Some Reiteration Theorems for , , , , , and Limiting Interpolation Spaces

Abstract

We consider the -interpolation methods involving slowly varying functions. We establish some reiteration formulae including so-called or limiting interpolation spaces as well as the , , , and extremal interpolation spaces. These spaces arise in the limiting situations. The proofs of most reiteration formulae are based on Holmstedt-type formulae. Applications to grand and small Lorentz spaces in critical cases are given.

1. Introduction

Let be a compatible couple of Banach or quasi-Banach spaces such that . For , Peetre’s -functional is given by

We refer to, e.g., [1] for basic concepts on interpolation theory and properties of the -functional. In recent years, the scale of interpolation spaces has been intensively studied, which is defined via the (quasi-)normswhere , , is a slowly varying function (see Definition 5 below), and is the usual (quasi-)norm in the Lebesgue space on the interval .

An important property of this scale is that the reiteration spaces with and belong to the same scale. (Here and below, “” stands for further parameters.) The consideration of the limiting cases leads normally to some new limiting interpolation spaces and . Following [2], we call them and spaces (see Definition 7 below). These limiting interpolation spaces occur naturally in many fields of analysis. For example, it is shown that the so-called grand and small Lorentz spaces can be described in terms of the and spaces [3, 4].

Naturally, the question arises about the description of the reiteration spaces for couples where one or both of the operands are or spaces. Such reiteration formulae under condition are established in the previous paper [5]. Some limiting cases , but not all, are also considered in [5]. For example, if , it has been shown in [5] that if and . However, the formula for has not been derived. The principal aim of this paper is to establish reiteration formulae for combinations of parameters, which are not included in [5].

The motivation for this work was the articles [6–8], where it has been shown that for some limiting combinations of parameters, new interpolation spaces are required. Following [6, 7], we call them , , , and extremal interpolation spaces (see Definition 8 below). For example, (see Theorem 19 below).

Our paper [5] and the articles [6–8] appeared practically at the same time. The results of the papers [6–8] demonstrate—mathematically speaking—a large intersection and a symmetric difference with the ones in [5] and in this paper. Fernández-Martínez and Signes work with the scale of interpolation spaces based on the family of rearrangement invariant Banach function spaces . They have introduced and studied this rich scale in [2, 9]. If and , the space coincides with the space . However, the spaces with do not belong to the scale . The same remark holds about and limiting interpolation spaces, too.

Note that in [5] and in this paper, the main parameters and for considered und spaces lie between 0 and 1. Limiting cases will be the topic of a future paper.

The paper is organized as follows. Sections 2 and 3 contain necessary notations, definitions, and technical results. In Sections 4–6, we establish reiteration formulae for the couples, where one of the operands is either or space. Reiteration formulae for the couples, where both operands are the or space, are established in Section 7. In Section 8, we consider couples where one of the operands is , , , or space. Note that latter combinations are not studied in [6–8]. Finally, in Section 9, we present interpolation results for the grand and small Lorentz spaces in critical cases as applications of our general reiteration theorems.

2. Preliminaries

Throughout the paper, we write for two (quasi-)normed spaces and to indicate that is continuously embedded in . We write if and . For and being positive functions, we write if , where the constant is independent of all significant quantities. Two functions and are considered equivalent () if and . We adopt the conventions and . The abbreviation LHS() (RHS()) will be used for the left- (right-) hand side of the relation (). By , we denote the characteristic function on an interval .

2.1. Slowly Varying Functions

In this section, we summarize some of the properties of slowly varying functions, which will be required later. For more details, we refer to, e.g., [2, 9–11].

Definition 1. We say that a positive Lebesgue measurable function is slowly varying on , notation , if, for each , the function is equivalent to an increasing function while the function is equivalent to a decreasing function.

Lemma 2. Let , , , , and .(i)Then, , , , and (ii)If , then (iii) and (iv)The functions and (if exist) belong to and and (v)We will often use these properties without referencing explicitly every time.

2.2. Hardy-Type Inequalities

The following Hardy-type inequalities will be applied later.

