A Note on Inclusion Properties of Weighted Orlicz Spaces

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1 A Note on Inclusion Properties of Weighted Orlicz Spaces Al Azhary Masta, Ifronika, Muhammad Taqiyuddin 3,3 Department of Mathematics Education, Universitas Pendidikan Indonesia, Jl. Dr. Setiabudi 9, Bandung 4054 ariv: v [math.fa] 7 Dec 08 Analysis and Geometry Group, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Jl. Ganesha 0, Bandung 403 alazhari.masta@upi.edu, ifronika@math.itb.ac.id, 3 taqi94@hotmail.com Abstract In this paper we present sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces which complete the Osançliol result in 04. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in. Keywords: Inclusion property, Weighted Lebesgue spaces, Weighted Orlicz spaces. MSC 00: Primary 46E30; Secondary 46B5, 4B35. Introduction Orlicz spaces are generalization of Lebesgue spaces which were firstly introduced by Z. W. Birnbaum and W. Orlicz in 93 (see [5, 4]. Let us first recall the definition of Orlicz spaces. Let : [0, [0, be a Young function [that is, is convex, lim (t = 0 = (0, left-continuous t 0 and lim (t = ], the Orlicz space L is the set of measurable functions f : R such t that R (a f(x dx < for some a > 0. The space L ( is a Banach space equipped with n the norm { ( } f(x f L := inf b > 0 : dx. b Meanwhile, for is a Young function, the weak Orlicz space wl is the set of all measurable functions f : R such that f wl := inf { b > 0 : sup (tµ t>0 ( { x : f(x > t} } <. b Now, we move to the weighted Orlicz spaces and weighted weak Orlicz spaces. Let be a Young function and u is a weight on (i.e u : (0, is a measurable function, the weighted

2 Orlicz space L u ( is the set of all measurable functions f : R such that uf L. Note that, the space L u (Rn is a Banach space equipped with the norm { f L u := uf L = inf b > 0 : ( } u(xf(x dx. b Similar with weighted Orlicz spaces, for a Young function and a weight u on, the weighted weak Orlicz space wl u (Rn is the set of all measurable functions f : R such that f wl u := uf wl <. Let u, u : (0,, we denote u u if there exists a constant C > 0 such that u (x Cu (x for all x. Note that, if u u then f u L (Rn C f L u (Rn and f wl u (Rn C f u wl. (Rn The study of Lebesgue spaces and Orlicz spaces has been studied by many researchers in the last few decades (see [3, 5, 6, 7,, 5], etc.. In 989, Maligranda [6] discussed inclusion properties of Orlicz spaces. Later in 06, Masta et al. [7] obtained sufficient and necessary conditions for inclusion relation between two Orlicz spaces and between two weak Orlicz spaces by using different technique from Maligranda. Moreover, they have found that two Orlicz spaces and two weak Orlicz spaces can be compared with respect to Young functions for any measurable set, although the Lebesgue space L p are not comparable with respect to the number p. On the other hand, Osançliol [3] have proved sufficient and necessary conditions for inclusion relation between two weighted Orlicz spaces, as in the following theorem. Theorem.. [3] Let be a continuous Young function satisfying the condition [that is, there exists K > 0 such that (t K(t for all t 0], and u, u are measurable functions such that u i (x + y u i (x u i (y for every x, y, where i =,. Then the following statements are equivalent: ( u u. ( L u (Rn L u (Rn. (3 There exists a constant C > 0 such that f L u (Rn C f L u (Rn, for every f Lu (Rn. Related result for weak type of Orlicz spaces can be found in [0]. In this paper, we are interested in studying the inclusion properties of weighted Orlicz spaces. In connection with Theorem., we shall prove inclusion relation between weighted Orlicz spaces with respect to Young functions, and weights u, u. To achieve our purpose, we will use the similar methods in [, 7, 8, 9, 3] which pay attention to the characteristic functions of open balls in. Next, we recall some lemmas which will be used later in next section. Lemma.. [] Suppose that is a Young function and (s := inf{r 0 : (r > s}. We have ( (0 = 0. ( (s (s for s s. (3 ( (s s ((s for 0 s <.

