A Note on Noneffective Weights in Variable Lebesgue Spaces

1 Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università di Napoli Federico II, Via Monteoliveto, 3, 80134 Napoli, Italy 2 Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sezione di Napoli, Consiglio Nazionale delle Ricerche, via Pietro Castellino, 111, 80131 Napoli, Italy 3 Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitnà 25, 115 67 Prague 1, Czech Republic


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 Δ 2 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 Δ 2 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 For brevity, we denote p − p − Ω and p p Ω .Given a weight w in Ω i.e., w : Ω → 0, ∞ , w ∈ L 1  loc Ω , we define the modular functional If |Ω ∞ | 0, then we set the last term equal to 0 w 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 w 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.

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 .
w 2 Ω holds if and only if there exist positive constants K 1 , K 2 , and h ∈ L 1 Ω , such that

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.
Proof.As above, we show only that, if In the first case, set 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 is a consequence of the following result, where in Ω \ Ω ∞ possibly empty an unbounded exponent is allowed.
and therefore, applying Theorem 2.1, there exist positive constants K 1 , K 2 , and for a.a.x ∈ Ω \ Ω ∞ and all t > 0, 2.7 and arguing similarly as before we get K ≤ w x , from which we get the existence of a constant K 0 such that K −p x −1 0 ≤ w x .Starting from the opposite inclusion, we get the existence of a constant K 3 such that w x ≤ K p x 1 3 . In conclusion, but this is equivalent to say that w x w Ω , and therefore the theorem is proved.

2.12
This can be checked also directly, in fact At first we consider the case p < ∞, where it is possible to show that the noneffective weights must be the trivial ones.L p • Ω , then by Theorem 2.1 and the boundedness of p • there exist c 1 > 0 and h 1 ∈ L 1 Ω such that t p x ≤ c 1 t p • w x h 1 x for a.a.x ∈ Ω and all t > 0, 1 • w 1 x h x for a.a.x ∈ Ω and all t > 0. 2.1 Theorem 2.2.Let Ω ⊂ R n , p • ∈ P Ω , and let w be weight.If p < ∞, then L p • Ω L a. x ∈ Ω.
and let w be weight.It isL p • Ω L 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 ∞ L p • Ω ∞ and then using Proposition 2.3.Vice versa, if 2.6 is true, the functionalρ w f Ω\Ω ∞ f x w x 1/p x p x dx fw L ∞ Ω ∞ 2.11can 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 1/p x ∼ constant for a.a.x ∈ Ω \ Ω ∞ ∞ | > 0, the second part of the statement, about Ω ∞ , follows analogously, from the fact that L p • w Ω p • 1/x dx ρ 2f , 2.13 and therefore f p • ,w 2 f p • .