The boundedness of classical operators on variable L-p spaces

Annales Academiæ Scien ia um Fennicæ Ma hema ica Volumen 31, 2006, 239–264 THE BOUNDEDNESS OF CLASSICAL OPERATORS ON VARIABLE LpSPACES D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez T ini y College, Depa men o Ma hema ics Ha o d, CT 06106-3100, U.S.A.; da id.c uzu ib[email p o ec ed] Uni e si ´a di Napoli, Dipa imen o di Cos uzioni e Me odi Ma ema ici in A chi e u a Via Mon eoli e o, 3, IT-80134 Napoli, I aly and Consiglio Nazionale delle Rice che, Is i u o pe le Applicazioni del Calcolo “Mau o Picone” Sezione di Napoli, ia Pie o Cas ellino, 111, IT-80131 Napoli, I aly; [email p o ec ed] Uni e sidad Au ´onoma de Mad id, Depa amen o de Ma em´a icas ES-28049 Mad id, Spain; [email p o ec ed] Uni e sidad de Se illa, Depa amen o de An´alisis Ma em´a ico Facul ad de Ma em´a icas, ES-41080 Se illa, Spain; ca losp[email p o ec ed] Abs ac . We show ha many classical ope a o s in ha monic analysis—such as maximal ope a o s, singula in eg als, commu a o s and ac ional in eg als—a e bounded on he a iable Lebesgue space Lp(·)whene e he Ha dy–Li lewood maximal ope a o is bounded on Lp(·). Fu he , we show ha such ope a o s sa is y ec o - alued inequali ies. We do so by applying he heo y o weigh ed no m inequali ies and ex apola ion. As applica ions we p o e he Calde ´on–Zygmund inequali y o solu ions o 4u= in a iable Lebesgue spaces, and p o e he Calde ´on ex ension heo em o a iable Sobole spaces. 1. In oduc ion Gi en an open se Ω ⊂Rn, we conside a measu able unc ion p: Ω −→ [1,∞), Lp(·)(Ω) deno es he se o measu able unc ions on Ω such ha o some λ > 0, ZΩ| (x)| λp(x) dx < ∞. This se becomes a Banach unc ion space when equipped wi h he no m k kp(·),Ω= in λ > 0 : ZΩ| (x)| λp(x) dx ≤1. These spaces a e e e ed o as a iable Lebesgue spaces o , mo e simply, as a i- able Lpspaces, since hey gene alize he s anda d Lpspaces: i p(x) = p0is 2000 Ma hema ics Subjec Classi ica ion: P ima y 42B25, 42B20, 42B15, 35J05. The hi d au ho is pa ially suppo ed by MEC G an MTM2004-00678, and he ou h au ho is pa ially suppo ed by DGICYT G an PB980106. 240 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez cons an , hen Lp(·)(Ω) equals Lp0(Ω). (He e and below we w i e p(·) ins ead o p o emphasize ha he exponen is a unc ion and no a cons an .) They ha e many p ope ies in common wi h he s anda d Lpspaces. These spaces, and he co esponding a iable Sobole spaces Wk,p(·)(Ω), a e o in e es in hei own igh , and also ha e applica ions o pa ial di e en ial equa ions and he calculus o a ia ions. (See, o example, [1], [12], [15], [19], [30], [39], [46] and hei e e ences.) In many applica ions, a c ucial s ep has been o show ha one o he clas- sical ope a o s o ha monic analysis—e.g., maximal ope a o s, singula in eg als, ac ional in eg als—is bounded on a a iable Lpspace. Many au ho s ha e con- side ed he ques ion o su icien condi ions on he exponen unc ion p(·) o gi en ope a o s o be bounded: see, o example, [13], [15], [27], [28], [29], [40]. Ou app oach is di e en . Ra he han conside es ima es o indi idual op- e a o s, we apply echniques om he heo y o weigh ed no m inequali ies and ex apola ion o show ha he boundedness o a wide a ie y o ope a o s ollows om he boundedness o he maximal ope a o on a iable Lpspaces, and om known es ima es on weigh ed Lebesgue spaces. In o de o p o ide he ounda ion o s a ing ou esul s, we discuss each o hese ideas in u n. The maximal ope a o . In ha monic analysis, a undamen al ope a o is he Ha dy–Li lewood maximal ope a o . Gi en a unc ion , we de ine he maximal unc ion, M , by M (x) = sup Q3x 1 |Q|ZQ | (y)|dy, whe e he sup emum is aken o e all cubes con aining x. I is well known ha Mis bounded on Lp, 1 <p<∞, and i is na u al o ask o which exponen unc ions p(·) he maximal ope a o is bounded on Lp(·)(Ω). Fo conciseness, de ine P(Ω) o be he se o measu able unc ions p: Ω −→ [1,∞) such ha p−= ess in {p(x) : x∈Ω}>1, p+= ess sup{p(x) : x∈Ω}<∞. Le B(Ω) be he se o p(·)∈P(Ω) such ha Mis bounded on Lp(·)(Ω). Theo em 1.1. Gi en an open se Ω⊂Rn, and p(·)∈P(Ω), suppose ha p(·)sa is ies (1.1) |p(x)−p(y)| ≤ C −log(|x−y|), x, y ∈Ω,|x−y| ≤ 1/2, (1.2) |p(x)−p(y)| ≤ C