DISTRIBUTIONAL ESTIMATES FOR THE BILINEAR HILBERT TRANSFORM

Transcription

1 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM DMITRIY BILYK AND LOUKAS GRAFAKOS Abstract We obtan sze estmates for the dstrbuton functon of the blnear Hlbert transform actng on a par of characterstc functons of sets of fnte measure, that yeld exponental decay at nfnty and blowup near zero to the power /3 modulo some logarthmc factors These results yeld all known L p bounds for the blnear Hlbert transform and provde new restrcted weak type endpont estmates on L p L when ether /p + / = 3/ or one of p, s equal to As a consequence of ths work we also obtan that the square root of the blnear Hlbert transform of two characterstc functons s exponentally ntegrable over any compact set Introducton Certan blnear sngular ntegral operators can be expressed as averages of blnear Hlbert transforms n a way analogous to that whch lnear sngular ntegrals can be wrtten as averages of lnear drectonal Hlbert transforms The blnear Hlbert transforms were ntroduced n the early sxtes to play exactly ths role by A Calderón n hs study of the frst commutator Propertes of these operators remaned elusve untl the appearance of the fundamental work of Lacey and Thele [0], [] n the late nnetes who establshed ther boundedness on certan products of Lebesgue spaces Ths work was based on a remarkable set of technques called tme-frequency analyss and revealed a fundamental and deep connecton wth almost everywhere convergence of Fourer seres and n partcular, the boundedness of the Carleson-Hunt operator, see Lacey and Thele []; on the latter the work of Fefferman [3] was nfluental The Carleson-Hunt operator s defned as +N Cfx = sup fξe πxξ dξ N>0 N where fξ = R fxe πxξ dx s the Fourer transform of the functon f on the lne Carleson [] answered a longstandng conjecture posed by Lusn by establshng the boundedness of the operator C on L A few years later, Hunt [8] obtaned ts L p boundedness for < p < as a consequence of the followng powerful dstrbutonal estmate { Cχ F > λ } C F { λ + log λ when λ e cλ when λ > Ths estmate holds for some fxed constants c, C and for all measurable sets F of fnte measure and all λ > 0 Usng standard nterpolaton, estmate easly mples the L p boundedness of C for < p < Moreover, recent extrapolaton technques by Antonov 000 Mathematcs Subject Classfcaton 46B70, 4B99 Key words and phrases multlnear operators, dstrbutonal estmates Both authors have been supported by the Natonal Scence Foundaton under grant DMS and by the Unversty of Mssour Research Councl

2 DMITRIY BILYK AND LOUKAS GRAFAKOS [] and ther refnement by Sjöln and Sora [4] show that estmate mples the boundedness of C on L log L log log log L of every compact set; ths mples the almost everywhere convergence of the partal Fourer ntegrals of functons locally n ths class The man purpose of ths artcle s to prove an estmate analogous to that n for the blnear Hlbert transform H α Ths operator s defned for a parameter α R by H α f, gx = π lm ε 0 t ε fx tgx + αt dt t, x R for functons f, g on the lne In the aforementoned work [0], [], Lacey and Thele proved that the operator H α maps L p L to L p, whenever < p,, p + = p, and 3 < p <, α Our approach uses the model sum reducton of Lacey and Thele [0],[], a tree analyss based on a selecton nspred by Lacey [9], and reles on an mproved energy estmate that appeared n the proof of by Grafakos, Tao, and Terwlleger [6] A varant of ths energy estmate had prevously appeared n the related work of Muscalu, Thele, and Tao [3] The man result of the artcle s the followng: Theorem Let < and α R \ {0, } Then there exst constants C = Cα,, c = cα, such that for all measurable sets F, F of fnte measure we have { H α χ F, χ F > λ} C F F {λ + + log + λ + when λ <, e c λ when λ Analogously, the followng estmate s vald for p < : 3 { H α χ F, χ F > λ} C F p F p p + {λ p p + + log λ p p + when λ <, e c λ when λ These estmates correspond to the lne segments { p, : p =, < } and { p, : p <, = } As a corollary we obtan the followng dstrbutonal estmate correspondng to the lne segment { p, : p,, p + = 3 } Corollary For any α R \ {0, } there exst constants C = Cα, c = cα such that for all measurable sets F, F of fnte measure we have 4 { H α χ F, χ F >λ} C { F F mn F, F λ 3 + log 3 λ 4 3 for λ< e c λ for λ Remark In the dstrbutonal estmate 4, the expresson F F mn F, F s domnated by F p F, where p j and p + = 3 Thus ths estmate up to a logarthmc term s smlar to a restrcted weak type estmate for such exponents Notce that the exponental decay at nfnty for the dstrbuton functon of H α s not as strong as n the case of the Carleson-Hunt operator and at the moment we don t know f t s sharp stmates, 3, and 4 not only capture the boundedness of H α on products of Lebesgue spaces but also yeld other crucal quanttatve nformaton such as local exponental ntegrablty of H α and also ts boundedness on other rearrangement nvarant spaces even at the endpont cases We state the exponental ntegrablty of H α n the form of corollary October 0, 005

