Sharp weighted inequalities for the vector-valued maximal function

T ans. Ame . Ma h. Soc. 352 (2000), 3265–3288. Sha p weigh ed inequali ies o he ec o – alued maximal unc ion Ca los P´e ez Depa men o de Ma em´a icas Uni e sidad Au ´onoma de Mad id 28049 Mad id, Spain e–mail: ca los.p[email p o ec ed] wo k pa ially suppo ed by DGICYT g an PB940192, Spain 1 1 Mo i a ion and desc ip ion o he main esul s The pu pose o his pape is o ob ain some sha p weigh ed inequali ies o he ec o – alued maximal unc ion Mqwhich a e no wi hin he scope o he s anda d Ap heo y o ec o – alued singula in eg als as can be ound in [RRT]. We s a wi h a e iew o some o he classical es ima es and hen we shall s a e he main esul s. 1.1 Backg ound Le Mbe he Ha dy–Li lewood maximal unc ion and le Mqbe he ec o – alued maximal ope a o de ined by Mq (x) = ∞ X i=1 (M i(x))q!1/q . This nonlinea ope a o was in oduced by C. Fe e man and E. M. S ein in [FS] as a gene aliza ion o bo h he (scala ) maximal unc ion Mand he classical in eg al o Ma cinkiewicz and since hen i has played an impo an ole in he de elopmen o mode n Ha monic Analysis. We ecall he wo basic es ima es ob ained in [FS] o 1 < q < ∞: •Le 1 < p < ∞, hen he e exis s a cons an Csuch ha ZRnMq (x)pdx ≤CZRn| (x)|p qdx. (1) •The ollowing weak ype (1,1) es ima e holds: he e exis s a cons an csuch ha sup λ>0 λ{x∈Rn:Mq (x)> λ}≤CZRn| (x)|qdx. (2) We a e using he e he no a ion | (x)|q= (P∞ i=1 | i(x)|q)1/q =k (x)k`q. Ano he undamen al gene aliza ion o he maximal heo em is due o B. Muckenhoup [M] who ga e a cha ac e iza ion o ollowing “weigh ed no m inequali y” ZRnM (x)pw(x)dx ≤cZRn| (x)|pw(x)dx, (3) in e ms o he Apcondi ion o Muckenhoup : he e exis s a posi i e cons an csuch ha o all cubes Q Ap 1 |Q|ZQ w(y)dy 1 |Q|ZQ w(y)1−p0dyp−1 ≤c. (4) I is also well known ha he Apcondi ion (4) also cha ac e izes all he weigh s w o which he weigh ed ec o – alued inequali y holds ZRnMq (x)pw(x)dx ≤CZRn| (x)|p qw(x)dx. (5) This esul is due o K. F. Ande sen and R. T. John [AJ], and o V. Kokilash ili [K]. The e a e by now h ee ways o p o ing (5): 2 •Re ining he a gumen o Fe e man and E. M. S ein in [FS] as done in [AJ] and [K]. •Looking a Mqas a ec o – alued singula in eg al wi h ope a o – alued ke nel sa is ying a poin wise g adien condi ion as can be ound in [RRT]. •By applying he ex apola ion heo y o J. Ga cia-Cue a and J. L. Rubio de F ancia as men ioned in [GCRdF] p. 521 which yields a simple p oo . In his pape we in es iga e he wo weigh p oblem o he ec o - alued maximal unc ion Mq ZRnMq (x)pw(x)dx ≤CZRn| (x)|p q (x)dx, (6) o which none o he abo e app oaches wo ks. Recall ha he e is a cha ac e iza ion due o E. Sawye [S] o he wo weigh p oblem in he scala si ua ion, he ollowing weigh ed inequali y: ZRnM (x)pw(x)dx ≤CZRn| (x)|p (x)dx, (7) holds i and only i he e exis s a cons an csuch ha o all cubes Q SpZQ M( 1−p0χQ)(y)pw(y)dy ≤cZQ (y)1−p0dy. (8) The ange p≤q o (6) is easy o handle since i coincides wi h he scala si ua ion. Indeed, i p≤qwe claim ha he Spcondi ion is necessa y and su icien o (6). I is clea ha condi ion (8) is necessa y. I we assume (8) we see ha (6) is immedia e o bo h q=pand q=∞. Then he case 1 <p<q<∞ ollows by in e pola ion o “linea izable ope a o s” in he ec o – alued con ex (c . he a gumen gi en in [GCRdF] p. 482). Al hough we shall gi e a ull cha ac e iza ion o (6) in Theo em 2.3 we a e mo e in e es ed in