Inclusion Properties of Weighted Weak Orlicz Spaces

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1 Inclusion Properties of Weighted Weak Orlicz Spaces Al Azhary Masta, Ifronika 2, Muhammad Taqiyuddin 3,2 Department of Mathematics, Institut Teknologi Bandung Jl. Ganesha no. 0, Bandung arxiv: v [math.fa] 2 Oct 207,3 Permanent Address: Department of Mathematics Education, Universitas Pendidikan Indonesia, Jl. Dr. Setiabudi 229, Bandung alazhari.masta@upi.edu, 2 ifronika@gmail.com, 3 taqi94@hotmail.com Abstract In this paper we discuss the structure of weighted weak Lebesgue spaces and weighted weak Orlicz spaces on R n. First, we present sufficient and necessary conditions for inclusion relation between weighted weak Lebesgue spaces. Next, we also obtain similar results on weighted weak Orlicz spaces. One of the keys to prove our results is to use the norm of the characteristic functions of the balls in R n. Keywords: Inclusion property, Weighted weak Lebesgue spaces, Weighted weak Orlicz spaces. MSC 200: Primary 46E30; Secondary 46B25, 42B35. Introduction Orlicz spaces as generalization of Lebesgue spaces were introduced by Z. W. Birnbaum and W. Orlicz in 93 (see [8,, 2, 22]. Many authors gave more attention in the study of Orlicz spaces (see [5, 7, 8, 9, 0,, 2, 9, 22, 23, 24]. In particular, Maligranda [2] discussed about inclusion properties of Orlicz spaces. In 206, Masta et al. [3] obtained inclusion relations between Orlicz spaces and between weak Orlicz spaces using different technique. On the other hand, Osançliol [2] have proved sufficient and necessary conditions for inclusion relation between two weighted Lebesgue spaces. Furthermore, he also obtained the sufficient and necessary conditions for inclusion relation between two weighted (strong Orlicz spaces. Motivated by these results, we are interested in studying the inclusion properties of weighted weak Lebesgue spaces and weighted weak Orlicz spaces. In particular, we shall give sufficient and necessary conditions for inclusion relations between weighted Lebesgue spaces and between weighted weak Orlicz spaces. For that purpose, we introduce some definitions. Let U be the set of all functions u : R n (0, such that u(x + y u(x u(y for all x, y R n. Let u, be the elements of U, we denote u if there exists a constant C > 0 such that u (x C (x for all x R n.

2 Now, let us give the definition of weighted weak Lebesgue spaces and weighted weak Orlicz spaces. For p < and u : R n (0,, the weighted weak Lebesgue space wl u p (Rn is the set of all measurable function f : R n R such that f wl u p (R n := sup t x R n : u(xf(x > t} p <. Let Φ : [0, [0, be a Young function [that is, s convex, lim = 0 = Φ(0, leftcontinuous and lim = ] and u : R n (0,, the weighted weak Orlicz space wl u t 0 t Φ(R n is the set of all measurable function f : R n R such that f wl u Φ (R n := inf b > 0 : sup x R n : u(xf(x b > t} Note that if there exists C > 0 such that u (x C (x for every x R n, then f wl u Φ (Rn C f wl Φ (Rn. } <. The space wl u Φ(R n is quasi-banach spaces equipped with the quasi-norm f wl u Φ (R n. If u(x = for every x R n, then wl u Φ (Rn is weak Orlicz space wl Φ (R n. Meanwhile, for = t p ( p <, we have wl u Φ (Rn = wl u p (Rn. To achieve our purpose, we will use the similar methods in [4, 3, 4, 2] which pay attention to the characteristic functions of open balls in R n. Next, we recall some lemmas which will be used later in next section. Lemma.. [7] Suppose that s a Young function and Φ (s := infr 0 : Φ(r > s}. We have ( Φ (0 = 0. (2 Φ (s Φ (s 2 for s s 2. (3 Φ(Φ (s s Φ (Φ(s for 0 s <. Lemma.2. [5] Let Φ, be Young functions. For any s > 0, if there exist constants C, C 2 > 0 such that Φ 2 (s C Φ (C 2 s, then we have Φ ( t C C 2 (t for t = Φ 2 (s. Lemma.3. Let u U and p <. If L x f is translation function, i.e L x f(y := f(y x for every x R n, then: ( For all f wl u p (Rn and for all x R n, we have L x f wl u p (Rn and L x f wl u p (R n u(x f wl u p (R n. (2 If f wl u p(r n and f 0, then there exist a constant C > 0 (depends on f such that u(x C L xf wl u p (R n Cu(x. 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 2, do not change in different lines. 2

