Research Article | Open Access
A Note on Noneffective Weights in Variable Lebesgue Spaces
We study noneffective weights in the framework of variable exponent Lebesgue spaces, and we show that if and only if in the set where , and in the set where .
The variable Lebesgue spaces generalize the classical Lebesgue spaces , where the constant exponent is replaced by a function . They fall within the scope of Musielak-Orlicz spaces (see ) 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 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 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. , and Samko .
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 , where the noneffective weights have been introduced and explored in the framework of Orlicz spaces . It has been proved that (we omit here notation and details) if a Young function satisfies the condition at infinity, a weight is noneffective (i.e., ) if and only if ; moreover, the fact that noneffective weights are exactly the trivial ones (i.e., the weights equivalent to a constant) may happen also if the condition at infinity is not satisfied, and a necessary and sufficient condition on 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. , or the pioneering papers [6, 7].
Given a set , an exponent function is a measurable function . We denote the set of all such functions by . Given , let , and for any , let
For brevity, we denote and .
Given a weight in (i.e., , ), we define the modular functional
If , then we set the last term equal to 0; if , then . We define the space to be the set of measurable functions such that, for some . This is a Banach function space when equipped with the norm
When , a constant, then with equality of norms. When , we will denote by the modular functional, the norm, and 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 .
Theorem 2.1. Let , , be exponents in , , , , , weights. The continuous embedding holds if and only if there exist positive constants , , and , such that
At first we consider the case , where it is possible to show that the noneffective weights must be the trivial ones.
Theorem 2.2. Let , , and let be weight. If , then if and only if .
Proof. If , then obviously , therefore we need to prove only the converse implication.
Since, in particular, , then by Theorem 2.1 and the boundedness of there exist and such that that is, from which, letting , 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 , as in the previous case, all noneffective weights must be trivial.
Proposition 2.3. Let , and let be weight. It is if and only if .
Proof. As above, we show only that, if , then . Arguing by contradiction, it is or . In the first case, set so that for all . Testing the inequality with , we get for all , which is absurd. In the second case we argue similarly, and the proposition is proved.
Theorem 2.4. Let , , and let be weight. If , then if and only if .
Theorem 2.4 is a consequence of the following result, where in (possibly empty) an unbounded exponent is allowed.
Theorem 2.5. Let , , and let be weight. It is if and only if
Proof. If , then the theorem follows from Proposition 2.3, therefore we may assume that . We now argue as in the proof of Theorem 2.2. Since , it is also , and therefore, applying Theorem 2.1, there exist positive constants , , and , such that and arguing similarly as before we get , from which we get the existence of a constant such that . Starting from the opposite inclusion, we get the existence of a constant such that . In conclusion, but this is equivalent to say that because If , the second part of the statement, about , follows analogously, from the fact that and then using Proposition 2.3.Vice versa, if (2.6) is true, the functional can be majorized and minorized with , where comes from assumption (2.6), and this implies the equivalence of the norms of and , and therefore the theorem is proved.
Example 2.6. Let , , . Then is unbounded, and, since , it is also noneffective: This can be checked also directly, in fact and therefore .
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.
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Copyright © 2012 Alberto Fiorenza and Miroslav Krbec. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.