Lemma 3 (see [10], Lemma 2.7). Let and .(i)The inequalityholds for all (Lebesgue) measurable nonnegative functions on if and only if .(ii)The inequalityholds for all (Lebesgue) measurable nonnegative functions on if and only if .

These inequalities will be used in the form of the following corollary.

Corollary 4. Let , , and . Then, for all (Lebesgue) measurable nonnegative functions on ,

Proof. We begin with the first estimate. If , using Lemma 3 with , , and , we getThe case can be considered similarly with . The case can be proved by Fubini’s theorem or by the exchange of the essential suprema (if ).
The second estimate follows from the first one if we take and .☐

3. Interpolation Methods

Below is a collection of crucial definitions and statements building a base of the real interpolation methods involving slowly varying functions. In the following, let be a compatible couple of (quasi-)Banach spaces such that . Peetre’s -functional on is given by (1).

3.1. Standard Interpolation Spaces

Definition 5 (see [10]). Let , , and . We put

Lemma 6 (see [10], Proposition 2.5). if and only if one of the following conditions is satisfied:(i)(ii) and (iii) and Moreover, if none of these conditions holds, then .
Spaces are referred to as standard interpolation spaces.

3.2. and Spaces

Definition 7 (see [10]). Let , , and . We putSimilarly,In the literature (e.g., [2, 5, 9, 12–14]), similar definitions are given alongside with properties of these spaces. As proposed in [2], we refer to the spaces and as and limiting interpolation spaces.

3.3. , , , and Spaces

We introduce here four interpolation spaces. We follow [6–8] where it has been shown that they appear in relation to the extreme reiteration results.

Definition 8. Let , , and . The space is the set of all for whichThe space is the set of all for whichThe space is the set of all for whichThe space is the set of all for whichWe refer to these spaces as , , , and extremal interpolation spaces.

3.4. Some Known Formulae, Auxiliary Lemmas, and Reiteration Theorems

Let , , and . Under suitable conditions, the following formulae hold ([1], Chap. 5, Proposition 1.2) [6–8, 10]:

Lemma 9 (cf. [5]; [6], Remark 2.6). Let and . Then, for all and ,Moreover,

Proof. The estimates (19) and (20) follow immediately from the monotonicity of the -functional. We prove the estimate (18). The estimate (17) can be proved similarly. For , because and is nondecreasing, we haveThus, (18) is proved for :If , using the last estimate, we getThis completes the proof.☐

The next lemma can be proved by repeating the proofs of [6] (Lemma 2.12) and [7] (Lemma 2.10).

Lemma 10. Let , , and . Then, for all and ,

Lemma 11. Let , , , and . Then, for all and ,

Proof. We prove the first estimate. The second one can be proved analogously. Because , the function is equivalent to an increasing function. It is known ([10], Theorem 3.1) that for all and ,Therefore, the function is equivalent to a decreasing function. Hence, for , we getThus, (28) is proved for :Denote . If , using the last estimate, we getThis completes the proof.☐

It is worth remarking that the estimates from Lemmas 9–11 can be used to establish different embedding theorems.

Lemma 12 (cf. [8], Lemma 4.2). Let , , and . Then, for all ,provided that , andprovided that .

Proof. We prove the first equivalence. The second one can be proved analogously. We haveHence,Let us prove the converse inequality.
Case. Because , by Corollary 4 (taking ), we getIn particular,Case. Because , using Lemma 9 and the last estimate, we getThis completes the proof.☐

Lemma 13. Let , , and . Then, for all ,

Proof. We prove the first equivalence. The second one can be proved analogously or using the first equivalence and symmetry argument. Denote .
Case. Because and , by Corollary 4 (taking ) and by Lemma 12, we getIn particular, it was shown thatCase. By Lemma 11 and (42), we getLet us prove the inverse estimate. By (27), we haveHence,This completes the proof.☐

Lemma 14. Let , , and .(i)If , then for all ,(ii)If , then for all ,

Proof. We prove the first equivalence. The second one can be proved analogously or using the symmetry argument. For , by (24), we haveHence, for each ,Let us prove the inverse estimate. Because and by Lemma 12, we getThis completes the proof.☐

The following lemma deals with the quasi-norms on the spaces and .