3 2 Results First, we present sufficient and necessary conditions for inclusion properties of weighted weak Lebesgue spaces in the following theorem. Teorema 2.. Let p < and u, : R n (0,. Then the following statements are equivalent: ( u. (2 wl p (R n wl u p (R n. (3 There exist a constant C > 0 such that for every f wl p (R n. Proof. f wl u p (R n C f wl p (R n, Assume that ( holds, then there exist a constant C > 0 such that u (x C (x for every x R n. Let f be an element of wl p (Rn. Now, take an arbitrary t > 0, observe that (by setting t = t C t x R n : u (xf(x > t} p t x R n : C (xf(x > t} p = t x R n : (xf(x > t } p C = Ct x R n : (xf(x > t } p C f wl p (R n. Since t > 0 is arbitrary, we have f wl u p (R n C f wl p (R n. Next, since (wl u p (R n, wl p (R n is a quasi-banach pair, as mentioned in [0, Appendix G], we are aware that [6, Lemma 3.3] still holds for quasi-banach spaces, so we have (2 and (3 are equivalent. Then we remain to show that (3 implies (. Assume that (3 holds. By Lemma.3, we have So, we can conclude that u. u (x C L u xf wl p (R n C L u xf wl 2 p (R n (x. Next, we will investigate the inclusion properties of weighted weak Orlicz spaces. For getting a result, we give attention to estimate the norm of the characteristic function of open ball in the following lemma. Lemma 2.2. [0, 3] Let Φ be a Young function, a R n, and r > 0 be arbitrary. Then we have χ B(a,r = χ u wl u B(a,r Φ (R n wlφ (R n = ( where B(a, r denotes the volume of open ball Φ B(a,r B(a, r centered at a R n with radius r > 0. 3

4 Now we come to the inclusion property of weighted weak Orlicz spaces wl u Φ (R n and wl u (R n with respect to Young functions Φ,. Given two Young functions Φ,, we write Φ if there exists a constant C > 0 such that Φ (t (Ct for all t > 0. Teorema 2.3. Let Φ, be Young functions and u : R n (0,. Then the following statements are equivalent: ( Φ. (2 wl u (R n wl u Φ (R n. (3 There exist a constant C > 0 such that f wl u Φ (R n C f wl u Φ2 (R n, for every f wl u (R n. Proof. Assume that ( holds. Let f wl u (R n, A Φ,u = b > 0 : sup Φ (t x R n : u(xf(x b and > t} } A Φ2,u = b > 0 : sup (Ct x R n : u(xf(x > t} } b = b > 0 : sup (s x R n : C u(xf(x } > s}. s>0 b Observe that, for arbitrary b A Φ2,u and t > 0, we have Φ (t x R n : u(xf(x b > t} (Ct x R n : u(xf(x > t} b sup (s x R n : C u(xf(x > s} s>0 b. Since t > 0 is arbitrary, we obtain f wl u Φ (R n := inf A Φ,u inf A Φ2,u = Cf wl u Φ2 (R n := C f wl u Φ2 (R n, which also proves that wl u (R n wl u Φ (R n. Using a similar argument in the proof of Theorem 2., we have (2 and (3 are equivalent. Assume now that (3 holds. By Lemma 2.2, we have χ ( = B(a,r χ Φ C B(a,r = u wl u Φ (R n u wl u Φ2 (R n B(a,r Φ 2 C ( B(a,r or CΦ ( B(a,r Φ 2 (, for arbitrary a B(a,r Rn and r > 0. By Lemma.2, we have Φ ( C t 0 (t 0, for t 0 = Φ 2 (. Since a B(a,r Rn and r > 0 are arbitrary, we conclude that Φ (t (Ct for every t > 0. 4

