MULTILINEAR EXTRAPOLATION AND APPLICATIONS TO THE BILINEAR HILBERT TRANSFORM

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1 MULTILINEAR EXTRAPOLATION AND APPLICATIONS TO THE BILINEAR HILBERT TRANSFORM MARÍA JESÚS CARRO, LOUKAS GRAFAKOS, JOSÉ MARÍA MARTELL, AND FERNANDO SORIA Absrac. We presen wo exrapolaion mehods for muli-sublinear operaors ha allow us o derive esimaes for general funcions from he corresponding esimaes on characerisic funcions. Of hese mehods, he firs is applicable o general muli-sublinear operaors while he second reuires woring wih he so-called ε, δ)-aomic operaors. Among he applicaions, we discuss some new endpoin esimaes for he bilinear Hilber ransform.. Inroducion For a variey of imporan operaors in analysis, i is easier o derive a resriced ype esimae, ha is an esimae on characerisic funcions of measurable ses, han o derive an esimae for general funcions. I is herefore ineresing o as wha ind of esimaes can be obained from a nown resriced ype esimae. This is, for example, he case for he Carleson operaor [5] Sfx) = sup D n f)x), n where f L T) and D n is he Dirichle ernel on T = {z C; z = }, for which he following esimae is nown see []) Sχ E ) L, C D E ) wih D) = + log + ). Anoher example of his sor appears in he case of he bilinear Hilber ransform Hf, g)x) = π lim ε ε fx )gx + ) d, for which he following resriced ype ineualiy has been proved in [3] see also [4]) { x R : Hχ E, χ F )x) > λ } ) C + log + 4/3 /3, E F min E, F )) λ 2/3 λ Dae: January 27, 29. Revised: April 8, Mahemaics Subjec Classificaion. 47A3, 47A63, 42A99, 42B35. Key words and phrases. Mulilinear exrapolaion, resriced wea ype esimaes, ε, δ)-aomic operaors, galb, bilinear Hilber ransform. The firs auhor is suppored by MTM27-65 and by 25SGR556. The second auhor is suppored by he Naional Science Foundaion under gran DMS The hird and fourh auhors are suppored by MEC Gran MTM and by UAM-CM Gran CCG6- UAM/ESP-286. The hird auhor is also suppored by CSIC PIE 285I5.

2 2 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA for all λ >. Wriing he las expression in erms of he decreasing rearrangemen of Hχ E, χ F ), one can easily see ha Hχ E, χ F ) X CD E, F ),.) where Ds, ) = s mins, ) ) /2 +log + s mins,)) 2 and X is he wea ype weighed Lorenz space Λ 2/3, w) defined by f Λ 2/3, w) = sup W ) 3/2 f ), > wih W ) = ws)ds and, in his case, W ) = + log+ /) 4/3. In.), he variables can be separaed since for any α, β [, ] wih α + β =, we have Ds, ) D s) D 2 ) where D s) = s +α 2 + log + ) 2, D2 ) = +β 2 + log + ) 2..2) s The preceding wo examples provide he main moivaion o invesigae he boundedness properies of linear or mulilinear operaors for which resriced esimaes are nown. In he linear or sublinear case we assume ha T saisfies T χ E ) X CD E ).3) for any measurable se E, E <, where D is increasing wih D) = and X is a general uasi-banach laice space. Analogously, in he bilinear or bi-sublinear case, T may saisfy an esimae of he form T χ E, χ E2 ) X CD E, E 2 ),.4) where D is a funcion which is increasing in boh variables wih D, ) = D, ) =. The analysis of m-linear or m-sublinear operaors for m 3 presens no significan differences and hus for simpliciy in our exposiion we may focus on he case m = 2. In order o inroduce he differen approaches ha we sudy in he presen paper, we give an overview of he exising resuls in he linear or sublinear) case. This sudy is moivaed by he need o undersand he a.e. convergence of Fourier series and hus he boundedness of he Carleson operaor on spaces near L. When X = L, and D is a concave funcion i is shown in [4] ha if T saisfies.3) hen T maps BD see [4] for he precise definiion of his space) o L,. The proof of his exrapolaion resul is based on decomposing each funcion f ino simple funcions o which he iniial hypoheses are applied. Here i is worhwhile o poin ou ha as X is a uasi-banach space, addiional issues appear since one needs o conrol he uasi-norm of a linear combinaion of funcions. In his paricular case one has ha if {g j } j is a seuence of funcions wih g j L, hen for any {c j } j R L c j g j {cj } llog j, l)..5) j These ingrediens appear in he adapaion of his scheme o he m-linear seing. The mehod inroduced in [4] applied o he Carleson operaor S wih D) = + log + ), gives ha he Fourier series of each funcion in L log L) log log L)

3 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 3 converges a.e.; his follows from he corresponding resul in BD since he laer space conains he former, see [4] for he precise deails. Concerning he a.e. convergence of Fourier series and he boundedness of he Carleson operaor, a closer space o L was obained by Anonov in [], namely, L log L) log log log L). The ideas in [] have been exploied in [3] see also [9] for relaed resuls) o obain ha Anonov s resul is a paricular example of a general exrapolaion resul: he mehod developed in [4] can be improved when applied o maximal operaors T fx) = sup j K j fx) where K j L similar resuls are obained for variable ernels, see [3], [9]). A furher exension of hese echniues is inroduced in [6] and [7] where a more general class of operaors, called ε, δ)-aomic see he definiion below), is considered. I is shown in [6] ha if an ε, δ)-aomic operaor T saisfies.3), which is an esimae for characerisic funcions, hen he same esimae holds for any funcion f L wih f : T f X CD f )..6) This means ha aing.6) as he iniial assumpion noe ha his wih f = χ E is.3)), in place of decomposing f ino simple funcions as in [4], one can use more general bounded funcions. This was used in [6] o give anoher proof of Anonov s resul: The Carleson operaor is ε, δ)-aomic and.6) holds wih X = L, and D) = + log + ). The ey idea in [] relies on decomposing each funcion f according o he level ses {d < f d } wih d = 2 2. Again, o deal wih linear combinaions in X one uses.5). This allows one o obain an esimae from Llog L) log log log L) o L,. Le us observe ha having aen he more naural seuence d = 2 would have led us o he smaller space Llog L) log log L). Moivaed by he aforemenioned resuls, in he presen paper we exend he wo approaches oulined above o he case of m-linear or m-sublinear operaors. We firs exend he approach in [4]: for general operaors saisfying.4) we decompose he given funcions ino simple funcions; he need o conrol he uasi-norm of linear combinaions of simple funcions reuires a subsiue for.5). Noe ha in his case i is naural o consider arge spaces X ha are uasi-banach spaces below L his is he case for he bilinear Hilber ransform). We use he concep inroduced by Turpin [5] of he GalbX) of a uasi-banach space X defined as follows { GalbX) = c n ) n ; } c n f n X, whenever f n X, n endowed wih he norm c GalbX) = sup fn X n c nf X n. This Galb space was sudied in [7] for he case of he weighed Lorenz spaces X = Λ w), for < <, and also for he wea spaces X = Λ, w). Nex, we inroduce ε, δ)-aomic operaors in he muli-variable seing. For hese, esimaes for characerisic funcions of he form.4) can be exended o L funcions bounded by as in.6)). Thus we ae as iniial assumpion he more general esimae T f,..., f m ) X CD f,..., f m ), for all funcions f,..., f m ) L L wih f j for j =,..., m. By decomposing general funcions no only ino sums of simple funcions bu also

