Interpolation of Generalized Gamma Spaces in a Critical Case

Abstract

We establish some interpolation formulae for generalized gamma spaces with double weights in a critical case. Our approach is based on identifying generalized gamma spaces as appropriate K-interpolation spaces with general weights and then applying the reiteration technique for K-interpolation spaces.

1 Introduction

The scale of generalized gamma spaces with double weights (see Sect. 2 for definitions) was introduced in [18] in order to characterize the following real interpolation spaces

$$\begin{aligned} (L^{p), \alpha }, L^{(q, \beta })_{\theta ,r} \end{aligned}$$

between grand Lebesgue spaces $L^{p), \alpha }$ (with $\alpha =1$) and small Lebesgue spaces $L^{(q, \beta }$ (with $\beta =1$) in the critical case $p=q$. Later on, it turned out (see [2, 4]) that the following real interpolation spaces (with appropriate conditions on $\alpha$ and $\beta$)

$$\begin{aligned} (L^{p)}, L^{q})_{\theta ,r}, \;\; (L^{p}, L^{(q})_{\theta ,r},\;\; (L^{p), \alpha }, L^{q), \beta })_{\theta ,r},\;\; (L^{(p, \alpha }, L^{(q, \beta })_{\theta ,r} \end{aligned}$$

also coincided with appropriate $G\Gamma$-spaces in the critical case $p=q$. Thus, it becomes imperative to investigate the interpolation properties of $G\Gamma$-spaces themselves in the critical case. The aim of the present paper is to pursue this goal. The main finding of our investigation is this: in our special critical case, the scale of $G\Gamma$-spaces remains stable under real interpolation method. We emphasize that this is not the case in non-critical cases as it is clear from the results in [3, 15,16,17,18].

Let us illustrate our special case critical. Consider the following real interpolation spaces

$$\begin{aligned} (\Lambda ^p(w_0), G\Gamma (q,m;v,w_1))_{\theta ,r} \end{aligned}$$

between classical Lorentz spaces $\Lambda ^p(w_0)$ and $G\Gamma$-spaces $G\Gamma (q,m;v,w_1)$. We characterize these interpolation spaces in the critical case $p=q$ with an extra restriction $w_0=w_1$ (see Theorem 6.1 below).

The key feature of our approach is to identify $G\Gamma$-spaces as K-interpolation spaces (with general weights) between the classical Lorentz and $L^\infty$ spaces. This is done in Sect. 3. Then, in order to apply the reiteration technique, we formulate appropriate reiteration theorems for K-interpolation spaces involving general weights (see Sect. 5). The proofs of these reiteration theorems are essentially based on certain Holmstedt-type estimates (from [1]) and weighted Hardy-type inequalities (presented in Sect. 4). The interpolation formulae for $G\Gamma$-spaces (our main results) are contained in Sect. 6. Finally, in Sect. 7, we single out some special cases from Sect. 6 in order to illustrate how our obtained results generalize/complement the existing results in previous papers [2,3,4, 9, 25].

2 Preliminaries

2.1 Notation

Throughout the paper we will stick to the following notations. We write $A\lesssim B$ or $B\gtrsim A$ for two non-negative quantities A and B to mean that $A\le c B$ for some positive constant c which is independent of appropriate parameters involved in A and B. If both the estimates $A \lesssim B$ and $B\lesssim A$ hold, we simply put $A\approx B.$ We let $\Vert \cdot \Vert _{q, (a,b)}$ denote the standard $L^q$-quasi-norm on an interval $(a,b)\subset \mathbb {R}$. We write $X\hookrightarrow Y$ for two quasi-normed spaces X and Y to mean that X is continuously embedded in Y. By a weight w on (0, 1), we always mean a positive locally integrable function on (0, 1). We let $\Omega$ denote a bounded Lebesgue measurable domain in $\mathbb {R}^{n}$ with measure 1. Finally, the symbol $f^*$ will denote the non-increasing rearrangement of a real-valued Lebesgue measurable function f on $\Omega$ (see, for instance, [7]).

2.2 Slowly Varying Functions

Following [22], we say a weight b is slowly varying on (0, 1) if for every $\varepsilon >0,$ there are positive functions $g_{\varepsilon }$ and $g_{-\varepsilon }$ on (0, 1) such that $g_{\varepsilon }$ is non-decreasing and $g_{-\varepsilon }$ is non-increasing, and we have

$$\begin{aligned} t^{\varepsilon }b(t)\approx g_{\varepsilon }(t)\; \; \text {and}\;\; t^{-\varepsilon }b(t)\approx g_{-\varepsilon }(t)\;\; \text {for all }\; t\in (0,1). \end{aligned}$$

We denote the class of all slowly varying functions by SV. The class SV contains, for example, positive constant functions, and the functions $t\mapsto (1-\ln t)$ and $t\mapsto 1+ \ln (1-\ln t).$ We collect in the next Proposition some properties of slowly varying functions. The proofs of these assertions can be carried out as in [22, Lemma 2.1] or [11, Proposition 3.4.33].

Proposition 2.1

Given b, $b_1$, $b_2\in SV$, the following are true:

• (i) $b_1b_2\in SV$ and $b^{r} \in SV$ for each $r\in \mathbb {R}.$

• (ii) If $0<k<1,$ then $b(kt)\approx b(t),\; 0<t< 1.$

• (iii) For $\alpha >0$, set $\tilde{b}(t)=b(t^\alpha ),\;0<t<1.$ Then $\tilde{b}\in SV$.

• (iv) If $\alpha >0$, then

  $$\begin{aligned} \int _{0}^{t}u^{\alpha }b(u)\frac{du}{u}\approx t^{\alpha }b(t),\qquad 0< t<1. \end{aligned}$$

• (v) If $\alpha >0$, then

  $$\begin{aligned} 1+\int _{t}^{1}u^{-\alpha }b(u)\frac{du}{u}\approx t^{-\alpha }b(t),\qquad 0< t<1. \end{aligned}$$

• (vi) Set

  $$\begin{aligned} \tilde{b}(t)=1+\int _{t}^{1}b(u)\frac{du}{u},\qquad 0< t<1. \end{aligned}$$

           Then $\tilde{b}\in SV,$ and $b(t)\lesssim \tilde{b}(t),\; 0< t<1.$

• (vii) Set

  $$\begin{aligned} \tilde{b}(t)=\sup \limits _{0<u<t}b(u),\qquad 0< t<1. \end{aligned}$$

   Then $\tilde{b}\in SV$.

2.3 K-Interpolation Spaces

Let $A_0$ and $A_1$ be two quasi-normed spaces. We say $(A_0,A_1)$ is a compatible couple if $A_0$ and $A_1$ are continuously embedded in the same Hausdorff topological vector space. For each $f\in A_0+A_1$ and $t>0$, the Peetre K-functional is defined by

$$\begin{aligned} K(t,f)= & {} K(t,f;A_0,A_1)\\= & {} \inf \{\Vert f_0\Vert _{A_0}+t\Vert f_1\Vert _{A_1}:\;f_0 \in A_0, \; f_1 \in A_1,\; f=f_0+f_1\}. \end{aligned}$$

Note that K(t, f) is, as a function of t, non-decreasing on $(0,\infty ).$ In the sequel, we will refer to this fact simply as monotonicity of K-functional.

In what follows, we always assume that the couple $(A_0,A_1)$ is ordered in the sense that $A_1 \hookrightarrow A_0$.

Let $0<q\le \infty ,$ and let w be a positive weight on (0, 1) satisfying the following condition

$$\begin{aligned} \Vert t^{1-1/q}w(t)\Vert _{q, (0,1)}<\infty . \end{aligned}$$

(2.1)

Then the K-interpolation space ${{\bar{A}}}_{w,q}=(A_{0},A_{1})_{w,q}$ is formed of those $f\in A_0$ for which the quasi-norm

$$\begin{aligned} \Vert f\Vert _{{\bar{A}}_{w,q}}=\Vert t^{-1/q }w(t) K(t,f)\Vert _{q,(0,1)} \end{aligned}$$

is finite; see, for instance, [1]. If $0<q<\infty$ and $w(t)=t^{-\theta }$ with $0<\theta < 1$, then we recover the classical real interpolation spaces ${{\bar{A}}}_{\theta ,q}$ (see [7, 8, 24, 27]).

