We study noneffective weights in the framework of variable exponent Lebesgue spaces, and we show that 𝐿𝑝()(Ω)=𝐿𝜔𝑝()(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 𝐿𝑝, 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 𝐿𝑝 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 𝐿𝑝 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 𝐿Φ(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., 𝐿Φ(𝑤)=𝐿Φ) if and only if 𝑤constant; 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 Ω𝑛, 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={𝑥Ω𝑝(𝑥)=}, and for any 𝐸Ω, let 𝑝(𝐸)=essinf𝑥𝐸𝑝(𝑥),𝑝+(𝐸)=esssup𝑥𝐸𝑝(𝑥).(1.1)

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

Given a weight 𝑤 in Ω (i.e., 𝑤Ω]0,[, 𝑤𝐿1loc(Ω)), we define the modular functional𝜌𝑤(𝑓)=ΩΩ||||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 |ΩΩ|=0, then 𝜌𝑤(𝑓)=𝑓𝑤𝐿(Ω). We define the space 𝐿𝑤𝑝()(Ω) to be the set of measurable functions such that, for some 𝜆>0,𝜌𝑤(𝑓/𝜆)<. This is a Banach function space when equipped with the norm𝑓𝑝(),𝑤=inf𝜆>0𝜌𝑤𝑓𝜆1.(1.3)

When 𝑝()=𝑝, a constant, then 𝐿𝑤𝑝()(Ω)=𝐿𝑝𝑤(Ω) with equality of norms. When 𝑤1, we will denote by 𝜌() the modular functional, 𝑓𝑝() the norm, and 𝐿𝑝()(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 Ω𝑛, 𝑝1(), 𝑝2() be exponents in 𝒫(Ω), (𝑝1)+<, (𝑝2)+<, 𝑤1, 𝑤2, weights. The continuous embedding 𝐿𝑝1𝑤()1(Ω)𝐿𝑝2𝑤()2(Ω) holds if and only if there exist positive constants 𝐾1, 𝐾2, and 𝐿1(Ω), such that 𝑡𝑝2(𝑥)𝑤2(𝑥)𝐾1𝐾2𝑡𝑝1()𝑤1(𝑥)+(𝑥)fora.a.𝑥Ωandall𝑡>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 Ω𝑛, 𝑝()𝒫(Ω), and let 𝑤 be weight. If 𝑝+<, then 𝐿𝑝()(Ω)=𝐿𝑤𝑝()(Ω) if and only if 𝑤constant.

Proof. If 𝑤constant, then obviously 𝐿𝑝()(Ω)=𝐿𝑤𝑝()(Ω), therefore we need to prove only the converse implication.
Since, in particular, 𝐿𝑤𝑝()(Ω)𝐿𝑝()(Ω), then by Theorem 2.1 and the boundedness of 𝑝() there exist 𝑐1>0 and 1𝐿1(Ω) such that 𝑡𝑝(𝑥)𝑐1𝑡𝑝()𝑤(𝑥)+1(𝑥)fora.a.𝑥Ωandall𝑡>0,(2.2) that is, 1𝑐1𝑤(𝑥)+1(𝑥)𝑡𝑝(𝑥)fora.a.𝑥Ωandall𝑡>0,(2.3) from which, letting 𝑡, 1𝑐1𝑤(𝑥)fora.a.𝑥Ω.(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 Ω𝑛, and let 𝑤 be weight. It is 𝐿(Ω)=𝐿𝑤(Ω) if and only if 𝑤constant.

Proof. As above, we show only that, if 𝐿𝑤(Ω)=𝐿(Ω), then 𝑤constant. Arguing by contradiction, it is 𝑤𝐿(Ω) or 𝑤1𝐿(Ω). In the first case, set 𝐸𝑛=||||{𝑥Ω𝑤(𝑥)>𝑛},𝑛,(2.5) so that |𝐸𝑛|>0 for all 𝑛. Testing the inequality 𝑓𝑤𝑐2𝑓 with 𝑓=𝜒𝐸𝑛, we get 𝑛<𝑤𝐿(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+(ΩΩ)< is in fact optimal.

Theorem 2.4. Let Ω𝑛, 𝑝()𝒫(Ω), and let 𝑤 be weight. If 𝑝+(ΩΩ)<, then 𝐿𝑝()(Ω)=𝐿𝑤𝑝()(Ω) if and only if 𝑤constant.

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

Theorem 2.5. Let Ω𝑛, 𝑝()𝒫(Ω), and let 𝑤 be weight. It is 𝐿𝑝()(Ω)=𝐿𝑤𝑝()(Ω) if and only if 𝑤(𝑥)1/𝑝(𝑥)constantfora.a.𝑥ΩΩ,𝑤constantfora.a.𝑥Ω.(2.6)

Proof. If |ΩΩ|=0, then the theorem follows from Proposition 2.3, therefore we may assume that |ΩΩ|>0. We now argue as in the proof of Theorem 2.2. Since 𝐿𝑤𝑝()(Ω)𝐿𝑝()(Ω), it is also 𝐿𝑤𝑝()(ΩΩ)𝐿𝑝()(ΩΩ), and therefore, applying Theorem 2.1, there exist positive constants 𝐾1, 𝐾2, and 1𝐿1(ΩΩ), such that 𝑡𝑝(𝑥)𝐾1(𝐾2𝑡)𝑝()𝑤(𝑥)+1(𝑥)fora.a.𝑥ΩΩandall𝑡>0,(2.7) and arguing similarly as before we get 𝐾11𝐾2𝑝(𝑥)𝑤(𝑥), from which we get the existence of a constant 𝐾0 such that 𝐾0𝑝(𝑥)1𝑤(𝑥). Starting from the opposite inclusion, we get the existence of a constant 𝐾3 such that 𝑤(𝑥)𝐾3𝑝(𝑥)+1. In conclusion, 𝑤(𝑥)1/(𝑝(𝑥)+1)constantfora.a.𝑥ΩΩ,(2.8) but this is equivalent to say that 𝑤(𝑥)1/𝑝(𝑥)constantfora.a.𝑥ΩΩ(2.9) because 𝑝(𝑥)𝑝(𝑥)+1constantfora.a.𝑥ΩΩ.(2.10) If |Ω|>0, the second part of the statement, about Ω, follows analogously, from the fact that 𝐿𝑤𝑝()(Ω)=𝐿𝑝()(Ω) and then using Proposition 2.3.Vice versa, if (2.6) is true, the functional 𝜌𝑤(𝑓)=ΩΩ||𝑓(𝑥)𝑤(𝑥)1/𝑝(𝑥)||𝑝(𝑥)𝑑𝑥+𝑓𝑤𝐿(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 𝐿𝑝()(Ω) and 𝐿𝑤𝑝()(Ω), and therefore the theorem is proved.

Example 2.6. Let Ω=(0,1), 𝑝(𝑥)=1/𝑥, 𝑤(𝑥)=21/𝑥. Then 𝑤 is unbounded, and, since 𝑤(𝑥)1/𝑝(𝑥)=2, it is also noneffective: 𝐿𝑤𝑝()(0,1)=L𝑝()(0,1).(2.12) This can be checked also directly, in fact 𝜌𝑤(𝑓)=10||||𝑓(𝑥)𝑝(𝑥)𝑤(𝑥)𝑑𝑥=10||||𝑓(𝑥)1/𝑥21/𝑥𝑑𝑥=𝜌(2𝑓),(2.13) and therefore 𝑓𝑝(),𝑤=2𝑓𝑝().


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.