Y. RAKOTONDRATSIMBA. Abstract. Sufficient (almost necessary) conditions are given on the weight functions u( ), v( ) for Φ 1 [C 1.

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1 GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 6, 996, WEIGHTED L Φ INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS Y. RAKOTONDRATSIMBA Abstact. Sufficiet almost ecessay) coditios ae give o the weight fuctios u ), v ) fo ) C M sf)x) ux)dx C Φ fx) )vx)dx Φ to hold whe Φ, Φ ae ϕ-fuctios with subadditive Φ, ad M s 0 s < ), is the usual factioal maximal opeato.. Itoductio Let N, 0 s <. The factioal maximal opeato M s of ode s is defied as } M s f)x) = sup { s fy) dy; is a cube with x, x, y. Thoughout this pape, will deote a cube with the sides paallel to the coodiate plaes. As i, a eal fuctio Φ ) defied o 0, is called a ϕ-fuctio if it is a odeceasig cotiuous fuctio which satisfies Φ0) = 0 ad lim s Φs) =. The ϕ-fuctio Φ ) is subadditive if Φt + t ) Φt ) + Φt ) fo all t, t 0,. Let u ), v ) be weight fuctios i.e., oegative locally itegable fuctios). I this pape we study the itegal iequality Φ C M s f)x) ) ux)dx Φ C Φ fx) )vx)dx ) 99 Mathematics Subject Classificatio. 4B5, 46E30. Key wods ad phases. Factioal maximal opeatos, Weighted iequalities, Olicz spaces X/96/ $09.50/0 c 996 Pleum Publishig Copoatio

2 584 Y. RAKOTONDRATSIMBA fo all fuctios f ). Hee Φ ), Φ ) ae ϕ-fuctios with subadditive Φ ) ), ad C, C ae oegative costats which do ot deped o each fuctio f ). Fo coveiece we also deote this iequality by M s : L Φ v L Φ u. As metioed i a ecet moogaph of Kokilashvili ad Kbec, this is a ope poblem to chaacteize a pai of weight fuctios u ), v ) fo which M s : Lv Φ L Φ u holds. Such a itegal iequality ca be useful i studyig the bouday value poblems fo quasiliea patial diffeetial equatios see 3). I the Lebesgue case, i.e., Φ t) = t p, Φ t) = t q with < p q <, iequality ) ca be ewitte as M s f) q x)ux)dx ) q C ) fx) p p vx)dx. ) Sawye 4 chaacteized the weight fuctios u ), v ) fo which ) held. Oe of the cucial keys he used to solve this poblem is to ote the equivalece of ) to M s σg) q x)ux)dx ) q C ) gx) p p σx)dx, 3) whee σ ) = v p ). Ideed, fσ R )x) p σx)dx = fx) p vx)dx. Thus i the Olicz settig, iequality 3) leads atually to the followig fist geealizatio: Φ C M s σg)x) ) ux)dx Φ C Φ gx) )σx)dx whee σ ) is some weight fuctio. Such itegal iequalities wee studied by L. isheg 5, ad, idepedetly, by the autho 6. Roughly speakig, iequality 4) is ot techically too fa fom the Lebesgue settig, ad so this poblem ca be hadled by Sawye s ideas 4. The secod geealizatio of ) is the two-weight modula iequality ) o M s : L Φ v L Φ u itoduced above. This iequality is moe difficult to study tha 4). Ideed, fom the latte, we caot easily deive ), sice cotay to the Lebesgue case, thee is o obvious coectio betwee Φ fx)σ )x) )σx)dx ad Φ fx) )vx)dx. The poblem M : Lw Φ L Φ w, whee M = M 0 is the classical Hady Littlewood maximal opeato, was studied by Kema ad Tochisky 7. They cosideed a N-fuctio Φ ) with Φ ), Φ ). Φt) Recall that a N-fuctio is a covex ϕ-fuctio satisfyig lim t 0 lim t t 4) t = Φt) = 0, ad the associated cojugate fuctio Φ ) is defied by

