The Distance to 𝐿^∞ in the Grand Orlicz Spaces

Abstract

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

1. Introduction

Let Ω be a bounded open subset of ℝ^𝑛 and let 1<𝑝<∞. The grand 𝐿^𝑝-space, denoted by 𝐿^𝑝)=𝐿^𝑝)(Ω), consists of functions ℎ∈⋂_{0<𝜀≤𝑝−1}𝐿^𝑝−𝜀(Ω) such that ‖ℎ‖_𝑝)=sup_{0<𝜀<𝑝−1}𝜀_Ω|ℎ(𝑥)|^𝑝−𝜀𝑑𝑥^{1/(𝑝−𝜀)}<∞, (1)

where ⨍_Ω=(1/|Ω|)∫_Ω denotes the average over Ω. Note that ‖⋅‖_𝑝) is a norm and 𝐿^𝑝)(Ω) is a Banach space. This space was introduced by Iwaniec and Sbordone in connection with the integrability of the Jacobian [1], and it comes into play in a various number of problems (see, e.g., [2–15]).

It is worth pointing out that 𝐿^∞(Ω) is not a dense subspace of 𝐿^𝑝)(Ω) (see [9]); it is proved in [16] that the distance to 𝐿^∞ in 𝐿^𝑝) is given by dist_𝐿^𝑝)ℎ,𝐿^∞=limsup_𝜀→0+𝜀_Ω|ℎ(𝑥)|^𝑝−𝜀𝑑𝑥^{1/(𝑝−𝜀)}. (2)

A generalization of the grand Lebesgue space is the grand Orlicz space 𝐿^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_𝑡→∞Φ(𝑡)=∞. The Orlicz space 𝐿^Φ(Ω) associated with Φ consists of all measurable functions 𝑢∶Ω→ℝ for which there exists 𝜆>0 such that _ΩΦ|𝑢(𝑥)|𝜆𝑑𝑥<∞. (3)

Let us introduce the Luxemburg functional defined as ‖𝑢‖_𝐿^Φ_(Ω)=inf𝜆>0∶_ΩΦ|𝑢(𝑥)|𝜆𝑑𝑥≤1. (4)

Because of the monotonicity of Φ we have Φ((1−𝛼)𝑠+𝛼𝑡)≤[Φ(𝑠)+Φ(𝑡)]forevery𝑠,𝑡∈(0,∞),𝛼∈(0,1), (5)

and among Orlicz functions we will consider the ones satisfying the following condition: Φ(𝛼𝑡)≤𝐶(𝛼)Φ(𝑡)forevery𝑡∈(0,∞),𝛼∈(0,1), (6) for some constant 𝐶(𝛼) such that 𝐶(𝛼)→0 as 𝛼→0. This will be done in order to ensure that the functional in (4) is a quasinorm. In what follows, we will lose no generality in assuming that Φ(1)=1. (7)

Suppose that ^∞₁Φ(𝑡)𝑡^𝑛+1𝑑𝑡=∞, (8) and let 𝑁_Φ∶[0,1]→[0,∞) be the increasing weight defined as 𝑁_Φ(𝜎)=1∫^∞₁[Φ(𝑡)]^1/(1+𝜎)/𝑡^𝑛+1.𝑑𝑡 (9)

Following a definition given in [18], we suppose that 𝑁_Φ is tempered; that is, 𝑐₁𝑁_Φ(𝜎)≤𝑁_Φ(2𝜎)≤𝑐₂𝑁_Φ(𝜎)forevery𝜎∈0,𝜎₀,𝜎₀∈10,2 (10) for some 𝑐₁,𝑐₂>0.

An example of function Φ satisfying (6)–(10) is Φ(𝑡)=𝑡^𝑛(1+log(1+𝑡))^−𝛼 for 0≤𝛼≤1, and in this case 𝑁_Φ(𝜎)≈𝜎^1−𝛼 as 𝜎→0⁺ when 0≤𝛼<1 and 𝑁_Φ(𝜎)≈|log𝜎|⁻¹ as 𝜎→0⁺ when 𝛼=1 (see Section 5 for details).

