# Reiteration Formulae for the Real Interpolation Method IncludingL or R Limiting Spaces

Authors: Leo R. Ya. Doktorski
Journal: Journal of Function Spaces
 (2020)

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## Abstract

We consider a real interpolation method defined by means of slowly varying functions. We present some reiteration formulae including so-calledL or R limiting interpolation spaces. These spaces arise naturally in reiteration formulae for the limiting cases θ=0 or θ=1. Applications to grand and small Lorentz spaces are given.


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## Body

## 1. Introduction

LetA¯≔A0,A1 be a compatible couple of (quasi-)Banach spaces such that A0⋂A1≠0. The Peetre K-functional on A0+A1 is given by(1)Kt,f;A¯≡Kt,f≔inff0A0+tf1A1f=f0+f1,fi∈Ai,i=0,1,t>0.The classical (Lions-Peetre) scale of interpolation spacesA¯θ,q is defined via the (quasi-)norms(2)fθ,q≔u−θ−1/qKu,fq,0,∞,where ∗q,a,b0<q≤∞,−∞≤a<b≤∞ is the usual (quasi-)norm in the Lebesgue space Lq on the interval (a,b). For further information about basic properties of the K-functional and the real interpolation method, we refer to, e.g., [1].Above definition requires0<θ<1 or q=∞ for θ∈0,1, but limiting real interpolation spaces play an important role in certain questions in analysis. See for example [2–13] and references therein. One of the possibilities to consider the limiting cases θ∈0,1 for q<∞ is to involve an additional factor au in the formula (2).(3)fθ,q;a≔u−θ−1/qauKu,fq,0,∞.For this factor, various functions have been considered: logarithms [2–4], broken-logarithmic functions [5], or more generally slowly varying functions [6–10] (see Definition 5 below). In this way, one gets a scale A¯θ,q;a, where the limiting cases θ∈0,1 for q<∞ have sense. An important property of this scale is that the reiteration spaces A¯θ0,q0,b0,A¯θ1,q1,b1θ,r,a with 0<θ0<θ1<1 and 0<θ<1 belong to the same scale. However, for limiting cases θ∈0,1, the reiteration spaces normally do not belong to this scale and new interpolation functors are needed to describe them. Thus, the limiting reiteration formulae lead in a natural way to new interpolation spaces. Following [7], we call them R and Lspaces (see Definition 6 below).This paper is principally aimed at establishing reiteration formulae for couples where one or both operands areL or Rspaces. This was motivated by articles [11–13], where it is shown that the so-called grand and small Lorentz spaces can be described in terms of the R and Lspaces. Other motivation was the paper [9], which gives a good technical tool to prove Holmstedt’s formulae for the 𝐾-functional, which are needed to establish the reiteration formulae.This paper is organized as follows. Section2 contains some notations, definitions, and technical results. In Section 3, we collect necessary definitions and statements dealing with the real interpolation method involving slowly varying functions. In Sections 4 and 5, we establish reiteration formulae for the couples, where one of the operands is either L or R space. Reiteration formulae for the couples, where both operands are the L or R space are established in Section 6. Section 7 discussed how to obtain from our general theorems the ones for an ordered couple of (quasi-)Banach spaces. Finally, in Section 8, we present some interpolation results for the grand and small Lebesgue spaces as applications of our general reiteration theorems.


## 2. Preliminaries

We writeX⊂Y for two quasi-normed spaces X and Y to indicate that X is continuously embedded in Y. The notation X≅Y means that X⊂Y and Y⊂X. If f and g are positive functions, we write f≺g if f≤Cg, where the constant C is independent on all significant quantities. We say that two functions f and g are equivalent (f≈g) if f≺g and g≺f. We write f↑ (f↓) if the positive function f is nondecreasing (nonincreasing).


### 2.1. Slowly Varying Functions

We will need definition and some basic properties of slowly varying functions. For further information about slowly varying functions, we refer to, e.g., [6–8].Definition 1.
We say that a positive Lebesgue measurable functionb is slowly varying on (0,∞) (or on (0,1)), notation b∈SV0,∞≡SV (or b∈SV0,1), if, for each ε>0, the function tεbt is equivalent to an increasing function while the function t−εbt is equivalent to a decreasing function.Lemma 2.
Letb,b1,b2∈SV, r>0, α>0, 0<q≤∞, and t∈0,∞.(i)
Then,br∈SV, b1/t∈SV, btrb1t∈SV, and b1b2∈SV(ii)
Iff≈g, then b∘f≈b∘g(iii)
uα−1/qbuq,0,t≈tαbt and u−α−1/qbuq,t,∞≈t−αbt(iv)
The functionsu−1/qbuq,0,t and u−1/qbuq,t,∞ (if exist) belong to SV and bt≺u−1/qbuq,0,t and bt≺u−1/qbuq,t,∞Remark 3.
Leta,b∈SV, λ>0, and ρt=tλat.(i)
We may assume without loss of generality that the functiona is continuous(ii)
Due to ([10], Lemma 3.1), there exists a strongly increasing, differentiable function σt≈ρt, such that σ+0=0, σ∞=∞, and(4)σ′t≈σt1tObviously, the functionσ has inverse σ−1 and it holds(5)σ−1′t≈σ−1t1t.(iii)
It is not difficult to show that for any compatible coupleA¯, it holds(6)Kρt,f;A¯≈Kσt,f;A¯for all f∈A0+A1 and t∈0,∞(iv)
Using ([6], Proposition 2.2 (ii)), it can be shown that b°σ∈SV and b°σ−1∈SV(v)
Let0≤θ≤1, 0<r≤∞. Consider the expression ρt−θt−1/rbρtK¯ρt,fr,0,∞. Equation (6) implies that for all f∈A0+A1(7)ρt−θt−1/raρtK¯ρt,fr,0,∞≈σt−θt−1/raσtK¯σt,fr,0,∞.
Using change of variablesu=σt and (4), we get(8)σt−θt−1/raσtK¯σt,fr,0,∞≈u−θ−1/rauK¯u,fr,0,∞.
Thus,(9)ρt−θt−1/rbρtK¯ρt,fr,0,∞≈u−θ−1/rauK¯u,fr,0,∞.
We will use the last property when making the change of variables.


### 2.2. Hardy-Type Inequalities

We need the following Hardy-type inequalities ([6], Lemma 2.7).Lemma 4.
Let1≤P≤∞ and b∈SV.(i)
The inequality(10)tν−1/Pbt∫0tguduP,0,∞≺tν+1−1/PbtgtP,0,∞holds for all (Lebesgue-) measurable nonnegative functions g on (0,∞) if and only if ν<0(ii)
The inequality(11)tν−1/Pbt∫t∞guduP,0,∞≺tν+1−1/PbtgtP,0,∞holds for all (Lebesgue-) measurable nonnegative functions g on (0,∞) if and only if ν>0.


## 2.1. Slowly Varying Functions

We will need definition and some basic properties of slowly varying functions. For further information about slowly varying functions, we refer to, e.g., [6–8].Definition 1.
We say that a positive Lebesgue measurable functionb is slowly varying on (0,∞) (or on (0,1)), notation b∈SV0,∞≡SV (or b∈SV0,1), if, for each ε>0, the function tεbt is equivalent to an increasing function while the function t−εbt is equivalent to a decreasing function.Lemma 2.
Letb,b1,b2∈SV, r>0, α>0, 0<q≤∞, and t∈0,∞.(i)
Then,br∈SV, b1/t∈SV, btrb1t∈SV, and b1b2∈SV(ii)
Iff≈g, then b∘f≈b∘g(iii)
uα−1/qbuq,0,t≈tαbt and u−α−1/qbuq,t,∞≈t−αbt(iv)
The functionsu−1/qbuq,0,t and u−1/qbuq,t,∞ (if exist) belong to SV and bt≺u−1/qbuq,0,t and bt≺u−1/qbuq,t,∞Remark 3.
Leta,b∈SV, λ>0, and ρt=tλat.(i)
We may assume without loss of generality that the functiona is continuous(ii)
Due to ([10], Lemma 3.1), there exists a strongly increasing, differentiable function σt≈ρt, such that σ+0=0, σ∞=∞, and(4)σ′t≈σt1tObviously, the functionσ has inverse σ−1 and it holds(5)σ−1′t≈σ−1t1t.(iii)
It is not difficult to show that for any compatible coupleA¯, it holds(6)Kρt,f;A¯≈Kσt,f;A¯for all f∈A0+A1 and t∈0,∞(iv)
Using ([6], Proposition 2.2 (ii)), it can be shown that b°σ∈SV and b°σ−1∈SV(v)
Let0≤θ≤1, 0<r≤∞. Consider the expression ρt−θt−1/rbρtK¯ρt,fr,0,∞. Equation (6) implies that for all f∈A0+A1(7)ρt−θt−1/raρtK¯ρt,fr,0,∞≈σt−θt−1/raσtK¯σt,fr,0,∞.
Using change of variablesu=σt and (4), we get(8)σt−θt−1/raσtK¯σt,fr,0,∞≈u−θ−1/rauK¯u,fr,0,∞.
Thus,(9)ρt−θt−1/rbρtK¯ρt,fr,0,∞≈u−θ−1/rauK¯u,fr,0,∞.
We will use the last property when making the change of variables.