3 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 3 Corollary 3 Let α R \ {0, } and c = cα be as n Corollary Then there s a constant C = C α such athat for any bounded measurable set K and for all measurable sets F, F of fnte measure the followng holds: e c H αχ F,χ F x dx C K + F F mn F, F 3 K for any 0 < c < c Decomposton of the blnear Hlbert transforms In the sequel we wll drop the dependence of H α on α and smply denote t by H We wll use the notaton A for the Lebesgue measure of a set A and f, g for the complex nner product fxgxdx For a number a > 0 and an nterval I we denote ai an nterval of length a I concentrc wth I and by a I the nterval [ap, aq] f I = [p, q] We wll use the notaton to express that a certan quantty s at most a constant multple of another one Our goal wll be to study the trlnear form f, f, f 3 Hf, f xf 3 xdx for three functons f, f, f 3 whch wll be characterstc functons of sets of fnte measure, e f = χ F, f = χ F, and f 3 = χ We fx L to be the smallest nteger greater than 0 max{ α, α, +α }3 The dependence of the bounds on α wll enter the proof through polynomal dependence on L We begn by notng that the dstrbuton pv t that appears n the defnton of H can be wrtten as c δ 0 + c γ for some constants c, c, where δ 0 s the Drac mass at the orgn and γ s another dstrbuton that satsfes γ = χ 0, Snce all the estmates that we are gong to be provng n ths paper are trval for δ 0, we may restrct our attenton to γ Let θ be a smooth functon whch s equal to on, L and 0 on 3L, Defne ψξ = θξ θξ Observe that ψ s nonzero and s supported n [L, 3L] For each nteger k we defne Then we have ψ k x = k ψ k x γ = k Z k ψk Indeed, f we look at the Fourer transform of the rghthand sde of the dentty above, we get a telescopc sum: N [ k ψk ξ = lm θ k ξ θ k+ ξ ] [ = lm θ N ξ θ N+ ξ ] = γ N N k Z N It clearly suffces to study the trlnear form 5 Λf, f, f 3 := k f x tf x + αtf 3 xψ k t dt dx k Z We can further break the functon ψ nto a sum of at most L functons ψ M such that ψ M s supported n the nterval [M, M + ] for L M L It would suffce to study each pece separately For notatonal convenence, we wll omt the dependence on M and wll just wrte ψ October 0, 005

4 4 DMITRIY BILYK AND LOUKAS GRAFAKOS For further decomposton we fx a Schwartz functon φ of L norm, wth Fourer transform supported n [, ], whch also has the property that for all ξ R we have φξ l/ C0 l Z for some constant C 0 > 0 Let u = I u ω u be a rectangle n R and set φ u x = I u φ x ciu I u e πcωux, where cj denotes the center of the nterval J For each k Z we consder the set of dyadc rectangles of scale k: S k = { k n, k n + k m/, k m/ + m, n Z} Then S = k S k s the set of all dyadc rectangles of area n R It s an easy calculaton to verfy that for all f L f = C 0 u S k f, φ u φ u where the convergence s n L Moreover, the seres also converges ae for all f L p, < p <, see [5] Usng ths decomposton of the dentty n the k th term of 5, as n [], we obtan 6 Λf, f, f 3 := C k,u,u,u 3 Λ k,u,u,u 3 f, f, f 3, k Z u,u,u 3 S k where C k,u,u,u 3 = C0 3 φ u x tφ u x + αtφ u3 xψ k t dt dx R R and Λ k,u,u,u 3 f, f, f 3 = k f, φ u f, φ u f 3, φ u3 We now take a closer look at the coeffcents C k,u,u,u 3 n two dfferent ways Frst, C k,u,u,u 3 x t C0 3 ciu x + αt ciu x ciu3 φ φ φ ψ k t I u I u I u3 k dt dx = C0 3 φ x t A φ x + αt A φ x A3 ψt dt dx, where A = ciu I u for =,, 3 these numbers are half-ntegers Observe that A A = x t A x + αt A + + αt, A 3 A = x t A x A 3 + t, A 3 A = x + αt A x A 3 αt Ths mples that at least one of the arguments n the last dsplayed double ntegral has to have sze at least 4L dam{a } Snce φ and ψ are Schwartz functons, t follows that, for any postve nteger m, there exsts a constant C m such that 7 C k,u,u,u 3 C m + dam{a m } = C m + max,j ci u ci uj m 4L k 4L October 0, 005