Lemma 15. Let and .(i)If and , then for all ,provided that . Furthermore,(ii)If and , then for all ,provided that . Furthermore,(iii)If and , then for all ,(iv)If and , then for all ,

Proof. We prove the first assertion. The second one can be proved analogously or using the symmetry argument. By Lemma 10, for , we haveThus, and the estimate (52) is proved. Let . Note thatHence, by Lemma 12, we getThe equivalences (iii) and (iv) follow immediately from Lemma 12.☐

In Section 8, we will use the following reiteration theorems. See [9, 14].

Theorem 16. Let , , and . Put .(i)If , thenwhere .(ii)If , thenwhere .(iii)If , then

Theorem 17. Let , , and . Put .(i)If , thenwhere .(ii)If , then(iii)If , thenwhere .

4. Limiting Interpolation between the Spaces on the Right and the Standard Interpolation Spaces

In this section, we establish some limiting reiteration formulae for the couples of the form (). They amend Theorems 16, 17, and 19 from [5]. Here, we need the relevant Holmstedt-type formulae. The proofs of all Holmstedt-type formulae in this paper are based on the paper [15]. Therefore, we adopt the notation from it.

Theorem 18 (cf. [6], Theorem 3.4). Let , , , and . Put .(i)Let . Then, for all and ,(ii)Let and . Then, for all and ,(iii)Let . Then, for all and ,

Proof. The formulae (i) and (iii) have been proved in [5]. Here, we prove the formula (ii). Let be the function space corresponding to :Consider the functions , , and .Thus,Let be the function space corresponding to :Consider the functions , , and .Because , we arrive atNow, we estimate the functions and .We see that and . Note thatHence, by [15] (Theorem 4, Case 1), we arrive atUsing Lemma 10, we getTherefore,This completes the proof.☐

Theorem 19 (cf. [6], Theorem 4.7 (c)). Let , , , , and . Put and . Then,

Proof. Denote , , , and . Using the change of variables (see, e.g., [5], Remark 3) and Theorem 18 (i), we can writewhereLemmas 12 and 10 imply thatThis completes the proof.☐

Theorem 20 (cf. [6], Theorem 4.8 (b, c)). Let , , , , and . Put and .(i)If , thenwhere .(ii)If , then

Proof. Denote , , and . First, we prove (i). Let . Using the change of variables and Theorem 18 (ii), we can writewhereBy Lemma 15 (i), we get . The formula (i) is proved.
Now, we prove (ii). Let . Using the change of variables and Theorem 18 (ii), we can writewhereBy Lemmas 12 and 10, we getThis completes the proof.☐

Theorem 21 (cf. [6], Theorem 4.9 (b)). Let , , , , and . Put and . Then,

Proof. Denote , , , and . Using the change of variables and Theorem 18 (iii), we can writeThis completes the proof.☐

5. Limiting Interpolation between the Spaces on the Left and the Standard Interpolation Spaces

In this section, we establish some limiting reiteration formulae for the couples of the form . They amend Theorems 11–13 from [5]. Here, we also need the relevant Holmstedt-type formulae.

Theorem 22 (cf. [7], Theorem 3.2). Let , , , and . Put .(i)Let . Then, for all and ,(ii)Let and . Then, for all and ,(iii)Let . Then, for all and ,

Proof. Recall that we adopt the notation from the paper [15]. Let be the function space corresponding to :Consider the functions , , and .Thus,First, we prove (i). Let be the function space corresponding to :Consider the functions , , and .Thus,Now, we estimate the functions and . Because , we getThus,Using [15] (Theorem 4, Case 4), we arrive at (93).
Proof of statement (ii). Put . Let be the function space corresponding to :Consider the functions , , and .Thus,Now, we estimate the functions and . Because , we getThus,Using [15] (Theorem 4, Case 4), we arrive at (94).
Proof of assertion (iii). Here, we use [15] (Theorem 1). Because , we getBy Lemma 9, we know thatSo, we arrive at (95).☐

Theorem 23 (cf. [7], Theorem 4.8). Let , , , , and . Put and . Then,where .