## 2.2. Hardy-Type Inequalities

We need the following Hardy-type inequalities ([6], Lemma 2.7).Lemma 4.
Let1≤P≤∞ and b∈SV.(i)
The inequality(10)tν−1/Pbt∫0tguduP,0,∞≺tν+1−1/PbtgtP,0,∞holds for all (Lebesgue-) measurable nonnegative functions g on (0,∞) if and only if ν<0(ii)
The inequality(11)tν−1/Pbt∫t∞guduP,0,∞≺tν+1−1/PbtgtP,0,∞holds for all (Lebesgue-) measurable nonnegative functions g on (0,∞) if and only if ν>0.


## 3. Real Interpolation Method Involving Slowly Varying Functions: A Short Overview

Here, we collect necessary definitions and statements dealing with the real interpolation method involving slowly varying functions. We consider a compatible couple of (quasi-)Banach spacesA¯=A0,A1 such that A0⋂A1≠0 and use the Peetre K-functional on A0+A1, given by (1).


### 3.1. Standard Interpolation Spaces

Definition 5 (see [6]).
Let0≤θ≤1, 0<q≤∞, and b∈SV. We write(12)A¯θ,q;b≡A0,A1θ,q;b≔f∈A0+A1:fθ,q;b≔u−θ−1/qbuKu,fq,0,∞<∞.
It is known [6] that the spaces A¯θ,q;b are intermediate spaces between A0 and A1 if and only if one of the following conditions is satisfied:(i)
0<θ<1(ii)
θ=0 and u−1/qbuq,1,∞<∞(iii)
θ=1 and u−1/qbuq,0,1<∞
We refer to the spacesA¯θ,q;b as standard interpolation spaces.


### 3.2.R and L Limiting Interpolation Spaces

Definition 6 (see [6]).
Let0<r,q≤∞, 0<σ<1, and a,b∈SV. We write(13)A¯σ,r,b,q,aL≡A0,A1σ,r,b,q,aL≔f∈A0+A1:fL;σ,r,b,q,a≔t−1/rbtatu−σ−1/qauKu,fq,0,tr,0,∞<∞.
Similarly,(14)A¯σ,r,b,q,aR≡A0,A1σ,r,b,q,aR≔f∈A0+A1:fR;σ,r,b,q,a≔t−1/rbtatu−σ−1/qauKu,fq,t,∞r,0,∞<∞.
It is known [6] that the spaces A¯σ,r,b,q,aL (A¯σ,r,b,q,aR) are intermediate spaces between A0 and A1 if and only if s−1/rbs/asr,1,∞<∞ (s−1/rbs/asr0,1<∞, respectively).Similar definitions can be found, for example, [2, 3, 5, 7, 8, 10]. In those papers, the reader can find further properties of these spaces. Following [7], we refer to the spaces A¯∗L and A¯∗R as L and R spaces (or L and R limiting interpolation spaces).


### 3.3. Some Known Formulae and Reiteration Theorems

Under suitable conditions, the following formulae hold ([1], Chap. 5, Proposition 1.2) [6]:(15)Kt,f;A0,A1=tKt−1,f;A1,A0,t>0,(16)A0,A1θ,q,b=A1,A01−θ,q,b~,(17)A0,A1σ,r,b,q,aL=A1,A01−σ,r,b~,q,a~R,whereb~t=bt−1.The following reiteration theorems in different modifications can be found, e.g., in [6, 8, 10, 13]. We formulate them in the form we will use them.Theorem 7.
(see ([6], (3.4))). Let 0<θ<1, 0<q0,q1,r≤∞, a,b0,b1∈SV, 0<θ0<θ1<1. Put ρt=tθ1−θ0b0t/b1t. Then,(18)A¯θ0,q0,b0,A¯θ1,q1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#=b01−θb1θa°ρ.Theorem 8.
(Cf. ([8], Theorems 5.10 and 5.12), ([13], Lemma 5.1.), [10]). Let 0<q,r≤∞ and a,b∈SV.(i)
Ift−1/qbtq,1,∞<∞ and 0<θ≤1, then(19)A¯0,q,b,A1θ,r,a≅A¯θ,r,a#,where a#t=s−1/qbsq,t,∞1−θats−1/qbsq,t,∞(ii)
Ift−1/qbtq,0,1<∞ and 0≤θ<1, then(20)A0,A¯1,q,bθ,r,a≅A¯θ,r,a#,where a#t=s−1/qbsq,0,tθat/s−1/qbsq,0,t.Theorem 9.
Let0<θ0<1, 0<q0,q1,r≤∞, and a,b0,b1∈SV.(i)
(Cf. ([7], Theorem 5.4) and ([6], (4.13))) If 0<θ≤1, then(21)A¯θ0,q0,b0,A1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ, a#t=b0t1−θaρt, and ρt=t1−θ0b0t(ii)
(Cf. ([8], Theorem 5.2)) If 0<θ<1 and s−1/q1b1sq1,0,1<∞, then(22)A¯θ0,q0,b0,A¯1,q1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ, a#t=b0t1−θs−1/q1b1sq1,0,tθaρt, and ρt=t1−θ0b0t/s−1/q1b1sq1,0,t.Theorem 10.
(Cf. ([6], (3.21) and Theorem 3.2) and ([8], (5.4))). Suppose that 0<θ0<θ1<1, 0<r0,q0,q1≤∞, a0,a1,b0∈SV, and s−1/r0b0s/a0sr0,1,∞<∞.(i)
Letσ be a strongly increasing, differentiable function such that σt≈t1−θ0a0t and a#=b0∘σ−1/a0∘σ−1. Then,(23)A¯θ0,q0,a0,A10,r0,a#≅A¯θ0,r0,b0,q0,a0L(ii)
Lets−1/q1a1sq1,0,1<∞, σ be a strongly increasing, differentiable function such that σt≈t1−θ0a0t/s−1/q1a1sq1,0,t and a#=b0∘σ−1/a0∘σ−1. Then,(24)A¯θ0,q0,a0,A1,q1,a10,r0,a#≅A¯θ0,r0,b0,q0,a0L(iii)
Letσ be a strongly increasing, differentiable function such that σt≈tθ1−θ0a0t/a1t and a#=b0∘σ−1/a0∘σ−1. Then,(25)A¯θ0,q0,a0,A¯θ1,q1,a10,r0,a#≅A¯θ0,r0,b0,q0,a0LProof.
We prove only the first statement. Letρt=t1−θ0a0t and σ≈ρ. Note that such function σ exists (see Remark 3). By ([6], (3.21)), we have to show that a0a#°ρ≈b0. Due to Lemma 2, a#°σ∈SV and a#°ρ≈a#°σ. Thus,(26)a0a#°ρt≈a0a#°σt=a0tb0σ−1σta0σ−1σt=b0t.
The statements (ii) and (iii) can be proved analogously.


## 3.1. Standard Interpolation Spaces

Definition 5 (see [6]).
Let0≤θ≤1, 0<q≤∞, and b∈SV. We write(12)A¯θ,q;b≡A0,A1θ,q;b≔f∈A0+A1:fθ,q;b≔u−θ−1/qbuKu,fq,0,∞<∞.
It is known [6] that the spaces A¯θ,q;b are intermediate spaces between A0 and A1 if and only if one of the following conditions is satisfied:(i)
0<θ<1(ii)
θ=0 and u−1/qbuq,1,∞<∞(iii)
θ=1 and u−1/qbuq,0,1<∞
We refer to the spacesA¯θ,q;b as standard interpolation spaces.


## 3.2.R and L Limiting Interpolation Spaces

Definition 6 (see [6]).
Let0<r,q≤∞, 0<σ<1, and a,b∈SV. We write(13)A¯σ,r,b,q,aL≡A0,A1σ,r,b,q,aL≔f∈A0+A1:fL;σ,r,b,q,a≔t−1/rbtatu−σ−1/qauKu,fq,0,tr,0,∞<∞.
Similarly,(14)A¯σ,r,b,q,aR≡A0,A1σ,r,b,q,aR≔f∈A0+A1:fR;σ,r,b,q,a≔t−1/rbtatu−σ−1/qauKu,fq,t,∞r,0,∞<∞.
It is known [6] that the spaces A¯σ,r,b,q,aL (A¯σ,r,b,q,aR) are intermediate spaces between A0 and A1 if and only if s−1/rbs/asr,1,∞<∞ (s−1/rbs/asr0,1<∞, respectively).Similar definitions can be found, for example, [2, 3, 5, 7, 8, 10]. In those papers, the reader can find further properties of these spaces. Following [7], we refer to the spaces A¯∗L and A¯∗R as L and R spaces (or L and R limiting interpolation spaces).