3 Lemma.3. [9] Let, be Young functions. For any s > 0, if there exists C, C > 0 such that (s C (C s, then we have ( t C C (t for t = (s. In this paper, the letter C will be used for constants that may change from line to line, while constants with subscripts, such as C, C, do not change in different lines. Results First,we will investigate the inclusion properties of weighted Orlicz spaces with respect to distinct Young functions and. For getting the result, we give attention to estimate the norm of the characteristic function of open ball in as in the following lemma. Lemma.. [4, 7] Let be a Young function, a, and r > 0 be arbitrary. Then we have χ B(a,r = χ = ( where B(a, r denotes the volume of open ball u L u B(a,r L B(a,r B(a, r centered at a with radius r > 0. Now we come to the inclusion relation between L u ( and L u ( with respect to Young functions,. Given two Young functions,, we write if there exists a constant C > 0 such that (t (Ct for all t > 0. Theorem.. Let, be Young functions and u : (0, be a measurable function. Then the following statements are equivalent: (. ( L u ( L u (. (3 There exists a constant C > 0 such that f L u C f L u (, for every f L u (. Proof. Assume that ( holds. Suppose that f L u (. Observe that u(xf(x dx C u(xf(x dx = u(xf(x dx. C f L u ( C f L u ( f L u ( By definition of L u, we have f L u L u (. C f L u (. This proves that L u ( Next, since (L u (, L u ( is a Banach pair, it follows from [, Lemma 3.3] that ( and (3 are equivalent. It thus remains to show that (3 implies (. Assume now that (3 holds. By Lemma., we have χ ( = B(a,r χ C B(a,r = u L u u L u ( B(a,r Since ( B(a,r C ( B(a,r a and r > 0, by Lemma.3, we have C ( B(a,r is equivalent to C ( B(a,r ( for arbitrary B(a,r ( t0 C (t 0, 3.

4 for t 0 = (. Since a B(a,r Rn and r > 0 are arbitrary, we conclude that (t (Ct for every t > 0. Remark.3. For u(x =, Theorem. reduces to Theorem.5 in [7]. Next, we also give the sufficient and necessary conditions for inclusion relation between weighted Orlicz spaces L u ( and L u ( with respect to Young functions, and weights u, u. To get the result, we need the following lemma. Lemma.4. Let u : (0, be a measurable function such that u(x + y u(x u(y for every x, y. If is a Young function, then: ( For all f L u ( and for all x, we have T x f L u u(x f L u, where T x f(y = f(y x. ( If f L u (Rn and f 0, then there exists a constant C > 0 (depends on f such that u(x C T xf L u Cu(x. Proof. ( Let f L u (Rn and T x f(y = f(y x, then Observe that (by setting v := y x, we have ( u(ytx f(y u(x f L u This shows that T x f L u u(x f L u. ( u(vf(v f L u dy = = =.. ( u(yf(y x u(x f L u ( u(v + xf(v u(x f L u ( u(vu(xf(v u(x f L u ( u(vf(v f L u ( Let f L u ( and f 0, then there exist a constant C > 0 (depends on f such that f L u C. By Lemma.4 (, then we have for every x. T x f L u φ, Cu(x, dy 4

5 Since f(x 0 for every x,we have T x f L u > 0. Observe that ( u(xf(v sup u( v T x f L u v ( =. u(xf(v u( v T x f L u ( u(v + xf(v T x f L u ( u(yf(y x T x f L u ( u(ylx f(y T x f L u dy dy This shows that u(x f L sup v u( v T x f L u. { Choose, C := max C, desired. sup u( v v f L }. Hence, we conclude that u(x C T x f L u Cu(x, as Now, we present the sufficient and necessary conditions for the inclusion properties of weighted Orlicz spaces L u ( and L u ( with respect to Young functions, and weights u, u. Theorem.5. Let, be Young functions such that and u, u are measurable functions such that u i (x + y u i (x u i (y for every x, y, where i =,. Then the following statements are equivalent: ( u u. ( L u ( L u (. (3 There exists a constant C > 0 such that f L u ( C f L u (, for every f Lu (. Proof. Assume that ( holds. Let f be an element of L u (. Since and u u, there exists constants C, C > 0 such that (t (C t for all t > 0 and u (x C u (x for every x. Using a similar argument in the proof of Theorem. we have f L u ( C f L u ( C C f L u (. As before, we have that ( and (3 are equivalent. It thus remains to show that (3 implies (. Assume that (3 holds. By Lemma.4, we have u (x C T u xf L (R n C T u xf L (R n Cu (x, for every x. So, we obtain u u. Note that, for (x = (x for every x, Theorem.5 reduces to Theorem.. 5

6 Remark.6. It follows from Theorems. and.5 that there cannot be an inclusion relation between L u p ( and L u ( for distinct values of p and. In spite of that, for finite measure set we can obtain inclusion relation between L u p ( and L u ( presented in the next section. 3 An additional case In the following, we will give sufficient condition for Hölder s inequality in weighted Orlicz spaces which will be used to obtain inclusion relation between L u p ( and L u (. Theorem 3.. (Hölder s inequality Let be a measurable set,,, 3 be Young functions and u, u, u 3 : R be measurable functions such that (t (t 3 (t for every t > 0 and u 3 (x u (xu (x for every x. If f L u ( and f L u (, then f f L u 3 3 ( with f f u L 3 ( f u 3 L ( f u L (. Proof. Let s, t 0. Without loss of generality, suppose that (s (t. By Lemma.(3, we obtain st ((s ((t ((t ((t 3 ((t. Hence 3 (st 3 ( 3 ( (t (t (t + (s. Since is a convex function, we have 3 u 3(xf (xf (x f u L ( f u L ( On the other hand, by Lemma. we obtain 3 u (xu (xf (xf (x f u L ( f dx u L ( dx 3 3 u (xf (x f L u ( u 3(xf (xf (x f u L ( f u L ( dx u (xu (xf (xf (x f u L ( f dx. u L ( dx + f (x f L u ( dx, whenever f L u ( and f L u (. By the definition of u L (, we have f f u L 3 3 f u L ( f u L (, as desired. 3 ( Corollary 3.. Let := B(a, r 0 for some a and r 0 > 0. If, are two Young functions, u, u : R are measurable functions and there are a Young function and a weight 0 < u(x for every x such that (t (t (t for every t 0 and u (x u(xu (x for every x, then with f L u ( ( L u ( L u ( f L u ( for f Lu B(a,r 0 (. 6

