On the Exact Value of Packing Spheres in a Class of Orlicz Function Spaces

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1 Jornal of Convex Analyi Volme 2004), No. 2, On the Exact Vale of Packing Sphere in a Cla of Orlicz Fnction Space Received Jne 30, 2003 Revied mancript received October 6, 2003 Ya Qiang Yan Department of Mathematic, Szho Univerity, Szho, Jiang, 25006, P. R. China yanyq@pb.z.jinfo.net Main relt: the Packing contant of Orlicz fnction pace L Φ) [0, ] and L Φ [0, ] with Lxembrg and Orlicz norm have the exact vale. i) ii) If F Φ t) tϕt)/φt) i decreaing, < C Φ < 2, then If F Φ t) i increaing, C Φ > 2, then where C Φ lim t F Φ t). 2/C Φ P L Φ) [0, ]) P L Φ [0, ]) ; /C Φ P L Φ) [0, ]) P L Φ [0, ]), + 2 /C Φ Keyword: Orlicz pace, Packing contant, Kottman contant 99 Mathematic Sbject Claification: 46E30. Introdction Definition. [5] [0]). The packing contant P X) of a Banach pace X i P X) p{ r > 0 : infinitely many ball of radi r are packed into the nit ball of X}. The Kottman contant of a infinite dimenional Banach pace X i defined [5] a { } KX) p inf x i x j : {x i } SX), i j where SX) i the nit phere of X. Clearly, KX) 2. The following relationhip wa offered by Kottman [5] cf. Ye [8]): Propoition.2. For a infinite dimenional Banach pace X, one ha P X) KX) 2 + KX). ) ISSN / $ 2.50 c Heldermann Verlag

2 392 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz... Hdzik [4] verified that KX) 2 if X i a nonreflexive Banach lattice, therefore P X) 2. If X i an infinite dimenional Hilbert pace, then Rankin [8] etablihed P X) + 2. Slightly later, Brlack, Rankin and Roberton [] generalized it a: P l p ), < p <. + 2 p Mch later, in 975, Well and William [5] howed P L p [0, ]) +2 p +2 p, < p 2,, 2 p < and that for p and p, the vale i /2. However, the calclation of packing contant of general Banach pace i a difficlt problem. Reearcher trned to tdy the Kottman contant in Orlicz pace. Let Φ) 0 ϕt)dt and Ψv) v 0 ψ)d be a pair of complementary N fnction, i.e., ϕt) i right contino, ϕ0) 0, and ϕt) a t to implify the dicion, we ame ϕ being continoly differential ). We call Φ 2 ), if there exit 0 > 0 and k > 2 ch that Φ2) kφ) for 0. The Orlicz fnction pace L Φ [0, ] i defined a L Φ [0, ] { xt) : ρ Φ λx) The Lxembrg norm and Orlicz norm are expreed a 0 } Φλ xt) )dt < for ome λ > 0. x Φ) inf{c > 0 : ρ Φ x c ) } and x Φ inf k>0 k [ + ρ Φkx)], repectively. For the Orlicz eqence pace eqipped with Lxembrg norm and Orlicz norm with Φ atifying the 2 -condition, Wang [4] and Ye [8] gave the expreion for Kottman contant. Latter on, the athor [7] gave the formlae for real comptation and anwered Rao and Ren [0] open problem which concerning the exact vale of ome Orlicz eqence pace. However, formlae for Orlicz fnction pace i till nknown in pite of the work of Cleaver [2], Ren [], [2]. Thi paper i trying to give a formla for packing contant in a cla of Orlicz fnction pace eqipped with both norm.