## 3.3. Some Known Formulae and Reiteration Theorems

Under suitable conditions, the following formulae hold ([1], Chap. 5, Proposition 1.2) [6]:(15)Kt,f;A0,A1=tKt−1,f;A1,A0,t>0,(16)A0,A1θ,q,b=A1,A01−θ,q,b~,(17)A0,A1σ,r,b,q,aL=A1,A01−σ,r,b~,q,a~R,whereb~t=bt−1.The following reiteration theorems in different modifications can be found, e.g., in [6, 8, 10, 13]. We formulate them in the form we will use them.Theorem 7.
(see ([6], (3.4))). Let 0<θ<1, 0<q0,q1,r≤∞, a,b0,b1∈SV, 0<θ0<θ1<1. Put ρt=tθ1−θ0b0t/b1t. Then,(18)A¯θ0,q0,b0,A¯θ1,q1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#=b01−θb1θa°ρ.Theorem 8.
(Cf. ([8], Theorems 5.10 and 5.12), ([13], Lemma 5.1.), [10]). Let 0<q,r≤∞ and a,b∈SV.(i)
Ift−1/qbtq,1,∞<∞ and 0<θ≤1, then(19)A¯0,q,b,A1θ,r,a≅A¯θ,r,a#,where a#t=s−1/qbsq,t,∞1−θats−1/qbsq,t,∞(ii)
Ift−1/qbtq,0,1<∞ and 0≤θ<1, then(20)A0,A¯1,q,bθ,r,a≅A¯θ,r,a#,where a#t=s−1/qbsq,0,tθat/s−1/qbsq,0,t.Theorem 9.
Let0<θ0<1, 0<q0,q1,r≤∞, and a,b0,b1∈SV.(i)
(Cf. ([7], Theorem 5.4) and ([6], (4.13))) If 0<θ≤1, then(21)A¯θ0,q0,b0,A1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ, a#t=b0t1−θaρt, and ρt=t1−θ0b0t(ii)
(Cf. ([8], Theorem 5.2)) If 0<θ<1 and s−1/q1b1sq1,0,1<∞, then(22)A¯θ0,q0,b0,A¯1,q1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ, a#t=b0t1−θs−1/q1b1sq1,0,tθaρt, and ρt=t1−θ0b0t/s−1/q1b1sq1,0,t.Theorem 10.
(Cf. ([6], (3.21) and Theorem 3.2) and ([8], (5.4))). Suppose that 0<θ0<θ1<1, 0<r0,q0,q1≤∞, a0,a1,b0∈SV, and s−1/r0b0s/a0sr0,1,∞<∞.(i)
Letσ be a strongly increasing, differentiable function such that σt≈t1−θ0a0t and a#=b0∘σ−1/a0∘σ−1. Then,(23)A¯θ0,q0,a0,A10,r0,a#≅A¯θ0,r0,b0,q0,a0L(ii)
Lets−1/q1a1sq1,0,1<∞, σ be a strongly increasing, differentiable function such that σt≈t1−θ0a0t/s−1/q1a1sq1,0,t and a#=b0∘σ−1/a0∘σ−1. Then,(24)A¯θ0,q0,a0,A1,q1,a10,r0,a#≅A¯θ0,r0,b0,q0,a0L(iii)
Letσ be a strongly increasing, differentiable function such that σt≈tθ1−θ0a0t/a1t and a#=b0∘σ−1/a0∘σ−1. Then,(25)A¯θ0,q0,a0,A¯θ1,q1,a10,r0,a#≅A¯θ0,r0,b0,q0,a0LProof.
We prove only the first statement. Letρt=t1−θ0a0t and σ≈ρ. Note that such function σ exists (see Remark 3). By ([6], (3.21)), we have to show that a0a#°ρ≈b0. Due to Lemma 2, a#°σ∈SV and a#°ρ≈a#°σ. Thus,(26)a0a#°ρt≈a0a#°σt=a0tb0σ−1σta0σ−1σt=b0t.
The statements (ii) and (iii) can be proved analogously.


