Weighted norm inequalities for general maximal operators

Publicacions Ma emá iques, Vol 35 (1991, 169-186 . WEIGHTED NORM INEQUALITIES FOR GENERAL MAXIMAL OPERATORS C . PÉREZ 1 . In oduc ion In [13] Muckenhoup p o ed he undamen al esul cha ac e izing all he weigh s o which he Ha dy-Li lewood maximal ope a o is bounded ; he su - p isingly simple necessa y and su icien condi ion is he so called A P -condi ion (see below) . A di e en app oach o his cha ac e iza ion was ound by Jaw- e h (c . [9]) . An ad an age wi h his app oach is ha i gene alizes o mo e gene al si ua ions ; o ins an e, o Ha dy-Li lewood ype maximal ope a o s, ob ained by eplacing he cubes by any collec ion o se s in R"°, and o spaces o homogeneous ype . Fo a gene al in oduc ion, and his o ical commen s we e e o [7] . The main pu pose o his pape is o use some o he esul s and echniques in [9] o u he in es iga e weigh ed no m inequali ies o Ha dy-Li lewood ype maximal ope a o s . We s a by in oducing some no a ion . By a basis !3 in Rn we mean a collec ion o open se s in R" . We say ha w is a weigh associa ed o he basis 13 i w is a non-nega i e measu able unc ion in Rn such ha w(B) = B w(y) dy < oo o each B in 13 . M 3,w is he co esponding maximal ope a o de ined by M ,w (x) = s u á w(B) Áj (y)jw(y)dy i x E UBE 3 and MB  , (x) = 0 o he wise . I w- 1, we jus w i e MB (x) . We say ha he weigh w belongs o he class A, ,j3, 1 < p < oo, i he e is a cons an c such ha IBI ~B w(y) dy)  ~BI JB w(J) 1-P' dy) P-1 <_ c o all B E 3 . p' will always deno e he dual o p, ha is 17 0  C . PÉREZ In he limi case p = 1 we ha e ha w belongs o he class A1,13 i o all B E 13 ; his is equi alen o saying almos e e ywhe e x E Rn . Fo he o he limi case, p = oo, we se I ollows om hese de ini ions and H&lde 's inequali y ha i 1<p<q<00 . In sec ion (5) we shall use also he ollowing no a ion . The weigh w belongs o he class A p u (dp), 1 < p < oo, i hese is a cons an c such ha and only i 11 1B w(y) dy) ess .SUPB(w -1 ) < e Maw(x) < cw(x) A,,~,a = Up>jA,,i3 . A p, a C A9,13 (p(B) 1B w(y) dp(y) / (p(B) 1B w(y) '-p' dp(y»p-' < c o all B EB . We also deno e by M B ,d, he maximal ope a o de ined by MB,wdp (x) = s up(wp)(B)  B J (y)j w(y)dp(y) i x E UBEa and Ma  ,d p (x) = 0 o he wise, wi h (wdF,)(B) = B w(y) dp(y) . One o he main esul s in [9] is he ollowing Theo em . Theo em 1 .1 (Jawe h) . Le¡ 1 <p < oo . Suppose ha Ci is a basis and ha w is a weigh , and se a =w l- p' . Then Ma : LP(w) -> LP(w) Ma : LP , (o) -~ LP'(u) wEA p a M8 , ,,, : LP'(w) -~ LP'(io) Ma, o : LP(o) --~ LP(o) . Theo eln 1 .1 includes Muckenhoup 's esul , men ioned abo e, ha o a ixed 1 <p<o0 M2 : LP(dp) -> Lp'(dp) WEIGHTED NORM INEQUALITIES  17 1 i and only i dlc = w(y)dy, wi h w E A P 2 . He e Q is he basis o all open cubes in R" . A key ac conce ning Theo em 1 .1 is ha he p oo comple ely a oids he (di icul ) "Re e se H6lde inequali y ." Acknowledgemen s . The con en o his pape is pa o my Washing on Uni e si y Ph . D . Thesis . I would like o exp ess my deepes g a i ude o my eache Bjó n Jawe h o his guidance and all his eaching . I also would like o hank R . Howa d and A . R . Schep om he Uni e si y o Sou h Ca olina, o se e al con e sa ions conce ning hei wo k in [8] . The e e ee has made se e al use ul obse a ions o which I am g a e ul . Finally, i is a g ea honou o me o dedica e his wo k o he memo y o José Luis Rubio de F ancia . He in oduced me o he ield o Ha monic Analysis, and, la e , always kindly suppo ed and encou aged me . 