Weighted estimates for the multilinear maximal function

a Xi :1304.5999 2 [ma h.CA] 9 Jul 2013 WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION WEI CHEN AND WENDOL´ IN DAMI´ AN Abs ac . A o mula ion o he Ca leson embedding heo em in he mul ilinea se ing is p o ed which allows o ob ain a mul ilinea analogue o Sawye ’s wo weigh heo em o he mul isublinea maximal unc ion Min oduced in [8]. A mul ilinea e sion o he Bp heo em om [6] is also ob ained and a mixed A~ P−W∞ ~ Pbound o Mis p o ed as well. 1. In oduc ion The beginning o he mode n heo y o weigh s was o igina ed in he wo ks o R. Hun , B. Muckenhoup , R. Wheeden, R. Coi man and C. Fe e man in he decade o he 70’s. In [12] B. Muckenhoup cha ac e ized he class o weigh s u, o which he ollowing weak inequali y holds (1.1) sup λ>0 λpZ{M >λ} u(x)dx ≤CZRn | (x)|p (x)dx, ∈Lp( ), whe e Mdeno es he Ha dy–Li lewood maximal ope a o and p≥1. This condi- ion on he weigh s is known as Apcondi ion, namely [u, ]Ap:= sup Q1 |Q|ZQ u(x)dx 1 |Q|ZQ (x)−1 p−1p−1 <∞, p > 1 whe e he sup emum is aken o e all he cubes in Rn. When p= 1, he e m (RQ 1 |Q| (x)−1 p−1)p−1mus be unde s ood as (in Q )−1. In he pa icula case when u= , Muckenhoup also p o ed ha he ollowing s ong es ima e ZRn (M (x))p (x)dx ≤CZRn | (x)|p (x)dx, ∈Lp( ), The i s au ho is suppo ed by he Na ional Na u al Science Founda ion o China (G an No. 11101353), he Na u al Science Founda ion o Jiangsu Educa ion Commi ee (G an No. 11KJB110018) and he Na u al Science Founda ion o Jiangsu P o ince (G an No. BK2012682). The second au ho is suppo ed by Jun a de Andaluc´ıa (G an No. P09-FQM-4745) 2010 Ma hema ics Subjec Classi ica ion. 42B25. Key wo ds and ph ases. Mul ilinea maximal ope a o , weigh ed bounds, e e se H¨olde ’s inequali y. 1 2 W. CHEN AND W. DAMI ´ AN holds i and only i sa is ies he Apcondi ion. Howe e , he p oblem o inding a condi ion on he weigh s u, sa is ying he s ong es ima e abo e was mo e com- plica ed. In [13] E. Sawye cha ac e ized he wo weigh inequali y, showing ha M:Lp( )−→ Lp(u) i and only i he pai (u, ) sa is ies he ollowing es ing condi ion known as Sawye ’s Spcondi ion (1.2) [u, ]Sp= sup Q RQM(χQσ)pudx σ(Q)!1/p <∞, whe e σ= 1−p′and 1 < p < ∞. Mo i a ed by hese esul s he heo y o weigh ed inequali ies de eloped apidly, no only o he Ha dy–Li lewood maximal ope a- o bu also o some o he main ope a o s in Ha monic Analysis like Calde ´on– Zygmund ope a o s. Much la e he in e es ocused in de e mining he sha p de- pendence o he Lp(w) ope a o no m in e m o he ele an cons an in ol ing he weigh s. On his poin , he p oblem o he Ha dy–Li lewood maximal ope a o was sol ed by S. Buckley [1] who p o ed (1.3) ||M||Lp(w)≤C p′[w] 1 p−1 Ap, whe e Cis a dimensional cons an . Mo i a ed by his esul and o he s, K. Moen ound in [11] a quan i a i e o m o E. Sawye ’s esul men ioned abo e in e ms o Sawye ’s condi ion (1.2), namely (1.4) ||M||Lp( )−→Lp(u)≈[u, ]Sp. Recen ly, T. Hy ¨onen and C. P´e ez in [6] (see also [7] o a be e esul and a simpli ied p oo ) imp o ed Buckley’s bound (1.3) eplacing a po ion o he Ap cons an by he weake A∞cons an as de ined by Fujii in [4] and la e used in he wo k o J.M. Wilson [14]. The A∞cons an is de ined as ollows (1.5) [w]A∞:= sup Q 1 w(Q)ZQ M(wχQ), whe e he sup emum is aken o e all he cubes Qin Rn. In [6] he au ho s show in a wo-weigh se ing and o p > 1 ha (1.6) ||M( σ)||Lp(w)≤Cp′(Bp[w, σ])1/p|| ||Lp(σ), and (1.7) ||M( σ)||Lp(w)≤Cp′([w]Ap[σ]A∞)1/p|| ||Lp(σ), whe e Cin bo h inequali ies is a dimensional cons an and WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 3 (1.8) Bp[w, σ] := sup Q1 |Q|ZQ w 1 |Q|ZQ σp exp 1 |Q|ZQ log σ−1, is known as he Bpcons an o he weigh s wand σ. This cons an clea ly sa is ies [w]Ap≤Bp[w, σ]≤[w]Ap[σ]′ A∞, whe e [σ]′ A∞deno es he A∞cons an in oduced by S. H usˇcˇe in [5] de ined as ollows [w]′ A∞= sup Q1 |Q|ZQ wexp 1 |Q|ZQ log w−1. In he one weigh se ing and by a s anda d change-o -weigh a gumen , (1.7) implies (1.9) ||M||Lp(w)≤Cp′([w]Ap[σ]A∞)1/p, whe e σ=w1−p′. The aim o his a icle is o gi e some mul ilinea analogues o some o he abo e men ioned esul s ollowing he spi i o he heo y o mul iple weigh s de eloped in [8]. The pape is o ganized as ollows. Some p elimina y de ini ions and esul s a e summa ized in Sec . 2. In Sec . 3we gi e he s a emen s o he main esul s on his pape and some ema ks on hem. Finally, in Sec . 4we gi e all he p oo s o ou esul s. Th oughou his pape , we will use he no a ion A.B o indica e ha he e is a cons an c, independen o he weigh cons an , such ha A≤cB. 2. P elimina ies Be o e s a ing and p o ing ou main esul s, we i s ecall some basics ela ed o he heo y o mul ilinea weigh ed inequali ies as well as we in oduce he de ini ion o some cons an s in ol ed in he mul iple heo y o weigh s and dyadic g ids. 2.1. Some basics on mul ilinea weigh ed inequali ies. One o he main ob- jec s o he heo y o mul iple weigh s is he ollowing ex ension o he classical Ha dy–Li lewood maximal unc ion. Gi en −→ = ( 1,..., m), we de ine ollowing [8] he mul i(sub)linea maximal ope a o Mby M(−→ )(x) = sup Q∋x m Y i=1 1 |Q|ZQ | i(yi)|dyi, whe e he sup emum is aken o e all cubes Qcon aining x. The impo ance o his ope a o s ems om he ac ha i con ols he class o mul ilinea Calde ´on– Zygmund ope a o s as i is shown in [8]. A pa icula example o his ela ionship is he class o weigh s cha ac e izing he weigh ed Lpspaces o which bo h ope a o s 4 W. CHEN AND W. DAMI ´ AN a e bounded. To de ine his class o weigh s we le −→ w= (w1,...,wm) and −→ P= (p1,...,pm) such ha 1 < p1,...,pm<∞. Se 1 p=1 p1+···+1 pmand ν−→ w= Qm i=1 wp/pi i. We say ha −→ wsa is ies he A−→ Pcondi ion i [−→ w]A−→ P= sup Q1 |Q|ZQ ν−→ wm Y i=1 1 |Q|ZQ w1−p′ i ip/p′ i<∞. I is easy o see ha in he linea case ( ha is, i m= 1) [−→ w]A−→ P= [w]Apis he