Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function

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 Ca los P´e ez J. Fou ie Analysis and Applica ions, 3(1997), 108–146. Depa men o de Ma em´a icas Uni e sidad Au ´onoma de Mad id 28049 Mad id, Spain e–mail: cp[email p o ec ed] wo k pa ially suppo ed by DGICYT g an PB940192, Spain 1 1 In oduc ion and desc ip ion o he main e- sul s The pu pose o his pape is o ob ain some sha p non s anda d weigh ed inequali ies o linea and nonlinea commu a o s o singula in eg al ope a o s. These es ima es p o ide a u he insigh in o he s uc u e o hese ope a o s and in pa icula hey e lec a highe deg ee o singula i y as compa ed wi h he s anda d Calde ´on– Zygmund singula in eg al ope a o s. Le Tdeno e a Calde ´on–Zygmund singula in eg al ope a o and le Mbe he Ha dy–Li lewood maximal unc ion. Acco ding o a esul o R. Coi man [C], T and Msa is y he ollowing a p io i es ima e: Le 0 < p < ∞and suppose ha w∈A∞(Rn). Then he inequali y ZRn|T (x)|pw(x)dx ≤C[w]p A∞ZRnM (x)pw(x)dx, (1) holds o e e y unc ion o which he le hand side is ini e. This es ima e plays a majo ole in he mode n heo y o weigh ed no m inequal- i ies since as i is well known i ollows ha Tis a bounded ope a o on Lp(w) when- e e w∈Apand p > 1. This ex ends he p e ious esul o R. A. Hun , B. Muck- enhoup and R. L. Wheeden in [HMW] whose me hod wo ks only o he Hilbe T ans o m. Fu he mo e, (1) makes explici he well known Calde ´on–Zygmund p inciple which es ablishes ha a singula in eg al ope a o is con olled by an app opia e maximal unc ion. The e is ano he aspec o Coi man’s es ima e ha we shall be exploi ing along his pape . I conce ns he wo weigh ed inequali y p oblem o singula in eg als, say he Hilbe ans o m, which is comple ely open. Combining (1) wi h ce ain sha p wo weigh ed inequali ies o Mwe can de i e wo weigh ed es ima es o T wi h no a p io i assump ion on he weigh w. As a sample we quo e he ollowing inequali y om [Wil] [P3]: Le Tbe a Calde ´on–Zygmund singula in eg al ope a o and le 1 < p < ∞. Then, he e exis s a cons an Csuch ha ZRn|T (x)|pw(x)dx ≤CZRn| (x)|pM[p]+1w(x)dx, (2) whe e Cis independen o wand . 2 1.1 Highe o de commu a o s In his pape we a e going o in es iga e gene aliza ions o abo e inequali ies (1) and (2) o a la ge amily o singula in eg al ope a o s. Fi s we shall conside he highe o de commu a o s in oduced by R. Coi man, R. Rochbe g and G. Weiss in [CRW]. These a e linea ope a o s de ined o app opia e unc ions band and o k= 0,1,2,· · · by Tk b (x) = Z(b(x)−b(y))kK(x, y) (y)dy which mus be unde s ood in he usual sense. When k= 1 he ope a o T1 bis usually deno ed by [Mb, T] = Mb◦T−T◦Mbwhe e Mbis he ope a o de ined by Mb =b , and bis usually called he “symbol” o he ope a o . These commu a o s ha e p o ed o be o in e es in many con ex s and in pa icula in he heo y o P.D.E. We shall only men ion he ecen esul s in he heo y o non di e gence ellip ic equa ions wi h discon inuous coe icien s [CFL1] [CFL2] [DiR]. The main esul om [CRW] is he ollowing: Le 1 < p < ∞and le b∈BMO, hen he e exis s a cons an Csuch ha   Tk b   Lp(Rn)≤Ckbkk BMO k kLp(Rn).