Note that, thanks to the condition (2.1), the spaces ${\bar{A}}_{w,q}$ are intermediate for the couple $(A_0,A_1)$, that is,

$$\begin{aligned} A_1\hookrightarrow {\bar{A}}_{w,q} \hookrightarrow A_0. \end{aligned}$$

Next let $f\in {\bar{A}}_{w,q}$. By monotonicity of K-functional and $K(1,f)\approx \Vert f\Vert _{A_0}$, we have

$$\begin{aligned} \Vert f\Vert _{{\bar{A}}_{w,q}}\lesssim \Vert f\Vert _{A_0}\Vert t^{-1/q}w(t)\Vert _{q, (0,1)}. \end{aligned}$$

Thus we can conclude that we always have to work under the following condition on w

$$\begin{aligned} \Vert t^{-1/q}w(t)\Vert _{q, (0,1)}=\infty , \end{aligned}$$

(2.2)

so that the trivial case ${\bar{A}}_{w,q}=A_0$ is excluded. If $w\in SV$, then the condition (2.1) is met thanks to Proposition 2.1 (iv) (if $0<q<\infty$) or to the very definition of a slowly varying function (if $q=\infty$).

2.4 Classical Lorentz Spaces

Let $0<q\le \infty$ and let w be weight on (0, 1). Assume that

• (c1) $w(2t)\lesssim w(t),\quad 0<t<1/2.$

• (c2) $\Vert t^{-1/q}w(t)\Vert _{q, (0,1)}<\infty .$

The classical Lorentz spaces $\Lambda ^q(w)=\Lambda ^q(w)(\Omega )$ consists of those real-valued Lebesgue measurable functions f on $\Omega ,$ for which the quasi-norm

$$\begin{aligned} \Vert f\Vert _{\Lambda ^q(w)}=\Vert t^{-1/q }w(t) f^*(t)\Vert _{q,(0,1)} \end{aligned}$$

is finite; see [26]. Thanks to the condition (c2), we always have $\Lambda ^q(w)\ne \{0\}$; more precisely, we have the embedding $L^\infty \hookrightarrow \Lambda ^q(w).$ The classical Lorentz spaces cover many well-known spaces: for instance, when $w(t)=t^{1/p}b(t)$ (with $0<p\le \infty$ and $b\in SV$) the spaces $\Lambda ^q(w)$ become the Lorentz–Karamata spaces $L_{p,q;b}$ (see, for instance, [20]). In particular, when $b(t)=(1-\ln t)^{\alpha }$, $\alpha \in \mathbb {R}$, we put $L^{p,q}(\log L)^{\alpha }=L_{p,q;b}$. The space $L^{p,q}(\log L)^{\alpha }$ is called the Lorent–Zygmund space and it was introduced by Bennett and Rudnick [6]. If $\alpha =0$, the Lorentz–Zygmund space $L^{p,q}(\log L)^{\alpha }$ coincides with the Lorentz space $L^{p,q}$ which becomes the Lebesgue space $L^p$ if $p=q$.

Remark 2.2

Let $f\in \Lambda ^\infty (w)$. Since $f^*$ is non-increasing, we can verify easily that

$$\begin{aligned} \sup \limits _{0<t<1}w(t)f^*(t)=\sup \limits _{0<t<1}\left[ \sup \limits _{0<s<t}w(s)\right] f^*(t). \end{aligned}$$

Thus, in the case $q=\infty$ we can assume that w is non-decreasing.

2.5 Generalized Gamma Spaces

We first introduce a notation. For $0<m,q\le \infty$, we say a pair (w, v) of weights is admissible if the following conditions are met:

• (d1) For all $0<t<1/2$, $w(2t)\lesssim w(t)$ and $v(2t)\lesssim v(t)$.

• (d2) $\Vert t^{-1/q}w(t)\Vert _{q, (0,1)}<\infty .$

• (d3) $\Vert t^{-1/m}v(t)\Vert _{m, (0,1)}=\infty .$

• (d4) $\left\| t^{-1/m}v(t)\Vert \tau ^{-1/q}w(\tau )\Vert _{q, (0,t)}\right\| _{m, (0,1)}<\infty .$

Definition 2.3

[18] Let $0<m,q\le \infty$ and (w, v) be a pair of admissible weights. The generalized gamma space $G\Gamma (q,m;v,w)=G\Gamma (q,m;v,w)(\Omega )$ consists of all those real-valued Lebesgue measurable functions f on $\Omega ,$ for which the quasi-norm

$$\begin{aligned} \Vert f\Vert _{G\Gamma (q,m;v,w)}= \left\| t^{-1/m}v(t)\Vert \tau ^{-1/q}w(\tau )f^*(\tau )\Vert _{q, (0,t)}\right\| _{m, (0,1)} \end{aligned}$$

is finite.

Remark 2.4

Let $f\in G\Gamma (q,m;v,w).$ Since $t\mapsto f^*(t)$ is non-increasing, we can check that the following function

$$\begin{aligned} t\mapsto \frac{1}{\Vert \tau ^{-1/q}w(\tau )\Vert _{q, (0,t)}}\Vert \tau ^{-1/q}w(\tau )f^*(\tau )\Vert _{q, (0,t)} \end{aligned}$$

is equivalent to a non-increasing function. Consequently (thanks to the Condition (d4)), it follows that

$$\begin{aligned} G\Gamma (q,m;v,w)\hookrightarrow \Lambda ^q(w). \end{aligned}$$

Moreover, the Condition (d3) guarantees that the converse embedding

$$\begin{aligned} \Lambda ^q(w)\hookrightarrow G\Gamma (q,m;v,w) \end{aligned}$$

does not hold. Thus, the trivial case $G\Gamma (q,m;v,w)=\Lambda ^q(w)$ is excluded. However, note that for $q=m$ the spaces $G\Gamma (q,m;v,w)$ again coincide with $\Lambda ^m(\tilde{w})$ for an appropriate weight $\tilde{w}$.

Remark 2.5

The scale of $G\Gamma (q,m;v,w)$ spaces is very general and covers many well-known scales of spaces. If we take $q=1$ and $w(t)=t$, then we recover the classical gamma spaces $\Gamma ^m(\tilde{v})$ (see [26]) for an appropriate weight $\tilde{v}$. Let $0<m,p,q<\infty$, $w(t)=t^{1/p}$ and $v\in SV$, then the spaces $G\Gamma (q,m;v,w)$ coincide with the small Lorentz spaces $L_{v}^{(p,q,m}$ from [3]. As a still more special case, if $\alpha >0$, $1<q<\infty$, $v(t)=(1-\ln t)^{-\frac{\alpha }{q}+\alpha -1}$, $w(t)= t^{1/q}$, $m=1$, the spaces $G\Gamma (q,m;v,w)$ become the small Lebesgue spaces $L^{(q,\alpha }$; see [18, 19]. Finally, since we also allow the case $m=\infty$ in our definition in contrast to [18], we observe that the spaces $S_{p,\alpha }$ considered in [12] are also a special case of the spaces $G\Gamma (q,m;v,w)$.

3 Generalized Gamma Spaces as K-Interpolation Spaces

In this section we characterize the generalized gamma spaces as K-interpolation spaces with general weights. To this end, we first need the following computation of K-functional for the couple $(\Lambda ^q(w), L^\infty )$. While this computation is a special case of a far more general formula in [13, p. 84], we present a simple proof for reader’s convenience.