log(e+|x|), x, y ∈Ω,|y| ≥ |x|. Then p(·)∈B(Ω), ha is, he Ha dy–Li lewood maximal ope a o is bounded on Lp(·)(Ω). The boundedness o classical ope a o s on a iable Lpspaces 241 Theo em 1.1 is independen ly due o C uz-U ibe, Fio enza and Neugebaue [10] and o Nek inda [35]. (In ac , Nek inda eplaced (1.2) wi h a sligh ly mo e gene al condi ion.) Ea lie , Diening [12] showed ha (1.1) alone is su icien i Ω is bounded. Examples show ha he con inui y condi ions (1.1) and (1.2) a e in some sense close o necessa y: see Pick and R˚uˇziˇcka [37] and [10]. See also he examples in [33]. The condi ion p−>1 is necessa y o M o be bounded; see [10]. Ve y ecen ly, Diening [14], wo king in he mo e gene al se ing o Musielak– O licz spaces, has gi en a necessa y and su icien condi ion on p(·) o M o be bounded on Lp(·)(Rn). His exac condi ion is somewha echnical and we e e he eade o [14] o de ails. Because ou p oo s ely on duali y a gumen s, we will no need ha he maximal ope a o is bounded on Lp(·)(Ω) bu on i s associa e space Lp0(·)(Ω), whe e p0(·) is he conjuga e exponen unc ion de ined by 1 p(x)+1 p0(x)= 1, x ∈Ω. Since |p0(x)−p0(y)| ≤ |p(x)−p(y)| (p−−1)2, i ollows a once ha i p(·) sa is ies (1.1) and (1.2), hen so does p0(·)—i.e., i hese wo condi ions hold, hen Mis bounded on Lp(·)(Ω) and Lp0(·)(Ω). Fu he mo e, Diening’s cha ac e iza ion o a iable Lpspaces on which he max- imal ope a o is bounded has he ollowing impo an consequence (see [14, The- o em 8.1]). Theo em 1.2. Le p(·)∈P(Rn). Then he ollowing condi ions a e equi - alen : (a) p(·)∈B(Rn). (b) p0(·)∈B(Rn) (c) p(·)/q ∈B(Rn) o some 1< q < p−. (d) p(·)/q0∈B(Rn) o some 1< q < p−. Weigh s and ex apola ion. By a weigh we mean a non-nega i e, locally in eg able unc ion w. The e is a as li e a u e on weigh s and weigh ed no m inequali ies; he e we will summa ize he mos impo an aspec s, and we e e he eade o [17], [21] and hei e e ences o comple e in o ma ion. Cen al o he s udy o weigh s a e he so-called Apweigh s, 1 ≤p≤ ∞. When 1 <p<∞, we say w∈Api o e e y cube Q, 1 |Q|ZQ w(x)dx 1 |Q|ZQ w(x)1−p0dxp−1 ≤C < ∞. 242 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez We say ha w∈A1i Mw(x)≤Cw(x) o a.e. x. I 1 ≤p < q < ∞, hen Ap⊂Aq. We le A∞deno e he union o all he Apclasses, 1 ≤p < ∞. Weigh ed no m inequali ies a e gene ally o wo ypes. The i s is (1.3) ZRn |T (x)|p0w(x)dx ≤CZRn | (x)|p0w(x)dx, whe e Tis some ope a o and w∈Ap0, 1 < p0<∞. (In o he wo ds, Tis de ined and bounded on Lp0(w).) The cons an is assumed o depend only on he Ap0cons an o w. The second ype is (1.4) ZRn |T (x)|p0w(x)dx ≤CZRn |S (x)|p0w(x)dx, whe e Sand Ta e ope a o s, 0 < p0<∞,w∈A∞, and is such ha he le - hand side is ini e. The cons an is assumed o depend only on he A∞cons an o w. Such inequali ies a e known o a wide a ie y o ope a o s and pai s o ope a o s. (See [17], [21].) Co esponding o hese ypes o inequali ies a e wo ex apola ion heo ems. Associa ed wi h (1.3) is he classical ex apola ion heo em o Rubio de F ancia [38] (also see [17], [21]). He p o ed ha i (1.3) holds o some ope a o T, a ixed alue p0, 1 < p0<∞, and e e y weigh w∈Ap0, hen (1.3) holds wi h p0 eplaced by any p, 1 < p < ∞, whene e w∈Ap. Recen ly, he analogous ex apola ion esul o inequali ies o he o m (1.4) was p o ed in [11]: i (1.4) holds o some p0, 0 < p0<∞and e e y w∈A∞, hen i holds o e e y p, 0 <p<∞. (Mo e gene al e sions o hese esul s will be s a ed in Sec ion 6 below.) 1.1. Main esul s. The p oo s o he abo e ex apola ion heo ems depend no on he p ope ies o he ope a o s, bu a he on duali y, he s uc u e o Apweigh s, and no m inequali ies o he Ha dy–Li lewood maximal ope a o . These ideas can be ex ended o he se ing o a iable Lpspaces o yield ou main esul , which can be summa ized as ollows: I an ope a o T, o a pai o ope a o s (T, S), sa is ies weigh ed no m inequali ies on he classical Lebesgue spaces, hen i sa is ies he co esponding inequali y in a a iable Lpspace on which he maximal ope a o is bounded. To s a e and p o e ou main esul , we will adop he app oach aken in [11]. The e i was obse ed ha since no hing is assumed abou he ope a o s in ol ed (e.g., linea i y o sublinea i y), i is be e o eplace inequali ies (1.3) and (1.4) wi h (1.5) ZRn (x)p0w(x)dx ≤CZRn g(x)p0w(x)dx, whe e he pai s ( , g) a e such ha he le -hand side o he inequali y is ini e. One impo an consequence o adop ing his app oach is ha ec o - alued in- equali ies ollow immedia ely om ex apola ion. The boundedness o classical ope a o s on a iable Lpspaces 243 He ea e Fwill deno e a amily o o de ed pai s o non-nega i e, measu able unc ions ( , g). Whene e we say ha an inequali y such as (1.5) holds o any ( , g)∈Fand w∈Aq( o some q, 1 ≤q≤ ∞), we mean ha i holds o any pai in Fsuch ha he le -hand side is ini e, and he cons an Cdepends only on p0and he Aqcons an o w. Finally, no e ha in he classical Lebesgue spaces we can wo k wi h Lpwhe e 0< p < 1. (Thus, in (1.4) o (1.5) we can ake p0<1.) We would like o conside analogous spaces wi h a iable exponen s. De ine P0(Ω) o be he se o measu able unc ions p: Ω −→ (0,∞) such ha p−= ess in {p(x) : x∈Ω}>0, p+= ess sup{p(x) : x∈Ω}<∞. Gi en p(·)∈P0(Ω), we can de ine he space Lp(·)(Ω) as abo e. This is equi - alen o de ining i o be he se o all unc ions such ha | |p0∈Lq(·)(Ω), whe e 0 < p0< p−and q(x) = p(x)/p0∈P(Ω). We can de ine a quasi-no m on his space by k kp(·),Ω= | |p0 1/p0 q(·),Ω. We will no need any o he p ope ies o hese spaces, so his de ini ion will su ice o ou pu poses. Theo em 1.3. Gi en a amily Fand an open se Ω⊂Rn, suppose ha o some p0,0< p0<∞, and o e e y weigh w∈A1, (1.6) ZΩ (x)p0w(x)dx ≤C0ZΩ g(x)p0w(x)dx, ( , g)∈F, whe e C0depends only on p0and he A1cons an o w. Le p(·)∈P0(Ω) be such ha p0< p−, and p(·)/p00∈B(Ω). Then o all ( , g)∈Fsuch ha ∈Lp(·)(Ω), (1.7) k kp(·),Ω≤Ckgkp(·),Ω, whe e he cons an Cis independen o he pai ( , g). We wan o call a en ion o wo ea u es o Theo em 1.3. Fi s , he conclusion (1.7) is an a p io i es ima e: ha is, i holds o all ( , g)∈Fsuch ha ∈ Lp(·)(Ω). In p ac ice, when applying his heo em in conjunc ion wi h inequali ies o he o m (1.3) o show ha an ope a o is bounded on a iable Lpwe will usually need o wo k wi h a collec ion o unc ions which sa is y he gi en weigh ed Lebesgue space inequali y and a e dense in Lp(·)(Ω). When wo king wi h inequali ies o he o m (1.3) he inal es ima e will hold o a sui able amily o “nice” unc ions. Second, he amily Fin he hypo hesis o and conclusion o Theo em 1.7 is he same, so he goal is o ind a la ge, easonable amily Fsuch ha (1.6) holds wi h a cons an depending only on p0and he A1cons an o w. 244 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez Rema k 1.4. In Theo em 1.3, (1.7) holds i p(·) sa is ies (1.1) and (1.2). By Theo em 1.1, se ing q(x) = p(x)/p0we ha e ha q(·)∈P(Ω) and |q0(x)−q0(y)| ≤ |p(x)−p(y)| p0(p−/p0−1)2. Rema k 1.5. When Ω = Rn, i 1 ≤p0< p−, hen by Theo em 1.2 he hy- po hesis ha p(·)/p00∈B(Rn) is equi alen o assuming ha p(·)∈B(Rn). As we will see below, his will allow us o conclude ha a a ie y o ope a o s a e bounded on Lp(·)(Rn) whene e he Ha dy–Li lewood maximal ope a o is. Rema k 1.6. Ou app oach using pai s o unc ions leads o an equi alen o mula ion o Theo em 1.3 in which he exponen p0does no play a ole. This can be done by de ining a new amily Fp0consis ing o he pai s ( p0, gp0) wi h ( , g)∈F. No ice ha in his case (1.6) is sa is ied by Fp0wi h p0= 1. Thus, he case p0= 1 will imply ha i 1 < p−and p(·)0∈B(Ω) hen (1.7) holds. The e o e, i we de ine (x) = p(x)p0, we ha e ha (·)∈P0(Ω), p0< −, ( (·)/p0)0∈B(Ω) and (1.7) holds wi h (·) in place o p(·). Bu his is exac ly he conclusion o Theo em 1.3. Rema k 1.7. We belie e ha a mo e gene al e sion o Theo em 1.3 is ue, one which holds o la ge classes o weigh s and yields inequali ies in weigh ed a iable Lpspaces. Howe e , p o ing such a esul will equi e a weigh ed e sion o Theo em 1.1, and e en he s a emen o such a esul has eluded us. Fo such a weigh ed ex apola ion esul he app op ia e class o weigh s is no longe A1, bu Ap(as in [38]) o A∞(as in [11]). We emphasize, hough, ha he class A1, which is he smalles among he Apclasses, is he na u al one o conside when a emp ing o p o e unweigh ed es ima es. Theo em 1.3 can be gene alized o gi e “o -diagonal” esul s. In he classi- cal se ing, he ex apola ion heo em o Rubio de F ancia was ex ended in his manne by Ha bou e, Mac´ıas and Sego ia [24]. Theo em 1.8. Gi en a amily