Inclusion Properties of Orlicz and Weak Orlicz Spaces

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1 J. Math. Fud. Sci., Vol. 48, No. 3, 06, Iclusio Properties of Orlicz ad Weak Orlicz Spaces Al Azhary Masta,, Hedra Guawa & Woo Setya Budhi Aalysis ad Geometry Group, Faculty of Mathematics ad Natural Scieces, Istitut Tekologi Badug, Jl. Gaesha 0, Badug 403, Idoesia Departmet of Mathematics Educatio, Uiversitas Pedidika Idoesia Jl. Dr. Setiabudi 9, Badug 4054, Idoesia Abstract. I this paper we discuss the structure of Orlicz spaces ad weak Orlicz spaces o. We obtai some ecessary ad sufficiet coditios for the iclusio property of these spaces. Oe of the keys is to compute the orm of the characteristic fuctios of the balls i. Keywords: Iclusio property; Lebesgue spaces; Orlicz spaces; Youg fuctio; weak Orlicz spaces. Itroductio Orlicz spaces were itroduced by Birbaum ad Orlicz i 93 []. Let Φ: 0, 0, be a Youg fuctio, that is, Φ is covex, left-cotiuous, Φ0 0, ad lim Φ. Give a measure space,, we defie the Orlicz space to be the set of measurable fuctios : such that X ( a f( x) ) dx for some 0. The space is a Baach space equipped with the orm f( x ) f L ( X) : if b 0: dx X b (see [,3]). Note that, if Φ: for some ad :, the, the Lebesgue space of -th itegrable fuctios o [4]. Thus, Orlicz spaces ca be viewed as a geeralizatio of Lebesgue spaces. Several authors have made importat observatios about Orlicz spaces (see [- 8], etc.). Here we are iterested i the iclusio property of these spaces. I [8], Wellad proved the followig iclusio property: Let be of fiite measure, ad Φ, Ψ be two Youg fuctios. If there is 0 such that Φ Ψ for every 0, the. Accordigly, if is of fiite measure, Φ, Ψ Received August 4 th, 05, Revised Jue 7 th, 06, Accepted for publicatio August 8 th, 06. opyright 06 Published by ITB Joural Publisher, ISSN: , DOI: 0.564/j.math.fud.sci

2 94 Al Azhary Masta, et al. are two Youg fuctios, ad there is 0 such that Ψ Φ Ψ for every 0, the we have. A refiemet of this result may be foud i [5], which states that if oly if there are 0 ad 0 such that Φ Ψ for every. Related results ca be foud i [6,9,0]. Motivated by these results, the purpose of this work is to get the iclusio property of Orlicz spaces ad exted the results to weak Orlicz spaces (see [-4]). (Here : has a ifiite measure.) The rest of this paper is orgaized as follows. The mai results are preseted i Sectios ad 3. I Sectio, we state the iclusio property of Orlicz spaces as Theorem.5, which cotais a ecessary ad sufficiet coditio for the iclusio property to hold. A aalogous result for the weak Orlicz spaces is stated as Theorem 3.3. To prove the results, we pay attetio to the characteristic fuctios of balls i ad use the iverse fuctio of Φ, which is give by Φ : if 0: Φ. The reader will fid the followig lemma useful. Lemma. Suppose that Φ is a Youg fuctio ad Φ if 0: Φ. We have () Φ 0 0. () Φ Φ for. (3) ΦΦ Φ Φ for 0. (4) Let 0. The Φ Φ if oly if Φ Φ, for every 0. (5) Let 0. The Φ Φ if oly if Φ Φ, for every 0. Proof. The proof of parts ()-(3) ca be foud i [5]. Now we will prove (4) ad (5). (4) Let 0. We will prove Φ Φ if oly if Φ Φ, for every 0. Take a arbitrary 0 such that Φ Φ for every 0. Let Φ if where if 0:Φ, ad write 0: Φ. Observe that if B if{ r 0 : ( r) t}

3 Iclusio Properties of Orlicz ad weak Orlicz Spaces 95 x if{ 0 : ( x) t} if{ x 0 : ( x) t} () t for. For a arbitrary, we have Φ Φ, ad thus it follows that. Hece we coclude that. Accordigly, we obtai Φ if if Φ. Now, suppose that Φ Φ, for 0 ad every 0. Observe that, by Lemma. (3) we have t ( ( t)) ( ) ( ) ( ( t)) ( ) ( ( ( t))) (). t As a result, we have Φ Φ or Φ Φ. (5) Let 0. We will prove Φ Φ if oly if Φ Φ, for every 0. Take a arbitrary 0 such that Φ Φ for every 0. Let Φ if where if 0:Φ, ad write 0: Φ. Observe that, for we have Φ Φ, so that. Hece we coclude that. Accordigly, we obtai Φ if if Φ. Now, suppose that Φ Φ, for 0 ad every 0. Observe that, by Lemma. (3) we have ( t) ( ( ( t)))