Lemma 3.1

Let $0<q\le \infty .$ Then, for all $f\in \Lambda ^q(w),$ we have

$$\begin{aligned} K(\tilde{w}(t),f; \Lambda ^q(w), L^\infty )\approx \Vert \tau ^{-1/q }w(\tau ) f^*(\tau )\Vert _{q,(0,t)},\quad 0<t<1, \end{aligned}$$

(3.1)

where

$$\begin{aligned} \tilde{w}(t)=\Vert \tau ^{-1/q }w(\tau ) \Vert _{q,(0,t)},\quad 0<t<1. \end{aligned}$$

Proof

Let $f=f_0+f_1$ be an arbitrary decomposition of f with $f_0\in \Lambda ^q(w)$ and $f_1\in L^\infty .$ Using the elementary inequality

$$\begin{aligned} f^*(\tau )\le f_0^*(\tau )+f^*_1(0), \quad 0<\tau <1, \end{aligned}$$

we get

$$\begin{aligned} \Vert \tau ^{-1/q }w(\tau ) f^*(\tau )\Vert _{q,(0,t)}\lesssim \Vert f_0\Vert _{\Lambda ^q(w)}+\tilde{w}(t) \Vert f_1\Vert _{L^\infty }, \end{aligned}$$

whence we get the estimate $``\gtrsim \hbox {''}$ in (3.1), by taking the infimum over all decompositions of f. To prove the converse estimate $``\lesssim \hbox {''}$, we fix $0<t<1$ and take the following particular decomposition of f:

$$\begin{aligned} g=(f-f^*(t) sgn f)\chi _E, \quad h=f-g, \end{aligned}$$

where $E=\{x\in \Omega : |f(x)|>f^*(t)\}.$ Then $g^*=(f^*-f^*(t))\chi _{(0,t)}$ and $h^*=f^*(t)\chi _{(0,t)}+f^*\chi _{(t,1)}.$ Therefore, we can check easily that

$$\begin{aligned} \Vert g\Vert _{ \Lambda ^q(w)}\le \Vert \tau ^{-1/q }w(\tau ) f^*(\tau )\Vert _{q,(0,t)}, \end{aligned}$$

and

$$\begin{aligned} \Vert h\Vert _{L^\infty }\le 2 f^*(t)\le \frac{2}{\tilde{w}(t)}\Vert \tau ^{-1/q }w(\tau ) f^*(\tau )\Vert _{q,(0,t)}. \end{aligned}$$

Thus, we arrive at

$$\begin{aligned} \Vert g\Vert _{ \Lambda ^q(w)}+ \tilde{w}(t)\Vert h\Vert _{L^\infty } \lesssim \Vert \tau ^{-1/q }w(\tau ) f^*(\tau )\Vert _{q,(0,t)},\quad 0<t<1, \end{aligned}$$

from which follows the estimate “$\lesssim$”. The proof is complete. $\square$

The next two results describe the characterization of $G\Gamma (q,m;v,w)$ spaces as K-interpolation spaces.

Theorem 3.2

Let $0<m \le \infty$, $0<q<\infty$ and (w, v) be a pair of admissible weights. Let $\phi$ be the inverse of the following function

$$\begin{aligned} \psi (t)=c\left( \int _{0}^{t}w^q(\tau )\frac{d\tau }{\tau }\right) ^{1/q}, \quad 0<t <1, \end{aligned}$$

where

$$\begin{aligned} 1/c=\left( \int _{0}^{1}w^q(\tau )\frac{d\tau }{\tau }\right) ^{1/q}. \end{aligned}$$

Moreover, define

$$\begin{aligned} \rho (t)=v(\phi (t))\left[ \frac{t}{w(\phi (t))}\right] ^{q/m}, \quad 0<t<1. \end{aligned}$$

Then

$$\begin{aligned} G\Gamma (q,m;v,w)={\left\{ \begin{array}{ll}(\Lambda ^q(w),L^\infty )_{\rho , m },\quad m<\infty ,\\ (\Lambda ^q(w), L^\infty )_{v(\phi ), m }, \quad m=\infty . \end{array}\right. } \end{aligned}$$

Proof

We give the argument only in the case $m<\infty$ since the other case $m=\infty$ is analogous. Set temporarily $X=(\Lambda ^q(w),L^\infty )_{\rho ,m}$, and let $f\in \Lambda ^q(w).$ In view of the simple fact that

$$\begin{aligned} K(c\psi ,f; \Lambda ^q(w), L^\infty )\approx K(\psi ,f; \Lambda ^q(w), L^\infty ), \end{aligned}$$

an application of Lemma 3.1 yields

$$\begin{aligned} \Vert f\Vert _{X}\approx \left( \int _{0}^{1}\rho ^m(t)\left( \int _{0}^{\phi (t)}\left[ w(\tau )f^*(\tau )\right] ^q\frac{d\tau }{\tau } \right) ^{m/q}\frac{dt}{t}\right) ^{1/m}, \end{aligned}$$

now making a change of variable $t=\psi (s)$, it turns out that

$$\begin{aligned} \Vert f\Vert _{X}\approx \left( \int _{0}^{1}\rho ^m(\psi (s))\left( \int _{0}^{s}\left[ w(\tau )f^*(\tau )\right] ^q\frac{d\tau }{\tau }\right) ^{m/q} \frac{\psi ^{\prime }(s)}{\psi (s)}ds\right) ^{1/m}, \end{aligned}$$

finally, the following simple computation

$$\begin{aligned} \rho ^m(\psi (s))\frac{\psi ^{\prime }(s)}{\psi (s)}\approx s^{-1}v^m(s), \quad 0<s<1, \end{aligned}$$

completes the proof. $\square$

We omit the proof of the next result since it can be carried out by using the same argument as in the proof of the previous theorem.

Theorem 3.3

Let $0<m\le \infty$. Suppose that (w, v) is a pair of admissible weights such that w is strictly increasing on (0, 1) with $\lim \limits _{t\rightarrow 0^+} w(t)=0$ and $\lim \limits _{t\rightarrow 1^-} w(t)=1.$ Then

$$\begin{aligned} G\Gamma (\infty ,\infty ;v,w)=(\Lambda ^\infty (w), L^\infty )_{v(w^{-1}), \infty }. \end{aligned}$$

If we assume additionally that w is differentiable on (0, 1), then

$$\begin{aligned} G\Gamma (\infty ,m ;v,w)=(\Lambda ^\infty (w), L^\infty )_{\rho , m}, \quad m\ne \infty , \end{aligned}$$

where

$$\begin{aligned} \rho (t)=v(w^{-1}(t))\left[ \frac{t}{w^{-1}(t)w^\prime \left( w^{-1}(t)\right) }\right] ^{1/m}, \quad 0<t<1. \end{aligned}$$

4 Weighted Hardy-Type Inequalities

The weighted Hardy-type inequalities presented in this section will be the key ingredients in the proofs of our reiteration theorems in the next section.

Theorem 4.1

[1, Lemma 3.2] Let $1< \alpha < \infty ,$ and assume that g and $\phi$ are non-negative functions on $(0,\infty )$. Put

$$\begin{aligned} v_1(t)=(g(t))^{1-\alpha }\left( \phi (t)\int _{t}^{\infty }g(u)du\right) ^{\alpha }. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{\infty }\left( \int _{0}^{t}\phi (u)h(u)du\right) ^{\alpha }g(t)dt \lesssim \int _{0}^{\infty }h^{\alpha }(t)v_1(t)dt \end{aligned}$$

holds for all non-negative functions h on $(0,\infty )$.

We also have the following variant of the previous result; see [3, Theorem 3.3].

Theorem 4.2

Let $1< \alpha < \infty ,$ and assume that g and $\phi$ are non-negative functions on $(0,\infty )$. Put

$$\begin{aligned} v_2(t)=(g(t))^{1-\alpha }\left( \phi (t)\int _{0}^{t}g(u)du\right) ^{\alpha }. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{\infty }\left( \int _{t}^{\infty }\phi (u)h(u)du\right) ^{\alpha }g(t)dt \lesssim \int _{0}^{\infty }h^{\alpha }(t)v_2(t)dt \end{aligned}$$

holds for all non-negative functions h on $(0,\infty )$.

