Inclusion Properties of Orlicz Spaces and Weak Orlicz Spaces Generated by Concave Functions

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1 IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Inclusion Properties of Orlicz Spaces and Weak Orlicz Spaces Generated y Concave Functions To cite this article: M Taqiyuddin and A A Masta 208 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. This content was downloaded from IP address on 27/09/208 at 8:5

2 The 2nd Annual Applied Science and Engineering Conference (AASEC 207) IOP Pulishing IOP Conf. Series: Materials Science and Engineering (208) 0203 doi:0.088/ x/288//0203 Inclusion Properties of Orlicz Spaces and Weak Orlicz Spaces Generated y Concave Functions M Taqiyuddin and A A Masta Department of Mathematics Education, Universitas Pendidikan Indonesia, Jl. Dr. Setiaudi no. 229, Bandung 4054, Indonesia *taqi94@hotmail.com Astract. In this paper, we introduce Orlicz spaces generated y concave functions and estalish some inclusion properties etween them. In addition, we also otain similar result for weak Orlicz spaces. One of the keys to prove our result is y estimating characteristic function of open alls in R n.. Introduction and preliminaries There is an ongoing discussion on Orlicz spaces over past decades. This space was firstly introduced y Z W Birnaum and W Orlicz in 93 [] and the discussion is still going on. As one of them, in 989, Maligandra [2] studied some inclusion properties on Orlicz spaces. In 206, Masta et al. [3] also reprove some inclusion properties of Orlicz spaces and weak Orlicz spaces. Furthermore, several oservations of the inclusion properties of these spaces have een done y other researchers. Osançliol [4] has proved sufficient and necessary conditions for inclusion relation etween two weighted Orlicz spaces. Related results aout inclusion properties of Orlicz spaces can e found in [5-6]. These results have attracted us to oserve further within the inclusion properties of these spaces and their weak type. It is a convenient way to associate Orlicz spaces with convex functions (see for example [3,4,7]). However, recently, some researchers associated Orlicz spaces with concave function. In particular, Zhou et al. [8] discussed the weak Martingale Orlicz-Karamata-Hardy spaces associated with concave functions, and Zhang and Zhang [9] studied the weak Orlicz spaces associated with concave functions. Motivated y this trend, we have decided to study the inclusion properties of Orlicz spaces and weak Orlicz spaces generated y concave functions. The rest of paper is organized as follows. In this section, we will recall and give some definitions and some lemmas which will e used in our proofs. More specifically, we will give the definitions of Orlicz spaces and weak Orlicz spaces generated y concave functions in this section. Then, we will present our results in section 2. The inclusion properties will e discussed there. Throughout this paper, the letter C will e used for constants that may change from line to line, while constants with suscripts, such as C do not change in different lines. Let G e the set of all funtionsφ: [0, ) [0, ) e an increasing, ijective, continuous, and concave function satisfying Φ(0) = 0, lim t + 0 Φ(t) = 0, and lim t Φ(t) =. We denote Φ Φ 2 for Φ, Φ 2 G if there is a constant C > 0 such that Φ (t) Φ 2 (Ct) for all t 0.For Φ G, the Orlicz space generated y concave functions L Φ (R n ) is defined as follows Content from this work may e used under the terms of the Creative Commons Attriution 3.0 licence. Any further distriution of this work must maintain attriution to the author(s) and the title of the work, journal citation and DOI. Pulished under licence y IOP Pulishing Ltd

