Hindawi Publishing Copoaion Jounal of Funcion Spaces and Applicaions Volume 22, Aicle ID 29363, pages doi:.55/22/29363 Reseach Aicle A Noe on Muliplicaion and Composiion Opeaos in Loenz Spaces Eddy Kwessi, Paul Alfonso, 2 Gealdo De Souza, 3 and Ashebe Abebe 3 Tiniy Univesiy, One Tiniy Place, San Anonio, TX 7822, USA 2 US Ai Foce Academy, Coloado Spings, CO 884, USA 3 Depamen of Mahemaics and Saisics, Aubun Univesiy, 22 Pake Hall, Aubun, AL 36849, USA Coespondence should be addessed o Gealdo De Souza, desougs@aubun.edu Received 6 Febuay 22; Acceped 2 June 22 Academic Edio: Gesu Ólafsson, Copyigh q 22 Eddy Kwessi e al. This is an open access aicle disibued unde he Ceaive Commons Aibuion License, which pemis unesiced use, disibuion, and epoducion in any medium, povided he oiginal wok is popely cied. we evisi he Loenz spaces L p, q fo p>,q > defined by G. G. Loenz in he nineeen fifies and we show how he aomic decomposiion of he spaces L p, obained by De Souza in 2 can be used o chaaceize he muliplicaion and composiion opeaos on hese spaces. These chaaceizaions, hough obained fom a compleely diffeen pespecive, confim he vaious esuls obained by S. C. Aoa, G. Da and S. Vema in diffeen vaians of he Loenz Spaces.. Inoducion In he ealy 95s, Loenz inoduced he now famous Loenz spaces L p, q in his papes, 2 as a genealizaion of he L p spaces. The paamees p and q encode he infomaion abou he size of a funcion; ha is, how all and how spead ou a funcion is. The Loenz spaces ae quasi-banach spaces in geneal, bu he Loenz quasi-nom of a funcion has bee conol ove he size of he funcion han he L p nom, via he paamees p and q, making he spaces vey useful. We ae mosly concened wih sudying he muliplicaion and composiion opeaos on Loenz spaces. These have been sudied befoe by vaious auhos in paicula by Aoa e al. in 3 6. In his pape, he esuls we obain ae in accodance wih wha hese auhos have found befoe. We believe ha he echniques and elaive simpliciy of ou appoach ae woh epoing o fuhe enich he opic. Ou esuls, found on he bounday of he uni disc due o he oiginal focus by De Souza in 7, will show how one
2 Jounal of Funcion Spaces and Applicaions can use he aomic chaaceizaion of he Loenz space L p, in he sudy of muliplicaion and composiion opeaos in he spaces L p, q. 2. Peliminaies Le X, μ be a measue space. Definiion 2.. Le f be a complex-valued funcion defined on X. The deceasing eaangemen of f is he funcion f defined on, by f inf { y>:d f, y }, 2. whee d f, y μ {x : f x >y} is he disibuion of he funcion f. Definiion 2.2. Given a measuable funcion f on X, μ and <p, q, define f L p,q q p sup > /p f q d /q, if q<, /p f, if q. 2.2 The se of all funcions f wih f L p,q < is called he Loenz space wih indices p and q and denoed by L p, q X, μ. We now conside he measue μ on X o be finie. Le g : X X be a μ-measuable funcion such ha μ g A Cμ A fo a μ-measuable se A, 2π and fo an absolue consan C. Heeg A efes o he peimage of he se A. Remak 2.3. I is impoan o noe ha g sup μ A / μ g A /μ A is no necessaily a nom. Definiion 2.4. Fo a given funcion g, we define he muliplicaion opeao T g on Loenz spaces as T g f f g and he composiion opeao C g as C g f f g. The following wo esuls ae used in ou poofs. The fis is a esul of De Souza 7 which gives an aomic decomposiion of L p,. The second is he Macinkiewicz inepolaion heoem see 8 which we sae fo compleeness of pesenaion. Theoem 2.5 see De Souza 7. A funcion f L p, fo p > if and only if f n c nχ An wih n c n μ /p A n <, wheeas μ is measue on X and A n ae μ-measuable ses in X. Moeove, f L p, inf n c n μ /p A n, whee he infimum is aken ove all possible epesenaions of f.