The grand Orlicz space 𝐿^Φ)(Ω) consists of all measurable functions 𝑢∶Ω→ℝ for which there exists 𝜆>0 such that sup_0<𝜎<𝜎0𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡<∞, (11)

where 𝑢^∗ is the decreasing rearrangement of 𝑢𝑢^∗(𝑡)=infℎ≥0∶𝜇_𝑢(ℎ)≤𝑡forevery𝑡∈[0,|Ω|] (12) and 𝜇_𝑢 is the distribution function of 𝑢𝜇_𝑢(ℎ)=|{𝑥∈Ω∶|𝑢(𝑥)|>ℎ}|foreveryℎ≥0. (13)

The quasinorm denoted by ‖⋅‖_𝐿^Φ)_(Ω) is defined as follows: ‖𝑢‖_𝐿^Φ)_(Ω)=inf𝜆>0∶sup_0<𝜎<𝜎0𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1. (14)

We address that if we take Φ(𝑡)=𝑡^𝑛 also the grand Orlicz space 𝐿^Φ)(Ω) reduces to the grand Lebesgue space 𝐿^𝑛)(Ω) (see [17, Proposition 2.6], [6]).

Our main result provides a formula for the distance of a function 𝑢∈𝐿^Φ)(Ω) to 𝐿^∞(Ω), defined by dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)=inf_{𝜑∈𝐿∞(Ω)}𝑢−𝜑_𝐿Φ)(Ω). (15)

Theorem 1. Let Ω be a bounded open set of ℝ^𝑛. Assume that Φ∶[0,∞)→[0,∞) is an Orlicz function verifying (6)–(10). For every function 𝑢∈𝐿^Φ)(Ω), one has dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)=inf≤1𝜆>0∶limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1. (16)

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 𝐿^Φ) can be characterized as interpolation or extrapolation spaces. We also refer to [5, 16, 20–24] for the problem of finding formulae for the distance to a subspace in a given function space.

Theorem 1 gives, as byproduct, a characterization of the closure of 𝐿^∞(Ω) in 𝐿^Φ)(Ω) with respect to the norm‖⋅‖_𝐿^Φ)_(Ω), which will be denoted by 𝐿_𝑏^Φ)(Ω).

Theorem 2. A function 𝑢 belongs to 𝐿_𝑏^Φ)(Ω) if and only if lim_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛽𝑢^∗(𝑡)𝑑𝑡=0∀𝛽>0. (17)

For the special choice Φ(𝑡)=𝑡^𝑛, Theorem 1 also provides new formula for the distance to 𝐿^∞ in 𝐿^𝑛) (see Theorem 5).

2. The Main Result

We start this section recalling few basic properties of the decreasing rearrangement 𝑢^∗ of a measurable function 𝑢∶Ω→ℝ defined in a bounded open set Ω of ℝ^𝑛. We refer the reader to [25, Propositions 1.7and 1.8] for details.

Lemma 3. Let 𝑢,𝑣∶Ω→ℝ be measurable functions defined in a bounded open set Ω of ℝ^𝑛: (𝑢+𝑣)^∗𝑡₁+𝑡₂≤𝑢^∗𝑡₁+𝑣^∗𝑡₂𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡₁,𝑡₂≥0𝑤𝑖𝑡ℎ𝑡₁+𝑡₂≤|Ω|,(𝑐𝑣)^∗=𝑐𝑣^∗𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦c>0,𝑣^∗(0)=‖𝑣‖_𝐿^∞_(Ω)𝑖𝑓𝑣∈L^∞(Ω). (18)(19)(20)

We need a technical result providing a useful property of the quantity (𝑢)_𝐿^Φ)_(Ω)=inf≤1𝜆>0∶limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1. (21)

We recall that the goal of Theorem 1 consists in proving that (𝑢)_𝐿^Φ)_(Ω) is equal to dist_𝐿^Φ)_(Ω)(𝑢,𝐿^∞(Ω)). We notice that if 𝜑∈𝐿^∞(Ω), then (𝜑)_𝐿^Φ)_(Ω)=0, because from (8) and (10) it is 𝑁_Φ(𝜎)→0 as 𝜎→0⁺ and the average remains bounded for every 𝜎>0.