3 The 2nd Annual Applied Science and Engineering Conference (AASEC 207) IOP Pulishing IOP Conf. Series: Materials Science and Engineering (208) 0203 doi:0.088/ x/288//0203 for some > 0, and its quasi-norm is L Φ (R n ) = {f R n : Φ ( f(x) )dx < }, R n f LΦ (R n ) inf { > 0: Φ ( f(x) ) dx }. R n By using similar argument on the Orlicz spaces with convex function (see [,2,4,7,0]) we can prove that the Orlicz space generated y concave functions L Φ (R n ) is a quasi-banach space with respect to the quasi-norm f LΦ Note that, if Φ(t): = t p for some 0 < p <, then L Φ (R n ) = L p On the other hand, for Φ G, the weak Orlicz spaces generated y concave functions wl Φ (R n ) is defined as follows wl Φ (R n ) = {f R n : f wlφ < }, Where f wlφ (R n ) inf { > 0: sup Φ(t) {x R n : f(x) > t} }. Weak Orlicz space generated y concave functions wl Φ (R n ) is also a quasi-banach space with respect to the quasi-norm f LΦ (R n ) (see [9]). For Φ(t): = t p (0 < p < ), wl Φ (R n ) = wl p (R n ) is the weak Leesgue space. Next, we recall some lemmas which will e used in our proofs. Lemma.0. Let G, a R n, and r > 0 aritrary. Then χ B(a,r) LΦ (R n ) = where B(a, r) denotes the volume of open all B(a, r). Φ ( ), Let a R n and r > 0. Oserve that χ B(a,r) LΦ (R n ) = inf { > 0: Φ ( ) A = { > 0: Φ ( ) }. Meanwhile, Φ ( ) = inf {r 0: Φ(r) > } = inf A, where } = inf E, where E = {r 0: Φ(r) > }. Choosing = Φ ( ), we have B(a, r) Φ ( ). So we have χ B(a,r) or LΦ (R n ) Φ ( ) < Φ ( ) < r χ B(a,r) LΦ (R n ) Φ ( ). Now, suppose that χ B(a,r) < LΦ (R n ) Φ ( ) χ B(a,r) LΦ (R n ) This contradicts the fact that r. By using the definition of Φ ( ), we can find r E such that. Since r E we otain r A, so we have r χ B(a,r) LΦ (R n ). χ B(a,r) LΦ (R n ). Hence we must have χ B(a,r) = LΦ (R n ) Φ ( ). Furthermore, we have similar result on weak Orlicz space generated y concave function presented in the following Lemma. Lemma..[9]Let G, a R n, and r > 0 is aritrary. Then we have χ B(a,r) = wlφ (R n ) where B(a, r) denotes the volume of open all B(a, r). Φ ( ) 2

4 The 2nd Annual Applied Science and Engineering Conference (AASEC 207) IOP Pulishing IOP Conf. Series: Materials Science and Engineering (208) 0203 doi:0.088/ x/288//0203 Lemma.2. [9] Let G, C <, and 0 t <. Then Φ(Ct) CΦ(t). Lemma.3. Suppose that Φ G and Φ is defined as Φ (t) = inf{ r 0: Φ (r) > t}. Then, the following statements are true: () Φ(Φ (s)) s Φ (Φ(s)), for 0 s <. (2) If, for some constants C, C 2 > 0, we have Φ 2 (s) C Φ (C 2 s), then Φ ( t C ) C 2 Φ 2 (t) fort = Φ 2 (s). We leave the proof of Lemma.3 () to the reader. Now, take aritrary s > 0, and set t = Φ 2 (s). Oserve that y using Lemma.3. (), we have Φ ( t ) = Φ C ( Φ 2 (s) ) Φ (Φ (C 2 s)) C 2 s C By t = Φ 2 (s) and the definition of Φ 2 (s) = inf{ r 0: Φ 2 (r) > s} we gets Φ 2 (t). Hence we can conclude that Φ ( t C ) C 2 Φ 2 (t). 2. Main results In this section, we present two main results which are stated as Theorem 2.. and Theorem 2.2. The following theorem contains a necessary and sufficient condition for the inclusion properties of Orlicz spaces generated y concave function. Theorem 2.. Let Φ, Φ 2 G. Then the following statements are equivalent: () Φ Φ 2. (2) L Φ2 (R n ) L Φ (3) There exist a constant C > 0 such that f LΦ (Rn ) C f LΦ 2 (Rn ), for every f L Φ2 Suppose that () holds. Let f L Φ2 (R n ), then there exist a > 0 such that Hence, we have Φ 2 R n (a f(x) )dx <. Φ ( a R n C f(x) ) dx Φ 2 ( a R n C C f(x) ) dx = Φ 2 (a f(x) )dx <. R n This proves L Φ2 (R n ) L Φ Next, since (L Φ2 (R n ), L Φ (R n )) is a quasi-banach pair, we can prove (2) and (3) are equivalent y similar argument in [3]. Now assume that (3) holds. By Lemma.0., we have Φ ( ) = χ B(a,r) C χ LΦ (Rn ) B(a,r) = LΦ 2 (Rn ) Φ 2 ( C ). In the other words, we have CΦ ( ) Φ 2 ( ) for aritrarya Rn and > 0. By Lemma.3. we have Φ ( t 0 C ) Φ 2(t 0 ) for t 0 = Φ 2 ( ). Since a Rn and > 0 are aritrary, we can conclude that Φ (t) Φ 2 (Ct) fort > 0. 3

