p oo o Theo em 1.2 o ge sup >0 p Q j∈Fβ 0 A u(Q j)≤CRn| |p dx. Two-weigh , Weak- ype No m Inequali ies 719 9Case 3: Cubes in Fβ i,1≤i≤k−1.Fix i; hena guingexac ly as wedid in Case 2, by Co olla y 3.5 he e exis ε>0 and a collec ion o disjoin dyadic cubes {¯ Q j}such ha i x∈¯ Q j, henM# δ1(Ci b )(x) > β ,whe eβ =ε1/δ1β, and such ha sup >0 p Q j∈Fβ i A u(Q j)≤Csup >0 p j A u(¯ Q j). We now apply Lemma 6.1 and epea he a gumen a he beginning o his p oo . When we do so we educe he deg ee o he highes o de commu a o appea ing omi oi−1. The e o e,a e epea ingou a gumen a ini enumbe o imes, we will educe o collec ions o cubes sa is ying condi ions such as hose in Case 1 and Case 2. Repea ing hose a gumen s will hen gi e us he desi ed inequali y. ❐ Acknowledgemen . We would like o hank E. Sawye o sha ing wi h he second au ho an unpublished manusc ip which sugges ed ou app oach. We wouldalsolike o hank he e e ee o poin ingou ane o inTheo em1.6. The i s au ho was suppo ed by a Fo d Founda ion ellowship; he second au ho by DGICYT G an PB40192, Spain. The second au ho is also g a e ul o he in i a ion and hospi ali y o he Cen e de Rece ca Ma em` a ica, Ba celona, whe e his wo k was comple ed. .REFERENCES [1] D.R. ADAMS,A no e on Riesz po en ials,DukeMa h.J.42 (1975), 765-778. [2] J. ALVAREZ,R.BAGBY,D.KURTZ &C. P´ EREZ,Weigh ed es ima es o commu a o s o linea ope a o s,S udiaMa h.104 (1993), 195-209. [3] J. ALVAREZ &C. P´ EREZ,Es ima es wi h A∞weigh s o a ious singula in eg al ope a o s,Bol- le ino U.M.I. (7) 8-A (1994), 123-133. [4] C. BENNETT &R. SHARPLEY,In e pola ion o Ope a o s, Academic P ess, New Yo k, 1988. [5] M. CAROZZA &A. PASSARELLI DI NAPOLI,Composi ion o maximal ope a o s,Publ.Ma . 40 (1996), 397-409. [6] M.CHRIST,Lec u esonSingula In eg alOpe a o s,CBMSRegionalCon e enceSe iesinMa h- ema ics, Volume 77, Ame . Ma h. Soc., P o idence, 1990. [7] R. COIFMAN,Dis ibu ion unc ion inequali ies o singula in eg als, P oc. Acad. Sci. U.S.A. 69 (1972), 2838–2839. [8] R. COIFMAN &Y. MEYER,Au del` adesop ´ e a eu s pseudo-diff´ e en iels,As ´ e isque 57 (1979), 1-84. [9] R. COIFMAN,R.ROCHBERG &G. WEISS,Fac o iza ion heo ems o Ha dy spaces in se e al a iables, Ann. o Ma h. 103 (1976), 611–635. 720 D. CRUZ-URIBE,SFO&C. P´ EREZ [10] D. CRUZ-URIBE,SFO&A. FIORENZA,The A∞p ope y o Young unc ions and weigh ed no m inequali ies,p ep in . [11] D.CRUZ-URIBE,SFO&C.P´ EREZ,Sha p wo-weigh ,weak- ypeno minequali ies o singula in eg al ope a o s,Ma h.Res.Le .6(1999), 417-428. [12] C. FEFFERMAN,Inequali ies o s ongly singula con olu ion ope a o s,Ac aMa h.124 (1970), 9-36. [13] C. FEFFERMAN &E.M. STEIN,Hpspaces in se e al a iables,Ac aMa h.129 (1972), 137- 193. [14] J. GARC´ IA-CUERVA &J.L. RUBIO DE FRANCIA,Weigh ed No m Inequali ies and Rela ed Topics, No h-Holland Ma h. S udies Volume 116, No h Holland, Ams e dam, 1985. [15] L.H¨ ORMANDER,Pseudo-diffe en ialope a o sandhypoellip icequa ions,P oc.Symp.Pu eMa h 10 (1967), 138-183. [16] J.