Sufficient and necessary conditions for generalized Ho lder s inequality in p-summable sequence spaces

Transcription

1 Sufficient and necessary conditions for generalized Ho lder s inequality in -suable sequence saces l Masta*, S Fatiah 2, rsisari 3, Y Y Putra 4 and F riani 5,2 Deartent of Matheatics Education, Universitas Pendidikan Indonesia, Jl Dr Setiabudi, Bandung, Indonesia 3,4,5 Deartent of Matheatics Education, Sekolah Tinggi Keguruan dan Ilu Pendidikan Muhaadiyah Bangka Belitung, Jl KH had Dahlan, Kabuaten Bangka Tengah, Bangka Belitung, Indonesia *Corresonding author s e-ail: alazhariasta@uiedu bstract In this aer, we discuss the generalized Ho lder s inequality in -suable sequence saces In articular, we shall rove sufficient and necessary conditions for generalized Ho lder s inequality in those saces One of the keys to rove our results is to estiate the nors of characteristic sequences in R Introduction In atheatics, Lebesgue saces are one of iortant toics, articularly in real and functional analysis There are two kinds of Lebesgue saces which are continuous Lebesgue saces denoted by L and - suable sequence saces denoted by l Many researchers have studied Lebesgue saces and its generalization over few decades (see [ - 2], etc) For exale, in 206, Masta, et al [2] resented the sufficient and necessary conditions for generalized Ho lder's inequality in continuous Lebesgue saces and in continuous Orlicz Morrey saces Recently, Ifronika, et al [3] also obtained the sufficient and necessary conditions for generalized Ho lder's inequality in continuous Morrey saces, in continuous generalized Morrey saces, and in their weak tye Motivated by these results, we are interested in discussing the -suable sequence saces In articular, we will rove the sufficient and necessary conditions for Ho lder s inequality in these saces First, we recall the definition of -suable sequence saces Let <, the -suable sequence l (R) is the set of sequences X = (x n ) R such that X l (R) ( x n ) < Note that, l (R) is a Banach sace with resect to the nor l (R) Next, for =, l (R) is defined as the set of sequences X = (x n ) R such that

2 X l (R) ax x n < n N Notice that, l (R) is also Banach saces equied with the nor l (R) The rest of this aer is organized as follows In Section 2, we resented soe leas which useful for obtain our results The ain results are resented in Section 3 In Section 3, we state the sufficient and necessary conditions for generalized Ho lder s inequality in -suable sequence saces 2 Methods To obtain the sufficient and necessary conditions for generalized Ho lder s inequality in -suable sequence saces, we use the nors of the characteristic sequences in R and soe leas as in the following Lea [4] Let Z and N {0,,2,3, }, write S,N { N,,, + N} Let ξ k,n {, if k S,N 0, otherwise, then there exists C > 0 (indeendent of and N) such that for every N {0,,2,3, } (2N + ) / ξ,n k l C(2N + )/ (R) Lea 2 [3] Let x i be ositive real nubers for i =,2,3,, If, 2,,, < satisfy the condition i= =, then wen have i ( x i ) i= x i i i Corollary 3 Let, 2, < satisfy the condition + 2 =, X= (x n) l (R) and Y = (y n ) l 2 If x n = y n 2 =, then x n y n Let + =, X = (x 2 n) l (R) and Y = (y n ) l 2 By using Lea 2, we have x n y n x n + y n 2 = + = Results and Discussion First, we resent sufficient and necessary conditions for Ho lder s inequality in l (R) sace in the following theore Theore 4 Let,, 2 < () If + 2 =, then XY l (R) X l (R) Y l 2 (R) for every X l (R) and Y l 2 (2) If XY l (R) X l (R) Y l 2 (R), for every X l (R) and Y l 2 (R), then + 2 i=