es ima es o he o m ZRnMq (x)pw(x)dx ≤CZRn| (x)|p qNw(x)dx, (9) whe e Nis an app opia e (scala ) maximal ype ope a o . Needless o say ha he p o o ypical es ima e ha we ha e in mind is he Fe e man–S ein weigh ed inequali y ZRnM (x)pw(x)dx ≤CZRn| (x)|pMw(x)dx, (10) which yields, as i is well known, he unweigh ed ec o – alued es ima e (1) when p>q. One o he main obse a ions ha ollows om ou esul s is ha Mqdoes no e i y a simila inequali y o (10) on he ange p > q (c . he i s ema k a e Theo em 1.1). Inequali ies o he ype (10) e lec how singula is he ope a o unde s udy. This can be seen o ins ance wi h he ollowing sha p inequali ies o singula in eg als ob ained in [P2] o p > 1 gene alizing some p e ious es ima es ob ained by M. Wilson in he ange 1 < p ≤2 [Wil2]: Le Tbe any Calde ´on–Zygmund ope a o , and le 1 <p<∞. Then he e exis s a cons an Csuch ha 3 ZRn|T (x)|pw(x)dx ≤CZRn| (x)|pM[p]+1w(x)dx, (11) wi h Cindependen o wand . Fu he mo e, he es ima e is sha p since i does no hold o M[p]. He e Mk=M◦(k) . . . ◦M k = 1,2,· · ·, deno es he Ha dy–Li lewood maximal ope a o Mi e a ed k imes. Ano he example which s esses ou poin o iew is ela ed o he classical A ea unc ion. This non linea ope a o is de ined by he in eg al Sϕ( )(x) = ZB (x) | ∗ϕ (y)|2d dy n+1 !1/2 , whe e ϕ∈C∞ 0wi h Rϕ= 0 and ϕ (x) = −nϕ(x ), > 0. Then he A ea unc ion sa is ies he ollowing inequali y: Le 1 < p ≤2, hen he e exis s a cons an Csuch ha ZRnSϕ( )(x)pw(x)dx ≤CZRn| (x)|pMw(x)dx. (12) Fu he mo e, his inequali y is alse o p > 2. The case p= 2 was i s ob ained by A. Chang, M. Wilson, and T. Wol in [CWW] and o 1<p<2 by S. Chanillo and R. Wheeden in [CW] as well as he coun e example o p > 2. See also he wo k by M. Wilson [Wil1]–[Wil4]. 1.2 Main esul s Mo i a ed by he Theo ems men ioned abo e we s a e now he main esul o he pape . THEOREM 1.1 Le 1< q < p < ∞. a)The e exis s a cons an Csuch ha ZRnMq (x)pw(x)dx ≤CZRn| (x)|p qM[p q]+1w(x)dx, (13) o all locally in eg able unc ions ,w≥0. b) Pa a) is sha p since he e exis s no cons an Csuch ha ZRnMq (x)pw(x)dx ≤CZRn| (x)|p qM[p q]w(x)dx, (14) o all locally in eg able unc ions ,w≥0. Likewise, he co esponding weak ype (p, p)es ima e is alse. We now make he ollowing ema ks. (a) I ollows om pa b) o he Theo em ha he ec o - alued analogue o he Fe e man-S ein inequali y (10) 4 ZRnMq (x)pw(x)dx ≤CZRn| (x)|p qMw(x)dx. (15) is alse in gene al in he ange p > q. (b) I we look a he p oo o (13) we see ha we can e ine such an inequali y by eplacing M[p q]+1 by ML(log L) p q−1+, > 0, o by MAwhe e Asa is ies Z∞ c A( )(p q)0−1d <∞. See Sec ion 3 o he app opia e de ini ion o he maximal ype unc ion MA. (c) We emphasize on he ac ha he e is no assump ion on wo he han local in eg abil y. In ac , i we assume ha w∈A∞ hen (15) holds being alse in gene al. Indeed, by he Lebesgue di e en a ion Theo em we ha e ZRnMq (x)pw(x)dx ≤ZRnMq (x)pMw(x)dx ≤CZRn| (x)|p qMw(x)dx, (16) whe e in he las inequali y we ha e used he Ap esul o Mq(5) since Mw ∈A1by s anda d esul s (see he las pa o Sec ion 4). Also, we may eplace M[p q]+1wby he A1weigh M(w )(x)1/ , > 1, by applying again (5). Howe e , he la e class o weigh s (essen ially he class A1) a e poin wise la ge han he non A∞ weigh s Mkwsince i may be shown using s anda d heo y ha Fo each in ege k= 1,2,· · ·, each > 1 and each locally in eg able unc ion , we ha e he ollowing poin wise inequali y o all x∈Rn: w(x)≤Mkw(x)≤[M(w )1/ ]k−1 A1M(w )(x)1/ . He e [w]A1deno es he “no m” o w∈A1, namely he smalles cons an Csuch ha Mw ≤C w. (d) Theo em 1.1 indica es ha Mqbeha es mo e as a singula in eg al ope a o a he han as a maximal ope a o . Howe e , we wan o emphasize he ac ha inequali y (13) does no i wi hin he scope o he heo y o ec o – alued singula in eg als as de eloped by J. L. Rubio de F ancia, F. J. Ruiz and J. L. To ea in [RRT] whe e he pionee ing wo k [BCP] was upda ed. In [RRT], he ope a o Mq, as well as many o he non–linea ope a o s such as he A ea unc ion Sϕ, a e seen as singula in eg als aking alues in an app opia e Banach space. Using his poin o iew, i is possible o ansla e o his mo e gene al con ex he one weigh scala Ap heo y a leas o any ec o – alued singula in eg al wi h su icien ly smoo h ke nel. Howe e , his is no he case o (13) (no o (12)) since he esul o he (scala ) Hilbe ans o m (11) is wo se han (13) indica ing ha he ope a o Mqis less singula han H. The p oo o he posi i e pa o Theo em 1.1 does no ollow he scheme used in [P2] o ea singula in eg als since we canno dualize (13). We shall de i e (13) as a consequence o a 5 cha ac e iza ion o he wo weigh p oblem gi en in Theo em 2.3. The condi ion we ob ain is a blend o Sawye ’s condi ion Sp oge he wi h Rubio de F ancia’s cha ac e iza ion o ec o – alued inequali ies o sublinea ope a o s as can be ound in [GCRdF] Chap e VI. I should be men ioned ha Y. Rako ond a simba has ob ained in [R] a di e en cha ac e iza- ion o he wo weigh p oblem which is much close in spi i o Sawye ’s condi ion Sp. 1.3 Sha p su icien condi ions close o Ap In his sec ion we ake up he wo weigh p oblem o Mq ha we w i e in he ollowing mo e con enien o m ZRn(w(x)Mq (x))pdx ≤CZRn( (x)| (x)|q)pdx. (17) The ask is o p o ide sha p su icien condi ions on he weigh s “close” in s uc u e o he Ap condi ion. Le us b ie ly e iew some esul s ela ed o he scala si ua ion ZRn(w(x)M (x))pdx ≤cZRn( (x)| (x)|)pdx. (18) I is well known ha he necessa y Apcondi ion o his p oblem 1 |Q|ZQ w(x)pdx1/p 1 |Q|ZQ (x)−p0dx1/p0 ≤c(19) is no su icien , and ha he co ec necessa y and su icien condi ion is, as we men ioned abo e, Sawye ’s condi ion which wi h ou no maliza ion on he weigh s has he ollowing o m ZQ (w(y)M( −p0χQ)(y))pdy ≤CZQ (y)−p0dy. The d awback o his condi ion is ha i in ol es he ope a o Mi sel , and i would be in e es ing o ob ain su icien condi ions close in o m o he Apcondi ion (19). Pe haps, he i s esul in ha di ec ion was ob ained by C. Neugebaue in [N]. He no iced ha i (w, ) is a couple o weigh s such ha o some > 1 1 |Q|ZQ w(y)p dy1/p 1 |Q|ZQ (y)−p0 dy1/p0 ≤c(20) o all cubes Q, hen ZRn(w(y)M (y))pdy ≤cZRn( (y)| (y)|)pdy. (21) In ac Neugebaue p o es ha (20) is equi alen o showing ha he e is an Apweigh inse ed (poin wise) be ween wpand pand he esul ollows i ially. This p oblem has been conside ed in [P1] whe e i is shown ha such a s ong condi ion is no needed. In pa icula i is no necessa y o “bump” he le weigh wand ha much less han a powe “bump” is equi ed on he igh weigh o ge he esul . We ex ac he ollowing esul om [P1]. Recall ha o a gi en Young unc ion Aand a cube Qon Rnwe de ined he A-a e age o a unc ion o e Qby k kA,Q = in {λ > 0 : 1 |Q|ZQ A| (y)| λdy ≤1}. 