Weighted inequalities for multilinear fractional integral operators

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1 Collect. Math. 60, 2 (2009, c 2009 Unverstat de Barcelona Weghted nequaltes for multlnear fractonal ntegral operators Kabe Moen Department of Mathematcs, Unversty of Kansas 405 Snow Hall 460 Jayhawk Blvd, Lawrence, Kansas , USA E-mal: moen@math.ku.edu Receved July 0, Revsed November 6, 2008 Abstract A weghted theory for multlnear fractonal ntegral operators and maxmal functons s presented. Suffcent condtons for the two weght nequaltes of these operators are found, ncludng power and logarthmc bumps and an A condton. For one weght nequaltes a necessary and suffcent condton s then obtaned as a consequence of the two weght nequaltes. As an applcaton, Poncaré and Sobolev nequaltes adapted to the multlnear settng are presented.. Introducton As t s well-known, Muckenhoupt [5] characterzed the weghts w, for whch the Hardy-Lttlewood maxmal operator, M, s bounded on L p (w for < p <. He showed that M s bounded on L p (w f and only f w belongs to the class A p,.e., ( ( p [w] Ap = sup w w <. Sawyer [2] characterzed the two weght nequalty, showng that M : L p (v L p (u f and only f the par (u, v satsfes the testng condton [u, v] Sp = sup M(χ σ p u < σ( Research supported n part by the Natonal Scence Foundaton under grant DMS Keywords: Fractonal ntegrals, maxmal operators, weghted norm nequaltes, multlnear operators. MSC2000: 26D0, 42B25. 23

2 24 Moen where σ = v. The Hardy-Lttlewood maxmal operator and the weghted estmates t satsfes play a very mportant role n harmonc analyss. In partcular M s ntmately related to the study of sngular ntegral operators. Also of mportance n harmonc analyss s the study of fractonal type operators and assocated maxmal functons. Recall the defnton of the fractonal ntegral operator or Resz potental, f(y I α f(x = dy, 0 < α < n. x y n α R n and the related maxmal functon, M α f(x = sup x α/n f(y dy, 0 α < n, where the supremum s taken over all cubes contanng x. Note that the case α = 0 corresponds to the Hardy-Lttlewood maxmal operator. A nce exposton about the propertes of the operators partcularly I α can be found n the books by Sten [24] and Grafakos [8]. Weghted estmates for I α have been studed as well. In Muckenhoupt and Wheeden [6], characterzed the one weght strong type nequalty, ( /q ( /p, (I α fw q C (fw p (. where, f 0, < p < n/α and q s defned by /q = /p α/n. They showed that (. holds f and only f w A p,q.e., ( /q ( /p [w] Ap,q = sup w q w <. These estmates are of nterest on ther own and they also have relevance to partal dfferental equatons and quantum mechancs. We refer the reader to Sawyer and Wheeden [23], Kerman and Sawyer [3] for further nformaton and applcatons. A characterzaton for the two weght nequalty for I α was gven by Sawyer [22]. Further results for these operators and more general potental operators were obtaned by Pérez [8, 9]. Of partcular nterest are the results n [9] where the power bump condton due to Neugebauer [7] s extended usng Banach functon spaces. See Cruz- Urbe, Martell, and Pérez, [4], for further references and hstorcal remarks. Multlnear maxmal functons appear naturally n connecton wth multlnear Calderón-Zygmund theory. Lerner, Ombros, Pérez, Torres, and Trujllo-González [4] developed a weghted theory for the mult(sublnear maxmal functon M( f(x = sup x f (y dy where f = (f,..., f m. They showed that for < p,..., p m <, and p gven by /p = /p + + /p m, Mf L p (ν w C f L p (w

3 Weghted nequaltes for multlnear fractonal ntegral operators 25 where ν w = m w p/p f and only f w A P.e., ( [ w] A P = sup ν w /p m ( /p w <. Ths lead to a development of a multlnear weghted theory for multlnear Calderón- Zygmund operators and other operators whch answered questons posed n earler works on the subject by Grafakos and Torres [0, ], and Pérez and Torres [20]. Motvated by the work n [4] we consder here the multlnear fractonal case. Multlnear fractonal ntegral operators were studed by Grafakos [7], Keng and Sten [2], Grafakos and Kalton [9]. These works present multlnear generalzatons of the blnear operator, f(x + tg(x t B α (f, g(x = R n t n α dt, 0 < α < n. They showed that B α maps L p L p 2 nto L q where /q = /p + /p 2 α/n. As a tool to understand B α, the operators I αf(x f (y f m (y m = d y (R n m ( x y + + x y m mn α 0 < α < nm were studed as well. We examne the one and two weght theory for these last operators and the correspondng mult(sublnear fractonal maxmal operators, M α f(x = sup x α/nm f (y dy. We obtan much of the multlnear counter part of the lnear results n [9]. The extenson to the multlnear settng, however, s not mmedate and new deas are requred. We also prove the multlnear analog of the result n [6] In partcular we show that the nequalty, ( /q ( I αf (Π m w q m ( C ( f w p f and only f w = (w,..., w m satsfes the A P,q condton, ( sup (Π m w q /q m /p ( /p w <. The general organzaton of the paper s as follows. Secton 2 contans some prelmnary defntons and the statements of the two weght results. Many of these are corollares of our man results Theorem 2.2 and Theorem 2.8. Secton 3 contans the statements of the one weght results. The proof of the two weght theorems are n Secton 4 whle the proof of the one weght theorems are n Secton 5. In Secton 6 we present a verson of Theorems 2.2 and 2.8 n the more general context of Banach functon spaces and gve examples. Fnally, we present n Secton 7 some applcatons of the theory, ncludng Poncaré and Sobolev nequaltes for products of functons.