Lemma 4. Let Φ∶[0,∞)→[0,∞) be an Orlicz function satisfying the assumptions of Theorem 1. Assume that 𝑢∈𝐿^Φ)(Ω) and 𝜑∈𝐿^∞(Ω). Then (𝑢)_𝐿^Φ)_(Ω)=𝑢−𝜑_𝐿Φ)(Ω). (22)

Proof. Let 𝜆>(𝑢)_𝐿^Φ)_(Ω) and let 𝜀∈(0,𝜆) so that 𝜆>𝜆−𝜀>(𝑢)_𝐿^Φ)_(Ω). (23)
We use (18) with 𝑡₁=𝑡∈[0,|Ω|] and 𝑡₂=0 and (20), and we get 𝑢−𝜑^∗(𝑡)≤𝑢^∗(𝑡)+𝜑_𝐿∞(Ω). (24)
We use (5) to get Φ𝑢−𝜑^∗(𝑡)𝜆𝑢≤Φ^∗(𝑡)𝜆+𝜑_𝐿∞(Ω)𝜆=Φ𝜆−𝜀𝜆𝑢^∗(𝑡)+𝜀𝜆−𝜀𝜆𝜑_𝐿∞(Ω)𝜀𝑢≤Φ^∗(𝑡)𝜑𝜆−𝜀+Φ^𝐿∞(Ω)𝜀. (25)
We multiply by 𝑡^𝜎 and we integrate over [0,|Ω|] to get ₀^|Ω|𝑡^𝜎Φ𝑢−𝜑^∗(𝑡)𝜆𝑑𝑡≤₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜑𝜆−𝜀𝑑𝑡+Φ^𝐿∞(Ω)𝜀₀^|Ω|𝑡^𝜎=𝑑𝑡_0|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜑𝜆−𝜀𝑑𝑡+Φ^𝐿∞(Ω)𝜀|Ω|^𝜎.1+𝜎 (26)
We multiply by 𝑁_Φ(𝜎), and since 𝑁_Φ(𝜎)→0 as 𝜎→0⁺, we have limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢−𝜑^∗(𝑡)𝜆𝑑𝑡≤limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆−𝜀𝑑𝑡. (27)
From (23) we get limsup_𝜎→0+𝑁_Φ(𝜎)Φ₀^|Ω|𝑡^𝜎𝑢−𝜑^∗(𝑡)𝜆𝑑𝑡≤1. (28)
We apply the definition of (𝑢−𝜑)_𝐿Ω)(Φ), and we have 𝑢−𝜑_𝐿Φ)(Ω)≤𝜆, (29) and then, passing to the limit as 𝜆→(𝑢)_𝐿^Φ)_(Ω), we have 𝑢−𝜑_𝐿Φ)(Ω)≤(𝑢)_𝐿^Φ)_(Ω). (30)
By replacing 𝑢 with 𝑢−𝜑 and 𝜑 with −𝜑 in (30), we obtain the converse inequality (𝑢)_𝐿^Φ)_(Ω)≤𝑢−𝜑_𝐿Φ)(Ω). (31)
Equality (22) is finally proved.

Now, we are in a position to prove Theorem 1.

