MARCINKIEWICZ SPACES, GARSIA RODEMICH SPACES AND THE SCALE OF JOHN NIRENBERG SELF IMPROVING INEQUALITIES

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1 Annales Academæ Scenarm Fenncæ Mahemaca Volmen 4, 26, 49 5 MARCINKIEWICZ SPACES, GARSIA RODEMICH SPACES AND THE SCALE OF JOHN NIRENBERG SELF IMPROVING INEQUALITIES Maro Mlman CONICET, Inso Argenno de Maemáca Saavedra 5, 83 Benos Ares, Argenna; maromlman@gmalcom Absrac We exend o n-dmensons a characerzaon of he Marcnkewcz L(p, spaces frs obaned by Garsa Rodemch n he one dmensonal case Ths leads o a new proof of he John Nrenberg self-mprovng neqales We also show a relaed resl ha provdes sll a new characerzaon of he L(p, spaces n erms of dsrbon fncons, reflecs he self-mprovng neqales drecly, and also characerzes L(,, he rearrangemen nvaran hll of BM O We show an applcaon o he sdy of ensor prodcs wh L(, spaces, whch complemens he classcal work of O Nel [9] and he more recen work of Asashkn [2] Inrodcon In her semnal paper [], John Nrenberg nrodced he space BMO and proved he celebraed John Nrenberg neqaly for fncons n BMO I s also well known, alhogh perhaps somewha less so, ha n he same paper, John Nrenberg showed ha he BMO self mprovemen neqaly can be refned and framed as a scale of neqales These neqales (or embeddngs are assocaed wh wha we nowadays call John Nrenberg spaces Ths resl of John Nrenberg, whch we now descrbe, s he sarng pon of or developmen n hs paper Le Q R n, be a fxed cbe, p < Le P(Q = {{Q } N : conable famles of sbcbes Q Q, The John Nrenberg spaces are defned by wh parwse dsjon nerors} JN p (Q = {f L (Q : JN p (f,q < }, where 2 JN p (f,q = sp {Q } P(Q { ( } p /p Q f f Q dx Q Q do:586/aasfm Mahemacs Sbjec Classfcaon: Prmary 42B35, 46E3 Key words: John Nrenberg neqaly, rearrangemen, BMO The ahor was parally sppored by a gran from he Smons Fondaon (#27929 o Maro Mlman A cbe n hs paper wll always mean a cbe wh sdes parallel o he coordnae axes 2 In wha follows, as sal, f Q = Q Q f dx

2 492 Maro Mlman Le s also recall ha, for a gven measre space, he Marcnkewcz L(p, spaces, p <, are defned by demandng 3 ha f L(p, <, where ( f L(p, = sp > {f ( /p } = sp{(λ f ( /p }; whle for p =, he space L(, (cf [3] s defned 4 by he condon f L(, <, where f L(, = sp{f ( f (}, > and f ( = > f (sds Then (cf [, Lemma 3], and also [2, Theorem 4, p 29] for a more dealed proof Theorem Le < p < Sppose ha f JN p (Q, hen f f Q L(p, (Q, and here exss a consan A(p,Q,n sch ha In parclar, f f Q L(p, (Q A(p,Q,nJN p (f,q f f Q r<pl r (Q The lmng condon defnngjn p (Q whenp = corresponds 5 obmo, and n hs case Theorem corresponds o a verson of he well known John Nrenberg neqaly [] In he one dmensonal case, Garsa and Rodemch [] mproved on Theorem To formlae he Garsa and Rodemch resl wll be convenen o nrodce a dfferen scale of spaces whch we shall erm Garsa Rodemch spaces I wll be sefl for laer se o gve he relevan defnons n he n-dmensonal case Le Q R n be a fxed cbe, le p <, and le p be defned by + = p p The Garsa Rodemch spaces GaRo p (Q are defned as follows We shall say ha f GaRo p (Q, f and only f f L (Q, and C > sch ha for all {Q } N P(Q we have (2 We le Q Then we have (cf [] Q ( f(x f(y dxdy C Q Q /p GaRo p (f,q = nf{c > : sch ha (2 holds} 3 Here f denoes he non-ncreasng rearrangemen of f and λ f s dsrbon fncon (cf [4] 4 Some ahors (ncldng somemes he ahor of hs paper se a dfferen noaon and le W denoe wha we call L(, a he same me ha hey se he noaon L(, = L 5 The JN (Q condon wold read sp {Q } P(Q Q Q f f Q dx <