3 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 585 Φ t) = sup s>0 {st Φs)}. The coditio Φ ) meas Φt) CΦt) fo all t > 0. As i the Lebesgue settig, the poblem M : L Φ v L Φ u fo u = v is completely diffeet. Usig the Lebesgue agumets 4, Che 8 ad Su 9 studied this poblem idepedetly. They obtaied the followig esult: Suppose Φ ) is a N-fuctio with Φ ), Φ ) ad thee is a weight fuctio σ ) satisfyig the exta-assumptio: sup Φ τ y Ñ τy στ y f) x) ) vx)dx C Φ fx) )vx)dx ) y fo all fuctios f ); the M : Lv Φ Lu Φ holds if ad oly if Φ MεσI )x) ) ux)dx c Φ εσi )x) ) vx)dx 5) fo all cubes ad ε > 0. Hee τ y f)x) = fx y), Ñ σ fx) = σx)n σ fσ )x), ad N σ is the maximal fuctio defied by { } N σ g)x) = sup gy) σy)dy; is a cube with x. σ Ufotuately, this esult has two mai dawbacks. Fistly, fo a geeal N-fuctio Φ ), o exact coditio is kow to be imposed o a weight fuctio v ) fo which thee is aothe weight fuctio σ ) satisfyig the exta-coditio ). Secodly, as i the Lebesgue settig, the Sawye s coditio 5) is ot easy to check sice it is expessed i tems of the maximal fuctio M itself. Thus it was a challegig poblem fo specialists i weighted iequalities to obtai a sufficiet almost ecessay) coditio o weight fuctios u ), v ) which esues M s : L Φ v L Φ u. A attempt i this diectio was made by the autho i 0. Fo completeess, we ecall hee the esult he obtaied. Let <, Ψ t) = Φ t ) be a N-fuctio ad Φ ) ) be subadditive. Moeove, let us assume Φ ), Ψ ). The M s : L Φ v holds if ad oly if L Φ u A ε s ) uy)dy A Φ ε) vy)dy) ) with costats A ad A fo all cubes ad all ε > 0, wheeve the weight fuctio v ) belogs to the Muckehoupt class A p fo < p.

4 586 Y. RAKOTONDRATSIMBA With the defiitio give below, w A p if ad oly if w, w) A 0, p, p). Although the test coditio ) is moe computable tha the above Sawye oe 5), the estictio o the weight fuctio v ) is a icoveiece. Theefoe ou mai pupose i this pape is to deive M s : L Φ v L Φ u by a simila test coditio without estictios o the weight fuctios u ), v ). This coditio is deoted by u, v) A s, Φ, Φ ) ) ad meas Φ ) s A Φ A ε ) ε εuy)dy εvx)) εvx)) dx 6) fo all cubes ad all ε > 0. The pesece of ε > 0 ca be explaied by the lack of homogeeities of the ϕ-fuctios Φ ) ad Φ ). Assumig that Φ ) is a N-fuctio fo some >, we will pove that u, v) A s, Φ, Φ ) is the sufficiet coditio i ode that M s : L Φ v L Φ u see Popositio 3). This sufficiet coditio we itoduce is almost ecessay i the sese that u, v) A s, Φ, Φ, ) is a ecessay coditio fo this embeddig see Popositio ). Ou esult icludes oe esult due to Peez. We wee eally ispied by this autho s techique, which we will develop i the Olicz settig. I ode to iclude the modula iequality 4), we also deal with the geeal embeddig M s : L Φ v w ) L Φ u w ), i.e., Φ Φ C w x)m s f)x) ) ux)dx Φ C Φ w x) fx) ) vx)dx. 7) Let us coside agai the iequality M s : Lv Φ L Φ u. The coespodig weak vesio M s : L Φ v L Φ, u is defied by Φ C λ) ux)dx C Φ fx) )vx)dx 8) {M s f)x)>λ} fo all fuctios f ) ad all λ > 0. Usig the ideas fom, we will pove that if Φ ) ) is subadditive ad Φ ) is a N-fuctio, the M s : L Φ v L Φ, u holds if ad oly if u, v) A s, Φ, Φ, ). This weak iequality was aleady solved ad cosideed by may authos, 3 fo the case s = 0 ad Φ ) = Φ ). We emphasize that i this pape we eve use the coditio fo the ϕ-fuctios Φ ) ad Φ ).

