A Note on Noneffective Weights in Variable Lebesgue Spaces

Abstract

We study noneffective weights in the framework of variable exponent Lebesgue spaces, and we show that 𝐿^𝑝(⋅)(Ω)=𝐿^{𝜔𝑝(⋅)}(Ω) if and only if 𝜔(𝑥)^1/𝑝(𝑥)∼constant in the set where 𝑝(⋅)<∞, and 𝜔(𝑥)∼constant in the set where 𝑝(⋅)=∞.

1. Introduction

The variable Lebesgue spaces generalize the classical Lebesgue spaces 𝐿_𝑝, where the constant exponent 𝑝 is replaced by a function 𝑝(⋅). They fall within the scope of Musielak-Orlicz spaces (see [1]) hence the general theory implies their basic properties. Nevertheless, they have appeared as an individual subject of study, also in other areas in recent years. One sees their formal similarity with the standard 𝐿_𝑝 spaces hence natural questions arise about their various common properties; further, interesting applications of them have been found in mathematical modelling of some physical processes. All this has triggered study of the variable exponent Lebesgue spaces oriented and related to the theory of integrable and weakly differentiable functions. Note in passing that at the very beginning one hits on a quite unpleasant fact that standard 𝐿_𝑝 techniques cannot be applied here in a straightforward manner; for example, the shift operator is never bounded, with the only exception when the exponent is constant. For more information we refer the reader to the surveys by Diening et al. [2], and Samko [3].

In this note we study the noneffective weights in the framework of variable exponent Lebesgue spaces. Our point of departure is the paper by Hudzik and Krbec [4], where the noneffective weights have been introduced and explored in the framework of Orlicz spaces 𝐿_Φ(𝑤). It has been proved that (we omit here notation and details) if a Young function Φ satisfies the Δ₂ condition at infinity, a weight 𝑤 is noneffective (i.e., 𝐿_Φ(𝑤)=𝐿_Φ) if and only if 𝑤∼constant; moreover, the fact that noneffective weights are exactly the trivial ones (i.e., the weights equivalent to a constant) may happen also if the Δ₂ condition at infinity is not satisfied, and a necessary and sufficient condition on Φ for the nonexistence of essentially unbounded noneffective weights is given.

To state our results, we first give some basic definitions. For information on the basic properties of the variable exponent Lebesgue spaces, see the recent book by Diening et al. [5], or the pioneering papers [6, 7].

Given a set Ω⊂ℝ^𝑛, an exponent function is a measurable function 𝑝(⋅)∶Ω→[1,∞]. We denote the set of all such functions by 𝒫(Ω). Given 𝑝(⋅)∈𝒫(Ω), let Ω_∞={𝑥∈Ω∶𝑝(𝑥)=∞}, and for any 𝐸⊂Ω, let 𝑝₋(𝐸)=essinf_𝑥∈𝐸𝑝(𝑥),𝑝₊(𝐸)=esssup_𝑥∈𝐸𝑝(𝑥).(1.1)

For brevity, we denote 𝑝₋=𝑝₋(Ω) and 𝑝₊=𝑝₊(Ω).

Given a weight 𝑤 in Ω (i.e., 𝑤∶Ω→]0,∞[, 𝑤∈𝐿^1loc(Ω)), we define the modular functional𝜌_𝑤(𝑓)=_Ω⧵Ω∞||𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥+‖𝑓(𝑤)‖_𝐿^∞_(Ω∞).(1.2)

If |Ω_∞|=0, then we set the last term equal to 0; if |Ω⧵Ω_∞|=0, then 𝜌_𝑤(𝑓)=‖𝑓𝑤‖_𝐿^∞_(Ω∞). We define the space 𝐿^{𝑤𝑝(⋅)}(Ω) to be the set of measurable functions such that, for some 𝜆>0,𝜌_𝑤(𝑓/𝜆)<∞. This is a Banach function space when equipped with the norm‖𝑓‖_{𝑝(⋅),𝑤}=inf𝜆>0∶𝜌_𝑤𝑓𝜆≤1.(1.3)

When 𝑝(⋅)=𝑝, a constant, then 𝐿^{𝑤𝑝(⋅)}(Ω)=𝐿^𝑝𝑤(Ω) with equality of norms. When 𝑤≡1, we will denote by 𝜌(⋅) the modular functional, ‖𝑓‖_𝑝(⋅) the norm, and 𝐿^𝑝(⋅)(Ω) the space.