3 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz In what follow, we will e Semenove and Simonenko indice of Φ): α Φ lim inf Φ ) Φ 2), β Φ lim p Φ ) Φ 2), 2) A Φ lim inf t tϕt) Φt), B Φ lim p t tϕt) Φt), 3) The ame indice can be applied to Ψv). The athor [7] obtained Rao and Ren [8] gave the following interrelation: If the index fnction F Φ t) tϕt) 2α Φ β Ψ 2α Ψ β Φ. 4) 2 A Φ α Φ β Φ 2 B Φ, 5) Φt) of, then the limit C Φ lim t + i monotonic increae or decreae) at a neighborhood tϕt) Φt) mt exit, and hence α Φ β Φ 2 C Φ. 6) Define G Φ c, ) Φ )/Φ c), c >, and G Φ G Φ 2, ). The athor [7] proved: Propoition.3. F Φ i increaing decreaing) on [Φ 0 ), + ) if and only if G Φ c, ) i increaing decreaing) on [ 0 /c, + ) for any c >. 2. Main relt We need only to oberve the Kottman contant for L Φ) [0, ] and L Φ [0, ] being reflexive, or eqivalently, Φ 2 ) 2 ), ince otherwie, the Kottman contant of a nonreflexive pace i 2 and hence the Packing phere contant mt be /2. Cleaver [2] and Ren [], [2] obtained the following relt: ) max, 2β Φ KL Φ) [0, ]), 7) α Φ ) max, 2β Ψ KL Φ [0, ]), 8) α Ψ and { KL Φ ) [0, ]), KL Φ [0, ]) } 2 2 9) for the interpolation of Orlicz pace. In view of the athor relt 4), the left ide of 7) and 8) are indeed the ame and greater than or eqal to 2. Therefore, the above relt can be refined for interpolation of pace a follow: Propoition 2. [2], [2], [9]). Let Φ be an N fnction, Φ 0 ) 2, and let Φ be the invere of Then Φ ) [ Φ ) ] [ Φ 0 ) ], 0 <, 0. ) max, 2β Φ { KL Φ) [0, ]), KL Φ [0, ]) } ) α Φ

4 394 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz... For any N fnction Φ with C Φ 2, we prodce a fnction M ch that Φ ) [ M ) ] [ Φ 0 ) ] ) for ome 0 < <, where Φ 0 ) 2. Then M i determined by M ) /2) [ Φ ) ] /). If < C Φ < 2, we take l ch that < l < C Φ and let 2C Φ l)/c Φ 2 l), then 0 < <. If C Φ > 2, we take l ch that C Φ < l < and let 2C Φ l)/c Φ 2 l), then we alo have 0 < <. It i important to how that M i an N fnction nder ome condition. Theorem 2.2. Let Φ be an N fnction, i) ii) If F Φ t) i decreaing, < C Φ < 2, then the fnction M determined by ) atifie: A) Mt)/t +. lim t + B) M i convex. If F Φ t) i increaing, C Φ > 2, then M determined by ) alo atifie A), B) in i). Proof. i) In thi cae, there i a 0 ch that C Φ Φ ) /C Φ. Therefore, Φ) for 0, or eqivalently, M ) 2 2 Φ ) ) 2 2 C Φ ) ) C Φ 2)l ) C Φ l 2) a +. Let M ) t, then Mt) and hence lim t Mt)/t +. To prove B), it ffice to prove M ) i concave. Oberve that M ) Φ ), 2) d d M Φ ) ϕ 2) Φ ) ϕ 2 Φ ) 2 2) ϕ ϕ Φ 2 2 Φ ) 2) ϕ 2 C Φ 2 l 2 Φ ) 2) ϕ > 0, C 2 Φ Φ ) 2 2) ϕ S) 2) Φ )