5 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 587 We leave ope the poblem I s : L Φ v L Φ u, whee I s 0 < s < ) is the factioal itegal defied by I s fx) = x y s fy)dy. A chaacteizatio of the weight fuctios u ), v ) fo which the coespodig weak iequality was poved was doe by the autho i 4. We state ou mai esults i, ad we will give the basic lemma i the ext sectio. Weak iequalities ae poved i 4, ad 5 is devoted to povig stog iequalities. The last sectio deals with ou test coditio 6).. The Results Weak Iequalities We fist chaacteize the weight fuctios u ), v ) fo which the weak iequality 8) holds. Popositio. Let 0 s <, Φ ) be a ϕ-fuctio ad Φ ) be a N-fuctio with subadditive Φ Φ ) ). Assume that the weak iequality M s : Lv Φ L Φ, u is satisfied fo some costats C, C > 0. The u, v) A s, Φ, Φ ) with the costats A = C ad A = C. Covesely, let u, v) A s, Φ, Φ ) fo some costats A, A > 0. The M s : Lv Φ L Φ, u is satisfied with the costats C = NA, C = CA. Hee N = N), C = Cs, ) ae oegative costats which deped espectively o ad s,. I fact, this weak iequality ca be cosideed as the paticula case of M s : L Φ w ), i.e., v w ) L Φ, u {M sf)x)>λ} C Φ Φ C λw x) ) ux)dx ) w x) fx) vx)dx fo all fuctios f ) ad all λ > 0. Hee v ), u ), w ), w ) ae weight fuctios. Recall see 4) that fo each N-fuctio Φ ) ad fo each weight fuctio w ), the quatity Φ,w defied by { } f ) Φ,w = sup fy)gy) wy)dy; Φ gy) )wy)dy yields a om i the Olicz space L Φ w which is the set of all measuable fuctios f ) satisfyig Φλ fy) )wy)dy < fo some λ > 0). Ad fo each cube we also defie f ) Φ,,w as f ) Φ, I w ). 9)

6 588 Y. RAKOTONDRATSIMBA Ou chaacteizatio of the weight fuctios u ), v ), w ), w ) fo the weak iequality 9) ca be stated as Theoem. Let 0 s <, Φ ) be a ϕ-fuctio ad Φ ) be a N-fuctio with subadditive Φ ) ). Suppose that the weak iequality M s : L Φ v w ) L Φ, u w ) is satisfied fo some costats C, C > 0. The u, v) A s, Φ, Φ, w, w ), i.e., Φ Φ A w x) s ) ux)dx Φ A ε w )εv ) Φ,,εv fo all cubes ad all ε > 0. Hee A = C, ad A = C. Covesely, if u, v) A s, Φ, Φ, w, w ), fo some costats A, A > 0, the M s : Lv Φ w ) L Φ, u w ) holds with the costats C = NA, C = CA. Fo w ) =, the coditio u, v) As, Φ, Φ, w, w ) ca be educed to the easy fom Φ s Φ A Φ A ε ) ε εvx)dx. εuy)dy w x)εvx) Ideed, fo each N-fuctio Ψ ) ad each weight fuctio w ) we have f Ψ,w equivalet to Ψ fx) )wx)dx. Obseve that u, v) A s, Φ, Φ,, ) is the same as the coditio u, v) A s, Φ, Φ ) itoduced above. Stog Iequalities Popositio 3. Let 0 s <, Φ ), Φ ) be ϕ-fuctios with subadditive Φ ) ). Suppose Φ ) is a N-fuctio ad the stog iequality M s : L Φ v L Φ u is satisfied fo some costats C, C > 0. The u, v) A s, Φ, Φ ) with the costats A = C ad A = C. Covesely, we assume that Φ ) is a N-fuctio fo some > ad u, v) A s, Φ, Φ ) fo some costats A, A > 0. The M s : L Φ v Lu Φ is satisfied with the costats C = N)A, C = cs, )A. I the Lebesgue case Φ t) = t p p > ) we take as = p p s) = ps p+ s s > ). Hee >, ad Φ t) = t p s). Thus ou popositio icludes a Peez esult 5. I fact, the stog iequality cosideed hee is the