-L. JOURN´ E,Zygmund Ope a o s, Pseudo-Diffe en ial Ope a o s and he Cauchy In eg al o Calde ´ on, Lec u e No es in Ma hema ics, Volume 994, Sp inge Ve lag, New Yo k, 1983. [17] M.A.KRASNOSEL’SKI˘ I&YA.B.RUTICKI˘ I,Con exFunc ionsandO liczSpaces,P.Noo dhoff, G ¨ oningen, 1961. [18] C.J. NEUGEBAUER,Inse ing Ap-weigh s, P oc. Ame . Ma h. Soc. 87 (1983), 644-648. [19] R. O’NEIL,F ac ional in eg a ion in O licz spaces,T ans.Ame .Ma h.Soc.115 (1963), 300- 328. [20] C. P´ EREZ,Weigh ed no m inequali ies o singula in eg al ope a o s,J.LondonMa h.Soc.49 (1994), 296-308. [21] ,Two weigh ed inequali ies o po en ial and ac ional ype maximal ope a o s, Indiana Ma h. J. 43 (1994), 663-683. [22] ,On sufficien condi ions o he boundedness o he Ha dy-Li lewood maximal ope a o be ween weigh ed Lp-spaces wi h diffe en weigh s, P oc. London Ma h. Soc. 71 (1995), 135-57. [23] ,Endpoin es ima es o commu a o s o singula in eg al ope a o s, J. Func. Anal. 128 (1995), 163-185. [24] ,Sha p es ima es o commu a o s o singula in eg als ia i e a ions o he Ha dy-Li lewood maximal unc ion,J.Fou ie Anal.Appl.3(1997), 743-756. [25] D.H. PHONG &E.M. STEIN,Hilbe in eg als, singula in eg als and Radon ans o ms,Ac a Ma h. 157 (1985), 99-157. [26] M.M. RAO &Z.D. REN,Theo y o O licz Spaces, Ma cel Dekke , New Yo k, 1991. [27] E.T. SAWYER,A wo weigh weak ype inequali y o ac ionalin eg als,T ans.Ame .Ma h.Soc. 281 (1984), 339-345. [28] E.T. SAWYER &R. WHEEDEN,Weigh ed inequali ies o ac ional in eg als on Euclidean and homogeneous spaces,Ame .J.Ma h.114 (1992), 813-874. [29] C.SEGOVIA&J.L.TORREA,Weigh edinequali ies o commu a o so ac ionalandsingula in eg als, Publ. Ma . 35 (1991), 209-235. [30] E.M. STEIN,No e on he class LlogL,S udiaMa h.32 (1969), 305-310. [31] ,Singula In eg alsandDiffe en iabili yP ope ieso Func ions,P ince onUni e si yP ess, P ince on, 1970. D. CRUZ-URIBE,SFO Depa men o Ma hema ics, T ini y College Ha o d, CT 06106-3100, U. S. A. EMAIL: [email p o ec ed] Two-weigh , Weak- ype No m Inequali ies 721 C. P´ EREZ Depa imen o de Ma em´ a icas Uni e sidad Au ´ onoma de Mad id 28049 Mad id, Spain EMAIL:ca los.pe [email protected] SUBJECT CLASSIFICATION: 42B20, 42B25 KEYWORDS: weigh s, weak- ype inequali ies, ac ional in eg al ope a o s, Calde ´ on-Zygmund op- e a o s, commu a o s Submi ed: Ap il 23 d, 1999, e ised: Ma ch 3 d, 2000.