(3) Th oughou he pape Mk=M◦(k) . . . ◦Mwill deno e he Ha dy–Li lewood maximal ope a o Mi e a ed k imes. Following he Calde ´on–Zygmund p inciple we shall show ha he maximal op- e a o which con ols he highe o de commu a o s Tk b when bis a BMO unc ion is Mk+1, namely in some sense we ha e ha Tk b≈M◦(k+1) . . . ◦M when b∈BMO. This can be made p ecise wi h he ollowing gene aliza ion o (1). 3 THEOREM 1.1 Le 0< p < ∞and le w∈A∞and b∈BMO. Then, he e exis s a cons an Csuch ha ZRn|Tk b (x)|pw(x)dx ≤Ckbkkp BMO [w](k+1)p A∞ZRnMk+1 (x)pw(x)dx. (4) This inequali y con ains he well known ac ha he highe o de commu a o s a e bounded on Lp(w), w∈Ap, by applying k+ 1 imes Muckenhoup ’s Theo em. As we said be o e (4) can be used as well o ge a gene aliza ion o inequali y (2). THEOREM 1.2 Le 1< p < ∞and le b∈BMO. Then, he e exis s a cons an Csuch ha o each weigh w ZRn|Tk b (x)|pw(x)dx ≤Ckbkkp BMO ZRn| (x)|pM[(k+1)p]+1w(x)dx. (5) We ema k ha he numbe o i e a ions o he maximal unc ion needed in bo h Theo ems a e op imal (see §5). In ac i ollows om he p oo o (5) ha he e is a sha pe es ima e: ZRn|Tk b (x)|pw(x)dx ≤Ckbkkp BMO ZRn| (x)|pML(log L)(k+1)p−1+(w)(x)dx whe e  > 0, being he esul alse o = 0. See §2 o he de ini ion o ML(logL)α. Obse e ha bo h es ima es (4) and (5) show ha he ope a o Tk bbecomes mo e singula wi h ksince he maximal unc ion on he igh hand side o he inequali ies needs mo e “i e a ions” o balance he inequali ies. Also obse e ha we canno ge he sha p case (5) i e a ing om he case k= 1. Be o e con inuing, le us poin ou ha M. Wilson [Wil] was he i s au ho who de i ed an es ima e such as (5) o singula in eg als o con olu ion ype T0 b=Tbu only on he ange 1 < p ≤2. Howe e , Wilson’s app oach is in e es ing because is di ec and based on sha p weigh ed es ima es o smoo h Li lewood–Paley squa e unc ions using as a key s ep a deep esul by T. Wol [CWW] conce ning he beha io o he squa e unc ions on L∞. Ou me hod is by duali y ha ing he ad an ages ha i s co e s he ull ange 1< p < ∞and second i is lexible enough o be applied o a wide class o ope a o s such as Tk b a he han T. Le us gi e an ou line o he p oo o Theo em 1.2 which is based on he ollowing s eps and which seems o be gene al enough o be applicable o o he (linea ) ope a o s: 4 1. Fo simplici y deno e Tk bby Tand [(k+ 1)p] + 1 by k(p). Now, ins ead o p o ing di ec ly (5) we conside he co esponding (equi alen ) dual inequali y, namely ZRn|T (x)|p0(Mk(p)w(x))1−p0dx ≤CZRn| (x)|p0 w(x)1−p0dx (6) since he adjoin ope a o o Tk bis essen ially he same. 2. A e obse ing ha (Mk(p)w)1−p0∈A∞(in ac i belongs