5 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz and that d 2 d M d 2 d {[ Φ ) 2 Φ ) ϕ ϕ Φ ) 2 ϕ Φ ) [ Φ ) 2 ϕ Φ ) 2 2) ϕ ϕ + Φ ) 2 ] [ ϕ 2 2) ϕ ϕ 2 2) + )} ] ) ϕ 2 2 2) Φ ) 2 2 ) ϕ 2) [ 2 Φ ϕ + 2 ) ] 4 Φ ) 2 ϕ 2 ϕ ϕ Φ 2 Φ ) 2 2) ) ϕ 2 2 2) ) [ 2C Φ l) l2 C Φ ) Φ ϕ + C Φ + l C Φ l 2C Φ 2 l) ] ϕ ϕ ) ) ] Φ 2 ϕ ϕ Φ ϕ Φ ) 2 )ϕ 2 2). Let Φ ) t, then Φt). It remain to check that ft) : 2C Φ l) tϕ l2 C Φ ) Φ + C ) ) 2 Φ + l C Φ l tϕ tϕ 2C Φ 2 l) Φ ϕ We firt prove hl) 2C Φ l) l2 C Φ ) for any definite < l < C Φ, or eqivalently, < 0. 2) C Φ + C ) Φ + l C Φ l C Φ ) 2 C Φ ) < 0, 3) 2C Φ 2 l) hl) C Φ [ C Φ ) 2 l + 4C Φ l + C Φ ) 2 4C Φ 4l + 4] l2 l)2 C Φ ) Let l +, then hl) 0. On the other hand, ince h l) < 0. C Φ [lc Φ 2) + 2] 2 8C Φ l ) 4 < 0, 2 C Φ 2l l 2 ) 2 we ee that hl) i decreaing on 0, C Φ ), and hence we dedce that hl) < h + ) 0 on 0, C Φ ). Secondly, note that the fnction gx) : 2C Φ l) l2 C Φ ) C Φ l) l2 C Φ ) [ CΦ + l C Φ l C Φ 2 l) x + C Φ + l C Φ l 2C Φ 2 l) x2 x 2 C Φ2 l) x ) C Φ l ) x ) ]

6 396 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz... i decreaing for x [C Φ l 2)] 2 /[2C Φ + l C Φ l)c Φ l)]. If C Φ < 2, then C Φ < [C Φ l 2)] 2 /[2C Φ + l C Φ l)c Φ l)] for a fficiently mall l > we will let l + in the following context). It follow that gf Φ t)) C Φ l) l2 C Φ ) hl) < 0 [ CΦ + l C Φ l C Φ 2 l) ince F Φ t) C Φ at a neighborhood of. Thirdly, By L Hopital theorem, we obtain C Φ ) 2 C ] Φ2 l) C Φ ) C Φ l tϕ t) lim t + ϕt) lim tϕt) t + Φt) C Φ. Therefore, for 0 < ε < hl), there i a fficiently large 0 ch that tϕ t) ϕt) > tϕt) Φt) ε 4) for 0. Coneqently, we have from 3) and 4) that [ )] tϕ tϕt) ft) gf Φ t)) ϕ Φt) hl) + ε < 0 and hence, we proved M i convex. ii) If F Φ t) i increaing and C Φ > 2, then for every fficiently mall ε > 0, there i 0, ch that /C Φ Φ ) < /C Φ+ε for > 0. Therefore, M ) 2 2 Φ ) ) 2 2 C Φ +ε ) C Φ 2)l ) C Φ l 2) a +. Therefore, lim t Mt)/t +, that i A) hold. ) ε To prove B), we are redced to prove 2). Since F Φ t) tϕt)/φt) i increaing from C Φ, we have ) F Φt) tϕ + ϕ)φ tϕ 2 tϕ ϕ + tϕ ϕ Φ 0. Φ 2 Φ Therefore, or Th, ft) 2C Φ l) l2 C Φ ) tϕ ϕ + tϕ Φ 0, tϕ ϕ tϕ Φ. tϕ Φ + C Φ + l C Φ l 2C Φ 2 l) ) ) 2 ) tϕ tϕ Φ Φ gf Φ t)). 5)