5 The 2nd Annual Applied Science and Engineering Conference (AASEC 207) IOP Pulishing IOP Conf. Series: Materials Science and Engineering (208) 0203 doi:0.088/ x/288//0203 We also have otained a necessary and sufficient condition for the inclusion properties of weak Orlicz spaces generated y concave function presented in the following theorem. Theorem 2.2.Let Φ, Φ 2 G. Then the following statements are equivalent: () Φ Φ 2. (2) wl Φ2 (R n ) wl Φ (3) There exist a constant C > 0 such that f wlφ (Rn ) C f wlφ 2 (Rn ), for every f wl Φ2 Suppose that () holds. Let f wl Φ2 (R n ), A Φ = { > 0: sup Φ (t) {x R n : f(x) > t} } and for s = Ct. A Φ2 = { > 0: sup Φ 2 (Ct) {x R n : f(x) > t} } = { > 0: sup Φ 2 (s) {x R n : C f(x) > s} }, Take an aritrary t > 0 and A Φ2, oserve that Φ (t) {x R n : f(x) > t} Φ 2 (Ct) {x R n : f(x) = Φ 2 (s) {x R n : C f(x) > s}. > t} Hence, we have f wlφ (Rn ) = inf A Φ inf A Φ2 = C f wlφ 2 (Rn ), which proves that wl Φ2 (R n ) wl Φ Next, y using similar argument in the Theorem 2.., we can get the equivalence of (2) and (3). Assume that (3) holds. By using Lemma.., we have Φ ( ) = χ B(a,r) C χ wlφ (Rn ) B(a,r) = C wlφ 2 (Rn ) Φ 2 ( ). In the other words, we have CΦ ( ) Φ 2 ( ) for aritrary a Rn and > 0. By Lemma.3., we have Φ ( t 0 C ) Φ 2(t 0 ) for t 0 = Φ 2 ( ). Since a Rn and > 0 are aritrary, we can conclude that Φ (t) Φ 2 (Ct) fort > Concluding remarks We have estalished the sufficient and necessary conditions for inclusion relations among Orlicz spaces and weakorliczspaces generated y concave function. Comining the results on Theorem 2. and Theorem 2.2, we can state that the inclusion relation etween Orlicz spaces generated y concave functions is equivalent to that etween their weak type. Furthermore, it follows from Theorem 2. and Theorem 2.2 that there cannot e an inclusion relation etween L p (R n ) and L p2 (R n )nor wl p (R n ) and wl p2 (R n ) for distinct values of p and p 2. 4

6 The 2nd Annual Applied Science and Engineering Conference (AASEC 207) IOP Pulishing IOP Conf. Series: Materials Science and Engineering (208) 0203 doi:0.088/ x/288//0203 References [] Orlicz W 992 Linear Functional Analysis (Series in Real Analysis Volume 4) World Scientific: Singapore [2] Maligranda L 989 Orlicz Spaces and Interpolation Departamento de Matema tica: Universidade Estadual de Campina [3] Masta A A, Gunawan H and Setya-Budhi W 206 An inclusion property of Orlicz dan weak Orlicz spaces J. Math. Fund. Sci [4] Osançliol A 204 Inclusion etween weighted Orlicz spaces J. Inequal. Appl [5] Masta A A, Gunawan H and Setya-Budhi W 207 An inclusion property of Orlicz-Morrey spaces to appear in J. Phys.: Conf. Ser [6] Masta A A, Gunawan H and Setya-Budhi W 207 An inclusion property of Orlicz-Morrey spaces Research Report ( arxiv: v) [7] Krasnosel skii M A and Rutickii Y A B 96 Convex Functions and Orlicz Spaces, P. Noordhoff LTD.: Groningen [8] Zhou D, Wu L and Jiao Y 207 Martingale weak Orlicz-Karamata-Hardy spaces associated with concave functions J. Math. Anal. Appl [9] Zhang X and Zhang C 20 Weak Orlicz spaces generated y concave functions International Conference on Information Science and Technology (ICIST) [0] Luxemurg W A J 955 Banach Function Spaces Thesis : Technische Hogeschool te Delft 5