7 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 589 paticula case of M s : L Φ v w ) L Φ u w ) o iequality 7). Thus the mai esult of this pape is Theoem 4. Let 0 s <, Φ ), Φ ) be ϕ-fuctios with subadditive Φ ) ). Let Φ be a N-fuctio ad suppose that the stog iequality M s : L Φ v w ) L Φ u w ) is satisfied fo some costats C, C > 0. The u, v) A s, Φ, Φ, w, w ), with the costats A = C ad A = C. Covesely, we assume that Φ ) is a N-fuctio fo some > ad u, v) A s, Φ, Φ, w, w ) fo some costats A, A > 0, i.e., Φ A w x) s ) w )εv )) Φ ux)dx ),,εv )) A ε fo all cubes ad all ε > 0. The M s : L Φ v w ) L Φ u w ) is satisfied with the costats C = N)A, C = cs, )A. Fo w ) =, the coditio u, v) A s, Φ, Φ, w, w ) ca be educed to the easy fom Φ ) s A Φ Aε ) ε εvx)) dx. εuy)dy w x)εvx)) Due to the Hölde iequality, it is clea that fo each > Φ Φ w )εv ),,εv ) w )εv ) ; ),,εv ) thus u, v) A s, Φ, Φ, w, w ) becomes a ecessay ad sufficiet coditio fo the stog iequality M s : L Φ v w ) Lu Φ w ) if fo some > Φ C w )εv ),,εv ) w )εv ) Φ. ),,εv ) The latte iequality was studied by Peez 5 i the Lebesgue case. As we have see i the fist pat of Popositio 3, u, v) A s, Φ, Φ ) is a ecessay coditio i ode that M s : L Φ v L Φ u. The agumet we will use to get Theoem 4 also ivolves

8 590 Y. RAKOTONDRATSIMBA Popositio 5. Let 0 s <, Φ ) be a N-fuctio, Φ ) a ϕ- fuctio with subadditive Φ Φ ) ). Suppose u, v) A s, Φ, Φ ). The C M s f)x) ) ux)dx C Φ fx) )vx)dx Φ fo all fuctios f ) satisfyig the evese Hölde iequality ω c ω ) Φ fx) )vx) ωx)dx ωx) whee > ad ω ) is a A weight fuctio. Φ fx) )vx) ωx)dx, 0) ωx) Of couse, C, C will deped o c, ad o the costats of the coditio A s, Φ, Φ ). Now we will discuss the meas fo checkig the test coditio u, v) A s, Φ, Φ, w, ). Fo simplicity we will coside oly the case whee Φ ) = Φ ) = Φ ), sice the geeal case ca be teated likewise with mio modificatios. Let us deote by A, ε, A, A the quatity Φ ) s A Φ A uε εvx)) w x)εvx)) dx. With this otatio, u, v) A s, Φ, Φ, w, ) if ad oly if thee ae A, A > 0 fo which sup ε>0 sup A, ε, A, A. It is clea that c 0 sup sup ε>0 R>0 AB0, R), ε, c A, c A sup ε>0 sup A, ε, A, A whee B0, R) is the ball ceteed at the oigi ad with adius R > 0, ad c 0, c, c ae oegative costats which deped o s ad. I applicatios it is easie to compute AB0, R), ε, c A, c A tha the expessio A, ε, A, A, especialy whe u ), v ) ae adial weight fuctios. Thus it is iteestig to kow whe the evese of the last iequality is satisfied. I ode to aswe this questio, we say that a weight fuctio w ) satisfies the gowth coditio C), whe thee ae costats c, C > 0 such that sup wx) C R< x R wy)dy y cr