Jounal of Funcion Spaces and Applicaions 3 Theoem 2.6 see Macinkiewicz. Assume ha fo < p / p, fo all q >, fo all measuable subses A of X, hee ae some consans <M, M < such ha fo a linea o quasi-linea opeao T g a T g χ A L p, M μ /p A. b T g χ A L p, M μ /p A. Then hee is some M> such ha T g f L p,q M f L p,q fo /p θ/p θ /p, < θ<. One implicaion of Theoem 2.5 is ha i can be used o pove and jusify a heoem of Sein and Weiss 9. Tha is, o show ha linea opeaos T : L p, B ae bounded, whee B is Banach space closed unde absolue value and saisfying f B f B, all one needs o show is ha Tχ A B Mμ /p A, p>. Theoem 2.6 will be used o show ha valid esuls on L p, ae also valid on L p, q. Definiion 2.7. We denoe by M p he se of eal-valued funcions defined on X, 2π such ha f p M sup x> px /p / f /p d <, 2.3 whee p <. We will show ha he space M p is equivalen o a weak L k space fo some k ha depends on p and and p M is quasinom. Lemma 2.8. M p a quasinom on M p. Poof. f by definiion. This implies ha f p M. Moeove, f p M implies ha x fo all <x 2π, f /p d/. Hence, we have f μ-a.e, hus, f since f is a epesenaive of an equivalence class. Now le k / be a eal consan, f M p,and x, 2π. Noing kf k f, he homogeneiy condiion kf p M k f p M follows ivially. Le f, g M p. Since f g f /2 g /2, fo any x, 2π, we have f g /p d 2 2p /2 x f /p d 2 f /p d g /p d 2 /2 x g /p d. 2.4
4 Jounal of Funcion Spaces and Applicaions Since a b / a / b / fo a, b >, we have f g p M 2 /p / f p M g M p, wih 2 /p / > fo, p >. 2.5 Theoem 2.9. M p L p,,whee,, / /. Poof. Suppose g M p. Thee is an absolue consan C such ha fo all x>, C px /p / g /p d g x px /p /p d / x /p g x. 2.6 Thus, sup x> x /p g x C implying ha g L p,. Convesely, le g L p,. Then hee is an absolue consan C such ha, g C /p. This implies ha g /p C /p.thus, sup x> px /p g /p d / sup x> C x /p d px /p / C /. 2.7 This implies ha g M p. Remak 2.. One can easily see fom Theoem 2.9 ha M p L p, and M p Moeove, g p M g. To see his, noe ha gm x p sup g /p d g x sup /p d g, x> px /p x> px /p g p M g x x /p d g x px /p L. 2.8 fo all x since g is deceasing. Taking he limi as x, we see ha g p M g g. 3. Main Resuls 3.. Muliplicaion Opeaos Theoem 3. see muliplicaion opeao on L p,. The muliplicaion opeao T g : L p, L p, fo p p> is bounded if and only g L. Moeove, T g g. Poof. I is convenien o use M p which is equivalen o L. Assume ha T g f L p, C f L p,. Then fo f χ,x whee x, 2π, Tg χ,x /p d g /p d C χ,x /p d Cpx /p. 3.
Jounal of Funcion Spaces and Applicaions 5 Muliplying and dividing he inegand on he lef by /p,wege g /p p p /pp d Cpx /p. 3.2 Since p p /pp is deceasing on,x and <x 2π, we have px /p g /p d C 2π p p /pp. 3.3 Taking he supemum ove all x>, we have ha g M p. Assume ha g M p and x>. Since p >pwe have x T g χ L p,x, g /p d g /p d. 3.4 And so, T g χ L p,x, M χ L p,,x, whee M sup x> px /p g /p d. 3.5 Using he aomic decomposiion of L p,,wege Tg f L p, M fl p, fo some posiive consan M. 3.6 To pove he second pa of he heoem, fis noe ha he expession in 3.5 gives ha T g g. Now ake f /x /p χ,x. We can easily see ha f L p, and T g f L p, g x fo x, 2π since g is deceasing. Now aking he sup ove f L p, and he limi as x gives T g g.thus, T g g. The following heoem, which is equivalen o Theoem. of 6, follows fom Theoems 2.6 and 3.. Theoem 3.2 see muliplicaion opeao on L p, q. The muliplicaion opeao T g : L p, q L p, q is bounded if and only if g L fo <p, <q. Moeove, T g g. Remak 3.3. Since, by Theoem 2.9, M p Mp fo >, he heoem implies ha if he muliplicaion opeao T g : L p, q L p, q defined by T g f g f is bounded, hen g M p fo p, q >. Remak 3.4. I is woh obseving ha he nom fl p, sup A X μ /p A μ A / μ A g /p d 3.7 can also be used o pove he pevious heoem, fis on L p, and hen on L p, q by means of eihe he Macienkiewiecz Inepolaion Theoem o Theoem 2.5. Acually, his nom
6 Jounal of Funcion Spaces and Applicaions was he oiginal moivaion fo he inoducion of he space M p. Fo sake of simpliciy and wihou loss of genealiy, we modified i by eplacing μ A, A X by x >. Theoem 3.5. If f L p,q, and g L p 2,q 2,whee <p,p 2,q,q 2 <, heng f L, s whee / = /p /p 2 and /s /q /q 2. Poof. Given <p,p 2,q,q 2 <, assume f L p,q and g L p 2,q 2.Le, s be such ha / /p /p 2 and /s /q /q 2. Since f g f g, we have f g / s d s s f /p g /p d 2. 3.8 Using Holde s inequaliy on he RHS wih s/q s/q 2, we have f g / s d f /p s/q q d 2π g /q 2 s/q2 q2 d. 3.9 Thus, we have g f L,s f L p,q g L p2,q 2. 3. Theoem 3.6. If g M p,hent g : L q, s L pq / p q,s is bounded, whee / / and fo s> and p, q >. Poof. Le g M p L p, Tg f s L k,s g f /k s d g /p f /q s d if k q q. 3. Theefoe, Tg f s L k,s sup g /p s > f /q s d. 3.2 Tha is, Tg f L k,s g M p f L q,s, whee k p q p q. 3.3 Noing ha M p L p,, /, i is easy o see ha Theoem 3.6 shows ha he esul of Theoem 3.5 exends o he case whee q 2.