3 The scale of John Nrenberg self mprovng neqales 493 Theorem 2 6 Le < p <, and le Q = I = [,] Then, as ses GaRo p (I = L(p, (I Remark The elemenary proof of he embeddng L(p, GaRo p olned n [] works n n-dmensons and acally shows ha (cf Theorem 5 par (, below (3 GaRo p (f,i p p 2 f L(p, (I By Theorem 7 we have JN p (I L(p, (I, herefore by Theorem 2 (cf Secon 2 below for a drec proof of he n dmensonal case follows ha (4 JN p (I GaRo p (I In conclson, Theorem 2 no only mproves on Theorem n he one dmensonal case, b also gves s an neresng characerzaon of he Marcnkewcz L(p, (I spaces, < p < Unfornaely, one par of he proof of Theorem 2 ses a nonrval rearrangemen neqaly, also de o Garsa Rodemch [], whch s only proved here n he one dmensonal 8 case In [], he ahors brefly sgges a possble dfferen mehod o prove Theorem 2 n n-dmensons, and who dmensonal consans, b no deals are provded 9 In hs noe we gve a new proof Theorem 2 ha s vald n n dmensons (cf Theorem 5 below Or approach s dfferen from he one gven n [], and does no se marngale echnqes Insead, or mehod s lmaely based on Calderón Zygmnd ype decomposons, followng classcal deas n [3] As we shall see (cf Secon 2 he verfcaon ha he John Nrenberg condons are sronger han he Garsa Rodemch condons (eg (4 s mmedae Therefore, he crcal aspec of hs approach o he John Nrenberg heorem s he fac ha he Garsa Rodemch spaces are he same as he Marcnkewcz L(p, spaces! Ths clarfes he self mprovemen resls of John Nrenberg Moreover, hese deas cold poenally be sefl n he nvesgaon of relaed sses, eg he dmensonal consans nvolved n he John Nrenberg embeddngs (cf [7] Now he classcal defnons of he L(p, spaces are gven n erms of growh condons on rearrangemens or dsrbon fncons (cf [4], [5], [8], [6], [2], ec The case p =, whch corresponds o L(, ( he rearrangemen nvaran hll of BMO, cf [3], also adms a smlar characerzaon hrogh he se of he oscllaon operaor f f, and ndeed one can fnd a characerzaon of all he 6 Here seems approprae o brng p he followng In hs paper [8], Dyson wres Professor Llewood, when he makes se of an algebrac deny always saves hmself he roble of provng ; he manans ha an deny, f re, can be verfed n a few lnes by anybody obse enogh o feel he need of verfcaon My objec n he followng pages s o confe hs asseron I s lef o reader o decde f he ahor of he presen paper s demonsrang hs own obseness 7 The fac ha he conanmen s src was shown n [] 8 See also [2] for relaed neqales 9 From [, p 5]: We wsh o pon o also ha sng he Marngale echnqes of [9] a proof of Theorem 2 can be obaned qe drecly and who dmensonal consans We hope o follow p hs sggeson elsewhere I has he drawback of conanng consans ha depend on he dmenson

4 494 Maro Mlman L(p, spaces, p (, ], n he same fashon, namely f # L(p, = sp {(f (s f (ss /p } < s Ths characerzaon, whle exremely sefl n many problems (cf [4], [7] s no always easy o mplemen, and does no reflec mmedaely he self mprovemen of he Garsa Rodemch consrcon In hs drecon, we fond a dfferen characerzaon of L(,, whch gves an mplc dfferenal neqaly reflecng he exponenal decay of he dsrbon fncon of elemens of L(,, va he se of dsrbon fncons (cf Secon 3 below Theorem 3 Le(Ω,µ be a measre space 2 Then,f L(, :=L(, (Ω f and only f here exss C > sch ha for all >, (5 λ f (sds Cλ f (, and f ## L(, := nf{c: sch ha (5 holds} = f L(, Ths characerzaon gves mmedaely he exponenal negrably of fncons n L(,, va he mplc dfferenal neqaly (5 In fac, s also welcome ha here s a smlar characerzaon for all L(p, spaces, < p < (6 Theorem 4 Le < p < Le (Ω, µ be a measre space Then, L(p, := L(p, (Ω = {f L loc (Ω : f L(p, = sp {f (ss /p } < } s = {f L loc (Ω : f L(p, = sp {f (ss /p } < }, s concdes wh he se of all f sch ha f ( =, and f # L(p, = sp {(f (s f (ss /p } <, s> whch n rn concdes wh he se of all f sch ha f ( =, and (7 f ## L(p, = sp { > (λ f ( /p λ f (sds} < If one combnes (7 wh he sal defnon of he spaces L(p, (cf (6, one readly obans a known characerzaon of he L(p, spaces whch was apparenly frs gven by O Nel [9] Corollary Le < p <, hen (8 f L(p, nf{c/p : λ f (sds C p } Remark 2 One dfference beween (7 and (8 s gven by he fac ha he former also works n he case p = Boh formlaons can be exended o more general Marcnkewcz spaces, M φ, where φ s a concave fncon In parclar, we refer o [9] for he correspondng heory of generalzed Marcnkewcz spaces M φ defned va (8 Noe however ha (f ( f ( = d d ( f ( 2 When dealng wh nfne measre spaces we shall always assme ha all fncons f consdered are sch ha her dsrbon fncons λ f ( are no eqal o, for all >

