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

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1 ISSN 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 Luxemburg norm. We show the exact nonsquare constant of it when the right derivative φ(t) of Φ(u) is convex or concave. Introduction Let X be a Banach space and S(X) ={x : x =,x X} denotes the unit sphere of X. The nonsquare constants in the sense of James J(X) and in the sense of Schaffer g(x) are defined as: J(X) =sup{min( x + y, x y ) :x, y S(X)}, () g(x) =inf{max( x + y, x y ) :x, y S(X)}. (2) Clearly, if dim X 2, then g(x) 2 J(X) 2. Ji and Wang [5] asserted g(x) J(X) =2 (3) for dim X 2. It is proved[] that J(X) =2ifX fails to be reflexive. Nonsquareness is an important geometric property of Banach spaces which expose the intrinsic construction of a space according to the shape of the unit ball of the spaces. Therefore, it is interesting to investigate it in classical Banach spaces, for example, Orlicz spaces. It is showed[5] that J(L p )= max(2 p, 2 p )( <p< ). However, examples for values of J(X) for X to be reflexive except L p remainsunknown. Inthispaper,wedeal with J(X) whenx is an Orlicz function space with Luxemburg norm Mathematics Subject Classification: 46E30. Servicio de Publicaciones. Universidad Complutense. Madrid,

2 Let Φ(u) = u 0 φ(t)dt be an N function,i.e., φ(0) = 0,φ is right continuous and φ(t) as t. The Orlicz function space L (Φ) [0, ) is defined to be the set { } L (Φ) [0, ) = x(t) :ρ Φ (λx) = Φ(λ x(t) )dt < for some λ>0. [0, ) The Luxemburg norm is expressed as x (Φ) =inf {c >0:ρ Φ ( x } c ). Φ(u) is said to satisfy the Δ 2 condition for for all u 0, in symbol Φ Δ 2, if there exists k>2 such that Φ(2u) kφ(u) foru 0. In what follows, we will frequently use Semenove indices of Φ(u): =inf u>0 Φ (2u), β Φ =sup u>0 Φ (2u). (4) 2 Main Results We first consider the lower bounds of L (Φ) [0, ).The following result is refined from Ren[8]. Theorem. Let Φ(u) be an N function. Then the nonsquare constant of L (Φ) [0, ), in the sense of James, satisfies ( ) max, 2β J(L (Φ) [0, )). (5) Φ Proof. To prove (5), we first show J(L (Φ) [0, )). (6) Take a real number u (0, ), choose measurable subsets G and G 2 in [0, ) such that G G 2 =. and μ(g )=μ(g 2 )= 2u. Put x(t) =Φ (2u)χ G (t) andy(t) =Φ (2u)χ G2 (t), 526

3 where χ G is the characteristic function of G. Note that χ G (Φ) = χ G2 (Φ) = We have x (Φ) = y (Φ) = and Φ ( μ(g ) ) = Φ (2u). x y (Φ) = x + y (Φ) = Φ (2u). Taking the supremum over u (0, ), since the function G Φ (u) = Φ (2u) is right continuous at 0 and takes value on [ 2, ], we deduce that J(L (Φ) [0, )) Finally we show Φ (2u) sup u (0, ) = sup Φ (2u) u [0, ) =. 2β Φ J(L (Φ) [0, )). (7) For every real number v>0, choose measurable subsets E,E 2 in [0, ) such that E E 2 = and μ(e )=μ(e 2 )= 2v. Put x(t) =Φ (v)[χ E (t)+χ E2 (t)] and y(t) =Φ (v)[χ E (t) χ E2 (t)], Then x (Φ) = y (Φ) = and x y (Φ) = x + y (Φ) = 2Φ (v) Φ (2v). Taking the supremum over v (0, ) wealsohave J(L (Φ) [0, )) 2β Φ. Hence (5) follows from (6) and (7). Assume Φ satisfies Δ 2 condition for all u. Ji and Wang([5],Theorem 3) offered a couple of formulas: (i) If φ(t) isaconcavefunction,then { g(l (Φ) [0, )) = inf k x > 0:ρ Φ ( 2x } )=2,ρ Φ (x) = ; (8) k x 527