## 4. Interpolation between theL Spaces and the Standard Interpolation Spaces

At first, we establish reiteration formulae for the couples in which theL space is the first operand.Theorem 11.
Let0<θ0<θ1<1, 0<q0,r0,r1,r≤∞, a,a0,b0,b1∈SV, and s−1/r0b0s/a0sr0,1,∞<∞. Put c0t=a0ts−1/r0b0s/a0sr0,t,∞ and ρt=tθ1−θ0c0t/b1t.(i)
If0<θ<1 then(27)A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t=c0t1−θb1tθaρt(ii)
If additionallys−1/rasr,0,1<∞, then(28)A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b11,r,a≅A→θ1,r,a#,r1,b1R,where a#t=b1taρtProof.
DenoteX0=A¯θ0,q0,a0, X1=A¯θ1,r1,b1, and Z=A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b1θ,r,a. Let σ be a strongly increasing, differentiable function such that σt≈tθ1−θ0a0t/b1t (see Remark 3 (ii)) and c=b0∘σ−1/a0∘σ−1. By Theorem 10 (iii), we obtain(29)A→θ0,r0,b0,q0,a0L≅A¯θ0,q0,a0,A¯θ1,r1,b10,r0,c=X¯0,r0,c.
Hence,Z≅X¯0,r0,c,X1θ,r,a. Using Theorem 8 (i), we conclude that for 0<θ≤1, it holds(30)Z≅X¯θ,r,d=A¯θ0,q0,a0,A¯θ1,r1,b1θ,r,d,where dt=s−1/r0csr0,t,∞1−θats−1/r0csr0,t,∞.
By change of variablesu=σ−1s (see Remark 3), we arrive at(31)s−1/r0csr0,σt,∞≈u−1/r0b0ua0ur0,t,∞.
Thus, becauseσtu−1/r0b0u/a0ur0,t,∞≈ρt,(32)dσt≈u−1/r0b0ua0ur0,t,∞1−θaρt.(i)
Let0<θ<1. By (31) and Theorem 7, it follows Z≅A¯η,r,a#, where a#t=a0t1−θb1tθdσt. Hence, by (33),(33)a#t≈a0tu−1/r0b0ua0ur0,t,∞1−θb1tθaρt=c0t1−θb1tθaρt.(ii)
Letθ=1. From (31), we have(34)A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b11,r,a≅A¯θ0,q0,a0,A¯θ1,r1,b11,r,d,where dt=ats−1/r0csr0,t,∞. By ([6], (4.7)), we conclude that(35)A¯θ0,q0,a0,A¯θ1,r1,b11,r,d≅A¯θ1,r,a#,r1,b1R,where by (33),(36)a#t=b1tdσt≈b1taρt.
This completes the proof.Theorem 12.
Let0<θ,θ0<1, 0<q0,r0,r1,r≤∞, a,a0,b0,b1∈SV, s−1/r0b0s/a0sr0,1,∞<∞, and s−1/r1b1sr1,0,1<∞. Put c0t=a0ts−1/r0b0s/a0sr0,t,∞, c1t=s−1/r1b1sr1,0,t, and ρt=t1−θ0c0t/c1t. Then,(37)A¯θ0,r0,b0,q0,a0L,A¯1,r1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ and a#t=c0t1−θc1tθaρt.Proof.
DenoteX0=A¯θ0,q0,a0,X1=A¯1,r1,b1, and Z=A¯θ0,r0,b0,q0,a0L,A¯1,r1,b1θ,r,a. Let σ be a strongly increasing, differentiable function such that σt≈t1−θ0a0t/s−1/r1b1sr1,0,t and c=b0∘σ−1/a0∘σ−1. By Theorem 10 (ii), we have(38)A¯θ0,r0,b0,q0,a0L≅A¯θ0,q0,a0,A¯1,r1,b10,r0,c=X¯0,r0,c.
Hence,Z≅X¯0,r0,c,X1θ,r,a. Using Theorem 8 (i), we conclude that(39)Z≅X¯θ,r,d=A¯θ0,q0,a0,A¯1,r1,b1θ,r,d,where dt=s−1/r0csr0,t,∞1−θats−1/r0csr0,t,∞. By Theorem 9 (ii), we derive that(40)Z≅A¯θ0,q0,a0,A¯1,r1,b1θ,r,d≅A¯η,r,a#,where by (33),(41)a#t=a0t1−θc1tθdσt=a0ts−1/r0csr0,σt,∞1−θc1tθaσts−1/r0csr0,σt,∞≈c0t1−θc1tθaρt.
This completes the proof.The following theorem can be proved analogously.Theorem 13.
Let0<θ0<1, 0<θ≤1, 0<q0,r0,r≤∞, a,a0,b0∈SV, and s−1/r0b0s/a0sr0,1,∞<∞. If θ=1, we additionally assume that s−1/rasr,0,1<∞. Put c0t=a0ts−1/r0b0u/a0ur0,t,∞ and ρt=t1−θ0c0t. Then,(42)A¯θ0,r0,b0,q0,a0L,A1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ and a#t=c0t1−θaρt.Now we establish reiteration formulae for the couples in which theL space is the second operand. Here, we need corresponding Holmstedt-type formulae (Theorem 14 and Theorem 17), which will be proved based on the theorems from [9]. Because of this, we will adopt the notation from that paper.Theorem 14.
Let0<θ1<1, 0<q1,r1≤∞, a1,b1∈SV, and s−1/r1b1s/a1sr1,1,∞<∞. Put c1t=a1ts−1/r1b1s/a1sr1,t,∞ and ρt=tθ1/c1t. Then, for all f∈A0+A¯θ1,r1,b1,q1,a1L and t>0,(43)Kρt,f;A0,A¯θ1,r1,b1,q1,a1L≈Kt,f+ρts−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞.Proof.
We will often be using the condition0<θ1<1 and the basic properties of slowly varying functions formulated in Lemma 2. Let Φ1 be the function space corresponding to A¯θ1,r1,b1,q1,a1L:(44)F∗Φ1=s−1/r1b1sa1su−θ1−1/q1a1uFuq1,0,sr1,0,∞.
Consider the functionsJt,f, g1t, and h1t.(45)Jt,f≔χt,∞∗K∗,fΦ1=s−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞,g1t≔tχt,∞∗Φ1=ts−1/r1b1sa1su−θ1−1/q1a1uq1,t,sr1,t,∞≤ts−1/r1b1sa1su−θ1−1/q1a1uq1,t,∞r1,t,∞≈t1−θ1c1t,h1t≔∗χ0,t∗Φ1=s−1/r1b1sa1su1−θ1−1/q1a1uq1,0,mint,sr1,0,∞≈s−1/r1b1sa1smint,s1−θ1a1mint,sr1,0,∞≈s1−θ1−1/r1b1sr1,0,t+t1−θ1a1ts−1/r1b1sa1sr1,t,∞≈t1−θ1b1t+t1−θ1c1t.
Becauseb1t=a1tb1t/a1t≺a1ts−1/r1b1s/a1sr1,t,∞=c1t, we get(46)g1t≺h1t≈t1−θ1c1t.
Thus, we arrive at(47)g1t+h1t≈t1−θ1c1t=tρt.
Hence, by ([9], Theorem 3), it yields(48)Kρt,f;A0,A¯θ1,r1,b1,q1,a1L≈Kt,f+ρts−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞.
This completes the proof.Remark 15.
Using the technique from the proof of [14, Lemma 2.12], it can be shown that the estimate (43) can be improved to the form (cf. [14, (3.20)])(49)Kρt,f;A0,A¯θ1,r1,b1,q1,a1L≈ρts−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞.Theorem 16.
Let0<θ1<1, 0≤θ<1, 0<q1,r,r1≤∞, a,a1,b1∈SV, and s−1/r1b1s/a1sr1,1,∞<∞. If θ=0, we additionally assume that s−1/rasr,1,∞<∞. Put c1t=a1ts−1/r1b1s/a1sr1,t,∞ and ρt=tθ1/c1t. Then,(50)A0,A¯θ1,r1,b1,q1,a1Lθ,r,a≅A¯η,r,a#,where η=θθ1 and a#t=c1tθaρt.Proof.
DenoteX0=A0, X1=A¯θ1,r1,b1,q1,a1L, K¯t,f=Kt,f;X¯, and Z=A0,A¯θ1,r1,b1,q1,a1Lθ,r,a. By Remark 3 (v) and Theorem 14, we can write(51)fZ=t−θ−1/ratK¯u,fr,0,∞≈ρt−θt−1/raρtK¯ρt,fr,0,∞=t−η−1/rc1tθaρtK¯ρt,fr,0,∞≈I1+I2,where(52)I1≔t−η−1/rc1tθaρtKt,fr,0,∞=fη,r;a#and(53)I2≔tθ1−η−1/rc1tθ−1aρt∗s−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞r,0,∞.
The proof of the theorem is completed if we show thatI2≺I1. We have(54)s−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞≤s−1/r1b1sa1sr1,t,∞u−θ1−1/q1a1uKu,fq1,t,∞.Denotedt=c1tθ−1aρts−1/r1b1s/a1sr1,t,∞. Then, we get(55)I2≤tθ1−η−1/rdtu−θ1−1/q1a1uKu,fq1,t,∞r,0,∞≔I3.It is enough to show thatI3≺I1.Caseq1=r. Because θ1−η>0 and dta1t=c1tθaρt, we have(56)I3=tθ1−η−1/rdtu−θ1−1/ra1uKu,fr,t,∞r,0,∞=tθ1−η−1/rdtr,0,uu−θ1−1/ra1uKu,fr,0,∞≈u−η−1/rdua1uKu,fr,0,∞=I1.Caseq1<r. In this case, using Lemma 4 (ii) (Hardy-type inequality), we also get I3≺I1. The details are left to the reader.Caser<q1. We need the following estimate:(57)u−θ1−1/q1a1uKu,fq1,t,∞≺u−θ1−1/ra1uKu,fr,t,∞.Foru>t, because Ku,f↑, we have(58)u−θ1a1uKu,f≈Ku,fs−θ1−1/ra1sr,u,∞≤s−θ1−1/ra1sKs,fr,u,∞≤s−θ1−1/ra1sKs,fr,t,∞.Thus, (57) is proved for q1=∞:(59)supt<u<∞u−θ1a1uKu,f≺u−θ1−1/ra1uKu,fr,t,∞.Ifr<q1<∞ using the last estimate, we get(60)u−θ1−1/q1a1uKu,fq1,t,∞=∫t∞u−θ1a1uKu,fq1duu1/q1≤supt<u<∞u−θ1a1uKu,fq1−r/q1∫t∞u−θ1a1uKu,frduu1/q1≺u−θ1−1/ra1uKu,fr,t,∞q1−r/q1u−θ1−1/ra1uKu,fr,t,∞r/q1=u−θ1−1/ra1uKu,fr,t,∞.Using now (57) and applying Fubini’s theorem, we conclude I3≺I1.Theorem 17.
Let0<θ,θ1<1, 0<r,r0,r1,q1≤∞, a,a1,b0,b1∈SV, s−1/r0b0sr0,1,∞<∞, and s−1/r1b1s/a1sr1,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞, c1t=a1ts−1/r1b1s/a1sr1,t,∞, and ρt=tθ1c0t/c1t. Then,(61)A¯0,r0,b0,A¯θ1,r1,b1,q1,a1Lθ,r,a≅A¯η,r,a#,where η=θθ1 and a#t=c0t1−θc1tθaρt.Proof.
DenoteX0=A0, X1=A¯θ1,r1,b1,q1,a1L, Z=A¯0,r0,b0,A¯θ1,r1,b1,q1,a1Lθ,r,a. Let σ be a strongly increasing, differentiable function such that σt≈tθ1/c1t. By Theorem 15, we have(62)A¯0,r0,b0≅A0,A¯θ1,r1,b1,q1,a1L0,r0,d,where d=b0°σ−1. Hence, Z≅X¯0,r0,d,X1θ,r,a. Using now Theorem 8 (i), we get Z≅X¯θ,r,c, where ct=s−1/r0dsr0,t,∞1−θats−1/r0dsr0,t,∞. By the change of variables s=σu, we arrive at(63)s−1/r0dsr0,σt,∞≈u−1/r0dσur0,t,∞=u−1/r0b0ur0,t,∞=c0t.
Becausec1tθcσt≈c1tθc0t1−θaσtc0t≈a#t, using once more Theorem 15, we obtain(64)Z≅A0,A¯θ1,r1,b1,q1,a1Lθ,r,c≅A¯η,r,a#.
This completes the proof.Theorem 18.
Let0<θ0<θ1<1, 0<q1,r0,r1≤∞, a1,b0,b1∈SV, s−1/r0b1s/a1sr0,1,∞<∞. Put c1t=a1ts−1/r1b1s/a1sr1,t,∞ and ρt=tθ1−θ0b0t/c1t. Then, for all f ∈ A¯θ0,r0,b0+A¯θ1,r1,b1,q1,a1L and all t>0,(65)Kρt,f;A¯θ0,r0,b0,A¯θ1,r1,b1,q1,a1L≈u−θ0−1/r0b0uKu,fr0,0,t+ρts−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞.Proof.
Recall, we adopted the notation from the paper [9]. Let Φ0 be the function space corresponding to A¯θ0,r0,b0:(66)F∗Φ0=u−θ0−1/r0b0uFur0,0,∞.
Let us calculate the functionsIt,f, g0t, and h0t.(67)It,f≔χ0,t∗K∗,fΦ0=u−θ0−1/r0b0uKu,fr0,0,t,g0t≔tχt,∞∗Φ0=tu−θ0−1/r0b0ur0,t,∞≈t1−θ0b0t,h0t≔∗χ0,t∗Φ0=u1−θ0−1/r0b0ur0,0,t≈t1−θ0b0t.
Thus,g0t+h0t≈h0t≈g0t≈t1−θ0b0t.
LetΦ1 be the function space corresponding to A¯θ1,r1,b1,q1,a1L:(68)F∗Φ1=s−1/r1b1sa1su−θ1−1/q1a1uFuq1,0,sr1,0,∞.
From the proof of Theorem14, we have the following:(69)Jt,f=s−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞,g1t≺t1−θ1c1t,h1t≈t1−θ1c1t.
Now we estimate the functionsgt and ht.(70)1gt≔χt,∞uug0u+h0uΦ1≈s−1/r1b1sa1suθ0−θ1−1/q1a1ub0uq1,t,sr1,t,∞≤s−1/r1b1sa1suθ0−θ1−1/q1a1ub0uq1,t,∞r1,t,∞≈tθ0−θ1a1tb0ts−1/r1b1sa1sr1,t,∞=tθ0−θ1c1tb0t,ht≔χ0,tuug1u+h1uΦ0≈uθ1−θ0−1/r0b0uc1ur0,0,t≈tθ1−θ0b0tc1t=ρt.
We see thath0t≈g0t, ht≈ρt≺gt, and h0t/h1t≈ρt. Hence, by ([9], Theorem 4, Case 4), we get (65).Theorem 19.
Let0<θ0<θ1<1, 0<q1,r,r1,r0≤∞, a,a1,b0,b1∈SV, and s−1/r1b1s/a1sr1,1,∞<∞. Put c1t=a1ts−1/r1b1s/a1sr1,t,∞ and ρt=tθ1−θ0b0t/c1t.(i)
If0<θ<1 then(71)A¯θ0,r0,b0,A¯θ1,r1,b1,q1,a1Lθ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t=b0t1−θc1tθaρt(ii)
If additionallys−1/rasr,1,∞<∞, then(72)A¯θ0,r0,b0,A¯θ1,r1,b1,q1,a1L0,r,a≅A¯θ0,r,a#,r0,b0L,where a#t=b0taρtProof.
DenoteX0=A¯θ0,r0,b0, X1=A¯θ1,r1,b1,q1,a1L, K¯t,f=Kt,f;X¯, and Z=A¯θ0,r0,b0,A¯θ1,r1,b1,q1,a1Lθ,r,a. We write for shortness et=b0t−θc1tθaρt. By Remark 3 (v) and Theorem 17, we can write(73)fZ=u−θ−1/rauK¯u,fr,0,∞≈ρt−θt−1/raρtK¯ρt,fr,0,∞=t−θθ1−θ0−1/retK¯ρt,fr,0,∞≈I1+I2,where(74)I1≔t−θθ1−θ0−1/retu−θ0−1/r0b0uKu,fr0,0,tr,0,∞,and(75)I2≔t−θθ1−θ0−1/retρts−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞r,0,∞.
LetI3≔t−θθ1−θ0−θ0−1/retb0tKt,fr,0,∞. Because Kt,f/t↓ and 1−θ0>0, we conclude(76)t−θ0b0tKt,f≈1tKt,fu1−θ0−1/r0b0ur0,0,t≤u−θ0−1/r0b0uKu,fr0,0,t.
Thus,I3≺I1.(i)
Let0<θ<1. We have(77)I3=t−η−1/rb0t1−θc1tθaρtKt,fr,0,∞=fη,r,a#.
Thus, it is enough to show thatI2≺I3 and I1≺I3. Consider I2. Note that(78)s−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞≤s−1/r1b1sa1sr1,t,∞u−θ1−1/q1a1uKu,fq1,t,∞.
Denote(79)dt=etb0tc1ts−1/r1b1sa1sr1,t,∞=b0t1−θc1tθ−1s−1/r1b1sa1sr1,t,∞aρt.
We get(80)I2≤t1−θθ1−θ0−1/rdtu−θ1−1/q1a1uKu,fq1,t,∞r,0,∞=tθ1−η−1/rdtu−θ1−1/q1a1uKu,fq1,t,∞r,0,∞≔I4.The estimatesI4≺I3 and I1≺I3 can be proved by repeating the corresponding arguments of the proof of Theorem 15, using Lemma 4 (ii) or Lemma 4 (i).(ii)
Let nowθ=0. In this case, we have(81)I1=t−1/raρtu−θ0−1/r0b0uKu,fr0,0,tr,0,∞=fL;θ0,r,a#,r0,b0,I2=tθ1−θ0−1/raρtb0tc1ts−1/r1b1sa1su−θ1−1/q1a1uKu,fq1,t,sr1,t,∞r,0,∞,and(82)I3=t−θ0−1/raρtb0tKt,fr,0,∞.BecauseI3≺I1, it is enough to show that I2≺I3. Denoting dt=aρtb0t/a1t, we get(83)I2≤tθ1−θ0−1/raρtb0tc1t∗s−1/r1b1sa1sr1,t,∞u−θ1−1/q1a1uKu,fq1,t,∞r,0,∞=tθ1−θ0−1/rdtu−θ1−1/q1a1uKu,fq1,t,∞r,0,∞≔I4.The estimateI4≺I3 can be proved as above by applying Hardy-type inequality and Fubini’s theorem.