2 . One-weigh heo y I is a undamen al ac ha M2, . is bounded in LP(w) o each 1 < p < oo, i he weigh w is doubling (c . [7] p .144 ) . In pa icula MQ, . is bounded i w is a A,,,2 weigh . R . Fe e man in [3] and B . Jawe h and A . To chinsky in [11] (also c . [7] p .463) p o ed ha he weigh ed s ong maximal ope a o M- R,,,, ha is he weigh ed maximal ope a o associa ed wi h he basis 13 = R o all ec angles in R'° wi h sides pa allel o he coo dina e axes, is also bounded in LP(w) whene e he weigh w belongs o he class A,, .- . The p oo o his esul is based on a geome ic co e ing lemma which goes back o he wo k o Có doba (c . [1]) . In his sec ion N e show ha hese esul s a e pa icula phenomena o a gene al ac . Theo em 2 .1 . Le Ci be a basis . The ollowing s a emenis a e equi alen . i) Fo each 1 < p< oo, a ad whene e w E A P ,5 (1)  Mi3 : LP(w) --, LP(w) ; ü) o each 1 < p < oc, and whene e w E A .,13 (2)  Mj3, . : L P (w) -+ LP(w) . P oo : Assume ha he basis 8 sa is ies ii) . Fix 1 < p < oo, and le w E A, j3 . By deno ing o =w 1- P', we ha e ha o E A P , L3 C A,, .j3, and hus M8,  , : L P' (w) -> LP'(w) Mj3 o : LP(o) -> LP(u) . Applying now Theo em 1 .1 we ge Ms : LP(w) -i LP(w) M 3 : L P' (o) - L P' (u), 17 2  C . PÉREZ which in pa icula gi es us i) . Assuming now i), we ix 1 < p < oo, and we ake w E A.. . Suppose ha w E A q , 1 < q < oo . The e a e wo cases . a)q<p' ; b) q > p' . In he i s case we ha e ha w E AP ,, which means ha w' - PE A p . Hence, by hypo hesis, Me : LP1(w) -> LP'(w) M13 : LP(w 1- P) - LP(wl-P) . And applying again Theo em 1 .1 we ob ain MB,w : LP(w) -> LP(w) . In he second case we ha e ha MB : Lq(w) -+ Lq(w) M 3 : Lq ' (w l- q ' ) -+ Lq'(wl-q') . since w l- 9 ' E A q , Now by using Theo em 1 .1 we ge Since we always ha e ha we can in e pola e o ge M 3,w : Lq'(w) -> Lq'(w) . M3, . : L'(w) -> L'(w), M 3,,, : LP(w) -+ LP(w), o e e y q' < p < oo, and hence p' < q . This concludes he p oo . Example 2 .2 . Le MR be he Có dóba-Zygmund maximal ope a o . The basis R de ining his maximal ope a o is o med by hese ec angles in R 3 wi h sides pa allel o he coo dina e axes whose sideleng hs a e o he o m {s, , s } . I has been shown by R . Fe e man (see o ins an e [4] ) ha MR is bounded in LP(w), o each 1 < p < oo, i and only i w E A P ,R . Hence, by Theo em 2 .1 MR, ,: L P (w) -+ LP(w) . o each 1 < p < oo, whene e w E A .,R In iew o all hese impo an examples wemake he ollowing de ini ion . De ini ion 2 .3 . We say ha he basis 13 is a Muckenhoup basis i o each 1<p<oo,ande e ywEA,a M,3 : LP(w) -> LP(w) . Wi h his de ini ion, Theo em 2 .1 can be s a ed as ollows Ci is a Muckenhoup basis i and only i WEIGHTED NORM INEQUALITIES  17 3 Ms, w : LP(w) --> LP(w), o each 1 < p < oo, and whene e w E A,,,s Nex esul is an ex ension o Lin's esul (c . [12]) o he s ong maximal ope a o o any