4 26 Moen The work [4] has spurred several efforts by other authors n the study of multlnear fractonal ntegrals and maxmal operators. After the research presented here was completed we were nformed of recent, smultaneous, and ndependent results by X Chen and ngyng Xue [2] and Alberto de la Torre [5]. In the one weght case, Chen and Xue have nvestgated suffcent condtons for boundedness of multlnear fractonal operators, arrvng to the same one we have. De la Torre has also found the same condton we found. He has a dfferent proof for the suffcency of the condton for the multlnear fractonal maxmal operator. We not only prove suffcency but also the necessty of the condton and our methods of proof are dfferent from those n [2] and [5]. The author would lke to thank Carlos Pérez for hs suggestons, and frutful nteracton. The author would also lke to thank Rodolfo Torres for hs advce and support. 2. Two weght results We recall some standard notaton. Throughout ths paper we consder cubes, usually denoted, wth sdes parallel to the axes. The sde length of a cube s denoted l(, and a, a > 0, denotes the cube concentrc wth and sde length al(. The set of dyadc cubes n R d, denoted D(R d or smply D when the dmenson s evdent, s the set of all half open cubes of the form m,k = 2 k (m + [0, d where k Z and m Z d. For convnence f D we wll say l( = 2 k for some k Z. In ths artcle a weght s smply a non-negatve measurable functon w. Gven a measurable set E, w(e wll denoted the weghted measure of E,.e., w(e = E w. Occasonally we wll use the notaton T : X Y to mean T s a bounded operator from X to Y,.e. T x Y C x X for all x n X. The multlnear verson, T : X X m Y means, T (x,..., x m Y CΠ x X for all (x,..., x m n X X m. For each q, q wll denote the dual exponent of q,.e., q = q/(q wth the usual modfcatons = and =. Fnally, gven a set of m exponents p,..., p m, P wll denote the m-tuple, (p,..., p m, and p wll often denote the number defned by the Hölder relatonshp p = p + + p m. Defnton 2. Let α be a number such that 0 < α < mn and f = (f,..., f m be a collecton of m functons on R n. We defne the multlnear fractonal ntegral as I αf(x f (y f m (y m = d y, (R n m ( x y + + x y m mn α where the ntegral s convergent f f S S. Our man result about I α s the followng. Theorem 2.2 Suppose that 0 < α < nm, < p,..., p m < and q s a number that satsfes /m < p q <. Suppose that one of the followng two condtons holds.

5 Weghted nequaltes for multlnear fractonal ntegral operators 27 q > and (u, v are weghts that satsfy sup l( α /q /p( u qr /qr m for some r >. q and (u, v are weghts that satsfy sup l( α /q /p( u q /q m ( /p v r r < (2. ( /p v r r < (2.2 for some r >. Then the nequalty, ( /q ( I αf u q m ( C ( f v p /p holds for all f L p (v p Lpm (v pm m. Remark 2.3 We notce that condtons (2. and (2.2 are two sded and one sded condtons respectvely. For q > we must requre a stronger norm.e., a power bump for u q and the v s. However for q we only need to power bump the v weghts. Ths s smlar to the condton for the operator M α below. Corollary 2.4 Suppose that 0 < α < nm, < p,..., p m < and q s such that /m < p q <. Further suppose that u, v,..., v m are weghts wth u q, v,..., v m m A, that satsfy, Then, sup l( α /q /p( u q /q m ( /q ( I αf u q m ( C ( f v p holds for all f L p (v p Lpm (v pm m. ( /p v <. /p The above corollary s a drect consequence of the fact that A weghts satsfy the reverse Hölder condton. We gve the proof of Theorem 2.2 n full detal n Secton 4. We stated the above results and proofs n the L p context for clarty n the presentaton and because t s what s needed for the one weght theory. There s however a better result that we obtan usng Banach functon spaces. In partcular we have the followng. Theorem 2.5 Suppose that 0 < α < nm, < p,..., p m < and q s a number that satsfes /m < p q <. Suppose that one of the followng two condtons holds.

6 28 Moen q > and (u, v are weghts that satsfy sup l( α /q /p u L q (log L q +δ (,/ v p < (2.3 L (log L p +δ (,/ for some δ > 0. q and (u, v are weghts that satsfy sup l( α /q /p( /q u q m for some δ > 0. Then the nequalty, v L p (log L p +δ (,/ < (2.4 ( /q ( I αf u q m ( C ( f v p /p holds for all f L p (v p Lpm (v pm m. We descrbe n Secton 6 how to modfy the proof of Theorem 2.2 so that t apples to the more abstract settng of Banach functon spaces. Also see Secton 6 for pertnent defntons n the above theorem. We now turn our attenton to the fractonal mult(sublnear maxmal operater. Defnton 2.6 For 0 α < nm and f = (f,..., f m L loc L loc, we defne the mult(sublnear maxmal operator M α by M α f(x = sup x l( α/m f (y dy. We wll refer to ths as the multlnear maxmal functon. Notce that the case α = 0 corresponds to the mult(sublnear maxmal functon M studed n [4]. In [4] t s shown that, for p,..., p m < the weak nequalty Mf L p, (u C f L p (v holds f and only f (u, v satsfes the condton, ( sup u /p m ( /p v <. (2.5 When p j =, ( v j j /p j s understood as (nf v j. There s a correspondng weak characterzaton for M α.