4 4 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA ino combinaions of bounded funcions, beer resuls can be obained following he ideas in [6] and [7]. Our main moivaion in he sudy of his problem is o obain esimaes for he bilinear Hilber ransform. For his purpose, i is naural o consider uasi-banach spaces wih GalbX) = l wih < < and funcions D, D 2 lie in.2). We denoe by L R n ) he class of Lebesgue measurable funcions ha are finie a.e. and by g ) = inf {s : µ g s) } he decreasing rearrangemen of g L, where µ g y) = {x R : gx) > y} is he disribuion funcion of g wih respec o he Lebesgue measure we refer he reader o [2] for furher informaion abou disribuion funcions and decreasing rearrangemens). For a measurable se E, χ E denoes is characerisic funcion and E is Lebesgue measure. For simpliciy of presenaion, we say ha an operaor T is sublinear if T λf) = λ T f and ) T f n T f n n N for all funcions f, f n and λ R. If we only have ha T f + f 2 ) T f + T f 2, hen o obain our conclusions we need o assume an addiional boundedness condiion on our operaor T such as n N T : L + L L, or assume some densiy propery of he spaces in uesion. For m-linear or m-sublinear operaors we sae and prove our resuls in he case m = 2, since he case wih more variables only presens rivial noaional changes. Given a funcion Ds, ), increasing in each variable wih D, ) = D, ) =, we wrie dd = dds, ) for he measure in [, ) 2 defined by dd [, a) [, b) ) = dds, ) = Da, b). [,a) [,b) Noe ha if D is smooh hen dd, s) = s D, s) d ds. 2. Decomposiions ino simple funcions and esimaes on Lorenz spaces Given an increasing funcion D such ha D) = and < <, he Lorenz space Λ dd) is given by f Λ dd) = ) f ) dd) λ D µ f λ) ) dλ λ I is nown ha his space is uasi-banach if and only if he funcion D saisfies he 2 -condiion; ha is D2 ) CD) for some consan C > and for every >, see [8]. 2.. Sublinear case. We sar wih he sublinear case which already conains many of he ideas ha will be used in he m-linear seing. We have he following resul. ).

5 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 5 Theorem 2.. Le T be a sublinear operaor, le X be a uasi-banach laice space and le D be an increasing funcion such ha D) =. Assume ha, for any measurable se E wih E <, we have Then, he following are valid: a) If GalbX) = l, hen T χ E ) X C D E ). 2.) T : Λ dd) X. b) If GalbX) = l p wih < p <, hen T : Λ p dd p ) X. c) If GalbX) = l log l) α wih α >, hen T : Λ αdd) X, where Λ αdd) is he subspace of Λ dd) defined by he funcional f Λ α dd) = λ D µ f λ) ) + log + f ) α Λ dd) λ D µ f λ) ) dλ λ λ D µf λ) ) ) dλ = f Λ dd) ϕ α f Λ dd) λ wih ϕ α ) = + log + /) α. We do no prove Theorem 2. here. In Theorem 2.6 below we obain similar resuls for bi-sublinear operaors and he argumens in ha proof can be easily adaped in he proof of Theorem 2.. Remar 2.2. We noice ha in a) and b) we do no lose informaion since we may recover he iniial assumpion by applying he obained esimae o characerisic funcions since χ E Λ p dd p ) = D E ). More precisely, given X such ha GalbX) = l p wih < p hen T χ E ) X D E ), E < T : Λ p dd p ) X. The same occurs in c) since and herefore χ E Λ α dd) = D E ) ) λ D E ) dλ ϕ α D E ) λ D E ), T χ E ) X D E ), E < T : Λ αdd) X. Remar 2.3. Le us observe ha Theorem 2. par a) wih D concave) is opimal in he sense ha one canno expec a space bigger han Λ dd) valid for every operaor T saisfying 2.): we ae X = Λ dd) which is a Banach space, T = Id and we observe ha T χ E ) X = D E ).

6 6 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA Example 2.4. Suppose ha X is any Banach space and hence GalbX) = l. If D) = / hen ΛdD) = L, and we have ha T χ E ) X E /, E < T : L, X. This is he bes esimae one can obain from he resriced ype assumpion on T. Naurally, his conclusion may no be opimal if T iself maps he bigger space L when ) ino X. The mehod of Theorem 2. does no use any specific propery of he operaor T. For insance, if we now ha T is a supremum of a seuence of linear operaors as in Moon s heorem [2]) or even more generally, ha T is aomic see he corresponding definiion in he nex secion), hen we are able o obain a beer conclusion. Le us examine a few more examples using he previous mehod. Example 2.5. In hese examples we se D) =. Le X = L, wih >, hence X is a Banach space. Then, we have for any sublinear operaor T, T χ E ) L, E, E < T : L L,. We noe ha his euivalence can be obained direcly by woring wih simple funcions. Le X = L, wih < <, hence GalbX) = l. In his case Λ dd ) = L, and, for any sublinear operaor T, T χ E ) L, E, E < T : L, L,. However, Moon s heorem [2] says ha under cerain condiions on T, one obains ha T maps L ino L, which is a sronger conclusion since L, L for < <. Le X = L,, hence GalbX) = l log l and ΛdD) = L. In his case we have Λ dd) = Bϕ see [4]) and hus T χ E ) L, E, E < T : Λ dd) L,. Ye anoher comparison wih Moon s heorem yields ha, under some condiions on T, i is bounded from L ino L, which is a sronger conclusion since f L = λ µ f λ) dλ ) λ λ µ f λ) + log + f L dλ λ µ f λ) λ = f Λ dd), and hence, Λ dd) L. To see ha his inclusion is proper we ae fx) = x log x log log x) 2 χ [e e, )x). We have ha f L bu one can easily see ha f Λ dd) =. Thus, Λ dd) is a proper subspace of L.