Proof. Denote , , , and . Using the change of variables and Theorem 22 (i), we can writewhereIt is enough to show that . By Lemmas 12 and 9, we getThis completes the proof.☐

Theorem 24 (cf. [7], Theorem 4.9). Let , , , , and . Put and .(i)If , thenwhere .(ii)If , thenwhere .

Proof. Denote , , and . First, we prove (i). Let . Using the change of variables and Theorem 22 (ii), we can writewhereThe estimate can be proved by repeating the corresponding part of the proof of Theorem 23.
Now, we prove (ii). Let . Using the change of variables and Theorem 22 (ii), we can writewhereUsing Lemma 15 (iv), we get . Hence, it is enough to show that . By Lemma 13, we getThis completes the proof.☐

Theorem 25 (cf. [7], Theorem 4.10). Let , , , , and . Put and . Then,where .

Proof. Denote , , , and . Using the change of variables and Theorem 22 (iii), we can writewhereThis completes the proof.☐

6. Limiting Interpolation between the Spaces and the Standard Interpolation Spaces

The theorems in this section amend theorems from [5] (Section 5). The proofs are based on the corresponding reiteration theorems from the previous sections and on the formulae (14)–(16). They are left to the reader. See, as an example, the proof of Theorem 34.

Theorem 26 (cf. [6], Theorem 4.4). Let , , , , and . Put and . Then,

Theorem 27 (cf. [6], Theorem 4.5). Let , , , , and . Put and .(i)If , then(ii)If , thenwhere .

Theorem 28 (cf. [6], Theorem 4.6). Let , , , , and . Put and . Then,

Theorem 29 (cf. [7], Theorem 4.5). Let , , , , and . Put and . Then,where .

Theorem 30 (cf. [7], Theorem 4.6). Let , , , , and . Put and .(i)If , thenwhere .(ii)If , thenwhere .

Theorem 31 (cf. [7], Theorem 4.7). Let , , , , and . Put and . Then,where .

7. Limiting Reiteration Formulae for Couples Formed Only by and Spaces

The reiteration theorems in this section amend theorems from [5] (Section 6). Here, we also need the relevant Holmstedt-type formulae.

Theorem 32 (cf. [8], Theorem 3.2). Let , , , , and . Put , , and . Then, for all and ,

Proof. Recall that we adopt the notation from the paper [15]. Let be the function space corresponding to :From the proof of Theorem 22, we know thatLet be the function space corresponding to :From the proof of Theorem 18, we know thatNow, we estimate the functions and . Because , we getThus,Therefore, using [15] (Theorem 4, Case 4), we arrive at (133).☐

Theorem 33 (cf. [8], Theorem 4.4). Let , , , , and . Put , , and .(i)If , thenwhere .(ii)If , then

Proof. Denote , , and . First, we prove (i). Let . Using the change of variables and Theorem 32, we can writewhereIt is enough to show that . Using Lemmas 12 and 9, we getStatement (i) is proved.
Now, we prove (ii). Let . Using the change of variables and Theorem 32, we can writewhereNote that due to Lemma 10, we haveHence, it is enough to show that and . Consider . By Lemma 13, we haveConsider . By Lemma 12, we getThis completes the proof.☐

The next assertion is a symmetric counterpart of Theorem 33.

Theorem 34 (cf. [8], Theorem 4.3). Let , , , , and . Put , , and .(i)If , then(ii)If , thenwhere .