6 THEOREM 1.2 [P1] Le 1<p<∞, and le Bbe a doubling Young unc ion such ha Z∞ c p0 B( )!p−1d <∞,(22) o some posi i e cons an c. Le (w, )be a couple o weigh s such ha he e is a posi i e cons an K o which 1 |Q|ZQ w(y)pdy1/p   −1 B,Q ≤K, (23) o all cubes Q. Then ZRn(w(y)M (y))pdy ≤cZRn( (y) (y))pdy (24) o all nonnega i e unc ions . As we may expec we need o conside s onge condi ions on he weigh s o ge co esponding esul s o Mqin he ange p > q. In pa icula we need o “bump” he le weigh was well since o he wise he esul is alse as he coun e example (w, Mw) in (15) shows. Indeed, obse e ha his pai o weigh s sa s ies (23) o any cube Qand any Young unc ion B: 1 |Q|ZQ w1/p   (Mw)−1/p  B,Q ≤1 |Q|ZQ w1/p     1 |Q|RQw−1/p    B,Q =1 |Q|ZQ w1/p 1 |Q|ZQ w−1/p k1kB,Q = 1 since 1 |Q|RQw≤Mw(x) o x∈Q. THEOREM 1.3 Le 1<q<p<∞, and le =p q. Le A, B be doubling Young unc ions such ha bo h Z∞ c A( ) 0−1d and Z∞ c q0 B( )!q−1d ,(25) a e ini e o some posi i e cons an c, ha is ¯ A∈B 0and ¯ B∈Bq. Le (w, )be a couple o weigh s such ha he e is a posi i e cons an K o which kwqk1/q A,Q   −1 B,Q ≤K, (26) o all cubes Q. Then he wo weigh ed ec o – alued inequali y   (P∞ i=0(w M i)q)1/q  Lp(Rn)≤C  (P∞ i=0 | i|q)1/q  Lp(Rn)(27) holds o all i. Some in e es ing examples a e gi en by A( )≈ (log ) −1+δand B( )≈ q0(log )q0−1+δwi h δ > 0. 7 1.4 Endpoin es ima es Al hough he ope a o Mqis, o some ex en , mo e closely ela ed o a singula in eg al his is no he case when we look a endpoin es ima es such as he ollowing. THEOREM 1.4 The e exis s a cons an Csuch ha o each weigh wand o all λ > 0 w({x∈Rn:Mq (x)> λ})≤C λZRn| (x)|qMw(x)dx. (28) This esul e lec s once again ha sha p esul s o Mqa e independen om he heo y o ec o – alued singula in eg als since we do no know whe he he (scala ) Hilbe ans o m H sa is ies w({x∈Rn:|H (x)|> λ})≤C λZRn| (x)|Mw(x)dx. (29) See [P2] o sha p esul s. The e exis s an in e es ing ela ionship be ween (28) and a possible ec o – alued e sion o he classical Besico i ch lemma. We shall o mula e his as a conjec u e. Mc wdeno es he weigh ed cen e ed maximal unc ion. CONJECTURE 1.5 w({x∈Rn: ∞ X i=1 (Mc w i(x))q!1/q > λ})≤C λZRn| (x)|qw(x)dx. (30) One can show (c . Sec ion 6) ha i he conjec u e we e ue hen he inequali y (28) ollows immedia ely. Acknowledgemen s. The au ho is e y g a e ul o A. Va gas o se e al con e sa ions con- ce ning he p oblems conside ed in his pape . 2 A cha ac e iza ion o he wo weigh p oblem The pu pose o his sec ion is o gi e a cha ac e iza ion o he wo weigh p oblem o he ec o – alued maximal unc ion Mq. We ecall ha he case 1 < p ≤qis cha ac e ized by means o Sawye ’s condi ion Sp. The main esul is Theo em 2.3. Fo he p oo o his Theo em i will be mo e e icien o wo k wi hin a mo e gene al con ex . Le Bbe a basis in Rn, and by his we mean a collec ion o open se s in Rn. We say ha wis a weigh associa ed o he basis Bi wis a non-nega i e measu able unc ion in Rnsuch ha w(B) = RBw(y)dy < ∞ o each Bin B.MB,w is he co esponding maximal ope a o de ined by MB,w (x) = sup x∈B 1 w(B)ZB | (y)|w(y)dy i x∈ ∪B∈B and MB,w (x) = 0 o he wise. I w≡1, we jus w i e MB (x). 