o RH∞) we apply he Calde ´on–Zygmund p inciple: we eplace he singula in eg al by a maximal ype ope a o , namely Mk+1 in ou case using Theo em 1.1: ZRn|T (x)|p0(Mk(p)w(x))1−p0dx ≤CZRnMk+1 (x)p0(Mk(p)w(x))1−p0dx. (7) 3. The e o e e e y hing is educed o showing a sha p wo weigh ed no m in- equali ies o he maximal ope a o Mk+1 ZRnMk+1 (x)p0(Mk(p)w(x))1−p0dx ≤CZRn| (x)|p0 w(x)1−p0dx. (8) 1.2 The Nonlinea commu a o The second commu a o ha we a e going o conside was in oduced by R. Rochbe g and G. Weiss in [RW]. This nonlinea ope a o is de ined o app op ia e unc ions by →N =T( log | |)−T log |T |. Nis homogeneous and can be w i en as a commu a o [Ω, T ] = T◦Ω−Ω◦Twhe e Ω deno es he ope a ion Ω = log | |. The e is a g owing in e es in s udying his ope a o due o i s ela ionship wi h he Jacobian mapping and wi h nonlinea P.D.E. as shown in [IS] [GI] (see also [M]). The main esul om [RW] is he ollowing: Le 1 < p < ∞, hen he e exis s a cons an Csuch ha kN kLp(Rn)≤Ck kLp(Rn).(9) 5 The heo y de eloped in [RW] is e y gene al. I shows, o ins ance, ha he singula in eg al Tmay be eplaced by any linea ope a o bounded on Lpi(Rn), i= 1,2, wi h 1 < p1<p<p2<∞. Howe e , o de i e Ap ype es ima es o N such a gene al amewo k does no seem o be sui able. We shall be using a di e en app oach based on eal a iable echniques and in pa icula on he heo y o Ap weigh s combined wi h some o he es ima es ob ained abo e o he linea commu- a o [Mb, T]. Fu he mo e and ying o ollow he Calde ´on–Zygmund p inciple again, we show ha he maximal ope a o which con ols Nis he Ha dy–Li lewood maximal unc ion i e a ed wice, namely N≈M◦M, exp ession which mo e p ecisely means he ollowing: THEOREM 1.3 Suppose ha 0<p<∞and ha w∈A∞. Then, he e exis s a cons an Csuch ha ZRn|N (x)|pw(x)dx ≤C[w]p A∞ZRnM2 (x)pw(x)dx, (10) As an immedia e consequence we ha e he ollowing co olla y. COROLLARY 1.4 Le 1< p < ∞and le w∈Ap. Then, he e exis s a cons an Csuch ha ZRn|N (x)|pw(x)dx ≤C[w]3p ApZRn| (x)|pw(x)dx, (11) Con a y o wha we did o he linea commu a o Tk bwe canno apply Theo em 1.3 o de i e o Na esul in he spi i o Theo em 1.2. The me hod ske ched abo e b eakdowns due o he nonlinea i y o N. Howe e and by a di ec app oach we can s ill deduce a co esponding es ima e. THEOREM 1.5 Suppose ha 1< p < ∞. Then, he e exis s a cons an Csuch ha o each weigh w ZRn|N (x)|pw(x)dx ≤CZRn| (x)|pM[2p]+1w(x)dx. (12) 6 To ge his es ima e whe show ha he e is a ela ionship be ween Nand he linea commu a o [Mb, T] and consequen ly wi h M◦M=M2. The obse a ion is ha Ncan be w i en using he linea i y o Tas ollows (see §4): N =T( log | | M )+[Mlog M , T]( )−T log |T | M =N1 +N2 +N3 . Obse e ha he symbol o he ope a o N2is he ope a ion b=b( ) = log M which is a BMO unc ion wi h a cons an independen o . 