5 Remark 2.4. For u(x =, Theorem 2.3 reduces to Theorem 3.3 in [3]. Next, we also give the sufficient and necessary conditions for inclusion relation between weighted weak Orlicz spaces wl u Φ (R n and wl (R n with respect to Young functions Φ, and weights u,. To get the result, we need some lemmas in the following. Lemma 2.5. Let Φ be a Young function. If f wl u Φ (Rn, then for arbitrary ǫ > 0 we have Proof. sup x R n : } u(xf(x f wl u Φ (R n + ǫ > t. Let f be an element of wl u Φ (Rn. Take an arbitrary ǫ > 0, then there exists b ǫ > 0 such that b ǫ f wl u Φ (R n + ǫ and sup x R n : u(xf(x b ǫ } > t. Because u(xf(x b ǫ u(xf(x, we have f wl u (R n Φ +ǫ x R n : } u(xf(x f wl u Φ (R n + ǫ > t x R n : u(xf(x } > t b ǫ for every t > 0. Since t > 0 is arbitrary, we can conclude that for every ǫ > 0. sup x R n : } u(xf(x f wl u Φ (R n + ǫ > t Lemma 2.6. Let u be element of U. If s a continuous Young function satisfying the 2 condition [that is, there exists K > 0 such that Φ(2t K for all t 0], then: ( For all f wl u Φ (Rn and for all x R n, we have L x f wl u Φ (Rn and L x f wl u Φ (R n u(x f wl u Φ (R n. (2 If f wl u Φ (Rn and f 0, then there exist a constant C > 0 (depends on f such that u(x C L xf wl u Φ (R n Cu(x. Proof. ( Let f wl u Φ (Rn and L x f(y = f(y x. For arbitrary t, ǫ > 0, observe that; 5

6 y R n : u(yl x f(y } u(x( f wl u Φ (R n + ǫ > t = y R n : = v R n : v R n : = v R n : u(yf(y x } u(x( f wl u Φ (R n + ǫ > t u(v + xf(v } u(x( f wl u Φ (R n + ǫ > t u(vu(xf(v } u(x( f wl u Φ (R n + ǫ > t u(vf(v } ( f wl u Φ (R n + ǫ > t for v := y x. Since t > 0 is arbitrary, we have y R n : sup u(yl x f(y u(x( f wl u Φ (R n + ǫ > t }. This shows that L x f wl u Φ (R n u(x( f wl u Φ (R n + ǫ for every ǫ > 0. Hence, we conclude that L x f wl u Φ (R n u(x f wl u Φ (R n. (2 Let f wl u Φ (Rn and f 0, then there exist a constant C > 0 (depends on f such that f wl u Φ (R n C. By Lemma 2.6 (, then we have L x f wl u Φ (R n Cu(x. Now, for every t, ǫ > 0, we have } v R n u(xf(v : sup u( v( L x f wl u v R n Φ (R n + ǫ > t v R n : v R n : y R n : = y R n : for y = v + x. So we obtain, } v R n u(xf(v : sup u( v( L x f wl u v R n Φ (R n + ǫ > t y R n : Since t > 0 is arbitrary, we also obtain u(xf(v u( v( L x f wl u Φ (R n + ǫ > t } u(v + xf(v L x f wl u Φ (R n + ǫ > t } u(yf(y x L x f wl u Φ (R n + ǫ > t } u(yl x f(y L x f wl u Φ (R n + ǫ > t } u(yl x f(y L x f wl u Φ (R n + ǫ > t }. u(x f wlφ (R n sup v R n u( v L x f wl u Φ (R n, 6

7 for f wlφ (R n := inf Choose C := max desired. C, b > 0 : sup sup u( v v R n f wl u (R n Φ x Rn : f(x > t} b }. }. Hence, we conclude that u(x C L xf wl u Φ (R n Cu(x, as Now, we present the sufficient and necessary conditions for the inclusion properties of weighted weak Orlicz spaces wl u Φ (R n and wl (R n with respect to Young functions Φ, and weights u,. Teorema 2.7. Let Φ, be continuous Young functions satisfying the 2 condition such that Φ and u, U. Then the following statements are equivalent: ( u. (2 wl (R n wl u Φ (R n. (3 There exist a constant C > 0 such that f wl u Φ (R n C f wl (R n, for every f wl (R n. Proof. Assume that ( holds. Let f be an element of wl (R n. Since Φ and u, there exists constant C, C 2 > 0 such that Φ (t (C t for all t > 0 and u (x C 2 (x for every x R n. Using a similar argument in the proof of Theorem 2.3 we have f wl u Φ (R n C f wl u (R n C C 2 f wl (R n. As before, we have (2 and (3 are equivalent. It thus remains to show that (3 implies (. Assume that (3 holds. By Lemma 2.6, we have u (x C L u xf wl Φ (R n C L u xf wl 2 Φ (R n 2(x, for every x R n. So, we obtain u. Remark 2.8. It follows from Theorems 2.3 and 2.7 that there cannot be an inclusion relation between wl u p (R n and wl (R n for distinct values of p and. In spite of that, for finite measure X we can obtained inclusion relation between wl u p (X and wl (X in the next section. 3 An additional case In the following, we will give sufficient condition for Hölder s inequality on weighted weak Orlicz spaces which will be used to obtain inclusion relation between wl u p (X and wl (X. Teorema 3.. (Hölder s inequality Let Φ,, Φ 3 be Young functions and u,, u 3 be weights such that Φ (tφ 2 (t Φ 3 (t for every t > 0 and u 3 (x u (x (x for every x X. If f wl u Φ (X and f 2 wl (X, then f f 2 wl u 3 Φ 3 (X with f f 2 wl u 3 Φ 3 (X 2 f wl u Φ (X f 2 wl (X. 7