8 PROPOSITION 2.1 Le 1< q < p < ∞, and le =p q. Suppose ha MB,σ :Lp `q(σ)→Lp `q(σ)whe e σ= 1−p0. Then he wo weigh ec o alued inequali y   (P∞ i=0(MB i)q)1/q  Lp(w)≤C  (P∞ i=0 | i|q)1/q  Lp( )(31) holds i and only i he e exis s a cons an csuch ha o each g∈L 0(Rn)we can ind G∈L 0(Rn) wi h kGkL 0(Rn)≤ kgkL 0(Rn)such ha ZΩ MB(σχΩ)(x)qw(x)1/ g(x)dx ≤cZΩ σ(x)1/ G(x)dx, (32) o e e y se Ωwhich is a union o se s in B. P oo : We i s show ha condi ion (32) is necessa y. Fi s obse e ha inequali y (31) is equi alen o    P∞ i=0(w1/p MB( i 1/p ))q1/q   Lp(Rn) ≤c  (P∞ i=0 | i|q)1/q  Lp(Rn). Now, by Rubio de F ancia’s heo em (c . [GCRdF] p. 555) his es ima e is equi alen o showing ha o each g∈L 0(Rn) he e exis s G∈L 0(Rn) wi h kGkL 0(Rn)≤ kgkL 0(Rn)and ZRn(w(y)1/pMB( 1/p )(y))qg(y)dy ≤CZRn| (y)|qG(y)dy, o all , o wha is he same ZRnMB( )(y)qw(y)1/ g(y)dy ≤CZRn| (y)|q (y)1/ G(y)dy, o all . Tes ing his inequali y wi h =σχΩ= 1−p0χΩgi es he necessa y condi ion (32). To p o e he su iciency o (32) we use ha L and L 0a e dual spaces. We adap he basic ideas om [GCRdF]. I we de ine Ias I=   P∞ i=0(w1/p MB i)q1/q    q Lp(Rn) , hen I= ∞ X i=0 ZRnMB( i)(y)qw(y)1/ g(y)dy o some g∈L 0(Rn) wi h uni no m. Fix i, and o each in ege kconside he se Ei k={y∈ Rn: 2k< MB i(y)≤2k+1}. F om he de ini ion o MB,Ei k⊂ ∪jBi k,j, whe e Bi k,j ∈ B sa is ies 2k<1 Bi k,jZBi k,j i(y)dy. De ine now Ei k,1=Bi k,1∩Ei k,and o j > 1Ei k,j =Bi k,j ∪s<j Bi k,s∩Ei k. Fo any ixed k, each o he se s Ei kis he disjoin union o he se s Ei k,j. We now can w i e 9 We also pos pone he p oo o his lemma un il he end o he p oo o he heo em. Now, using (49) and (51) we can es ima e he le side o (43) as ollows ZRnMB (y)pw(y)dy =X kZΩk−Ωk+1 MB (y)pw(y)dy (52) ≤apX k akpw(Ωk)≤CX k,j akpw(3Qk,j)≤ ≤CX k,j k kp B,Qk,j w(3Qk,j) = CX k,j k kp B,Qk,j w(3Qk,j) |3Qk,j||Qk,j| ≤CX k,j      w(3Qk,j ) |3Qk,j|1/p     p B,Qk,j |Ek,j| ≤CX k,j ZEk,j MB( (Mw)1/p)(y)pdy ≤CZRnMB( (Mw)1/p)(y)pdy ≤CZRn (y)pMw(y)dy, since we a e assuming ii). This p o es iii). Le us assume ha iii) holds. Obse e ha (44) is equi alen wi h ZRnM( g)(y)pw(y) [M¯ B(g)(y)]pdy ≤cZRn (y)pMw(y)dy, o all nonnega i e unc ions ,g, and w. Then i ) ollows immedia ely om (43) a e an appli- ca ion o he inequali y M( g)(y)≤MB (y)M¯ Bg(y)y∈Rn which is a consequence o he gene alized H¨olde ’s inequali y (41). To p o e ha i ) implies i) we le w= 1 in (44) ob aining ZRnM (y)p1 [M¯ B(u1/p)(y)]pdy ≤cZRn (y)p1 u(y)dy, o all nonnega i e unc ions , and u. Tes ing his inequali y wi h =u=χQ(0,1) , whe e Q(x, ) deno es he cube cen e ed a x∈Rnand wi h sideleng h equal o , we ha e ZRnM (y)p1 [M¯ B( )(y)]pdy ≤C. (53) On he o he hand we ha e o la ge x ha M¯ B( )(x)≈1 ¯ B−1(1 |x|n). The e o e we ge 16 ZRnM (y)p1 [M¯ B( )(y)]pdy ≥CZ|y|>c 1 |y|np 1 ¯ B−1(1 |y|n)pdy =CZ∞ c 1 np 1 ¯ B−1(1 n)p nd ≈Z∞ c B( ) p d . This es ima e combined wi h (53) shows ha i ) ⇒i). To conclude he p oo o he Theo em, apa om he p oo s o Lemmas 3.3 and 3.4, we need o show ha i) ⇔ ). Tha i) is necessa y is i ial since ) implies he scala case, namely ii). To show ha i) is su icien obse e ha he case p<s ollows by in e pola ion om he cases s=p, s=∞. Now he case p > s ollows om he weigh ed inequali y iii) by s anda d a gumen s. 2 P oo o Lemma 3.3: The p oo is a simple adap a ion o a gumen s in [GCRdF] Ch. 2. Since is bounded wi h compac suppo , say supp ⊂K, k kB,Q ≤ k kL∞kχKkB,Q =k kL∞ 1 B−1|Q| |Q∩K|, and i ollows ha k kB,Q →0 as Q↑Rn. Hence, i he e a e any dyadic cubes Qwi h k kB,Q > , hey a e con ained in cubes o his ype which a e maximal wi h espec o inclusion. We le C ={Pj}be he amily o he dyadic maximal nono e lapping cubes sa is ying < k kB,Pj. Le P0 jbe he only dyadic cube con aining Pjwi h sideleng h wice ha o Pj. Then < k kB,Pj≤2nk kB,P 0 j . The las inequali y can easily be deduced om he de ini ion o he Luxembu g no m using he ac ha →B( ) is non dec easing. Hence by he maximali y o he cubes {Pj}we ge < k kB,Pj≤2n . (54) Obse e ha om his discussion i is clea ha {y∈Rn:Md B (y)> }=∪jPj.