maximal ope a o whose basis is a Muckenhoup basis . Co olla y 2 .4 . Suppose ¡ha¡ 8 is a Muckenhoup basis . Le 1 < p < oo, and suppose ha w E A,,,s, hen he ollowing Fe e man-S ein inequali y holds (4)  n Ms (y) P w(y)dy C c  n (y)PMs w(y)dy . IR  R P oo .. Suppose ha w E A,,s, o some 1 < < oo . The ollowing poin - wise inequali y hen ollows easily om Hólde 's inequali y and om he A, ., s- condi ion (5)  MB(XE)(x) <_ c(Ms,w(XE)(X))l1 . Since l3 is a Muckenhoup basis, Theo em 2 .1 yields ( 6 )  Ms,w : LP(w) --> LP(w), and oge he wi h (5) we easily conclude ha o each measu able E and each 0 < A < oo he ollowing inequali y holds : w (Ms(XE) > A) < c(A)w(E) . Since also Ms : LP(R") -> LP(R'), we a e now in a posi ion whe e we can p oceed as in he p oo o Lemma 7 .1 in [9], o conclude he p oo o he Co olla y . Rema k 2 .5 . We poin ou ha o (4) o hold, we do no need o assume ha w E A,, .,s ; (7) o some 0 < A < 1 would be su icien . Fo he sake o comple ness we obse e ha Muckenhoup bases sa is y Jones' ac o iza ion heo em . We jus s a e he esul and e e he eade o [9], Co olla y 6 .1, o he p oo unde a weake condi ion on he basis . P oposi ion 2 .6 . Suppose ha Ci is a Muckenhoup basis, and le 1 < p < oo . Suppose ha w E A P ,s . Then he e a e weigh .s w1,w2 E Al,s such ha w= WJw2-P . 3 . Vec o - alued inequali ies I is well known by now ha he e is an in ima e connec ion be ween ec o alued inequali ies and weig hed no m inequali ies (c . [7] Chap e 5) . In his sec ion we shall use he esul s om he p e ious one o ob ain an ex ension o he classical Fe e man-S ein ec o - alued inequali y (c . [2]) lIq  lIq IMQ ilq~  < CP,q P  ~~ I i q q-O  4O  [,P whe e 1 < p < oo and 1 < q <_ oo, o any maximal ope a o ML3 whose basis is a Muckenhoup basis (c . he de ini ion abo e) . We would like o poin ou ha (8) has played a undamen al ole in he analysis made in he ecen wo ks [5] and [10] . We shall assume h oughou he sec ion ha X3 is a Muckenhoup basis . Fo a ix 1 < p < oc, and mo i a ed by he me hod in oduced by J . L . Rubio de F ancia (c . sec ion 5 .5 in [7] and sec ion 6 in [9] ), we deno e by R13 he ope a o Rp : LP(R') -> LP(R") °°  , . -~ 1 : MB (2K)' , whe e MI = Id and M,' 3 is he i h i e a e o he ope a o M,3 . K is he no m o M13, as an ope a o on LP(R") . Al hough RL3 is poin wise la ge han ML3, i p ese es mos o i s p ope ies, namely i) u<RL3u IIRi3 UJILP(R^) :~ 2IIuJILP(R^)' Fu he mo e, RB has he p ope y ha iii) Ri3 E A l , 3 o each ELP(R") . The las s a emen is no ue o MB as he ollowing a gumen shows . Conside he Ha dy-Li lewood maximal ope a o 117 =MQ, and ake any posi i e unc ion cp E L l (R") . We shall show ha D lW is no e en an A,,,, weigh . Indeed, suppose ha Mcp E A,, o some 1 < p < oo . Then (Mco) 1-P ' E A p , and, by Muckenhoup 's heo em, M (y)" M~p(y) 1-P' dy C c~  (y) " MW(y)dy . R^  R^ and o each > n _1 . WEIGIITED NORM INEQUALITIES  175 By aking = cp we see by Lebesgue's di e en ia ion heo em, ha he igh hand side o he inequali y is ini e, while he le hand side is in ini e l . Lemma 3 .1 . Leí 1 < p, < oo, and leí B be a Muckenhoup basis . Then o each nonnega i e unc ion u E Us>1L 9 (R n ) we ha e I  M[3 (y) Pu(y)dy ~ c  n (y) P Reu(y)dy, R  R Rn MI3 (y)P (Ri3(u )) (y)1/' dy < c %Rn (y)P u(J)dy, P oo .. The i s inequali y ollows om abo e ema ks and (4) . Fo he second we obse e ha (RL3 U)-11 = [(R5 u ) 1 / (P -1 )J 1-P EAP, 1, and his, in u n, ollows om he ac ha i w E A1,8 hen w óEA1, 3, i 0 _< 6 < 1, and om he easy pa o (2 .6) . Finally, (9) ollows om he de ini ion o Muckenhoup basis and om abo e ema ks abou RB . Rema k 3 .2 . We poin ou ha o he case B= Q he ollowing inequali y holds : (10)  IR- M (y)P M(u)(y)1/ < cJR- (y) P u y , o 1 < p < oo and > 1 1 . The esul is alse i = P 1 1 , (c . [14]) . P_- We may hink o R ; as being he igh subs i u e in he gene al case . Once we ha e his lemma hen he ec o - alued inequali y o M ; ollows om Theo em 5 .2 Chap e 5 in [7] . Theo em 3 .3 . Leí 1 < p < oo, and 1 < q _< oo . I B is a Muckenhoup basis, hen 1/q  1/q ~~ I MCi i I ql  < C P,q  ~~ I i q q-o  La i-o Ly Rema k 3 .4 . Al hough we ha e men ioned ha he heo em ollows om Lemma 3 .1, i is o be men ioned ha we jus need he i s hal o i . Indeed, his and a s anda p ocedu e would gi e he case q < p . Now, since he heo em is ob ious o q = p, and also o q = oo, he case 1 <p < q < oo is ob ained by in e pola ion . In he same spi i we make he ollowing obse a ion abou he class A 1 ,1 ; . 1 Following he me hod in [16] i is possible o p o e ha Mcp may no be e en doubling . We a e indeb ed o F . So ia o showing his o us . 176  C . PÉREZ Lemma 3 .5 . Assume ha B is a Muckenhoupi oasis . Le¡ 0 < q <_ 1, and le w E UP>1LP(Rn) . Then i and only i he e is a posi i e unc ion gE Up>1LP(Rn), such ha o some la ge enough cons an A (12) Since ob iously w.IZZ~ w< w E Al,B 0 00 ( Mb9 )9 1/y E Ai P oo : By i e a ion Mgw <_ c'w, o some cons an c . Then pu ing A c2 1 / 9 and g =w we ge < 2w . (12) ollows . To p o e he con e se, le G deno e he igh hand side o (12) . Since w ',Z~ G, i is enough o deal wi h G . We i s show ha G is in UP>1LP(Rn) . Indeed, suppose ha g E LP(R n ), 1 < p < oo . Then IpliiD = I  (Mgg)9I  = ~ 0  Ai  ~ Lp/4 = E  n( mÁi(Y) ) u(y)dy, R o some u E L(Pl9)'(Rn) wi h uni no m . The e o e, by he abo e ema ks and by i e a ing (4), he las exp ession is domina ed by 00 C M ~ (y) ) Reu(y)dy< g(y)9 RBu(y)dy . R .n i i=0  i=0 He e F deno es he smalles cons an o which (4) holds . Finally, by aking A = 2 1 / 9 F, using Hdlde 's inequali y, and ii) abo e ob ain ha JIGIlL,  < 2 1 /9 11911LI , We shall p o e now ha G E A 1 ,13 . Fo each B E Ci we shall see ha 1  G (y)  <AGx IBI e  (y) y _  ( ), WEIGIITED NORM INEQUALITIES  177 a . e . x E B . Indeed, by Minkowski's in eg al inequali y wi h = 9 > 1 _  l 1/ = [ 1  ¡  -  Msg(y) q ( 1  B G(y) dyl  ¡B¡ JB  1  A  )~ 1  7 Még(y)  ` 1/  oo ( M~ +1  x l 9 = <  B  As  dy  < A9  As g() A G(x )9 . $=o  ;=o 4 . An al e na i e o mula ion o Muckenhoup 's heo em In his sec ion we gi e a di e en c i e ion o decide whe he he ope a o M 13 is bounded on LP(w), assuming ha he basis is a Muckenhoup basis . In pa icula his esul applies o he case X3 = Q, p o iding a di e en cha ac- e iza ion o Muckenhoup 's heo em . This app oach is inspi ed by he esul s in [8] . Theo em 4 .1 . Le¡ 1 < p < oo, and suppose ha 13 is a Muckenhoup basis . Le w be a weigh o 13 .  