5 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 5 Secondly, we set F x, t = φ u x tφ u x+αt, F x, t = φ u3 xψ k t These are Schwartz functons of two varables We have F ξ, τ = + α φ αξ τ ξ + τ u φ u, + α + α F ξ, τ = + α φ u3 ξ ψ k τ Thus, applyng the two-dmensonal Plancherel formula, we obtan 8 C k,u,u,u 3 C αξ τ ξ + τ φ + α + α B φ + α B φξ B 3 ψτ dξdτ, where B = cωu ω u = k cω u notce that ths s an nteger or a half-nteger Assume that the ntegral above s not zero Then we must have αξ τ + α B whch mply 9 B and 0 B [, ], [ α + α B 3 + α ξ + τ + α B [ + α B α [, + α + + α M, + α + + α M, + α ], ξ B 3 [, ] [, τ M, M + ], α + α B 3 ] + α + + α M +, + α + α + α B 3 + ] + + α M + + α + α Ths means that the trple of parameters B, B, B 3 really depends only on the parameter B 3 as for each value of B 3, the quanttes B and B can take only a fnte number of values dependng on α Also, 9 and 0 show that B + B = B 3 up to an error that can only take a fnte number of nteger values dependng on α We ntroduce parameters ν, ν, µ, µ by settng A = A 3 + ν, A = A 3 + ν, B = α α + B 3 + µ, B = α + B 3 + µ We also set ν = max ν We am to reduce the sum over u, u, u 3 S k as the rapdly convergng sum over ν, ν, µ, µ of the sum over the tles u 3 For N suffcently large we have Λf, f, f 3 C N + ν 4L N ν=0 ν,ν : max ν =ν ε ν,ν,µ,µ,u 3 Λ k,u,u,u 3 f, f, f 3 µ µ u 3 S k k Z where u = u u 3 and u = u u 3 are unquely determned by u 3 n terms of ν, ν, µ, µ, ε ν,ν,µ,µ,u 3 s a constant of modulus at most, and µ Z α +α Z and µ Z +α Z range n the ntervals [ µ + α + + α M, ] + α + + α M +, + α + α + α + α µ [ + α + + α M, + α + + α M + α + α ] October 0, 005

6 6 DMITRIY BILYK AND LOUKAS GRAFAKOS Thus µ and µ take only a fnte number of values dependng on α Note that ε ν,ν,µ,µ,s 3 s the rato of C k,u,u,u 3 by C N + max ν 4L N It wll clearly suffce to study the boundedness of the expresson nsde the absolute values n and to obtan bounds ndependent of µ and polynomal n ν, snce for each ν, there are of the order of ν pars ν, ν wth max ν = ν Next, we further separate the trples n such a way that for two trples u, u, u 3 and u, u, u 3 from the same group the followng condtons hold: f k k, then k k > L 0, 3 f A 3 A 3, then A 3 A 3 > νl 0, 4 f B 3 B 3, then B 3 B 3 > L 0 Obvously, the number of such groups s polynomal n L and ν To facltate the study of the sums above, we ntroduce tr-tles A tr-tle s a rectangle s = I s ω s and three subrectangles s, s, s 3 bult n the followng way: Let u, u, u 3 be a trple of rectangles partcpatng n the sum n Defne I s = I s = I u3 Defnng the frequency projectons requres a lttle bt more work, as we cannot just use the dyadc grd We want these projectons to satsfy the followng propertes: 5 J = s S ωs ω s ω s ω s3 s a grd 6 If ω s J for some J J, then ω sj J for some J J for all j =,, 3 7 ω s ω sj for j We buld these ntervals by nducton on the cardnalty of the set U of trples of rectangles If ths set s nonempty, we pck the trple u, u, u 3, such that k, where ω u3 = k, s maxmal Let U = U \ u, u, u 3 By nducton we fnd the ntervals ω s, ω s =,, 3, correspondng to the elements of U If there s an element u U such that ω u 3 = ω u3, then we defne ω s, ω s =,, 3 to be the same as the correspondng ntervals for u Otherwse we defne for =,, or 3 ω s to be the convex hull of the nterval C ω u C = +α α, C = + α, C 3 = and all all sets ω s that ntersect t Note that because of the separaton of scales what we get s only slghtly smaller than the nterval tself Next, we defne ω s as follows: take [a, b] to be the convex hull of ω s =,, 3, then set ω s to be the convex hull of [a, b] and all ntervals ω s that ntersect [a, b] Propertes 5 and 6 are obvous n vew of 4 and Also ω s and ω s are comparable to k wth a factor dependng on L Property 7 follows from 9, 0, and the separaton of scales We defne the functons adapted to the tr-tle s wth parameters ν, ν, µ, µ as follows: ϕ ν,µ,α s x = I s x cis x φ ν e π α α+ cωs +θs ωs = φ u x, I s ϕ ν,µ,α s x = I s x cis x φ ν e π α+ cωs +θs ωs = φ u x, I s ϕ α s 3 x = I s φ x cis I s e π cω s3 +θ s3 ω s3 x = φ u3 x, where the error terms θ s n the modulatons are chosen so that α+ cω s +θ s ω s = cω u, α+ cω s + θ s ω s = cω u, and cω s + θ s3 ω s3 = cω u3 Obvously, θ s CL October 0, 005 α