Proof. We prove the second formula. The first one can be proved similarly. Recall that , and we use the denotation . Note thatUsing Theorem 33 (i) and formulae (14) and (15), we getwhereAdditionally, we haveHence, . Therefore,and by (16),This completes the proof.☐

Theorem 35 (cf. [8], Theorem 3.4). Let , , , , and . Put , , and . Then, for all and ,

Proof. As usual, we use the notation from the paper [15]. Let be the function space corresponding to :From the proof of Theorem 22, we know thatLet be the function space corresponding to :Consider the functions , , and .Thus,Now, we estimate the functions and . Because , we getThus, . Therefore, using [15] (Theorem 4, Case 4), we arrive at (158).☐

Theorem 36 (cf. [8], Theorem 4.6). Let , , , , and . Put , , and .(i)If , thenwhere .(ii)If , thenwhere .

Proof. Denote , , and . We prove (i). Statement (ii) can be proved analogously. Let . Using the change of variables and Theorem 35, we can writewhereIt is enough to show that and . By Lemmas 12 and 9, we getUsing Lemmas 13 and 10, we arrive atThis completes the proof.☐

Theorem 37 (cf. [8], Theorem 3.3). Let , , , , and . Put , , and . Then, for all and ,

Proof. Let be the function space corresponding to :Consider the functions , , and .Thus,Let be the function space corresponding to :From the proof of Theorem 18, we know thatNow, we estimate the functions and .Because , using [15] (Theorem 4, Case 2), we arrive atBy Lemma 10, we getThus, (171) is proved.☐

Theorem 38 (cf. [8], Theorem 4.5). Let , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , then(iii)If , then

Proof. Denote , , , and (). Using the change of variables and Theorem 37, we can writewhereFirst, let . In this case, and . By Lemma 14 (i), we getSo, it is enough to show that . Note thatThus, by Lemma 12, we getNow, we prove (ii). In this case,It is enough to show that . Using Lemmas 12 and 10, we haveThe proof of (iii) follows similar steps.☐

Remark 39. Assertion (i) of Theorem 38 coincides with [5] (Theorem 29). But there it has been wrongly stated that it can be proved based on symmetry arguments. The correct proof is presented above.

8. Reiteration Formulae for Couples Where One of the Operands Is , , , or Space

In this section, we establish reiteration formulae for the couples in which one of the operands is , , , or space. Here, we have chosen an indirect method of the proofs. Compared to the direct method, i.e., calculation of the corresponding Holmstedt-type formulae, we subsequently apply known formulae but unfortunately often arrive at weaker results (embeddings instead of isomorphisms).

Theorem 40. Let , , , , and . Put and .(i)If , thenwhere and .(ii)If , then(iii)If , then

Proof. Denote , , and . Let be a strongly increasing, differentiable function such that and . Additionally, denote , , and (). By the change of variables , we get . Hence, andBy Theorem 19, we obtainSo, . Using Theorem 17, we conclude that for , it holdsIf , by [5] (Theorem 19) (i) and (194), we get , whereIf , by [5] (Theorem 19) (ii), we getwhile due to (194), . Note that .
If , using Theorems 17 and 19, we conclude thatThis completes the proof.☐

Theorem 41. Let , , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , thenwhere .(iii)If , then

Proof. Denote , , and . Let be a strongly increasing, differentiable function such that and . Additionally, denote , , and (). By the change of variables , we get andBy Theorem 20 (ii), we obtainHence, . Using Theorem 17, we conclude that for , it holdsIf , by [5] (Theorem 17) and (203), we getwhere and .
If , using Theorem 20 (i), we getwhere .
If , using Theorems 17 and 20, we conclude thatThis completes the proof.☐

Theorem 42. Let , , , , and . Put and .(i)If , thenwhere and .(ii)If , then(iii)If , then

Proof. Denote , , and . Let be a strongly increasing, differentiable function such that and. Additionally, denote , , and (). By the change of variables , we get andBy Theorem 21, we obtainHence, .
If , by Theorem 17, [5] (Theorem 16), and (212), we getwhere and .
If , using Theorems 17 and 21, we conclude thatThis completes the proof.☐

Theorem 43. Let , , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , thenwhere .(iii)If , then