7 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM Bi-sublinear case. Before discussing a bi-sublinear exension of Theorem 2. we inroduce some noaion. Given a funcion of wo variables D such ha i is increasing in each variable and D, ) = D, ) =, le Λ p dd p ) be he funcion space given by f, f 2 ) Λ p dd p ) = f s ) p f2 s 2 ) p dd p s, s 2 ) s p s p 2 D µ f s ), µ f2 s 2 ) ) p ds ds 2 s s 2. Noice ha if Ds, s 2 ) = D s ) D 2 s 2 ) hen dd p s, s 2 ) = dd p s ) dd p 2s 2 ) and Λ p dd p ) = Λ p dd p ) Λ p dd p 2) since f, f 2 ) Λ p dd p ) = f Λ p dd p ) f 2 Λ p dd p 2 ). Analogously, we inroduce he funcion space Λ αdd) given by he funcional: f, f 2 ) Λ α dd) = = D µ f s ), µ f2 s 2 ) )[ ] f +log +, f 2 ) α Λ dd) s s 2 D µ f s ), µ f2 s 2 ) ) ds ds 2 s s 2 D µ f s ), µ f2 s 2 ) ) ) ds ds 2 = f, f 2 ) Λ dd) ϕ α, f, f 2 ) Λ dd) s s 2 wih ϕ α ) = + log + /) α. In his case, if Ds, s 2 ) = D s ) D 2 s 2 ) we have ha Λ αdd ) Λ αdd 2 ) Λ αdd) since f, f 2 ) Λ α dd) f Λ α dd ) f 2 Λ α dd 2 ). We now sae a bi-sublinear exension of Theorem 2.: Theorem 2.6. Le T be a bi-sublinear operaor and le X be a uasi-banach space. Le D be a wo-variable funcion increasing in each variable wih D, ) = D, ) =. Assume ha for all measurable ses E, E 2 wih E, E 2 <, we have T χ E, χ E2 ) X D E, E 2 ). 2.2) Then he following are valid: a) If GalbX) = l, hen T : Λ dd) X. b) If GalbX) = l p wih < p <, hen T : Λ p dd p ) X. c) If GalbX) = l log l) α wih α >, hen T : Λ αdd) X. As an immediae conseuence of his resul and he discussion above, in he paricular case Ds, ) = D s) D 2 ), we obain he following resul. Corollary 2.7. Le T be a bi-sublinear operaor and le X be a uasi-banach space. Le D, D 2 be increasing funcions ha vanish a he origin. Assume ha for all measurable ses E, E 2 wih E, E 2 < we have T χ E, χ E2 ) X D E ) D 2 E 2 ). 2.3) a) If GalbX) = l, hen T : Λ dd ) Λ dd 2 ) X. b) If GalbX) = l p wih < p <, hen T : Λ p dd p ) Λ p dd p 2) X.

8 8 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA c) If GalbX) = l log l) α wih α >, hen T : Λ αdd ) Λ αdd 2 ) X. We poin ou ha his corollary is also a conseuence of he corresponding onevariable resul: we freeze one variable in T and apply Theorem 2. o he resuling sublinear operaor, hen we freeze he oher variable and apply again Theorem 2.. Proof of Theorem 2.6. Assume wihou loss of generaliy ha f, g. By [4, Lemma 4], fx) = 2 χ E,j x) a.e.; gx) = 2 χ F,j x) a.e., Z j Z j where and E,j {x : fx) > 2 +j } = µ f 2 +j ), F,j {x : gx) > 2 +j } = µ g 2 +j ). Thus, using 2.2) we have T f, g) X 2 2 T χe,j, χ F,j ) X j,j, { 2 2 D µ f 2 j+ ), µ g 2 j + ) )} j,j,, GalbX). We sar wih a) and b) in which case GalbX) = l p wih < p. Then, T f, g) p X 2 p 2 p D µ f 2 j+ ), µ g 2 j + ) ) p, Z j,j j,j = j,j 2 j p 2 j p s p p D µ f s 2 j ), µ g 2 j ) ) p ds s s p p D µ f s), µ g ) ) p ds s d s p p D µ f s), µ g ) ) p ds s d d f, g) p Λ p dd p ). Le us esablish c) in which case GalbX) = l log l) α wih α >. Following [4], we wrie ϕ α ) = + log + /) α and we observe ha given a non-rivial seuence of non-negaive numbers a = {a } we have a l log l) α a + log a ) α l = Nα a). a We wrie F j = 2 j f, G j = 2 j g and β,,j,j = 2 2 Dµ f 2 j+ ), µ g 2 j + )) = 2 2 Dµ Fj 2 ), µ Gj 2 )). As in [4, p. 239] here, he compuaions are done wih α = bu he argumen adaps rivially o an arbirary α > ) we obain T f, g) X N α {β,,j,j },,j,j ) N α{ Nα {β,,j,j }, )} j,j ).

9 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 9 Then, for any j, j, β,,j,j = 2 2 Dµ Fj 2 ), µ Gj 2 )),, and F j, G j ) Λ dd) = 2 j 2 j f, g) Λ dd) N α {β,,j,j }, ), β,,j,j + log 2 j 2 j s D µ Fj s), µ Gj ) ) ds d s f, g) Λ dd) β,,j,j s D µ Fj s), µ Gj ) ) + log + 2 j 2 j f, g) Λ dd) s D µ Fj s), µ Gj ) ) = 2 j 2 j f, g) Λ α dd). ) α ) α ds d s Therefore, { } ) T f, g) X N α 2 j 2 j f, g) Λ α dd) j,j = f, g) Λ α dd) N α {2 j 2 j } j,j ) = C f, g) Λ α dd). Remar 2.8. As already observed in Remar 2.2, in Theorem 2.6 and hus in Corollary 2.7) we do no lose informaion and we recover he iniial assumpion by applying he obained esimae o characerisic funcions since χ E, χ F ) Λ p dd p ) = D E, F ) and χ E, χ F ) Λ α dd) = D E, F ) ϕ α s ) ds d s Therefore, given X such ha GalbX) = l p wih < p, D E, F ). T χ E, χ F ) X D E, F ), E, F < T : Λ p dd p ) X and, given X such ha GalbX) = l log l) α, T χ E, χ F ) X D E, F ), E, F < T : Λ αdd) X. 3. Aomic operaors 3.. Aomic and one-variable case. Le us recall firs some definiions and resuls from [6] and [7]. We wor in R n, and Q represens a cube wih sides parallel o he coordinae axes. The resuls can be exended in he naural way o T N idenifying T N wih [, ) N ). In [6], he following definiions were inroduced: Definiion 3.. Given δ >, a funcion a L R n ) is called a δ-aom if i saisfies he following properies: a) ax) dx =. R n b) There exiss a cube Q such ha Q δ and supp a Q.