7 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 7 Then the expresson nsde the absolute values n becomes exactly I s εν s 3 S,ν,µ,µ,s f, ϕ ν,µ,α s f, ϕ ν,µ,α s f 3, ϕ α s 3 S k k Z Ths expresson needs to be controlled wth bounds that grow polynomally n the parameters ν, ν, and are ndependent of µ, µ We wll work wth sums over fnte sets of tr-tles and get bounds ndependent of the choce of the fnte set, whch s clearly suffcent by a lmtng argument Note that f ω u and ω u were not dsjont, then nether are ω s and ω s Thus ϕ sj f ω s ω s =, then ϕ s, ϕ s = 0 For notatonal convenence, n the sequel we wll suppress the dependence of the functons on the parameters ν, ν, µ, µ Notce that ϕ sk x C + x ci s 0 ν k C + x ci s 0 I s I s + ν 0 3 stmates for the model sums The case I s Ω Let S be a fnte set of tr-tles wth fxed data ν, ν, µ, and µ Then we defne the model sum assocated wth S as follows: We set Ω = H S f, f x = s S { x : Mχ F x > 8 mn, F I s εs f, ϕ s f, ϕ s ϕ s3 x } { x : Mχ F x > 8 mn, F where M s the Hardy-Lttlewood maxmal functon Snce M f of weak type, wth constant at most, t s easy to see that Ω < We now set = \ Ω Obvously, then The man purpose of ths artcle s to obtan a good estmate for the expresson H S χ F, χ F xdx = H S χ F, χ F, χ To do so we wll break the model sum nto two parts: the sum over those s S for whch I s Ω easer case and the sum over tles wth I s Ω We begn wth the easer case For a dyadc nterval J we set x cj 5 ωx = + J and S J = {s S : I s = J} We have the followng nequaltes for =,, 3: 8 f, ϕ s l S J + ν 0 J f L ω 9 s S J α s ϕ s L ω + ν 0 α s l S J 0 f, ϕ s l S J + ν 0 f L ω }, October 0, 005

8 8 DMITRIY BILYK AND LOUKAS GRAFAKOS Indeed, to prove 8, for any s S J we have f, ϕ s = fx J x cj ϕ ν e πxc cω s +θ ω s dx J R C + ν 0 J f L ω Next, we prove 9, whch s an analogue of Bessel s nequalty Although the functons ϕ s are no longer orthogonal n the weghted space L ω, we wll see that they are almost orthogonal n ths space It s straghtforward to check that ϕ s, ϕ s ω = ϕy ν + y 5 K cω s cω s + θ s θ s ω s J C + ν 0 + K cω s cω s J 0, snce ϕy ν +y 5 and ts Fourer transform s a Schwartz functon here K = K = α+, K 3 = Now we have α s ϕ s α s α s ϕ s, ϕ s L s S ω ω J s,s S J C + ν 0 k,m C + ν 0 k Z C + ν 0 α k l α k α m + k m 0 αk + k m 0 m Z α α+, Note that 0 s the dual statement of 9 Let M be the Hardy-Lttlewood maxmal functon and M f = Mf / We prove the followng estmate: Lemma 3 For A > we have H SJ χ F, χ F L AJ c + ν 0 C M A M J nf x J Mχ F x nf x J M χ F x Proof If we wrte H SJ χ F, χ F = H SJ χ F, χ F ω ω and use Hölder s nequalty, we obtan: H SJ χ F, χ F L AJ c ω L AJ c H SJ χ F, χ F L ω C A M J J L χf, ϕ s χ F, ϕ s ϕ s3 s S ω J C A M χ F, ϕ s χ F, ϕ s l S J C A M χ F, ϕ s l S J χ F, ϕ s l S J C + ν 0 A M J χf L ω χ F L ω C + ν 0 A M J J nf Mχ F x J nf M χ F x x J x J In the last estmate we have used the fact that x cj x θ + + J J for all θ J October 0, 005