4 96 Al Azhary Masta, et al. ( ( ( t))) (). t As a result, we have Φ Φ. More lemmas (ad their proofs) will be preseted i the ext sectios. Iclusio Property of Orlicz Spaces Let us first recall several Lemmas below. Lemma. If Φ is a Youg fuctio, the Φ Φ for 0 ad 0. Lemma. [] Let Φ be a Youg fuctio ad. If 0, the f( x) f L ( ) dx. Furthermore, if oly if Φ. orollary.3 Let Φ, Ψ be Youg fuctios. If there exists 0 such that Φ Ψ for 0, the with for every. Proof. Suppose that. Observe that f x f x f x Φ dx Ψ dx Ψ dx f f. f L L L By the defiitio of, we have. This proves that, as desired. Remark. From orollary.3, we ote that if Φ Ψ, the with for every. As we shall see below, the coverse of this statemet also holds. We eed the followig lemma. Lemma.4 [7] Let Φ be a Youg fuctio,, ad 0. The, where, deotes the volume of,.,,

5 Iclusio Properties of Orlicz ad weak Orlicz Spaces 97 Theorem.5 Let Φ, Ψ be Youg fuctios. The the followig statemets are equivalet: () Φ Ψ for every 0. (). (3) For every, we have.. Proof. We have see that () implies (). Next, sice, is a Baach pair, it follows from [6, Lemma 3.3] that () ad (3) are equivalet. It thus remais to show that (3) implies (). Now assume that (3) holds. By Lemma.4, we have,,, Bar L Bar Φ LΨ Bar, Bar, or Φ, Ψ, for every,, 0. By Lemma. (4), we obtai Φ Ψ. Sice 0 is arbitrary, we coclude that,, Φ Ψ for every 0.. A Special ase Oe may ask whether from iclusio relatios betwee Orlicz spaces we may deduce some kow fact of those i Lebesgue spaces. The aswer is affirmative; we eed the followig lemma for this purpose. Lemma.6 Let Φ,Φ, ad Φ be Youg fuctios such that Φ Φ Φ for every 0. If ad, the with fg f g. L ( ) ( ) ( L L ) 3 Proof. Let, 0. Without loss of geerality, suppose that Φ Φ. By Lemma. (3), we obtai stφ Φ s Φ Φ t Φ Φ t Φ Φ t Φ Φ t. 3 Hece Φ Φ Φ Φ Φ Φ Φ. From Lemma. we have

6 98 Al Azhary Masta, et al. f xgx f xgx Φ 3 dx Φ 3 dx f g. f g L L L L O the other had, by Lemma. we obtai f xgx f x gx Φ 3 dx Φ dx Φ dx f g f g L L L L wheever ad. By usig the defiitio of, we have, as desired. orollary.7 Let :, for some ad 0. If Φ,Φ are two Youg fuctios ad there is a Youg fuctio Φ such that () t () t () t for every 0, the with f f ( ) Bar (, ) for. L ( X) L ( ) X 0 Proof. Let. By Lemma.4 ad choosig :,, we obtai f Φ 0 0 Φ f,, Φ f Bar L X Bar L X L X LΦ X. Ba, r0 This shows that. orollary.8 Let :, for some ad 0. If, the. Proof. Let Φ :,Φ :, ad Φ: (0). Sice, we have. Thus, Φ,Φ, ad Φ are three Youg fuctios. Observe that, usig the defiitio of Φ ad Lemma., we have

7 Iclusio Properties of Orlicz ad weak Orlicz Spaces 99 p p p p pp t t t t t t (), (),ad (). Moreover, Φ Φ orollary.7 that. Φ, ad so it follows from,, ad therefore Remark. Of course we ca prove the iclusio of property of Lebesgue spaces o a fiite measure space directly via Hölder s iequality. What we showed here is that we ca obtai the result through the les of Orlicz spaces. 3 Iclusio Property of Weak Orlicz Spaces First, we recall the defiitio of weak Orlicz spaces [4]. Let Φ be a Youg fuctio. We defie the weak Orlicz spaces to be the set of measurable fuctios : such that, where :if 0:sup Φ :. Remark. Note that defies a quasi-orm i, ad that (, ) forms a quasi-baach space (see [,]). The relatio betwee weak Orlicz spaces ad (strog) Orlicz spaces is clear, as preseted i the followig theorem. Theorem 3. [,7] Let Φ be a Youg fuctio. The with for every. Proof. The proof of this theorem ca be foud i [,7]. We rewrite the proof here for coveiece. Give, let, : 0: Φ x f AΦ, w : { b0:sup Φ t { x : t} } ad t0 b. The if, ad if,. Observe that, for arbitrary, ad 0, we have f( x) f( x) () t { x : t} f ( x) dx { x : t} b b b