7 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz Oberving hl) defined in 3), we fond that hl) < 0 for l > C Φ. lim hl) 0, and l + h l) > 0. Therefore, Since the fnction gx) i increaing for x [C Φ l 2)] 2 /[2C Φ + l C Φ l)c Φ l)]. If C Φ > 2, then C Φ > [C Φ l 2)] 2 /[2C Φ + l C Φ l)c Φ l)] for a fficiently big l. We dedce that ft) gf Φ t)) C Φ l) l2 C Φ ) hl) < 0 [ CΦ + l C Φ l C Φ 2 l) C Φ ) 2 C ] Φ2 l) C Φ ) C Φ l ince F Φ t) C Φ at a neighborhood of. Th, we proved M i convex. The proof i finihed. Theorem 2.3. Let Φ be an N fnction. i) If F Φ t) tϕt)/φt) i decreaing, < C Φ < 2, then KL Φ) [0, ]) KL Φ [0, ]) 2 C Φ. 6) ii) If F Φ t) tϕt)/φt) i increaing, C Φ > 2, then KL Φ) [0, ]) KL Φ [0, ]) 2 C Φ. 7) Proof. i) When F Φ t) i decreaing and < C Φ < 2, it follow from 0) and ) that ) max, 2β Φ {KL Φ) [0, ]), KL Φ [0, ])} 2 2 8) α Φ Since F Φ t) i decreaing, α Φ 2 C Φ by 6). On the other hand, in 8) let l +, then 2 )/2 /C Φ. Therefore, 6) hold. ii) Similar to i), 7) follow from 2 lim l 2 2 C Φ 2β Φ. Remark 2.4. In view of the eqation ), 6) and 7) are eqivalent to P L Φ) [0, ]) P L Φ [0, ]) 2/C Φ /C Φ and P L Φ) [0, ]) P L Φ [0, ]), + 2 /C Φ repectively. Therefore, we obtained the expreion of Packing contant in a cla of Orlicz fnction pace. It i obvio to ee that 6) alo hold for C Φ, and 7) hold for C Φ ince the pace generated by Φ i nonreflexive. One can eaily dedce the Kottman contant of L p [0, ], that i, ) KL p [0, ]) max 2 p, 2 p.

8 398 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz... ince C Φ p for Φ) p. The athor have ed the method in thi paper to tdy the other geometric contant in Orlicz fnction pace a well a eqence pace. However, whether the formlae 6) and 7) i fit for C Φ 2 i till nknown. Corollary 2.5. Let Φ be an N fnction. i) ii) If F Φ t) tϕt)/φt) i decreaing, ϕ i concave, then 6) hold; If F Φ t) i increaing, ϕ i convex, then 7) hold. Proof. i) Note that ϕ0) 0 ince Φ i an N fnction. When ϕ i concave, ϕt) t 0 ϕ )d + ϕ0) t 0 ϕ )d tϕ t). Therefore, [tϕt) 2Φt)] tϕ ϕ 0, and hence tϕt) 2Φt) 0, in other word, F Φ t) 2 which mean C Φ 2. If C Φ < 2, then 6) hold by Theorem 2.3. If C Φ 2, then tϕt)/φt) 2 ince F Φ t) i decreaing from at a neighborhood of, and hence,tϕt)/φt) 2 at a neighborhood of. Thi mean that Φt) at 2 a > 0), which generate the Hilbert pace L 2 [0, ], o 7) hold by the well known relt. Analogoly we can prove ii). Example 2.6. Let N fnction be Φ) 2 p + 2p, p >. Then and C Φ 2p. Therefore, we have ) F Φ t) tφ t) t p Φt) 2p + 2p t p + 2 KL Φ) [0, ]) KL Φ [0, ]) 2 C Φ 2 2p. 9) Let 0 <, then we can prodce Φ by We have the exact vale: Φ ) + ) p 2. KL Φ ) [0, ]) KL Φ [0, ]) 2 C Φ 2 2p 2. 20) In fact, α Φ β Φ lim Φ Φ ) 2) lim ) + p 2 + ) ) 2 2p In view of Propoition.3 we ee that F Φ t) i increaing on 0, + ) althogh it i impoible to expre, ince it i eay to check that the fnction + )/ c + ) i increaing on 0, + ) for any c >. Th, C Φ 2p + 2 2, + ).