The next result is a simple consequence of [1, Lemma 3.3].

Theorem 4.3

Let $0<\alpha <1$, and assume that g and $\phi$ are non-negative functions on $(0,\infty ).$ Put

$$\begin{aligned} v_{3}(t)=\phi (t)\left( \int _{t}^{\infty }\phi (u)du\right) ^{\alpha -1}\int _{0}^{t }g(u)du. \end{aligned}$$

Then

$$\begin{aligned} \int _{0}^{\infty }\left( \int _{t}^{\infty }\phi (u)h(u)du\right) ^{\alpha }g(t)dt\lesssim \int _{0}^{\infty }h^{\alpha }(t)v_3(t)dt \end{aligned}$$

holds for all non-negative and non-decreasing functions h on $(0,\infty )$.

Theorem 4.4

[23, Theorem 3.3 (b)] Let $0<\alpha <1.$ Assume that g and v are non-negative functions on (0, 1), and $\psi$ is a non-negative function on $(0,1)\times (0,1).$ Then

$$\begin{aligned} \int _{0}^{1}\left( \int _{0}^{1}\psi (t,u)h(u)du \right) ^{\alpha }g(t)d{t}\lesssim \int _{0}^{\infty }h^{\alpha }(t)v(t)dt \end{aligned}$$

(4.1)

holds for all non-negative and non-decreasing functions h on (0, 1) if and only if

$$\begin{aligned} \int _{0}^{1}\left( \int _{x}^{1}\psi (t,u)du\right) ^{\alpha }g(t)d{t}\lesssim \int _{x}^{1}v(t)dt \end{aligned}$$

(4.2)

holds for all $0<x<1.$

5 Reiteration

First of all, we recall (from Sect. 2.3) that a weight w appearing in the K-interpolation space ${{\bar{A}}}_{w,q}$ has to satisfy the conditions (2.1) and (2.2) so that both the trivial cases ${{\bar{A}}}_{w,q}=\{0\}$ and ${{\bar{A}}}_{w,q}=A_0$ are excluded.

For convenience we introduce a further notation: for $0<m<\infty$, we say a weight w satisfies the condition ($H_m$) if the following estimate holds:

$$\begin{aligned} t^{-1}\left( \int _{0}^{t}u^mw^m(u)\frac{du}{u}\right) ^{1/m}\lesssim \left( 1+\int _{t}^{1}w^m(u)\frac{du}{u}\right) ^{1/m},\quad 0<t<1. \end{aligned}$$

Moreover, we say a weight w satisfies the condition ($H_\infty$) if the following estimate holds:

$$\begin{aligned} t^{-1}\sup \limits _{0<u<t}uw(u) \lesssim w(t), \quad 0<t<1. \end{aligned}$$

Remark 5.1

Let $w\in SV.$ Then, by Proposition 2.1 (iv)–(vi), w satisfies ($H_m$). Clearly, by the very definition of a slowly varying function, w also satisfies ($H_\infty$).

Theorem 5.2

Let $0<m, r<\infty$ , $0<\theta <1$, and let w satisfy ($H_m$). Then

$$\begin{aligned} (A_0, {\bar{A}}_{w,m})_{\theta ,r}={\bar{A}}_{\tilde{w},r}, \end{aligned}$$

where

$$\begin{aligned} \tilde{w}(t)=\left( 1+\int _{t}^{1}w^{m}(u)\frac{du}{u}\right) ^{\theta /{m}-1/r}w^{m/r}(t), \quad 0<t< 1. \end{aligned}$$

Proof

Set $X=(A_0, {\bar{A}}_{w,m})_{\theta ,r}$, $Y={\bar{A}}_{\tilde{w},r}$ and

$$\begin{aligned} \rho (t)=\left( 1+\int _{t}^{1}w^m(u)\frac{du}{u}\right) ^{-1/m}, \quad 0<t<1. \end{aligned}$$

Note that $\rho$ is increasing with $\lim \limits _{t\rightarrow 0^+}\rho (t)=0$ (thanks to (2.2)) and $\lim \limits _{t\rightarrow 1^-}\rho (t)=1.$ Next define

$$\begin{aligned} W(t)=\left\{ \begin{array}{cc} w(t), &{} 0<t<1,\\ &{} \\ t^{-1}, &{} t\ge 1, \end{array} \right. \end{aligned}$$

and note that

$$\begin{aligned} \left( \int _{t}^{\infty }W^m(u)\frac{du}{u}\right) ^{1/m}\approx \left( 1+\int _{t}^{1}w^m(u)\frac{du}{u}\right) ^{1/m} , \quad 0<t<1. \end{aligned}$$

Let $f\in A_0$. Since w satisfies ($H_m$), we can apply the estimate (2.19) in [1] to obtain

$$\begin{aligned} K(\rho (t),f; A_0,{\bar{A}}_{w,m})\approx \rho (t)\left( \int _{t}^{\infty }W^{m}(u)K^m(u,f)\frac{du}{u}\right) ^{1/m}, \quad 0<t<1, \end{aligned}$$

whence, by an appropriate change of variable, we get

$$\begin{aligned} \Vert f\Vert ^r_X\approx \int _{0}^{1}\rho ^{(1-\theta )r}(t)\left( \int _{t}^{\infty }W^{m}(u)K^m(u,f)\frac{du}{u}\right) ^{r/m}\frac{\rho ^\prime (t)}{\rho (t)}dt. \end{aligned}$$

(5.1)

In view of monotonicity of K-functional, it follows immediately from (5.1) that

$$\begin{aligned} \Vert f\Vert ^r_X\gtrsim \int _{0}^{1}\rho ^{(1-\theta )r}(t)K^r(t,f)\left( \int _{t}^{\infty }W^{m}(u)\frac{du}{u}\right) ^{r/m}\frac{\rho ^\prime (t)}{\rho (t)}dt, \end{aligned}$$

from which it follows that $\Vert f\Vert _X\gtrsim \Vert f\Vert _Y$ since

$$\begin{aligned} \rho ^\prime (t)\approx t^{-1}w^m(t)\rho ^{1+m}(t), \quad 0<t<1/2. \end{aligned}$$

Next we establish the converse estimate $\Vert f\Vert _X\lesssim \Vert f\Vert _Y$. To this end, we note that, from (5.1), we have

$$\begin{aligned} \Vert f\Vert ^r_X\approx I_1+I_2, \end{aligned}$$

where

$$\begin{aligned} I_1=\int _{0}^{1}\rho ^{(1-\theta )r}(t)\left( \int _{t}^{1}w^{m}(u)K^m(u,f)\frac{du}{u}\right) ^{r/m}\frac{\rho ^\prime (t)}{\rho (t)}dt, \end{aligned}$$

and

$$\begin{aligned} I_2=\int _{0}^{1}\rho ^{(1-\theta )r}(t)\left( \int _{1}^{\infty }u^{-m}K^m(u,f)\frac{du}{u}\right) ^{r/m}\frac{\rho ^\prime (t)}{\rho (t)}dt. \end{aligned}$$

In view of

$$\begin{aligned} K(t,f)\approx \Vert f\Vert _{A_0},\quad t\ge 1, \end{aligned}$$

and

$$\begin{aligned} \int _{0}^{1}\rho ^{(1-\theta )r}(t)\frac{\rho ^\prime (t)}{\rho (t)}dt<\infty , \end{aligned}$$

we get that $I_2 \approx \Vert f\Vert ^r_{A_0}.$ Since $Y\hookrightarrow A_0$, it follows that $I_2\lesssim \Vert f\Vert ^r_{Y}$. Thus, it remains to establish that $I_1\lesssim \Vert f\Vert ^r_{Y}$. The case $r=m$ immediately follows from Fubini’s theorem. For the case $r\ne m$, we take $\alpha =r/m$, $h(t)=K^m(t,f)$, $\phi (t)=t^{-1}W^m(t)$ and $g=\rho ^{(1-\theta )r-1}\rho ^\prime \chi _{(0,1)}$, and apply Theorem 4.2 (if $r>m$) or Theorem 4.3 (if $r<m$). It is not hard to verify that

$$\begin{aligned} v_2(t)\approx v_3(t)\approx t^{-1}[\tilde{w}(t)]^r,\quad 0<t<1, \end{aligned}$$

and consequently, the estimate $I_1\lesssim \Vert f\Vert ^r_{Y}$ holds. The proof is complete. $\square$

Remark 5.3

If we take $w(t)=t^{-\theta _1}$, $0<\theta _1<1$, then we get back the classical result from [24]. If we take $w\equiv 1$ and $m=1$, then we recover the first assertion in [21, Theorem 3.21]. The case when $w\in SV$ also follows from [5, Theorem 11]. The particular case when w is a logarithmic function has earlier been considered in [10, Theorem 4 (a)].

