Reiteration Theorems for Two-Parameter Limiting Real Interpolation Methods

Abstract

We introduce a limiting real interpolation method involving two scalar parameters. We derive Holmstedt-type estimates for this method that are applied to establish the reiteration theorems.

1. Introduction

Let be a compatible couple of quasi-normed spaces; that is, we assume that both and are continuously embedded in some common quasi-normed space. Peetre’s -functional is defined, for each and , bywhere the infimum extends over all representations of with and . Let be a quasi-normed function space with a monotone quasi-norm on and Haar measure , such that The general real interpolation space consists of those for which the following quasi-norm is finite. We refer to [1–4] for a full development of the real interpolation method.

Let and ; then the classical scale of real interpolation spaces is obtained when is the weighted Lebesgue space defined by the quasi-norm (When , the integral should be replaced by appropriate supremum.) The reiteration theorem for this scale states that (see [1])where , and Moreover, for extreme cases, we haveWe note that the scale makes no sense for unless However, several authors (see, e.g., [5–7]) have investigated the limiting reiteration (with or ) by taking the parameter to be more general weighted Lebesgue space , with weight being of the form , where is a broken-logarithmic function or, more generally, a slowly varying function. See [8] for more general weights.

Recently, Cobos and Segurado [9] have defined a new limiting interpolation method , corresponding to limiting values , in a different way. Namely, if is given by the quasi-norm where , then . The reiteration spaces and have been characterized in [10].

In this paper we extend the limiting interpolation method by considering the parameter space to be where We denote the resulting real interpolation method by without using the subindex for convenience. Clearly, In addition, note that we do not use the notation for our method in order to avoid notational confusion (see, e.g., [2]). Since, for and/or , coincides with (see [9, Lemma ]), we only pay attention to the case when

The motivation for introducing the two-parameter limiting spaces mainly stems from the fact the sum of the limiting spaces and , introduced by Cobos et al. [11] in connection with the interpolation over the unit square, is precisely . This fact is established in [9, Proposition ] for , and the same argument also works for arbitrary values of and . We further note that two different parameters have already been used in defining certain -limiting spaces (see [12, Definition ]).

The main goal of this paper is to characterize the reiteration spaces Moreover, the assertions of [10, Theorem ] have been extended by identifying the spaces and .

The classical identities (5)-(6) are based on the estimates which relate the -functionals of the interpolated couples , , and with that of the original couple (see [13]). The main ingredient of our proofs will be the similar estimates for the limiting spaces . These estimates are derived in the next section as corollaries of more general Holmstedt-type estimates. Some Hardy-type inequalities, along with two other useful results, are given in Section 3. Finally, the reiteration theorems are established in Section 4.

2. Holmstedt-Type Estimates of the -Functional

Let be a positive weight on , that is, a positive locally integrable function on , and let Then, by we will mean the real interpolation space , where has the quasi-normNote that for

In this section we present Holmstedt-type estimates for the real interpolation spaces , and we omit the proofs as they can be done as in [8, Section ], where these estimates have been obtained for the case

First, we formulate the results for the case . For this purpose we introduce some notations:

Subsequently, we will use the notation for nonnegative quantities to mean that for some positive constant which is independent of appropriate parameters involved in and If and , we will put

Theorem 1. For every and , one hasIf , then

Next, we present the estimates for the case To this end, we denote

Theorem 2. For every and , one hasIf , then

Finally, we state the results for the case . SetWe assume that the weights and are such that and are finite.

Theorem 3. For any weight , all , and all , one has

Next we apply the above general results to obtain estimates for our limiting spaces.

Corollary 4. Let Set Then, for all ,and, for all ,

Proof. We apply Theorem 1 to the weight given byWe can easily compute thatHence, (24) follows from (13), and (25) follows from (14).

Corollary 5. Let Set Then, for all ,and, for all ,

Proof. This time we apply Theorem 2 to the weight given byWe see thatTherefore, (30) and (31) are consequences of (17) and (16), respectively.

Corollary 6. Let , and Set Then, for all , and, for all ,

Proof. We apply Theorem 3 to the weights and defined in (26) and (32), respectively, and it will suffice to derive the following:Now , ,  , and are given by (27), (28), (33), and (34), respectively. Firstly, we establish the estimate “” in (38). For this we note the following, since is nondecreasing in “” and is nonincreasing in “” (see, e.g., [3, Proposition ]): In view of , and , the previous two estimates yieldFurthermore, from (39), we can writeTherefore, if we take in (20) then “” in (38) follows, for , from (41) and (43), and for , from (40) and (42).
In order to obtain the reverse estimate, we exploit (21) and (22), and for this we need to compute and , defined by (18) and (19), respectively. Since so, for all , as . For all , where the first integral is convergent thanks to the condition . Therefore, Noting (where the convergence of the first and third integrals is being implied by the condition , we get Therefore, Hence, Similarly, we find that Consequently, “” in (38) results from (21) and (22). The proof is finished.

