The Distance to L ∞ in the Grand Orlicz Spaces

We establish a formula for the distance to from the grand Orlicz space introduced in Capone et al. (2008). A new formula for the distance to from the grand Lebesgue space introduced in Iwaniec and Sbordone (1992) is also provided.

It is worth pointing out that  ∞ (Ω) is not a dense subspace of  ) (Ω) (see [9]); it is proved in [16] that the distance to  ∞ in  ) is given by dist  ) (ℎ,  ∞ ) = lim sup A generalization of the grand Lebesgue space is the grand Orlicz space  Φ) (Ω), introduced by Capone et al. in [17].Let us recall that Φ : [0, ∞) → [0, ∞) is called an Orlicz function if it is continuous, strictly increasing, and satisfies Φ(0) = 0 and lim  → ∞ Φ() = ∞.The Orlicz space  Φ (Ω) associated with Φ consists of all measurable functions  : Ω → R for which there exists  > 0 such that Let us introduce the Luxemburg functional defined as Because of the monotonicity of Φ we have
For every function  ∈  Φ) (Ω), one has Our theorem is in the framework of the results of paper [19], which cannot be directly applied to our context, without a preliminary check that the grand Orlicz spaces  Φ) can be characterized as interpolation or extrapolation spaces.We also refer to [5,16,[20][21][22][23][24] for the problem of finding formulae for the distance to a subspace in a given function space.
Theorem 2. A function  belongs to  Φ)  (Ω) if and only if For the special choice Φ() =   , Theorem 1 also provides new formula for the distance to  ∞ in  ) (see Theorem 5).

The Main Result
We start this section recalling few basic properties of the decreasing rearrangement  * of a measurable function  : Ω → R defined in a bounded open set Ω of R  .We refer the reader to [25, Propositions 1.7 and 1.8] for details.
We need a technical result providing a useful property of the quantity We recall that the goal of Theorem 1 consists in proving that ( 8) and ( 10) it is  Φ () → 0 as  → 0 + and the average remains bounded for every  > 0.
Proof of Theorem 1. From Lemma 4 we know that for every  ∈  ∞ (Ω).This clearly proves that In order to achieve the claimed inequality, we prove that if then Without loss of generality we may assume that  ∉  ∞ (Ω).
Let us observe that while Using the fact that the distribution function is decreasing, we easily see that for every  ∈ (0,  0 ).Hence, we make use of (38) if  ∈ (0,   ] and of (41 In particular, sup Since (50) holds for every  > 0, we obtain that its lefthand side is smaller than 1, and therefore ‖ −   ‖  Φ) (Ω) ≤ .We get Hence (36) is established.Since  0 is any arbitrary number for which (35) holds, we may pass to the limit as  0 approaches Combining ( 52) with (33) we obtain ( 16) as desired.

Theorem 5.
Let Ω be a bounded open set of R  .For every function  ∈  ) (Ω), one has Proof.First we prove that lim sup ≤ lim sup To this aim, we consider ,  > 0 and  > 1 such that Using Hölder's inequality we have which in turn implies that Since () / → 1 as  → 0 + , we deduce from (61) that lim sup Since  is any number strictly greater than 1, (62) immediately implies (58).We wish to prove the converse inequality lim sup which proves (63).

Few Properties of the Distance
In this concluding section we provide certain properties of the functional (⋅)  Φ) (Ω) .
Lemma 6.Let Φ : [0, ∞) → [0, ∞) be an Orlicz function satisfying the assumptions of Theorem 1 and let V ∈  Φ) (Ω).Assume that lim sup for some constants positive  and .Then, there exists a positive constant  0 depending only on  such that Proof.Let () be the constant appearing in (6).We may take  0 ∈ (0, 1) such that since () → 0 as  → 0. We use (6) to get We take the lim sup as  → 0 + and use (68) to get lim sup Therefore, from (72) and (70) we have lim sup The desired constant  0 is obtained by setting  0 = 1/ 0 .We address that  0 is independent of V, and thus the proof is completed.
We follow closely the lines of Example 3.6 in [17].