10 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA Definiion 3.2. a) A sublinear operaor T is ε, δ)-aomic if, for every ε >, here exiss δ > such ha, for every δ-aom a, T a L +L ε a. b) A sublinear operaor T is ε, δ)-aomic approximable if here exiss a seuence T j ) j of ε, δ)-aomic operaors such ha, for every measurable se E, T j χ E ) T χ E ) and, for every f L such ha f, and every >, T f) ) lim j inft j f) ) In paricular, any maximal operaor of he form sup j K j f, where K j L p j for some p j <, is ε, δ)-aomic approximable see [6] for more examples of his ind of operaors). Also, i was proved in [7] ha operaors bounded on L p wih < p < are no ε, δ)-aomic approximable. Theorem 3.3 [7]). Le T be a sublinear, ε, δ)-aomic approximable operaor. Le X be a uasi-banach rearrangemen invarian funcion space. Assume ha, for every measurable se E, T χ E ) X D E ), 3.) for some posiive funcion D. Then, for every funcion f L wih f we have T f X D f ). 3.2) As a conseuence of his resul, we can improve Theorem 2. when D) =. Corollary 3.4 [7]). Le T be a sublinear, ε, δ)-aomic approximable operaor and le X be a uasi-banach r.i. space. Assume ha for any measurable se E wih E < we have T χ E ) X C E. 3.3) Then, and hus T : L X. T f X C f, f L L, This resul exends Moon s heorem since i includes a wider class of operaors and holds for any uasi-banach space X. Proof. Le f L L and wrie f = f/ f such ha f. Thus, 3.3) and Theorem 3.3 imply T f X = f T f X C f f = C f. To complee he proof le us give he densiy argumen. Le f L. As L L is dense in L, here exiss a seuence {f } L L such ha f f in L as. As T is sublinear we have T fj T f X T f j f ) X C f j f L. Thus {T f } is a Cauchy seuence on X and hence is convergen in X. This allows us o define T f and o conclude ha T maps L ino X.

11 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM We compare Theorem 2. wih Corollary 3.4. Noe ha Theorem 2. par a) and Corollary 3.4 give he same esimae since ΛdD) = L in he laer we assume ha T is ε, δ)-aomic approximable). In Theorem 2. par b) resp. c)) we obain ha T maps Λ p dd p ) = L,p resp. Λ dd)) ino X. These are improved in Corollary 3.4 since L,p L for < p < and Λ dd) L. Therefore, he aomiciy assumpion allows one o obain beer esimaes. This reflecs ha, in principle, he esimaes in Theorem 2. can be improved when he operaor T is ε, δ)-aomic approximable. We noice ha, once we now ha 3.2) holds, we can ae his as our iniial assumpion. This conains in paricular he resriced ype esimae 3.) and as we sar wih a more general esimae more decomposiions of he funcions are allowed. We follow his approach in Secion 3.3 where we consider he funcions D) =, D) = +log + /) α, ec. Evenually we apply hese resuls o he bilinear Hilber Transform. Remar 3.5. When X is a Banach space and D is concave hen 3.) implies 3.2), wheher or no T is aomic. To see his, le f L wih f. As X is a Banach space GalbX) = l. This fac, Theorem 2. par a) and he concaviy of D yield T f X C f Λ dd) = C D µ f λ) ) dλ C D ) µ f λ) dλ = C D ) f. This means ha he fac ha a given operaor is aomic only maers when X is a uasi-banach space Aomic and muli-variable case. Definiion 3.6. Given δ >, a pair of funcions a, a 2 ) L R n ) L R n ) is called a δ-aom if i saisfies he following properies: a) a x ) a 2 x 2 ) dx dx 2 =. R n R n b) There exis cubes Q, Q 2 wih Q, Q 2 δ such ha supp a Q, supp a 2 Q 2. Definiion 3.7. a) A bi-sublinear operaor T is ε, δ)-aomic if, for every ε >, here exiss δ > such ha for every δ-aom a, a 2 ), T a, a 2 ) L +L ε a a 2. b) A bi-sublinear operaor T is ε, δ)-aomic approximable if here exiss a seuence T n ) n of ε, δ)-aomic operaors such ha, for all measurable ses E, E 2 T n χ E, χ E2 ) T χ E, χ E2 ) and, for all f, f 2 ) L L such ha f, f 2, and every >, T f, f 2 )) ) lim n inft n f, f 2 )) )

12 2 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA c) A muli-bilinear operaor is ieraive ε, δ)-aomic approximable), if for every f L wih f, he sublinear operaors T g = T g, f ) and T 2 g = T f, g) are ε, δ)-aomic approximable). Example 3.8. Consider T f, f 2 )x) = Kx, y, y 2 ) f y ) f 2 y 2 ) dy dy 2. R n R n Assume ha i is well defined for f, f 2 ) L L and he ernel K saisfies ha lim K, y, y 2 ) K, x, x 2 ) L +L =, 3.4) y,y 2 ) x,x 2 ) uniformly in x, x 2 ) R n R n, hen T is ε, δ)-aomic. To see his, observe ha if a, a 2 ) is a δ-aom, hen T a, a 2 ) L +L = K, y, y 2 ) a y ) a 2 y 2 ) dy dy 2 R n R n L +L = K, y, y 2 ) K, x Q, x Q2 )) a y ) a 2 y 2 ) dy dy 2 R n R n L +L K, y, y 2 ) K, x Q, x Q2 ) L +L a y ) a 2 y 2 ) dy dy 2, Q 2 Q wih x Qj being he cener of he cube Q j where a j is suppored. Therefore, given ε, we can choose δ in such a way ha he above uaniy is bounded by ε a a 2. In paricular we have he following examples: A) For funcions f, f 2 on R n define heir ensor on R 2n by f f 2 )x, y) = f x)f 2 y) for x, y R n. If K L p R n R n ) for some p < and T f, f 2 )x) = K f f 2 ))x, x), x R n, hen 3.4) holds since lim K y, y 2 )) K x, x 2 )) L +L lim... L p =. y,y 2 ) x,x 2 ) y,y 2 ) x,x 2 ) B) Consider a family of ernels {K j } j saisfying 3.4) for each j N. Le T m f, f 2 )x) = sup K j x, y, y 2 ) f y ) f 2 y 2 ) dy dy 2, j m R n R n where m N, hen T m is ε, δ)-aomic. Conseuenly, T f, f 2 )x) = sup K j x, y, y 2 ) f y ) f 2 y 2 ) dy dy 2, j N R n R n is ε, δ)-aomic approximable. In general, T f, f 2 )x) = sup n T n f, f 2 )x), where T n is ε, δ)-aomic, is ε, δ)-aomic approximable. We sae our main resul concerning bi-sublinear aomic operaors: Theorem 3.9. Le T be a bi-sublinear operaor ha is ε, δ)-aomic approximable or ieraive ε, δ)-aomic approximable. i) Assume ha for all measurable ses E, E 2, T χe, χ E2 ) ) ) h; E, E 2 ), 3.5)