usual Apcons an . In [8] he ollowing mul ilinea ex ension o he Muckenhoup Ap heo em o he maximal unc ion was ob ained: he inequali y (2.1) kM(−→ )kLp(ν−→ w)≤C m Y i=1 k ikLpi(wi) holds o e e y −→ i and only i −→ wsa is ies he A−→ Pcondi ion. Ve y ecen ly in [3] A. Le ne , C. P´e ez and he second au ho p o ed a mul ilinea e sion o Buckley’s esul as well as a ull analogue o (1.7). In his wo k he au ho s ound ha he mul ilinea e sion o (1.7) is sha p when m≥1, al hough is much mo e complica ed o do he same o Buckley’s esul . In his case, se e al pa ial esul s we e ob ained in [3] which ha e been imp o ed in [9]. 2.2. Some cons an s on mul iple weigh heo y. Nex we s a e he no a ion ha we will ollow in he sequel ela ed o some cons an s in ol ed in he mul iple heo y o weigh s. To de ine hese cons an s, le w1,...,wmand be weigh s and le us deno e −→ w= (w1,...,wm). Also le 1 < p1,...,pm<∞and pbe numbe s such ha 1 p=1 p1+···+1 pmand deno e −→ P= (p1,...,pm). We say ha ( , −→ w) sa is ies he A−→ Pcondi ion i (2.2) [ , −→ w]A−→ P:= sup Q1 |Q|ZQ m Y i=1 1 |Q|ZQ w1−p′ i ip/p′ i<∞. In pa icula when =ν−→ w:= Qm i=1 w p pi i, we will w i e [ν−→ w,−→ w]A−→ Pas [−→ w]A−→ P. Nex we de ine he mul ilinea analogues o he A∞cons an de ined by Fujii in [4], he Bpcons an de ined by Hy ¨onen and P´e ez in [6] and he Spcons an de ined by Sawye in [13], espec i ely. We say ha (1) −→ wsa is ies he W∞ −→ Pcondi ion i [−→ w]W∞ −→ P= sup QZQ m Y i=1 M(wiχQ)p pidxZQ m Y i=1 w p pi idx−1 <∞. WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 5 (2) ( , −→ w) sa is ies he B−→ Pcondi ion i [ , −→ w]B−→ P:= sup Q (Q) |Q|m Y i=1 wi(Q) |Q|pexp 1 |Q|ZQ log m Y i=1 w−p pi idx<∞. (3) ( , −→ w) sa is ies he S−→ Pcondi ion i [ , −→ w]S−→ P= sup QZQ M(−−→ σχQ)p dx1 pm Y i=1 σi(Q)1 pi−1 <∞, whe e −−→ σχQ= (σ1χQ,...,σmχQ) and σi=w1−p′ i i o all i= 1,...,m and all he sup ema in he abo e de ini ions a e aken o e all cubes Qin Rn. Addi ionally we de ine a mul iple Re e se H¨olde condi ion ha we will use in he ollowing. We say ha −→ wsa is ies he RH−→ Pcondi ion i he e exis s a posi i e cons an Csuch ha (2.3) m Y i=1 ZQ σidxp pi≤CZQ m Y i=1 σ p pi idx, whe e σi=w1−p′ i i o i= 1,...,m. We deno e by [−→ w]RH−→ P he smalles cons an C in (2.3). 2.3. Dyadic g ids. Recall ha he s anda d dyadic g id Din Rnconsis s o he cubes 2−k([0,1)n+j), k ∈Z, j ∈Zn. By a gene al dyadic g id Dwe mean a collec ion o cubes wi h he ollowing p ope ies: (1) Fo any Q∈Di s sideleng h ℓQis 2k, k ∈Z (2) Q∩R∈ {Q, R, ∅} o any Q, R ∈D. (3) The cubes o a ixed sideleng h 2k o m a pa i ion o Rn. We say ha {Qk j}is a spa se amily o cubes i : (1) The cubes Qk ja e disjoin in j, wi h k ixed. (2) I Ωk=∪jQk j, hen Ωk+1 ⊂Ωk. (3) |Ωk+1 ∩Qk j| ≤ 1 2|Qk j|. Wi h each spa se amily {Qk j}we associa e he se s Ek j=Qk j Ωk+1. Obse e ha he se s Ek ja e pai wise disjoin and |Qk j| ≤ 2|Ek j|. In he sequel we will use he ollowing