(55) Le x∈Ω . By de ini ion, he e is a cube Rcon aining xsuch ha < k kB,R .(56) 17 Le kbe he unique in ege such ha 2−(k+1)n<|R| ≤ 2−kn. The e is some dyadic cube wi h side leng h 2−k, and a mos 2no hem, {Ji:i= 1, . . . , n}, mee he in e io o R. I is easy o see ha o one o hese cubes, say J1, 2n<  χJ1   B,R .(57) This can be seen as ollows. I o each i= 1, . . . , 2nwe had   χJi   B,R ≤ 2n, we would ge since R⊂ ∪2n i=1Ji ha k kB,R =    χ∪2n i=1Ji    B,R ≤ 2n X i=1   χJi   B,R ≤2n 2n= , con adic ing (56). Since |R| ≤ |J1|<2n|R|one can also show 4n<k kB,J1.(58) By le ing C /(4)n={Qj}, we ha e by (54) ha 4n<k kB,Qj≤ 2n,(59) o each j, yielding (46). Equa ion 48) also ollows since {y∈Rn:Md B (y)> 4n}=∪jQj. Also, we see om (58) ha J1⊂Qk, o some k, and hen R⊂3J1⊂3Qk. This gi es Ω ⊂ ∪j3Qj, which is (45). Now, by he le side o he inequali y (59), and he de ini ion o k kB,Q we ge |Ω | ≤ CX j |Qj| ≤CX jZQj B4n (y) dy ≤CZRnB (y) dy. (60) To ob ain (47) we jus use he s anda d idea o w i ing as = 1+ 2, whe e 1(x) = (x) i (x)> 2, and 1(x) = 0 o he wise. Then MB (x)≤MB 1(x) + MB 2(x)≤MB 1(x) + 2. Finally, since (60) holds o each ≥0, > 0 we ha e |Ω | ≤ {y∈Rn:MB 1(y)> 2}≤CZRnB 1(y) dy =CZ{y∈Rn: (y)> /2} B (y) dy, concluding he p oo o Lemma 3.3. 2 18 We now conclude he p oo o he Theo em by p o ing Lemma 3.4. P oo o Lemma 3.4: The amily Ek,j is clea ly disjoin . We no e ha (49) and he de ini ion o he Luxembu g no m implies ha 1<1 |Qk,j|ZQk,j B4n ak (y)dy, and 1 |Qk,j|ZQk,j B2n ak (y)dy ≤1. Hence by s anda d p ope ies o he dyadic cubes we can es ima e wha po ion o Qk,j is co e ed by Dk+1 as in [GCRdF] p. 398 |Qk,j ∩Dk+1| |Qk,j|=X i |Qk,j ∩Qk+1,i| |Qk,j|=X i:Qk+1,i⊂Qk,j |Qk+1,i| |Qk,j| <X i:Qk+1,i⊂Qk,j 1 |Qk,j|ZQk+1,i B4n ak+1 (y)dy ≤2n a 1 |Qk,j|ZQk,j ∩∪iQk+1,i B2n ak (y)dy ≤2n a. He e we ha e used ha B(2n a )≤2n aB( ), > 0, since 2n a<1, and because →B( ) is inc easing. This gi es (50). Finally |Ek,j| |Qk,j|>1−2n a>0, comple ing he p oo o he Lemma and hence ha o Theo em 3.2. 2 4 P oo o he main Theo em In his sec ion we gi e he p oo o Theo em 1.1. We s a wi h he p oo o pa a), he posi i e pa . We apply Theo em 2.3 by e i ying condi ion (37). The weigh wis ixed and will be chosen ina a momen . Recall ha =p qand ha σ= 1−p0. We need o show ha he e exis s a cons an csuch ha o a bi a y g∈L 0(Rn) he e is G∈L 0(Rn) wi h kGkL 0(Rn)≤ kgkL 0(Rn)and such ha o all cubes Q ZQ M(σχQ)(x)qw(x)1/ g(x)dx ≤cZQ σ(x)1 G(x)dx. (61) We use he Fe e man-S ein inequali y (10) oge he wi h he gene alized H¨olde ’s inequali y (41) o es ima e he le hand side o (61) by a mul iple o ZQ σ(x)qM(w1/ g)(x)dx ≤ZQ σ(x)qMB(w1/ )(x)M¯ B(g)(x)dx =ZQ σ(x)q[(MB(w1/ )(x)) ]1/ M¯ B(g)(x)dx. 19 I we le = (MB(w1/ )) we ha e ha ZQ M(σχQ)(x)qw(x)1/ g(x)dx ≤CZQ σ(x)1/ M¯ B(g)(x)dx. To conlude he p oo all we ha e o do know is o choose Bsuch ha ¯ B∈B 0, namely ha Z∞ c B( ) 0−1d <∞ since by he cha ac e iza ion in Theo em 3.2 M¯ B:L 0(Rn)→L 0(Rn). I we le  M¯ B be he no m o his ope a ion we can ake G=M¯ B(g)  M¯ B  such ha kGkL 0(Rn)≤ kgkL 0(Rn)and we ha e ZQ M(σχQ)(x)qw(x)1/ g(x)dx ≤C M¯ B(g) ZQ σ(x)1/ G(x)dx. Finally we a e le wi h showing ha we can pick Bsuch ha = (MB(w1/ )) ≤M[ ]+1w. Indeed, le B( )≈ (log(1 + ))[ ], hen B( 1/ )≈ (log(1 + )[ ]and (MB(w1/ )) =ML(log L)[ ](w). Now, we make he ollowing obse a ion. Le k= 1,2,3,· · ·, hen he e exis s a cons an C=Cnsuch ha o all bounded unc ions wi h suppo con ained in Q k kL(log L)k,Q ≤C |Q|ZQ Mk (y)dy. Indeed, by homogenei y we can assume ha he igh hand side is one. Then by he de ini ion o he Luxembu g no m i is enough o p o e 1 |Q|ZQ (y)(1 + log+( (y)))kdy ≤C, which is a consequence o i e a ing he ollowing well known inequali y o E.M. S ein: ZQ (y)(1 + log+( (y)))kdy ≤CZQ M (y)(1 + log+(M (y)))k−1dy, (62) wi h k= 1,2,3,· · ·. The e o e we inally ha e ha =ML(log L)[ ](w)≤M[ ]+1wconcluding he p oo o he Theo- em. This concludes he p oo o he i s pa , o he coun e example we ake n= 1 and we le N be a la ge posi i e in ege and =p q>1. Se w=χ(0,1) and de ine i(x) = (log x)−1/qχ(ei,ei+1)(x) o each i= 1,· · · , N −1, and i= 0 o i≥N. Then,   (P∞ i=1 | i|q)1/q   p Lp(M[ ]w)=ZR N−1 X i=1 (log x)−1χ(ei,ei+1)(x)! M[ ]w(x)dx 20 ≈ZeN e (log x)− (log x)[ ]−1dx x. When is an in ege his is compa able o log N, and when is no an in ege he in eg al is a cons an independen o Nsince [ ]− + 1 >0. In any case, i is less han a cons an imes log N. On he o he hand   (P∞ i=1(M i)q)1/q   p Lp(w)=Z1 0 N−1 X i=1 (M i(x))q! dx ≥Z1 0 N−1 X i=1 1 i! dx ≈(log N) , since o 0 < x < 1 and i= 1,· · · , N −1, M i(x)≥1 ei+1 Zei+1 0 (log y)−1/qχ(ei,ei+1)(y)dy ≈1 i1/q . To conclude, obse e ha (log N) ≤Clog Ndoes no make sense o la ge Nwi h Cinde- penden o N.2 We conclude he sec ion by gi ing he ollowing simple a gumen showing ha Mw ∈A1 assuming ha w∈A∞which was used o p o e inequali y (16). Indeed since ws ais ies o some > 1 he e e se H¨olde inequali y 1 |Q|ZQ w 1/ ≤C |Q|ZQ w wi h Cindependen o he cube Q. Now o ixed Qand x∈Qwe ha e 1 |Q|ZQ Mw ≤1 |Q|ZQ M(wχ2Q) + 1 |Q|ZQ M(wχRn 2Q) ≤1 |Q|ZQ M(wχ2Q) 1/ +Cin QM(w)≤C1 |2Q|Z2Q w 1/ +M(w)(x) ≤C |2Q|Z2Q w+M(w)(x)≤C M(w)(x). He e we ha e used ha M(χRn 2Qw)(y)≈M(χRn 2Qw)(z) o each y, z ∈Q, [GCRdF] p. 159 and he L boundedness o M. This means ha Mw ∈A1. 5 P oo o he sha p su icien condi ions We wan o poin ou ha we do no know how o p o e his heo em di ec ly om he cha ac e i- za ion gi en in Theo em 2.3. We a e going o modi y and combine he p oo o his Theo em wi h he esul s in Theo em 3.2 which in ac con ains he key es ima e. P oo o Theo em 1.3: 21   (P∞ i=0(wM i)q)1/q   q Lp(Rn)= ∞ X i=0 ZRnM i(y)qw(y)qg(y)dy (63) o some g∈L 0(Rn) wi h uni no m. Le ibe ixed. Fo each in ege k, and o any a bi a y cons an a > 2nwe le Ωi k, and Di kbe he se s Ωi k={x∈Rn:ak< M i(x)}, Di k={x∈Rn:Md i(x)>ak 4n}. By he classical Calde ´on–Zygmund decomposi ion (c . [GCRdF] p. 137) he e exis s a amily o maximal nono e lapping dyadic cubes {Qi k,j} o which Ωi k⊂ ∪j3Qi k,j,Di k=∪jQi k,j, and ak 4n<1 Qi k,jZQi k,j i(y)dy ≤ak 2n.(64) We can now es ima e he in eg al in (63) as ollows ZRnM i(y)qw(y)qg(y)dy =X kZΩi k−Ωi k+1 M i(y)qw(y)qg(y)dy (65) ≤aqX k akq(wqg)(Ωi k)≤CX k,j akq(wqg)(3Qi k,j) ≤CX k,j   1 Qi k,jZQi k,j i(y)dy  q (wqg)(3Qi k,j) =CX k,j   1 Qi k,jZQi k,j i(y) (y) (y)−1dy  q (wqg)(3Qi k,j) ≤CX k,j   1 3Qi k,jZ3Qi k,j i(y) (y) (y)−1dy  q1 3Qi k,jZ3Qi k,j w(y)qg(y)dy Qi k,j. Fo each in ege k, j we se Ei k,j =Qi k,j −Qi k,j ∩Di k+1. Then {Ei k,j}is a disjoin amily o se s, and by Lemma 3.3 wi h B( ) = , he e is a posi i e cons an βsuch ha o each k, j Qi k,j< β Ei k,j. This oge he wi h he gene alized H¨olde ’s inequali y (41) allows o domina e he las sum by a mul iple o X k,j