Proof of Theorem 1. From Lemma 4 we know that (𝑢)_𝐿^Φ)_(Ω)=𝑢−𝜑_𝐿Φ)(Ω), (32) for every 𝜑∈𝐿^∞(Ω). This clearly proves that dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)≥(𝑢)_𝐿^Φ)_(Ω) (33) since (𝑢)_𝐿^Φ)_(Ω)=(𝑢−𝜑)_𝐿Φ)(Ω)≤‖𝑢−𝜑‖_𝐿Φ)(Ω) for every 𝜑∈𝐿^∞(Ω).
Now, we want to show that dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)≤(𝑢)_𝐿^Φ)_(Ω). (34)
In order to achieve the claimed inequality, we prove that if 𝜆₀>(𝑢)_𝐿^Φ)_(Ω), (35) then dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)<𝜆₀. (36) Without loss of generality we may assume that 𝑢∉𝐿^∞(Ω). From (35) we find 𝜆∈(0,𝜆₀) such that limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1. (37) For each 𝜀>0 there exists 𝜎_𝜀∈(0,𝜎₀) such that 𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1+𝜀forevery𝜎∈0,𝜎_𝜀. (38) Let ℎ^∗>0 be such that 𝜇_𝑢ℎ^∗<1, (39) and let 𝜎∈(𝜎_𝜀,𝜎₀). From (38), we find some constant ℎ_𝜆 (depending on 𝜆), with ℎ_𝜆>ℎ^∗, such that 𝑁_Φ(1)|Ω|^{𝜇𝑢(ℎ𝜆)}₀𝑡^𝜎𝜀Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1. (40) Using the monotonicity of weight 𝑁_Φ, the fact that ℎ_𝜆>ℎ^∗, and (39), we deduce from (40) that 𝑁_Φ(𝜎)|Ω|^{𝜇𝑢(ℎ𝜆)}₀𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1forevery𝜎∈𝜎_𝜀,𝜎₀. (41)
We set 𝑢_𝜆(𝑥)=𝑢(𝑥) if |𝑢(𝑥)|≤ℎ_𝜆 and, 𝑢_𝜆(𝑥)=0 if |𝑢(𝑥)|>ℎ_𝜆, and we show that 𝑢−𝑢_𝜆^∗(𝑡)=𝑢^∗(𝑡)for𝑡∈0,𝜇ℎ_𝜆,𝑢−𝑢_𝜆^∗(𝑡)=0for𝑡∈𝜇ℎ_𝜆,|Ω|. (42)
Let us observe that 𝜇_{𝑢−𝑢𝜆}(ℎ)=𝜇_𝑢ℎ_𝜆if0<ℎ≤ℎ_𝜆, (43) while 𝜇_{𝑢−𝑢𝜆}(ℎ)=𝜇_𝑢(ℎ)ifℎ≥ℎ_𝜆. (44)
Using the fact that the distribution function is decreasing, we easily see that 𝜇_{𝑢−𝑢𝜆}(ℎ)≤𝜇_𝑢ℎ_𝜆foreveryℎ≥0. (45)
Therefore, if we let 𝑡∈[𝜇(ℎ_𝜆),|Ω|], we see that condition 𝜇_{𝑢−𝑢𝜆}(ℎ)≤𝑡 (46) is verified for all ℎ≥0. Thus (𝑢−𝑢_𝜆)^∗(𝑡)=0 for 𝑡∈[𝜇(ℎ_𝜆),|Ω|]. On the other hand, if we let 𝑡∈[0,𝜇(ℎ_𝜆)), we see that condition (46) is the same as requiring 𝜇_𝑢(ℎ)≤𝑡. (47)
Thus (𝑢−𝑢_𝜆)^∗(𝑡)=𝑢^∗(𝑡) holds if 𝑡∈[0,𝜇(ℎ_𝜆)), and (42) is proved.
It follows directly from (42) that 𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢−𝑢_𝜆^∗(𝑡)𝜆=𝑁𝑑𝑡_Φ(𝜎)|Ω|^{𝜇𝑢(ℎ𝜆)}₀𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡, (48) for every 𝜎∈(0,𝜎₀). Hence, we make use of (38) if 𝜎∈(0,𝜎_𝜀] and of (41) if 𝜎∈(𝜎_𝜀,𝜎₀) to conclude that 𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢−𝑢_𝜆^∗(𝑡)𝜆𝑑𝑡≤1+𝜀forevery𝜎∈0,𝜎₀. (49)
In particular, sup_0<𝜎<𝜎0𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢−𝑢_𝜆^∗(𝑡)𝜆𝑑𝑡≤1+𝜀. (50)
Since (50) holds for every 𝜀>0, we obtain that its left-hand side is smaller than 1, and therefore ‖𝑢−𝑢_𝜆‖_𝐿Φ)(Ω)≤𝜆. We get dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)≤𝑢−𝑢_{𝜆𝐿Φ)(Ω)}≤𝜆<𝜆₀. (51) Hence (36) is established. Since 𝜆₀ is any arbitrary number for which (35) holds, we may pass to the limit as 𝜆₀ approaches (𝑢)_𝐿^Φ)_(Ω) in (36) to get dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω)≤(𝑢)_𝐿^Φ)_(Ω). (52)
Combining (52) with (33) we obtain (16) as desired.

Proof of Theorem 2. As a consequence of Theorem 1, it is clear that 𝑢∈𝐿_𝑏^Φ)(Ω) if and only if limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡≤1∀𝜆>0. (53)
We fix an arbitrary 𝛼∈(0,1) and we set 𝜆=𝛼/𝛽. Using (6) we have 𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛽𝑢^∗(𝑡)𝑑𝑡=𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛼𝑢^∗(𝑡)𝜆(𝛼)𝑁𝑑𝑡≤𝐶_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡)𝜆𝑑𝑡. (54)
Hence, using (53) we have limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛽𝑢^∗(𝑡)𝑑𝑡≤𝐶(𝛼), (55)
and (17) follows since 𝐶(𝛼)→0 as 𝛼→0.