## 5. Interpolation between theR Spaces and the Standard Interpolation Spaces

The proofs of all theorems in this section are based on the corresponding reiteration theorems from the previous section and on the formulae (16) and (17).Theorem 20.
Let0<θ0<θ1<1, 0<q1,r0,r1,r≤∞, a,a1,b0,b1∈SV, and s−1/r1b1s/a1sr1,0,1<∞. Put c1t=a1ts−1/r1b1s/a1sr1,0,t and ρt=tθ1−θ0b0t/c1t.(i)
If0<θ<1 then(84)A¯θ0,r0,b0,A¯θ1,r1,b1,q1,a1Rθ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t≈b0t1−θc1tθaρt(ii)
If additionallys−1/rasr,1,∞<∞, then(85)A¯θ0,r0,b0,A¯θ1,r1,b1,q1,a1R0,r,a≅A→θ0,r,a#,r0,b0L,where a#t≈b0taρtTheorem 21.
Let0<θ,θ1<1, 0<q1,r0,r1,r≤∞, a,a1,b0,b1∈SV, s−1/r1b1s/a1sr1,0,1<∞, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞, c1t=a1ts−1/r1b1s/a1sr1,0,t, and ρt=tθ1c0t/c1t. Then,(86)A¯0,r0,b0,A¯θ1,r1,b1,q1,a1Rθ,r,a≅A¯η,r,a#,where η=θθ1 and a#t=c0t1−θc1tθaρt.Theorem 22.
Let0<θ1<1, 0≤θ<1, 0<q1,r1,r≤∞, a,a1,b1∈SV, and s−1/r1b1s/a1sr1,0,1<∞. If θ=0, we additionally assume that s−1/rasr,1,∞<∞. Put c1t=a1ts−1/r1b1s/a1sr1,0,t and ρt=tθ1/c1t. Then,(87)A¯0,A¯θ1,r1,b1,q1,a1Rθ,r,a≅A¯η,r,a#,where η=θθ1 and a#t=c1tθaρt.Theorem 23.
Let0<θ0<1, 0<θ≤1, 0<q0,r,r0≤∞, a,a0,b0∈SV, and s−1/r0b0s/a0sr0,0,1<∞. If θ=1, we additionally assume that s−1/rasr,0,1<∞. Put c0t=a0ts−1/r0b0s/a0sr1,0,t and ρt=t1−θ0c0t. Then,(88)A¯θ0,r0,b0,q0,a0R,A1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ and a#t=c0t1−θaρt.Theorem 24.
Let0<θ,θ0<1, 0<q0,r,r0,r1≤∞, a,b0,b1,a0∈SV, s−1/r0b0s/a0sr0,0,1<∞, and s−1/r1b1sr1,0,1<∞. Put c0t=a0ts−1/r0b0s/a0sr0,0,t, c1t=s−1/r1b1sr1,0,t, and ρt=t1−θ0c0t/c1t. Then,(89)A¯θ0,r0,b0,q0,a0R,A¯1,r1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θ and a#t=c0t1−θc1tθaρt.Theorem 25.
Let0<θ0<θ1<1, 0<q0,r,r1,r0≤∞, a,a0,b0,b1∈SV, and s−1/r0b0s/a0sr0,0,1<∞. Put c0t=a0ts−1/r0b0s/a0sr0,0,t and ρt=tθ1−θ0c0t/b1t.(i)
If0<θ<1 then(90)A¯θ0,r0,b0,q0,a0R,A¯θ1,r1,b1θ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t=c0t1−θb1tθaρt(ii)
If additionallys−1/rasr,0,1<∞, then(91)A¯θ0,r0,b0,q0,a0R,A¯θ1,r1,b11,r,a≅A¯θ1,r,a#,r1,b1R,where a#t=b1taρt


## 6. Reiteration Formulae for Couples Where Both Operands AreL or RSpaces

Theorem 26.
Let0<θ0<θ1<1, 0<θ<1, 0<r,r0,r1,q0,q1≤∞, a,a0,a1,b0,b1∈SV, s−1/r0b0s/a0sr0,1,∞<∞, and s−1/r1b1s/a1sr1,1,∞<∞. Put c0t=a0ts−1/r0b0s/a0sr0,t,∞, c1t=a1ts−1/r1b1s/a1sr1,t,∞, and ρt=tθ1−θ0c0t/c1t. Then,(92)A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b1,q1,a1Lθ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t=c0t1−θc1tθaρt.Proof.
LetX0=A¯θ0,q0,a0, X1=A¯θ1,r1,b1,q1,a1L, Z=A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b1,q1,a1Lθ,r,a. Let σ be a strongly increasing, differentiable function such that σt≈tθ1−θ0a0t/c1t and ct=b0σ−1t/a0σ−1t. By Theorem 18 (ii), we have(93)A¯θ0,r0,b0,q0,a0L≅A¯θ0,q0,a0,A¯θ1,r1,b1,q1,a1L0,r0,c=X¯0,r0,c.
Hence,Z≅X¯0,r0,c,X1θ,r,a. Because s−1/r0b0s/a0sr0,1,∞<∞, it holds s−1/r0ctr0,1,∞<∞. This means that we can apply Theorem 8 (i). So, we get Z≅X¯θ,r,d, where dt=s−1/r0ctr0,t,∞1−θats−1/r0ctr0,t,∞. By Theorem 18 (i), we obtain(94)Z≅A¯θ0,q0,a0,A¯θ1,r1,b1,q1,a1Lθ,r,d≅A¯η,r,a#,where a#t=a0t1−θc1tθdσt. By the change of variables u=σ−1s, we arrive at(95)s−1/r0csr0,σt,∞=s−1/r0b0σ−1sa0σ−1sr0,σt,∞≈u−1/r0b0ua0ur0,t,∞.
Thus, becauseσts−1/r0b0s/a0sr0,t,∞≈ρt, we conclude(96)dσt=s−1/r0csr0,σt,∞1−θaσts−1/r0csr0,σt,∞≈s−1/r0b0sa0sr0,t,∞1−θaρt.
Hence,a#t≈c0t1−θc1tθaρt.Repeating the previous proof with Theorem19 instead of Theorem 18 leads us to the following statement.Theorem 27.
Let0<θ0<θ1<1, 0<θ<1, 0<r,r0,r1,q0,q1≤∞, a,a0,a1,b0,b1∈SV, s−1/r0b0s/a0sr0,1,∞<∞, and s−1/r1b1s/a1sr1,0,1<∞. Put c0t=a0ts−1/r0b0s/a0sr0,t,∞, c1t=a1ts−1/r1b1s/a1sr1,0,t, and ρt=tθ1−θ0c0t/c1t. Then,(97)A¯θ0,r0,b0,q0,a0L,A¯θ1,r1,b1,q1,a1Rθ,r,a≅A¯η,r,a#,
whereη=1−θθ0+θθ1 and a#t=c0t1−θc1tθaρt.Using the symmetry arguments given by the formulae (16) and (17), the two following theorems can be proved.Theorem 28.
Let0<θ0<θ1<1, 0<θ<1, 0<r,r0,r1,q0,q1≤∞, a,a0,a1,b0,b1∈SV, s−1/r0b0s/a0sr0,0,1<∞, and s−1/r1b1s/a1sr1,0,1<∞. Put c0t=a0ts−1/r0b0s/a0sr0,0,t, c1t=a1ts−1/r1b1s/a1sr1,0,t, and ρt=tθ1−θ0c0t/c1t. Then,(98)A¯θ0,r0,b0,q0,a0R,A¯θ1,r1,b1,q1,a1Rθ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t=c0t1−θc1tθaρt.Theorem 29.
Let0<θ0<θ1<1, 0<θ<1, 0<r,r0,r1,q0,q1≤∞, a,a0,a1,b0,b1∈SV, s−1/r0b0s/a0sr0,0,1<∞, and s−1/r1b1s/a1sr1,1,∞<∞. Put c0t=a0ts−1/r0b0s/a0sr0,0,t, c1t=a1ts−1/r1b1s/a1sr1,t,∞, and ρt=tθ1−θ0c0t/c1t. Then,(99)A¯θ0,r0,b0,q0,a0R,A¯θ1,r1,b1,q1,a1Lθ,r,a≅A¯η,r,a#,where η=1−θθ0+θθ1 and a#t=c0t1−θc1tθaρt.