4 (ii) If φ(t) isconvex,then { J(L (Φ) [0, )) = sup k x > 0:ρ Φ ( 2x } )=2,ρ Φ (x) =. (9) k x We now extend the above representatives and deduce the upper bounds. Theorem 2. Suppose φ(t) be the right derivative of Φ(u). We have (i) If φ(u) is concave, then (ii)if φ(u) is convex, then J(L (Φ) [0, )) ; (0) J(L (Φ) [0, )) 2β Φ. () Proof. If Φ Δ 2, which is equivalent to β Φ =, then L (Φ) [0, ) is nonreflexive and hence J(L (Φ) [0, ))=2accordingtotheresultsin Chen[] or Hudzik[4]. Since φ(t) isconcaveimpliesφ Δ 2 (see Krasnoselskiĭ and Rutickiĭ[6],p.26), we only need to check () when φ(t) is convex, but this is trivial since J(l (Φ) )=2=2β 0 Φ =2 β Φ. Therefore it suffices for us to prove (0) and () for Φ Δ 2. We first prove (0) for Φ(u) Δ 2, which is equal to g(l (Φ) [0, )) 2 (2) when φ(t) is concave in view of (3) and (8). Let H Φ (u) = Φ (2u), then Φ (2u) =H Φ (u). Put x =, then u =Φ(x) and 2Φ(x) =Φ[H Φ (Φ(x)) x]. (3) Therefore, when u =Φ(x(t)) 0wehave ( ) ( ) ( ) ρ 2x(t) Φ 2 = ρ x(t) Φ ρ Φ (2u) Φ x(t) = ρ Φ [H Φ (u) x(t)] = 2ρ Φ (x(t)) = 2 528

5 for ρ Φ (x(t)) =. It follows that (2) and hence (0) holds. One can prove () analogously by (9). We obtain the main result from the above theorems: Theorem 3. Let Φ(u) be an N function, φ(t) be the right derivative of Φ(u). Then (i)if φ(t) is concave, then (ii)if φ(t) is convex, then J(L (Φ) [0, )) = ; (4) J(L (Φ) [0, )) = 2β Φ. (5) Remark 4. If the index function G Φ (u) = Φ (u) Φ (2u) is decreasing or increasing on interval [0, ), then the indices and β Φ take the values at either end of the interval. The author[0] found that if F Φ (t) = Φ(t) is increasing(decreasing) on (0, Φ (u 0 )] then G Φ (u) is also increasing(decreasing) on (0, u 0 2 ], respectively. Rao and Ren[7] gave interrelations between Semenove and Simonenko indices: 2 A Φ β Φ 2 B Φ, 2 A 0 Φ α 0 Φ β 0 Φ 2 B 0 Φ, where and A Φ = lim inf t A 0 Φ = lim inf t 0 = lim inf u α 0 Φ = lim inf u 0 Φ(t), Φ(t), Φ (2u), Φ (2u), B Φ =limsup t B0 Φ =limsup t 0 β Φ =limsup u β0 Φ =limsup u 0 Φ(t) ; Φ(t) ; Φ (2u) ; Φ (2u). 529

6 When the index function F Φ (t) is monotonic, the limits C Φ = lim t F Φ (t) and C 0 Φ = lim t 0 F Φ(t) mustexistandwehave = β Φ = lim G Φ(u) =2 u C Φ, αφ 0 = βφ 0 = lim G Φ (u) =2 C Φ 0. u 0 (6) This makes it easier to calculate the indices in Theorem 3. Example 5. Observe the N function(see Gallardo[2]) Φ p,r (u) = u p ln r ( + u ), p<, 0 <r<. It is easy to check the right derivative of Φ p,r (u), φ(t) is convex when p<, 2 r<. The index function F Φp,r (t) = tφ p,r(t) Φ p,r (t) = p + rt ( + t)ln(+t) is decreasing from p + r to p on [0, ) since d dt Φ p,r(t) = r[ln( + t) t] ( + t) 2 ln 2 ( + t) < 0. So C 0 Φ p,r (t) = lim t 0 F Φp,r(t) =p+r. According to (6) in the above remark and Theorem 3 we have J(L (Φp,r) [0, )) = 2β Φp,r =2β 0 Φ p,r =2 2 p+r =2 p+r. (7) References [] S. T. Chen, Nonsquareness of Orlicz spaces, Chinese Ann. Math., 6A (985), [2] D. Gallardo, Orlicz spaces for which the Hardy-Litllewood maximal operator is bounded, Publications Matematiques, 32 (988), [3] J. Gao and K. S. Lau, On the geometry of spheres in normed linear spaces, J. Austral. Math. Soc., 48A (990), 0-2. [4] H. Hudzik, Uniformly non-l n () Orlicz spaces with Luxemburg norm, Studia. Math., 8 (985),

7 [5] D.H.JiandT.F.Wang,Nonsquareconstantsofnormedspaces,Acta. Sci. Math.(Szeged), 59 (994), [6]M.A.KrasnoselskiĭandYa.B.Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 96. [7] M. M. Rao and Z. D. Ren, Packing in Orlicz sequence spaces, Studia Math., 26 (997), [8] Z. D. Ren, Nonsquare constants of Orlicz spaces, Lecture Notes in Pure and Applied Mathematics, 86 (997), [9] Y. W. Wang and S. T. Chen, Non squareness B-convexity and flatness of Orlicz spaces, Comment. Math. Prace. Mat., 28 (988), [0] Y. Q. Yan, Some results on packing in Orlicz sequence spaces, Studia Math., 47() (200), Department of Mathematics, Suzhou University Suzhou, Jiangsu, 25006, P. R. China Recibido: 28 de Junio de 200 Revisado: de Marzo de