3 () Let + =, X = (x 2 n) l (R) and Y = (y n ) l 2 First, suose that x n = and y n 2 = B By setting x n = x n and y / n =, we have x B / 2 n and y n 2 = By using Corollary 3, we have = y n 2 B Since B2 x n y n = x ny n B 2 y n = x n y n 2 B x n y n is equivalent to x n y n ( x n y n ) / ( x n ) ( y n 2) So, we have XY l (R) X l (R) Y l 2 (R) = x n 2 B = 2 we obtain (2) Now, assue that XY l (R) X l (R) Y l 2 (R) holds, for every X l (R) and Y l 2 Take X = Y = ξ k,n, by using Lea, we have (2N + ) / ξ,n k l (R) ξ,n k l (R) ξ,n k l2 (R) C2 (2N + ) + 2 or (2N + ) ( + 2 ) C 2 for every N {0,, 2, 3, } Hence, we can conclude that + 2 Theore 5 Let, 2, 3,,, < () If i= =, then X i= i l (R) i= X i li (R), for every X i l i i (2) If i= X i l (R) i= X i li (R), for every X i l i (R), then i= () Let i= =, X i = (x n,i ) l i (R) for i =,2,3,, First, suose that x n,i i = i i for every i =,2,, By setting x n,i Lea 4, we have x n,i i= = x n,i i / i, we have x n, + x n,2 2 2 = = i i= i x n,i i = x n,i i i + + x n, = By using

4 So we have, x n,i i= = x n,i i= i i = i= i i= x n,i Since i i= i ( i= x n,i ) is equivalent to i= x n,i i= i, we obtain So, we have i= ( x n,i ) X i i= l (R) / C X i i= ( x n,i i ) i= l (R) i C X i li (R),N (2) Now, assue that i= X i l (R) i= X i li (R) holds for every X i l i Take X i = ξ k for every i =,2,3,,, by using Lea, we have or (2N + ) (2N + ) / ξ,n k l (R) ξ,n k li (R) ( i i= i= C (2N + ) i= i i= ) C for every N {0,, 2, } Hence, we can conclude that i i= i 4 Conclusion We have shown the sufficient and necessary conditions for generalized Ho lder s inequality in l (R) sace, we can state that the condition i= is a necessary conditions for generalized Ho lder s i inequality in l (R) sace 5 References [] Carothers N L 2000 Short Course on Banach Sace Theory, Deartent of Matheatics and Statistics, Bowling Green State University [2] Masta l, Gunawan H and Setya-Budhi W 207 n inclusion roerty of Orlicz-Morrey saces J Phys: Conf Ser, [3] Ifronika, Idris M, Masta l and Gunawan H 208 Generalized Holder s Inequality on Morrey Saces, Mat Vesnik, 70-4, [4] Gunawan H, Kikianty E and Schwanke C 207 Discrete Morrey saces and their inclusion roerties Math Nachr 4 [5] Castillo R E and Rafeiro H 206 n Introductory Course in Lebesgue Saces, CMS books in Matheatics, Sringer: Canadian Matheatical Society [6] Masta l, Gunawan H and Setya-Budhi W 207 On inclusion roerties of two versions of Orlicz Morrey saces Mediterr J Math, 4-6,

5 [7] Masta l, Gunawan H and Setya-Budhi W 205 n inclusion roerty of Orlicz saces ProcThe 5 th nnual Basic Science International Conference 205, vol 5 ISSN: , [8] Masta l, Gunawan H and Setya-Budhi W 206 n inclusion roerty of Orlicz and weak Orlicz saces J Math Fund Sci [9] Masta l, Suiaty E, Taqiyuddin M,and Pradita I 208 The sufficient condition for inclusion roerties of discrete weighted Lebesgue saces, J Phys: Conf Ser [0] Nekvinda 2007 Ebeddings between discrete weighted Lebesgue saces with variable exonents Math Ineq l0-, [] Osançliol 204 Inclusion between weighted Orlicz saces J Inequal l 204, [2] Taqiyuddin M and Masta l 208 Inclusion roerties of Orlicz saces and weak Orlicz saces generated by concave function IOP Conf Ser: Mater Sci Eng, [3] Li H and X-M Gu X-M 207 The weighted a-g inequality is equivalent to the Ho lder s inequality research reort [htts://arxivorg/df/ df] cknowledgeent The first and second authors is suorted by Hibah Penguatan Koetensi UPI 208