Journal of Function Spaces
Volume 2018 (2018), Article ID 3298582, 5 pages
https://doi.org/10.1155/2018/3298582

Research Article

Real Interpolation of Small Lebesgue Spaces in a Critical Case

Irshaad Ahmed, Aneesa Hafeez, and Ghulam Murtaza

Department of Mathematics, Government College University, Faisalabad, Pakistan

Correspondence should be addressed to Irshaad Ahmed; quantized84@yahoo.com

Received 19 June 2018; Accepted 1 August 2018; Published 8 August 2018

Academic Editor: Alberto Fiorenza

Copyright © 2018 Irshaad Ahmed et al. 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.

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 Then

holds for all positive and nondecreasing functions on if and only if

holds 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 that

and

Since , 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 have

where

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).

References

A. Fiorenza, “Duality and reflexivity in grand Lebesgue spaces,” Universitat de Barcelona. Collectanea Mathematica, vol. 51, no. 2, pp. 131–148, 2000.

A. Fiorenza and J. M. Rakotoson, “New properties of small Lebesgue spaces and their applications,” Mathematische Annalen, vol. 326, no. 3, pp. 543–561, 2003.

J. M. Rakotoson, “Some new applications of the pointwise relations for the relative rearrangement,” Advances in Differential Equations, vol. 7, no. 5, pp. 617–640, 2002.

O. Domínguez, “Tractable embeddings of Besov spaces into small Lebesgue spaces,” Mathematische Nachrichten, vol. 289, no. 14-15, pp. 1739–1759, 2016.

O. Domínguez, “Sharp embeddings of Besov spaces with logarithmic smoothness in sub-critical cases,” Analysis Mathematica, vol. 43, no. 2, pp. 219–240, 2017.

A. Fiorenza, M. Krbec, and H.-J. Schmeisser, “An improvement of dimension-free Sobolev imbeddings in r.i. spaces,” Journal of Functional Analysis, vol. 267, no. 1, pp. 243–261, 2014.

A. Fiorenza, M. R. Formica, and A. Gogatishvili, “On grand and small Lebesgue and Sobolev spaces and some applications to PDE's,” Differential Equations & Applications, vol. 10, no. 1, pp. 21–46, 2018.

G. Di Fratta and A. Fiorenza, “A direct approach to the duality of grand and small Lebesgue spaces,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 70, no. 7, pp. 2582–2592, 2009.

C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1988.

A. Fiorenza and G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs,” Zeitschrift für Analysis und ihre Anwendungen. Journal for Analysis and Its Applications, vol. 23, no. 4, pp. 657–681, 2004.

A. Fiorenza, M. R. Formica, A. Gogatishvili, T. Kopaliani, and J. M. Rakotoson, “Characterization of interpolation between Grand, small or classical Lebesgue spaces,” Nonlinear Analysis: Theory, Methods & Applications, 2017.

I. Ahmed, G. E. Karadzhov, and A. Raza, “General Holmstedts formulae for the K-functional,” Journal of Function Spaces, vol. 2017, Article ID 4958073, 9 pages, 2017.

J. Bergh and J. Lofstrom, Interpolation Spaces, Springer, New York, NY, USA, 1976.

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978.

I. Ahmed, D. E. Edmunds, W. D. Evans, and G. E. Karadzhov, “Reiteration theorems for the K-interpolation method in limiting cases,” Mathematische Nachrichten, vol. 284, no. 4, pp. 421–442, 2011.

H. Heinig and L. Maligranda, “Weighted inequalities for monotone and concave functions,” Studia Mathematica, vol. 116, no. 2, pp. 133–165, 1995.

A. Fiorenza and J. M. Rakotoson, “Compactness, interpolation inequalities for small Lebesgue-Sobolev spaces and applications,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 2, pp. 187–203, 2006.

A. Fiorenza and J. M. Rakotoson, “Some estimates in spaces,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 793–805, 2008.

A. Fiorenza, J. M. Rakotoson, and L. Zitouni, “Relative rearrangement method for estimating dual norms,” Indiana University Mathematics Journal, vol. 58, no. 3, pp. 1127–1149, 2009.