Endpoint estimates from restricted rearrangement inequalities

Transcription

1 Rev. Ma. Iberoamericana 2 24, 3 5 Endpoin esimaes from resriced rearrangemen inequaliies María J. Carro and Joaquim Marín Absrac Le T be a sublinear operaor such ha Tf h, f for some posiive funcion h, s and every funcion f such ha f. Then, we show ha T can be exended coninuously from a logarihmic ype space ino a weighed weak Lorenz space. This ype of resul is conneced wih he heory of resriced weak ype exrapolaion and exends a recen resul of Arias-de-Reyna concerning he poinwise convergence of Fourier series o a much more general conex.. Inroducion Le S be he Carleson maximal operaor see [6] Sfx =sup S n fx, n where S n fx =D n fx, being D n he Dirichle kernel on T = {z C; z =} and f L T. Then, i was proved in [6] and [3] he following resriced weak-ype esimae: sup y> yλ SχA y /p C p A /p, for every <p 2 and every measurable se A, wihc independen of p, where λ g y = {x T : gx >y} is he disribuion funcion of g wih respec o he Lebesgue measure. Using his esimae and Yano s exrapolaion heorem see [22] and [7], one can easily see ha S : Llog L 2 L is bounded. 2 Mahemaics Subjec Classificaion: 46M35, 47A3. Keywords: rearrangemen inequaliy, real inerpolaion, Banach couples, exrapolaion heory, Carleson s operaor.

2 32 M. J. Carro and J. Marín However, in [8], his boundedness was improved by using ha if we ake he infimum in p in he above inequaliy hen, for every measurable se E T,. Sχ E E +log +, E g =inf{s : λ g s } is he decreasing rearrangemen and, proving ha his esimaes also holds by he so called special funcions. Then, he boundedness of S : L log L log log L L, was proved. Some years laer, F. Soria in [2] improves he above exrapolaion resul by showing ha S : B ϕ L, is bounded wih ϕ = + log + / andb ϕ a block ype space such ha L log L log log L B ϕ. In 996, Anonov see [] proved he following lemma: Lemma. Anonov LeS N fx =sup n N S n fx. Then, for every ε>, every N N and every fx, here exiss a measurable se F such ha F = f and S N f χ F ε. Using his resul and he above esimae on characerisic funcions one can conclude ha.2 Sf f +log +, f for every f L such ha f, and from i, Anonov proves ha S : L log L log log log L L, is bounded. Quie recenly, i has been proved by Arias-de-Reyna in [2] ha S : QA L, is bounded where QA is a rearrangemen invarian space such ha L log L log log log L QA and B ϕ QA. Moreover, QA is sricly bigger han boh spaces and herefore QA is, up o now, he bigges space where he poinwise convergence of he Fourier series is known o hold.

3 Endpoin esimaes from resriced rearrangemen inequaliies 33 Anonov s lemma has been exended in [9] o more general operaors, namely o any maximal operaor of he form Tfx =sup K j fx, j where K j L, and herefore,.2 holds for any operaor T of he above form such ha T saisfies.. Examples of such operaors are given in [9] in he seing of differeniaion of inegrals and he Halo conjecure. In paricular, and his is he connecion wih he weak exrapolaion heory, see [4] and [2] if T is an operaor such ha, for every <p 2, Tf /p p f m p, hen, for every f L such ha f, Tf f /p, p m /p and aking he infimum in p, we conclude ha.3 Tf f m +log +. f Our main purpose see Theorem 3. is o show ha if T is a sublinear operaor saisfying Tf h, f, for some posiive funcion h and every f, hen T : Q D MR is bounded, where h, s DsR, Q D = {f; f = k e k f k, f k, f QD < }, wih { f QD =inf and k e k D f k +log ; a k k f MR := sup > f R. a k =,a k,f= k e k f k }, In paricular, if Ds =s +log + s and T = S, hen our space Q D coincides wih he space of Arias-de Reyna QA.