9 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 59 fo all R > 0. May of the usual weight fuctios w ) satisfy this gowth coditio, sice both oiceasig ad odeceasig adial fuctios ae admissible. This is also the case whe the weight w ) is essetially costat o auli, i.e., wy) cwx) fo y x y. Now we ca state Popositio 6. Let <, ad Φ ) ) be a N-fuctio. Assume that u ) is a) weight satisfyig C). Also suppose that the weights Φ ) λ v ) w )v ) satisfy the gowth coditio C) uifomly i λ > 0. The thee ae oegative costats c 0, c, c such that sup ε>0 sup A, ε, A, A c 0 sup sup AB0, R), ε, c A, c A. ε>0 R>0 Ideed, c 0, c, c deped o the costats i the gowth coditios ivolvig u ), v ), w ) but ot o these idividual weights. We oly have to pove Theoems, 4, ad Popositios 5, 6. We fist begi by the basic esult udelyig the poof of these esults. 3. A Basic Lemma Lemma 7. Let, Φ ) be a N-fuctio. Suppose u, v) A s, Φ, Φ, w, w ) fo some costats A, A > 0. The Φ C w x) s C ) fy) dy ux)dx Φ w x) fx) )vx) ) dx ) fo all cubes ad all locally itegable fuctios f ) with suppots cotaied i. Hee C = A, C = A. Covesely, iequality ) implies u, v) A s, Φ, Φ, w, w ) with the costats A = C, A = C. Let Φ ) be a N-fuctio ad w ) a weight fuctio. The Hölde iequality assets that fy)gy) wy)dy f Φ,w g, Φ,w whee f Φ,w = if{s > 0; Φλ fy) )wy)dy }. Poof of Lemma 7. Suppose the coditio u, v) A s, Φ, Φ, w, w ) is satisfied fo some costats A, A > 0.

10 59 Y. RAKOTONDRATSIMBA Let be a cube ad f ) a locally itegable fuctio whose suppot is cotaied i. Without loss of geeality we ca assume 0 < Φ w x) fx) )vx) dx <. Note that w )v ) is ot idetically zeo o. The by the Hölde iequality we have fx) dx = w x) fx) ) ) εvx)) w x)εvx)) dx w )f ) Φ,,εv) w )εv )) fo all ε > 0. Choosig ε > 0 such that Φ w x) fx) )εvx) ) dx = o ε = Φ w x) fx) )vx) dx), we have w )f ). Φ,,εv) Φ ),,εv) Next, usig the coditio u, v) A s, Φ, Φ, w, w ) ad the above two iequalities, we get M = Φ Φ A w x) s ) fy) dy ux)dx Φ A w x) s ) w )εv )) Φ ux)dx ),,εv) A ε = A Φ w x) fx) )vx) ) dx. Theefoe iequality ) is satisfied with the costats C = A, C = A. Covesely, suppose that iequality ) is satisfied fo some costats C, C > 0. Let be a cube ad ε > 0. By the defiitio of the Olicz s om, thee is a oegative fuctio g ) such that Φ w x)gx))εvx) dx