lemmas ha could be ound in [6] and [3], espec i ely. Lemma 2.1. The e a e 2ndyadic g ids Dαsuch ha o any cube Q⊂Rn he e exis s a cube Qα∈Dαsuch ha Q⊂Qαand ℓQα≤6ℓQ. 6 W. CHEN AND W. DAMI ´ AN Lemma 2.2. Fo any non-nega i e in eg able i, i = 1,...,m, he e exis spa se amilies Sα∈Dαsuch ha o all x∈Rn, M(−→ )(x)≤(2 ·12n)m 2n X α=1 ADα,Sα(−→ )(x), whe e −→ = ( 1,..., m)and gi en a spa se amily S={Qk j}o cubes om a dyadic g id D, he ope a o AD,Sis gi en by AD,S(−→ ) = X j,k m Y i=1 ( i)Qk j!χQk j. 3. Main esul s In his sec ion we summa ize he main esul s on his wo k. Fi s ly we s a e he main ool o his pape . This lemma ex ends o he mul ilinea se ing a nons anda d o mula ion o he (dyadic) Ca leson embedding heo em p o ed in [6] and i will allow us o p o e ou main esul s. Lemma 3.1. Suppose ha he nonnega i e numbe s {aQ}Qsa is y (3.1) X Q⊂R aQ≤AZR m Y i=1 σ p pi idx, ∀R∈D whe e σia e weigh s o i= 1,...,m. Then o all 1< pi<∞and p∈(1,∞) sa is ying 1 p=1 p1+···+1 pmand o all i∈Lpi(σi), X Q∈D aQm Y i=1 1 σi(Q)ZQ i(yi)σi(yi)dyip!1/p ≤A||Md −→ σ(−→ )||Lp(ν−→ σ) ≤A m Y i=1 p′ i|| i||Lpi(σi), (3.2) whe e Md −→ σ(−→ ) = sup Q∋x Q∈D m Y i=1 1 σi(Q)ZQ | i(yi)|σi(yi)dyi. Nex we es ablish a gene aliza ion o Sawye ’s heo em o he mul ilinea se ing. Ve y ecen ly i was shown in [10] a mul ilinea e sion o Sawye ’s heo em using a kind o mono one p ope y on he weigh s. We es ablish he e ano he condi ion ha is a so o e e se H¨olde inequali y in he mul ilinea se ing (see Sec . 2 o de ini ion) and ha was used by he i s au ho in [2] in he se ing o ma ingale spaces. When m= 1 his e e se H¨olde condi ion is supe luous and we eco e he linea esul o Moen (1.4). WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 7 Theo em 3.2. Le 1< pi<∞,i= 1,...,m and 1 p=1 p1+...+1 pm. Le and wibe weigh s. I we suppose ha −→ w∈RH−→ P hen he e exis s a posi i e cons an C such ha (3.3) ||M(−→ σ)||Lp( )≤C m Y i=1 || i||Lpi(σi), i∈Lpi(σi), whe e σi=w1−p′ i i, i and only i ( , −→ w)∈S−→ P. Mo eo e , i we deno e he smalles cons an Cin (3.3)by ||M||, we ob ain (3.4) [ , −→ w]S−→ P.||M|| .[ , −→ w]S−→ P[−→ w]1/p RH−→ P . He e we make some ema ks ela ed o he p e ious heo em. Rema k 3.3. In he pa icula case when =ν−→ w, he ollowing s a emen s a e equi alen : (1) −→ w∈A−→ P. (2) σi=w1−p′ i i∈Amp′ i, o i= 1,...,m and ν−→ w∈Amp. (3) (ν−→ w,−→ w)∈S−→ P. (4) The e exis s a posi i e cons an Csuch ha (3.5) ||M(−→ )||Lp(ν−→ w)≤C m Y i=1 || i||Lpi(wi), i∈Lpi(wi). Indeed, he equi alence be ween 1., 2.and 4.was p o ed in [8, Th. 3.6, Th. 3.7]. I can be easily seen ha in his pa icula case [ν−→ w,−→ w]S−→ P.||M|| whe e ||M|| deno es he smalles cons an in (3.5) and [−→ w]A−→ P.[ν−→ w,−→ w]p S−→ P. The e o e we ha e ha 4.implies 3.and 3.implies 1.. So we ha e ob ained ha all he s a emen s a e equi alen . Addi ionally, ollowing [3, Th. 1.1], we also ha e ha ||M|| .[−→ w]1/p A−→ PQm i=1[σi] 1 pi ∞. So, we ha e ob ained (3.6) [−→ w]1/p A−→ P.