7 Theorem 2.7 Weghted nequaltes for multlnear fractonal ntegral operators 29 Suppose that 0 α < nm, p,..., p m < and q s a number satsfyng /m < p q <. Then the nequalty, M αf L q, (u C f L p (v holds f and only f the weghts (u, v satsfy, sup l( α /q /p( /q m u ( /p v <. Here ( v j j /p j s understood as (nf v j when p j =. We now state the man theorem for M α. For the next results, we preform the normalzaton u u q and v v p. Ths smplfes matters for the extensons to Banach functon spaces n Secton 6 and makes computatons n the one weght case easer. Theorem 2.8 Suppose 0 α < nm, < p,..., p m <, and q s a number such that /m , then ( /q (M αfu q m ( C ( f v p holds for all f L p (v p Lpm (v pm m. Remark 2.9 /p As before ths s a one-sded condton and the prevous remarks can be adapted to the multlnear fractonal maxmal functon,.e., we may assume that v are A weghts and the same concluson holds. Once agan we obtan a better result usng Banach functon spaces. Theorem 2.0 Suppose 0 α < nm, < p,..., p m <, and q s a number such that /m 0, then ( /q (M αfu q m ( C ( f v p holds for all f L p (v p Lpm (v pm m. /p

8 220 Moen Remark 2. For α = 0 and p = q, Theorems 2.8 and 2.0 gve new results for the operator, M, studed n [4]. As already mentoned, n [4] necessary and suffcent condtons for two weght weak bounds of M were found, whle Theorems 2.8 and 2.0 gve suffcent condtons for the strong boundedness. We note, however, that the two weght condtons n [4] are not suffcent for the strong boundedness of M. See Remark 7.3 n Secton One weght theory We now turn our attenton to the mult-lnear one vector weght case. As n the lnear case we also have that M α s a smaller operator that I α, more specfcally, M α f CI α f for f 0. However we also have the reverse nequalty n norm. We obtan the followng theorem relatng I α and M α as an applcaton of the extrapolaton theorem of Cruz-Urbe, Pérez, and Martell [3]. Theorem 3. Suppose that 0 < α < mn, then for every w A and all 0 < q <, we have, I αf(x q w(x C M αf(x q w(x for all functons f wth f bounded wth compact support. If we assume, say v A, then the two weght characterzaton for the fractonal maxmal functon becomes, f and only f ( (M α ( /q fu q m ( C ( f v p sup α/n+/q /p( u q /q m Notce that when we have the Sobolev relatonshp, we obtan from (3. ( sup u q /q m q = p α n, /p ( /p v <. (3. ( /p v <. In ths stuaton, the Lesbegue dfferentaton theorem gves then u C v. Wth ths motvaton we defne a one weght condton as follows.

9 Weghted nequaltes for multlnear fractonal ntegral operators 22 Defnton 3.2 Let < p,..., p m < and q be a number /m < p q <. We say that a vector of weghts w = (w,..., w m s n the class A P,q, or that t satsfes the A P,q condton, f ( sup (Π m w q /q m ( /p w <. Remark 3.3 have, If p q and /q = /q + + /q m then by Hölder s nequalty we ( (Π w q /q m ( /q ( q w ( /p w /p w, and hence A p,q A P,q, (3.2 q,...,q m where the unon s over all q p that satsfy /q = /q + + /q m. We wll show that ths contanment s strct. See Remark 7.5. We examne the propertes of the weghts n the class A P,q. We have the follow theorem whch s a varant of Theorem 3.6 n [4]. Theorem 3.4 Suppose, < p,..., p m <, and w A P,q, then ( Π m w q Amq and w A mp. We now state the man theorem for these weghts. In the one weght stuaton we obtan necessary and suffcent condtons for the boundedness of I α and M α. Theorem 3.5 Suppose that 0 < α < nm and < p,..., p m < are exponents wth /m < p < n/α and q s the exponent defned by /q = /p α/n. Then the nequalty ( Rn ( Iαf (Π w q /q m ( C R n ( f w p /p holds for every f L p (w p Lpm (w pm m f and only f w satsfes the A P,q condton. In lght of Theorem 3.4 and Corollary 2.4 the suffcency of the A P,q condton follows from the two weght case wth u = Π w and v = w. The necessty of the A P,q condton follows from Theorem 2.7 and the fact that I α s a bgger operator than M α.