3. The Case of the Grand Lebesgue Space 𝐿^𝑛)

We denote by (𝑢)_𝐿^𝑛)_(Ω) the functional (𝑢)_𝐿^Φ)_(Ω) as in (21) when Φ(𝑡)=𝑡^𝑛. In this case, (𝑢)_𝐿^𝑛)_(Ω) takes the form (𝑢)_𝐿^𝑛)_(Ω)=limsup_𝜎→0+𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛. (56)

Our next result proves that the distance given by formula (2) reduces to (𝑢)_𝐿^𝑛)_(Ω).

Theorem 5. Let Ω be a bounded open set of ℝ^𝑛. For every function 𝑢∈𝐿^𝑛)(Ω), one has dist_𝐿^Φ)_(Ω)𝑢,𝐿^∞(Ω=)limsup_𝜎→0+𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛. (57)

Proof. First we prove that limsup_𝜀→0+𝜀_Ω|𝑢(𝑥)|^𝑛−𝜀𝑑𝑥^{1/(𝑛−𝜀)}≤limsup_𝜎→0+𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛. (58)
To this aim, we consider 𝜀,𝜎>0 and 𝑘>1 such that 𝑛𝑛−𝜀=.1+𝑘𝜎 (59)
Using Hölder's inequality we have 𝜀_Ω|𝑢(𝑥)|^𝑛−𝜀=𝑑𝑥𝑛𝑘𝜎1+𝑘𝜎₀^|Ω|𝑡^{𝜎/(1+𝑘𝜎)}𝑢^∗(𝑡)^{𝑛/(1+𝑘𝜎)}𝑡^{−𝜎/(1+𝑘𝜎)}≤𝑑𝑡𝑛𝑘𝜎1+𝑘𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^{1/(1+𝑘𝜎)}₀^|Ω|𝑡^−1/𝑘𝑑𝑡^{𝑘𝜎/(1+𝑘𝜎)} (60) which in turn implies that 𝜀_Ω|𝑢(𝑥)|^𝑛−𝜀𝑑𝑥^{1/(𝑛−𝜀)}≤𝑘(𝑘−1)|Ω|^{1/𝑘𝑘𝜎/𝑛}𝑛𝑘𝜎1+𝑘𝜎^{(1+𝑘𝜎)/𝑛}×₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛=𝑘(𝑘−1)|Ω|^{1/𝑘𝑘𝜎/𝑛}×𝑛𝑘𝜎1+𝑘𝜎^{(1+𝑘𝜎)/𝑛}1+𝜎𝑛𝜎^1/𝑛×𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛=𝑘(𝑘−1)|Ω|^{1/𝑘𝑘𝜎/𝑛}×𝑘1+𝑘𝜎^{(1+𝑘𝜎)/𝑛}(1+𝜎)^1/𝑛(𝑛𝜎)^{𝑘𝜎/𝑛}×𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛. (61) Since (𝑛𝜎)^{𝑘𝜎/𝑛}→1 as 𝜎→0⁺, we deduce from (61) that limsup_𝜀→0+𝜀_Ω|𝑢(𝑥)|^𝑛−𝜀𝑑𝑥^{1/(𝑛−𝜀)}≤𝑘^1/𝑛limsup_𝜎→0+𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛. (62)
Since 𝑘 is any number strictly greater than 1, (62) immediately implies (58).
We wish to prove the converse inequality limsup_𝜀→0+𝜀_Ω|𝑢(𝑥)|^𝑛−𝜀𝑑𝑥^{1/(𝑛−𝜀)}≥limsup_𝜎→0+𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛. (63)
For each 𝑡∈[0,|Ω|], we have _Ω|𝑢(𝑥)|^{𝑛/(1+𝜎)}𝑑𝑥=₀^|Ω|𝑢^∗(𝑠)^{𝑛/(1+𝜎)}≥1𝑑𝑠|Ω|^𝑡₀𝑢^∗(𝑠)^{𝑛/(1+𝜎)}≥1𝑑𝑠|Ω|𝑡𝑢^∗(𝑡)^{𝑛/(1+𝜎)}. (64)
Thus ₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡=₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^{𝑛𝜎/(1+𝜎)}𝑢^∗(𝑡)^{𝑛/(1+𝜎)}≤|Ω|𝑑𝑡_𝜎0|Ω|𝑢^∗(𝑡)^{𝑛/(1+𝜎)}𝑑𝑡^1+𝜎. (65)
We consider 𝜀,𝜎>0 such that 𝑛𝑛−𝜀=.1+𝜎 (66)
Then 𝑛𝜎1+𝜎₀^|Ω|𝑡^𝜎𝑢^∗(𝑡)^𝑛𝑑𝑡^1/𝑛≤𝑛𝜎1+𝜎^−𝜎|Ω|^𝜎𝑛𝜎1+𝜎₀^|Ω|𝑢^∗(𝑡)^{𝑛/(1+𝜎)}𝑑𝑡^1+𝜎1/𝑛=𝑛𝜎1+𝜎^−𝜎/𝑛|Ω|^𝜎/𝑛𝑛𝜎1+𝜎₀^|Ω|𝑢^∗(𝑡)^{𝑛/(1+𝜎)}𝑑𝑡^{(1+𝜎)/𝑛}=𝑛𝜎1+𝜎^−𝜎/𝑛|Ω|^𝜎/𝑛𝜀_Ω|𝑢(𝑥)|^𝑛−𝜀𝑑𝑥^{1/(𝑛−𝜀)}, (67) which proves (63).