4 34 M. J. Carro and J. Marín Our proof urns ou o be very simple and is based in he following basic resul see [9]: Lemma.2 Basic resul Le f = n f n and le c n > be such ha n c n =. Then f 3 fn+ f nsds. n c n From i, he main resul of his paper, which covers as a paricular case he resul of Arias-de-Reyna, can be immediaely obained. The poin now is ha he space Q D is difficul o handle and, herefore, i is convenien for he applicaions o find spaces of Logarihmic ype L such ha L Q D. As was menioned before, i was proved, in [2], ha he space L log L log log log LT QA. We shall exend his resul o our general conex. Anoher siuaion we consider in his work is he following: Le Ω be any domain in R n,lew,p Ω be he classical Sobolev space and se W,p Ω he closure of C Ω in W,p Ω, under he norm f W,p Ω = f p + f p, where f is he gradien of f. LeT be a sublinear operaor such ha T : W,p Ω L p, is bounded wih consan C p for every p I [,. Then, for every f such ha f + f, i holds ha /p /p. Tf /p C p f p + f p d Cp f + f d Consequenly, Tf inf p I f W, C p Ω /p := h, f W, Ω. Then we show ha he echnique developed in Secion 2 can also be exended o cover his siuaion and, in fac, our heory can be presened in he seing of compaible pairs of Banach spaces Ā =A,A using some of he ideas developed in [8]; ha is, our operaor T will be a sublinear operaor acing on elemens of he sum space A +A and aking values on he se of measurable funcions: T : A + A L µ. Our firs ask is o exend he noion of characerisic funcions o he seing of pairs. This will be done in Secion 3.

5 Endpoin esimaes from resriced rearrangemen inequaliies 35 As usual, he symbol f g will indicae he exisence of a universal posiive consan C independen of all parameers involved so ha /Cf g Cf, while he symbol f g means ha f Cg. M,µ will be a oally σ finie resonan measure space and we shall denoe by L µ he class of measurable funcions ha are finie µ a.e., endowed wih he opology of he convergence in measure. We wrie g p o denoe g L p µ, λ µ g y =µ {x M: gx >y} is he disribuion funcion of g wih respec o he measure µ and gµ = inf { s : λ µ g s } is he decreasing rearrangemen we refer he reader o [3] for furher informaion abou disribuion funcions and decreasing rearrangemens. In wha follows we shall omi he indices µ whenever i is clear he measureweareworkingwih. 2. Main resuls Firs of all, given a posiive concave funcion D such ha D+ =, we define he space { } ΛD = f; f ΛD = Dλ f y dy = f sdds. Then, we have ha he following properies holds: Lemma 2. Given a posiive concave funcion D such ha D+ =, we have ha ΛD L + L, and Q D ΛD, wih coninuous embeddings. Proof: The firs embedding is well known, since min,s Ds and hence f L +L = f sds = minλ f y, dy Dλ f y dy = f ΛD. For he second embedding, le us observe ha if f, hen f ΛD = Dλ f ydy D λ f ydy = D f, and hence, if f = k e kf k,wih f k, we obain ha f ΛD k e k f k ΛD k e k D f k f QD.

6 36 M. J. Carro and J. Marín Now we are ready o formulae our firs main resul: Theorem 2. Le T be a sublinear operaor such ha T : L µ+l µ L µ is bounded, and le us assume ha, for every f L L wih f, Tf h, f, for some posiive funcion h :,,, such ha for every >, he funcion h, is increasing and, for every s>, h, s is also an increasing funcion in he variable. Then,ifD and R are such ha h, s DsR, and D is a concave funcion saisfying D+ =, we have ha is bounded. T : Q D MR Alhough no condiions are assumed on R, i is clear ha since h, s is increasing in he variable, we can assume wihou loss of generaliy ha his condiion also holds for R. Proof: Le f Q D and le us wrie f = k e kf k wih f k. Then, by he previous lemma, we have ha he convergence of he series is in L +L and herefore, we can conclude ha. Tf e k Tf k Using now he basic resul ogeher wih he hypohesis, we obain ha, for every sequence a k of posiive numbers such ha k a k =, Tf 3 k k k e k Tf k + e k Tf k u du k k a k e k h, f k + e k hu, f k du. And, using he properies of he funcion h, we conclude ha a k Tf 3 k e k D f k R3+R3 k e k D f k log a k, and hence, Tf MR =sup Tf R f QD.