8 00 Al Azhary Masta, et al. f( x) dx. b Sice 0 is arbitrary, we have sup Φ :, ad,,. Hece, with. Remark. As the strog ad weak Orlicz spaces cotai the strog ad weak Lebesgue spaces respectively, the iclusio i the above theorem is proper. See [8] for a couterexample. I additio to Lemma.4, we have the followig lemma for the characteristic fuctios of balls i weak Orlicz spaces. Lemma 3. [3] Let Φ be a Youg fuctio,, ad 0 be arbitrary. The we have,., Now we come to the iclusio property of weak Orlicz spaces. Theorem 3.3 Let Φ, Ψ be Youg fuctios. The the followig statemets are equivalet: () Φ Ψ for every 0. (). (3) For every, we have. Proof. Assume that () holds, ad let. Put ad, 0:sup Φ :, 0:sup Ψ : 0:sup Ψ :, for s t. The f if A wl Φ, w ad f Φ for arbitrary, ad 0, we have Φ : if A Ψ wl Ψ : Ψ, w.. Observe that,

9 Iclusio Properties of Orlicz ad weak Orlicz Spaces 0 Thus, sup Φ :. Hece it follows that,, ad so we coclude that,,. Accordigly, we obtai if, if,, which also proves that. As metioed i [0, Appedix G], we are aware that [6, Lemma 3.3] still holds for quasi-baach spaces, ad so () ad (3) are equivalet. Assume ow that (3) holds. By Lemma 3., we have or, 0, 0 Bar wlφ Bar wlψ Ba, r0 Ba, r0,,, Φ, Ψ for arbitrary ad 0. By Lemma., we have Φ Ψ.,, Sice ad 0 are arbitrary, we coclude that Φ Ψ for every 0. 4 ocludig Remarks We have proved the iclusio property of (strog) Orlicz spaces ad of weak Orlicz spaces. Both proofs use the orm of the characteristic fuctios of the balls i. As our fial coclusio, we have the followig corollary which states that the iclusio property of (strog) Orlicz spaces are equivalet to that of weak Orlicz spaces, ad both ca be observed just by comparig the associated Youg fuctios. To be precise, if Φ, Ψ are two Youg fuctios, the the followig statemets are equivalet: () Φ Ψ for every 0. (). (3) For every, we have. (4). (5) For every, we have.

10 0 Al Azhary Masta, et al. Ackowledgemets The first ad secod authors are supported by ITB Research ad Iovatio Program 05. Refereces [] Orlicz, W., Liear Fuctioal Aalysis (Series i Real Aalysis Volume 4), World Scietific, Sigapore, 99. [] Léoard,., Orlicz Spaces, Preprit, (August 7 th, 05). [3] Luxemburg, W.A.J., Baach Fuctio Spaces, Thesis, Techische Hogeschool te Delft, 955. [4] Masta, A.A., O Uiform Orlicz Spaces, Thesis, Badug Istitute of Techology, 03. [5] Kufer, A., Joh, O. & Fučik, S., Fuctio Spaces, Noordhoff Iteratioal Publishig, zechoslovakia, 977. [6] Osaçliol, A., Iclusio betwee Weighted Orlicz Spaces, J. Iequal. Appl., 04(390), pp. -8, 04. [7] Rao, M.M. & Re, Z.D., Theory of Orlicz Spaces, Moographs ad Textbooks i Pure ad Applied Mathematics, 46, Marcel Dekker, Ic., New York, 99. [8] Wellad, R., Iclusio Relatios amog Orlicz Spaces, Proc. Amer. Math. Soc., 7(), pp , 966. [9] Alotaibi, A., Mursalee, M. & Sharma, S.K., Double Sequece Spaces Over N-ormed Spaces Defied by a Sequece of Orlicz Fuctios, J. Iequal. Appl., 04(), pp. -, 04. [0] Mursalee, M., Raj, K. & Sharma, S.K., Some Spaces of Differece Sequeces ad Lacuary Statitical overgece i N-ormed Space Defied by Sequece of Orlicz Fuctios, Miskolc Mathematical Notes, 6(), pp , 05. [] Bekja, T.N., he, Z., Liu, P. & Jiao, Y., Nocommutative Weak Orlicz Spaces ad Martigale Iequalities, Studia Math., 04(3), pp. 95-, 0. [] Jiao, Y., Embeddigs betwee Weak Orlicz Martigale Spaces, J. Math. Appl., 378, pp. 0-9, 0. [3] Liu N. & Ye, Y., Weak Orlicz Space ad Its overgece Theorems, Acta Math. Sci. Ser., B30(5), pp , 00. [4] Nakai, E., Orlicz-Morrey Spaces ad Some Itegral Operators, Research Report, (August 7 th, 05). bitstream/433/6035//399-3.pdf

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