Then (13)  Mí3 : LP(w) --> LP(w) i and only i (14) Me (w(M6go) P-1 ) < c w go 1,-1, o some nonnega i e measu able unc ion g o , wi h B g o (y) dy < co o each B in 13, and o some posi i e cons an c . P oo .. Assuming (13) i is s anda d o see ha w E A P L3, and hence o = W1-P' E A P , , 3 . Since Li is a Muckenhoup basis Mg : LP(w) -> LP(w), and M,3 : LP , (o) ---> LP'(o) . Hence and which implies Si : LP(Rn) -> LP(Rn) --> w1/P M13 (W-11P ) S 2 : LP'(Rn) , LP'(Rn) ~, w-1/P A4 - L ; (w 1 IP ), T : LP'(R') -> LP'(Rn) 184  C . PÉREZ whe e and We iew he sum Ek j Pk jgk , j , as an in eg al on a measu e space (X, ) buil o e he se X = {k, j}, assigning o each (k, j) he measu e [¿k,j . Fo A > 0, se Then We can es ima e p(F(, )) as ollows 9 Mk,j - (Ek,j) (IBkj ~ ~ gk,i ~(y)dy) l 1 '   (Y)  d  p 9k,j = (a(Bk j)  B k , i -(Y)  (y) yl P(A) = {(k, .7) : gk,j > A}, G(, ) = U(k,j)E (a)Bk,j- gk /.i F k j =  Aglp  dA k, j Pk,j (k,j)EF(a)  MI ; (o-XB,k,i )(J) 9 (J)dy < _ (k,j)E (A) Ekj < cu(G(A))q/p < c ({J E R' : DIB, ( la)(J) p > , })9/p He e we ha e used he hypo hesis on AIB Q in he hi d inequali y . Finally by making a change o a iables we ob ain J  AIB (J) g (J)dy < < e  Ao , ({J E R n : 11IB o( w)(J) > A})1/pl9  _ = lo ( CIIMB,o( lU)II L(p,9)' concluding he p oo . As a consequence o his heo em we can deduce he ollowing cha ac e iza- ion . (24)  Mí3 : LP(u) -> Lq( ) i and only i (25) WEIGHTED NORM INEQUALITIES  185 Co olla y 6 .2 . Le 1 < p < q < oo, and le ( , u) be a couple o weigh s . Suppose ha MB o : LP(o) -> LP(o), whe e o = u 1- ' . Then 1/q U MH (oXG(a))(y)q (y)dy)  < cu(G(ñ)1/P o e e y se G which is a union o se s in 8 . P oo .. By se ing = UXG(a) in (24) we eadily ge (25) . To p o e he con e se we use Theo em 6 .1, ha p < q, and ou hypo hesis on Mí 3 , , IIMS IIL , ( ,, ) :5 V II MI3,o ( l~)II L ,a( o ) < < CIIMI3,o( IQ)IILP(o) <_ CII lUIILp(o) -CII lILP(u) As a consequence o his esul we can ob ain Sawye 's cha ac e iza ion o hose couple o weigh s ( , u) o which he Ha dy-Li lewood is bounded om LP(u) o Lq( ) . We jus s a e he esul since he p oo is like he gi en o he case p = q in [7] p .432, wi h some ob ious modi ica ions . Co olla y 6 .3 .  La 1 < p < q < oo, and le ( , u) be a couple o weigh s, ando,= u' - P' . Then (26)  M : LP(u) -> Lq( ) i and only i 1/q (27)  (IQ M(-XQ)(y) q (y)dy)  < co(Q)1/P o e e y cabe Q . Re e ences 1 .  A . CÓRDOBA, On he Vi a co e ing p ope ies o a di e en ia ion oasis, S udia Ma h . 57 (1976), 91-95 . 2 .  C . 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SAWYER, A cha ac e iza ion o a wo weigh no m weigh inequali y o maximal ope a o s, S udia Ma h . 75 (1982), 1-11 . 16 .  F . SORIA, A ema k on A l -weigh s heo y o he s ong maximal unc ion, P oc . Ame . Ma h . Soc . 10 0 (1987), 46-48 . 17 . J.-O . STRUBERG AND R .L . WIIEEDEN, F ac ional in eg als on weigh ed HP and LP spaces, T ans . Ame . Ma h . Soc . 287 (1983), 293-321 . Depa men o Ma hema ics Campus Box 1146 Washing on Uni e si y Sain Louis, Missou i 63130 U .S .A . cu en add ess : Depa men o Ma hema ical Sciences New Mexico S a e Uni e si y Las C uces, NEW MEXICO 88003