7 Endpoin esimaes from resriced rearrangemen inequaliies 37 As was menioned in he inroducion, he poin now is o analyze he space Q D o make i useful for he applicaions. To his end, we have o inroduce he following logarihmic spaces: Definiion 2. Le ϕ be a posiive and concave funcion such ha ϕ + =. The space L log log L ϕ is defined as he se of measurable funcions f such ha 2. f L log log L ϕ := f s+log log s + e dϕs <. 2 The space L log log Lϕ is defined as he se of measurable funcions f such ha 2.2 f L log log Lϕ := f s +log + log + dϕs <. s 3 The space L log log log Lϕ is defined as he se of measurable funcions f such ha 2.3 f L log log log Lϕ := f s +log + log + log + dϕs <. s We also need he wo following echnical lemmas: Lemma 2.2 Le Φs =s + log + and le f be such ha f s Λϕ =. Then Φf sϕs dϕs ϕs Φsϕ λ f s ds s f L log log L ϕ. Proof: To show he firs equivalence, le H = f ϕ s. Then one has ha λ H s =ϕλ f s and, by Proposiion 4.3 of [2], we have ha Φsλ H s ds s ΦsHs ds s. A simple change of variable ends he proof of he firs par. For he second par, le us consider he ses { E = s<:ϕsf s > log 2 }, s + e and E = { s :ϕsf s > log s + e 2}.

8 38 M. J. Carro and J. Marín Then, we can wrie Φ f sϕs dϕs ϕs = = Now, E + I =,\E + E f s f s E +, \E +log + +2log Φf sϕs dϕs ϕs = I + I 2 + I 3 + I 4. dϕs f sϕs log s + e dϕs 2 f L log log L ϕ. On he oher hand, since Φ is increasing, dϕs ϕs/sds and = f Λϕ f L log log L ϕ, we obain ha I 2 +2log log s dϕs +2log log log 2 ϕs s s log + e 2 ds s s f L log log L ϕ. Similarly, and I 3 I 4 f s + 2 log log s + e dϕs f L log log L ϕ, + 2 log log s + e s log s + e 2 ds f L log log L ϕ. Lemma 2.3 [] Le w be a posiive and measurable funcion and le ϕ be a posiive and concave funcion such ha ϕ + =.Then f s ϕ λ f s wsds = wd dϕs. Theorem 2.2 Le D be any posiive and concave funcion such ha D + =. Then, L log log L D Q D. 2 If s Ds, hen L log log LD Q D. 3 If s Ds and, for every s, Ds 2 sds, hen L log log log LD Q D.

9 Endpoin esimaes from resriced rearrangemen inequaliies 39 Proof: Le f L log log L D be such ha f ΛD =andleus wrie f = 2 i+ f i, i Z where f i = fχ 2 i+ {2 i < f 2 i+ }. Then, for every sequence of posiive number a i i such ha i Z a i =,wehaveha f QD 2 i D f i +log ai 2 i Dλ f 2 i +log ai. i Z i Z Taking now a i = 2i Dλ f 2 i i 2i D λ f 2 i, we conclude ha f QD Dλ f s +log ds, sdλ f s and he resul now follows by Lemma Since s Ds wehavehallog log LD ΛD L. Le f L log log LD be such ha f ΛD =, and decompose f as f = fχ { f } + 2 i+ f i, where f i = fχ 2 i+ {2 i < f 2 i+ }. Then, for every a i i such ha i a i =, f QD D f + 2 i Dλ f 2 i +log ai, i i and aking a i i as in, we ge f QD + Dλ f s To esimae I, i follows, by Lemma 2.2, ha I f s +log {f } +log ds +I. sdλ f s f sds dds, and since sλ f s, we ge ha λ f s ifs. Hence, {f } [, ] and using he same argumen han in he proof of Lemma 2.2, i follows ha I f s +log dds f f sds L log log LD.