8 Proof. Let f i be elements of wl u i (X, i =,2. By Lemma 2.5, for every k N we have Φ (t x X : > t} and (t x X : for every t > 0. u (xf (x (+ k f wl u Φ (X For each x X and k N, let From ( for i =, 2. Hence 2 and i= ( u (xf (x M(x, k := max (Φ ( + f u k wl (X Φ u i (xf i (x (+ k f i u wl i (X u i (xf i (x ( + k f i wl u i (X u i (xf i (x (+ k f i wl u i (X, ( M(x, k and Lemma. (3, we have ( Φ u i (xf i (x i ( ( + f u k i wl i (X Φ (M(x, kφ 2 (M(x, k Φ 3 (M(x, k Φ 3 ( 2 i= u i (xf i (x ( + k f i wl u i (X On the other hand, we have M(x, k 2 ( Therefore Φ 3 (t x X : 2 i= u 3 (x f i (x ( + k f i wl u i (X i= (xf 2 (x (+ k f 2 wl (X (xf 2 (x ( + f. u k 2 wl 2 (X Φ 3 (Φ 3 (M(x, k M(x, k. u i (xf i (x (+ k f i u wl i (X } ( 2 > t = Φ 3 i=. u 3 (x t 0 f i (x Φ i (M(x, k, ( + f u k i wl i (X ( 2 t0 u i (xf i (x Φ 3 i= ( + f u k i wl i (X 2 ( t0 u i (xf i (x i= ( + f u k i wl i (X > t} x X : > t 0 } x X : > t 0 } x X : > t 0 } where t 0 = t(+ k 2 f wl u Φ (X f 2 wl (X u 3 (xf (xf 2 (x. 8

9 Next, we also have ( ( t0 u i (xf i (x ( + f x X : > t 0 } = (t i x X : u k i wl i (X = (t i x X : u i (xf i (x ( + f u k i wl i (X u i (xf i (x ( + f u k i wl i (X 2 > t 2 i } > t i } where t i = t 0 u i (xf i (x (+ k f i u wl i (X So, we obtain Φ 3 (t x X : On the other hand, we have Φ 3 (t x X : 2 i=, for i =, 2. 2 i= u 3 (x f i (x (+ k f i wl u i (X u 3 (x f i (x 2( + f u k i wl i (X > t} sup > t} 2. Φ 3 (t x X : i= s = sup Φ 3( x X : s>0 2 sup s>0. Since t > 0 is an arbitrary positive real number, we get 2 Φ 3(s x X : 2 u 3 (x f i (x ( + f > 2t} u k i wl i (X 2 u 3 (x f i (x i= ( + f > s} u k i wl i (X Φ i 2 u 3 (x f i (x i= ( + k f i wl u i (X > s} sup Φ 3 (t x X : u 3 (xf (xf 2 (x 2( + k 2 f wl u Φ (X f 2 wl (X > t}. This shows that f f 2 u wl 3 (X 2(+ Φ 3 k 2 f u wl (X f u 2 Φ wl 2 (X and this is true for every k N. We can conclude that f f 2 u wl 3 (X 2 f u Φ wl 3 Φ (X f 2 u wl 2 (X, as desired. Corollary 3.2. Let X := B(a, r 0 R n for some a R n and r 0 > 0. If Φ, are two Young functions, u, are weights and there are a Young function Φ and a weight 0 < u(x for every x X such that Φ (tφ (t Φ 2 (t for every t 0 and u (x u(x (x for every x X, then wl u Φ (X wl (X with f wl Φ (X 2 2 Φ ( f wl u B(a,r 0 Φ (X for f wlu Φ (X. Proof. Since 0 < u(x for every x X, we have f wl (X f u wl (X. Let f wlu Φ (X, 9