10 222 Moen Theorem 3.6 Suppose that 0 < α < nm and < p,..., p m < are exponents wth /m < p < n/α and q s the exponent defned by /q = /p α/n. Then the nequalty ( Rn ( Mα ( f(π w q /q m ( C R n ( f w p /p holds for every f L p (w p Lpm (w pm m f and only f w satsfes the A P,q condton. Once agan the suffcency of the A P,q condton follows from the two weght case Theorem 2.8 and the necessty follows from the weak characterzaton n Theorem 2.8. We do note, however, that Theorem 3.6 combned wth Theorems 3. and 3.4 gves a dfferent proof of the suffcency of the A P,q condton n Theorem 3.5. When α = 0 (so p = q we recover the result from [4]. 4. Proof of the two weght theorems Proof of Theorem 2.2 We frst treat the case q. We wsh to show that ( Rn ( Iαf(x u(x q /q m ( C R n ( f (x v (x p /p. Equvalently, snce I α s a postve operator, t s enough to show that Rn ( /q I αf(xu(xg(x m ( ( C g(x q f (x v (x p /p for all g L q (R n, wth g 0, and all f 0, bounded wth compact support. We apply a dscretzaton technque smlar to that used n [8] for the operator I α. For a fxed x R n and l Z there s a unque dyadc cube of sde length 2 l that contans x. Hence we have I αf(x f (y f m (y m = d y 2 ν < x y 2 ν ( x y + + x y m nm α ν Z = χ (x ν Z D(R n l(=2 ν C C D(R n D(R n C D l( α 3 m l( α m l( α m l(/2< x y l( x y l( sup x y l( f (y f m (y m d y ( x y + + x y m nm α f (y f m (y m d y χ (x f (y f m (y m d y χ (x. (3 m f (y f m (y m d y χ (x.

11 Weghted nequaltes for multlnear fractonal ntegral operators 223 Let g be a non-negatve functon n L q (R n, then I α l( α f(xu(xg(x C R n 3 m f (y f m (y m d y w(xg(x. D (3 m Further, defne M 3D h(x = sup x D h (y dy, 3 3 to be the maxmal functon wth the bass of trples of dyadc cubes. Notce that M 3Df Mf. Let M be the constant from the L L L /m, nequalty for M, a > 6 n M and D k = { x R d : M 3D f(x > a k }. If D k s non-empty we can fnd a dyadc cube wth x and 3 m f (y f m (y m d y > a k. (3 m Snce f s bounded wth compact support we can fnd a dyadc cube that satsfes ths condton and s maxmal wth respect to ncluson. Thus, we get D k = j k,j where, for each k the cubes k,j are maxmal, dsjont, dyadc cubes that satsfy a k < 3 k,j m (3 k,j m f (y f m (y m d y 2 nm a k. Fx k,j, we compute the part of k,j covered by D k+. We have, Snce x k,j the supremum n M 3D f(x = k,j D k+ = { x k,j : M 3D f(x > a k+ }. sup x P D 3P m (3P m f (y f m (y m d y > a k+. s taken over all dyadc cubes that contan k,j or are contaned n k,j. maxmalty of k,j mples 3P m f (y f m (y m d y a k P k,j. (3P m But the It now follows that f x k,j and M 3D f(x > a k+, then M 3D (f χ 3k,j,..., f m χ 3k,j (x > a k+. We have, k,j D k+ = {x k,j : M 3D f(x > a k+ } {x k,j : M 3D (f χ 3k,j,..., f m χ 3k,j (x > a k+ } {x R n : M(f χ 3k,j,..., f m χ 3k,j (x > a k+ } ( M /m a k+ f (y dy 3 k,j ( M /m 3k,j a k+ 3 k,j m f (y f m (y m d y (3 k,j m 6n M /m a /m k,j.

12 224 Moen Thus, k,j D k+ β k,j for some 0 < β <. If E k,j = k,j \D k+ then {E k,j } k,j s a dsjont famly of sets that satsfy k,j C E k,j for some C > 0. Let, C k = { D : a k < } 3 m f (y f m (y m d y a k+. (3 m Then, I αf(xu(xg(x R n C l( α 3 m f (y f m (y m d y u(xg(x D (3 m C l(α 3 m f (y f m (y m d y u(xg(x k Z C k (3 m C a k+ α l( u(xg(x k Z j Z C k k,j Ca a k l( k,j α u(xg(x. k Z j Z k,j C l(3 k,j α m f (y v (yv (y dy 3 k,j k,j 3 k,j u(xg(x k,j. 3 k,j 3 k,j Usng now Hölder s nequalty repeatedly and replacng k,j wth the dsjont E k,j we have, C l(3 k,j α m ( (f v (rp 3 k,j k,j 3 k,j ( /qr ( u qr 3 k,j 3 k,j 3 k,j CK ( /(qr g (qr 3 k,j k,j 3 k,j ( (f v (rp 3 k,j 3 k,j CK ( k,j ( g (qr 3 k,j 3 k,j /(rp ( 3 k,j g (qr 3 k,j /(rp E k,j /p+/q /p p /(qr E k,j p /q v rp 3 k,j /(qr k,j /rp (4.