13 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 3 where, for all s,s 2 >, h; s, s 2 ) is coninuous as a funcion of >. Then, for all f, f 2 L such ha f, f 2, we have ha T f, f 2 )) ) h; f, f 2 ). 3.6) ii) Le X be a uasi-banach r.i. space and assume ha, for all measurable ses E, E 2, T χ E, χ E2 ) X D E, E 2 ), 3.7) where D is increasing in each variable and D, ) = D, ) =. Then, for all f, f 2 L such ha f, f 2, we have T f, f 2 ) X D f, f 2 ). 3.8) Proof. When T is ieraive ε, δ)-aomic approximable, he desired esimaes follow by applying wo imes he sublinear case see [6] and [7]): each ime we freeze one of he variables. In he oher case, we use he ideas in [6] and [7] wih he appropriae changes. Firs of all, le us assume ha T is ε, δ)-aomic. Le {a i, a 2)} i, be a collecion of δ-aoms. For every s > we have ) T a i, a 2) s) s T a i s, a2) ) ) d i, s i, T a i s, a 2)) ) d maxs, ) T a i, a 2) L +L i, i, maxs, ) ε a i a ) i Le f, f 2 ) L L be a pair of posiive funcions such ha f j. Given ε >, le δ be he number associaed o ε by he propery ha T is ε, δ)-aomic. Le F δ be any collecion of pairwise disjoin cubes {Q i } i such ha i Q i = R n, and Q i = δ for every i. Given Q i F δ and j = or 2, le f i j = f j χ Q i. Then, R n f i jx)dx Q i, and hence, we can ae a se Q i j Q i his se can be empy) such ha Q i j = fjx) i dx = f j x) dx. R n Q i We define gj i = fj i χ Qi j, which clearly has vanishing inegral, and saisfies ha gj i f j x) dx + Q i j = 2 f j x) dx. Q i Q i Therefore, gj i 2 f j, i χ Qi j = i i Q i j = f j. 3.)

14 4 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA Noe ha f j = i f i j = i gi j + χ Ej = A j + χ Ej, where E j = i Qi j. Then, by sublineariy T f, f 2 ) T f, A 2 ) + T f, χ E2 ) T A, A 2 ) + T χ E, A 2 ) + T A, χ E2 ) + T χ E, χ E2 ) i, T g i, g 2) + i, T χ Qi, g 2) + i, T g i, χ Q 2 ) + T χ E, χ E2 ) and herefore, T f, f 2 )) ) T g, i g2) ) α ) + T χ Qi, g2) ) α 2 ) i, i, + T g, i χ Q 2 ) ) α 3 ) + T χ E, χ E2 )) α 4 ), 3.) i, for all α j >, j 4, wih 4 j= α j =. Le us poin ou ha g, i g2), χ Qi, g2) and g, i χ Q 2 ) are δ-aoms. We firs prove i). Using 3.), 3.9), 3.), and 3.5) we have T f, f 2 )) ) 3 j= 3 j= )) 4 max α j, ε f f 2 + T χ E, χ E2 )) α 4 ) )) 4 max α j, ε f f 2 + hα 4 ; E, E 2 ), and, since E j = f j by 3.), we obain 3 )) T f, f 2 )) ) 4 max α j, ε f f 2 + hα 4 ; f, f 2 ). j= Leing firs ε and hen α 4, we obain he desired esimae for T. Nex we obain ii). We ae α = α 2 = α 3 = /3 N 2 ) and α 4 = /N 2 wih N 2. Then, for /N, N) we have ha /N /N 2 ) = α 4 and R N ) = T χ E, χ E2 )) α 4 ) χ /N,N) ) T χ E, χ E2 )) /N) χ /N,N) ). This yields ha R N ) T χ E, χ E2 )) ) for every >. Le X be he uasi- Banach r.i. space given by he Luxemburg represenaion heorem such ha h X = h X. Then, using 3.7) and ha E j = f j we have R N X T χ E, χ E2 )) X = T χ E, χ E2 ) X D E, E 2 ) = D f, f 2 ). On he oher hand by 3.), 3.9) and 3.) we obain for every /N, N) ) ) T f, f 2 )) )... /3 N 3 )) +... /3 N 3 )) i, i, ) +... /3 N 3 )) + R N ) i,

15 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 5 Therefore, 24 N 3 ε f f 2 + R N ). T f, f 2 )) χ /N,N) X N 3 ε f f 2 χ /N,N) X + R N X N 3 ε f f 2 χ /N,N) X + D f, f 2 ). Leing firs ε and hen N, we deduce he desired esimae as a conseuence of he Faou propery for X h N h a.e. implies h N X h X ). To finish, we consider he case when T is ε, δ)-aomic approximable. Le T n ) n be he corresponding seuence of ε, δ)-aomic operaors given in Definiion 3.7. To obain i) we observe ha T n χ E, χ E2 )) ) T χ E, χ E2 )) ) h, E, E 2 ), and hence T n f, f 2 )) ) h, f, f 2 ), for all pairs of posiive funcions f, f 2 ) such ha f j. Using T f, f 2 )) ) lim inf n T n f, f 2 )) ), he desired esimae for T follows a once. To derive ii) we noice ha T n χ E, χ E2 ) X T χ E, χ E2 ) X D E, E 2 ), and we deduce ha T n saisfies 3.8). Thus we conclude he same esimae for T using ha T f, f 2 )) ) lim inf n T n f, f 2 )) ) and he Faou propery. As a conseuence of Theorem 3.9, we can improve Theorem 2.6 and also Corollary 2.7) when D, s) = s. Corollary 3.. Le T be a bi-sublinear operaor ha is ε, δ)-aomic approximable or ieraive ε, δ)-aomic approximable. Le X be uasi-banach r.i. space. Assume ha for all measurable ses E, E 2 wih E, E 2 < we have Then, and hus T : L L X. T χ E, χ E2 ) X C E E ) T f, f 2 ) X C f f 2, f, f 2 L L, This gives a bilinear and hus mulilinear) version of Moon s heorem improving he resul in [], where only he case X = L, wih > was considered Decomposiions ino level ses and esimaes on Orlicz spaces. If T is an operaor as in Theorem 3.3, hen 3.) implies 3.2). A his poin, we mae his laer condiion our saring assumpion. Tha is, from now on we will be woring wih sublinear operaors T for which 3.2) holds. Wheher his condiion follows from he assumpion ha T is aomic or no plays no role in he argumens below. Le us emphasize ha in 3.) we only allow characerisic funcions while in 3.2) a wider a class of funcions is considered f L wih f ). Noice ha in Theorem 2., funcions are decomposed as linear combinaions of characerisic funcions. Saring wih 3.2) we can use more general decomposiions: characerisic funcions can be replaced by L -funcions bounded by. In he following argumen we will use decomposiions based on he level ses of he funcions. As