3. Auxiliary Results

In order to prove our main results in the next section, we need certain Hardy-type inequalities. We will derive them by verifying the sufficient conditions, for particular weights, for the general weighted Hardy-type inequalities. For the next two results we refer the reader to [14, Section ]. For , put

Lemma 7. Let . Then the inequality holds for all nonnegative functions on if and only if holds for all

Lemma 8. Let . Then the inequalityholds for all nonnegative functions on if and only if holds for all

For the proof of the next result, we refer to [15, Theorem (b)].

Lemma 9. Let , and let Then the inequalityholds for all nonnegative nondecreasing functions on if and only if holds for all

Corollary 10. Let ; then holds for all nondecreasing nonnegative functions on

Proof. For , the inequality follows by interchanging the order of integration on the left side of the inequality. For , the result follows from Lemma 7, applied with and Finally, putting and , we obtain the result, for , from Lemma 9.

Corollary 11. Let ; then holds for all nonincreasing nonnegative functions on

Proof. The proof follows by applying the previous lemma to the nondecreasing function on

Corollary 12. Let ; then holds for all nondecreasing nonnegative functions on

Proof. By interchanging the order of integration, we get the result for For , the estimate “” results from Lemma 8, applied with and For , it follows from Lemma 9, by taking and The other estimate “” follows fromand the fact that and are asymptotically the same as The proof is finished.

Corollary 13. Let ; then holds for all nonincreasing nonnegative functions on

Proof. Apply the previous lemma to the nonincreasing function on

In order to facilitate certain change of variables in the first two theorems of the next section, we will make use of the next two lemmas concerning slowly varying functions. Here we say that a positive Lebesgue-measurable function is slowly varying on if, for all , the function is equivalent to a nondecreasing function and is equivalent to a nonincreasing function. By symmetry, we say that is slowly varying on if the function is slowly varying on Finally, is slowly varying on if it is slowly varying on both and For example, is slowly varying on for every real number We refer to [16] for details on slowly varying functions.

Lemma 14. Let be a positive function on of the following form: where and is a slowly varying function. Then there exists a positive function on having the following properties:(i);(ii) is strictly increasing, locally absolutely continuous;(iii);(iv), and

Proof. Let be a nondecreasing function such that Set where ; satisfies (i)–(iv) (see [8, Lemma ]).

Similarly, we can establish the following lemma.

Lemma 15. Let be a positive function on of the following form: where and is a slowly varying function. Then there exists a positive function on having the following properties: (i);(ii) is strictly increasing, locally absolutely continuous;(iii);(iv), and

4. Reiteration Theorems

Finally, we derive the reiteration theorems for our two-parameter limiting spaces by using the results of the previous two sections.

Theorem 16. Let Then one has with equivalent norms where

Proof. PutBy Corollary 4, whereWe note that is nonincreasing since it is an integral average (with respect to the measure ) of a nonincreasing function Consequently, which gives Next we apply Corollary 10 to the nondecreasing function to conclude that and apply Corollary 13 to the nonincreasing function to find thatAltogether, it follows thatOn the other hand, in (66), we replace by an equivalent function on (as obtained by Lemma 14) and make change of variable to find that And making change of variable in (67), we obtain Combining the previous two estimates, we achievewhich completes the proof in view of (73).

Remark 17. By taking , we get back the first assertion of [10, Theorem ].

Theorem 18. Let Then we have with equivalent norms where

Proof. This time, puttingwe obtain, by Corollary 5, that where In view of the fact that is nondecreasing, it follows that Moreover, by Corollaries 11 and 12, we get and Collecting all the previous estimates yieldsNext we make change of variables in (79) to obtain that As for change of variables in (80), we replace by an equivalent function (obtained this time by Lemma 15) and put to get thatHence,which, along with (85), completes the proof.

Remark 19. The above theorem provides an extension of the second assertion of [10, Theorem ].

In order to describe our final result, we need two more scales of real interpolation spaces and . Namely, and , where The reader is referred to [8] for details on and . Recall that the quasi-norm on the intersection of two quasi-normed spaces and is given by

Theorem 20. Let , and Then one has with equivalent normswhere , , , and

Proof. Putting and making appropriate change of variables, we observe thatOn the other hand, by Corollary 6, we obtain whereNote that and Furthermore, we havethanks to Corollaries 12 and 13, respectively. Thus, . Making use of the fact that are, respectively, nonincreasing and nondecreasing, we can further deduce that and . Collecting all these estimates, we get at the following: Combining this estimate with (94) finishes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.