13 Weghted nequaltes for multlnear fractonal ntegral operators 225 ( k,j ( CK CK k,j ( k,j ( k,j ( (f v (rp 3 k,j 3 k,j ( g (qr 3 k,j 3 k,j ( (f v (rp 3 k,j 3 k,j E k,j M (rq (g(x q ( Rn CK M (rq (g(x q q /(qr E k,j p/(rp E k,j /q p /(rp E k,j /p /p /q m ( M (rp E (f v (x p k,j /q m ( k,j R n M (rp (f v (x p /p /p where K s the constant n (2. and M s s the operator M s f = M( f s /s. Notce that snce r > we have, have M (rq : L q (R n L q (R n M (rp : Lp (R n L p (R n. M (rp m : Lpm (R n L pm (R n. (4.2 Hence usng these boundedness propertes above we obtan ( I αf(xu(xg(x C R n g(x q R n /q m ( ( f (x v (x p R n /p whch concludes the case q >. Now suppose /m < p q, then we work drectly wth the norm I α fw L q. Usng the same dscrtzaton technque as above, and q we obtan I αf(x q C ( l( α qχ 3 m f (y f m (y m d y (x. D (3 m Multplyng by u q and ntegratng, ( /q (I αf(xu(x q R n ( C ( l( α q 3 m f (y f m (y m d y D (3 m u(x q /q. Performng the same decomposton as above we obtan { k,j } k,j and construct {E k,j }

14 226 Moen satsfyng the same propertes. Thus, ( C ( l( α q /q 3 m f (y f m (y m d y u(x q D (3 m ( C C a (k+q /q αq l( u(x q k Z j Z C k k,j ( l(3 k,j αq ( C u(x q /q q k,j f (y v (y v (y dy. 3 k,j 3 k,j 3 k,j 3 k,j k,j Usng Hölder s nequalty and condton (2.2, (4.3 CK k,j ( m ( (f v (rp 3 k,j 3 k.j q/(rp k,j q/p /q. Usng p q, replacng the k,j s wth E k,j s and multlnear Hölder s nequalty agan we have, ( m CK k,j ( CK CK CK k,j ( ( ( (f v (rp 3 k,j 3 k.j ( (f v (rp 3 k,j 3 k,j R n M (rp (f v p R n (f (y v (y p /p dy /p. p/(rp E k,j p /(rp E k,j /p /p Ths concludes the proof of Theorem 2.2. Remark 4. A close examnaton of the above proof yelds that the operator norm denoted I α has the dependence, I α CK where C s a dmensonal constant and K s the constant from (2.. Proof of Theorem 2.7 The proof s smlar to that of the weak nequalty gven n [4]. We only present the case where p,..., p m > as a the case when some p j = s a mnor modfcaton of the lnear case. Suppose that M α s weakly bounded.e. u ( {x R n : M α f(x > λ} ( C λ q f L p (v

15 Weghted nequaltes for multlnear fractonal ntegral operators 227 for all λ > 0. Let f 0 and fx a cube wth Π α/nm f > 0. Notce that for x we have l( α/m f M α (f χ,..., f m χ (x. Hence, f λ < Π α/nm f M α (f χ,..., f m χ (x we have Thus, { x R n : M α (f χ,..., f m χ (x > λ }. u( u ( {x R n : M α (f χ,..., f m χ (x > λ} ( C λ Snce ths holds for all λ < Π α/nm f t follows that If we set f = v m α/n m u( /q ( we get m α/n m u( /q ( f C ( v C ( ( q /p p f v. f p v /p. v /p whch gves the A P,q condton. Conversely, suppose that (u, v A P,q and assume for the moment that for all m f L p (v =. We wll also use the centered fractonal multlnear maxmal functon M c α where the supremum s taken over all cubes centered at x. Clearly M α M c α. Gven x fx a cube,, centered at x. Then Hölder s nequalty yelds l( α/m m f = α/n m α/n m m ( f v /p v /p m = α/n m u( /q ( Cu( /q m ( f p /p ( /p v v v f p v /p. /p m u( /q ( f p v /p Now, snce we are assumng that f L p (v =, we have ( f p v /p.

16 228 Moen Moreover, snce p/q we have l( α/m ( f C u( /q f p /p v ( m ( p/q C u( /q f p /p v ( ( p/q = C u( /p f p /p v ( m ( p/q = C f p v u /p u u( ( m p/q. C Mu( f c p v /u(x /p Hence, M c ( m p/q. αf(x C Mu( f c p v /u(x /p Usng a weak-type Hölder s nequalty we have, M c α f L q, (u C (Π M c u( f p v /u /p p/q L q, (u = C Π Mu( f c p v /u /p p/q L p, (u ( m C Mu( c f p v /u L p, (u = C C = C ( m ( m ( m p/q Mu c ( f p v /u /p p/q L, (u f p v /u /p p/q L (u f L p (v p/q = C. For general f the result follows f we replace f f / f L p (v. Proof of Theorem 2.8 We frst prove the boundedness for the dyadc verson, M d l( α αf(x = sup f (y dy. D:x Let a be a constant satsfyng a > 2 nm and let D k = { x R n : M d αf(x > a k}. If D k s non-empty then we can wrte D k = j k,j where each k,j s a maxmal dyadc cube satsfyng a k < k,j α/(nm f (y dy < 2 mn α a k 2 mn a k. k,j