10 4 M. J. Carro and J. Marín 3 In his case, we ake f L log log log LD such ha f ΛD =andwe wrie f = fχ { f 2} + 2 2i+ f i, where Then, if i a i =, f QD + + and since D is concave, f QD + i= i= i= i= f i = 2 2i+ fχ {2 2 i < f 2 2i+ }. 2 2i+ D f i +log ai 2 i+ 2 2i+ D 2 j λ f 2 j +log ai, 2 2i+ j=2 i 2i+ 2 2i+ j=2 i D 2 j λ f 2 j +log ai. 2 2i+ Now, using Ds/s decreases, and ha 2 i j<2 i+, we obain ha 2 j 2 2i+ D λ f 2 j 2 j 2 λf 2 j D. 2 2i+ 2 j Now we ake a i =6/π 2 i + 2, and hence f QD + i= 2 i+ j=2 i 2 i+ 2 j 2 λf 2 j D 2 j i= j=2 i λf s sd s Using ha sλ f s, we ge ha and since sd/s increases λf s sd s 2 j 2 λf 2 j D + logi + 2 j + log + log + log + 2 j + log + log + log + s ds. s λ f s λ f s 2 λ f s D λ f s 2.

11 Endpoin esimaes from resriced rearrangemen inequaliies 4 Moreover, since λ f s sλ f s, if s andds sds, λ f s D λ f s 2 D λ f s. Using his esimae and Lemma 2.3 we ge I = D λ f s +log + log + log + s ds f s +log + log + log + d dds. Now, since +log + log + log + is increasing and sf s I f s +log + log + log + f s dds f s +log + log + log + dds = f L log log log LD. s Le us now define he space G = {f; f G < }, where { c k+ } c f G =inf c k+ c k D k λ f ydy logk +2<, c k+ c k k= where he infimum exends over all sequences increasing c k k such ha c = and lim k c k =. Proposiion 2. I holds ha G Q D. Proof: Le f G and le c k k be a sequence such ha ck+ c c k+ c k D k λ f ydy logk +2<. c k+ c k k= Then we wrie f = c k+ c k f k +min f,c, k where f k = min f,c k+ min f,c k, c k+ c k and since ck+ c f k = k λ f ydy c k+ c k we conclude ha f Q D and f QD f G.

12 42 M. J. Carro and J. Marín Remark 2. By aking c k =2 2k, one can easily see ha under he condiions of Theorem 2.2, 3, we have ha L log log log LD G. If T is he Carleson maximal operaor S, hen one can immediaely see ha we can ake Ds =s +log + and R =. In his paricular case, he s above resul has been recenly obained by Arias-de-Reyna in [2]. Also, for such funcion D, iisveryeasyoseeha and he boundedness L log log log LD =L log L log log log LT, S : L log L log log log LT L,, was obained previously by Anonov in [], and for oher more general operaors, as menioned in he inroducion, in [9]. 3. Exension o arbirary compaible pairs Le Ā = A,A beacompaible pair of Banach spaces, ha is, we assume ha here is a opological vecor space U such ha A i U,i=,, coninuously. In wha follows we drop he erms compaible and Banach and refer o a compaible Banach pair simply as a pair. The Peere K funcional see [3], [4] and [5] associaed wih a pair Ā is defined, for each a A + A and >, by Ka, =Ka, ; Ā =inf{ a A + a A : a = a + a,a i A i }. I is easy o see ha K, a is a nonnegaive and concave funcion of >, and hus also coninuous. Therefore Ka, ; Ā =Ka, + ; Ā+ ka, s; Ā ds, where he k funcional, ka, s; Ā =ka, s, is a uniquely defined, nonnegaive, decreasing and righ-coninuous funcion of s>. In order o find he analogue of he se {f L ; f } in he seing of pairs, le us recall ha he Gagliardo compleion à and à of a pair Ā is defined by see [3] a à = supk, a; Ā <, a à = sup K, a; Ā <.