9 C + ν 0 F F DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 9 The man concluson s the followng: Lemma 3 s:i s Ω where C ν C + ν 0 I s χf, φ s χ F, φ s φ s3 x dx C νmn F, F F F Proof Snce the roles of F and F are symmetrc, t wll suffce to prove that holds wth the expresson C F F on the rght hand sde of the nequalty We organze all dyadc ntervals J Ω nto sets F k k 0 n the followng way: We note that F k = {J : k J Ω, k+ J Ω} J 4 Ω J F k Indeed, assume J max s a maxmal element of F k wth respect to ncluson If J J max and J < J max, then J must have a common endpont wth J max otherwse, we would have k+ J = k J k J max Ω, thus J / F k Thus, for each partcular scale, J max may contan at most ntervals belongng to F k Therefore J k+ J max 4 J max J F k,j J max k=0 Snce the maxmal elements of F k are dsjont, summng over them we obtan the requred concluson Also, for any J F k we have Ω c k J c Thus we have: H {Is Ω}χ F, χ F dx H SJ χ F, χ F dx J Ω = H SJ χ F, χ F dx k=0 J F k H SJ χ F, χ F L k J c k=0 J F k C M + ν 0 J km k=0 J F k C M + ν 0 km C0 k+ k=0 C + ν 0 k=0 km C k+ 0 nf Mχ F nf M χ F x J x J J F k J J F J F k nf Mχ F nf M χ F k+ J k+ J F October 0, 005

10 0 DMITRIY BILYK AND LOUKAS GRAFAKOS 4 stmates for model sums The case I s Ω We wll now deal wth the harder case I s Ω Ths part of the proof s based on an adaptaton of the L L L, estmate n [9] We denote by P the set of all tr-tles s S, for whch I s Ω Tr-tles admt a partal order We say that s < s f I s I s and ω s ω s We note that s and s ntersect as rectangles f and only f they are comparable under < The separaton of scales allows to say that f s < s, then ω s ω s for some =,, 3 or t s dsjont wth all ω s s We say that a collecton of tr-tles T s a tree wth top t f for all s T, s < t very fnte collecton of tr-tles S s a unon of trees Indeed, f we denote by S the set of all elements n S whch are maxmal under <, and, for each t S, T t s the maxmal tree n S wth top t, then S = t S T t We refne the noton of the tree by sayng that T s a j-tree j =,, 3 f T s a tree wth top T and for every s T, ω sj ω t = For a tree T, s T, s t, at most one of the ntervals ω s can ntersect ω t Thus f we denote T k = {s T : ω sk ω t }, k =,, 3, then T k s a j-tree for j k there are also elements such that ω s ω t = for all =,, 3, but those may be added to any of the T k s Then T = 3 k= T k, e any tree s a unon of at most three subtrees whch are j-trees for at least two choces of j For a k-tree T we set T, k = I t f k, ϕ sk f k s T and we defne the k-energy of a fnte set of tles S by k S = sup T, k, where the supremum s taken over all k-trees T S Note that a sngleton {s} s a k-tree for all k, so for all s S, I s fk, ϕ sk k S f k Now fx some j =,, 3 and let T be a k-tree for k j Applyng the above estmate and the Cauchy-Schwarz nequalty, we deduce 3 H T f, f, f 3 f j, ϕ sj f s T I s k, ϕ sk k j j S f j f k, ϕ sk s T k j j S f j I t k j T, k f k I t 3 j S f j j= Ths s crucal estmate on a sngle tree that wll be used n conjuncton wth the dea that any tree can be wrtten as a unon of three trees of the above type Next, we state the man lemma whch wll allow us to obtan the estmates for the model sums cf [9] October 0, 005

11 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM Lemma 4 Let S be a fnte set of tr-tles Then S can be wrtten as a unon of two sets S = S S, whch have the followng propertes Let S be the set of elements whch are maxmal n S under < e S s a unon of trees wth tops n S We then have 4 I t C + ν 0 k S, t S 5 k S ks Ths lemma only yelds weak-type estmates from L L nto L, But the fact that we are now workng wth the set of tles P = {s S : I s Ω} and all functons are characterstc of some sets gves us an advantage quantfed by the followng energy estmate whch appeared n [6], [4], and s essentally contaned n [3]: Lemma 4 For k =, and f k = χ Fk, there exsts a constant C > 0, such that the followng estmate s vald: ] 6 k P C mn [ Fk, Fk Wth these two lemmata at hand we can derve an estmate of the model sum for the case I s Ω n the followng way We construct nductvely the sequence of parwse dsjont sets P j such that P = n 0 j= and the followng propertes are satsfed: k P j j+ for k =,, 3 P j s a unon of trees T jk such that k I topt jk C 0 + ν 0 j for all j n 0 3 k P \ P n0 P j j for k =,, 3 Ths sequence s constructed n the followng way: We start the nducton at the number j = n 0 such that k n 0 for k =,, 3 We set P n0 = Then propertes,, and 3 are clearly satsfed Assumng that we have already constructed the set P n, we construct P n as follows Let S = P \ P n0 P n Frst, f S > n, then apply Lemma 4 to S wth k =, thus obtanng the sets S wth a control of the sum of the tops and S wth small -energy, otherwse just skp ths step e S = Then, n the same fashon, f S > n, we apply ths lemma to S obtanng the set S and S otherwse agan skppng ths step, S = And, fnally, we apply Lemma 4 for the thrd tme wth k = 3 to the set S to obtan S 3 and S 3 we also skp ths step, f S n We set P n = S S S 3 Observe that f all three steps were skpped, then P n = We have to verfy that propertes -3 ndeed hold Frst, for k =,, 3: k P \ P n0 P n kp \ P n0 P n n by Lemma 4 and the fact that we just skpped the correspondng step f ths was already so for some k, thus verfyng 3 Then, k P n k P \ P n0 P n n = n +, P j October 0, 005