17 Weghted nequaltes for multlnear fractonal ntegral operators 229 Also, each D k+ D k and each k+,l s contaned n k,j for some j by propertes of dyadc cubes and we have k,j D k+ 2n a /m k,j. Hence the sets E k,j = k,j \( k,j D k+ are dsjont and satsfy for some β >. Thus, we have ( (M d /q R n αf(xu(x q ( = (M d αf(xu(x q D k \D k+ a a k ( k ( a (k+q D k u q (x k,j ( ( m k,j k,j < β E k,j /q /q a kq u q (x k,j /q l( k,j α/m q /q f (y v (y v (y dy u q (x. k,j k,j k,j Ths equaton s the same as (4.3 n the proof of Theorem 2.2 and the dyadc verson of the theorem follows. The non-dyadc verson follows from the nequalty M k αf(x q C α,n (τ t M d α τ t ( f(x B k q dt (4.4 B k for all x R n and f 0. Where B k = [ 2 k+2, 2 k+2 ] n, M k α f s the maxmal functon wth the supremum taken over cubes of sde length less than 2 k, τ t g(x = g(x t, τ t f = (τt f,..., τ t f m. The nequalty (4.4 holds for all 0 < q <, and a proof for the lnear case can be found n [6, p. 43] and the multlnear case s a slght modfcaton. From (4.4 t follows that M αfu L q sup τ t M d α τ tfu L q. t Noe that f (u, v satsfy condton (2.6, then (τ t u, τ t v satsfy the condton (2.6 ndependent of t. By the dyadc case we have, (τ t M d α τ t fu L q = (M d α τ t fτ t u L q C τ t f τ t v L p = C f L p (v, where the constant C s ndependent of t. It now follows that, M αfu L q C sup τ t M d α τ tfu L q C f v L p. t

18 230 Moen 5. Proof of the one weght theorems Proof of Theorem 3. In lght of the extrapolaton theorem n [3] we just need to show that the result holds for q = and all w A. Usng the same decomposton as n Theorem 2.2 wth g = we have, R n I α fw c k,j l(3 k,j α 3 k,j Snce w A and k,j C E k,j we have, w( k,j Cw(E k,j. 3 k,j f (y dy w( k,j. Hence, R n I α fw C k,j l(3 k,j α 3 k,j C M αfw k,j E k,j M αfw. R n 3 k,j f (y dy w(e k,j Proof of Theorem 3.4 We use technques smlar to those n [4]. Snce p q, f we let q = qp /p then, q p and /q = /q + + /q m. Further, we have q + + q m = m q, and hence Hölders nequalty wth r = (m /qq can be appled to get ( (Π w q /mq( ( (Π w q /mq ( We now use Hölder s wth p /q > to get (Π w q/(mq (mq /mq ( /q w q /m. ( ( (Π w q /q ( /p w /m. Ths shows that Π w q A mq. Now to show that w A mp, for ths fx m, then the A mp condton s, ( sup /mp w ( p w /(mp (mp /mp <.

19 Weghted nequaltes for multlnear fractonal ntegral operators 23 If we set r = p (m + ( = p m p p and r j = p j r p j p = p j p r j m. Then notce < r j < and m j= = ( + r j r j p j m j p =. Further ( p w /(mp (mp /mp = = ( r /mp p/r w ( ( Π m p/r j= w j Π j w /r r /mp j. We use Hölder s nequalty wth exponents r,..., r m to get ( = = r /mp ( p/r Πj w j Π j w /r r /mp j [ ( ( ( r /mp ( ( ( ( ( p /r ( Πj w j j ( p /p ( Πj w j ( Πj w j p /p j ( Πj w j q /q The second to last nequalty follows snce r p = m p j j r /mp w /rj j j w /p /m j j j ( w /p /m j j j ( w /p /m j j j. = p +, p j m j and the last nequalty follows from Hölder s wth q/p. Thus we arrve at the nequalty, ( /mp w ( p w /(mp (mp /mp ( ( (Π j w j q /q m ( w /p /m j j j. j= Ths shows that w A mp.

20 232 Moen 6. Banach functon spaces Suppose X s a Banach functon space over R n wth respect to Lebesgue measure. We refer the reader to [] for a detaled account of Banach functons spaces. X has an assocate Banach functon space X for whch the generalzed Hölder nequalty, R n f(xg(x f X g X, holds. Examples of Banach functon spaces are the Lebesgue L p spaces, Lorentz spaces, and Orlcz spaces whch we shall descrbe next. The Orlcz space L B = L B (R n s defned by a Young functon B (see [9] wth and conssts of all measurable functons f such that ( f(y B dy < R n λ for some λ > 0. The space s then equpped wth a norm gven by { } ( f(y f B = nf λ > 0 : dy. λ R n B As n [9], for a functon f X and a cube R n we defne the X average of f over to be f X, = δ l( (fχ X, where for a > 0, δ a f(x = f(ax. Observe that f X = L r then ( /r f X, = f r and f X = L B then, { } ( f(y f B, = nf λ > 0 : B dy. λ We defne the maxmal operator assocated to the Banach functon space X to be M X f(x = sup f X,. x When X s the Orlcz space L B we denote M X by M B. Notce that f M s the Hardy- Lttlewood maxmal operator, then M L = M and M L rf(x = M r f = M(f r /r. If Y,..., Y m are Banach functon spaces we defne the mult(sublnear maxmal functon to be M Y f(x = sup f Y,. x Notce that M Y f(x m M Y f (x. Hence f p,..., p m and M Y : L p L p then by Hölder s nequalty M Y : L p L pm L p. We have have the followng generalzed verson of Theorem 2.2.

