The Packing Spheres Constant for a Class of Separable Orlicz Function Spaces

Few results have been obtained on the packing spheres constant or exact formula for separable Orlicz function spaces (Yang, 2002, P.895-899, Ye, 1987, P.487-493). In this paper, by using the continuity of ideal space norm, we firstly proved that simple function class is dense in L∗ Φ function space. This is a necessary condition of interpolation theorem. Hence, the exact value of packing sphere for a class of sparable Orlicz function spaces (with two kinds of norm) is obtained. Secondly, for the space L∗ Φ [0, 1] discussed in (Yang, 2002, P.895-899), we propose the following conjecture: the L∗ Φ [0, 1] space is actually the Lp[0, 1] space, therefore, the results obtained there is actually the proved results in Lp space.


Preliminary material
It is well known that only finite number of sphere can be packed in a finite dimensional space if the spheres have the same radius but uncrossed, no matter how small is the radius.However, for infinite dimensional Banach space X, there exist a constant Λ(X) such that infinite number of disjoint spheres can be packed in a unit sphere B(X) if the radius less than Λ(X).Whereas, only finite number of disjoint sphere can be packed in sphere B(X) if the radius larger than Λ(X).This constant is referred to as packing sphere constant.From the 50s last century, researchers begin to investigate the packing spheres problem in Banach space.In 1970, Kottman (1970) finally determined the range of packing sphere value Λ(X) is [ 1 3 , 1 2 ] for general normed linear spaces.In 1932, Orlicz introduced Orlicz space, from then people begin to study the packing sphere problem for this specific Banach space (Ye, 1987, P.487-493).In (Rao, 1997, P.235-251, Wang, 1990, P.197-203, Ye, 1991, P.203-216, Han, 2002, P.1155-1158, Wang, 1987, P.508-513), the authors investigated the packing sphere problem for a class of separable Orlicz function spaces, and obtained the exact packing spheres value; moreover, they also proposed their points of view for the space L * Φ [0, 1] studied in (Yang, 2002, P.895-899).In this paper, by using the continuity of ideal space norm (Matin, 1997), we firstly proved that simple function class is dense in L * Φ function space.This is a necessary condition of interpolation theorem.For the functions satisfying the following conditions: From the above two points, we prove that the interpolation function Φ s (u) constructed from Φ(u) is N-function.By using this property, we can construct the interpolation inequality for subspace L * Φ , and hence obtain the exact value of packing spheres for Orlicz function subspace satisfying this properties.In the last section, we propose a conjecture for paper (Yang, 2002, P.895-899).
Subsequently, we outline some useful definitions and theorems.
Definition 1.1.The packing sphere constant Λ(X) for Banach space is defined by where B(X) is the unit ball in Banach space X.
¢ www.ccsenet.org/jmrISSN: 1916-9795 Definition 1.2.The Kottman constant for infinite dimensional Banach space X is defined by where S (X) is the unit spheres in Banach space X.
Property 1.7.Suppose Ψ(y) is a N-function, (Ω, Σ, μ) is finite complete nonnegative measurable space, then we can define measurable function set Definition 1.11.Suppose (T, Σ, μ) is a measurable space, and S is the set of measurable function defined in this space.Suppose X ⊂ S is a normed linear space of measurable function, then we call X a semi-ideal space, if ∀x ∈ X, y ∈ S that satisfy |y(s)| ≤ |x(s)|, we have y ∈ S , and ||y|| ≤ ||x|| where || • || is a norm defined in X.If X is complete, then we call X an ideal space.
Proposition 2.1.Suppose Φ is a N-function, for n ≥ 2 we have From Φ ∈ Δ 2 (∞), we have Otherwise, ∀u(t) ∈ L * Φ , suppose u(t) > 0, a.e.take the trunction function of u(t) as follows, Then we have u n (t) ↑ u(t).For nonnegative measurable function u n (t), there exist simple function series {ϕ n,k (t)} k such that ϕ n,k n (t) ↑ u n (t), a.e.(k → ∞).Subsequently, we will show that there exist a subseries of We know that there exist k n > 0 such that for almost all t we have 0 As From Property 1.13 and Property 1.14, we can get the above result.
Lemma 2.4.For function space L * Φ [0, 1] normalized by Luxemburg norm, its packing sphere value can be bounded by max{ We just need to prove the case with n = 2. From the definition of β (Φ) , we have Thus the result is true. (3).

Remarks.
(1).From the above process we can see that in the case of 1 ≤ p < 2, the condition lim u→∞ F(u) = p can be weaken as lim u→∞ sup F(u) = p.The results are still true.Of course the results are also true for the stronger condition that the right derivation of Φ(u), i.e. ϕ(u) is a N-function.