Fand an open se Ω⊂Rn, assume ha o some p0and q0,0< p0≤q0<∞, and e e y weigh w∈A1, (1.8) ZΩ (x)q0w(x)dx1/q0 ≤C0ZΩ g(x)p0w(x)p0/q0dx1/p0 ,( , g)∈F. Gi en p(·)∈P0(Ω) such ha p0< p−≤p+< p0q0/(q0−p0), de ine he unc ion q(·)by (1.9) 1 p(x)−1 q(x)=1 p0 −1 q0 , x ∈Ω. I q(x)/q00∈B(Ω), hen o all ( , g)∈Fsuch ha ∈Lq(·)(Ω), (1.10) k kq(·),Ω≤Ckgkp(·),Ω. Rema k 1.9. As be o e, (1.10) holds i p(·) sa is ies (1.1) and (1.2). The boundedness o classical ope a o s on a iable Lpspaces 245 We can gene alize Theo em 1.3 by combining i wi h he wo ex apola ion heo ems discussed abo e. This is possible since A1⊂Ap, 1 < p ≤ ∞. This has wo ad an ages. Fi s , i makes clea ha he hypo heses which mus be sa is ied co espond o hose o he known weigh ed no m inequali ies; see, in pa icula , he applica ions discussed in Sec ion 2 below. Second, as in [11], we a e able o p o e ec o - alued inequali ies in a iable Lpspaces wi h essen ially no addi ional wo k. All such inequali ies a e new. Co olla y 1.10. Gi en a amily Fand an open se Ω⊂Rn, assume ha o some p0,0< p0<∞, and o e e y w∈A∞, (1.11) ZΩ (x)p0w(x)dx ≤C0ZΩ g(x)p0w(x)dx, ( , g)∈F. Le p(·)∈P0(Ω) be such ha he e exis s 0< p1< p−wi h p(·)/p10∈B(Ω). Then o all ( , g)∈Fsuch ha ∈Lp(·)(Ω), (1.12) k kp(·),Ω≤Ckgkp(·),Ω. Fu he mo e, o e e y 0<q<∞and sequence {( j, gj)}j⊂F, (1.13)    X j ( j)q1/q   p(·),Ω ≤C   X j (gj)q1/q   p(·),Ω . Co olla y 1.11. Gi en a amily Fand an open se Ω⊂Rn, assume ha (1.11) holds o some 1< p0<∞, o e e y w∈Ap0and o all ( , g)∈F. Le p(·)∈P(Ω) be such ha he e exis s 1< p1< p−wi h p(·)/p10∈B(Ω). Then (1.12) holds o all ( , g)∈Fsuch ha ∈Lp(·)(Ω). Fu he mo e, o e e y 1<q<∞and {( j, gj)}j∈F, he ec o - alued inequali y (1.13) holds. The es o his pape is o ganized as ollows. To illus a e he powe o ou esul s, we i s conside some applica ions. In Sec ion 2 we gi e a numbe o examples o ope a o s which a e bounded on Lp(·). These esul s a e immedia e consequences o he abo e esul s and he heo y o weigh ed no m inequali ies. Some o hese ha e been p o ed by o he s, bu mos a e new. We also p o e ec o - alued inequali ies o hese ope a o s, all o which a e new esul s. In Sec ion 3 we p esen an applica ion o pa ial di e en ial equa ions: we ex end he Calde ´on– Zygmund inequali y (see [5], [22]) o solu ions o 4u= wi h ∈Lp(·)(Ω). In Sec ion 4 we gi e an applica ion o he heo y o Sobole spaces: we show ha he Calde ´on ex ension heo em (see [2], [4]) holds in a iable Sobole spaces. In Sec ion 5 we p o e Theo ems 1.3 and 1.8. Ou p oo is adap ed om he a gumen s gi en in [11]. Finally, in Sec ion 6 we p o e Co olla ies 1.10 and 1.11. Th oughou his pape , we will make use o he basic p ope ies o a iable Lpspaces, and will s a e some esul s as needed. Fo a de ailed discussion o hese spaces, see Ko ´aˇcik and R´akosn´ık [30]. As we no ed abo e, in o de o emphasize ha we a e dealing wi h a iable exponen s, we will always w i e p(·) ins ead o p o deno e an exponen unc ion. Th oughou , Cwill deno e a posi i e cons an whose exac alue may change a each appea ance. 246 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez 2. Applica ions: Es ima es o classical ope a o s on Lp(·) In his sec ion we gi e a numbe o applica ions o Theo ems 1.3 and 1.8, and Co olla ies 1.10 and 1.11, o show ha a wide a ie y o classical ope a o s a e bounded on he a iable Lpspaces. In he ollowing applica ions we will impose di e en condi ions on he exponen s p(·) o gua an ee he co esponding es ima es. In mos o he cases, i will su ice o assume ha p(·)∈B(Rn), o in pa icula ha p(·) sa is ies (1.1) and (1.2). As we no ed in he ema ks ollowing Theo em 1.3, o p o e hese applica ions we will need o use densi y a gumen s. In doing so we will use he ollowing ac s: (1) L∞ c, bounded unc ions o compac suppo , and C∞ c, smoo h unc ions o compac suppo , a e dense in Lp(·)(Ω). See Ko ´aˇcik and R´akosn´ık [30]. (2) I p+<∞and ∈Lp+(Ω) ∩Lp−(Ω), hen ∈Lp(·)(Ω). This ollows om he ac ha | (x)|p(x)≤ | (x)|p+χ{| (x)|≥1}+| (x)|p−χ{| (x)|<1}. 