11 ad theefoe INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 593 w )εv )) Φ ),,εv) = gy)dy. Now usig iequality ) with such a fuctio g ) we obtai M = Φ Φ C w x) s ) Φ w )εv )) ux)dx ),,εv) Φ C s ) gy)dy w ux)dx C Φ w x)gx))vx) ) dx C ε. Thus u, v) A s, Φ, Φ, w, w ) with the costats A = C, A = C. 4. Poof of Theoem Assume that Φ ) is a N-fuctio, ad the weak iequality M s : L Φ v w ) L Φ, u w ) is satisfied with the costats C, C > 0. Let be a cube, ad f ) be a locally itegable fuctio whose suppot is i. Sice M s f)x) s fy) dy)i x), by takig the eal λ = s fy) dy) i the above weak iequality we get iequality ) with the costats C ad C. Theefoe by the fist pat of Lemma 7, u, v) A s, Φ, Φ, w, w ) with the costats A = C, A = C. To pove the covese, we follow the same lies we used i. Thus, suppose u, v) A s, Φ, Φ, w, w ) with some costats A, A > 0. Let N be a oegative itege ad Ms N the tucated maximal opeato defied by Ms N f)x) = sup { s fy) dy; is a cube with x ad N Let λ > 0, f ) be a locally itegable fuctio ad Ω λ,n = {Ms N f) ) > λ}. Sice {M ux)dx = lim sf)x)>λ} N Ω λ,n ux)dx, it is sufficiet to pove C λw x) ) ) ux)dx Φ C w x) fx) vx)dx, Ω λ,n Φ Φ }.

12 594 Y. RAKOTONDRATSIMBA whee C, C, do ot deped o the itege N. Fo evey x Ω λ,n, thee is a cube x) ceteed at x such that x) s ) fy) dy > cλ, x) x) whee c = cs, ) > 0 depeds oly o s ad. Sice sup{ x) ; x Ω λ,n } <, by the classical Besicovitch coveig theoem we ca choose fom the set {x); x Ω λ,n } a sequece of cubes k ) k satisfyig ) Ω λ,n k ; I k K)I k ; k s fy) dy > cλ. k k k k By the secod pat of Lemma 7, the coditio u, v) A s, Φ, Φ, w, w ) implies iequality ). Usig the latte ad the fact that Φ Φ ) ) is subadditive, we obtai ca λw x) ) ux)dx ca λw x) ) ux)dx Ω λ,n Φ k k Φ A w x) k s k Φ A k Φ A Φ k k Φ k Φ C)A k Φ k Φ k k fy) dy ) ux)dx w x) fx) ) vx)dx w x) fx) ) vx)dx w x) fx) ) vx)dx, ad theefoe M s : L Φ v w ) L Φ, u w ) holds with the costats C = C)A, C = cs, )A. 5. Poofs of Theoem 4 ad Popositio 5 Poof of the ecessity pat of Theoem 4. Assume that Φ is a N-fuctio ad suppose the stog iequality M s : L Φ v w ) L Φ u w ) is satisfied fo some costats C, C > 0. This implies the weak iequality M s : L Φ v w ) Lu Φ, w ) with the same costats C, C > 0. Thus, as i the poof of Theoem, u, v) As, Φ, Φ, w, w ) with the costats A = C, A = C.

13 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 595 To pove the covese, to get M s : L Φ v w ) L Φ u w ), we fist do some pelimiaies so as to discetize the opeato M s. Pelimiaies fo the poof of sufficiecy pat of Theoem 4. Recall that the dyadic vesio Ms d of the maximal opeato M s is defied as } Ms d f)x) = sup { s fy) dy; is a dyadic cube with x. We eed Lemma 8. Let f ) be a fuctio with a compact suppot ad λ > 0. If {M s f) ) > λ} is a oempty set the we ca fid a family of oovelappig maximal dyadic cubes j ) j fo which { Ms f) ) > λ } ) 3 j ), s λ < j s fy) dy λ, j j { M d s f) ) > s λ } = j j. j This lemma is the stadad oe ad was poved i 5. As a cosequece we obtai Lemma 9. Let f ) be a fuctio with a compact suppot. itege k let Fo each Ω k = { M s f) ) > a k}, Γ k = { M d s f) ) > a k}, whee a = 3. Thee is a family of maximal oovelappig dyadic cubes kj ) j fo which Ω k 3 kj ), Γ k = kj, j j ) a k < kj s fy) dy s a k. kj kj ) Let E k,j = kj \ kj Γ k+ ). The E k,j ) k,j is a disjoit family of sets fo which k,j < c E k,j with c = cs, ) = +s) +s). 3) The fist pat of this esult is a immediate cosequece of Lemma 8, fo λ = s a k, while the secod oe follows fom the estimate k,j Γ k+ < +s) k,j.