2. Noneffective Weights for Variable Exponent Lebesgue Spaces

We begin by stating the following general embedding theorem, which is a consequence of a classical result by Ishii [8].

Theorem 2.1. Let Ω⊂ℝ^𝑛, 𝑝₁(⋅), 𝑝₂(⋅) be exponents in 𝒫(Ω), (𝑝₁)₊<∞, (𝑝₂)₊<∞, 𝑤₁, 𝑤₂, weights. The continuous embedding 𝐿^{𝑝1𝑤(⋅)}₁(Ω)⊂𝐿^{𝑝2𝑤(⋅)}₂(Ω) holds if and only if there exist positive constants 𝐾₁, 𝐾₂, and ℎ∈𝐿¹(Ω), such that 𝑡^𝑝2(𝑥)𝑤₂(𝑥)≤𝐾₁𝐾₂𝑡^𝑝1(⋅)𝑤₁(𝑥)+ℎ(𝑥)fora.a.𝑥∈Ωandall𝑡>0.(2.1)

At first we consider the case 𝑝₊<∞, where it is possible to show that the noneffective weights must be the trivial ones.

Theorem 2.2. Let Ω⊂ℝ^𝑛, 𝑝(⋅)∈𝒫(Ω), and let 𝑤 be weight. If 𝑝₊<∞, then 𝐿^𝑝(⋅)(Ω)=𝐿^{𝑤𝑝(⋅)}(Ω) if and only if 𝑤∼constant.

Proof. If 𝑤∼constant, then obviously 𝐿^𝑝(⋅)(Ω)=𝐿^{𝑤𝑝(⋅)}(Ω), therefore we need to prove only the converse implication.
Since, in particular, 𝐿^{𝑤𝑝(⋅)}(Ω)⊂𝐿^𝑝(⋅)(Ω), then by Theorem 2.1 and the boundedness of 𝑝(⋅) there exist 𝑐₁>0 and ℎ₁∈𝐿¹(Ω) such that 𝑡^𝑝(𝑥)≤𝑐₁𝑡^𝑝(⋅)𝑤(𝑥)+ℎ₁(𝑥)fora.a.𝑥∈Ωandall𝑡>0,(2.2) that is, 1≤𝑐₁𝑤(𝑥)+ℎ¹(𝑥)𝑡^𝑝(𝑥)fora.a.𝑥∈Ωandall𝑡>0,(2.3) from which, letting 𝑡→∞, 1𝑐₁≤𝑤(𝑥)fora.a.𝑥∈Ω.(2.4) The upper bound follows, in the same way, from the opposite inclusion.

We consider now the case of unbounded exponents.

Let us first remark that, if 𝑝≡∞, as in the previous case, all noneffective weights must be trivial.

Proposition 2.3. Let Ω⊂ℝ^𝑛, and let 𝑤 be weight. It is 𝐿^∞(Ω)=𝐿^∞𝑤(Ω) if and only if 𝑤∼constant.

Proof. As above, we show only that, if 𝐿^∞𝑤(Ω)=𝐿^∞(Ω), then 𝑤∼constant. Arguing by contradiction, it is 𝑤∉𝐿^∞(Ω) or 𝑤⁻¹∉𝐿^∞(Ω). In the first case, set 𝐸_𝑛=||{𝑥∈Ω∶𝑤(𝑥)>𝑛}||,𝑛∈ℕ,(2.5) so that |𝐸_𝑛|>0 for all 𝑛. Testing the inequality ‖𝑓𝑤‖_∞≤𝑐₂‖𝑓‖_∞ with 𝑓=𝜒_𝐸𝑛, we get 𝑛<‖𝑤‖_𝐿^∞_(𝐸𝑛)≤𝑐₂ for all n, which is absurd. In the second case we argue similarly, and the proposition is proved.

Proposition 2.3 is a special case of the following more general result, which in fact includes also Theorem 2.2. We will see (Theorem 2.5) that the assumption 𝑝₊(Ω⧵Ω_∞)<∞ is in fact optimal.

Theorem 2.4. Let Ω⊂ℝ^𝑛, 𝑝(⋅)∈𝒫(Ω), and let 𝑤 be weight. If 𝑝₊(Ω⧵Ω_∞)<∞, then 𝐿^𝑝(⋅)(Ω)=𝐿^{𝑤𝑝(⋅)}(Ω) if and only if 𝑤∼constant.

Theorem 2.4 is a consequence of the following result, where in Ω⧵Ω_∞ (possibly empty) an unbounded exponent is allowed.