2.2. The Ha dy–Li lewood maximal unc ion. I is well known ha o 1 <p<∞and o w∈Ap, ZRn M (x)pw(x)dx ≤CZRn (x)pw(x)dx. F om Co olla y 1.11 wi h he pai s (M , | |), we ge ec o - alued inequali ies o Mon Lp(·), p o ided he e exis s 1 < p1< p−wi h p(·)/p10∈B(Rn); by Theo em 1.2, his is equi alen o p(·)∈B(Rn). To apply Co olla y 1.11 we need o es ic he pai s o unc ions ∈L∞ c, bu since hese o m a dense subse we ge he desi ed es ima e o all ∈Lp(·)(Rn). Co olla y 2.1. I p(·)∈B(Rn), hen o all 1< q < ∞,    X j (M j)q1/q   p(·),Rn ≤C   X j | j|q1/q   p(·),Rn . Rema k 2.2. F om Co olla y 1.11 we also ge one o he implica ions o Theo em 1.2: i p(·)/p10∈B(Rn) hen p(·)∈B(Rn). I is e y emp ing o specula e ha all o Theo em 1.2 can be p o ed ia ex apola ion, bu we ha e been unable o do so. 2.2. The sha p maximal ope a o . Gi en a measu able unc ion and a cube Q, de ine Q=1 |Q|ZQ (y)dy, and he sha p maximal ope a o by M# (x) = sup x3Q 1 |Q|ZQ | (y)− Q|dy. The boundedness o classical ope a o s on a iable Lpspaces 247 The sha p maximal ope a o was in oduced by Fe e man and S ein [20], who showed ha o all p, 0 <p<∞, and w∈A∞, ZRn M (x)pw(x)dx ≤CZRn M# (x)pw(x)dx. (Also see Jou n´e [26].) The e o e, by Co olla y 1.10 wi h he pai s (M , M# ), ∈L∞ c(Rn), and by Theo em 1.2 we ha e he ollowing esul . Co olla y 2.3. Le p(·)∈P0(Rn)be such ha he e exis s 0< p1< p− wi h p(·)/p1∈B(Rn). Then, (2.1) kM kp(·),Rn≤CkM# kp(·),Rn, and o all 0<q<∞, (2.2)    X j (M j)q1/q   p(·),Rn ≤C   X j (M# j)q1/q   p(·),Rn . Rema k 2.4. Co olla y 2.3 gene alizes esul s due o Diening and R˚uˇziˇcka [15, Theo em 3.6] and Diening [14, Theo em 8.10], who p o ed (2.1) wi h M eplaced by on he le -hand side and unde he assump ions ha p(·) and p0(·)∈B(Rn) wi h 1 < p−≤p+<∞in he i s pape and p(·)∈B(Rn) in he second. No ice ha ou esul is mo e gene al since we allow p(·) o go below 1 and we only need p(·)/p10∈B(Rn) o some small alue 0 < p1< p−. Fu he mo e, we au oma ically ob ain he ec o - alued inequali ies gi en in (2.2). 2.3. Singula in eg al ope a o s. Gi en a locally in eg able unc ion K de ined on Rn {0}, suppose ha he Fou ie ans o m o Kis bounded, and K sa is ies (2.3) |K(x)| ≤ C |x|n,|∇K(x)| ≤ C |x|n+1 , x 6= 0. Then he singula in eg al ope a o T, de ined by T (x) = K∗ (x), is a bounded ope a o on weigh ed Lp. Mo e p ecisely, gi en 1 <p<∞, i w∈Ap, hen (2.4) ZRn |T (x)|pw(x)dx ≤CZRn | (x)|pw(x)dx. (Fo de ails, see [17], [21].) F om Co olla y 1.11, we ge ha Tis bounded on a iable Lpp o ided he e exis s 1 < p1< p−wi h p(·)/p10∈B(Rn); by Theo em 1.2 his is equi alen o p(·)∈B(Rn). Again, o apply he co olla y we need o es ic ou sel es o a sui able dense amily o unc ions. We use he ac ha C∞ cis dense in Lp(·)(Rn), and he ac ha i ∈C∞ c, hen T ∈T1<p<∞Lp⊂Lp(·)(Rn). 254 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez We begin wi h a ew de ini ions and a lemma. Gi en p(·)∈P(Ω) and a na u al numbe k, de ine he a iable Sobole space Wk,p(·)(Ω) o be he se o all unc ions ∈Lp(·)(Ω) such ha X |α|≤k kDα kp(·),Ω<+∞, whe e he de i a i es a e unde s ood in he sense o dis ibu ions. Gi en a unc ion which is wice di e en iable (in he weak sense), we de ine o i= 1,2, Di =X |α|=i (Dα )21/2 . We need he ollowing auxilia y esul whose p oo can be ound in [30]. Lemma 3.1. I Ω⊂Rnis a bounded domain, and i p(·), q(·)∈P(Ω) a e such ha p(x)≤q(x),x∈Ω, hen k kp(·),Ω≤(1 + |Ω|)k kq(·),Ω. Theo em 3.2. Gi en an open se Ω⊂Rn,n≥3, suppose p(·)∈P(Ω) wi h p+< n/2sa is ies (1.1) and (1.2). I ∈Lp(·)(Ω), hen he e exis s a unc ion u∈Lq(·)(Ω), whe e (3.1) 1 p(x)−1 q(x)=2 n, such ha (3.2) 4u(x) = (x),a.e. x∈Ω. Fu he mo e, kD2ukp(·),Ω≤Ck kp(·),Ω,(3.3) kD1uk (·),Ω≤Ck kp(·),Ω,(3.4) kukq(·),Ω≤Ck kp(·),Ω,(3.5) whe e 1 p(x)−1 (x)=1 n. In pa icula , i Ωis bounded, hen u∈W2,p(·)(Ω). P oo . Ou p oo oughly ollows he p oo in he se ing o Lebesgue spaces gi en by Gilba g and T udinge [22], bu also uses his esul in key s eps. Fix ∈Lp(·)(Ω); wi hou loss o gene ali y we may assume ha k kp(·),Ω= 1. Decompose as = 1+ 2= χ{x:| (x)|>1}+ χ{x:| (x)|≤1}. The boundedness o classical ope a o s on a iable Lpspaces 255 No e ha | i(x)| ≤ | (x)|and so k ikp(·),Ω≤1. Fu he , we ha e ha 1∈ Lp−(Ω) and 2∈Lp+(Ω) since, by he de ini ion o he no m in Lp(·)(Ω) and since k kp(·),Ω= 1, ZΩ 1(x)p−dx =Z{x∈Ω:| (x)|>1} | (x)|p−dx ≤ZΩ | (x)|p(x)dx ≤1, ZΩ 2(x)p+dx =Z{x∈Ω:| (x)|≤1} | (x)|p+dx ≤ZΩ | (x)|p(x)dx ≤1. Thus, we can sol e Poisson’s equa ion wi h 1and 2(see [22]): mo