Jounal of Funcion Spaces and Applicaions 7 3.2. Composiion Opeaos Theoem 3.7. The composiion opeao C g : L p, q L p, q is bounded if and only if hee is an absolue consan C such ha μ g A Cμ A, 3.4 fo all μ-measuable ses A, 2π and fo <p, q. Moeove, C g g /p. Poof. We will pove his heoem fo L p, and use he inepolaion heoem o conclude fo L p, q. Fis assume ha C g : L p, L p, is bounded ha is, hee is an absolue consan C such ha Cg f L p, C f L p,. 3.5 Le A be a μ-measuable se in, 2π and le f χ A. Then, 3.5 is equivalen o C g χ L p, A C χ L p, A Cg χ /p A d p C p χ A /p d; 3.6 ha is, 2π χa g /p d C χ,μ A /p d. p p 3.7 Since χ A g χ g A, hen χ A g χ,μ g A. Theefoe, he pevious inequaliy gives p μ g A /p d C p μ A /p d. 3.8 And hence, μ g A C p μ A. 3.9
8 Jounal of Funcion Spaces and Applicaions On he ohe hand, assume ha hee is some consan C> such ha μ g A Cμ A. Then, Cg χ L p, A χa g /p d p p χ,μ g A /p d /p μ g A C /p μ A /p. 3.2 Consequenly, Cg χ A L p, C /p μ A /p. 3.2 As a consequence of Theoem 2.5 o he esul by Weiss and Sein in 9, we have C g f L p, C /p f L p,. 3.22 To pove he second pa of he heoem, noe ha fom he above, we have Cg sup f L p, Cg f L p, f L p, C /p. 3.23 Bu inf{c : μ g A Cμ A } g. Thus, C g g /p. To obain he ohe inequaliy, le f / μ A /p χ A. This gives f L p, and Cg f L p, { μ g A } /p. 3.24 μ A Thus, Cg sup Cg f { } μ g /p A L p, sup g /p. 3.25 f L p, μ A / μ A
Jounal of Funcion Spaces and Applicaions 9 Now o show he esul fo L p, q, noe ha he opeao C g is linea on L p, q and ha fo p and p such ha p <p<p, we have C g χ A L p, M μ A /p and C g χ A L p, M μ A /p. Since L p i, L p i,, i,, hen fo some absolue consans C and C we have Cg χ L p A, C /p μ A, Cg χ L p A, C /p μ A. 3.26 ha Hence, by he inepolaion heoem we conclude ha hee is a consan C> such Cg f L p,q C f L p,q fo p <p<p, q and f L p, q. 3.27 Remak 3.8. The necessay and sufficien condiion 3.4 makes inuiive sense if we conside a vaiey of measues. Le us conside wo of hem. If μ is he Lebesgue measue and X happens o be an ineval, hen i suffices o ake g as he lef muliplicaion by an absolue consan a o achieve 3.4. 2 If insead μ is he Haa measue, by aking X,, he locally compac opological goup of nonzeo eal numbes wih muliplicaion as opeaion, hen fo any Boel se A X, we have μ A A d. Hence 3.4 is achieved fo a measuable funcion g such ha g A A. The lef muliplicaion by he ecipocal of an absolue consan a would be enough. Remak 3.9. The esuls in Theoems 3.6 and 3.7 ae in accodance wih he esuls of Aoa e al. in 5, 6. In fac, even hough hey obained hei esuls in a moe geneal vesion of Loenz spaces, hei necessay and sufficien condiions fo boundedness of he muliplicaion and composiion opeaos ae he same as ous. 4. Discussion The space L p,, p >, seems o be undeuilized in analysis despie he fac ha in he 95s Sein and Weiss 9 showed ha fo a sublinea opeao T and a Banach space X if Tχ A X cμ A /p, hen Tf X c f L p, ;hais,t can be exended o he whole L p,. De Souza 7 showed ha he eason fo his is he naue of L p, in ha f L p, if and only if f n c nχ An wih n c n μ A n /p <. This aomic decomposiion of L p, povides us wih a echnique o sudy opeaos on L p, q and in paicula L p. In ohe wods, o sudy opeaos on L p, q, all we need is o sudy he acions of he opeao on chaaceisic funcions which can hen be lifed o L p, q hough he use of inepolaion heoems. Alhough we only consideed muliplicaion and composiion opeaos, ohe opeaos Hady-Lilewood maximal, Caleson maximal, Hankel, ec. can be sudies likewise. To conclude, he goal of he pesen pape is o show ha a simple aomic decomposiion of L p, spaces shows he boundedness of opeaos on L p, q saighfowad.