4. Few Properties of the Distance

In this concluding section we provide certain properties of the functional (⋅)_𝐿^Φ)_(Ω).

Lemma 6. Let Φ∶[0,∞)→[0,∞) be an Orlicz function satisfying the assumptions of Theorem 1 and let 𝑣∈𝐿^Φ)(Ω). Assume that limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑣^∗(𝑡)𝜆𝑑𝑡≤𝐿, (68)for some constants positive 𝜆 and 𝐿. Then, there exists a positive constant 𝐶₀ depending only on 𝐿 such that (𝑣)_𝐿^Φ)_(Ω)≤𝐶₀𝜆. (69)

Proof. Let 𝐶(𝛼) be the constant appearing in (6). We may take 𝛼₀∈(0,1) such that 𝐶𝛼₀𝐿≤1 (70) since 𝐶(𝛼)→0 as 𝛼→0. We use (6) to get 𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛼₀𝑣^∗(𝑡)𝜆𝛼𝑑𝑡≤𝐶₀𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝑣^∗(𝑡)𝜆𝑑𝑡. (71)
We take the limsup as 𝜎→0⁺ and use (68) to get limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛼₀𝑣^∗(𝑡)𝜆𝑑𝑡≤𝐶𝛼₀𝐿. (72)
Therefore, from (72) and (70) we have limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ𝛼₀𝑣^∗(𝑡)𝜆𝑑𝑡≤1. (73)
The desired constant 𝐶₀ is obtained by setting 𝐶₀=1/𝛼₀. We address that 𝐶₀ is independent of 𝑣, and thus the proof is completed.

Remark 7. It is clear from the definition of (𝑢)_𝐿^Φ)_(Ω) that we can pick 𝐶₀=1 if 𝐿=1.

Our next lemma provides a sort of triangle inequality involving the functional (⋅)_𝐿^Φ)_(Ω).

Lemma 8. Let Φ∶[0,∞)→[0,∞) be an Orlicz function satisfying the assumptions of Theorem 1 and let 𝑢,𝑣∈𝐿^Φ)(Ω). Then, there exists a constant 𝐶₁ depending only on Φ such that (𝑢+𝑣)_𝐿^Φ)_(Ω)≤𝐶₁(𝑢)_𝐿^Φ)_(Ω)+(𝑣)_𝐿^Φ)_(Ω). (74)