14 596 Y. RAKOTONDRATSIMBA The latte oe ca be obtaied fom ) as follows, k,j Γ k+ = k,j k+,i k+,i i i; k+,i k,j a k+ k+,i s fy) dy i; k+,i k,j s a k+ s a k+ k,j s k,j s i; k+,i k,j k,j fy) dy k+,i k+,i s a fy) dy k,j = +s) k,j. Now we ca poceed to Poof of the sufficiecy pat of Theoem 4. We assume that Φ ) is a N-fuctio fo some > ad u, v) A s, Φ, Φ, w, w ) fo some costats A, A > 0. We have to pove M s : L Φ v w ) L Φ u w ), i.e., Φ C w x)m s f)x) ) ux)dx Φ C Φ w x) fx) ) vx)dx fo all locally itegable fuctios f ). We assume that f ) is bouded ad it has a compact suppot. We do ot lose geeality, sice the estimates we obtai do ot deped o the boud of f ), ad the mootoe covegece theoem yields the coclusio. With the otatios of Lemma 9, we have S = Φ = k k,j k,j 3 3 A w x)m s f)x) ) ux)dx = Ω k Ω k+ Φ 3 k,j Φ 3 k,j Φ 3 3 A w x)m s f)x) ) ux)dx 3 3 A a k+ w x) ) ux)dx 3 A w x) k,j s k,j k,j ) fy) dy ux)dx

15 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 597 see estimate ) i Lemma 9) Φ A w x) 3 k,j s ) fy) dy ux)dx 3 k,j k,j 3 k,j 3 k,j Φ A 3 k,j Φ w x) fx) ) vx) ) dx 3 k,j k,j 3 k,j by the fist pat of Lemma 7) Φ A k,j 3 k,j 3 k,j 3 k,j Φ w x) fx) ) vx) ) dx ecall that Φ is subadditive) = Φ 3 A k,j 3 k,j k,j Φ 3 cs, )A E k,j 3 k,j 3 k,j k,j see estimate 3) i Lemma 9) Φ 3 cs, )A 3 k,j Φ w x) fx) ) vx) ) dx k,j E k,j Φ w x) fx) ) vx) ) dx ) MΦ w f )v x)dx M = M 0 is the Hady Littlewood maximal opeato) Φ 3 cs, )A M Φ w f )v ) x)dx E k,j s ae disjoit sets) Φ 3 cs, )A Φ w x) fx) ) vx)dx by the well kow maximal theoem > )). Thus the iequality is fulfilled with the costats C = 3 +s) +s) A, C = 3 3 A.

16 598 Y. RAKOTONDRATSIMBA Poof of popositio 5. Sice this esult ca be obtaied by usig a few chages i the above estimates, we will outlie the essetial agumets. With the above otatio, the coditio ω ) A implies the existece of a costat c = cω) > 0 such that E kj ω c kj ω. Now, as above, by Lemma 7 ecall that u, v) A s, Φ, Φ )) we obtai k,j S = Φ Φ c A E kj ω 3 kj ω Φ Φ c A E kj ω 3 kj ω k,j Φ Φ cc A cc A Φ k,j E kj c A M s f)x) ) ux)dx 3 k,j 3 k,j Φ fx) )vx) ωx)dx) ωx) ) Φ fx) )vx) ωx)dx ωx) M ω Φ f )v ) x)ωx)dx ω M ω Φ f )v ω cc A Φ ) x)ωx)dx fx) ) vx)dx. 6. Poof of Popositio 6 Let ε > 0 ad be a cube ceteed at x 0 ad havig a side with legth R > 0. Fist we suppose x 0 R. The fo a costat c which depeds oly o the dimesio we obtai B0, c R), which esults i A, ε, A, A c ) c R) B0,c R) Φ ) c R) s c ) s A εvx)) Φ c ) w A x)εvx)) dx. c R) B0,c εuy)dy R)