## 7. Ordered Couples

In this section, we assume that the coupleA0,A1 is ordered in the sense that A1⊂A0. This is the case for example if A0=L1Ω and A1=L∞Ω where Ω⊂Rn with Ω<∞. We briefly outline how the previous results need to be amended for this particular case.SinceA1⊂A0, it holds(100)Ku,f≈f0A0,u>1.Based on (100), it is natural to define interpolation spaces with integration over (0,1) (see the arguments in ([6], Section 7)).Definition 30.
LetA1⊂A0, 0≤θ≤1, 0<q≤∞, and b∈SV0,1. We put(101)A¯θ,q;b0,1≡A0,A1θ,q;b0,1≔f∈A0+A1:fθ,q;b;0,1=u−θ−1/qbuKu,fq,0,1<∞.Definition 31.
LetA1⊂A0, 0<r,q≤∞, 0<σ<1, and a,b∈SV0,1. We put(102)A¯σ,r,b,q,aL;0,1≔f∈A0+A1:fL;σ,r,b,q,a;0,1≔t−1/rbtatu−σ−1/qauKu,fq,0,tr,0,1<∞.
Similarly,(103)A¯σ,r,b,q,aR;0,1≔f∈A0+A1:fR;σ,r,b,q,a;0,1≔t−1/rbtatu−σ−1/qauKu,fq,t,1r,0,1<∞.
Using (99), it is not difficult to show that(i)
A¯θ,q;b≅A¯θ,q;b0,1,(ii)
ifs−1/qbsq,0,1<∞, then A¯0,q;b0,1≅A0,(iii)
A¯σ,r,b,q,aL;0,1⊂A¯σ,q;a0,1,(iv)
ifs−1/rbs/asr,0,1<∞, then A¯σ,r,b,q,aL;0,1≅A¯σ,q;a0,1(v)
ifs−1/rbs/asr0,1=∞, then A¯σ,r,b,q,aR;0,1=0(vi)
ifs−1/rbs/asr0,1<∞, then A¯σ,q;b0,1⊂A¯σ,r,b,q,aR;0,1,(vii)
ifs−1/rbs/asr,1,∞<∞, then A¯σ,r,b,q,aL≅A¯σ,r,b,q,aL;0,1,(viii)
ifs−1/rbs/asr0,1<∞, then A¯σ,r,b,q,aR≅A¯σ,r,b,q,aR;0,1
In the properties (i), (vii), and (viii), we assume that the functions fromSV0,1 are extended to suitable functions from SV0,∞.These properties show how all main theorems from previous sections can be reformulated for the case whenA1⊂A0. For example, Theorem 28 can be reformulated as follows.Theorem 32.
LetA1⊂A0, 0<θ0<θ1<1, 0<θ<1, 0<r,r0,r1,q0,q1≤∞, a,a0,a1,b0,b1∈SV0,1, s−1/r0b0s/a0sr0,0,1<∞, and s−1/r1b1s/a1sr1,0,1=∞. Put c0t=a0ts−1/r0b0s/a0sr0,0,t, c1t=a1t1+s−1/r1b1s/a1sr1,t,1, and ρt=tθ1−θ0c0t/c1t (0<t<1). Then,(104)A¯θ0,r0,b0,q0,a0R;0,1,A¯θ1,r1,b1,q1,a1L;0,1θ,r,a≅A¯η,r,a#0,1,where η=1−θθ0+θθ1 and a#t=c0t1−θc1tθaρt.


## 8. Applications

Here, we present some interpolation results for the grand and small Lebesgue spaces as applications of our general reiteration theorems. LetΩ,μ denote a totally σ-finite measure space with a nonatomic measure μ. We consider functions f from the set MΩ,μ of all μ-measurable functions on Ω. As conventional (see e.g., [1]), f∗tt>0 denotes the nonincreasing rearrangement of f.


### 8.1. The Lorentz–Karamata Spaces,LL and LR Spaces, and the Grand and Small Lorentz Spaces

Definition 33 (see ([6], Definition 5.1)).
Let0<p,q≤∞ and b∈SV. The Lorentz–Karamata space Lp,q;b≡Lp,q;bΩ is the set of all f∈MΩ,μ such that(105)fp,q;α≔t1/p−1/qbtf∗tq,0,∞<∞.
The Lorentz–Karamata spaceLp,q;1Ω coincides with the Lorentz space Lp,qΩ; it becomes the Lebesgue space LpΩ if b=1 and p=q.Definition 34.
(Cf. ([6], (5.21), (5.33)) ([3], Definition 7.1)). Let 0<p<∞, 0<q,r≤∞, and a,b∈SV. The spaces Lp,r,b,q,aL≡Lp,r,b,q,aLΩ and Lp,r,b,q,aR≡Lp,r,b,q,aRΩ are the sets of all f∈MΩ,μ such that(106)fLp,r,b,q,aL≔t−1/rbtatu1/p−1/qauf∗uq,0,tr,0,∞<∞or(107)fLpr,b,q,aR≔t−1/rbtatu1/p−1/qauf∗uq,t,∞r,0,∞<∞,correspondently.
We will require thatt−1/rbt/atr,1,∞<∞ for Lp,r,b,q,aL and t−1/rbt/atr,01<∞ for Lp,r,b,q,aR. Otherwise, the corresponding space consists only of the null element. Similar definitions can be found in [4, 5, 8]. We refer to the spaces L∗L and L∗R as LL and LR spaces, respectively.In order to be able to compare our results with the results from [8, 11, 13], we introduce the analogues of the grand and small Lorentz spaces. They find many different important applications and have been well studied by many authors (see [8, 11, 13] and the references therein). One can find different definitions of grand and small Lorentz spaces. Normally, they are defined for a bounded Lebesgue measurable domain Ω in Rn with measure 1. In [13], it is required that the functions are real valued. We do not require that μΩ=1 and that the functions are real valued. Note, that using arguments of Section 7, all corollaries below can be easy reformulated for the case when μΩ=1.Definition 35.
Let0<p<∞, 0<q,r≤∞, and b∈SV. The small Lorentz space Lbp,q,r is the set of all f∈MΩ,μ such that(108)fLbp,q,rΩ≔t−1/rbtu1/p−1/qf∗uq,0,tr,0,∞<∞.
The grand Lorentz spaceLbp,q,r is the set of all f∈MΩ,μ such that(109)fLbp,q,rΩ≔t−1/rbtu1/p−1/qf∗uq,t,∞r,0,∞<∞.
It is clear, thatLbp,q,r=Lp,r,b,q,1L and Lbp,q,r=Lp,r,b,q,1R.For simplicity, we consider below only interpolation spaces betweenL1 and L∞. Lemma 36 and Lemma 37 characterise the LL andLR spaces as appropriate L and R limiting interpolation spaces. Lemma 36 is a special case of ([6], Lemma 5.4). Lemma 37 can be proved analogously.Lemma 36.
Let1<p<∞, 0<q,r≤∞, a,b∈SV, and θ=1−1/p. Then,(110)Lp,r,b,q,aL≅L1,L∞θ,r,b,q,aL.
In particular,Lbp,q,r≅L1,L∞θ,r,b,q,1L.Lemma 37.
(Cf. ([3], Theorem 7.1) and ([6], Lemma 5.9)). Let 1<p<∞, 0<q,r≤∞, a,b∈SV, and θ=1−1/p. Then,(111)Lp,r,b,q,aR≅L1,L∞θ,r,b,q,aR,
In particular,Lbp,q,r≅L1,L∞θ,r,b,q,1R.Using Lemma36 and Lemma 37, we are able to characterise interpolation spaces lying between LL and LR spaces and Lorentz–Karamata spaces. In corollaries below, we restrict ourselves to grand and small Lorentz spaces. Moreover, for 0<θ<1, we formulate the formulae only for the classical case ∗,∗θ,r.