Next we treat the case $m=\infty .$ In this regard, an elementary but important observation is made in the next remark.

Remark 5.4

Let $(A_0, A_1)$ be a compatible couple of quasi-normed spaces. Using monotonicity of K-functional, we observe that the following identity

$$\begin{aligned} \sup \limits _{0<t<1}w(t)K(t,f)=\sup \limits _{0<t<1}\left[ \sup \limits _{t<s<1}w(s)\right] K(t,f), \end{aligned}$$

holds for every $f\in A_0.$ Therefore, while working with ${{\bar{A}}}_{w,\infty },$ we can always assume, without loss of generality, that w is non-increasing.

Theorem 5.5

Let $0< r<\infty$ , $0<\theta <1$, and suppose w is strictly decreasing and differentiable on (0, 1) and satisfies $(H_\infty ).$ Put $\rho =1/w$, and assume that $\lim \limits _{t\rightarrow 1^-}\rho (t)=1.$ Then we have

$$\begin{aligned} (A_0, {\bar{A}}_{w,\infty })_{\theta ,r}={\bar{A}}_{\tilde{w},r}, \end{aligned}$$

where

$$\begin{aligned} \tilde{w}(t)=w^{\theta }(t)\left[ tw(t)\rho ^\prime (t)\right] ^{1/r}, \quad 0<t< 1. \end{aligned}$$

Proof

Put $X=(A_0, {{\bar{A}}}_{w, \infty })_{\theta , r}$ and $Y={\bar{A}}_{\tilde{w},r}$. Next, in view of (2.2), we observe that $\lim \limits _{t\rightarrow 0^+}\rho (t)=0.$ Let $f\in A_0$. Since w satisfies $(H_\infty )$, we can apply the estimate (2.19) in [1] to obtain

$$\begin{aligned} K(\rho (t),f; A_0, {{\bar{A}}}_{w, \infty })\approx \rho (t)\sup \limits _{t\le u< 1}w(u)K(u,f),\quad 0<t<1, \end{aligned}$$

whence we arrive at

$$\begin{aligned} \Vert f\Vert ^{r}_{X}\approx \int _{0}^{1} \rho ^{(1-\theta )r}(t)\left[ \sup \limits _{t \le u < 1}{w}^r(u)K^r(u,f)\right] \frac{\rho ^\prime (t)}{\rho (t)} dt. \end{aligned}$$

(5.2)

Now the estimate $\Vert f\Vert _{X}\gtrsim \Vert f\Vert _{Y}$ follows immediately from (5.2). Next we establish the converse estimate $\Vert f\Vert _{X} \lesssim \Vert f\Vert _{Y}.$ Put

$$\begin{aligned} I=\int _{0}^{1} \rho ^{(1-\theta )r}(t)\left[ \sup \limits _{t \le u < 1}{w}^r(u)K^r(u,f)\right] \frac{\rho ^\prime (t)}{\rho (t)} dt, \end{aligned}$$

and noting

$$\begin{aligned} \int _{t}^{1}w^r(u)\frac{\rho ^\prime (u)}{\rho (u)}du= \frac{1}{r} \left( w^{r}(t)-1\right) ,\quad 0<t<1, \end{aligned}$$

we can write

$$\begin{aligned} I\lesssim I_1+I_2 \end{aligned}$$

where

$$\begin{aligned} I_1= \int _{0}^{1} \rho ^{(1-\theta )r}(t) \sup \limits _{t\le u <1}K^r(u,f) \left[ \int _{u}^{1} w^r(\tau )\frac{\rho ^\prime (\tau )}{\rho (\tau )}d\tau \right] \frac{\rho ^\prime (t)}{\rho (t)} dt, \end{aligned}$$

and

$$\begin{aligned} I_2=\int _{0}^{1}\rho ^{(1-\theta )r}(t)\sup \limits _{t\le u <1}K^r(u,f)\frac{\rho ^\prime (t)}{\rho (t)}dt. \end{aligned}$$

Now by monotonicity of K-functional, we obtain

$$\begin{aligned} I_1\le \int _{0}^{1} \rho ^{(1-\theta )r}(t) \left[ \sup _{t\le u <1} \int _{u}^{1} w^r(\tau )\frac{\rho ^\prime (\tau )}{\rho (\tau )}K^r(\tau ,f) d\tau \right] \frac{\rho ^\prime (t)}{\rho (t)} dt, \end{aligned}$$

and

$$\begin{aligned} I_2=\int _{0}^{1}\rho ^{(1-\theta )r}(t)K^r(1,f)\frac{\rho ^\prime (t)}{\rho (t)}dt, \end{aligned}$$

whence we get

$$\begin{aligned} I_1\le \int _{0}^{1} \rho ^{(1-\theta )r}(t) \left[ \int _{t}^{1} w^r(\tau )\frac{\rho ^\prime (\tau )}{\rho (\tau )}K^r(\tau ,f) d\tau \right] \frac{\rho ^\prime (t)}{\rho (t)} dt, \end{aligned}$$

and

$$\begin{aligned} I_2\approx \Vert f\Vert ^r_{A_0}. \end{aligned}$$

Now an application of Fubini’s theorem gives

$$\begin{aligned} I_1 \lesssim \int _{0}^{1} w^r(\tau )\frac{\rho ^\prime (\tau )}{\rho (\tau )}K^r(\tau ,f)\rho ^{(1-\theta )r}(\tau ) d\tau , \end{aligned}$$

which shows that $I_1\lesssim \Vert f\Vert ^r_{Y}.$ Since $Y\hookrightarrow A_0$, we also have $I_2\lesssim \Vert f\Vert ^r_{Y}.$ Altogether, we arrive at $\Vert f\Vert ^{r}_{X}\approx I\lesssim \Vert f\Vert ^r_{Y}$ which completes the proof. $\square$

Remark 5.6

To the best of our knowledge, the assertion of Theorem 5.5 is new. We note that the particular case when w is a general slowly varying function is entirely missing from [1, 5, 20], and also not covered by [14, Theorem 5.5].