17 INTEGRAL INEUALITIES FOR MAXIMAL OPERATORS 599 Next we coside the case R < x 0. The x x 0 wheeve x, ad B0, c 3 x 0 ) with a costat c 3 which depeds oly o. Thus, the gowth coditio o u ) yields uy)dy c 4, u) c 3 x 0 ) B0,c 3 x 0 ) uy)dy. ) Sice the family of weights Φ ) λ v ) w )v ) i λ) the gowth coditio C), we obtai A, ε, A, A Φ ) c 3 x 0 ) s c A Φ c A c 3 x 0 ) s c A Φ c A c 3 x 0 ) B0,c εuy)dy 3 x 0 ) supφ ) x c 3 x 0 ) B0,c εuy)dy 3 x 0 ) c 5 c 3 x 0 ) c 3 x 0 ) s c A Φ c A B0,c 3 x 0 ) c 3 x 0 ) B0,c εuy)dy 3 x 0 ) Fially, by these estimates we get sup ε>0 sup A, ε, A, A c 0 sup sup ε>0 R>0 satisfies uifomly εvx)) w x)εvx)) dx w x)εvx)) Φ ) εvx)) w x)εvx)) εvx)) dx. AB0, R), ε, c A, c A. Refeeces. J. Musielak, Olicz spaces ad modula spaces. Lectue Notes i Math. 034, Spige, Beli etc., V. Kokilashvili ad M. Kbec, Weighted iequalities i Loetz ad Olicz spaces. Wold Scietific, Sigapoe, 99.

18 600 Y. RAKOTONDRATSIMBA 3. M. Schechte, Potetial estimates i Olicz spaces. Pacific J. Math. 33)988), E. Sawye, A chaacteizatio of a two weight om iequality fo maximal opeatos. Studia Math. 7598),. 5. L. isheg, Two weight mixed Φ-iequalities fo the Hady opeato ad the Hady Littlewood maximal opeato. J. Lodo Math. Soc. 46)99), Y. Rakotodatsimba, Iégalités à poids pou des opéateus maximaux et des opéateus de type potetiel. Thèse de Doctoat. Uiv. Oléas, Face, R. Kema ad A. Tochisky, Itegal iequality with weights fo the Hady maximal fuctio. Studia Math ), J. C. Che, Weights ad L Φ -boudedess of the Poisso itegal opeato. Isael J. Math. 8993), Su, Weighted om iequalities o spaces of homogeeous type. Studia Math. 03)99), Y. Rakotodatsimba, Weighted stog iequalities fo maximal fuctios i Olicz spaces. Pepit Uiv. Oléas, Face, 99.. C. Peez, Two weighted iequalities fo potetial ad factioal type maximal opeatos. Idiaa Uiv. Math. J ), Y. Rakotodatsimba, Weighted weak iequalities fo some maximal fuctios i Olicz spaces. Pepit Uiv. Oléas, Face, L. Pick, Two-weight weak type maximal iequalities i Olicz classes. Studia Math. 003)99), Y. Rakotodatsimba, Weighted weak iequalities fo factioal itegals i Olicz spaces. Pepit Uiv. Oléas, Face, 99. Autho s addess: Uevesité d Oléas Depatemet de Mathematiques U.F.R. Faculté des Scieces B.P Oléas cedex Face Cuet addess: Received ) Cete Polytecique St-Louis Ecole de Physique et de Mathematiques Idustielles 3, boulevad de l Hautil Cegy-Potoise cedex Face