13 Endpoin esimaes from resriced rearrangemen inequaliies 43 Definiion 3. Given a pair Ā, we say ha a is a characerisic elemen of Ā if a à à and a Ã. The collecion of characerisic elemens of a pair Ā will be denoed by CĀ. The following lemma was proved in [8] and i is fundamenal for our purpose. Lemma 3. Given an elemen a A +A such ha Ka, + ; Ā =,here exis a consan γ depending only on Ā and a collecion of characerisic elemens a i i Z such ha a = γ i Z 2 i a i convergence in A + A, and a i à λ ka, 2 i. We say ha a = γ i= 2i a i is a dyadic decomposiion of a. Definiion 3.2 [8]GivenapairĀ =A,A and a concave funcion ϕ, he minimal Lorenz space, Λϕ; Ā, isheseofelemensa A + A such ha Ka, + ; Ā =and a Λϕ;Ā = ka, s; Ā dϕs <. If Ā is he classical pair L ν,l ν, hen ka, s =f s and hence Λϕ; Ā =Λϕ is he classical Lorenz spaces defined in he previous secion. Definiion 3.3 Given a pair Ā, and a quasi-banach laice B Λϕ, we define Bϕ; Ā as { } 3. Bϕ; Ā = a Λϕ; Ā; a Bϕ;Ā := ka, B <. Remark 3. Obviously, L log log L ϕ; Ā L log log Lϕ; Ā L log log log Lϕ; Ā, and he above embeddings are, in general, sric. However, if Ā is an ordered pair, ha is A A henka, =if >, and hence L log log L ϕ; Ā = L log log Lϕ; Ā.

14 44 M. J. Carro and J. Marín Definiion 3.4 Le h :,,, be such ha for every >, he funcion h, is increasing and, for every s>, h, s is also an increasing funcion in he variable. We say ha a sublinear coninuous operaor T : A + A L µ, saisfies a resriced h rearrangemen inequaliy if, for every >and every characerisic elemen a of Ā, 3.2 Ta h, a Ã. Examples: If Ā =L ν,l ν, hen CĀ ={f L ; f }, and hence, any sublinear operaor saisfying 3.2, saisfies he condiion assumed in he previous secion. 2 Le Ω be any domain in R n and le W,p Ω be he classical Sobolev space f W,p Ω = f p + f p, where f is he gradien of f. SeW,p Ω he closure of C Ω in W,p Ω., Then i is known, see [2], ha if Ā =W Ω,W, Ω, K, f; f Ā + f,, and herefore CĀ ={f W Ω; f + f }. Hence, if T is a sublinear operaor such ha T : W,p Ω L p, is bounded wih consan C p for every p I [,, hen, f Tf W, inf C p p I Ω /p := h, f W, Ω. 3 Le us now consider, for example, he pair Ā =Λ w,l, where Λ w is he weighed Lorenz space inroduced by Lorenz in [5] and defined by /p f Λ p w = f p w d <. Le us recall ha he weak ype version of hese spaces are defined by f Λ p, W =sup f W /p <. >

15 Endpoin esimaes from resriced rearrangemen inequaliies 45 Consider a sublinear operaor T such ha, for some weighs w and W, T :Λ p w Λ p, W, wih consan less han or equal o C p. Then, since i is known ha K, f; Ā = f w s ds, we can conclude ha CĀ ={ f Λ w : f }, and herefore, for every characerisic elemen, f Λ Tf inf C w /p p := h, f p W Λ w. Le us now define he space { Q D Ā = a = } e k a k ; a k Ã, a QD Ā <, k where a QD Ā { =inf k e k D a k à +log ck ; k Then, we have he following exension of Theorem 2.: c k =,c k,a= k e k a k }. Theorem 3. Le T : A + A L µ be a sublinear operaor saisfying a resriced h rearrangemen inequaliy. Then, if D and R are wo posiive funcions such ha D is concave, D + =and 3.3 h, s DsR, we have ha is bounded. T : Q D Ā MR Proof: Given a ΛD; Ā such ha a ΛD;Ā =, we can decompose a as in Lemma 3. a = γ 2 i a i. i Z Then, if a N herefore, = γ N i= N 2i a i,wehavehata N Ta in measure, and 3.4 Ta N Ta a.e. >.