Remark 5.7

Let $0<m<\infty .$ Suppose that $w_0$ and $w_1$ are two weights such that $w_0/w_1$ is non-decreasing. Then it is not hard to check that ${\bar{A}}_{w_1,m}\hookrightarrow {\bar{A}}_{w_0,m}.$ If we assume, additionally, that

$$\begin{aligned} \frac{w_0(t)}{w_1(t)}\le 1,\quad 0<t<1, \end{aligned}$$

then we also have

$$\begin{aligned} \frac{w_0(t)}{w_1(t)}\le \left( \frac{1+\int _{t}^{1} w^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} w^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m},\quad 0<t<1. \end{aligned}$$

Theorem 5.8

Let $0<m,r<\infty$ and $0<\theta <1$. Suppose that $w_0$ and $w_1$ are two weights such that $\rho =w_0/w_1$ is strictly increasing on (0, 1) with $\lim \limits _{t\rightarrow 0^+ }\rho (t)=0$ and $\lim \limits _{t\rightarrow 1^- }\rho (t)=1$. Assume further that $w_1$ satisfies ($H_m$) and that there exists $c_1\in (1,\infty )$ and $c_2\in (0, 1)$ such that

$$\begin{aligned} \left( \frac{1+\int _{t}^{1} w^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} w^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m}< c_1 \rho (t),\quad 0<t<1, \end{aligned}$$

(5.3)

and

$$\begin{aligned} \rho (t)< c_2 \left( \frac{1+\int _{t}^{1} w^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} w^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m},\quad 0<t<1/2. \end{aligned}$$

(5.4)

Then we have

$$\begin{aligned} \left( {\bar{A}}_{w_0,m}, {\bar{A}}_{w_1,m}\right) _{\theta ,r}= {\bar{A}}_{\tilde{w},r}, \end{aligned}$$

where

$$\begin{aligned} \tilde{w}(t)=[\rho (t)]^{(1-\theta )}w^{m/r}_1(t)\left( 1+\int _{t}^{1}w_1^{m}(u)\frac{du}{u}\right) ^{1/m-1/r}, \quad 0<t< 1. \end{aligned}$$

Proof

Set $X=\left( {\bar{A}}_{w_0,m}, {\bar{A}}_{w_1,m}\right) _{\theta ,r}$, $Y={\bar{A}}_{\tilde{w},r}$ and

$$\begin{aligned} W_1(t)=\left\{ \begin{array}{cc} w_1(t), &{} 0<t<1,\\ &{} \\ t^{-1}, &{} t\ge 1. \end{array} \right. \end{aligned}$$

Let $f\in A_{0},$ and put

$$\begin{aligned} \sigma (t)=\left( \frac{1+\int _{t}^{1} w^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} w^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m},\quad 0<t<1. \end{aligned}$$

In view of Remark 5.7 and (5.3), we have $\rho \approx \sigma$ on (0, 1). Moreover, since $\rho$ is strictly increasing, we have in fact $\rho < \sigma$ on (0, 1). As a consequence, we obtain $\sigma ^\prime >0$ on (0, 1), that is, $\sigma$ is also strictly increasing on (0, 1). Now, according to the estimates (2.30) and (2.35) in [1], for all $0<t<1$ we have

$$\begin{aligned} {\begin{matrix} K\left( \sigma (t),f,{\bar{A}}_{w_0,m}, {\bar{A}}_{w_1,m} \right) \lesssim &{} \; I(t,f)+ \sigma (t)J(t,f) \\ &{} + \frac{\sigma (t)}{\sigma _1(t)}K(t,f)+\frac{\rho _0(t)}{t}K(t,f), \end{matrix}} \end{aligned}$$

(5.5)

and

$$\begin{aligned} K\left( \sigma (t),f,{\bar{A}}_{w_0,m}, {\bar{A}}_{w_1,m} \right) \gtrsim I(t,f)+ \sigma (t)J(t,f), \end{aligned}$$

(5.6)

where

$$\begin{aligned} I(t,f)= & {} \left( \int _{0}^{t} w^{m}_{0}(u) K^{m}(u,f) \frac{du}{u} \right) ^{\frac{1}{m}}, \\ J(t,f)= & {} \left( \int _{t}^{\infty } W^{m}_{1}(u) K^{m}(u,f) \frac{du}{u} \right) ^{\frac{1}{m}}, \\ \sigma _1(t)= & {} t \left( \int _{0}^{t}u^m w^{m}_{1}(u)\frac{du}{u}\right) ^{-1/m}, \end{aligned}$$

and

$$\begin{aligned} \rho _0(t)=t \left( 1+\int _{t}^{1} w^{m}_{0}(u)\frac{du}{u}\right) ^{1/m}. \end{aligned}$$

By monotonicity of K-functional, we get

$$\begin{aligned} J(t,f)\ge K(t,f) \left( 1+\int _{t}^{1} w^{m}_{1}(u)\frac{du}{u}\right) ^{1/m}, \end{aligned}$$

from which it follows that

$$\begin{aligned} \sigma (t)J(t,f)\gtrsim \frac{\rho _0(t)}{t}K(t,f). \end{aligned}$$

Since $w_1$ satisfies ($H_m$), we also have

$$\begin{aligned} J(t,f)\gtrsim \frac{1}{\sigma _1(t)}K(t,f). \end{aligned}$$

Altogether, (5.5) reduces to

$$\begin{aligned} K\left( \sigma (t),f,{\bar{A}}_{{w_0,m}}, {\bar{A}}_{w_1,m} \right) \lesssim I(t,f)+ \sigma (t)J(t,f). \end{aligned}$$

(5.7)

Thus, from (5.6) and (5.7), we have the following two-sided Holmstedt-type estimate

$$\begin{aligned} K\left( \sigma (t),f,{\bar{A}}_{w_0,m}, {\bar{A}}_{w_1,m} \right) \approx I(t,f)+ \sigma (t)J(t,f), \quad 0<t<1, \end{aligned}$$

whence it turns out that

$$\begin{aligned} \Vert f\Vert _{X}^r\approx I_1+I_2, \end{aligned}$$

where

$$\begin{aligned} I_1=\int _{0}^{1}[\rho (t)]^{-\theta r}\left( \int _{0}^{t}w_0^m(u)K^{m}(u,f)\frac{du}{u}\right) ^{r/m}\frac{\sigma ^\prime (t)}{\rho (t)}dt, \end{aligned}$$

and

$$\begin{aligned} I_2=\int _{0}^{1}[\rho (t)]^{(1-\theta )r}\left( \int _{t}^{\infty } W^{m}_{1}(u) K^{m}(u,f) \frac{du}{u} \right) ^{\frac{r}{m}}\frac{\sigma ^\prime (t)}{\rho (t)}dt. \end{aligned}$$

Next, using (5.4), we can compute that

$$\begin{aligned} \frac{\sigma ^\prime (t)}{\sigma (t)}\approx t^{-1} \frac{w_{1}^{m}(t)}{1+\int _{t}^{1} w^{m}_{1}(u)\frac{du}{u}},\quad 0<t<1/2. \end{aligned}$$

(5.8)

Now following the same line of argument which we used while estimating the quantity on right hand side of (5.1), we can show that $\Vert f\Vert ^r_{Y}\approx I_2.$ Thus it remains to establish the estimate $I_1\lesssim \Vert f\Vert ^r_{Y}.$ In the case when $r\ge m$, this desired estimate follows from Fubini’s theorem (if $r=m$) or from Theorem 4.1 (if $r>m$). For the remaining case $r<m$, we apply Theorem 4.4 with $\alpha =r/m,$ $h(t)=K^m(t,f)$, $g=\rho ^{-\theta r-1}\sigma ^\prime$, $\psi (t,u)=u^{-1}w_0^{m}(u)\chi _{(0,t)}(u)$ and $v(t)=t^{-1}[\tilde{w}(t)]^r.$ Observe that (4.2) holds trivially for $1/2<x<1$, and for $0<x<1/2$ we have

$$\begin{aligned} \int _{0}^{1}\left( \int _{x}^{1}\psi (t, u)du\right) ^{\alpha }g(t)dt= & {} \int _{x}^{1}\left( \int _{x}^{t}w_0^{m}(u)\frac{du}{u}\right) ^{r/{m}}g(t)dt\\\le & {} \left( \int _{x}^{1}w_0^{m}(u)\frac{du}{u}\right) ^{r/{m}} \int _{x}^{1}g(t)dt\\\lesssim & {} \left( \int _{x}^{1}w_0^{m}(u)\frac{du}{u}\right) ^{r/{m}}[\rho (x)]^{-\theta r}, \end{aligned}$$

and

$$\begin{aligned} \int _{x}^{1}v(t)dt\gtrsim & {} [\rho (x)]^{r(1-\theta )} \int _{x}^{1}[w_1(t)]^{m}\left( \int _{t}^{1}w_1^{m}(u)\frac{du}{u}\right) ^{r/m-1}\frac{dt}{t}\\\approx & {} \left( \int _{x}^{1}w_0^{m}(u)\frac{du}{u}\right) ^{r/{m}}[\rho (x)]^{-\theta r}. \end{aligned}$$

Thus, (4.2) is valid. Hence, the estimate $I_1\lesssim \Vert f\Vert ^r_{Y}$ follows from Theorem 4.4 in the case $r<m$. This completes the proof. $\square$

Remark 5.9

The particular case, when $w_j(t)=(1-\ln t)^{-\alpha _j}$ ($j=0,1$) with $\alpha _1< \alpha _0< 0$, has earlier been considered in [10, Corollary 1].