5 The scale of John Nrenberg self mprovng neqales 495 In hs expansve work [9], O Nel sed he formlae (8 o sdy ensor prodcs of L(p,q spaces (cf also [2] and [3] The space L(, was nrodced laer (cf [3], and conseqenly was no consdered n [9] In he las secon of hs paper we gve an applcaon of (5 o show ha (cf Theorem 6 n Secon 4 below (9 L(, (Ω L (Ω 2 L(, (Ω Ω 2 Whle we hnk ha (9 cold be sefl n esablshng oher embeddngs of ensor prodcs nvolvng L(,, sch an nderakng falls osde he scope of hs noe In conclson, we shold menon ha hs paper s par of seres of papers by he ahor on BMO, self mprovemen and nerpolaon, ha go back a leas o [4], [5], [6], wh he mos recen ops beng [7], o whch we refer for backgrond nformaon and frher references 2 John Nrenberg spaces and Garsa Rodemch spaces I s easy o see he connecon of he John Nrenberg spaces wh BMO Fx a cbe Q R n and le { } f BMO(Q = sp f f Q dx: Q sbcbe of Q Q Then, for p <, Q JN p (f,q f BMO(Q Q /p Indeed, f {Q } N P(Q, hen we clearly have { ( } p /p { } ( /p p Q f f Q dx Q f Q BMO(Q Q The prpose of hs secon s o prove he followng f BMO(Q Q /p Theorem 5 Le < p <, and le Q R n be a fxed cbe Then ( JN p (Q GaRo p (Q, n fac (2 GaRo p (f,q 2JN p (f,q ( GaRo p (Q = L(p, (Q, n fac we have Q GaRo p (f,q 2p p f L(p, and sp /p (f ( f ( 2 n/p + GaRo p (f,q + ( /p 4 f Q L Proof ( Sppose ha {Q } N P(Q Then for all Q, N, we have, f(x f(y dxdy f(x f Q dxdy + f Q f(y dxdy Q Q Q Q Q = 2 Q f f Q dx Q

6 496 Maro Mlman Therefore, f(x f(y dxdy 2 f f Q dx Q Q Q Q = 2 Q /p Q /p f f Q dx Q Q ( /p { ( } p /p 2 Q Q f f Q dx, Q Q and (2 follows ( We show frs ha L(p, (Q GaRo p (Q Le {Q } N P(Q, hen Q Q f(x f(y dxdy ( f(x + f(y dxdy Q Q Q Q Q 2 f(x dx 2 f (d Q 2 f L(p, Q = 2p p f L(p, /p d ( Q /p Conseqenly, GaRo p (f,q 2p p f L(p, To show he remanng nclson, GaRo p (Q L(p, (Q, we arge as n [4, Chaper 5] We provde all he deals for he sake of compleeness To show ha a fncon f belongs o L(p, (Q s eqvalen o show ha f L(p, (Q, herefore, snce GaRo p ( f,q GaRo p (f,q, o show ha f GaRo p (Q belongs o L(p, (Q, we can assme who loss ha f Le f GaRo p (Q, f Fx >, sch ha < Q /4, and le E = {x Q : f(x > f (} By defnon, E < Q /4, conseqenly, we can fnd a relavely open sbse of Q, Ω, say, sch ha E Ω and Ω 2 Q /2 By [4, Lemma 72, p 377] we can fnd a seqence of cbes {Q } N, wh parwse dsjon nerors, sch ha ( Ω Q 2 Q Ω c Q, =,2, ( Ω Q Q, N ( Ω N Q 2 n+ Ω