Proof. Denote , , and . Let be a strongly increasing, differentiable function such that and . Additionally, denote , , and . By the change of variables , we get andBy Theorem 33 (ii), we obtainHence, . Using Theorem 17, we conclude that for , it holdsIf , by [5] (Theorem 26) (i) and (219), we get , where .
If , by Theorem 33 (i), we getwhere .
If , using Theorems 17 and 33 (ii), we conclude thatThis completes the proof.☐

Theorem 44. Let , , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , then(iii)If , then

Proof. Denote , , and . Let be a strongly increasing, differentiable function such that and . Additionally, denote , , and . By the change of variables , we get andBy Theorem 38 (iii), we obtainHence, . Using Theorem 17, we conclude that for , it holdsIf , by Theorem 38 (i) and (227), we get , where .
If , by Theorem 38 (ii), we getwhere .
If , using Theorems 17 and 38 (iii), we conclude thatThis completes the proof.☐

From the previous theorems in this section, using the symmetry arguments given by the formulae (14)–(16), the following five theorems can be proved.

Theorem 45. Let , , , , and . Put and .(i)If , thenwhere and .(ii)If , then(iii)If , then

Theorem 46. Let , , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , then(iii)If , thenwhere .

Theorem 47. Let , , , , and . Put and .(i)If , thenwhere and .(ii)If , then(iii)If , then

Theorem 48. Let , , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , then(iii)If , thenwhere .

Theorem 49. Let , , , , , and . Put , , and .(i)If , thenwhere and .(ii)If , then(iii)If , then

The technique used in the section allows considering also some couples, where one of the operands is or space. In these cases, though, only “reiteration embedding” can be proved.

Theorem 50. Let , , , , , and . Put and .(i)If , thenwhere and .(ii)If , then

Proof. Denote , , and . Let be a strongly increasing, differentiable function such that and . Additionally, denote , , and . By the change of variables , we get andBy Theorem 23, we obtainThus, . Using Theorem 16, we conclude that for , it holdsIf , by [5] (Theorem 19) (i) and (249), we get , where .
Let . Using [5] (Theorem 11) (ii), we arrive at☐

Similar theorems can be formulated and proved also for the couples , , , , , , , , and . We leave this to the reader.

9. Applications

Here, we demonstrate how our general reiteration theorems can be used to establish limiting interpolation results for the grand and small Lorentz spaces. For the sake of shortness, we present only some possible results.

Let denote a -finite measure space with a nonatomic measure . We consider functions from the set of all -measurable functions on . As conventional (see, e.g., [1]), () denotes the nonincreasing rearrangement of and the maximal function of is defined by

For simplicity, we consider below only interpolation spaces between and . For functions from these spaces, Peetre has shown that ([1], Theorem V.1.6)

The following assertion is a modification of [10] (Lemma 5.2) and can be proved similarly.

Lemma 51. Let , , , and . If , we additionally suppose that . Then, for all and ,

9.1. Function Spaces

First, we define function spaces under consideration.

Definition 52. Let and . The Lorentz–Karamata space is the set of all such thatThe Lorentz–Karamata spaces form an important scale of spaces. It contains, e.g., the Lebesgue space , Lorentz space , Lorentz–Zygmund, and the generalized Lorentz–Zygmund spaces. For further information about Lorentz–Karamata spaces, we refer to, e.g., [10, 11, 16]. They have found many different important applications in analysis (see, e.g., [1, 4, 10–12, 14, 16–19] and the references therein).

Lemma 51 implies the following interpolation result.

Lemma 53 (cf. [10], Corollary 5.3; [9]). Let , , , and . Then, .

Definition 54 (cf. [10], (5.21), (5.33)). Let , , and . The spaces and are the sets of all such thatorcorrespondingly.
We will require that for and for . Otherwise, the corresponding space consists only of the null element. Similar definitions can be found in [8, 9, 17]. We refer to the spaces and as and spaces, respectively. Note that the spaces are a special case of the generalized gamma space with double weights [3].