12 DMITRIY BILYK AND LOUKAS GRAFAKOS whch proves And fnally, usng Lemma 4, we have wth the conventon that S 0 = S: 3 I toptjk C + ν 0 k S k 3C + ν 0 n, k k= snce the sum actually ranges over those values of k for whch k S k > n, otherwse the correspondng part of P n s empty Takng nto account the above famles P j, we obtan the followng: 7 H P χ F, χ F, χ H Tjk χ F, χ F, χ j= C k j= C j= = C j= k I toptjk χ F, S j χ F, S j 3 χ, S j F F j mn F F,, j mn F F,, j j F F j mn F F,, j mn F F,, j F F where we used the estmate on a sngle tree 3 and the mproved energy estmate 6 We control 7 n dfferent cases: A Suppose F F Then 7 s bounded by log F j= j + log F j=log F F + F F j=log F j F F F F + log F F So, by symmetry, n the case F, F the expresson 7 can be controlled by 8 mn F, F F F + log F F We may also note that n ths case log F F log F F B Suppose that F F and F F In ths case we can bound 7 by log F j= j + October 0, 005 log F j=log F F + F F j=log F j F F F F + log F F,

13 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 3 Thus, by symmetry, n the case when s between F and F and F F we obtan that 7 s bounded by 9 mn F, F F F + log F F The other cases work n a smlar way: C If s between F and F, but F F, the bound s 30 mn F, F + log F F D For F, F, we obtan the bound 3 mn F, F + log F F Combnng the four cases A, B, C, and D we obtan the followng nequalty for the case when the tles s satsfy I s Ω: H χ {s: Is Ω} F, χ F dx 3 C mn F, F F F mn, + log F log F As a consequence of the results so far we deduce the followng: Proposton 43 There exsts a constant C such that, for any sets,f,f wth the property that F F there exsts a set wth such that for any set of tr-tles S we have the followng estmate: 33 H S χ F, χ F xdx C mn F, F F F + log F F Ths estmate s also vald for the blnear Hlbert transform H Proof The result for H S follows from the estmates and 3 Note that the constructon of dd not depend on the choce of the set of tr-tles, so s the same for any S, and by an averagng argument ths estmate s also vald for H It s clear that, snce both adjonts of H S, are essentally the same operators, the same estmate wth dfferent constants also holds for them 5 L r L r L r boundedness of the model sums In ths secton we wll show that estmates and 3 mply boundedness of the model sum operator H S from L r L r to L r for r + r = r, r, r >, r > 3 We nclude ths secton for the sake of completeness as we wll use ths result n the sequel, but we pont out that the reader may wsh to skp t and cte the results of Lacey and Thele [0],[] Take some p, such that p + = 3 and p, > We wll show that H S s of restrcted weak type r, r, r where r = p ε, r = ε and r = 3 ε By nterpolaton t follows that H S s bounded from L r L r to L r when r r < / and /3 < r < The boundedness of H S n the remanng range of exponents follows by dualty Note that October 0, 005

14 4 DMITRIY BILYK AND LOUKAS GRAFAKOS the same concluson may be obtaned usng the nterpolaton theorem of Grafakos and Tao [7] as the operator H S has bounded kernel whenever S s a fnte set We recall that a blnear operator T s of restrcted weak type r, r, r f and only f the followng s vald: For any sets, F, F of fnte measure there exsts a set wth, such that T χ F, χ F xdx F r F r r Take arbtrary sets, F, F of fnte postve measure It follows from and 3 that H S χ F, χ F dx F p F + log F + log F We wll use the fact that + log x x ε for x In the case when max F, F we can estmate the rghthand sde of 35 by the expresson F p F + log + log F ε p F ε F F ε = F r F r r Now consder the case F F as the case F F s symmetrc Fx some ε > ε Put α = p ε + ε ε and ε have to be chosen small enough, so that α and β = ε + ε thus β also We have α + β = 3 Thus, smlarly to 35, we obtan: H S χ F, χ F dx F α F β F α F β + log F F ε F + log F ε ε = F ε p F ε = F r F r ε r The remanng case s mn F, F We observe that n ths case the set Ω s empty, snce Mχ F We therefore only need to use 7 whch for small yelds: H S χ F, χ F dx mn F, F + log F + log F F p F F ε F ε = F ε p F ε = F r F r ε r Thus, for any measurable sets and F, F, H S satsfes 34 and ths mples that t s of restrcted weak type r, r, r The strong type estmates for the same range of exponents can now be obtaned by varyng r and r and usng the result on nterpolaton between adjont operators cf [7] October 0, 005