16 46 M. J. Carro and J. Marín By he sublineariy of T we ge ha N Ta N γ 2 i Ta i γ 2 i Ta i, and hence i= N Ta γ i= i= 2 i Ta i a.e. >. The proof now follows as in Theorem 2.. We also have and analogue o Theorem 2.2: Theorem 3.2 Le D be any posiive and concave funcion D such ha D + =.Then, L log log L D; Ā QDĀ. 2 If s Ds, hen L log log LD; Ā QDĀ. 3 s Ds and, for every s, Ds 2 sds, hen L log log log LD; Ā QDĀ. Proof: In his case, given a ΛD; Ā such ha a ΛD;Ā =,we decompose a as in Lemma 3. a = γ 2 i a i, i Z and coninue as in he proof of Theorem 2.2,. 2 Since s Ds wehavehallog log LD; Ā ΛD; Ā Ã. Le a ΛD; Ā such ha a ΛD;Ā =, and decompose a as a = γ 2 i a i + 2 i a i = γ a + 2 i a i. i< i i Then since a CĀ, and a i à λ ka, 2 i, we have ha a QD Ā D a à + Dλ ka, s +log ds sdλ ka, s = I + I 2. Obviously I D a à D a ΛD;Ā =D D a L log log LD;Ā, and o esimae I 2, we follow as in he proof of Theorem 2.2, 2.

17 Endpoin esimaes from resriced rearrangemen inequaliies 47 3 In his case, given a L log log log LD; Ā such ha a ΛD;Ā =, le a = γ i Z 2i a i be a dyadic decomposiion. Then, if, for every k N, d k = 2 k+ i=2 2 i, we obain ha k 2k+ a = 2 i a i + d k 2 i a i = a + d k A k, d i= k k= i=2 k k= where, i is immediae o see ha boh a and A k are characerisic elemens. Then, for every k c k =, a QD Ā D a à + d k D A k à +log ck = D a à +I. k= Since a i CĀ andd is subaddiive, we have ha 2k+ 2 i 2 k+ d k D A k à d k a i à d k d k i=2 k D and he proof now follows as in Theorem 2.2, Applicaions i=2 k D 2 i λ ka, 2 i, d k Le T be a sublinear operaor saisfying a resriced h-rearrangemen inequaliy, where h, s = s +log + m s wih m>, as i happens wih he examples we have menioned in he inroducion. Then, h, s s +log + m s +log + m s s + log+ m, and we can ake Ds =s +log s + m and R = + log+ m in our Theorems 3. and 3.2 o conclude he following resul. Theorem 4. If T : A +A L µ saisfies a resriced h-rearrangemen inequaliy wih h, s = s +log + m s, T can be exended coninuously T : Q D Ā MR, where D = +log + m,andr = + log+ m. In paricular, T : L log log log LD; Ā MR is bounded, where L log log log LD; Ā ={a A + A ; ka, Llog L m log log log L}, wih a L log log log LD; Ā = ka, Llog L m log log log L.

18 48 M. J. Carro and J. Marín Examples I If Ā =L T,L T, kf, =f and we recover he resul of Secion 2., II If Ā =W Ω,W, Ω where Ω has finie measure, and T : W,p Ω L p, is bounded wih consan say /p, hen, applying Theorem 4., we obain ha T : W Ω L, is bounded, where W Ω is he closure of C Ω in W Ω wih W Ω = {f; f + f L log L log log log L}. III In all our previous applicaions we have considered sublinear operaors wih values in L p, where he consan blows up when p ends o. This was he unique ineresed case since if p end o p wih p,henwecan subsiue Tf by Tf wihou a change in he behaviour of he consan and hence we can apply he srong ype exrapolaion heory sudied in [7] and [8] insead of he heory developed in his work o obain beer resuls. However his is no he general case. Our hird applicaion deals wih he heory of weighed Lorenz spaces and wih a sublinear operaor T wih values in spaces of he form Λ p, W where f can no be, in general, subsiued by f even if p. Lew and W be weighs in, andlet be a sublinear operaor such ha is coninuous and, for every p>2, T :Λ w +L L R n T :Λ p w Λ p, W is bounded wih consan p see, [6], [7], [] o find examples of operaors T saisfying he above condiion; ha is /p Tf W /p p f sw s ds. Now, if we ake Ā =Λ w,l, we have ha à =Λ w and hence, i follows, aking he infimum in p>2, ha Tf hw, a Ã, where h, s =inf p>2 ps/ /p s/ /2 + log + s/. Therefore, we can deduced he following resul.