e p ecisely, de ine u1(x) = (Γ ∗ 1)(x), u2= (Γ ∗ 2)(x), whe e Γ is he New onian po en ial, Γ(x) = 1 n(2 −n)ωn |x|2−n, and ωnis he olume o he uni ball in Rn. Since p−and q−also sa is y (3.1), by he Calde ´on–Zygmund inequali y on classical Lebesgue spaces, u1∈Lq−(Ω). Simila ly, since p+and q+sa is y (3.1), u2∈Lq+(Ω). Le u=u1+u2; hen u∈Lq−(Ω) + Lq+(Ω). Since u1and u2a e solu ions o Poisson’s equa ion, 4u(x) = 4u1(x) + 4u2(x) = 1(x) + 2(x) = (x),a.e. x∈Ω. We show ha u∈Lq(·)(Ω) and ha (3.5) holds: by inequali y (2.12), kukq(·),Ω≤ ku1kq(·),Ω+ku2kq(·),Ω =1 n(2 −n)ωnkI2 1kq(·),Ω+kI2 2kq(·),Ω ≤Ck 1kp(·),Ω+k 2kp(·),Ω ≤C=Ck kp(·),Ω; he las equali y holds since k kp(·),Ω= 1. Simila ly, a di ec compu a ion shows ha o any mul i-index α,|α|= 1, |DαΓ(x)| ≤ 1 n ωn |x|1−n. The e o e, |Dαu(x)| ≤ |Dα(Γ ∗ 1)(x)|+|Dα(Γ ∗ 2)(x)| =|(DαΓ∗ 1)(x)|+|(DαΓ∗ 2)(x)| ≤1 n ωnI1(| 1|)(x) + I1(| 2|)(x). 256 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez So again by inequali y (2.12) we ge kDαuk (·),Ω≤Ck 1kp(·),Ω+k 2kp(·),Ω≤C, which yields inequali y (3.4). Gi en a mul i-index α,|α|= 2, ano he compu a ion shows ha DαΓ is a singula con olu ion ke nel which sa is ies (2.3). The e o e, he ope a o Tαg(x) = (DαΓ∗g)(x) = Dα(Γ ∗g)(x) is singula in eg al ope a o , and as be o e (3.3) ollows om inequali y (2.5) and Rema k 2.6 applied o 1and 2. Finally, i Ω is bounded, since p(x)≤q(x) and p(x)≤ (x), x∈Ω, by Lemma 3.1 we ha e ha u∈W2,p(·)(Ω). Rema k 3.3. In he p e ious es ima es we could ha e wo ked di ec ly wi h . Had we done so, howe e , we would ha e had o check ha all he in eg als appea ing we e absolu ely con e gen . The ad an age o decomposing as 1+ 2 is ha we did no need o pay a en ion o his since 1∈Lp−(Ω), 2∈Lp+(Ω). We also wan o s ess ha u1and u2, as solu ions o Poisson’s equa ion wi h 1∈Lp−(Ω) and 2∈Lp+(Ω), sa is y Lebesgue space es ima es. Fo ins ance, as no ed abo e, u∈Lq−(Ω) + Lq+(Ω). Howe e , we ha e ac ually p o ed mo e, since Lq(·)(Ω) is a smalle space. Simila ema ks hold o he i s and second de i a i es o u. 4. The Calde ´on ex ension heo em In his sec ion we s a e and p o e he Calde ´on ex ension heo em o a iable Sobole spaces. Ou p oo ollows closely he p oo o he esul in he classical se - ing; see, o example, R. Adams [2] o Calde ´on [4]. Fi s , we gi e wo de ini ions and a lemma. De ini ion 4.1. Gi en a poin x∈Rn, a ini e cone wi h e ex a x,Cx, is a se o he o m Cx=B1∩ {x+λ(y−x) : y∈B2, λ > 0}, whe e B1is an open ball cen e ed a x, and B2is an open ball which does no con ain x. De ini ion 4.2. An open se Ω ⊂Rnhas he uni o m cone p ope y i he e exis s a ini e collec ion o open se s {Uj}(no necessa ily bounded) and an associa ed collec ion {Cj}o ini e cones such ha he ollowing hold: (1) he e exis s δ > 0 such ha Ωδ={x∈Ω : dis (x, ∂Ω) < δ} ⊂ S j Uj; (2) o e e y index jand e e y x∈Ω∩Uj,x+Cj⊂Ω. The boundedness o classical ope a o s on a iable Lpspaces 257 An example o a se Ω wi h he uni o m cone p ope y is any bounded se whose bounda y is locally Lipschi z. (See Adams [2].) Finally, in gi ing ex ension heo ems o a iable Lpspaces, we mus wo y abou ex ending he exponen unc ion p(·). The ollowing esul shows ha his is always possible, p o ided ha p(·) sa is ies (1.1) and (1.2). Lemma 4.3. Gi en an open se Ω⊂Rnand p(·)∈P(Ω) such ha (1.1) and (1.2) hold, he e exis s a unc ion ˜p(·)∈P(Rn)such ha : (1) ˜psa is ies (1.1) and (1.2); (2) ˜p(x) = p(x),x∈Ω; (3) ˜p−=p−and ˜p+=p+. Rema k 4.4. Diening [13] p o ed an ex ension heo em o exponen s p(·) which sa is y (1.1), p o ided ha Ω is bounded and has Lipschi z bounda y. I would be in e es ing o de e mine i e e y exponen p(·)∈B(Ω) can be ex ended o an exponen unc ion in B(Rn). P oo . Since p(·) is bounded and uni o mly con inuous, by a well-known e- sul i ex ends o a con inuous unc ion on Ω. S aigh o wa d limi ing a gumen s show ha his ex ension sa is ies (1), (2) and (3). The ex ension o p(·) on Ω o ˜p(·) de ined on all o Rn ollows om a cons uc ion due o Whi ney [45] and desc ibed in de ail in S ein [42, Chap e 6]. Fo ease o e e ence, we will ollow S ein’s no a ion. We i s conside he case when Ω is unbounded; he case when Ω is bounded is simple and will be ske ched below. When Ω is unbounded, (1.2) is