16 6 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA menioned before in Remar 3.5, only he case where X is uasi-banach maers, since being aomic or no maes a difference. Nex we explain he general scheme ha we are going o follow: Sep. We sar from T f X C D f ), f. 3.3) Sep. Given f and an increasing seuence of non-negaive numbers {d } such ha d as and d + as +, we wrie f = d f, f = d f χ {d <f d }. Le us observe ha in some cases we will ae d = for and d =. Thus, he summaion runs for and f = f χ {f }. Sep 2. We use 3.3) as f ) and he definiion of he Galb: T f X d T f { d D f )} GalbX) = {A } GalbX). X Sep 3. We pic a non-negaive funcion ϕ ha i is essenially consan in he inervals [d, d ] and wrie c = ϕd ). Then, seing a = d <f d f ϕf) we have ) ) a ) A = d D f d D f ϕf) = d D. d d <f d d c d <f d d c Sep 4. We show ha for every non-negaive seuence {a } l wih {a } l =, we have { a d D. 3.4) d c )} GalbX) Sep 5. If we are able o chec all he seps in his procedure, hen we will ge for all f such ha = a = T f X f ϕf) = f ϕf). d <f d R n Therefore, T maps L ψ ino X where L ψ is he Orlicz-ype space defined by he funcion ψ) = ϕ). Looing a all hese seps, he sraegy consiss in finding an appropriae funcion ϕ as in Sep 3 such ha he esimae in Sep 4 holds. The choice of ϕ should depend on he seuence {d } as ϕ has o be essenially consan in he inervals defined by he seuence) and also on he funcion D and GalbX). Moivaed by he resriced

17 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 7 esimaes for he bilinear Hilber ransform we consider uasi-banach spaces X wih GalbX) = l for < <. We sar wih funcions D) = + log + /) α. For every we wrie ε = + log ) +ε wih ε >. Example 3.. Le D) = + log + /) α and GalbX) = l wih < <. We ae d = 2 for, d = and d = for. We pic ϕ) = + log + ) α + log + ) + log + log + ) +ε) wih ε >. Then, c α + log ) +ɛ) a ) d D Da ) + 2 D d c GalbX) = α ε. We have o esimae a ) + ΣI + Σ 2 α ε )/ II, where Σ I, Σ II are he corresponding sums where he indices run over he following ses I = { : a ε }, II = { : a > ε }. Then, Σ I 2 ) ) D + log + 2 α ε / )α Also, ε. 2 α ε / { a Σ II α ε )/ { a α ε )/ { a ε )/ 2 2 α ε / + log + 2 α ε )/ } + log + 2 α ε / II a =. l ) α a } II ) α } II l Thus we have shown 3.4) and herefore for every ε > we obain α+ +ε) T : L log L) log log L) X. Le us observe ha aing a lile bigger seuence in l, ha is, ε = +ε we can α+ replace he firs space by L log L) +ε). Example 3.2. We proceed as in he previous example bu now we choose a differen seuence d. Le D) = + log + /) α and GalbX) = l wih < <. We ae d = 2 2 for, d = and d = for. We pic ϕ) = + log + ) α + log + log + ) + log + log + log + ) +ε) wih β >. Then, c 2 α ε. The ses I and II are he same and we esimae Σ I and Σ II : { Σ I 2 2 D a α ε )/ )} I l { 2 2 D l α ε / )} l

18 8 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA and { Σ II ε, a 2 α ε )/ a =. + log α ε )/ a ) α } II l {a } ε II l Thus we have shown 3.4) and herefore, for every ε >, we obain T : L log L) α log log L) +ε) log log log L) X. As before, he space of origin can be replaced by L log L) α log log L) +ε) or even more by L log L) α+ε. Noe ha hese improve wha was obained in he previous example, hus he seuence 2 2 gives beer esimaes han 2. Le us observe ha aing d = 2 22 hen + log + ) α is no essenially consan on he inerval [d, d ]. In some sense, as we have sared wih a resriced wea ype associaed wih he space Llog L) α we should ae seuences d for which he funcion + log + ) α is essenially consan on he inervals [d, d ]. In he following wo examples we wan o illusrae how his mehod behaves wih respec o differen logarihms. We give he final resuls leaving he deails o he ineresed reader. Example 3.3. Le D) = + log + /) α + log + log + /) β and GalbX) = l wih < <. We ae d = 2 for, d = and d = for. The α+ β++ε) ideas used before lead o he space L log L) log log L). A beer resul is proved by choosing he seuence d = 2 2 for, d = and d = for in which case one ges L log L) α β+ +ε) log log L) log log log L). Le us emphasize ha as before we canno ae d = 2 22 since + log + ) α is no essenially consan on he inerval [d, d ]. Example 3.4. Le D) = +log + log + /) α and GalbX) = l wih < <. The previous ideas indicae ha one should find seuences for which he funcion + log + log + ) α is essenially consan in he inerval [d, d ]. In his way, we ae d = 2 22 for noice ha we canno wor wih d = ), and hen he space obained by his mehod is L log log L) α log log log L) +ε) log log log log L). Noe ha if we had aen he seuences d = 2, d = 2 2, we would have obained he smaller spaces, respecively L log L) α++ε) log log L), L log log L) α+ +ε) log log log L). As in he case on he bilinear Hilber ransform, we also have funcions of he form D) = /p + log + /) α wih < p <, and we invesigae wha spaces one obains via his mehod. Noe ha his funcion is associaed wih an Orlicz

19 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 9 space near L p indeed L p log L) α p ), hus we loo for seuences d for which p is essenially consan in he inervals [d, d ], ha is, d 2. Noice ha in he previous examples he Orlicz funcions saisfy ψ) = for. Thus, i was no necessary o decompose he funcion f χ {f } ino level ses. Here, as he Orlicz funcion is going o be near p, we invesigae wheher decomposing his funcion leads or no o a beer esimae. We sar wih he simpler case α =. Example 3.5. Le D) = /p wih < p < and GalbX) = l wih < <. We firs ae d = 2 for, d = and d = for. We ae ϕ) = for and ϕ) = p + log + ) p/ + log + log + ) p/ ) +ε). Then c 2 p ) ε p/ d D a d c ). We have o esimae GalbX) Da ) + 2 a D 2 p ε p/ ) + a /p ε /p and we spli he sum in he righ-hand side as Σ I + Σ II wih he same definiion of I and II). Noe ha we rivially have ha Σ I ε. On he oher hand, since < < < p, we have ha Σ II a =. Then we obain ha T maps L ψ ino X where ψ) = for and ψ) = p + log + ) p/ + log + log + p/ ) +ε) ) for. Nex, we ae d = 2 for every Z. Consider he funcion ϕ) = p + log ) p/ + log + p/ ) +ε) log ) and hen c 2 p ) ε p/ ) for, and c =. Since ϕ) = ϕ) for, we only have o esimae he erms. Proceeding as before now we compare a wih /ε ) we conclude ha d D a d c ) + Then we obain ha T maps L eψ ino X where a /p ε /p. ψ) = p + log ) p/ + log + log ) p/ ) +ε). Le us observe ha ψ) ψ) = for as p > ) and also ha ψ) = ψ) for. Thus, L ψ L eψ. Therefore, decomposing f χ {f } leads o a beer esimae. Example 3.6. Le D) = /p + log + /) α wih < p < and GalbX) = l wih < <. We ae d = 2 for every Z. Consider he funcion ϕ) = p + log + ) α + log ) p/ + log + log ) p/ ) +ε) and hen c 2 p ) max, ) α ε p/ ) for, and c =. The same ideas allow us o show ha T is bounded from L ψ ino X where ψ) = p + log + ) α + log ) p/ + log + log ) p/ ) +ε).