15 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 5 6 Dstrbutonal estmates correspondng to the case p =, < Fx < There are some mnor dfferences n the treatment of the cases = and > In the case = for the moment we shall assume that F F CAS: =, 3 F F, F F Snce 3 F F and F F, we have F F Usng estmate 33 we obtan 36 Hχ F, χ F dx C F F 3 + log F F We note that ths estmate s also vald f max F, even when F F We wll use ths estmate n the nductve procedures below CAS: >, + F F, F Let α = > 0, β = > 0 Snce F we must have F F Usng 33 we obtan Hχ F, χ F x dx C mn F, F F F + log F F F F p α C + log + 37 F F F + log + F F F F, + log β F β snce the functon fx = x α + log x β s bounded on [0, ] when α > 0 here x = F CAS:, + F F, F whch mples F In ths case we wll obtan an estmate va an teratve procedure whch conssts of two parts Let us denote by H the adjont of H wth respect to the second varable At frst, we set F 0 = F We wll contnue ths part of the teraton untl the frst nteger n such that F n At the jth step, accordng to the estmates above, we choose a subset S j of F j wth S j F j, such that: F j S j H χ F, χ xdx F + log F j + F F + log F + F Then we set F j+ = F j \ Sj Obvously, for the number of steps n we have n + log F Thus, we have Hχ F, χ F dx F + log F + F F F + log + + log F F F + + Hχ F, χ F n dx Hχ F, χ F n dx October 0, 005

16 6 DMITRIY BILYK AND LOUKAS GRAFAKOS In the last lne we have used the followng smple nequalty wth a = F, b = F : For a, b, such that ab we have 38 To prove 38 we note that f b whle when a b + log ab + + log b a, then log a b + log a + log ab + + log b + log a a, then b b log a = log a and we have + log a + log ab + + log b + log b b It remans to estmate the term Hχ F, χ F n dx b b, In the second part of the teraton process we proceed n a smlar manner, only now we wll be splttng ether F or, dependng on whch one s larger n sze We set n = At the j th step, f j F j, we choose Sj j such that S j j and Hχ F, χ F j dx S j F F j + log j + j F F j + log j F F j F j + log F, F j where we have once agan made use of the fact that fx = x log x x = F j p j p + log F s bounded on [0, ] In the other case, when F j j, we choose S j F j wth Sj F j such that H χ F, χ j dx F j + log F j + S j F j F j An dentcal calculaton and the fact that F j show that ths can also be domnated by F + log F In the frst case we set F j+ = F j, j+ = j \ S j In the second case we set F j+ = F j \ Sj, j+ = j We proceed untl the frst nteger m such that both m, F m F October 0, 005

17 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 7 Obvously, the number of steps n the second part m + log F We now have Hχ F, χ F n dx = Hχ F, χ F n dx n+ S n Hχ F, χ F n dx + Hχ F, χ F n+ dx S n n+ F + log + F Hχ F, χ F n+ dx n+ m F + log + F Hχ F, χ F m dx m F + log + m F 4 F m F 4 F F + log +, F F where we made use of the boundedness of H on L 4 L 4 L and the followng nequalty: For any a,b, such that ba we have + log b + logba a, wth a = F, b = F The proof of ths nequalty s smlar to that n 38 and s omtted CAS:, + F F = Here we wll need the followng lemma We are stll assumng that F F when Lemma 6 Let < For all measurable sets, F, F of fnte measure satsfyng + F F and also F F when = we have Hχ F, χ F xdx + log F F + Proof Let us denote F 0 = F for =, We shall now employ an nductve procedure smlar to the one descrbed above At the j th step among the sets F j and F j we choose the one whch has greater sze and denote t by Fmax j and the other one by F j mn By H max we shall denote the expresson H χ, χ F j n the case when F j max = F j and H χ F j, χ n the other case By 37 or 36 appled to the respectve adjont of H wth the roles of and F j max nterchanged, we can choose S j F j max such that S j F j max and 39 H max x dx F j mn + log F max j + S j Fmax j F j mn We defne F j+ = F j \ Sj for all =,, where we set Sj = Sj f Fmax j = F j and S j = otherwse Let us examne the rghthand sde of the nequalty above If F j mn, t s October 0, 005