7 The scale of John Nrenberg self mprovng neqales 497 Now, o esmae /p (f ( f (, wll be more convenen, by homogeney, o consder (f ( f ( frs Then, we have (f ( f ( = E{f(x f (}dx {f(x f (}dx N E Q = ( {f(x f Q }dx+ E Q {f Q f (} N E Q ( {f(x f Q }dx+ E Q {f Q f (} N Q = (I+(II Le J = {: f Q > f (}, hen (II = N E Q {f Q f (} J E Q {f Q f (} Lkewse, Ω Q {f Q f (} Ω c Q {f Q f (} J J = {f Q f (}dx {f Q f(x}dx (snce Ω c E c J Ω c Q J Ω c Q f Q f(x dx f(y f(x dxdy J Q Q J Q Q ( /p GaRo p (f,q Q (I = N N N {f(x f Q }dx = (f(x f(ydxdy Q Q N Q Q ( f(x f(y dxdy GaRo p (f,q Q Q Q Q Combnng he neqales we have obaned, ( (f ( f ( 2GaRo p (f,q Q Therefore, N /p N 2GaRo p (f,q (2 n+ /p 2 /p /p sp /p (f ( f ( 2 n/p + GaRo p (f,q Q /4 To deal wh > Q /4, we noe ha (f ( f ( = f ( λ f (sds λ f (sds = f L ; /p

8 498 Maro Mlman herefore, /p (f ( f ( /p f L ( /p 4 f Q L Ths, ( /p 4 sp /p (f ( f ( 2 n/p + GaRo p (f,q + f Q L, and he desred resl follows by Theorem 4 3 Anoher characerzaon of he L(p, spaces, < p The prpose of hs secon s o gve a proof of Theorem 3 and Theorem 4 We sar wh he former Proof Le f be sch ha here exss C > sch ha (5 holds for all > Then, we have Ths, (f ( f ( = f ( λ f (sds Cλ f (f ( C f L(, nf{c: (5 holds} = f ## L(, Conversely, sppose ha f L(, Then, for all >, we have, Therefore, f ( λ f (sds = (f ( f ( f L(, f (λ f ( Now, snce f (λ f (, we have Conseqenly, concldng he proof λ f (sds λ f ( f L(, λ f (sds λ f ( f L(, f ## L(, f L(,, We proceed wh he proof of Theorem 4 Proof We rvally have f # L(p, f L(p, Moreover, f f ( =, hen /p f ( /p = (f (s f (s ds s = /p Conseqenly, /p f # L(p, s /pds s = p f # L(p, f L(p, p f # L(p, (f (s f (ss /p s /pds s The las par of he resl follows exacly as he proof of Theorem 3 (he casep = For example, from λ f (sds f ## L(p, (λ f( /p

9 The scale of John Nrenberg self mprovng neqales 499 we ge and herefore Conversely, for all >, Ths, (f ( f ( = f ( λ f (sds f ## L(p, /p f # L(p, f ## L(p, f # L(p, /p (f ( f ( /p λ f ( f # L(p, (λ f( /p and he desred resl follows f (λ f ( f ( λ f (sds /p λ f (sds (λ f ( λ f (sds, Remark 3 Observe ha when p =, he prevos consderaons provde a characerzaon of L, no of L(, Indeed, he correspondng resl for p = s sp f ( = sp f (sds = f L = sp In oher words, L s characerzed by he condon sp > Noe ha n hs case (λ f ( / = λ f (sds C To conclde hs secon we prove Corollary Lemma Le < p < Then Proof Noe ha f ## L(p, nf{c/p : λ f (sds C p } λ f (sds (3 (λ f ( /p f L(p, λ f ( f p L(p, p Sppose ha f L(p, By he prevos Theorem, Conversely, sppose ha Then, snce λ f decreases, λ f ( /p 2 /2 λ f (sds f ## L(p, (λ f( /p λ f (s /p ds = f ## L(p, f p( /p L(p, p( /p (by (3 C f p L(p, p /2 λ f (sds C p λ f (sλ f (s /p ds λ f (/2 /p /p λ f (/2 λ f (sds λ f (/2 /p C2 p p /2 /2 λ f (sds