equi alen o he exis ence o a cons an p∞, p−≤p∞≤p+, such ha o all x∈Ω, |p(x)−p∞| ≤ C log(e+|x|). De ine a new unc ion (·) by (x) = p(x)−p∞. Then (·) is s ill bounded ( hough no longe necessa ily posi i e), s ill sa is ies (1.1) on Ω and sa is ies (4.1) | (x)| ≤ C log(e+|x|). We will ex end o all o Rn. I we de ine ω( ) = 1/log(e/2 ), 0 < ≤1/2, and ω( ) = 1 o ≥1/2, hen a s aigh o wa d calcula ion shows ha ω( )/ is a dec easing unc ion and ω(2 )≤C ω( ). Fu he , since log(e/2 )≈log(1/ ), 0< < 1/2, and since is bounded, | (x)− (y)| ≤ Cω(|x−y|) o all x, y ∈Ω. The e o e, by Co olla y 2.2.3 in S ein [42, p. 175], he e exis s a unc ion ˜ (·) on Rnsuch ha ˜ (x) = (x), x∈Ω, and such ha ˜ (·) sa is ies (1.1). Fo x∈Rn Ω, ˜ (x) is de ined by he sum ˜ (x) = X k (pk)ϕ∗ k(x), 258 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez whe e {Qk}a e he cubes o he Whi ney decomposi ion o Rn Ω, {ϕ∗ k}is he pa i ion o uni y subo dina e o his decomposi ion, and each poin pk∈Ω is such ha dis (pk, Qk) = dis (Ω, Qk). I ollows immedia ely om his de ini ion ha o all x∈Rn, −≤˜ (x)≤ +. Howe e , ˜ (·) need no sa is y (4.1) so we mus modi y i sligh ly. To do so we need he ollowing obse a ion: i 1, 2a e unc ions such ha | i(x)− i(y)| ≤ Cω(|x−y|), x, y ∈Rn,i= 1,2, hen min( 1, 2) and max( 1, 2) sa is y he same inequali y. The p oo o his obse a ion consis s o a numbe o e y simila cases. Fo ins ance, suppose min 1(x), 2(x)= 1(x) and min 1(y), 2(y)= 2(y). Then 1(x)− 2(y)≤ 2(x)− 2(y)≤Cω(|x−y|), 2(y)− 1(x)≤ 1(y)− 1(x)≤Cω(|x−y|). Hence, min 1(x), 2(x)−min 1(y), 2(y)=| 1(x)− 2(y)| ≤ Cω(|x−y|). I ollows immedia ely om his obse a ion ha s(x) = maxmin(˜ (x), C/ log(e+|x|),−C/ log(e+|x|) sa is ies (1.1) and (4.1). The e o e, i we de ine ˜p(x) = s(x) + p∞, hen (1), (2) and (3) hold. Finally, i Ω is bounded, we de ine (x) = p(x)−p+and epea he abo e a gumen essen ially wi hou change. Theo em 4.5. Gi en an open se Ω⊂Rnwhich has he uni o m cone p ope y, and gi en p(·)∈P(Ω) such ha (1.1) and (1.2) hold, hen o any na u al numbe k he e exis s an ex ension ope a o Ek:Wk,p(·)(Ω) →Wk,p(·)(Rn), such ha Eku(x) = u(x), a.e. x∈Ω, and kEkukp(·),Rn≤C(p(·),k,Ω)kukp(·),Ω. The p oo o Theo em 4.5 in a iable Sobole spaces is nea ly iden ical o ha in he classical se ing. (See Adams [2].) The p oo , beyond calcula ions, equi es he ollowing ac s which ou hypo heses insu e a e ue. The boundedness o classical ope a o s on a iable Lpspaces 259 – By Lemma 4.3, p(·) immedia ely ex ends o an exponen unc ion on Rn. – Func ions in C∞(Ω) a e dense in Wk,p(·)(Ω). By ou hypo heses, he max- imal ope a o is bounded on Lp(·)(Ω), and he densi y o C∞(Ω) ollows om his by he s anda d a gumen (c . Zieme [47]). Fo mo e de ails, see Diening [12] o C uz-U ibe and Fio enza [9]. – I ϕis a smoo h unc ion on Rn {0}wi h compac suppo , and i he e exis s ε > 0 such ha on Bε(0), ϕis a homogeneous unc ion o deg ee k, k > −n, hen kϕ∗ kp(·),Ω≤Cp(·), ϕk kp(·),Ω. This again ollows om he ac ha he maximal ope a o is bounded on Lp(·)(Ω), and om he well- known inequali y |ϕ∗ (x)| ≤ CM (x). Fo mo e de ails, see C uz-U ibe and Fio enza [9]. – Singula in eg al ope a o s wi h ke nels o he o m K(x) = G(x) |x|n, whe e Gis bounded on Rn {0}, has compac suppo , is homogeneous o deg ee ze o on BR(0) {0} o some R > 0, and has RSRG dx = 0, a e bounded on Lp(·)(Ω). Such ke nels a e essen ially he same as hose gi en by (2.9), and as discussed abo e, ou hypo heses imply ha hey a e bounded. Rema k 4.6. I p(·) sa is ies (1.1), hen C∞ c(Rn) is dense in Wk,p(·)(Rn). (See [9], [41].) Hence, i he hypo heses o Theo em 4.5 hold, hen i ollows immedia ely ha he se {uχΩ:u∈C∞ c(Rn)}is dense in Wk,p(·)(Ω). Howe e his esul is ue unde much weake hypo heses; see [9], [18], [19], [25], [46] o de ails. 5. P oo o Theo ems 1.3 and 1.8 Since Theo em 1.3 is a pa icula case o Theo em 1.8 wi h p0=q0, i su ices o p o e he second esul . We need wo ac s abou a iable Lpspaces. Fi s , i p(·), q(·)∈P0(Ω) and p(x)/q(x) = , hen i ollows om he de ini ion o he no m ha (5.1) k k p(·),Ω= | |  q(·),Ω. Second, gi en p(·)∈P(Ω), we ha e he gene alized H¨olde ’s inequali y (5.2) ZΩ | (x)g(x)|dx ≤1 + 1 p− −1 p+k kp(·),Ωkgkp0(·),Ω, and he “duali y” ela ionship (5.3) k kp(·),Ω≤sup