18 8 DMITRIY BILYK AND LOUKAS GRAFAKOS easy to check that F j max + F j mn F j mn F j max + p + Fmn F max + p + Thus, n ths case we can estmate the rghthand sde by + In the case when F j mn, we have F + max F max + F j mn p F mn + So, n ths case the rghthand sde of the nequalty can be estmated by F j mn Fmax j + log F max j F j mn, snce the functon fx = x + log x s bounded for x [0, ] Thus, n each case we get H max xdx C + log F F p S j + p We proceed by nducton and we stop at the frst nteger n such that + p F n F n Such an nteger always exsts snce the quantty F n F n F F p log F F p + gets smaller by at least a factor of when j s replaced by j + Obvously, the number of steps n + log F F p + We can now estmate Hχ F, χ F dx = H, χ S 0 + χ F max, dx H max x dx + Hχ F S 0, χ F dx + log F F + + H, χ S + χ F max, dx October 0, 005 n + log F F + + log F F C + + log F F, + Hχ F n, χ F n dx p + + θ F n θ F n θ p

19 DISTRIBUTIONAL STIMATS FOR TH BILINAR HILBRT TRANSFORM 9 where n the second lne from the bottom we have used the Hölder nequalty and the fact that H s of strong type θ, θ, θ + for some large θ In the case > we obtan the followng estmate : For any sets F, F, and of fnte measure we can fnd wth such that [ ][ 40 Hχ F, χ F dx mn, F F + log + p ] + p F F p We now remove the assumpton that F F when = For =, we can consder the symmetrc case when F F, proceed as above wth the roles of F and F nterchanged and puttng together the two estmates we obtan: For any sets F, F, and of fnte measure we can fnd a set wth such that [ ][ 4 Hχ F, χ F dx mn, mn F F F 3 + log ] 3 mn F F F 7 dstrbutonal estmates for the blnear Hlbert transform We can now prove Theorem Proof For a gven λ > 0, we set + λ = {Hχ F, χ F > λ}, λ = {Hχ F, χ F < λ} Suppose that + λ + > F F Then by 40 there s a subset S + λ of + λ half ts measure so that λ + λ S + λ Hχ F, χ F dx C 3 F F + λ + log + λ + F F Ths mples that 4 + λ C 4 F F + λ + + log + λ of at least But then ths mples that there s a λ 0 > 0 such that for λ > λ 0 we have + λ + F F Thus for λ > λ 0, + λ + F F holds and estmate 40 gves λ + λ C 5 + λ + log F F from whch one easly deduces that + λ λ C e c λ F F + Suppose now that λ λ 0 As shown, f + λ + > F F, then 4 s vald If + λ + F F then 43 holds whch s even stronger An dentcal argument yelds the same result for λ wth the same λ 0 For = we run the same argument for estmate 4 and n the end domnate the expresson mn F, F F F by F F October 0, 005

20 0 DMITRIY BILYK AND LOUKAS GRAFAKOS Replacng the constants C, c by dfferent ones we may take λ 0 = and thus estmate s now proved stmate 3 s proved lkewse Fnally, Corollares and 3 are easy consequences of these estmates References [] N Y Antonov, Convergence of Fourer seres, Proceedngs of the XXth Workshop on Functon Theory Moscow 995 ast J Approx 996, [] L Carleson, On convergence and growth of partals sums of Fourer seres, Acta Math 6 966, [3] C Fefferman, Pontwse convergence of Fourer seres, Ann of Math , [4] L Grafakos, Classcal and Modern Fourer Analyss, Prentce Hall, Pearson ducaton, Upper Saddle Rver NJ, 003 [5] L Grafakos and C Lennard, Characterzaton of L p R n usng Gabor frames, J Fourer Anal and Appl 7 00, 0 6 [6] L Grafakos, T Tao, and Terwlleger, L p bounds for a maxmal dyadc sum operator, Math Zet , [7] L Grafakos and T Tao, Multlnear nterpolaton between adjont operators, J Funct Anal , [8] R A Hunt, On the convergence of Fourer Seres, Orthogonal xpansons and ther Contnuous Analogues Proc Conf dwardsvlle, IL 967, D T Hamo ed, Southern Illnos Unv Press, Carbondale IL, [9] M T Lacey On the blnear Hlbert transform, Doc Math 998, xtra Vol II, [0] M T Lacey and C M Thele, L p bounds for the blnear Hlbert transform, < p <, Ann Math , [] M T Lacey and C M Thele, On Calderón s conjecture, Ann of Math , [] M T Lacey and C M Thele, A proof of boundedness of the Carleson operator, Math Res Lett 7 000, [3] C Muscalu, T Tao, and C Thele, Mult-lnear operators gven by sngular multplers, J Amer Math Soc 5 00, [4] P Sjöln and F Sora, Remarks on a theorem by NYu Antonov, Studa Math , Dmtry Blyk, Department of Mathematcs, Unversty of Mssour, Columba, MO 65, USA -mal address: blykd@mathmssouredu Loukas Grafakos, Department of Mathematcs, Unversty of Mssour, Columba, MO 65, USA -mal address: loukas@mathmssouredu October 0, 005