[ −→ w,−→ w]S−→ P.||M|| .[−→ w]1/p A−→ P m Y i=1 [σi] 1 pi ∞. Rema k 3.4. As we ha e obse ed in he p e ious ema k, RH−→ Pcondi ion is no necessa y when =ν−→ win Theo em 3.2. We a e no su e i his condi ion can be emo ed in he gene al case. Making use o he analogue o he Bpcons an wi hin he mul ilinea se ing al eady de ined in Sec . 2, we ob ain an ex ension o (1.6). 8 W. CHEN AND W. DAMI ´ AN Theo em 3.5. Le 1< pi<∞,i= 1,...,m and 1 p=1 p1+...+1 pm. Le and wi be weigh s. Then (3.7) ||M(−→ σ)||Lp( ).[ , −→ σ]1/p B−→ P m Y i=1 || i||Lpi(σi), i∈Lpi(σi), whe e σi=w1−p′ i i,−→ σ= (σ1,...,σm)and −→ σ = ( 1σ1,..., mσm). And inally, using he gene aliza ion o he Fujii–Wilson A∞cons an [−→ w]W∞ −→ Pand he wo-weigh cons an [ , −→ w]A−→ Pde ined in Sec . 2, we ge a mixed A−→ P−W∞ −→ P bound o M ha ex ends (1.7) o he mul ilinea se ing. Theo em 3.6. Le 1< pi<∞,i= 1,...,m and 1 p=1 p1+...+1 pm. Le and wi be weigh s. Then (3.8) ||M(−→ σ)||Lp( ).([ , −→ w]A−→ P[−→ σ]W∞ −→ P)1/p m Y i=1 || i||Lpi(σi), i∈Lpi(σi), whe e σi=w1−p′ i i,−→ σ= (σ1,...,σm)and −→ σ = ( 1σ1,..., mσm). 4. P oo s We s a p o ing Lemma 3.1, ha is, he mul ilinea e sion o he dyadic Ca leson embedding heo em. I ollows a scheme o p oo simila o he one used by Hy ¨onen and P´e ez in [6]. Lemma 3.1.Le us see he sum X Q∈D aQ m Y i=1 1 σi(Q)ZQ i(yi)σi(yi)dyi!p as an in eg al on a measu e space (D,2D, µ) buil o e he se o dyadic cubes D, assigning o each Q∈D he measu e aQ. Thus X Q∈D aQ m Y i=1 1 σi(Q)ZQ i(yi)σi(yi)dyi!p = =Z∞ 0 pλp−1µ(Q∈D: m Y i=1 1 σi(Q)ZQ i(yi)σi(yi)dyi> λ) =: Z∞ 0 pλp−1µ(Dλ)dλ. WEIGHTED ESTIMATES FOR THE MULTISUBLINEAR MAXIMAL FUNCTION 9 Le us deno e by D∗ λ he se o maximal dyadic cubes Rwi h he p ope y ha Qm i=1 1 σi(Q)RR i(yi)σi(yi)dyi> λ. Then he cubes R∈D∗ λa e disjoin and hei union is equal o he se {Md −→ σ(−→ )> λ}. Thus µ(Dλ) = X Q∈Dλ aQ≤X R∈D∗ λX Q⊂R aQ ≤AX R∈D∗ λZR m Y i=1 σ p pi idx =AZ{Md −→ σ(−→ )>λ} m Y i=1 σ p pi idx. Then we ob ain X Q∈D aQ m Y i=1 1 σi(Q)ZQ i(yi)σi(yi)dyi!p ≤AZ∞ 0 pλp−1Z{Md −→ σ(−→ )>λ} m Y i=1 σ p pi idxdλ =AZRn Md −→ σ(−→ )p m Y i=1 σ p pi idx ≤AZRn m Y i=1 ((Md σi( i))piσi)p pidx ≤A m Y i=1 ZRn (Md σi( i))piσidxp pi ≤A m Y i=1 (p′ i)pZRn | i|piσidxp pi, whe e we ha e used ha Md −→ σ(−→ )≤Qm i=1 Md σi( i), H¨olde ’s inequali y and he boundedness p ope ies o Md σi( i) in Lpi(σi).  Nex we p o e Theo em 3.2 making use o Lemma 3.1. Theo em 3.2.I is clea ha (3.3) implies he S−→ Pcondi ion wi hou using ha ( , −→ w)∈RH−→ P. Thus, i emains o p o e ha ( , −→ w)∈S−→ Pimplies (3.3) o comple e he p oo o he heo em. By Lemma 2.1, i su ices o p o e he heo em o he dyadic maximal ope a o s MDα. Since he p oo is independen o he pa icula dyadic g id, wi hou loss o gene ali y we conside Md aken wi h espec o he s anda d dyadic g id D. Nex we p oceed as in he p oo o Lemma 2.2. Le a= 2m(n+1) and o k∈Zconside he ollowing se s