10 by Theorem 3. and choosing g := χ B(a,r0, we obtain This shows that wl u Φ (X wl (X. f u wl 2 Φ (X = fχ u B(a,r 0 2 wl 2 Φ (X 2 fχ B(a,r0 u u wl 2 Φ (X 2 χ 2 B(a,r0 f u u wl u wl Φ (X 2 = Φ ( f wl u (X. Φ B(a,r 0 We shall now discuss the inclusion properties of weighted weak Lebesgue spaces wl u p (X and wl (X with respect to distinct values of p and as well as u and. Corollary 3.3. Let X := B(a, r 0 for some a R n and r 0 > 0. If < p < and u (x (x for every x X, then wl u p (X wl (X. (t 0. Since < p <, we have >. Thus, Φ,, and Φ are Young functions. Now, define u(x = u (x for every x X. (x Observe that, using the definition of Φ and Lemma., we have Proof. Let Φ (t := t p, (t := t, and := t p p p p Moreover, Φ (tφ (t = t p t p p = t it follows from Corollary 3.2 that f u wl 2 conclude that wl u p (X wl (X. Φ (X Φ (t = t p, Φ 2 (t = t, and Φ (t = t p p. = Φ 2 (t and u (x = u (x 2 (X Φ ( u (x 2(x = u(x (x. So f wl u p B(a,r 0 (X, and therefore we can 4 Concluding Remarks We have shown the sufficient and necessary conditions for the inclusion relation between weighted weak Lebesgue spaces and between weighted weak Orlicz spaces. The inclusion properties of weighted weak Orlicz spaces generalize the inclusion properties of weak Orlicz spaces in [3]. In the proof of the inclusion property we used the norm of characteristic function on R n and estimate the norm of the translation functions in R n. As a corollary of Theorem 2.7 and Theorem 2.22 in [2], we have that the inclusion properties of weighted weak Orlicz spaces are equivalent with the inclusion properties of weighted Orlicz spaces. References [] R.E. Castillo, F.A.V. Narvaez, and J.C.R. Fernándes, Multiplication and composition operators on weak L p spaces, Bull. Malays. Math. Sci. Soc (205,

11 [2] S. Gala, Y. Sawano, and H. Tanaka, A remark on two generalized Orlicz-Morrey spaces, J. Approx. Theory 98 (205, 9. [3] L. Grafakos, Classical Fourier Analysis, Springer, New York, [4] H. Gunawan, D.I. Hakim, K.M. Limanta, A.A. Masta, Inclusion properties of generalized Morrey spaces, Math. Nachr. 290 (207, doi:0.002/mana [5] Y. Jiao, Embeddings between weak Orlicz martingale spaces, J. Math. Appl. 378 (20, [6] 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., 982. [7] M. A. Krasnosel skii and Y. Z. B. Rtuickii, Convex Fuctions and Orlicz Spaces, first edition, Noordhoff, Groningen, 96. [8] A. Kufner, O. John, and S. Fu cik, Function Spaces, Noordhoff International Publishing, Czechoslovakia, 977. [9] C. Léonard, Orlicz spaces, preprint. [ leonard/papers/orlicz.pdf, accessed on August 7, 205.] [0] N. Liu and Y. Ye, Weak Orlicz space and its convergence theorems, Acta Math. Sci. Ser. B 30-5 (200, [] W.A.J. Luxemburg, Banach Function Spaces, Thesis, Technische Hogeschool te Delft, 955. [2] L. Maligranda, Orlicz Spaces and Interpolation, Departamento de Matemática, Universidade Estadual de Campinas, 989. [3] A.A. Masta, H. Gunawan, and W. Setya-Budhi, Inclusion property of Orlicz and weak Orlicz spaces, J. Math. Fund. Sci (206, [4] A.A. Masta, H. Gunawan, W.S. Budhi, An inclusion property of Orlicz-Morrey spaces, to appear in J. Phys.: Conf. Ser. [5] A.A. Masta, H. Gunawan, W.S. Budhi, On Inclusion Properties of Two Versions of Orlicz- Morrey Spaces, Research Report ( [6] C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (938, [7] E. Nakai, On Orlicz-Morrey spaces, research report [ accessed on August 7, 205.] [8] E. Nakai, Orlicz-Morrey spaces and some integral operators, research report [ pdf, accessed on August 7, 205.] [9] W. Orlicz, Linear Functional Analysis (Series in Real Analysis Volume 4, World Scientific, Singapore, 992. [20] 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.

12 [2] A. Osançliol, Inclusion between weighted Orlicz spaces, J. Inequal. Appl (204, 8. [22] 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. [23] R. Welland, Inclusion relations among Orlicz spaces, Proc. Amer. Math. Soc. 7- (966, [24] X. Zhang and C. Zhang, Weak Orlicz spaces generated by concave functions, International Conference on Information Science and Technology (ICIST 20,