Jounal of Funcion Spaces and Applicaions In fac, we showed ha unlike ohe echniques in he lieaue, he boundedness of hese opeaos on chaaceisic funcions is enough o genealize o he whole Loenz space. This echnique is no new a all, since i was fis used by Sein and Weiss in 9 o exend he Macinkiewicz inepolaion heoem. The boade quesion is if he he same echnique can be exended o Loenz-Bochne spaces and even Loenz-maingale spaces. If answeed posiively, he echnique poposed in ou pape will conas he ones by Yong e al. in, which we believe ae no as saighfowad as ous. I has been shown in he lieaue ha opeaos such as he ceneed Hady opeao, he Hilbe opeao unde Δ 2 condiion ae all bounded on Loenz spaces. Usually he poofs of hese facs ae no ivial, so by fis finding an aomic decomposiion on Loenz spaces L p, q, <p, q<, i would be easie o ge anohe poof of he boundedness of hese opeaos, wihou having o eso o he Benne and Shapley inequaliy in. I is impoan o noe ha, aomic decomposiion on geneal Banach spaces has been found in 2, unde he same line of eseach as ous. Because chaaceisic funcions ae easy o manipulae, an even boade quesion would be o find he class of Banach spaces whose aomic decomposiion can be expessed in ems of chaaceisic funcions only. Refeences G. G. Loenz, Some new funcional spaces, Annals of Mahemaics, vol. 5, pp. 37 55, 95. 2 G. G. Loenz, On he heoy of spaces, Pacific Jounal of Mahemaics, vol., pp. 4 429, 95. 3 S. C. Aoa, G. Da, and S. Vema, Composiion opeaos on Loenz spaces, Bullein of he Ausalian Mahemaical Sociey, vol. 76, no. 2, pp. 25 24, 27. 4 S. C. Aoa, G. Da, and S. Vema, Muliplicaion and composiion opeaos on Olicz-Loenz spaces, Inenaional Jounal of Mahemaical Analysis, vol., no. 25 28, pp. 227 234, 27. 5 S. C. Aoa, G. Da, and S. Vema, Composiion opeaos on Loenz spaces, Bullein of he Ausalian Mahemaical Sociey, vol. 76, no. 2, pp. 25 24, 27. 6 S. C. Aoa, G. Da, and S. Vema, Muliplicaion and composiion opeaos on Loenz-Bochne spaces, Osaka Jounal of Mahemaics, vol. 45, no. 3, pp. 629 64, 28. 7 G. De Souza, A new chaaceizaion of he Loenz spaces L p, fo p> and applicaions, in Poceedings of he Real Analysis Exchange, pp. 55 58, 2. 8 L. Gafakos, Classical Fouie analysis, vol. 249 of Gaduae Texs in Mahemaics, Spinge, New Yok, NY, USA, 2nd ediion, 28. 9 E. M. Sein and G. Weiss, An exension of a heoem of Macinkiewicz and some of is applicaions, vol. 8, pp. 263 284, 959. J. Yong, P. Lihua, and L. Peide, Aomic decomposiions of Loenz maingale spaces and applicaions, Jounal of Funcion Spaces and Applicaions, vol. 7, no. 2, pp. 53 66, 29. S. Benne and R. Shapley, Inepolaion of Opeaos, vol. 29 of Pue and Applied Mahemaics, Academic Pess, Boson, Mass, USA, 988. 2 H. G. Feichinge and G. Zimmemann, An exoic minimal Banach space of funcions, Mahemaische Nachichen, vol. 239/24, pp. 42 6, 22.