19 Endpoin esimaes from resriced rearrangemen inequaliies 49 Theorem 4.2 Le T be a sublinear operaor as above. exended coninuously Then, T can be T : L log log L D; Ā MR, where Ds =s /2 + log + s, andr =W /2 +log + W. Open Quesion: When is is rue ha he space L log log log log LD; A Q D Ā or in general, L logm LD; A Q D Ā? References [] Anonov, N. Y.: Convergence of Fourier Series. Eas J. Approx , no. 2, [2] Arias-de-Reyna, J.: Poinwise Convergence of Fourier Series. J. London Mah. Soc , no., [3] Benne, C. and Sharpley, R.: Inerpolaion of Operaors. Academic Press, Boson, 988. [4] Bergh, J. and Lofsröm, J.: Inerpolaion spaces. An inroducion. Springer, New York, 976. [5] Brudnyi, Yu. A. and Krugljak, N. Ya.: Inerpolaion Funcors and Inerpolaion Spaces. Norh-Holland, Amserdam, 99. [6] Carleson, L.: On convergence and growh of parial sums of Fourier series. Aca Mah , [7] Carro, M. J.: New exrapolaion esimaes. J. Func. Anal. 74 2, no., [8] Carro, M. J. and Marín, J.: An absrac exrapolaion heory for he real inerpolaion mehod. Collec. Mah , no. 2, [9] Carro, M. J. and Marín, J.: A useful esimae for he decreasing rearrangemen of a sum of funcions. To appear in Quar. J. Mah. [] Carro, M. J., Raposo, J. A. and Soria, J.: Recen developmens in he heory of Lorenz spaces and weighed inequaliies. Preprin 2. [] Carro, M. J. and Soria, J.: Weighed Lorenz spaces and he Hardy operaor. J. Func. Anal , [2] DeVore, R. and Scherer, K.: Inerpolaion of linear operaors on Sobolev spaces. Ann. of Mah , [3] Hun, R. A.: On he convergence of Fourier series. In Orhogonal Expansions and heir Coninuous Analogues Proc. Conf., Edwardsville, Ill., 967, Souhern Illinois Univ. Press, Carbondale, Ill., 967. [4] Jawerh, B. and Milman, M.: Exrapolaion Theory wih Applicaions. Mem. Amer. Mah. Soc , no. 44.

20 5 M. J. Carro and J. Marín [5] Lorenz, G. G.: On he heory of spaces Λ. Pacific J. Mah. 95, [6] Muckenhoup, B.: Weighed norm inequaliies for he Hardy-Lilewood maximal funcion. Trans. Amer. Mah. Soc , [7] Sawyer, E.: Boundedness of classical operaors on classical Lorenz spaces. Sudia Mah , [8] Sjölin, P.: An inequaliy of Paley and convergence a.e. of Walsh-Fourier series. Ark. Ma , [9] Sjölin, P. and Soria, F.: Remarks on a heorem by N. Yu. Anonov. Sudia Mah , no., [2] Soria, F.: On an exrapolaion heorem of Carleson-Sjölin wih applicaions o a.e. convergence of Fourier series. Sudia Mah , [2] Soria, F.: Characerizaions of classes of funcions generaed by blocks and associaed Hardy spaces. Indiana Univ. Mah. J , [22] Yano, S.: Noes on Fourier analysis. XXIX. An exrapolaion heorem. J.Mah.Soc.Japan3 95, Recibido: de abril de 22 María J. Carro Deparamen de Maemàica Aplicada i Anàlisi Universia de Barcelona E 87 Barcelona, Spain carro@ma.ub.es Joaquim Marín Deparamen de Maemàiques Universia Auònoma de Barcelona Bellaerra, E 893, Barcelona, Spain jmarin@ma.uab.es This work has been parially suppored by he CICYT BFM and by CURE 2SGR 69. The second auhor has also been suppored by Programa Ramón y Cajal, MCYT.