Proof. Take 𝜆₁>(𝑢)_𝐿^Φ)_(Ω),𝜆₂>(𝑣)_𝐿^Φ)_(Ω). (75)
Let 𝑡∈[0,|Ω|]. We use (18) with 𝑡₁=𝑡₂=𝑡/2, the monotonicity of Φ, to obtain Φ(𝑢+𝑣)^∗(𝑡)𝜆₁+𝜆₂𝜆≤Φ¹𝜆₁+𝜆₂𝑢^∗(𝑡/2)𝜆₁+𝜆₂𝜆₁+𝜆₂𝑣^∗(𝑡/2)𝜆₂≤Φ𝑢^∗(𝑡/2)𝜆₁𝑣+Φ^∗(𝑡/2)𝜆₂. (76)
Fix 𝜎∈(0,1). We multiply by 𝑡^𝜎 and we integrate over [0,|Ω|] to get ₀^|Ω|𝑡^𝜎Φ(𝑢+𝑣)^∗(𝑡)𝜆₁+𝜆₂𝑑𝑡≤₀^|Ω|𝑡^𝜎Φ𝑢^∗(𝑡/2)𝜆₁+𝑑𝑡_0|Ω|𝑡^𝜎Φ𝑣^∗(𝑡/2)𝜆₂𝑑𝑡. (77)
With the aid of two changes of variables in the integrals appearing at the right-hand side of (77) we have ₀^|Ω|𝑡^𝜎Φ(𝑢+𝑣)^∗(𝑡)𝜆₁+𝜆₂𝑑𝑡≤2^𝜎+1₀^|Ω|/2𝑠^𝜎Φ𝑢^∗(𝑠)𝜆₁+𝑑𝑠_0|Ω|/2𝑟^𝜎Φ𝑣^∗(𝑟)𝜆₂𝑑𝑟, (78)
which in turn implies ₀^|Ω|𝑡^𝜎Φ(𝑢+𝑣)^∗(𝑡)𝜆₁+𝜆₂𝑑𝑡≤4₀^|Ω|𝑠^𝜎Φ𝑢^∗(𝑠)𝜆₁𝑑𝑠+₀^|Ω|𝑟^𝜎Φ𝑣^∗(𝑟)𝜆₂𝑑𝑟. (79)
We multiply both sides of (79) by 𝑁_Φ(𝜎), and we take the limsup as 𝜎→0⁺ and use (75) to get limsup_𝜎→0+𝑁_Φ(𝜎)₀^|Ω|𝑡^𝜎Φ(𝑢+𝑣)^∗(𝑡)𝜆₁+𝜆₂𝑑𝑡≤8. (80)
We appeal to Lemma 6 to conclude that there exists a constant 𝐶₁ such that (𝑢+𝑣)_𝐿^Φ)_(Ω)≤𝐶₁𝜆₁+𝜆₂. (81)
Finally, (74) follows letting 𝜆₁→(𝑢)_𝐿^Φ)_(Ω) and 𝜆₂→(𝑣)_𝐿^Φ)_(Ω), respectively.

5. An Example

In this section we study the behaviour of weight 𝑁_Φ(𝜎) as 𝜎→0⁺ when Φ(𝑡)=𝑡^𝑛(1+log(1+𝑡))^−𝛼 with 0≤𝛼≤1. We follow closely the lines of Example 3.6 in [17].

Example 9. Let Φ(𝑡)=𝑡^𝑛(1+log(1+𝑡))^−𝛼 and let 0≤𝛼≤1. We start by proving that 𝑁_Φ(𝜎)≈𝜎^1−𝛼as𝜎→0⁺when0≤𝛼<1. (82)
To see this, let 𝜃>0. Then ^∞₁Φ(𝑡)^1−𝜃𝑡^𝑛+1𝑑𝑡≈^∞₁𝑡^{−𝑛𝜃−1}1+log𝑡^{−𝛼(1−𝜃)}≈𝑑𝑡_∞1𝑒^{−𝑛𝜃𝑢}𝑢^{−𝛼(1−𝜃)}≈𝑑𝑢_∞𝜃𝑒^−𝑛𝑣𝑣𝜃^{−𝛼(1−𝜃)}𝑑𝑣𝜃≈𝜃^𝛼−1∞_𝜃𝑒^−𝑛𝑣𝑣^{−𝛼(1−𝜃)}𝑑𝑣≈𝜃^𝛼−11+¹_𝜃𝑣^{−𝛼(1−𝜃)}𝑑𝑣. (83)
We pick 𝜃=𝜎/(1+𝜎) in such a way that 𝑁_Φ(𝜎)=1∫^∞₁[Φ(𝑡)]^1/(1+𝜎)/𝑡^𝑛+1𝑑𝑡≈𝜎^1−𝛼. (84)
A similar argument leads to 𝑁_Φ(𝜎)≈log𝜎⁻¹as𝜎→0⁺when𝛼=1. (85)

Acknowledgment

The research of the first author has been supported by the 2008 ERC Advanced Grant 226234 “Analytic Techniques for Geometric and Functional Inequalities.”