6 Interpolation Formulae

Finally, we are in a position to describe the interpolation properties of generalized gamma spaces. In view of well-known reiteration technique, our interpolation formulae are rather straightforward consequences of reiteration theorems (from previous section) and characterization of generalized gamma spaces as K-interpolation spaces (Theorems 3.2 and 3.3). Thus, we illustrate how the reiteration technique works only in a single case, and omit the proofs of the remaining assertions.

Throughout this section, $\psi$ and $\phi$ are same as defined in Theorem 3.2.

Theorem 6.1

Let $0< m, q\le \infty$, $0<r <\infty$ , $0<\theta <1$, and let (w, v) be a pair of admissible weights.

(a) Let $0<q, m<\infty$, and put

$$\begin{aligned} \eta _1(t)=v(\phi (t))\left[ \frac{t}{w(\phi (t))}\right] ^{q/m},\quad 0<t< 1. \end{aligned}$$

Assume that $\eta _1$ satisfies ($H_m$). Then

$$\begin{aligned} (\Lambda ^q(w), G\Gamma (q,m;v,w))_{\theta ,r}=G\Gamma (q,r;V_1,w), \end{aligned}$$

where

$$\begin{aligned} V_1(t)=\left( 1+\int _{\psi (t)}^{1}\eta _1^m(u)\frac{du}{u}\right) ^{\theta /{m}-1/r}v^{m/r}(t), \quad 0<t< 1. \end{aligned}$$

(b) Let $0<m<\infty$ and $q=\infty$. Assume that w is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+} w(t)=0$ and $\lim \limits _{t\rightarrow 1^-} w(t)=1.$ Put

$$\begin{aligned} \eta _2(t)=v(w^{-1}(t))\left[ \frac{t}{w^{-1}(t)w^\prime \left( w^{-1}(t)\right) }\right] ^{1/m}, \quad 0<t<1, \end{aligned}$$

and assume that $\eta _2$ satisfies ($H_m$). Then

$$\begin{aligned} (\Lambda ^\infty (w), G\Gamma (\infty ,m;v,w))_{\theta ,r}=G\Gamma (\infty ,r;V_2,w), \end{aligned}$$

where

$$\begin{aligned} V_2(t)=\left( 1+\int _{w(t)}^{1}\eta _2^m(u)\frac{du}{u}\right) ^{\theta /{m}-1/r}v^{m/r}(t), \quad 0<t< 1. \end{aligned}$$

(c) Let $m=\infty$ and $0<q<\infty$. Assume that $\rho =1/v$ is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+}\rho (\phi (t))=0$ and $\lim \limits _{t\rightarrow 1^-}\rho (\phi (t))=1.$ Assume further that $v(\phi )$ satisfies $(H_\infty ).$ Then

$$\begin{aligned} (\Lambda ^q(w), G\Gamma (q,\infty ;v,w))_{\theta ,r}=G\Gamma (q,r;V_3,w), \end{aligned}$$

where

$$\begin{aligned} V_3(t)=v^{\theta }(t)\left[ tw^q(t)\psi ^{-q}(t)v(t)\rho ^\prime (t)\right] ^{1/r}, \quad 0<t< 1. \end{aligned}$$

(d) Let $m=q=\infty$. Assume that w is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+} w(t)=0$ and $\lim \limits _{t\rightarrow 1^-} w(t)=1.$ Assume further that $v(w^{-1})$ satisfies $(H_\infty )$ and that $\rho =1/v$ is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+}\rho (t)=0$ and $\lim \limits _{t\rightarrow 1^-}\rho (t)=1.$ Then

$$\begin{aligned} (\Lambda ^\infty (w), G\Gamma (\infty ,\infty ;v,w))_{\theta ,r}=G\Gamma (\infty ,r;V_4,w), \end{aligned}$$

where

$$\begin{aligned} V_4(t)=v^{\theta }(t)\left[ t^2v(t)(w(t))^{-1}w^\prime (t)\rho ^\prime (t)\right] ^{1/r}, \quad 0<t< 1. \end{aligned}$$

Proof

We give the argument only in the first case. Using Theorem 3.2, we can write

$$\begin{aligned} (\Lambda ^q(w), G\Gamma (q,m;v,w))_{\theta ,r}=(\Lambda ^q(w), (\Lambda ^q(w), L^\infty )_{\eta _1,m})_{\theta ,r}, \end{aligned}$$

now an application of Theorem 5.2 yields

$$\begin{aligned} (\Lambda ^q(w), G\Gamma (q,m;v,w))_{\theta ,r}= (\Lambda ^q(w), L^\infty )_{\tilde{\eta }_1,r}, \end{aligned}$$

where

$$\begin{aligned} \tilde{\eta }_1(t)=V_1(\phi (t))\left[ \frac{t}{w(\phi (t))}\right] ^{q/r},\;\; 0<t<1. \end{aligned}$$

Temporarily set $X=(\Lambda ^q(w), L^\infty )_{\tilde{\eta }_1,r}$ and take $f\in \Lambda ^q(w).$ Then

$$\begin{aligned} \Vert f\Vert ^r_X=\int _{0}^{1}\tilde{\eta }^r_1(t)K^r(t,f;\Lambda ^q(w), L^\infty )\frac{dt}{t}, \end{aligned}$$

now a change of variable $t=\psi (s)$ gives

$$\begin{aligned} \Vert f\Vert ^r_X=\int _{0}^{1}\tilde{\eta }^r_1(\psi (s))K^r(\psi (s),f;\Lambda ^q(w), L^\infty )\frac{\psi ^\prime (s)}{\psi (s)}ds, \end{aligned}$$

next using Lemma 3.1, we arrive at

$$\begin{aligned} \Vert f\Vert ^r_X\approx \int _{0}^{1}\tilde{\eta }^r_1(\psi (s))\left( \int _{0}^{s}[w(\tau )f^*(\tau )]^q\frac{d \tau }{\tau }\right) ^{r/q}\frac{\psi ^\prime (s)}{\psi (s)}ds, \end{aligned}$$

or,

$$\begin{aligned} \Vert f\Vert ^r_X\approx \int _{0}^{1}V^r_1(s)\left( \int _{0}^{s}[w(\tau )f^*(\tau )]^q\frac{d \tau }{\tau }\right) ^{r/q}\frac{ds}{s}, \end{aligned}$$

whence we get $X=G\Gamma (q,r;V_1,w)$ as desired. $\square$

Theorem 6.2

Let $0< m, r< \infty$ and $0<q\le \infty$ , $0<\theta <1$, and let $(w,v_0)$ and $(w,v_1)$ be two pairs of admissible weights.