gZΩ (x)g(x)dx≤1 + 1 p− −1 p+k kp(·),Ω, 260 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez whe e he sup emum is aken o e all g∈Lp0(·)(Ω) such ha kgkp0(·),Ω= 1. Fo p oo s o hese esul s, see Ko ´aˇcik and R´akosn´ık [30]. The p oo o Theo em 1.8 begins wi h a e sion o a cons uc ion due o Rubio de F ancia [38] (also see [11], [21]). Fix p(·)∈P0(Ω) such ha p−> p0, and le ¯p(x) = p(x)/p0. De ine q(·) as in (1.9), and le ¯q(x) = q(x)/q0. By assump ion, he maximal ope a o is bounded on L¯q0(·)(Ω), so he e exis s a posi i e cons an Bsuch ha kM k¯q0(·),Ω≤Bk k¯q0(·),Ω. De ine a new ope a o Ron L¯q0(·)(Ω) by Rh(x) = ∞ X k=0 Mkh(x) 2kBk, whe e, o k≥1, Mk=M◦M◦· · ·◦Mdeno es ki e a ions o he maximal ope - a o , and M0is he iden i y ope a o . I ollows immedia ely om his de ini ion ha : (a) i his non-nega i e, h(x)≤Rh(x); (b) kRhk¯q0(·),Ω≤2khk¯q0(·),Ω; (c) o e e y x∈Ω, M(Rh)(x)≤2BRh(x), so Rh∈A1wi h an A1cons an ha does no depend on h. We can now a gue as ollows: by (5.1) and (5.3), k kq0 q(·),Ω=k q0k¯q(·),Ω≤sup ZΩ (x)q0h(x)dx, whe e he sup emum is aken o e all non-nega i e h∈L¯q0(·)(Ω) wi h khk¯q0(·),Ω= 1. Fix any such unc ion h; i will su ice o show ha ZΩ (x)q0h(x)dx ≤Ckgkq0 p(·),Ω wi h he cons an Cindependen o h. Fi s no e ha by (a) abo e we ha e ha (5.4) ZΩ (x)q0h(x)dx ≤ZΩ (x)q0Rh(x)dx. By (5.2), (b), and since ∈Lq(·)(Ω), ZΩ (x)q0Rh(x)dx ≤Ck q0k¯q(·),ΩkRhk¯q0(·),Ω ≤Ck kq0 q(·),Ωkhk¯q0(·),Ω ≤Ck kq0 q(·),Ω<∞. The boundedness o classical ope a o s on a iable Lpspaces 261 The e o e, we can apply (1.8) o he igh -hand side o (5.4) and again apply (5.2), his ime wi h exponen ¯p(·): ZΩ (x)q0Rh(x)dx ≤CZΩ g(x)p0Rh(x)p0/q0dxq0/p0 ≤Ckgp0kq0/p0 ¯p(·),Ωk(Rh)p0/q0kq0/p0 ¯p0(·),Ω =Ckgkq0 p(·),Ωk(Rh)p0/q0kq0/p0 ¯p0(·),Ω. To comple e he p oo , we need o show ha k(Rh)p0/q0kq0/p0 ¯p0(·),Ωis bounded by a cons an independen o h. Bu i ollows om (1.9) ha o all x∈Ω, ¯p0(x) = p(x) p(x)−p0 =q0 p0 q(x) q(x)−q0 =q0 p0 ¯q0(x). The e o e, k(Rh)p0/q0kq0/p0 ¯p0(·),Ω=kRhk¯q0(·),Ω≤Ckhk¯q0(·),Ω=C. This comple es ou p oo . 6. P oo o Co olla ies 1.10 and 1.11 The p oo s o Co olla ies 1.10 and 1.11 equi e he mo e gene al e sions o he ex apola ion heo ems discussed in he in oduc ion. Fo he con enience o he eade we s a e hem bo h he e. Theo em 6.1. Gi en a amily Fand an open se Ω⊂Rn, assume ha o some p0,0< p0<∞, and o e e y w∈A∞, (6.1) ZΩ (x)p0w(x)dx ≤C0ZΩ g(x)p0w(x)dx, ( , g)∈F. Then o all 0<p<∞and w∈A∞, (6.2) ZΩ (x)pw(x)dx ≤C0ZΩ g(x)pw(x)dx, ( , g)∈F. Fu he mo e, o e e y 0< p, q < ∞,w∈A∞, and sequence {( j, gj)}j⊂F, (6.3)    X j ( j)q1/q   Lp(w,Ω) ≤C   X j (gj)q1/q   Lp(w,Ω) . Theo em 6.2. Gi en a amily Fand an open se Ω⊂Rn, assume ha o some p0,1< p0<∞, and o e e y w∈Ap0,(6.1) holds. Then o e e y 1< p < ∞and w∈Ap,(6.2) holds. Fu he mo e, o e e y 1< p, q < ∞, w∈Ap, and sequence {( j, gj)}j⊂F,(6.3) holds. 262 D. C uz-U ibe, SFO, A. Fio enza, J. M. Ma ell and C. P´e ez Theo em 6.1 is p o ed in [11]. The o iginal s a emen o Theo em 6.2 is only o pai s o he o m (|T |, ), and does no include he ec o - alued es i- ma e (6.3). (See [17], [21], [38].) Howe e , an examina ion o he p oo s shows ha hey hold wi hou change when applied o pai s ( , g)∈F. Fu he mo e, as we no ed be o e, his app oach immedia ely yields he ec o - alued inequali ies: gi en a amily Fand 1 <q<∞, de ine he new amily Fq o consis o he pai s (Fq, Gq), whe e Fq(x) = X j ( j)q1/q , Gq(x) = X j (gj)q1/q ,{( j, gj)}j⊂F. Clea ly, inequali y (6.1) holds o Fqwhen p0=q, so by ex apola ion we ge (6.3). Co olla y 1.10 ollows immedia ely om Theo ems 1.3 and 6.1. Since (1.11) holds o some p0, by Theo em 6.1 i holds o all 0 <p<∞and o all w∈A∞. The e o e, we can apply Theo em 1.3 wi h p1in place o p0 o ob ain (1.12). To p o e he ec o - alued inequali y (1.13), no e ha by (6.3) we can apply Theo em 1.3 o he amily Fqde ined abo e, again wi h p1in place o p0. In exac ly he same way, Co olla y 1.11 ollows om Theo ems 1.3 and 6.2. Re e ences [1] Ace bi, E., and G. Mingione: Regula i y esul s o s a iona y elec o heological luids. - A ch. Ra ion. Mech. Anal. 164, 2002, 213–259. [2] Adams, R.: Sobole Spaces. - Academic P ess, New Yo k, 1975. [3] Al a ez, J., and C. 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