9 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz Particlarly, let /2, then we obtain the N fnction defined a: We have the vale: Example 2.7. Let defining an N fnction. Then Indeed, we have Φ /2 ) + ) 2p 4. KL Φ /2) [0, ]) KL Φ /2 [0, ]) 2 4p 4. 2) F Φp p Φ p ) Example 2.8. Conider the N fnction Then Again for t > 0, we have Example 2.9. Let F Φp,r p + p lne + ), p > 2 KL Φ p) [0, ]) KL Φ p [0, ]) 2 p. 22) t e + t) lne + t) p C Φ p, t + ). Φ p,r ) p ln r + ), < p < 2. KL Φ p,r) [0, ]) KL Φ p,r [0, ]) 2 p. 23) rt + t) ln + t) p C Φ p,r t + ). Φ p ) p C + ln ), < p < 2, c 2p pp ) It wa introdced by Gribanow and revied by Maligranda. Since F Φp p + C + ln t > p, t >, we have Coneqently, C Φp lim t + F Φ p p. KL Φ p) [0, ]) KL Φ p [0, ]) 2 p. 24)

10 400 Y. Q. Yan / On the Exact Vale of Packing Sphere in a Cla of Orlicz... Reference [] J. A. C. Brlack, R. A. Rankin, A. P. Roberton: The packing of phere in the pace l p, Proc. Glagow Math. Aoc ) [2] C. E. Cleaver: Packing phere in Orlicz pace, Pacific J. Math ) [3] J. Han, X. L. Li: Exact vale of packing contant in cla of Orlicz fnction pace, Jornal of Tongji Univerity ) [4] H. Hdzik: Every nonreflexive Banach lattice ha the packing contant eqal to /2, Collect. Math ) [5] C. A. Kottman: Packing and reflexivity in Banach pace, Tran. Amer. Math. Soc ) [6] M. A. Kranoelkiĭ, Ya. B. Rtickiĭ: Convex Fnction and Orlicz Space, Noordhoff, Groningen 96). [7] J. Lindentra, L. Tzafriri: Claical Banach Space, I) and II), Springer, Berlin 977) and 979). [8] R. A. Rankin: On packing of phere in Hilbert pace, Proc. Glagow Math. Aoc., 2 955) [9] M. M. Rao, Z. D. Ren: Application of Orlicz Space, Marcel Dekker, New York 2002). [0] M. M. Rao, Z. D. Ren: Packing in Orlicz eqence pace, Std. Math ) [] Z. D. Ren: Packing in Orlicz fnction pace with Lxembrg norm, J. Xiangtan Univ. 985) [2] Z. D. Ren: Packing in Orlicz Fnction Space, Ph. D. Diertation, Univerity of California, Riveride 994). [3] Z. D. Ren, S. T. Chen: Jng contant of Orlicz fnction pace, Collect. Math ) [4] T. F. Wang: Packing contant of Orlicz eqence pace, Chinee Ann. Math., Ser. A 8 987) in Chinee). [5] J. H. Well, L. R. William: Imbedding and Extenion Problem in Analyi, Springer 975). [6] C. X. W, S. T. Chen, Y. W. Wang: Geometric character of reflexivity and flatne of Orlicz eqence pace, Northeat. Math. J. 2) 986) in Chinee). [7] Y. Q. Yan: Some relt on packing in Orlicz eqence pace, Std. Math. 47) 200) [8] Y. N. Ye: Packing phere in Orlicz pace, Chinee Ann. Math., Ser. A 4 983) in Chinee).

ON THE NONSQUARE CONSTANTS OF L (Φ) [0, + ) Y. Q. YAN

http://dx.doi.org/0.5209/rev_rema.2002.v5.n2.696 ISSN 39-38 ON THE NONSQUARE CONSTANTS OF L (Φ) [0, + ) Y. Q. YAN Abstract Let L (Φ) [0, + ) be the Orlicz function space generated by N function Φ(u) with