21 Theorem 6. Weghted nequaltes for multlnear fractonal ntegral operators 233 Suppose 0 < α < nm, < p,..., p m <, wth /p = /p + + /p m, and Y,..., Y m are Banach functon spaces over R n such that M Y : L p (R n L pm (R n L p (R n (6. where M Y s the multlnear maxmal functon assocated to Y,..., Y m. Let q be an exponent satsfyng /m , X s a Banach functon space that satsfng and (u, v are weghts that satsfy sup q and (u, v are weghts that satsfy M X : L q (R n L q (R n, (6.2 l( α /q /p m u X, v Y, <. sup l( α /q /p( /q u q m v Y, <. Then the nequalty ( /q ( I αf u q m ( C ( f v p /p holds for all f L p (v p Lpm (v pm m. Remark 6.2 Theorem 2.2 s the specfc case of Theorem 6. when X = L qr and Y = L rp for some r >. In ths case the boundedness of the maxmal functons n (6. and (6.2 are automatc. The proof of Theorem 6. s very smlar to the proof Theorem 2.2. The man ngredents are the generalzed Hölder nequalty whch s used n place of equaton (4. and the assumed boundedness of the maxmal functons M Y n (6. and M X n (6.2 are used n place of (4.2. For the Orlcz spaces, L B, the boundedness of the correspondng maxmal functons M B has been developed by Pérez [9]. He showed that f and only f there exsts c > 0 such that M B : L s L s Thus we have the followng theorem. c B(t dt t s t <.

22 234 Moen Theorem 6.3 Suppose 0 < α < nm, < p,..., p m <, wth /p = /p + + /p m, q s an exponent wth /m 0. Let q be an exponent satsfyng /m < p q < and assume that Then the nequalty sup l( α /q /p m u Ψ, v Φ, <. ( /q ( I αf u q m ( C ( f v p holds for all f L p (v p Lpm (v pm m. /p We notce that the functons Ψ(t = t q (log( + t q +δ and Φ = t p (log( + t p +δ satsfy (6.3 and (6.4 respectvely f δ > 0. From here we obtan Theorem 2.5. Smlary we extend Theorem 2.8. Theorem 6.4 Suppose 0 α < nm, < p,..., p m <, wth /p = /p + + /p m, and q s an exponent satsfyng /m < p q <, and Y,..., Y m are are translaton nvarant Banach functon spaces wth If (u, v are weghts that satsfy then the nequalty M Y : L p (R n L pm (R n L p (R n. sup l( α /q /p( /q u q m ( /q (M αfu q m ( C ( f v p holds for all f L p (v p Lpm (v pm m. v Y, < /p Remark 6.5 Once agan the proof Theorem 6.4 s almost dentcal to the proof of Theorem 2.8 usng the generalzed Hölder nequalty and the boundedness of M Y n the rght places. The translaton nvarance s used to pass from the dyadc verson va equaton (4.4. Theorem 2.8 s also a partcular case of Theorem 6.4 where Y = L rp. In the context of Orlcz space we have the followng theorem for M α.

23 Theorem 6.6 Weghted nequaltes for multlnear fractonal ntegral operators 235 Suppose 0 < α < nm, < p,..., p m <, wth /p = /p + + /p m, q s an exponent wth /m 0. Let q be an exponent satsfyng /m 0 we obtan Theorem Applcatons and examples Wth the mult-lnear fractonal ntegral operator we have some Poncaré and Sobolev type nequaltes for products of functons. We do the estmates wth two functons but the nterested reader may generalze these nequaltes to m functons. Theorem 7. Suppose that < r, s < wth /p = /r + /s and /2 0 such that fgν L q C( ( fu L r gv L s + fu L r ( gv L s for all f, g C c (R n. Proof. Gven y, y 2 R n Denote y R 2n by y = (y, y 2 = (y,,..., y,n, y 2,,..., y 2,n. and 2n = (,,,,n, 2,,..., 2,n be the gradent n R 2n. Snce f, g have compact support and are smooth we have, see Sten [24, p. 25] we have 2n f(x y g(x y 2 f(xg(x C R 2n y 2n d y C(I ( f, g (x + I ( f, g (x, where f and g are the gradents of f and g n R n. It now follows that, fgν L q!c( I ( f, g ν L q + I ( f, g ν L q C( ( fu L r gv L s + fu L r ( gv L s. For x R n let = n 2 2 denote the Laplacan operator n R n and for x x = (x, x 2 R 2n let 2n = 2 nj= 2 denote the Laplacan operator n R 2n. x 2,j

24 236 Moen Theorem 7.2 Suppose that n > and < r, s < wth /p = /r + /s and /2 0 such that fgν L q C( ( fu L r gv L s + f L r ( gv L s for all f, g C c (R n. Proof. Snce f, g C c (R 2n we can wrte 2n f(y g(y 2 f(xg(x = C R 2n ( x y 2 + x y 2 2 d y. (2n 2/2 Notce we are restrctng the ntegral to a set of measure zero, however ths s lgtmate snce t s an absolutely convergent ntegral. Then we have Thus we have 2n f(y g(y 2 f(xg(x = C R 2n ( x y 2 + x y 2 2 d y (2n 2/2 g(y 2 f(y + f(y g(y 2 = C R 2n ( x y 2 + x y 2 2 d y. (2n 2/2 f(xg(x CI 2 ( f, g (x + CI 2 ( f, g (x. Usng the boundedness of I 2 we obtan, fgν L q C( I 2 ( f, g ν L q + I 2 ( f, g ν L q C( ( fu L r gv L s + fu L r ( gv L s. Remark 7.3 Condton (2.5 s not suffcent for the strong boundedness of M. Ths uses an arguement smlar to the lnear case. We requre the followng lemma whch us a multlnear verson of the fact w A p mples w A p n the lnear stuaton. Lemma 7.4 Gven < p,..., p m < and (u, v,..., v m that satsfy (2.5. If we set P = (p,..., p m, and q = p/(pm so Then the weghts satsfy (2.5 wth respect to P. q = m p = p + + p m. (v q/p v q/pm m, u,..., u m