### 8.2. Interpolation between the Small Lorentz Space and the Lorentz Spaces

Corollary 38.
(Cf. ([13], Theorem 5.3)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0≤∞, b0∈SV, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞. Then,(112)Lb0p0,q0,r0,Lp1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θ.Proof.
LetA¯=L1,L∞, θi=1−1/pi, and η=1−θθ0+θθ1. By Lemma 36, we have(113)Lb0p0,q0,r0≅L1,L∞θ0,r0,b0,q0,1L=A¯θ0,r0,b0,q0,1L.
It is known ([6], Corollary 5.3) that A¯θ1,r1≅Lp1,r1. Using now Theorem 11 (i) and ([6], Corollary 5.3), we get(114)Lb0p0,q0,r0,Lp1,r1θ,r≅A¯θ0,r0,b0,q0,1L,A¯θ1,r1θ,r≅A¯η,r,a#≅Lp,r,a#.
This completes the proof.All corollaries below in this and the next subsections can be proved analogously using theorems from Sections4 and 5.Corollary 39.
Let1<p0<p1<∞, 0<r,r0,r1,q0≤∞, a,b0∈SV, s−1/rasr,0,1<∞, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞ and ρt=t1/p1−1/p0c0t. Then,(115)Lb0p0q0,r0,Lp1,r11,r;a≅La#p1,r1,r,where a#t=aρt.Corollary 40.
(Cf. ([8], Corollary 7.8)). Let 0<θ<1, 1<p0<∞, 0<r,r0,q0≤∞, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞. Then,(116)Lb0p0,q0,r0,L∞θ,r≅Lp,r,a#,where 1/p=1−θ/p0 and a#t=c0t1−θ.Corollary 41.
Let0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q1≤∞, b1∈SV, and s−1/r1b1sr1,1,∞<∞. Put c1t=s−1/r1b1sr1,t,∞. Then,(117)Lp0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c1tθ.Corollary 42.
Let1<p0<p1<∞, 0<r,r0,r1,q1≤∞, a,b1∈SV, s−1/r1b1sr1,1,∞<∞, and s−1/rasr,1,∞<∞. Put c1t=s−1/r1b1sr1,t,∞ and ρt=t1/p1−1/p0/c1t. Then,(118)Lp0,r0,Lb1p1,q1,r10,r;a≅La#p0,r0,r,where a#t=aρt.Corollary 43.
Let0<θ<1, 1<p1<∞, 0<r,r1,q1≤∞, and s−1/r1b1sr1,1,∞<∞. Put c1t=s−1/r1b1sr1,t,∞. Then,(119)L1,Lb1p1q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ+θ/p1 and a#t=c1tθ.


### 8.3. Interpolation between the Grand Lorentz Spaces and the Lorentz Spaces

Corollary 44.
Let0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q1≤∞, b1∈SV, and s−1/r1b1sr1,0,1<∞. Put c1t=s−1/r1b1sr1,0,t. Then,(120)Lp0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c1tθ.Corollary 45.
Let1<p0<p1<∞, 0<r,r0,r1,q1≤∞, b1∈SV, s−1/rasr,1,∞<∞, and s−1/r1b1sr1,0,1<∞. Put c1t=s−1/r1b1sr1,0,t and ρt=t1/p1−1/p0/c1t. Then,(121)Lp0,r0,Lb1p1,q1,r10,r;a≅La#p0,r0,r,where a#t=aρt.Corollary 46.
(Cf. ([8], Corollary 7.7)). Let 0<θ<1, 1<p1<∞, 0<r,r1,q1≤∞, b1∈SV, and s−1/r1b1sr1,0,1<∞. Put c1t=s−1/r1b1sr1,0,t. Then,(122)L1,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ+θ/p1 and a#t=c1tθ.Corollary 47.
(Cf. ([13], Theorem 7.3)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0≤∞, b0∈SV, and s−1/r0b0sr0,0,1<∞. Put c0t=s−1/r0b0sr0,0,t. Then,(123)Lb0p0,q0,r0,Lp1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θ.Corollary 48.
(Cf. ([11], Theorem 1.1)). Let 1<p0<p1<∞, 0<r,r0,r1,q0≤∞, a,b0∈SV, s−1/rasr,0,1<∞, and s−1/r0b0sr0,0,1<∞. Put c0t=s−1/r0b0sr0,0,t and ρt=t1/p1−1/p0c0t. Then,(124)Lb0p0,q0,r0,Lp1,r11,r;a≅La#p1,r1,r,where 1/p=1−θ/p0+θ/p1 and a#t=aρt.Corollary 49.
Let0<θ<1, 1<p0<∞, 0<r,r0,q0≤∞, b0∈SV, and s−1/r0b0sr0,0,1<∞. Put c0t=s−1/r0b0sr0,0,t. Then,(125)Lb0p0,q0,r0,L∞θ,r≅Lp,r,a#,where 1/p=1−θ/p0 and a#t=c0tθ.


### 8.4. Interpolation Formulae for Couples Where Both Operands Are Small or Grand Lorentz Spaces

Corollary 50.
(Cf. ([11], Theorem 3.4)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,1,∞<∞, and s−1/r1b1sr1,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞ and c1t=s−1/r1b1sr1,t,∞. Then,(126)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.Proof.
LetA¯=L1,L∞, θi=1−1/pi, and η=1−θθ0+θθ1. By Lemma 36, we have(127)Lb0p0,q0,r0≅L1,L∞θ0,r0,b0,q0,1L=A¯θ0,r0,b0,q0,1Land(128)Lb1p1,q1,r1≅L1,L∞θ0,r1,b0,q1,1L=A¯θ0,r1,b0,q1,1L.
Using now Theorem25 and ([6], Corollary 5.3), we get(129)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅A¯θ0,r0,b0,q0,1L,A¯θ0,r1,b0,q1,1Lθ,r≅A¯η,r,a#≅Lp,r,a#.
This completes the proof.All corollaries below can be proved analogously using theorems from Section6.Corollary 51.
(Cf. ([11], Theorem 5.1) and ([13], Theorem 6.5)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,1,∞<∞, and s−1/r1b1sr1,0,1<∞. Put c0t=s−1/r0b0sr0,t,∞ and c1t=s−1/r1b1sr1,0,t. Then,(130)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.Corollary 52.
(Cf. ([11], Theorem 1.2) and ([13], Theorem 8.3)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,0,1<∞, and s−1/r1b1sr1,0,1<∞. Put c0t=s−1/r0b0sr0,0,t and c1t=s−1/r1b1sr1,0,t. Then,(131)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.Corollary 53.
Let0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,0,1<∞, and s−1/r1b1sr1,1,∞<∞. Put c0t=s−1/r0b0sr0,0,t and c1t=s−1/r1b1sr1,t,∞. Then,(132)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.


## 8.1. The Lorentz–Karamata Spaces,LL and LR Spaces, and the Grand and Small Lorentz Spaces

Definition 33 (see ([6], Definition 5.1)).
Let0<p,q≤∞ and b∈SV. The Lorentz–Karamata space Lp,q;b≡Lp,q;bΩ is the set of all f∈MΩ,μ such that(105)fp,q;α≔t1/p−1/qbtf∗tq,0,∞<∞.
The Lorentz–Karamata spaceLp,q;1Ω coincides with the Lorentz space Lp,qΩ; it becomes the Lebesgue space LpΩ if b=1 and p=q.Definition 34.
(Cf. ([6], (5.21), (5.33)) ([3], Definition 7.1)). Let 0<p<∞, 0<q,r≤∞, and a,b∈SV. The spaces Lp,r,b,q,aL≡Lp,r,b,q,aLΩ and Lp,r,b,q,aR≡Lp,r,b,q,aRΩ are the sets of all f∈MΩ,μ such that(106)fLp,r,b,q,aL≔t−1/rbtatu1/p−1/qauf∗uq,0,tr,0,∞<∞or(107)fLpr,b,q,aR≔t−1/rbtatu1/p−1/qauf∗uq,t,∞r,0,∞<∞,correspondently.
We will require thatt−1/rbt/atr,1,∞<∞ for Lp,r,b,q,aL and t−1/rbt/atr,01<∞ for Lp,r,b,q,aR. Otherwise, the corresponding space consists only of the null element. Similar definitions can be found in [4, 5, 8]. We refer to the spaces L∗L and L∗R as LL and LR spaces, respectively.In order to be able to compare our results with the results from [8, 11, 13], we introduce the analogues of the grand and small Lorentz spaces. They find many different important applications and have been well studied by many authors (see [8, 11, 13] and the references therein). One can find different definitions of grand and small Lorentz spaces. Normally, they are defined for a bounded Lebesgue measurable domain Ω in Rn with measure 1. In [13], it is required that the functions are real valued. We do not require that μΩ=1 and that the functions are real valued. Note, that using arguments of Section 7, all corollaries below can be easy reformulated for the case when μΩ=1.Definition 35.
Let0<p<∞, 0<q,r≤∞, and b∈SV. The small Lorentz space Lbp,q,r is the set of all f∈MΩ,μ such that(108)fLbp,q,rΩ≔t−1/rbtu1/p−1/qf∗uq,0,tr,0,∞<∞.
The grand Lorentz spaceLbp,q,r is the set of all f∈MΩ,μ such that(109)fLbp,q,rΩ≔t−1/rbtu1/p−1/qf∗uq,t,∞r,0,∞<∞.
It is clear, thatLbp,q,r=Lp,r,b,q,1L and Lbp,q,r=Lp,r,b,q,1R.For simplicity, we consider below only interpolation spaces betweenL1 and L∞. Lemma 36 and Lemma 37 characterise the LL andLR spaces as appropriate L and R limiting interpolation spaces. Lemma 36 is a special case of ([6], Lemma 5.4). Lemma 37 can be proved analogously.Lemma 36.
Let1<p<∞, 0<q,r≤∞, a,b∈SV, and θ=1−1/p. Then,(110)Lp,r,b,q,aL≅L1,L∞θ,r,b,q,aL.
In particular,Lbp,q,r≅L1,L∞θ,r,b,q,1L.Lemma 37.
(Cf. ([3], Theorem 7.1) and ([6], Lemma 5.9)). Let 1<p<∞, 0<q,r≤∞, a,b∈SV, and θ=1−1/p. Then,(111)Lp,r,b,q,aR≅L1,L∞θ,r,b,q,aR,
In particular,Lbp,q,r≅L1,L∞θ,r,b,q,1R.Using Lemma36 and Lemma 37, we are able to characterise interpolation spaces lying between LL and LR spaces and Lorentz–Karamata spaces. In corollaries below, we restrict ourselves to grand and small Lorentz spaces. Moreover, for 0<θ<1, we formulate the formulae only for the classical case ∗,∗θ,r.