In order to be able to compare our results with the results from [3, 4, 7–9], we introduce the generalized grand and small Lorentz spaces. Note that the interpolation results in [3, 4] do not contain limiting cases .

Definition 55. Let , , and . We define the small Lorentz space as the set of all such thatWe define the grand Lorentz space as the set of all such thatIt is clear that and . Grand and small Lebesgue and Lorentz spaces find many different important applications and have been intensively studied by many authors (see [3, 4, 6–9] and the references therein). Normally, they are defined on a bounded domain in with measure 1. In [4], it is required that the functions are real-valued. We do not require that and neither that the functions are real-valued.

The following lemma characterizes the and spaces as appropriate and limiting interpolation spaces. It follows from the corresponding definitions, formula (254), and Lemma 51.

Lemma 56 (see [10], Lemmas 5.4 and 5.9). Let , , , and . Then, and .
In particular, and .

Definition 57. Let , , and . The spaces , , , and are the sets of all such thatorcorrespondingly.

Remark 58. Because of Lemma 51, if in the formulae (257), (259), and (261) will be replaced with , one gets equivalent (quasi-)norms for corresponding spaces.

In view of Lemma 51, Remark 58, and (254), the following analogue of Lemma 56 holds.

Lemma 59. Let , , , and . Then,

9.2. Limiting Interpolation between the Small or Great Lorentz Spaces and the Lorentz–Karamata Spaces

Using results from Sections 4–6 and Lemmas 56 and 59, we are able to characterize some limiting interpolation spaces lying between or spaces and Lorentz–Karamata spaces. In the corollaries below, we restrict ourselves to the grand and small Lorentz spaces.

Corollary 60 (cf. [9], Corollary 7.8; [7], Corollary 5.12). Let , , , and . Put and .(i)If , thenwhere .(ii)If , then

Proof. Let . By Lemmas 56 and 59 and Theorem 25, we haveSimilarly, by [5] (Theorem 13) and Lemma 53, we getThis completes the proof.☐

Corollary 61 (cf. [7], Corollary 5.10). Let , , , and . Put and . Then,(i)If , thenwhere .(ii)If , then

Proof. Let and . By Lemmas 56, 53, and 59 and Theorem 23, we haveSimilarly, by [5] (Theorem 11), we getThis completes the proof.☐

Analogously, by using Theorems 28 and 26 and [5] (Theorem 23), the next two corollaries can be proved.

Corollary 62. (cf. [6], Corollary 5.9). Let , , , and . Put . Then,(i)If , then(ii)If , then

Corollary 63 (cf. [6], Corollary 5.5). Let , , , , and . Put . Then,

Four corollaries above amend Corollaries 38–40 and 47–49 from [5]. Other corollaries from Sections 8.2 and 8.3 of [5] can be similarly amended.

9.3. Limiting Interpolation Formulae for Couples Where Both Operands Are Small or Grand Lorentz Spaces

Four corollaries below amend corollaries from Section 8.4 of [5]. We prove only the first one. Three further corollaries can be proved analogously using theorems from Section 7.

Corollary 64 (cf. [6], Corollary 5.12; [8], Corollary 5.7). Let , , , , and . Put , , and .(i)If , thenwhere .(ii)If , then

Proof. Let and . Using Lemmas 56 and 59 and Theorem 33, we getSimilarly,This completes the proof.☐

Corollary 65 (cf. [6], Corollary 5.7; [8], Corollary 5.5). Let , , , , and . Put , , and .(i)If , then(ii)If , thenwhere .

Corollary 66 (cf. [7], Theorem 5.7; [8], Corollary 5.10). Let , , , , and . Put , , and .(i)If , thenwhere .(ii)If , thenwhere .

Corollary 67 (cf. [8], Corollary 5.9). Let , , , , and . Put , , and .(i)If , then(ii)If , then

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

The author is grateful to Pedro Fernández-Martínez and Teresa Signes for the possibility to read the paper [8] before it was posted in arXiv and for the productive discussion about our papers. We address a grateful thanks to Teresa Signes for pointing out a wrong statement in [5]. See Remark 39.