k ikq ¯ B,3Qi k,j   −1  q B,3Qi k,j kwqkA,3Qi k,j kgk¯ A,3Qi k,j Ei k,j ≤KqX k,j ZEi k,j M¯ B( i)(y)qM¯ A(g)(y)dy ≤CZRnM¯ B( i)(y)qM¯ A(g)(y)dy. since he se s {Ei k,j}a e pai wise disjoin when iis ixed. Hence, by H¨olde ’s inequali y wi h exponen s and 0we can es ima e (63) by 22   (P∞ i=0(wM i)q)1/q   q Lp(Rn)≤C  P∞ i=0(M¯ B( i ))q1/q   q Lp(Rn) M¯ A(g) L 0(Rn) ≤C  (P∞ i=0( i)q)1/q   q Lp(Rn)kgkL 0(Rn)=  (P∞ i=0( i)q)1/q   q Lp(Rn) since M¯ B:Lp `q(Rn)→Lp `q(Rn) and M¯ A:L 0(Rn)→L 0(Rn) by Theo em 3.2. 2 6 Endpoin es ima es and he Besico i ch lemma P oo o Theo em 1.4: I is enough o conside ≥0 in he sense ha i≥0 o all i. Le Ω = {x∈Rn:Md(| |q)(x)> λ}=∪Q, whe e he dyadic cubes Qa e maximal nono e lapping sa is ying λ < 1 |Q|ZQ | (x)|qdx ≤2nλ, (66) and | (x)|q≤λ, a.e. x∈Rn Ω. W i e = ·χRn−Ω+ ·χΩ=g+b. Since Mq ≤Mqg+Mqb, i is su icien o es ima e he dis ibu ion se o Mqgand Mqbsepa e ly. Obse ing ha |g(x)|q≤λwe ge by (10) w({x∈Rn:Mqg(x)> λ/2})≤C λqZRnMqg(x)qw(x)dx ≤C λqX iZRn(Mgi)(x)qw(x)dx ≤C λqX iZRngi(x)qMw(x)dx =C λqZRn|g(x)|q qMw(x)dx ≤C λZRn| (x)|qMw(x)dx. Fo bwe spli he dis ibu ion se o Mqbse as ollows. Le ˜ Ω = ∪˜ Q,˜ Q= 3Q. Then w({x∈Rn:Mqb(x)> λ/2})≤w({x∈Rn ˜ Ω : Mqb(x)> λ/2}) + w(˜ Ω). The second e m is es ima ed by he le hand side o (66): w(˜ Ω) ≤C λX Q w(˜ Q)1 |Q|ZQ | (x)|qdx ≤C λX QZQ | (x)|qMw(x)dx ≤C λZRn| (x)|qMw(x)dx. Fo he i s e m we use he a gumen in [FS] p. 110 which shows ha Mbi(x)≤M(¯ bi)(x), x∈Rn ˜ Ω whe e ¯ biis he unc ion 23 ¯ bi(x) = n0x∈Rn Ω1 |Q|RQ i(y)dy x ∈Q w({x∈Rn ˜ Ω : Mqb(x)> λ/2} ≤ 1 λqX iZRn ˜ Ω (Mbi(x))qw(x)dx ≤1 λqX iZRn ˜ Ω (M(¯ bi)(x))qw(x)dx ≤C λqX iZRn ¯ bi(x)qM(wχRn ˜ Ω)(x)dx =C λqX iX Q1 |Q|ZQ i(y)dyqZQ M(wχRn ˜ Q)(x)dx ≤C λqX Q"X i1 |Q|ZQ i(y)dyq#1 qqZQ M(wχRn ˜ Q)(x)dx ≤C λqX Q1 |Q|ZQ | (y)|qdyq in QM(w)|Q| ≤ C λX Q 1 |Q|ZQ | (y)|qdy in QM(w)|Q| ≤C λX QZQ | (x)|qMw(x)dx ≤C λZRn| (x)|qMw(x)dx. 2 As we men ioned in he in oduc ion he e exis s a close connec ion be ween he weigh ed scala inequali y w({x∈Rn:M (x)> λ})≤C λZRn| (x)|Mw(x)dx (67) and he he Besico i ch co e ing lemma [dG]. Indeed, he i s obse a ion is ha (67) is equi alen o w({x∈Rn:M( w Mw)(x)> λ})≤C λZRn| (x)|w(x)dx. The second is ha we i ially ha e he poin wise inequali y M( w Mw)(x)≤cnMc w (x), whe e Mc wis he weigh ed cen e ed maximal unc ion Mc w (x) = sup >0 1 w(B (x)) ZB (x) | (y)|w(y)dy. (68) The e o e (67) ollows om w({x∈Rn:Mc w (x)> λ})≤C λZRn| (x)|w(x)dx which is a consequence o he Besico i ch co e ing lemma. 24 We can epea his a gumen wi h M eplaced by he ec o – alued maximal ope a o Mq excep o he ac ha he e is no ec o – alued analogue o he Besico i ch lemma, namely w({x∈Rn: ∞ X i=1 (Mc w i(x))q!1/q > λ})≤C λZRn| (x)|qw(x)dx. Combining his es ima e oge he wi h (68) would yield a di e en p oo o Theo em 1.4. Re e ences [AJ] K. F. Ande sen and R. T. John, Weigh ed inequali ies o ec o – alued maximal unc ions and singula in eg als, S udia Ma h. 69 (1980), 97–101. [BCP] A. Benedek, A. P. Calde ´on, R. Panzone, Con olu ion ope a o s on Banach space alued unc ions, P oc. Na . Acad. Aci. USA 48, (1962), 356–365. [CWW] S. Y. A. Chang, J. M. Wilson, and T. H. Wol , Some weigh ed no m inequali ies conce ning he Sch ¨odinge ope a o s, Commen . Ma h. Hel e ici 60 (1985), 217–286. [CW] S. Chanillo and R. Wheeden, Some weigh ed no m inequali ies o he A ea in eg al, Indiana Uni . 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