## 8.2. Interpolation between the Small Lorentz Space and the Lorentz Spaces

Corollary 38.
(Cf. ([13], Theorem 5.3)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0≤∞, b0∈SV, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞. Then,(112)Lb0p0,q0,r0,Lp1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θ.Proof.
LetA¯=L1,L∞, θi=1−1/pi, and η=1−θθ0+θθ1. By Lemma 36, we have(113)Lb0p0,q0,r0≅L1,L∞θ0,r0,b0,q0,1L=A¯θ0,r0,b0,q0,1L.
It is known ([6], Corollary 5.3) that A¯θ1,r1≅Lp1,r1. Using now Theorem 11 (i) and ([6], Corollary 5.3), we get(114)Lb0p0,q0,r0,Lp1,r1θ,r≅A¯θ0,r0,b0,q0,1L,A¯θ1,r1θ,r≅A¯η,r,a#≅Lp,r,a#.
This completes the proof.All corollaries below in this and the next subsections can be proved analogously using theorems from Sections4 and 5.Corollary 39.
Let1<p0<p1<∞, 0<r,r0,r1,q0≤∞, a,b0∈SV, s−1/rasr,0,1<∞, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞ and ρt=t1/p1−1/p0c0t. Then,(115)Lb0p0q0,r0,Lp1,r11,r;a≅La#p1,r1,r,where a#t=aρt.Corollary 40.
(Cf. ([8], Corollary 7.8)). Let 0<θ<1, 1<p0<∞, 0<r,r0,q0≤∞, and s−1/r0b0sr0,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞. Then,(116)Lb0p0,q0,r0,L∞θ,r≅Lp,r,a#,where 1/p=1−θ/p0 and a#t=c0t1−θ.Corollary 41.
Let0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q1≤∞, b1∈SV, and s−1/r1b1sr1,1,∞<∞. Put c1t=s−1/r1b1sr1,t,∞. Then,(117)Lp0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c1tθ.Corollary 42.
Let1<p0<p1<∞, 0<r,r0,r1,q1≤∞, a,b1∈SV, s−1/r1b1sr1,1,∞<∞, and s−1/rasr,1,∞<∞. Put c1t=s−1/r1b1sr1,t,∞ and ρt=t1/p1−1/p0/c1t. Then,(118)Lp0,r0,Lb1p1,q1,r10,r;a≅La#p0,r0,r,where a#t=aρt.Corollary 43.
Let0<θ<1, 1<p1<∞, 0<r,r1,q1≤∞, and s−1/r1b1sr1,1,∞<∞. Put c1t=s−1/r1b1sr1,t,∞. Then,(119)L1,Lb1p1q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ+θ/p1 and a#t=c1tθ.


## 8.3. Interpolation between the Grand Lorentz Spaces and the Lorentz Spaces

Corollary 44.
Let0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q1≤∞, b1∈SV, and s−1/r1b1sr1,0,1<∞. Put c1t=s−1/r1b1sr1,0,t. Then,(120)Lp0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c1tθ.Corollary 45.
Let1<p0<p1<∞, 0<r,r0,r1,q1≤∞, b1∈SV, s−1/rasr,1,∞<∞, and s−1/r1b1sr1,0,1<∞. Put c1t=s−1/r1b1sr1,0,t and ρt=t1/p1−1/p0/c1t. Then,(121)Lp0,r0,Lb1p1,q1,r10,r;a≅La#p0,r0,r,where a#t=aρt.Corollary 46.
(Cf. ([8], Corollary 7.7)). Let 0<θ<1, 1<p1<∞, 0<r,r1,q1≤∞, b1∈SV, and s−1/r1b1sr1,0,1<∞. Put c1t=s−1/r1b1sr1,0,t. Then,(122)L1,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ+θ/p1 and a#t=c1tθ.Corollary 47.
(Cf. ([13], Theorem 7.3)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0≤∞, b0∈SV, and s−1/r0b0sr0,0,1<∞. Put c0t=s−1/r0b0sr0,0,t. Then,(123)Lb0p0,q0,r0,Lp1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θ.Corollary 48.
(Cf. ([11], Theorem 1.1)). Let 1<p0<p1<∞, 0<r,r0,r1,q0≤∞, a,b0∈SV, s−1/rasr,0,1<∞, and s−1/r0b0sr0,0,1<∞. Put c0t=s−1/r0b0sr0,0,t and ρt=t1/p1−1/p0c0t. Then,(124)Lb0p0,q0,r0,Lp1,r11,r;a≅La#p1,r1,r,where 1/p=1−θ/p0+θ/p1 and a#t=aρt.Corollary 49.
Let0<θ<1, 1<p0<∞, 0<r,r0,q0≤∞, b0∈SV, and s−1/r0b0sr0,0,1<∞. Put c0t=s−1/r0b0sr0,0,t. Then,(125)Lb0p0,q0,r0,L∞θ,r≅Lp,r,a#,where 1/p=1−θ/p0 and a#t=c0tθ.


## 8.4. Interpolation Formulae for Couples Where Both Operands Are Small or Grand Lorentz Spaces

Corollary 50.
(Cf. ([11], Theorem 3.4)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,1,∞<∞, and s−1/r1b1sr1,1,∞<∞. Put c0t=s−1/r0b0sr0,t,∞ and c1t=s−1/r1b1sr1,t,∞. Then,(126)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.Proof.
LetA¯=L1,L∞, θi=1−1/pi, and η=1−θθ0+θθ1. By Lemma 36, we have(127)Lb0p0,q0,r0≅L1,L∞θ0,r0,b0,q0,1L=A¯θ0,r0,b0,q0,1Land(128)Lb1p1,q1,r1≅L1,L∞θ0,r1,b0,q1,1L=A¯θ0,r1,b0,q1,1L.
Using now Theorem25 and ([6], Corollary 5.3), we get(129)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅A¯θ0,r0,b0,q0,1L,A¯θ0,r1,b0,q1,1Lθ,r≅A¯η,r,a#≅Lp,r,a#.
This completes the proof.All corollaries below can be proved analogously using theorems from Section6.Corollary 51.
(Cf. ([11], Theorem 5.1) and ([13], Theorem 6.5)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,1,∞<∞, and s−1/r1b1sr1,0,1<∞. Put c0t=s−1/r0b0sr0,t,∞ and c1t=s−1/r1b1sr1,0,t. Then,(130)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.Corollary 52.
(Cf. ([11], Theorem 1.2) and ([13], Theorem 8.3)). Let 0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,0,1<∞, and s−1/r1b1sr1,0,1<∞. Put c0t=s−1/r0b0sr0,0,t and c1t=s−1/r1b1sr1,0,t. Then,(131)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.Corollary 53.
Let0<θ<1, 1<p0<p1<∞, 0<r,r0,r1,q0,q1≤∞, b0,b1∈SV, s−1/r0b0sr0,0,1<∞, and s−1/r1b1sr1,1,∞<∞. Put c0t=s−1/r0b0sr0,0,t and c1t=s−1/r1b1sr1,t,∞. Then,(132)Lb0p0,q0,r0,Lb1p1,q1,r1θ,r≅Lp,r,a#,where 1/p=1−θ/p0+θ/p1 and a#t=c0t1−θc1tθ.


## 9. Final Remarks

After this work has been completed, the author learned about the papers of Fernández-Martínez and Signes [14, 15], which were made available practically at the same time. The results of this papers demonstrate – mathematically speaking – a large intersection and a symmetric difference with the ones of Sections4 and 5. Fernández-Martínez and Signes work with the scale A0,A1θ,E,b of interpolation spaces based on the family of rearrangement invariant Banach function spaces E. They have introduced and studied this rich scale in [7, 8]. If E=Lq and 1≤q≤∞, the space A0,A1θ,E,b coincides with the space A0,A1θ,q;b. However, the spaces A0,A1θ,q;b with 0<q<1 do not belong to the scale A0,A1θ,E,b. The same remark holds about R and L limiting interpolation spaces, too.



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