(a) Let $0<q<\infty .$ For $j=0,1$, put

$$\begin{aligned} \sigma _j(t)=v_j(\phi (t))\left[ \frac{t}{w(\phi (t))}\right] ^{q/m},\quad 0<t< 1, \end{aligned}$$

and assume that $\rho =\sigma _0/\sigma _1$ is strictly increasing on (0, 1) with $\lim \limits _{t\rightarrow 0^+ }\rho (t)=0$ and $\lim \limits _{t\rightarrow 1^- }\rho (t)=1$. Assume further that $\sigma _1$ satisfies ($H_m$) and that there exists $c_1\in (1,\infty )$ and $c_2\in (0, 1)$ such that

$$\begin{aligned} \left( \frac{1+\int _{t}^{1} \sigma ^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} \sigma ^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m}< c_1 \rho (t),\quad 0<t<1, \end{aligned}$$

(6.1)

and

$$\begin{aligned} \rho (t)< c_2 \left( \frac{1+\int _{t}^{1} \sigma ^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} \sigma ^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m},\quad 0<t<1/2. \end{aligned}$$

(6.2)

Then

$$\begin{aligned} (G\Gamma (q,m;v_0,w), G\Gamma (q,m;v_1,w))_{\theta ,r}=G\Gamma (q,r;V_1,w), \end{aligned}$$

where

$$\begin{aligned} V_1(t)=\left[ \frac{v_0(t)}{v_1(t)}\right] ^{(1-\theta )}v^{m/r}_1(t)\left( 1+\int _{\psi (t)}^{1} \sigma _1^{m}(u)\frac{du}{u}\right) ^{1/m-1/r}, \quad 0<t< 1. \end{aligned}$$

(b) Let $q=\infty$. Assume that w is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+} w(t)=0$ and $\lim \limits _{t\rightarrow 1^-} w(t)=1.$ For $j=0,1$, put

$$\begin{aligned} \delta _j(t)=v_j(w^{-1}(t))\left[ \frac{t}{w^{-1}(t)w^\prime \left( w^{-1}(t)\right) }\right] ^{1/m}, \quad 0<t<1 \end{aligned}$$

and assume that $\rho =\delta _0/\delta _1$ is strictly increasing on (0, 1) with $\lim \limits _{t\rightarrow 0^+ }\rho (t)=0$ and $\lim \limits _{t\rightarrow 1^- }\rho (t)=1$. Assume further that $\delta _1$ satisfies ($H_m$) and that there exists $c_1\in (1,\infty )$ and $c_2\in (0, 1)$ such that

$$\begin{aligned} \left( \frac{1+\int _{t}^{1} \delta ^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} \delta ^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m}< c_1 \rho (t),\quad 0<t<1, \end{aligned}$$

and

$$\begin{aligned} \rho (t)< c_2 \left( \frac{1+\int _{t}^{1} \delta ^{m}_{0}(u)\frac{du}{u}}{1+\int _{t}^{1} \delta ^{m}_{1}(u)\frac{du}{u}}\right) ^{1/m},\quad 0<t<1/2. \end{aligned}$$

Then

$$\begin{aligned} (G\Gamma (\infty ,m;v_0,w), G\Gamma (\infty ,m;v_1,w))_{\theta ,r}=G\Gamma (\infty ,r;V_2,w), \end{aligned}$$

where

$$\begin{aligned} V_2(t)=\left[ \frac{v_0(t)}{v_1(t)}\right] ^{(1-\theta )}v^{m/r}_1(t)\left( 1+\int _{w(t)}^{1} \delta _1^{m}(u)\frac{du}{u}\right) ^{1/m-1/r}, \quad 0<t< 1. \end{aligned}$$

7 Special Cases

Throughout this section, we let $0<\theta <1$ and $0<r <\infty$.

(i) Let $0<p, q, m<\infty ,$ and let $v\in SV.$ Take $w(t)=t^{1/p}$, then $\psi (t)\approx t^{1/p}.$ Now, in view of Proposition 2.1 (vi), we can see easily that $\eta _1$ satisfies ($H_m$). Thus, according to Theorem 6.1 (a), we have

$$\begin{aligned} (L^{p,q}, L_{v}^{(p,q,m})_{\theta ,r}=L_{V}^{(p,q,r}, \end{aligned}$$

(7.1)

where

$$\begin{aligned} V(t)=\left( 1+\int _{t}^{1}v^m(u)\frac{du}{u}\right) ^{\theta /{m}-1/r}v^{m/r}(t), \quad 0<t< 1. \end{aligned}$$

Here $L_{v}^{(p,q,m}$ is the small Lorentz space considered in [3] (see Remark 2.5). Thus, interpolation formula (7.1) provides a limiting version of the interpolation formula contained in [3, Theorem 5.3]. In addition, if we take $1<p=q<\infty$, $m=1$, and $v(t)= (1-\ln t)^{-1/p}$ in (7.1), then we recover the interpolation formula in [2, Corollary 3.2].

(ii) Let $0<p,q<\infty$ and let $\alpha , \beta \in \mathbb {R}$ with $\alpha < \beta .$ Take $w(t)=t^{1/p}(1-\ln t)^{\alpha }$, $v(t)=(1-\ln t)^{\beta -\alpha -1/q}$ and $m=q$. Now we have $\psi (t)\approx t^{1/p}(1-\ln t)^{\alpha }$. Thus, we can check easily that

$$\begin{aligned} \left[ \psi (t)\right] ^{-1}\left( \int _{0}^{t}\left[ \psi (u)\right] ^qv^q(u)\frac{du}{u}\right) ^{1/q}\lesssim \left( 1+\int _{t}^{1}v^q(u)\frac{du}{u}\right) ^{1/q},\quad 0<t<1. \end{aligned}$$

Consequently, $\eta _1$ satisfies ($H_q$). Thus, we can apply Theorem 6.1 (a) to obtain the following description of interpolation spaces between Lorentz–Zygmund spaces (in a limiting case):

$$\begin{aligned} \left( L^{p,q}(\log L)^{\alpha }, L^{p,q}(\log L)^{\beta }\right) _{\theta ,r}=G\Gamma (q,r, {V}, w), \end{aligned}$$

(7.2)

where

$$\begin{aligned} V(t)=(1-\ln t)^{\theta (\beta -\alpha )-1/r}, \quad 0<t< 1. \end{aligned}$$

The interpolation spaces $\left( L^{p,q}(\log L)^{\alpha }, L^{p,q}(\log L)^{\beta }\right) _{\theta ,r}$ have already been characterized in [25, Theorem 6 (c)], but the description given there is theoretical and complicated. On the other hand, the formula (7.2) provides a concrete description in terms of generalized gamma spaces.

(iii) Assume that $w_0$ is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+} w_0(t)=0$ and $\lim \limits _{t\rightarrow 1^-} w_0(t)=1.$ Let $w_1$ be another weight such that $\rho =w_0/w_1$ is strictly increasing and differentiable on (0, 1) with $\lim \limits _{t\rightarrow 0^+}\rho (t)=0$ and$\lim \limits _{t\rightarrow 1^-}\rho (t)=1.$ We can check easily that $v(w^{-1}_0)$ satisfies $(H_\infty )$. Thus, by Theorem 6.1 (d), we have

$$\begin{aligned} (\Lambda ^\infty (w_0), \Lambda ^\infty (w_1))_{\theta ,r}=G\Gamma (\infty ,r;V,w_0), \end{aligned}$$

(7.3)

where

$$\begin{aligned} V(t)=\left[ \frac{w_1(t)}{w_0(t)}\right] ^{\theta }\left[ t^2w_1(t)(w_0(t))^{-2}w_0^\prime (t)\rho ^\prime (t)\right] ^{1/r}, \quad 0<t< 1. \end{aligned}$$

The interpolation formula (7.3) complements the diagonal case ($r=\infty$) considered in [9, Theorem 4.4].

(iv) If we take $1<q<\infty$, $m=1$, $w(t)=t^{1/q}$ and $v_j(t)= (1-\ln t)^{-\alpha _j/q+\alpha _j-1}$ ($j=0,1$) with $0<\alpha _0< \alpha _1<\infty$ in Theorem 6.2 (a), then we recover the interpolation formula in [4, Theorem 7]. Indeed, all the conditions necessary to apply Theorem 6.2 (a) are trivially met as $\sigma _j\approx v_j$.