20 2 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA Remar 3.7. We would lie o emphasize ha he mehod used in his secion, of picing he seuence d = a, a >, canno improve Theorem 2.. Indeed, going bac o Sep 2, we need o esimae {A } GalbX). Noice ha if a is big enough, {A } GalbX) = { a D f )} GalbX) { a D { f a } ) } GalbX) { a D µ f a ) ) } GalbX). We observe ha he las uaniy is a discreized version of he norms appearing in Theorem 2.. Therefore, he approach developed in his secion becomes meaningful when d growhs faser when d = 2 2, 2 2,... he previous uaniies are no longer comparable since d d + ). This shows ha he spaces obained in Examples 3.5, 3.6 are worse han he ones ha follow from Theorem 2.. We will use his when woring wih he bilinear Hilber ransform The muli-variable case. As observed before, having some exra informaion abou he operaor leads us, in some cases, o beer esimaes. Thus we will sudy differen cases for which he previous argumens in he linear case can be also exploied. For simpliciy we firs consider he case where D can be broen up ino wo funcions. We sar wih a bi-sublinear operaor saisfying T f, g) X C D f ) D 2 g ), f, g. This occurs when T is ε, δ)-aomic approximable or when i is ieraive ε, δ)-aomic approximable. Once we have he las ineualiy we will no use hese properies anymore. For hese operaors we can freeze one of he variables and wor wih he oher one. Thus, here is no difference wih he -sublinear case considered before. In he case general case where D is no spli we have o wor wih wo seuences a he same ime, one for each variable. We ae {d } and c = ϕ d ) wih ϕ essenially consan in he inervals [d, d ]. This seuence is relaed o he funcion f. For he funcion g we ae δ j and η j = ϕ 2 δ j ) wih ϕ 2 essenially consan in he inervals [δ j, δ j ]. We define A, a and B j, b j as in Sep 3 A is for f, d, c ; B j is for g, δ j, η j ). Everyhing reduces o show he following analog of Sep 4: for all non-negaive seuences {a }, {b j } j l wih {a } l = {b j } j l = we have { a d δ j D, d c b j. 3.5) δ j η j )},j GalbX) If we are able o show his, we obain ha T maps L ψ L ψ 2 ino X, where ψ ) = ϕ ) and ψ 2 ) = ϕ 2 ). 4. The Bilinear Hilber Transform We sar wih he basic esimae proved in [3] see also [4]): sup Φ) H ) ) χ E, χ E2 D E, E 2 ), 4.) where Φ) = 3/2 + log + ) 2, Ds, ) = s mins, ) ) /2 + log + ) 2. s mins, )

21 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 2 Noice ha for any α, β [, ] wih α + β =, we have Ds, ) D s) D 2 ) where D s) = s +α 2 + log + ) 2, D2 ) = +β 2 + log + ) 2. s We define X o be he space given by he uasi-norm I is nown see [7]) ha GalbX) = l 2 3. f X = sup Φ) f ). > 4.. Esimaes on Lorenz spaces. Applying Corollary 2.7 we obain Hf, g) X f g. Λ 2/3 dd 2/3 ) Λ 2/3 dd 2/3 2 ) I remains o idenify hese Λ-Lorenz spaces. We have: f 2/3 Λ 2/3 dd 2/3 ) = = f 2 3 f ) 2/3 dd 2/3 ) 2 L +α), 23 log L) 4 3. f ) +α 2 + log + Considering he exreme cases α = and β =, or vice versa, we obain H : L, 2 3 log L) 4 3 L 2, 2 3 log L) 4 3 X, H : L 2, 2 3 log L) 4 3 L, 2 3 log L) 4 3 X. ) 2 ) 2 3 d We see below ha in he exreme case α = and β = he previous esimae can be improved exploiing he fac ha he bilinear Hilber ransform is aomic. We can also use Theorem 2.6 wih he original funcion D and hen Hf, g) X s 2/3 2/3 D µ f s), µ g ) ) 2/3 ds d s f s) 2/3 g ) 2/3 dd 2/3 s, ), wih Ds, ) = s mins, ) ) /2 + log + ) 2. s mins, ) As observed in Remar 2.8, here we do no lose any informaion in he following sense: Hχ E, χ F ) X D E, F ), E, F < H : Λ 2/3 dd 2/3 ) X Aomiciy and esimaes on Orlicz spaces. We show ha he following runcaions of H H N f, g) = fx ) gx + ) d = fx ) gx + ) N ) d R /N< <N are ieraive ε, δ)-aomic. Le g L be such ha g and consider he -linear operaor T N f defined by T N fx) = H N f, g). We obain ha T N is ε, δ)- aomic he oher case in which f is frozen can be obained in he same manner).

22 22 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA We wrie T N in he following way, T N fx) = f) g2 x ) N x ) d = R R f) K N x, ) d. Given ε >, le a be a δ-aom wih δ > o be chosen. Then supp a I for some inerval I wih I δ. Le be he cener of I. We firs show ha here exiss δ = δε, g, N) such ha KN, s) K N, s ) L ε, for all s, s R wih s s < δ/2. 4.2) Using ha g i follows ha KN, s) K N, s ) L g2 x s) N x s) g2 x s ) N x s ) dx R N x s) g2 x s) g2 x s ) dx R + g2 x s ) N x s) N x s ) dx R N gx + s s )) gx) dx + N x + s s )) N x) dx. R Thus, since g, N L R), using properies of he ranslaion operaor in L R), here exiss δ = δε, g, N) such ha for every < δ N gx + ) gx) dx + N x + ) N x) dx N ε 2 N + ε 2 = ε. R R Applying his wih = s s, we obain 4.2). In his way, using ha a has vanishing inegral and 4.2), we conclude ha T N a L +L T Na a) K N, ) d R = a) K N, ) K N, )) d R a) KN, ) K N, ) L d ε a. <δ/2 Therefore, we have shown ha T N is ε, δ)-aomic and H N is ieraive ε, δ)-aomic. We observe ha H N saisfies 4.) uniformly in N. Thus, by ii) in Theorem 3.9 we conclude ha for all f, g, ) ) H N f, g) X = sup Φ) H N f, g D f, g ), where he consans involved are uniform in N, and Φ) = 3/2 + log + ) 2, Ds, ) = s mins, ) ) /2 + log + ) 2. s mins, ) Noice ha for any α, β [, ] wih α + β =, we have Ds, ) D s) D 2 ) where D s) = s +α 2 + log + ) 2, D2 ) = +β 2 + log + ) 2. s R