2 Some p elimina ies and no a ion We shall in oduce in his sec ion some o he necessa y ools ha we need o p o e ou esul s. Recall ha a unc ion B: [0,∞)→[0,∞) is called a Young unc ion i i is con inuous, con ex and inc easing sa is ying B(0) = 0 and B( )→ ∞ as → ∞. We de ine he B–a e age o a unc ion o e a cube Qby means o he Luxembu g no m k kB,Q = in {λ > 0 : 1 |Q|ZQ B | (y)| λ!dy ≤1},(13) and ecall he ollowing gene aliza ion o H¨olde ’s inequali y: 1 |Q|ZQ| (y)g(y)|dy ≤ k kB,Q kgk¯ B,Q ,(14) whe e ¯ Bis he complemen a y Young unc ion associa ed o B. The e is a u he gene aliza ion which u ns ou o be use ul o ou pu poses (see [O1]): Le A,B, Cbe Young unc ions such ha A−1( )·B−1( )≤C−1( ), hen k gkC,Q ≤2k kA,Q k kB,Q (15) We de ine a na u al maximal ope a o associa ed o he Young unc ion associ- a ed o B. 7 DEFINITION 2.1 Fo each locally in eg able unc ion he maximal ope a o MBis de ined by MB (x) = sup x∈Q k kB,Q , whe e he sup emum is aken o e all he cubes con aining x. The main examples ha we a e going o be using a e B( ) = (1 + log+ )α, α > 0, wi h maximal unc ion deno ed by ML(logL)α. The complemen a y Young unc ion is gi en by ¯ B( )≈e 1/α wi h co esponding maximal unc ion deno ed by Mexp(L1/α). The boundedness p ope ies o MBwill play a cen al ole o de i e sha p wo weigh ed es ima es. We need he ollowing class o Young unc ions. DEFINITION 2.2 Le 1< p < ∞. We say ha a doubling Young unc ion B sa is ies he Bpcondi ion i he e is a posi i e cons an csuch ha Z∞ c B( ) p d ≈Z∞ c p0 ¯ B( )!p−1d <∞. This condi ion p o ides wi h a cha ac e iza ion o hose maximal ope a o s MB which a e bounded on Lp(Rn), 1 < p < ∞. In ac , we ha e he ollowing Theo em whose p oo can be ound in [P1]. THEOREM 2.3 Le 1<p<∞. Suppose ha Bis a doubling Young unc ion. Then he ollowing a e equi alen . i) B∈Bp; (16) ii) he e is a cons an csuch ha ZRnMB (x)pdx ≤cZRn| (x)|pdx (17) o all unc ions ; iii) he e is a cons an csuch ha ZRnMB (x)pw(x)dx ≤cZRn| (x)|pMw(x)dx (18) 8 o all unc ions and all weigh s w; i ) he e is a cons an csuch ha ZRnM (x)pw(x) [M¯ B(u1/p)(x)]pdx ≤cZRn| (x)|pMw(x) u(x)dx, (19) o all unc ions and all weigh s wand u. In he p oo o Theo em 1.2 and o p > 1 we shall be wo king wi h Young unc- ions o he o m B( )≈ p(log )−1−which sa is y he Bpcondi ion and he e o e he associa ed maximal ope a o s MLp(log L)−1−a e bounded on Lp(Rn). We conclude his sec ion wi h a co olla y o his Theo em ha will be used la e on. The esul can be seen as a weigh ed inequali y “dual” o he classical Fe e man–S ein inequali y ZRnM (x)pw(x)dx ≤cZRn| (x)|pMw(x)dx. I Mwe e a linea ope a o his inequali y would imply ZRnM (x)p0Mw(x)1−p0dy ≤cZRn| (x)|p0 w(x)1−p0dx, which is alse in gene al, howe e we ha e he ollowing sha p eplacemen . COROLLARY 2.4 Le 1<p<∞and le w, u be weigh s. Then he e exis s a cons an Cindependen o he weigh s such ha ZRnM (x)p0u(x) M[p]+1w(x)p0−1dx ≤cZRn| (x)|p0Mu(x) w(x)p0−1dx (20) o all . In pa icula i u∈A1 ZRnM (x)p0u(x) M[p]+1w(x)p0−1dx ≤c[u]A1ZRn| (x)|p0u(x) w(x)p0−1dx (21) P oo : By pa i ) o he Theo em we ha e ha B∈Bp0i and only i ZRnM (x)p0w(x) [M¯ B(u(p0−1)/p0)(x)]p0dx ≤cZRn| (x)|p0Mw(x) u(x)p0−1dx, 9 ≤C[w]p A∞ZRn(M2 )pw. Fo he las e m we spli Rnin wo disjoin se s Aand Bwhe e A={y∈Rn:|T (y)| ≤ M (y)}and B={y∈Rn:|T (y)|> M (y)}. W i ing N3 =M |T | M log |T | M = Using in B ha log ≤C  , > 1,  > 0 we ha e he ollowing ZRn|N3 |pw≤CZA(M )pw+CZB|T |p(+1) (M )− p w We would like o apply again Theo em 1.1 wi h k= 0. To do his we mus show ha w(M )− p w∈A∞ o small enough and wi h a cons an independen o . ( ecall ha is s ill a ailable). Indeed, since w∈A∞w∈Aq o some q > 1 and by he ac o iza ion (c . [GCRdF] p. 436) heo em w=w1w1−q 2whe e w1and w2 a e A1weigh s. Then w(M )− p =w1w1−q 2(M )− p =w1(w2(M ) p q−1)1−q. By he ac o iza ion heo em i is enough o show ha w2(M ) p q−1∈A1 o small enough. To do his we ix a cube Q, and a bi a y a.e. x∈Q. Then 1 |Q|ZQ w2(M ) p q−1≤(1 |Q|ZQ w 2)1/ (1 |Q|ZQ(M ) 0p q−1)1/ 0.(27) Now, since w2∈A∞we can pick > 1 such ha we can con inue wi h C |Q|ZQ w2(1 |Q|ZQ(M ) 0p q−1)1/ 0, and i we pick wi h 0 <  < q−1 p 0 hen (M ) 0p q−1∈A1and hen his is less o equal han C |Q|ZQ w2(M (x))  p q−1≤C w2(x) (M (x))  p q−1. An impo an obse a ion is ha he A∞no m does no depend on .2 P oo o Theo em 1.5 P oceeding as be o e we ha e 16 kN kLp(w)≤ kN1 kLp(w)+kN2 kLp(w)+kN3 kLp(w), and he p oo o he wo i s pieces a e simila o he p e ious case. Recall ha he e is no assump ion on w. Fo N1we combine Theo em 1.2 o k= 0, he ac ha | log | ≤ 1 e, 0 < ≤1, oge he wi h he classical Fe e man–S ein inequali y ZRn(M )pw≤cZRn| |pMw o ob ain ZRn|N1 |pw=ZRn|T(M M log(| | M )| p w ≤CZRn(M )pM[p]+1w≤CZRn| (y)|pM[p]+2w ≤CZRn| |pM[2p]+1w. Fo N2we use Theo em 1.2 since de BMO no m o log M is independen o ZRn|N2 |pw=ZRn|[log M , T] |pw ≤ klog M k2p BMO ZRn| |pM[2p]+1w≤CZRn| |pM[2p]+1w. Fo he las e m N3we s a as abo e wi h ZRn|N3 |pw≤CZRn(M )pw+CZRn|T |p(+1) (M )− p w, ≤CZRn| |pMw +CZRn|T |p(+1) (M )− p w. The key poin o he p oo is o unde s and he las e m. We a e emp ed in applying Theo em 1.1 wi h k= 0 eplacing Tby Mwhich would inish he p oo o he Theo em; howe e , he e is no assump ion on win such a way ha we canno say ha he weigh on he igh hand side, namely (M )− p w, is an A∞weigh . We may a gue as ollows. Le p=p(+ 1), hen he e exis a unc ion g∈L(p)0wi h uni no m such ha ZRn|T |p(M )− p w1 p=  T (M )− +1 w1 p  Lp(Rn)=ZRnT (M )− +1 w1 pg. 17 Since he adjoin ope a o T∗is also a Calde ´on– Zygmund ope a o bounded on all he Lpspaces as well we can equal las exp ession o ZRn T∗(g(M )− +1 w1 p) = ZRn (Mkw)1 p (M ) +1 T∗(g(M )− +1 w1 p)(M ) +1 (Mkw)1 p ≤ ZRn| |pMkw (M ) +1 p!1 p ZRn|T∗(g(M )− +1 w1 p)|(p)0(M ) +1 (p)0 (Mkw)(p)0 p  1 (p)0 =I×II, whe e kis an in ege o be chosen in a momen . To es ima e Iwe simply use he Lebesgue di e en ia ion