7 Proof. Since 0 < u(x for every x, we have f L u ( f u L u (. Let f Lu (, by Theorem 3. and choosing g := χ B(a,r0, we obtain This shows that L u ( L u (. f L u ( = fχ B(a,r 0 L u ( = fg u L ( fg u u L ( g f u u L u L ( = ( f L u B(a,r 0 ( (. We shall now discuss the inclusion properties of weighted weak Lebesgue spaces L u p ( and L u ( with respect to distinct values of p and as well as u and u. Corollary 3.3. Let := B(a, r 0 for some a and r 0 > 0. If < p < and u, u : R are measurable functions such that u (x u (x for every x, then L u p ( L u (. Proof. Let (t := t p, (t := t, (t := t p p for every t 0. Since < p <, we have p p >. Thus,,, and are Young functions. Now, define u(x = u (x u for every (x x. Observe that, using the definition of and Lemma., we have Moreover, (t (t = t p t p p = t (t = t p, (t = t, and (t = t p p. = (t and u (x = u (x u (x u (x = u(xu (x. So it follows from Corollary 3. that f u L ( B(a, r 0 p p p f u L p (, and therefore we can conclude that L u p ( L u (. 4 Concluding Remarks We have shown the inclusion properties of weighted Orlicz spaces for distinct Young functions, and weights u, u. The inclusion properties of weighted Orlicz spaces are generalization of inclusion properties of Orlicz spaces in [7] and inclusion properties of weighted Lebesgue spaces. In the proof of our results, we used the norm of characteristic function on and estimated the norm of the translation functions in. Furthemore, from Theorem.5 and Lemma., Theorem.8 in [0], we also have the following inclusion relations L u L u ց wl u wl u 7

8 for and u u, where the arrows mean contained in or embedded into. Acknowledgement. The first author is supported by Hibah Penguatan Kompetensi UPI 08. The authors thank the referee for his/her useful remarks on the earlier version of this paper. References [] H. Gunawan, D.I. Hakim, K.M. Limanta, and A.A. Masta, Inclusion properties of generalized Morrey spaces, Math. Nachr. 90 (07, [DOI: 0.00/mana ] [] S.G Kreǐn, Yu.Ī Petunīn, and E.M. Semënov, Interpolation of Linear Operators, Translation of Mathematical Monograph vol. 54, American Mathematical Society, Providence, R.I., 98. [3] A. Kufner, O. John, and S. Fu cik, Function Spaces, Noordhoff International Publishing, Czechoslovakia, 977. [4] N. Liu and Y. Ye, Weak Orlicz space and its convergence theorems, Acta Math. Sci. Ser. B 30-5 (00, [5] W.A.J. Luxemburg, Banach Function Spaces, Thesis, Technische Hogeschool te Delft, 955. [6] L. Maligranda, Orlicz Spaces and Interpolation, Departamento de Matemática, Universidade Estadual de Campinas, 989. [7] A.A. Masta, H. Gunawan, and W. Setya-Budhi, Inclusion property of Orlicz and weak Orlicz spaces, J. Math. Fund. Sci (06, [DOI: [8] A.A. Masta, H. Gunawan, and W. Setya-Budhi, An inclusion property of Orlicz-Morrey spaces, J. Phys.: Conf. Ser., (07, 8 [DOI: [9] A.A. Masta, H. Gunawan, and W. Setya-Budhi, On Inclusion Properties of Two Versions of Orlicz-Morrey Spaces, Mediterr. J. Math., 4-6 (07, [DOI: [0] A.A. Masta, Ifronika, and M. Taqiyuddin, Inclusion properties of weighted weak Orlicz spaces, research report ( 07. [] E. Nakai, On Orlicz-Morrey spaces, research report [ accessed on August 7, 05.] [] W. Orlicz, Linear Functional Analysis (Series in Real Analysis Volume 4, World Scientific, Singapore, 99. [3] A. Osançliol, Inclusion between weighted Orlicz spaces, J. Inequal. Appl (04, 8 [DOI: [4] M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, volume 46 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 99. 8

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