10 5 Maro Mlman Therefore, and, conseqenly, The desred resl follows λ f (/2 /p+ λ f ( /p C p λ f ( /p+ λ f ( /p C p 4 Fnal remarks 4 Tensor prodcs wh L(, The new formla we presened for he compaon of he norm L(, has several applcaons Here, followng O Nel [9] (cf also [2], [3], we shall brefly consder ensor prodcs wh L(, I s no or prpose o develop he mos general resls, b o gve a flavor of he deas nvolved Theorem 6 Le (Ω,µ,(Ω 2,µ 2, be measre spaces Then, (4 L(, (Ω L (Ω 2 L(, (Ω Ω 2, wh f g L(, (Ω Ω 2 f L(, (Ω g L (Ω 2 Proof Le f L(, (Ω,g L (Ω 2 The dsrbon fncon of f g s comped n [9, Lemma 7 (2, p 97] ( z (42 λ f g (z = λ f d( λ g (, z > Therefore, on accon ha g L, we have (43 λ f g (z = g L (Ω2 Then, by Tonnell s heorem, for all >, λ f g (zdz = Hence, by Theorem 3, = = g L (Ω2 g L (Ω2 g L (Ω2 λ f ( z g L (Ω 2 f ## L(, (Ω d( λ g ( ( z λ f d( λ g (dz ( z λ f dzd( λ g ( λ f (rdrd( λ g ( g L (Ω2 λ f ( = g L (Ω 2 f ## L(, (Ω λ f g( (by (43 f g ## L(, (Ω Ω 2 f ## L(, (Ω g L (Ω 2 d( λ g ( 42 More problems I seems o s ha or approach o prove he second par of Theorem 5 can be modfed o sdy he rearrangemen neqaly of Garsa Rodemch [, Theorem 73] n he n-dmensonal case 2 I wold be of neres o follow p on he sggeson of Garsa Rodemch and prove a verson of Theorem 5 sng he mehods of [9] 3 I wold be of neres o complee he sdy of ensor prodcs wh L(,

11 The scale of John Nrenberg self mprovng neqales 5 Acknowledgemen I am graefl o Professor Mchael Cwkel for a nmber of sggesons ha allowed me o clarfy and mprove he presenaon of he paper References [] Aalo, D, L Berkovs, OE Kansanen, and H Ye: John Nrenberg lemmas for a doblng measre - Sda Mah 24, 2, 2 37 [2] Asashkn, S V: Tensor prodc n symmerc fncon spaces - Collec Mah 48, 997, [3] Benne, C, R DeVore, and R Sharpley: Weak-L and BMO - Ann of Mah (2 3, 98, 6 6 [4] Benne, C, and R Sharpley: Inerpolaon of operaors - Academc Press, 988 [5] Calderón, AP: Spaces beween L and L and he heorem of Marcnkewcz - Sda Mah 26, 966, [6] Cwkel, M, and P Nlsson: Inerpolaon of Marcnkewcz spaces - Mah Scand 56, 985, [7] Cwkel, M, Y Sagher, and P Shvarsman: A new look a he John Nrenberg and John Srömberg heorems for BMO - J Fnc Anal 263, 22, [8] Dyson, FD: Some gesses n he heory of parons - Ereka 8, 944, 5 [9] Garsa, A M: Marngale neqales: Semnar noes on recen progress - Mahemacs Lecre Noes Seres, W A Benjamn, Inc, Readng, Mass-London-Amserdam, 973 [] Garsa, A M, and E Rodemch: Monooncy of ceran fnconal nder rearrangemens - Ann Ins Forer (Grenoble 24, 974, 67 6 [] John, F, and L Nrenberg: On fncons of bonded mean oscllaon - Comm Pre Appl Mah 4, 96, [2] Marn, J, and M Mlman: Fraconal Sobolev neqales: symmerzaon, sopermery and nerpolaon - Asérsqe 366, 24 [3] Mlman, M: Some new fncon spaces and her ensor prodcs - Bll As Mah Soc 9, 978, [4] Mlman, M: Rearrangemens of BMO fncons and nerpolaon - In: Lecre Noes n Mah 7, 984, [5] Mlman, M: A noe on Gehrng s lemma - Ann Acad Sc Fenn Mah 2, 996, [6] Mlman, M: A noe on nerpolaon and hgher negrably - Ann Acad Sc Fenn Mah 23, 998, 69 8 [7] Mlman, M: BMO: Oscllaons, self-mprovemen, Gaglardo coordnae spaces and reverse Hardy neqales - In: Harmonc Analyss, Paral Dfferenal Eqaons, Banach Spaces, and Operaor Theory Celebrang Cora Sadosky s Lfe Volme (eded by S Marcanognn, M C Pereyra, A Sokolos and W Urbna, AWM-Sprnger Seres (o appear [8] Oklander, E T: Inerpolacón, espacos de Lorenz y eorema de Marcnkewcz - Crsos y Semnaros de Maemácas 2, Unv Benos Ares, 965 [9] O Nel, R: Inegral ransforms and ensor prodcs on Orlcz spaces and L(p,q spaces - J Anal Mah 2, 968, [2] Psylnk, E: On some properes of generalzed Marcnkewcz spaces - Sda Mah 44, 2, [2] Torchnsky, A: Real varable mehods n harmonc analyss - Academc Press, 986 Receved 2 Ags 25 Acceped 9 Ocober 25