Abstract
We establish an interpolation formula for small Lebesgue spaces in a critical case.
1. Introduction
The small Lebesgue spaces were introduced by A. Fiorenza in [1], and they have found applications, for example, in boundary value problems [2, 3], Besov embeddings [4, 5], and dimension-free Sobolev embeddings [6]. The reader is referred to a recent survey paper [7] for more details.
Let be a bounded domain in with measure , and let and . The small Lebesgue space is formed by all those real-valued Lebesgue measurable functions on , for which the norm is finite; see [8]. Here denotes the nonincreasing rearrangement of (see, for instance, [9]). For , the spaces coincide with the classical small Lebesgue spaces (see [10, Corollary 3.3]).
Let , , , and The real interpolation spaces have been recently characterized in [11, Theorem 3.4] as the well-known Lorentz-Zygmund spaces. Here denotes the classical real interpolation method (see Section 2). The critical case has not been considered in [11], and the goal of the present paper is to fill this gap. More precisely, we will characterize the interpolation spaces under an appropriate condition on and . Our approach is essentially different from that of [11] and allows the parameter to lie also in the interval The key ingredient of our method is a general Holmstedt-type estimate from a recent paper [12].
The plan of the paper is simple. We collect the necessary background, along with certain weighted Hardy-type inequalities, in Section 2. The main result is contained in Section 3.
2. Preliminaries
Throughout the paper, we will write or for two nonnegative quantities and to mean that for some positive constant which is independent of appropriate parameters involved in and We put , if and . Moreover, we will write for two quasi-normed spaces and to mean that is continuously embedded in
Let be a compatible couple of quasi-normed spaces; that is, we assume that both and are continuously embedded in the same Hausdorff topological vector space. Peetre’s -functional is defined, for each and , by
In what follows, we always assume that the couple is ordered in the sense that
Let , , and The real interpolation space consists of all for which the quasi-norm is finite (with usual modification when ). When and , we put . Since , we have Combining this with the fact that, as a function of , is nonincreasing, we can easily check that Thus, the spaces are in fact the classical Lions-Peetre spaces (see [9, 13, 14]).
In the sequel, we will work with the limiting spaces and the classical spaces It is not hard to verify that the limiting spaces are intermediate, without any condition on and , for the couple , that is, However, in order to exclude the trivial case , we have to work under the assumption (if ) or (if ). This observation immediately follows from the elementary fact that, as a function of , is nondecreasing.
We close this section with the following weighted Hardy-type inequalities which will be needed in order to prove our main result in the next section.
Lemma 1 ([15, Lemma 3.2]). Let , and assume that and are nonnegative functions on . Put Then holds for all nonnegative functions on .
By an obvious change of variable, we get the following variant of the previous result.
Lemma 2. Let , and assume that and are nonnegative functions on . Put Then holds for all nonnegative functions on .
Lemma 3 ([16, Theorem 3.3 (b)]). Let . Assume and are positive functions on , and is a positive function on Thenholds for all positive and nondecreasing functions on if and only ifholds for all
3. Interpolation Formula
First we compute the -functional for the couple , To this end, we need the following Holmstedt-type estimate.
Theorem 4. Let , and let Then where and
Proof. Let . Put and According to case in [12, Theorem 4], we have provided thatandSince , then Moreover, note that Finally, the condition implies that both (18) and (19) hold. The proof is complete.
Remark 5. If , then we have Thus, in this case, we trivially have
Corollary 6. Let , , and let . Then, for all , we have where and
Proof. First observe that as has finite measure. We put and Since (see [13, Theorem 5.2.1]) it follows immediately that , where Thus, the proof simply follows by applying Theorem 4 to , , , and
Following [4], we define small Lebesgue spaces in a slightly general way as follows. Let , , and The small Lebesgue space is formed by all those real-valued Lebesgue measurable functions on , for which the quasi-norm is finite. The spaces are a particular case of more general spaces introduced and studied in [17–19]. Note that if , , and with , then we recover the spaces
The following interpolation formula is the main contribution of this paper.
Theorem 7. Let , , , and Then where
Proof. First note that as Put and . Let Assume that By making an appropriate change of variable and using Corollary 6, we arrive at where and with Thus, the proof will be complete, in the case , if we show that and . Since is nondecreasing, we get or whence we obtain as and are asymptotically the same as
Next we establish the estimates and . To this end, we distinguish two cases: and . Assume first that . Then, follows from Lemma 1, applied with , , , and . Similarly, follows from Lemma 2. Next assume that . Since is nondecreasing, we can apply Lemma 3 with , , , , and Note that (10) turns into . Thus, we have to verify (11) for all Observe that (11) holds trivially for all , and thus we may assume that Let us put and . Now we have which proves the validity of (11). Hence, the estimate follows from (10). Similarly, we can obtain from Lemma 3. This completes the proof in the case Next we assume that This time we havewhere and In order to estimate , we note that whence we get Similarly, we can show that On the other hand, again using the fact that is nondecreasing, we get Altogether, we have which gives in view of (34). The proof of the theorem is complete.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
The authors have been partially supported by a research grant from Higher Education Commission of Pakistan (Grant 5687/Punjab/NRPU/R&D/HEC/2016).