Theo em I= ZRn| |p  | |pMkw (M )p  !1 p ≤ZRn| |pMkw1 p, whe e kis s ill a ailable. Fo he las e m II we a e going o eplace T∗by he Ha dy–Li lewood maximal unc ion using again Theo em 1.1 wi h k= 0 and since T∗is also a Calde ´on–Zygmund ope a o . All we ha e o do is o show ha he weigh u=(M ) +1 (p)0 (Mkw)(p)0 p = (M ) +1 (p)0(Mkw)1−(p)0 is an A∞weigh wi h cons an independen o . To do his obse e i s ha (M ) +1 (p)0∈A1since  +1 (p)0= +1/p0<1 and (Mkw)1−(p)0∈RH∞by Lemma 4.2 whe e he cons an s a e independen o bo h and w. The e o e u∈A∞by Lemma 4.1 abo e and we ha e applying Theo em 1.1 ha II ≤C ZRnM(g M − +1 w1 p)(p)0(M ) +1 (p)0 (Mkw)(p)0−1! 1 (p)0 . Finally we can apply Co olla y 2.4 wi h k= [p] + 1 using as we poin ed ou abo e ha (M ) +1 (p)0∈A1. Then II ≤C ZRn|g|(p)0(M )− +1 (p)0w(p)0 p(M ) +1 (p)0 w(p)0−1! 1 (p)0 18 =CZRn|g|(p)01 (p)0 =C. Combining all hese inequali ies we ge ha ZRn|T |p(+1) M − p w≤CZRn| |pM[p]+1w=CZRn| |pM[p]+1w wi h small enough. The e o e we ha e ZRn|N3 (y)|pw(y)dy ≤ZRn| (y)|pM[p]+1w(y)dy which combined wi h he es ima es o N1and N2yield he inal esul . Obse e ha he piece co esponding o N3beha es mo e as he usual singula in eg al and ha he “wo s ” piece co esponds o N2.2 5 A coun e example We end he pape by showing ha Theo em 1.2, and consequen ly he o he s, is op imal. Conside he classical Hilbe ans o m H (x) = p ZR (y) x−ydy, and le m= 1,2,· · · be he la ges exponen o which he ollowing inequali y does no hold ZRn|Hk b (x)|pw(x)dx ≤Ckbkkp BMO ZRn| (x)|pMmw(x)dx. (28) By duali y his is equi alen o showing ZRn|Hk b (x)|p0 Mmw(x)1−p0dx ≤Ckbkkp0 BMO ZRn| (x)|p0 w(x)1−p0dx Conside he BMO unc ion b(x) = log |x|and le =w=χ(0,1) so ha he igh hand is equal o a ini e cons an . Fo he le hand side we use ha o x>e |Hk b (x)| ≈ (log x)k x≈Mk+1 (x). 19 Then ZR |Hk b (x)|p0 Mmw(x)1−p0dx ≥ ≥Zx>e (log x)k x!p0 (log x)m−1 x!1−p0 dx =Zx>1 xk p0−(m−1)(p0−1)+1 dx x, which becomes unbounded when m≤[(k+ 1)p]. The e o e (28) is alse when m= [(k+ 1)p] and hen inequali y (5) in Theo em 1.2 is op imal. Re e ences [CWW] S. Y. A. Chang, J. M. Wilson, and T. H. Wol , Some weigh ed no m in- equali ies conce ning he Sch ¨odinge ope a o s, Commen . Ma h. Hel e ici 60 (1985), 217–286. [CFL1] F. Chia enza, M. F asca y P. Longo, In e io W2,p es ima es o non di e - gence ellip ic equa ions wi h discon inuous coe icien s, Riche che Ma . 40 (1991), 149–168. [CFL2] F. Chia enza, M. F asca y P. Longo, W2,p–sol abili y o he Di ichle p oblem o nondi e gence ellip ic equa ions wi h VMO coe icien s, T ans. Ame . Ma h. Soc. 334 (1993), 841–853. [C] R. Coi man, Dis ibu ion unc ion inequali ies o singula in eg als, P oc. Acad. Sci. U.S.A. 69 (1972), 2838–2839. [CRW] R. Coi man, R. Rochbe g and 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. [DiR] G. Di Fazio y M. A. 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