Theorem 2.5. Let Ω⊂ℝ^𝑛, 𝑝(⋅)∈𝒫(Ω), and let 𝑤 be weight. It is 𝐿^𝑝(⋅)(Ω)=𝐿^{𝑤𝑝(⋅)}(Ω) if and only if 𝑤(𝑥)^1/𝑝(𝑥)∼constantfora.a.𝑥∈Ω⧵Ω_∞,𝑤∼constantfora.a.𝑥∈Ω_∞.(2.6)

Proof. If |Ω⧵Ω_∞|=0, then the theorem follows from Proposition 2.3, therefore we may assume that |Ω⧵Ω_∞|>0. We now argue as in the proof of Theorem 2.2. Since 𝐿^{𝑤𝑝(⋅)}(Ω)⊂𝐿^𝑝(⋅)(Ω), it is also 𝐿^{𝑤𝑝(⋅)}(Ω⧵Ω_∞)⊂𝐿^𝑝(⋅)(Ω⧵Ω_∞), and therefore, applying Theorem 2.1, there exist positive constants 𝐾₁, 𝐾₂, and ℎ₁∈𝐿¹(Ω⧵Ω_∞), such that 𝑡^𝑝(𝑥)≤𝐾₁(𝐾₂𝑡)^𝑝(⋅)𝑤(𝑥)+ℎ₁(𝑥)fora.a.𝑥∈Ω⧵Ω_∞andall𝑡>0,(2.7) and arguing similarly as before we get 𝐾¹⁻¹𝐾^{2−𝑝(𝑥)}≤𝑤(𝑥), from which we get the existence of a constant 𝐾₀ such that 𝐾^{0−𝑝(𝑥)−1}≤𝑤(𝑥). Starting from the opposite inclusion, we get the existence of a constant 𝐾₃ such that 𝑤(𝑥)≤𝐾^{3𝑝(𝑥)+1}. In conclusion, 𝑤(𝑥)^{1/(𝑝(𝑥)+1)}∼constantfora.a.𝑥∈Ω⧵Ω_∞,(2.8) but this is equivalent to say that 𝑤(𝑥)^1/𝑝(𝑥)∼constantfora.a.𝑥∈Ω⧵Ω_∞(2.9) because 𝑝(𝑥)𝑝(𝑥)+1∼constantfora.a.𝑥∈Ω⧵Ω^∞.(2.10) If |Ω_∞|>0, the second part of the statement, about Ω_∞, follows analogously, from the fact that 𝐿^{𝑤𝑝(⋅)}(Ω_∞)=𝐿^𝑝(⋅)(Ω_∞) and then using Proposition 2.3.Vice versa, if (2.6) is true, the functional 𝜌_𝑤(𝑓)=∫_Ω⧵Ω∞||𝑓(𝑥)𝑤(𝑥)^1/𝑝(𝑥)||^𝑝(𝑥)𝑑𝑥+‖𝑓𝑤‖_𝐿^∞_(Ω∞)(2.11) can be majorized and minorized with 𝜌(𝜆𝑓), where 𝜆>0 comes from assumption (2.6), and this implies the equivalence of the norms of 𝐿^𝑝(⋅)(Ω) and 𝐿^{𝑤𝑝(⋅)}(Ω), and therefore the theorem is proved.

Example 2.6. Let Ω=(0,1)⊂ℝ, 𝑝(𝑥)=1/𝑥, 𝑤(𝑥)=2^1/𝑥. Then 𝑤 is unbounded, and, since 𝑤(𝑥)^1/𝑝(𝑥)=2, it is also noneffective: 𝐿^{𝑤𝑝(⋅)}(0,1)=L^𝑝(⋅)(0,1).(2.12) This can be checked also directly, in fact 𝜌_𝑤(𝑓)=¹⁰||𝑓(𝑥)||^𝑝(𝑥)𝑤(𝑥)𝑑𝑥=¹⁰||𝑓(𝑥)||^1/𝑥2^1/𝑥𝑑𝑥=𝜌(2𝑓),(2.13) and therefore ‖𝑓‖_{𝑝(⋅),𝑤}=2‖𝑓‖_𝑝(⋅).

Acknowledgments

The research of the second author has been carried out in the framework of the Institutional Research Plan no. AV0Z10190503 of the Institute of Mathematics, Academy of Sciences of the Czech Republic. This author also gratefully acknowledges support of the Grant Agency of the Czech Republic under no. 201/10/1920 and of the Nečas Center for Mathematical. Modeling, LC 06052.