23 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 23 As menioned before GalbX) = l 2 3. We have already observed ha he mehod presened in Secion 3.3 is useful when D i ) is of he form + log + ) 2. Hence, we fix α =, β = and hen D s) = s + log + ) 2, D2 ) = 2 + log + ) 2. s Woring wih he firs variable and applying Example 3.2 wih = 2/3 and α = 2, we deduce ha he domain space for f is L log L) 2 log log L) 2 log log log L) 2 +ε for any ε >. We can ae smaller spaces such as L log L) 2 log log L) 2 +ε or L log L) 2+ε for any ε >. For he oher variable, we use he non-aomic approach and obain ha he domain space is L 2, 2 3 log L) 4/3. Thus, using he symmery of he problem, we have H N : L log L) 2 log log L) 2 log log log L) 2 +ε L 2, 2 3 log L) 4 3 X, H N : L 2, 2 3 log L) 4 3 L log L) 2 log log L) 2 log log log L) 2 +ε X. From here one can inerpolae by he complex mehod o conclude some oher esimaes. Noice ha all hese esimaes are uniform in N. Nex we are going o show how o derive hese esimaes for H. By densiy in he domain spaces), i suffices o consider Schwarz funcions f, g. In ha case we have lim N H N f, g) = Hf, g) a.e. and conseuenly H f, g ) ). lim infn H N f, g Then, for any < < we have ) ) ) ) f, g lim inf Φ) H N f, g Φ) H f, g ) ) lim inf N Φ) H N = lim inf N H Nf, g) X. Taing he supremum for < < we conclude ha N sup Hf, g) X lim inf N H Nf, g) X. This, he uniform esimaes obained before for H N and a sandard densiy argumen lead us o H : L log L) 2 log log L) 2 log log log L) 2 +ε L 2, 2 3 log L) 4 3 X, H : L 2, 2 3 log L) 4 3 L log L) 2 log log L) 2 log log log L) 2 +ε X. We finish his secion by comparing he differen spaces ha we have obained using he wo approaches. When α = and β =, he wo mehods have led us o he following spaces X = L, 2 3 log L) 4 3, X2 = L log L) 2 log log L) 2 log log log L) 2 +ε. We see ha X, X 2 are no comparable. Our firs funcion is given by h ) = + log + /) χ 7/2,e )). ee Then, h 2 3 X = h ) + log + / ) 2 2) d e e e 3 = d + log /) =.

24 24 M. J. CARRO, L. GRAFAKOS, J. M. MARTELL, AND F. SORIA On he oher hand, we have ha X 2 = L log L) 9/4 X 2 and herefore h X2 h ex2 = = e e e h ) + log + / ) 9/4 d + log /) 5/4 d <. Nex, we consider a second funcion h ) = A j χ aj+,a j )), A j = e ej4 j 3, a j = e ej4 +2 j 4). j= Noice ha A j is increasing and a j decreasing. We se m) = + log + /) 2 and noice ha ma j ) e ej4 +2 j 4) e j4 + 2 j 4 ) 2 e ej4 = A j j 3 Then, h 2 3 X = A 2 aj 3 j m) 2 d 3 a j+ A 2 3 j ma j ) 2 3 j <. 2 j= On he oher hand, le us observe ha X 2 X 2 = L log L) 2 log log L) /2. We wrie ϕ) = m) + log + log + /) /2. Observe ha a j e a j+ and aj Therefore, a j+ ϕ) d aj a j /e j= ϕ) d ϕa j/e) ma j ) log + log + a j ) /2 A j j. h X2 h bx2 = aj A j j= a j+ j= ϕ) d j =. These wo examples show ha he symmeric difference of X and X 2 is nonempy and hence he wo approaches developed in he presen paper give independen esimaes. Conseuenly, combining boh mehods, we obain esimaes for funcions in he larger space X + X 2, ha is, he bilinear Hilber ransform H saisfies H : X + X 2 ) L 2, 2 3 log L) 4 3 X, H : L 2, 2 3 log L) 4 3 X + X 2 ) X. References [] N.Y. Anonov, Convergence of Fourier series, Proceedings of he XXh Worshop on Funcion Theory Moscow, 995), Eas J. Approx ), [2] C. Benne and R.C. Sharpley, Inerpolaion of Operaors, Pure and Appl. Mah. 29, Academic Press, 988. [3] D. Bily and L. Grafaos, Disribuional esimaes for he bilinear Hilber ransform, Journal of Geom. Anal. 6 26), no. 4, [4] D. Bily and L. Grafaos, A new way of looing a disribuional esimaes; applicaions for he bilinear Hilber ransform, Proceedings of he 7h Inernaional Conference on Harmonic Analysis and Parial Differenial Euaions El Escorial, 24), Collec. Mah. 26, [5] L. Carleson, Convergence and growh of parial sums of Fourier series, Aca Mah ), j=

25 MULTILINEAR EXTRAPOLATION AND THE BILINEAR HILBERT TRANSFORM 25 [6] M.J. Carro, From resriced wea ype o srong ype esimaes, J. London Mah. 7 24), [7] M.J. Carro, L. Colzani, and G. Sinnamon, From resriced wea ype o srong ype esimaes on uasi-banach rearrangemen invarian spaces, Sudia Mah ), no., 27. [8] M.J. Carro and J. Soria, Weighed Lorenz spaces and he Hardy operaor, J. Func. Anal ), [9] L. Grafaos, J.M. Marell, and F. Soria, Weighed norm ineualiies for maximally modulaed operaors, Mah. Ann ), no. 2, [] L. Grafaos and M. Masy lo, Resriced wea ype versus wea ype, Proc. Amer. Mah. Soc ), [] R. Hun, On he convergence of Fourier series, Orhogonal Expansions and Their Coninuous Analogues Edwardsville, Ill., 967), , D. T. Haimo ed.), Souhern Illinois Univ. Press, Carbondale IL, 968. [2] K. H. Moon, On resriced wea ype, ), Proc. Amer. Mah. Soc ), [3] P. Sjölin and F. Soria, Remars on a heorem by N.Y. Anonov, Sudia Mah ), [4] F. Soria, On an exrapolaion heorem of Carleson-Sjölin wih applicaions o a.e. convergence of Fourier series, Sudia Mah ), [5] P. Turpin, Convexiés dans les espaces vecoriels opologiues généraux, Disseraiones Mah ). María Jesús Carro, Deparamen de Maemàica Aplicada i Anàlisi, Universia de Barcelona, E-87 Barcelona, Spain address: carro@ub.edu Louas Grafaos, Deparmen of Mahemaics, Universiy of Missouri, Columbia, MO 652, USA address: louas@mah.missouri.edu José María Marell, Insiuo de Ciencias Maemáicas CSIC-UAM-UC3M-UCM, Consejo Superior de Invesigaciones Cieníficas, C/ Serrano 2, E-286 Madrid, Spain address: chema.marell@uam.es Fernando Soria, Deparameno de Maemáicas, C-XV, Universidad Auónoma de Madrid, E-2849 Madrid, Spain address: fernando.soria@uam.es