25 Weghted nequaltes for multlnear fractonal ntegral operators 237 Proof. The condton for (v q/p vm q/pm, u,..., u m s ( sup v q/p vm q/pm Usng Hölder s nequalty, /q m ( v q/p vm q/pm ( v /p ( /p (u < /q m /p ( ( /p (u u /p. Thus the lemma follows from ths nequalty. Now notce that (u, Mu,..., Mu satsfy condton 2.5. By the lemma wth q = p/(mp we have (Mu q p, u,..., u m satsfyng condton 2.5 wth respect to P. If condton 2.5 were suffcent then we would have ( R n M(f,..., f m q Mu q p /q m ( C f p u R n /p. But settng f = = f m = u, and usng the fact that M(u,..., u = (Mu m we have whch s a clear contradcton. ( Mu R n ( C u, R n Remark 7.5 In general we have strct contanment n (3.2,.e. A p,q A P,q. q,...,q m Take for example, n =, m = 2, p = p 2 = 2, and q = 3/2. We use a smlar example to the one gven n [4] let w (x = { x /2 x [0, 2] otherwse and w 2 (x = x /2. Then (w w 2 q s n A and nf (w w 2 q (nf w q (nf w q 2 but for any power r 2 w r / L loc and hence cannot be n A r,2 for any such r. References. C. Bennet and R. Sharply, Interpolaton of Operators, Academc Press, New York, X. Chen and. Xue, Weghted estmates for a class of multlnear fractonal type operators, (preprnt 2008.

26 238 Moen 3. D. Cruz-Urbe, J.M. Martell, and C. Pérez, Extrapolaton from A weghts and applcatons, J. Funct. Anal. 23 (2004, D. Cruz-Urbe, J.M. Martell, and C. Pérez, Sharp two-weght nequaltes for sngular ntegrals, wth applcatons to the Hlbert transform and the Sarason conjecture, Adv. Math. 26 (2007, A. De la Torre, personal communcaton, J. Garca-Cuerva, J.L. Rubo de Franca, Weghted Norm Inequaltes and Related Topcs, North- Holland Mathematcs Studes 6, North-Holland Publshng Co., Amsterdam, L. Grafakos, On multlnear fractonal ntegrals, Studa Math. 02 (992, L. Grafakos, Classcal and Modern Fourer Analyss, Prentce Hall, L. Grafakos and N. Kalton, Some remarks on multlnear maps and nterpolaton, Math. Ann. 39 (200, L. Grafakos and R.H. Torres, Multlnear Calderón-Zygmund theory, Adv. Math. 65 (2002, L. Grafakos and R.H. Torres, Maxmal operator and weghted norm nequaltes for multlnear sngular ntegrals, Indana Unv. Math. J. 5 (2002, C.E. Keng and E.M. Sten, Multlnear estmates and fractonal ntegraton, Math. Res Lett. 6 (999, R. Kerman and E. Sawyer, The trace nequalty and egenvalue estmates for Schrödnger operators, Ann. Inst. Fourer (Grenoble 36 (986, A. Lerner, S. Ombros, C. Pérez, R.H. Torres, and R. Trujllo-González, New maxmal functons and multple weghts for the multlnear Calderón-Zygmund theory, Adv. Math., to appear. 5. B. Muckenhoupt, Weghted norm nequaltes for the Hardy maxmal functon, Trans. Amer. Math. Soc. 65 (972, B. Muckenhoupt and R. Wheeden, Weghted norm nequaltes for fractonal ntegrals, Trans. Amer. Math. Soc. 92 (974, C. Neugebauer, Insertng A p -weghts, Proc. Amer. Math. Soc. 87 (983, C. Pérez, Two weght norm nequaltes for Resz potentals and unform L p -weghted Sobolev nequaltes, Indana Unv. Math. J. 39, (990, C. Pérez, Two weght nequaltes for potental and fractonal type maxmal operators, Indana Unv. Math. J. 43, (994, C. Pérez and R.H. Torres, Sharp maxmal functon estmates for commutators of sngular ntegrals, Harmonc analysys at Mount Holyoke (South Hadley, MA, 200, , Contemp. Math. 320, Amer. Math. Soc., Provdence, RI, E.T. Sawyer, A characterzaton of a two-weght norm nequalty for maxmal operators, Studa Math. 75 (982,. 22. E.T. Sawyer, A characterzaton of a two-weght norm nequalty for fractonal and posson ntegrals, Trans. Amer. Math. Soc. 308, (988, E.T. Sawyer and R.L. Wheeden, Weghted nequltes for fractonal ntegrals on Eucldean and homogeneous spaces, Amer. J. Math. 4 (992, E.M. Sten, Sngular Integrals and Dfferentablty Propertes of Functons, Prnceton Mathematcal Seres 30, Prnceton Unversty Press, Prnceton, NJ., 970.

THE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION

THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton