Measure of non-compactness and limiting interpolation with slowly varying functions

Abstract

We give estimates for the measure of non-compactness of an operator interpolated by the limiting methods involving slowly varying functions. As applications we establish estimates for the measure of non-compactness of operators acting between Lorentz–Karamata spaces.

1 Introduction

The real interpolation method $(A_0,A_1)_{\theta ,q}$ has found important applications in Operator Theory, Approximation Theory, Function Spaces and Harmonic Analysis. See, for example, the monographs by Butzer and Berens [8], Bergh and Löfström [4], Triebel [43, 44], König [32] and Bennett and Sharpley [3]. The real method is very flexible, admitting several equivalent definitions, what is very useful in applications.

The real method applied to the couple of Lebesgue spaces $(L_1,L_{\infty })$ yields Lorentz spaces $L_{p,q}$. It is possible to obtain more general spaces if we modify the definition of the real method. So, logarithmic perturbations of the real method produce Lorentz–Zygmund spaces $L_{p,q}(\log {L})_a$ (see [20, 27, 28]) and perturbations involving slowly varying functions $(L_1,L_{\infty })_{\theta ,q;b}$ give Lorentz–Karamata spaces $L_{p,q;b}$(see [29]).

We are interested here in the limit cases when $\theta =0,1$ of the perturbations with slowly varying functions $(A_0,A_1)_{\theta ,q;b}$. These spaces are very close to $A_0$ when $\theta =0$ and to $A_1$ when $\theta =1$. They have received attention from a number of authors either to study limiting embeddings between function spaces or to establish limiting properties of operators (see, for example, [26, 29, 36]).

Among the classical problems for any interpolation method, a prominent one is to describe the behavior of properties that operators may have. First of all boundedness but then other useful properties of operators. For example, techniques used by Davis, Figiel, Johnson and Pelczyński [23] in the proof of their famous factorization theorem for weakly compact operators motivated the investigation on the behavior of weak compactness under interpolation (see, for example, [1, 18, 31, 34, 35]).

The behavior under interpolation of compactness have been also deeply studied (see [9, 17, 22] and the references given there). Quantitative estimates in terms of the measure of non-compactness have been also established. Concerning the real method, the first result in this direction is due to Edmunds and Teixeira [42]. They assume an approximation condition for the couple in the target. The case of general Banach couples has been studied by Cobos, Fernández-Martínez and Martínez [15]. Results for the real method with a function parameter and $0<\theta <1$ are due to Cordeiro [21], Szwedek [41] and Cobos, Fernández-Cabrera and Martínez [11]. Besides, the case of limiting methods involving logarithms have been considered by Cobos, Fernández-Cabrera and Martínez [12, 14] and Besoy and Cobos [5].

Our aim here is to establish estimates for the measure of non-compactness of operators interpolated by the limiting perturbations of the real method involving slowly varying functions. As applications we derive estimates for the measure of non-compactness of operators acting between certain Lorentz–Karamata spaces. In particular, one of our results can be considered as a quantitative extension of a compactness result of Edmunds and Opic [26] for operators acting between Lorentz–Zygmund spaces.

We work with quasi-Banach couples $(A_0,A_1).$ Our techniques are based on the vector-valued sequence spaces that come up with the definition of $(A_0,A_1)_{0,q;b}$ and with its description as a J-space. These ideas originated in the papers on compactness by Cobos and Peetre [19] and Cobos, Kühn and Schonbek [17]. In the context of the measure of non-compactness, they were developed by Cobos, Fernández-Martínez and Martínez [15], Cobos, Fernández-Cabrera and Martínez [14] and Besoy and Cobos [5] among other authors.

2 Limiting real interpolation spaces

Let $(A,\Vert \cdot \Vert _A)$ be a quasi-Banach space and let $c_A\ge 1$ be its constant in the quasi-triangle inequality. Let $0<p\le 1$ such that $c_A=2^{1/p-1}$. According to the Aoki-Rolewicz theorem (see [33, Section 15.10]) there is another quasi-norm $|||\cdot |||$ on A which is equivalent to $\Vert \cdot \Vert _A$ and such that ${|||\cdot |||}^p$ satisfies the triangle inequality. Then $(A,|||\cdot |||)$ is called a p-Banach space. Note that if $0<r<p$, then $(A,|||\cdot |||)$ is also an r-Banach space and that any p-Banach space satisfies the quasi-triangle inequality with constant $2^{{1}/{p}-1}$.

If B is another quasi-Banach space, we write $A=B$ if $A\hookrightarrow B$ and $B\hookrightarrow A$, where $\hookrightarrow$ means continuous embedding.

For $0<q\le \infty$, let $\ell _q$ be the space of q-summable sequences with $\mathbb {Z}$ as index set. If $(w_m)_{m\in Z}$ is a sequence of positive numbers, we denote by $\ell _q(w_m)$ the space of all scalars sequences $(\xi _m)$ such that $(w_m\xi _m)\in \ell _q$.

Let $(W_m)$ be a sequence of quasi-Banach spaces with the same constant in the quasi-triangle inequality. We put

$$\begin{aligned} \ell _q(w_m\mathrm {W_m})&=\{\textrm{w}=\mathrm {(w_m)}:\mathrm {w_m}\in \mathrm {W_m} \ \text {and} \\&\quad \Vert \textrm{w}\Vert _{\ell _q(w_mW_m)}=\Vert (w_m\Vert \mathrm {w_m}\Vert _{\mathrm {W_m}})\Vert _{\ell _q}<\infty \}. \end{aligned}$$

A quasi-Banach space $(\Gamma ,\Vert \cdot \Vert _{\Gamma })$ of real valued sequences with $\Gamma \hookrightarrow \ell _q+\ell _q(2^{-m})$ is said to be a quasi-Banach sequence lattice if $\Gamma$ contains all the sequences with only finitely many non-zero coordinates and whenever $(\eta _m)\in \Gamma$ and $|\xi _m|\le |\eta _m|$ for each $m\in \mathbb {Z}$, then $(\xi _m)\in \Gamma$ and $\Vert (\xi _m)\Vert _{\Gamma }\le \Vert (\eta _m)\Vert _{\Gamma }$.

We define $\Gamma (\mathrm {W_m})$ as the collection of all sequences $\textrm{w}=\mathrm {(w_m)}$ such that $\mathrm {w_m}\in \mathrm {W_m}$ and $\Vert \textrm{w}\Vert _{\Gamma (\mathrm {W_m})}=\Vert (\Vert \mathrm {w_m}\Vert _{\mathrm {W_m}})\Vert _{\Gamma }<\infty$.

Subsequently, if b and v are non-negative functions on $(0,\infty )$, we say that b and v are equivalent (and write $b(t)\approx v(t))$ if there are positive constants $c,\textrm{C}$ such that $c b(t)\le v(t) \le \textrm{C} b(t)$ for any $t>0$.

A positive, finite and Lebesgue-measurable function b on $(0,\infty )$ is said to be slowly varying $(b\in SV(0,\infty ))$ if, for each $\varepsilon >0,\,t^{\varepsilon }b(t)$ is equivalent to a positive non-decreasing measurable function and $t^{-\varepsilon }b(t)$ is equivalent to a positive non-increasing measurable function. Important examples of slowly varying functions are powers of iterated logarithms and broken logarithmic functions $v(t)=\ell ^{\mathbb {A}}(t)$ where $\ell (t)=(1+|\log (t)|), \mathbb {A}=(\alpha _0, \alpha _\infty ) \in \mathbb {R}^2, \ell ^{\mathbb {A}}(t)=\ell ^{\alpha _0}(t)$ if $0<t\le 1$ and $\ell ^{\mathbb {A}}(t)=\ell ^{\alpha _\infty }(t)$ if $1<t < \infty$.

We refer to [29] for properties of slowly varying functions. We only recall here that if $\varepsilon >0$, then there are positive constant $c_{\varepsilon },\,C_{\varepsilon }$ such that

$$\begin{aligned} c_{\varepsilon }\min \{s^{-\varepsilon },s^{\varepsilon } \}b(t)\le b(st)\le \textrm{C}_{\varepsilon }\max \{s^{\varepsilon },s^{-\varepsilon }\} b(t) \ \text {for every }s,t>0, \end{aligned}$$

(2.1)

(see [29, Proposition 2.2]). Put

$$\begin{aligned} \overline{b}(s)= \sup \limits _{t>0} \frac{b(st)}{b(t)}\,. \end{aligned}$$

The function $\overline{b}$ satisfies that $\overline{b}(st)\le \overline{b}(s)\overline{b}(t)$. Moreover, using (2.1) with $\varepsilon =1/2$, we have

$$\begin{aligned} s\bar{b}(s)\le {\textrm{C}}_{1/2} s ^{1/2}\rightarrow 0 \ \text {as} \ s \rightarrow 0. \end{aligned}$$

(2.2)

Another consequence of (2.1), this time with $\varepsilon =1$, is that

$$\begin{aligned} c_1 /2 \le \bar{b}(s)\le 2{\textrm{C}}_{1} \ \text {for any} \ 1/2 \le s \le 1. \end{aligned}$$

(2.3)

Let $\mathbb {A}=(\alpha _0, \alpha _\infty ) \in \mathbb {R}^2, v(t)=\ell ^{\mathbb {A}}(t)$ and $\mathbb {B}=(\alpha _0 ^+ + (-\alpha _\infty )^+, \alpha _\infty ^+ + (-\alpha _0)^+ )$ with $\alpha ^+ =\max \{0, \alpha \}$. It follows from [14, Lemma 2.1] and [5, (2.6)] that $\bar{v}(s) \le \ell ^{\mathbb {B}}(s), s\in (0,\infty )$.

For $0<q\le \infty$ and $b\in SV (0,\infty )$, the quasi-Banach sequence space $\ell _q(b(2^m))$ will be of special interest for us.

If $k\in \mathbb {Z}$, the shift operator $\tau _k$ is defined by $\tau _k\xi =(\xi _{m+k})_{m\in \mathbb {Z}}$ for $\xi =(\xi _m)$. We have

$$\begin{aligned} \Vert \tau _k\xi \Vert _{\ell _q(b(2^m))}=\Vert (b(2^m)|\xi _{m+k}|)\Vert _{\ell _q}\le \overline{b}(2^{-k})\Vert (b(2^{m+k})|\xi _{m+k}|)\Vert _{\ell _q}. \end{aligned}$$

Hence $\tau _k:\ell _q(b(2^m))\rightarrow \ell _q(b(2^m))$ is bounded with

$$\begin{aligned} \Vert \tau _k\Vert _{\ell _q(b(2^m)),\ell _q(b(2^m))}\le \bar{b}(2^{-k}). \end{aligned}$$

(2.4)

We say that $\overline{A}=(A_0,A_1)$ is a (p-Banach) quasi-Banach couple if $A_0$ and $A_1$ are (p-Banach) quasi-Banach spaces which are continuously embedded in the same Hausdorff topologic vector space.

For $t>0$ and $a\in A_0+A_1$, the Peetre’s K-functional is given by

$$\begin{aligned} K(t,a)=K(t,a;A_0,A_1)=\inf \{\Vert a_0\Vert _{A_0}+t\Vert a_1\Vert _{A_1}:a=a_0+a_1,a_j\in A_j\}. \end{aligned}$$

If $a\in A_0\cap A_1$, the J-functional of Peetre is

$$\begin{aligned} J(t,a)=J(t,a;A_0,A_1)=\max \{\Vert a\Vert _{A_0},t\Vert a\Vert _{A_1}\}. \end{aligned}$$

Note that $K(1,\cdot )$ and $J(1,\cdot )$ are the quasi-norms of $A_0+A_1$ and $A_0\cap A_1$, respectively.

If $(A_j,\Vert \cdot \Vert _{A_j})$ is a p-Banach space for $j=0,1$, then $J(t,\cdot )$ is also a p-norm, as well as

$$\begin{aligned} K_p(t,a)&=K_p(t,a;A_0,A_1)\\&=\inf \{(\Vert a_0\Vert ^p_{A_0}+t^p\Vert a_1\Vert ^p_{A_1}) ^{1/p}:a=a_0+a_1,a_j\in A_j\}. \end{aligned}$$

This last functional is equivalent to the K-functional. In fact

$$\begin{aligned} K(t,a)\le K_p(t,a)\le 2^{1/p-1}K(t,a),\quad a\in A_0+A_1. \end{aligned}$$

(2.5)

Note that if $\xi =(\xi _m)\in \ell _p+\ell _p(2^{-m})$ then

$$\begin{aligned} K_p(2^r ,\xi ;\ell _p,\ell _p(2^{-m})) =\left( \sum ^{\infty }_{m=-\infty }[\min \{1,2^{r -m}\}|\xi _m|]^p\right) ^{1/p}. \end{aligned}$$

This expression will be useful later.

A quasi-Banach space A is said to be an intermediate space with respect to the couple $\overline{A}$ if $A_0\cap A_1\hookrightarrow A\hookrightarrow A_0+A_1$. We write $A^{\circ }$ for the closure of $A_0\cap A_1$ in A. The fundamental lemma (see [4, Lemma 3.3.2] and [37, Lemma 2.4]) yields that

$$\begin{aligned}&a\in (A_0+A_1)^{\circ } \ \text {if and only if} \ \min \left\{ 1,\frac{1}{t}\right\} K(t,a)\rightarrow 0\nonumber \\&\ \text {as} \ t\rightarrow 0 \ \text {and as} \ t\rightarrow \infty . \end{aligned}$$

(2.6)

For $0\le \theta \le 1,\,0<q\le \infty$ and $b\in SV(0,\infty )$, the space ${\bar{A}_{\theta ,q;b}}=(A_0,A_1)_{\theta ,q;b}$ consists of all those $a\in A_0+A_1$ that have a finite quasi-norm

$$\begin{aligned} \Vert a\Vert _{{\bar{A}_{\theta ,q;b}}}=\Vert a\Vert _{(A_0,A_1)_{\theta ,q;b}}=\left( \sum \limits _{m=-\infty }^{\infty }[2^{-\theta m}b(2^m)K(2^m,a)]^q\right) ^{1/q} \end{aligned}$$

(the sum should be replaced by the supremum when $q=\infty$). See [29, 37]. If $b\equiv 1$ and $0<\theta <1$, we recover the classical real interpolation space $(A_0,A_1)_{\theta ,q;b}$ (see [3, 4, 8, 43]). If $0<\theta <1$ then $(A_0,A_1)_{\theta ,q;b}$ is a special case of the real method with function parameter (see [30, 40]). If $\theta =0,1,\, \alpha _0,\alpha _{\infty }\in \mathbb {R}$ and

$$\begin{aligned} b(t)= {{\left\{ \begin{array}{ll} (1+|\log {t}|)^{\alpha _0}&{}\text {if }0<t\le 1,\\ (1+|\log {t}|)^{\alpha _{\infty }}&{}\text {if }1<t<\infty , \end{array}\right. }} \end{aligned}$$

then we recover the logarithmic interpolation spaces $\bar{A}_{\theta ,q;( \alpha _0,\alpha _{\infty })}$ (see [10, 16, 20, 27, 28]).

We are mainly interested here in the limiting spaces $\overline{A}_{0,q;b}$ and $\overline{A}_{1,q;b}$. Since $K(t,a;A_0,A_1)=tK(t^{-1},a;A_1,A_0)$, they are related by the equality

$$\begin{aligned} (A_0,A_1)_{0,q;b}=(A_1,A_0)_{1,q;v} \ \text {where} \ v(t)=b({1}/{t}). \end{aligned}$$

(2.7)

Note that v is also slowly varying on $(0,\infty )$. Due to equality (2.7), in what follows we focus on the case $\theta =0$.

As it is shown in [28], $(A_0,A_1)_{0,q;b}$ is an intermediate space with respect to $\overline{A}$ if and only if

$$\begin{aligned} \left( \int _1^{\infty }b(t)^q\,\textrm{d}t/t\right) ^{1/q}<\infty . \end{aligned}$$

(2.8)

Let $\overline{B}=(B_0,B_1)$ be another quasi-Banach couple. We write $T\in \mathcal {L} (\overline{A},\overline{B})$ to mean that T is a bounded linear operator from $A_0+A_1$ into $B_0+B_1$ such that the restrictions $T:A_j\rightarrow B_j$ are bounded for $j=0,1$. Then the restriction

$$\begin{aligned} T:(A_0,A_1)_{0,q;b}\rightarrow (B_0,B_1)_{0,q;b} \end{aligned}$$

is also bounded. Indeed, if $M_j$ is bigger than or equal to the norm of $T:A_j\rightarrow B_j$, $j=0,1$, then

$$\begin{aligned} K(t,Ta;B_0,B_1)\le M_0K\left( \frac{tM_1}{M_0},a;A_0,A_1\right) . \end{aligned}$$

Therefore, if $M_1\le M_0$, we obtain that $\Vert T\Vert _{\bar{A}_{0,q;b},\bar{B}_{0,q;b}}\le M_0.$ If $M_0<M_1$ then we can find $r\in \mathbb {N}\cup \{0\}$ such that $2^r\le M_1/M_0<2^{r+1}.$ Hence

$$\begin{aligned} \Vert Ta \Vert _{\bar{B}_{0,b;q}}&\le \left( \sum \limits _{m=-\infty }^{\infty }[b(2^m)M_0K(2^{(m+r+1)},a)]^q\right) ^{1/q}\\&\le M_0\,\bar{b}(2^{-r-1})\Vert a\Vert _{\bar{A}_{0,b;q}}\\&\le cM_0\,\bar{b}\left( \frac{M_0}{M_1}\right) \Vert a\Vert _{\bar{A}_{0,b;q}} \end{aligned}$$

where we have used (2.3) in the last inequality. Therefore

$$\begin{aligned} \Vert T\Vert _{\bar{A}_{0,q;b},\bar{B}_{0,q;b}}\le {{\left\{ \begin{array}{ll} M_0 &{}\text {if } M_1\le M_0,\\ cM_0\bar{b}\left( \frac{M_0}{M_1}\right) &{}\text {if }M_0<M_1,\\ \end{array}\right. }} \end{aligned}$$

(2.9)

where $c>0$ is a constant depending only on b.

If $(T_n)\subseteq \mathcal {L}(A_0+A_1,B_0+B_1)$ with

$$\begin{aligned} \sup {\{\Vert T_n\Vert _{A_1,B_1}:n\in \mathbb {N}\}}<\infty \text { and } \lim \limits _{n\rightarrow \infty }\Vert T_n\Vert _{A_0,B_0}=0, \end{aligned}$$

then it follows from (2.9) and (2.1) that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert T_n\Vert _{\bar{A}_{0,q;b},\bar{B}_{0,q;b}}=0. \end{aligned}$$

(2.10)

Next we show a sufficient condition on b for the inclusion $(A_0,A_1)_{0,q;b}\subseteq (A_0+A_1)^\circ$. Let $0<q\le \infty$ and take any $a\in (A_0,A_1)_{0,q;b}.$ Then

$$\begin{aligned} \left( \sum \limits _{n=-\infty }^{\infty }\left[ \frac{b(2^n)}{\min {\{1,2^{-n}\}}}\min {\{1,2^{-n}\}}K(2^n,a)\right] ^q\right) ^{1/q}=\Vert a\Vert _{\bar{A}_{0,q;b}}<\infty . \end{aligned}$$

(2.11)

Since tb(t) is equivalent to a non-decreasing function, we have $\left( \sum \nolimits ^{\infty }_{n=0}[2^nb(2^n)]^q\right) ^{1/q}=\infty$. Hence, from (2.11) it follows that

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{1}{t}K(t,a)=\lim _{n\rightarrow \infty } \frac{1}{2^n}K(2^n,a)=0. \end{aligned}$$

On the other hand, if we assume

$$\begin{aligned} {\left\{ \begin{array}{ll} (\int _0^1 b(t)^q\,\textrm{d}t/t)^{1/q}=\infty &{}\text {if }0<q<\infty , \\ \lim \limits _{t\rightarrow 0 } b(t)=\infty &{}\text {if } \ q=\infty , \end{array}\right. } \end{aligned}$$

(2.12)

then we also have that $\lim \nolimits _{t\rightarrow 0} K(t,a)=0$. Having in mind (2.6), it turns out that if (2.12) is satisfied then $(A_0,A_1)_{0,q;b}\subseteq (A_0+A_1)^\circ$.

The Gagliardo completion $A_{j}^{\sim }$ of $A_j$ consists of all those $a\in A_0+A_1$ having a finite quasi-norm

$$\begin{aligned} \Vert a\Vert _{A_j^{\sim }}=\sup \{t^{-j}K(t,a):t>0\}\quad \text{(see } \text{[4,3]). } \end{aligned}$$

We have that $A_j\hookrightarrow A_j^{ \sim }$ for $j=0,1$. The quasi-Banach couple $\overline{A}$ is called mutually closed if $A_j=A_j^{\sim }$ for $j=0,1$.

If $\Gamma$ is a quasi-Banach sequence lattice and $\overline{A}=(A_0,A_1)$ is a p-Banach couple, then the J-space $\overline{A}_{\Gamma ;J}=(A_0,A_1)_{\Gamma ;J}$ is formed by all sums $a=\sum \nolimits ^{\infty }_{m=-\infty }u_m$ (convergence in $A_0+A_1)$, where $(u_m)\subseteq A_0\cap A_1$ and $(J(2^m,u_m))\in \Gamma$. We endow $\overline{A}_{\Gamma ;J}$ with the quasi-norm

$$\begin{aligned} \Vert a\Vert _{\overline{A}_{\Gamma :J}}=\Vert a\Vert _{(A_0,A_1)_{\Gamma ;J}}=\inf \left\{ \Vert (J(2^m,u_m))\Vert _{\Gamma }:a=\sum \limits ^{\infty }_{m=-\infty }u_m\right\} \end{aligned}$$

(see [37]).

Next, we give a description of $(A_0,A_1)_{0,q;b}$ by means of the J-functional.

Theorem 2.1

Let $\overline{A}=(A_0,A_1)$ be a mutually closed p-Banach couple $(0<p\le 1)$. Let $0<q\le \infty$ and let $b\in SV(0,\infty )$ satisfying (2.8) and (2.12). Put $\Lambda =(\ell _p,\ell _p(2^{-m}))_{0,q:b}$. Then we have with equivalent quasi-norms

$$\begin{aligned} (A_0,A_1)_{0,q:b}=(A_0,A_1)_{\Lambda ;J}. \end{aligned}$$

Proof

Let $a\in (A_0,A_1)_{0,q:b}$. By the assumption on b, we know that $(A_0,A_1)_{0,q:b}\subseteq (A_0+A_1)^\circ$. Hence, according to [37, Theorem 3.2], there exists $(u_m)\subseteq A_0\cap A_1$ such that $a=\sum \nolimits ^{\infty }_{m=-\infty }{u_m}$ (in $A_0+A_1$) and

$$\begin{aligned} \left( \sum \limits ^{\infty }_{m=-\infty }\min (1, 2^{k-m})^pJ(2^m,u_m)^p\right) ^{1/p}\le cK(2^k,a),\quad k\in \mathbb {Z}, \end{aligned}$$

where c only depends on p. Whence

$$\begin{aligned} \Vert a\Vert _{(A_0,A_1)_{\Lambda ;J}}&\le \Vert (J(2^m,u_m))\Vert _{(\ell _p,\ell _p(2^{-m}))_{0,q;b}}\\&\le \left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)K_p(2^k,((J(2^m,u_m));\ell _p,\ell _p(2^{-m}))\right] ^q\right) ^{1/q}\\&\le \left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)\left( \sum ^\infty _{m=-\infty }\min (1,2^{k-m})^p J(2^m,u_m)^p\right) ^{1/p}\right] ^q\right) ^{1/q}\\&\le c\left( \sum ^{\infty }_{k=-\infty }[b(2^k)K(2^k,a)]^q\right) ^{1/q}\\&=c\Vert a\Vert _{(A_0,A_1)_{0,q;b}}\,. \end{aligned}$$

Conversely, take any $a\in (A_0,A_1)_{\Lambda ;J}$. We can find a J-representation $a=\sum \nolimits ^{\infty }_{m=-\infty }{u_m}$ with $\Vert (J(2^m,u_m))\Vert _\Lambda \le 2\Vert a\Vert _{(A_0,A_1)_{\Lambda ;J}}$. Since

$$\begin{aligned} K_p(2^k,a)\le \left( \sum ^{\infty }_{m=-\infty }\min (1,2^{k-m})^pJ(2^m,u_m)^p\right) ^{1/{p}},\quad k\in \mathbb {Z}, \end{aligned}$$

we obtain that

$$\begin{aligned}&\Vert a\Vert _{(A_0,A_1)_{0,q;b}}\\&\le \left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)\left( \sum ^{\infty }_{m=-\infty }\min (1,2^{k-m})^pJ(2^m,u_m)^p\right) ^{1/p}\right] ^q\right) ^{1/q}\\&\le 2^{1/p-1}\Vert (J(2^m,u_m))\Vert _{(\ell _p,\ell _p(2^{-m}))_{0,q;b}}\\&\le 2^{1/p}\Vert a\Vert _{(A_0,A_1)_{\Lambda ;J}}. \end{aligned}$$

$\square$

In Theorem 2.1, the sequence space that defines $(A_0,A_1)_{0,q;b}$ as a J-space is not explicitly described, it appears as the interpolation space $\Lambda =(\ell _p,\ell _p(2^{-m}))_{0,q:b}$ instead, what is enough for our aims here. Assuming extra conditions on the couple $(A_0,A_1)$ and on the function b, there are several papers in the literature where the sequence space $\Lambda$ is explicitly described. More precisely, in the case of logarithmic interpolation spaces, explicit descriptions as J-spaces have been obtained by Cobos and Kühn [16] for the case of ordered Banach couples, by Cobos and Segurado [20] and Besoy, Cobos and Fernández–Cabrera [7] for general Banach couples and by Besoy and Cobos [6] for quasi-Banach couples. If $(A_0,A_1)$ is a Banach couple and $1\le q\le \infty$, an explicit description of $(A_0,A_1)_{0,q;b}$ as a J-space has been recently established by Grover and Opic [38].

The following estimate for the norm of the shift operator $\tau _k$ on $\Lambda$ will be useful later.

Lemma 2.2

Let $0<p\le 1$, $0<q\le \infty$ and let $b\in SV(0,\infty )$ satisfying (2.8). Put $\Lambda =(\ell _p,\ell _p(2^{-m}))_{0,q;b}$. Then, for any $k\in \mathbb {Z},$ we have

$$\begin{aligned} \Vert \tau _k\Vert _{\Lambda ,\Lambda }\le 2^{1/p-1}\overline{b}(2^{-k}). \end{aligned}$$

Proof

Given any $\xi =(\xi _m)\in \Lambda$, we have

$$\begin{aligned} \Vert \tau _k\xi \Vert _\Lambda&\le \left( \sum ^{\infty }_{n=-\infty }\left[ b(2^n)K_p(2^n,\tau _k\xi ;\ell _p,\ell _p(2^{-m}))\right] ^q\right) ^{1/q}\\&=\left( \sum ^{\infty }_{n=-\infty }\left[ b(2^n)\left( \sum ^{\infty }_{m=-\infty }\min ({1,2}^{n-m})^p|\xi _{m+k}|^p\right) ^{1/p}\right] ^q\right) ^{1/q}\\&\le \overline{b}(2^{-k})\left( \sum ^{\infty }_{n=-\infty }\left[ b(2^{n+k})\left( \sum ^{\infty }_{m=-\infty } \min (1,2^{n+k-m})^p|\xi _m |^p\right) ^{1/p}\right] ^q\right) ^{1/q}\\&\le 2^{1/p-1}\overline{b}(2^{-k})\left( \sum \limits _{n=-\infty }^{\infty }[b(2^n)K(2^n,\xi )]^q\right) ^{1/q} \\&=2^{1/p-1}\overline{b}(2^{-k})\Vert \xi \Vert _\Lambda \end{aligned}$$

where we have used (2.5) in the penultimate inequality. $\square$

3 Measure of non-compactness

Let A, B be quasi-Banach spaces and $T\in \mathcal {L}(A,B).$ The (ball) measure of non-compactness $\mathbf {\beta }(T)=\mathbf {\beta }(T:A\rightarrow B)$ is defined to be the infimum of the set of numbers $\sigma >0$ for which there is a finite subset $\{z_1,\ldots ,z_n\}\subseteq B$ such that

$$\begin{aligned} T(U_A)\subseteq \bigcup \limits _{j=1}^n\{z_j+\sigma U_B\}. \end{aligned}$$

Here $U_A,\,U_B$ are the closed unit balls of A and B, respectively. See [24] for details on the measure of non-compactness. Note that $\mathbf {\beta }(T)\le \Vert T\Vert _{A,B}$ and that $\mathbf {\beta }(T)=0$ if and only if T is compact. That is, $\mathbf {\beta }(T)=0$ means that T transforms each bounded set of A into a set whose closure is compact in B.

If $T_1$ is another operator belonging to $\mathcal {L}(A,B)$, then it is not difficult to check that

$$\begin{aligned}\beta (T+T_1: A\rightarrow B)\le c_B \big (\beta (T:A \rightarrow B) + \beta (T_1:A \rightarrow B)\big ).\end{aligned}$$

If we assume that E, X are other quasi-Banach spaces and that $S\in \mathcal {L} (B,E)$ and $R\in \mathcal {L} (X,A)$, then we have

$$\begin{aligned} \beta (STR:X\rightarrow E)\le \Vert R\Vert _{X,A}\beta (T:A\rightarrow B)\Vert S\Vert _{B,E}. \end{aligned}$$

Furthermore, if $\Vert Sb\Vert _E=\Vert b\Vert _B$ for all $b\in B$, then

$$\begin{aligned} \beta (T:A\rightarrow B)\le 2c_E\beta (ST:A\rightarrow E). \end{aligned}$$

If for any $a\in A$ with $\Vert a\Vert _A<1$, there is $x\in X$ with $\Vert x \Vert _{X}<1$ such that $Rx=a$, then

$$\begin{aligned} \beta (T:A\rightarrow B)\le \beta (TR:X\rightarrow B). \end{aligned}$$

We will use freely these properties in our later computations.

Next we establish the main result of the paper. It shows an estimate for the measure of non-compactness of an operator interpolated using parameters 0, q, b.

Theorem 3.1

Let $\overline{A}=(A_0,A_1)$, $\overline{B}={(B}_0,B_1)$ be quasi-Banach couples and let $T\in \mathcal {L} (\overline{A},\overline{B})$. Let $0<q\le \infty$ and $b\in SV(0,\infty )$ satisfying (2.8) and (2.12). Then we have

• (i) $\beta (T:\overline{A}_{0,q;b}\rightarrow {\overline{B}}_{0,q;b})=0$ if $\beta (T: A_0\rightarrow B_0)=0$,

• (ii)
  $$\begin{aligned} \beta (T:&\overline{A}_{0,q;b}\rightarrow {\overline{B}}_{0,q;b}) \le C\beta (T:A_0\rightarrow B_0) \quad \text {if} \\ {}&0 \le \beta (T:A_1\rightarrow B_1) < \beta (T: A_0\rightarrow B_0),\end{aligned}$$
• (iii)
  $$\begin{aligned} \beta (T:&\overline{A}_{0,q;b}\rightarrow {\overline{B}}_{0,q;b}) \\ {}&\le C \max \Big \{\beta (T:A_0\rightarrow B_0), \beta (T:A_0\rightarrow B_0)\overline{b}\left( \frac{\beta (T:A_0\rightarrow B_0)}{\beta (T:A_1\rightarrow B_1)}\right) \Big \} \end{aligned}$$

  if $0 < \beta (T:A_0\rightarrow B_0) \le \beta (T: A_1\rightarrow B_1)$.

Here C is a constant independent of T.

Proof

Step 1. Consider the mutually closed quasi-Banach couples

$\overline{A^{\sim }}=(A^{\sim }_0,A^{\sim }_1),\,\overline{B^{\sim }}=(B^{\sim }_{0},B^{\sim }_1)$. The arguments of [3, Theorem V.1.5] may be modified to give that

$$\begin{aligned} K(t,a;A^{\sim }_0,A^{\sim }_1) \le K(t,a;A_0,A_1)\le \max \{c_{A_0},c_{A_1}\} K(t,a;A^{\sim }_0,A^{\sim }_1). \end{aligned}$$

Therefore,

$$\begin{aligned} (A_0,A_1)_{0,q;b}=(A^{\sim }_0,A^{\sim }_1)_{0,q;b}\quad \text {and}\quad (B_0,B_1)_{0,q;b}=(B^{\sim }_0,B^{\sim }_1)_{0,q;b}\,. \end{aligned}$$

Besides, $T\in \mathcal {L} (\overline{A^{\sim }},\overline{B^{\sim }})$ and, according to [5, Lemma 3.1],, we have

$$\begin{aligned} \beta (T:A^{\sim }_j\rightarrow B^{\sim }_j)\le \max \{c_{B_0},c_{B_1}\}\beta (T:A_j\rightarrow B_j),\quad j=0,1. \end{aligned}$$

Consequently, without loss of generality we may assume in the following that the couples $\overline{A}=(A_0,A_1)$ and $\overline{B}=(B_0,B_1)$ are mutually closed. We may also assume that the spaces $A_0,A_1,B_0,B_1$ are p-Banach for some $0<p\le 1$. Therefore, we can use Theorem 2.1.

Step 2. In this step we will introduce vector-valued sequence spaces and projections which will allow to split the operator T.

Let $\Lambda =(\ell _p,\ell _p(2^{-m}))_{0,q;b}.$ By Theorem 2.1, we know that $(A_0,A_1)_{0,q;b}=(A_0,A_1)_{\Lambda ;J}.$ Consider the vector-valued sequence space $\Lambda (G_m)$ where $G_m=(A_0\cap A_1, J(2^m,\cdot )),\,m\in \mathbb {Z}.$ Let $\pi :\Lambda (G_m)\rightarrow (A_0,A_1)_{\Lambda ;J}$ be the linear operator defined by $\pi (u_{m})=\sum \nolimits _{m=-\infty }^{\infty }u_m$ (convergence in $A_0+A_1$). Then $\pi$ is surjective and induces the quasi-norm of $(A_0,A_1)_{\Lambda ;J}.$ Note also that $\pi \in \mathcal {L}(\ell _p(2^{-mj}G_m),A_j),\,j=0,1,$ and its norm is less than or equal to 1.

Put $\overline{\ell _p(G)}=(\ell _p(G_m),\ell _p(2^{-m}G_m))$. The following projections will be useful. For $n\in \mathbb {N}$ and $u=(u_m)\in \ell _p(G_m)+\ell _p(2^{-m}G_m)$ let

$$\begin{aligned}&P_nu=(\dots ,0,0,u_{-n},u_{-n+1},\dots ,u_{n-1},u_n,0,0,\dots ),\\&P^+_nu=(\dots ,0,0,u_{n+1},u_{n+2},u_{n+3},\dots ),\\&P^-_nu=(\dots ,u_{-n-3},u_{-n-2},u_{-n-1},0,0,\dots ). \end{aligned}$$

Then the identity operator I on $\ell _p(G_m)+\ell _p(2^{-m}G_m)$ can be decomposed as $I=P_n+P^+_n+P^-_n$, $n\in \mathbb {N}$. These projections are bounded from $\ell _p(2^{-mj}G_m)$ into $\ell _p(2^{-mj}G_m)$ with norm less than or equal to 1 for $j=0,1$, and the same happens on $\Lambda (G_m)$. Furthermore,

$$\begin{aligned} \Vert P^+_n\Vert _{\ell _p(G_m),\ell _p(2^{-m}G_m)}=2^{-(n+1)} =\Vert P^-_n\Vert _{\ell _p(2^{-m}G_m),\ell _p(G_m)}\,. \end{aligned}$$

(3.1)

Write $F_m=(B_0+B_1,K(2^m,\cdot )),\,m\in \mathbb {Z}.$ Then the linear operator $\iota b=(....,b,b,b,....)$ is a metric injection from $(B_0,B_1)_{0,q;b}$ into $\ell _q(b(2^m)F_m).$ Consider the couple $\overline{\ell _{\infty }(F)}=(\ell _{\infty }(F_m),\ell _{\infty }(2^{-m}F_m))$. Note that $\iota :B_j\rightarrow \ell _{\infty }(2^{-mj}F_m)$ is bounded with norm less than or equal to 1. On $\overline{\ell _{\infty }(F)}$ we can consider the corresponding sequences of projections that we denote by $(Q_n)$, $(Q^+_n)$, $(Q^-_n)$. They enjoy analogous properties as $(P_n)$, $(P^+_n)$ and $(P^-_n)$. In particular, we have

$$\begin{aligned} \Vert Q^+_n\Vert _{\ell _{\infty }(F_m),\ell _{\infty }(2^{-m}F_m)}=2^{-(n+1)}=\Vert Q^-_n\Vert _{\ell _{\infty }(2^{-m}F_m),\ell _{\infty }(F_m)}\,. \end{aligned}$$

(3.2)

The following diagram illustrates the situation

$$\begin{aligned}\frac{ \begin{array}{c} \ell _p(G_m){{{\mathop {\longrightarrow }\limits ^{\pi }}}}A_0{\mathop {\longrightarrow }\limits ^{T}}B_0{\mathop {\longrightarrow }\limits ^{\iota }}\ell _{\infty }(F_m) \\ \ell _p(2^{-m}G_m){\mathop {\longrightarrow }\limits ^{\pi }}A_1{\mathop {\longrightarrow }\limits ^{T}}B_1{\mathop {\longrightarrow }\limits ^{\iota }}\ell _{\infty }(2^{-m}F_m)\end{array} }{\Lambda (G_m){\mathop {\longrightarrow }\limits ^{\pi }}\overline{A}_{0,q;b}{\mathop {\longrightarrow }\limits ^{T}}{\overline{B}}_{0,q;b}{\mathop {\longrightarrow }\limits ^{\iota }}\ell _q(b(2^m)F_m)}. \end{aligned}$$

In this diagram, the first three spaces of the last line are obtained by interpolation of the couple above and the fourth space contains the corresponding interpolation space. That is to say, we have

$$\begin{aligned} (\ell _p(G_m),\ell _p(2^{-m}G_m))_{0,q;b}=\Lambda (G_m) \ \text {and} \end{aligned}$$

$$\begin{aligned} (\ell _{\infty }(F_m),\ell _{\infty }(2^{-m}(F_m))_{0,q;b}\hookrightarrow \ell _q(b(2^m)F_m). \end{aligned}$$

(3.3)

To establish the first formula we proceed as in the case when $b(t)=(1+|\log t|)^{\mathbb {A}}$ (see [5, Lemma 3.2]). Take any $u=(u_m)\in (\ell _p(G_m),\ell _p(2^{-m}G_m))_{0,q;b}$. For any $k\in \mathbb {Z}$ and $0<\varepsilon <1$, there are $u^j=(u_{j,m})\in \ell _p(2^{-mj}G_m)$ such that $u=u^0+u^1$ and

$$\begin{aligned} \Vert u^0\Vert _{\ell _p(G_m)}+2^k\Vert u^1\Vert _{\ell _p(2^{-m}G_m)} \le (1+\varepsilon )K(2^k,u;\ell _p(G_m),\ell _p(2^{-m}G_m)). \end{aligned}$$

Then

$$\begin{aligned} \left( \sum ^{\infty }_{m=-\infty } \min (1,2^{k-m})^p\Vert u_m\Vert _{G_m}^p\right) ^{1/p}&\le (\Vert u^0\Vert ^p_{\ell _p(G_m)}+2^{kp}\Vert u^1\Vert ^p_{\ell _p(2^{-m}G_m)})^{1/p}\\&\le 2^{1/p-1}(1 + \varepsilon )K(2^k,u;\ell _p(G_m),\ell _p(2^{-m}G_m)) \end{aligned}$$

and thus

$$\begin{aligned} \Vert u\Vert _{\Lambda (G_m)}&=\left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)K(2^k,(\Vert u_m\Vert _{G_m});\ell _p,\ell _p(2^{-m}))\right] ^q\right) ^{1/q}\\&\le \left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)\left( \sum ^{\infty }_{m=-\infty }( \min (1, 2^{k-m})\Vert u_m\Vert _{G_m})^p\right) ^{1/p}\right] ^q\right) ^{1/q}\\&\le 2^{1/p-1}(1+\varepsilon )\left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)K(2^k,u;\ell _p(G_m),\ell _p(2^{-m}G_m))\right] ^q\right) ^{1/q}\\&\le 2^{1/p}\Vert u\Vert _{(\ell _p(G_m),\ell _p(2^{-m}G_m))_{0,q;b}}. \end{aligned}$$

Reciprocally, if $u=(u_m)\in \Lambda (G_m)$, given any $k\in \mathbb {Z}$ we can decompose $u=u^0+u^1$ with

$$\begin{aligned}u_{0,m}={\left\{ \begin{array}{ll} u_m&{}\text {if} \ m\le k, \\ 0&{}\text {if} \ m>k, \end{array}\right. },\quad u_{1,m}={\left\{ \begin{array}{ll} 0&{}\text {if} \ m\le k, \\ u_m&{}\text {if} \ m>k. \end{array}\right. } \end{aligned}$$

Then $u^0 \in \ell _p(G_m), u^1 \in \ell _p(2^{-m}G_m))$ and we have

$$\begin{aligned}&K(2^k,u;\ell _p(G_m),\ell _p(2^{-m}G_m))\\&\le \left( \sum ^k_{m=-\infty }\Vert u_m\Vert ^p_{G_m}\right) ^{1/p}+2^k\left( \sum ^{\infty }_{m=k+1}(2^{-m}\Vert u_m\Vert _{G_m})^p\right) ^{1/p}\\&\le 2\left( \sum ^{\infty }_{m=-\infty }\min (1, 2^{k-m})^p\Vert u_m\Vert ^p_{G_m}\right) ^{1/p}. \end{aligned}$$

Consequently,

$$\begin{aligned}&\Vert u\Vert _{(\ell _p(G_m),\ell _p(2^{-m}G_m))_{0,q;b}}\\&=\left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)K(2^k,u;\ell _p(G_m),\ell _p(2^{-m}G_m))\right] ^q\right) ^{1/q}\\&\le 2\left( \sum ^{\infty }_{k=-\infty } \left[ b(2^k)\left( \sum ^{\infty }_{m=-\infty }\min (1, 2^{k-m})^p\Vert u_m\Vert ^p_{G_m}\right) ^{1/p}\right] ^q\right) ^{1/q}\\&=2\left( \sum ^{\infty }_{k=-\infty }\left[ b(2^k)K_p\left( 2^k,(\Vert u_m\Vert _{G_m});\ell _p,\ell _p(2^{-m}) \right) \right] ^q\right) ^{1/q}\\&\le 2^{1/p}\Vert u\Vert _{\Lambda (G_m)}. \end{aligned}$$

To establish the second embedding in (3.3), take any

$x=(x_m)\in (\ell _{\infty }(F_m),\ell _{\infty }(2^{-m}F_m))_{0,q;b}$. Give any decomposition $x=x^0+x^1$ with $x^0=(x_{0,m})\in \ell _{\infty }(F_m)$ and $x^1=(x_{1,m})\in \ell _\infty (2^{-m}F_m)$, and any $k\in \mathbb {Z}$, we obtain

$$\begin{aligned} \Vert x_k\Vert _{F_k}&\le c(\Vert x^0_k\Vert _{F_k}+\Vert x^1_k\Vert _{F_k})\\&\le c(\Vert x^0\Vert _{\ell _{\infty }(F_m)}+2^k\Vert x^1\Vert _{\ell _{\infty }(2^{-m}F_m)}). \end{aligned}$$

It follows that

$$\begin{aligned} \Vert x_k\Vert _{F_k}\le cK(2^k,x;\ell _{\infty }(F_m),\ell _{\infty }(2^{-m}F_m)),\quad k\in \mathbb {Z}. \end{aligned}$$

Therefore, $\Vert x\Vert _{\ell _q(b(2^m)F_m)}\le c\Vert x\Vert _{(\ell _{\infty }(F_m),\ell _{\infty }(2^{-m}F_m))_{0,q;b}}$ as we wanted.

Put $\hat{T}=\iota T\pi .$ Since

$$\begin{aligned} \beta (T:\bar{A}_{0,q;b}\rightarrow \bar{B}_{0,q;b})&\le c_1\beta (\iota T:\bar{A}_{0,q;b}\rightarrow \ell _q(b(2^m)F_m))\\&\le c_2\beta (\hat{T}:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m)), \end{aligned}$$

it suffices to estimate the measure of non-compactness of $\hat{T}$ acting between the vector-valued sequence spaces. With this aim, for $n\in \mathbb {N}$ we decompose $\hat{T}$ as

$$\begin{aligned} \hat{T}&=\hat{T}P_n+\hat{T}(P_n^++P_n^-)\\&=\hat{T}P_n+Q_n\hat{T}(P_n^++P_n^-)+Q_n^-\hat{T}P_n^++Q_n^+\hat{T}P_n^-+Q_n^-\hat{T}P_n^-+Q_n^+\hat{T}P_n^+ \end{aligned}$$

and we proceed to estimate the measure of non-compactness of each of these six operators acting from $\Lambda (G_m)$ into $\ell _q(b(2^m)F_m).$

Step 3. We start with $Q^-_n\hat{T}P^+_n$. We are going to show that $\beta (Q^-_n\hat{T}P^+_n:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m))$ tends to 0 as $n\rightarrow \infty$.

Using the factorization

$$\begin{aligned} \ell _p(G_m){{{\mathop {\longrightarrow }\limits ^{P^+_n}}\ell _p\left( 2^{-m}G_m\right) }}{\mathop {\longrightarrow }\limits ^{\hat{T}}}\ell _{\infty }(2^{-m}F_m){\mathop {\longrightarrow }\limits ^{Q^-_n}}\ell _{\infty }(F_m) \end{aligned}$$

and (3.1) and (3.2), we get

$$\begin{aligned} \Vert Q^-_n\hat{T}P^- _n:\ell _p(G_m)\rightarrow \ell _{\infty }(F_m)\Vert \le 2^{-(n+1)}\Vert T:A_1\rightarrow B_1\Vert 2^{-(n+1)}\rightarrow 0 \ \text {as }n\rightarrow \infty . \end{aligned}$$

In addition, the factorization

$$\begin{aligned} \ell _p(2^{-m}G_m){{{\mathop {\longrightarrow }\limits ^{P^+_n}}\ell _p(2^{-m}G_m){{{\mathop {\longrightarrow }\limits ^{\hat{T}}}\ell _{\infty }\left( 2^{-m}F_m\right) {{{\mathop {\longrightarrow }\limits ^{Q^-_n}}\ell _{\infty }\left( 2^{-m}F_m\right) }}}}}} \end{aligned}$$

yields that

$$\begin{aligned} \Vert Q^-_n\hat{T}P^+_n:\ell _p(2^{-m}G_m)\rightarrow \ell _{\infty }(2^{-m}F_m)\Vert \le \Vert T:A_1\rightarrow B_1\Vert \ \text {for any }n\in \mathbb {N}. \end{aligned}$$

Therefore, by formulae (3.3) and (2.10), we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }&\beta (Q^-_n\hat{T} P^+_n:\Lambda (G_m) \longrightarrow \ell _q(b(2^m)F_m)) \\&\le c_1\lim _{n\rightarrow \infty } \Vert Q^-_n\hat{T}P^+_n\Vert _{ \overline{\ell _p(G)}_{0,q;b}, \overline{\ell _{\infty }(F)}_{0,q;b}} =0\,. \end{aligned}$$

Step 4. Consider $Q^+_n\hat{T}P^-_n$. Using the factorizations

and having in mind estimates (3.1), (3.2) and formulae (3.3) we get that $\Vert P^-_n:\Lambda (G_m)\rightarrow \ell _p(G_m)\Vert \le c_2$ and $\Vert Q^+_n:\ell _{\infty }(F_m)\rightarrow \ell _q(b(2^m)F_m)\Vert \le c_3$. Hence, with the help of the diagram

we derive

$$\begin{aligned} \beta (Q^+_n\hat{T}P_n^-:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m))&\le c_2c_3\beta (\hat{T}:\ell _p(G_m)\rightarrow \ell _{\infty }(F_m))\\&\le c_2c_3\beta (T:A_0\rightarrow B_0). \end{aligned}$$

Step 5. Now we proceed with $Q^-_n\hat{T}P^-_n$. Take any $\sigma _j>\beta (T:A_j\rightarrow B_j)$, $j=0,1$. First we are going to compare $\Vert Q_n^-\hat{T}P_n^-\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}$ with $\sigma _1.$ We have

$$\begin{aligned} \Vert Q_n^-\hat{T}P_n^-\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}\le \Vert \hat{T}P_n^-\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)} \end{aligned}$$

and

$$\begin{aligned} \Vert \hat{T}P_1^-\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}\ge \Vert \hat{T}P_2^-\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}\ge \ldots \ge 0. \end{aligned}$$

Therefore, the sequence $(\Vert \hat{T}P_n^-\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)})$ is convergent, say, to $\tau \ge 0$. Let $(v_n)\subseteq U_{\ell _p(2^{-m}G_m)}$ such that $\lim \nolimits _{n\rightarrow \infty }\Vert \hat{T}P_n^-v_n\Vert _{\ell _{\infty }(2^{-m}F_m)}=\tau$. To relate $\tau$ and $\sigma _1$, let $\{z_1,\ldots ,z_r\}\subseteq B_1$ such that

$$\begin{aligned} T\pi (U_{\ell _{p}(2^{-m}{G_m})})\subseteq \cup _{k=1}^r \{z_k + \sigma _1 U_{B_1}\}. \end{aligned}$$

We can find a subsequence $(v_{n'})$ of $(v_n)$ and some $1\le k\le r$ such that $\Vert T\pi P_{n'}^- v_{n'}-z_k\Vert _{B_1}\le \sigma _1$ for all $n'$. Then, for any $s\in \mathbb {Z},$ we have

$$\begin{aligned} K(2^s,z_k)&\le \Vert T\pi P_{n'}^-v_{n'}\Vert _{B_0}+2^s\Vert z_k-T\pi P_{n'}^-v_{n'}\Vert _{B_1}\\&\le \Vert P_{n'}^-v_{n'}\Vert _{\ell _p(G_m)}\Vert T\Vert _{A_0,B_0}+2^s\sigma _1\\&\le 2^{-n'}\Vert T\Vert _{A_0,B_0}+2^s\sigma _1\rightarrow 2^s\sigma _1\text { as }n'\rightarrow \infty . \end{aligned}$$

It follows that

$$\begin{aligned} \Vert \iota z_k\Vert _{\ell _{\infty }(2^{-m}F_m)}=\sup \limits _{s\in \mathbb {Z}}\{2^{-s}K(2^s,z_k)\}\le \sigma _1. \end{aligned}$$

Hence,

$$\begin{aligned} \tau&=\lim \limits _{n'\rightarrow \infty } \Vert \hat{T}P^-_{n'}v_{n'}\Vert _{\ell _{\infty }(2^{-m}F_m)}\\&\le \max \{c_{B_0},c_{B_1}\}\sup \limits _{n'} \{\Vert \hat{T}P^-_{n'}v_{n'}-\iota z_k\Vert _{\ell _{\infty (2^{-m}F_m)}}+\Vert \iota z_k\Vert _{\ell _{\infty }(2^{-m}F_m)}\}\\&\le \max \{c_{B_0},c_{B_1}\}\sup \limits _{n'} \{\Vert T\pi P^-_{n'}v_{n'}-z_k\Vert _{B_1}+\sigma _1\}\\&\le 2\max \{c_{B_0},c_{B_1}\} \sigma _1. \end{aligned}$$

Since the sequence $(\Vert \hat{T}P^-_n\Vert _{\ell _{p}{(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}})$ is decreasing, we conclude that there exists $N_1\in \mathbb {N}$ such that if $n\ge {N_1}$ then

$$\begin{aligned} \Vert Q_n^-\hat{T}P^-_n\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}&\le \Vert \hat{T}P^-_n\Vert _{\ell _p(2^{-m}G_m),\ell _{\infty }(2^{-m}F_m)}\\&\le 3\max \{c_{B_0},c_{B_1}\}\sigma _1. \end{aligned}$$

Next we compare $\Vert Q^-_n\hat{T}P^-_n\Vert _{\ell _p(G_m),\ell _{\infty }(F_m)}$ with $\sigma _0$. Since sequences having a finite number of coordinates different from 0 are dense in $\ell _p(G_m)$, we can find $\{d_1,\dots ,d_s\}\subseteq U_{\ell _p(G_m)}$ such that each $d_k$ has a finite number of coordinates different from 0 and with

$$\begin{aligned}\hat{T}(U_{\ell _p(G_m)})\subseteq \bigcup \limits ^s_{k=1}\{\hat{T}d_k+c_4\sigma _0 U_{\ell _{\infty }(F_m)}\} \end{aligned}$$

where $c_4=3\max \{c_{B_0},c_{B_1}\}^2$. We can also find $N_2\in \mathbb {N}$ such that if $n\ge N_2$ we have

$$\begin{aligned}\Vert Q^-_n\hat{T}d_k\Vert _{\ell _{\infty }(F_m)}\le 2^{-(n+1)}{\Vert \hat{T}d_k\Vert }_{\ell _{\infty }(2^{-m}F_m)}\le \sigma _0 \text { for any }1\le k\le s. \end{aligned}$$

Take any $n\ge N_2$ and any $u\in U_{\ell _p(G_m)}$. Then $P^-_nu\in U_{\ell _p(G_m)}$ and so there is $1\le k\le s$ such that $\Vert \hat{T}P^-_nu-\hat{T}d_k\Vert _{\ell _{\infty }(F_m)}\le c_4\sigma _0$. Therefore, $\Vert Q^-_n\hat{T}P^-_nu\Vert _{\ell _{\infty }(F_m)}\le \Vert Q^-_n\hat{T}P^-_nu-Q^-_n\hat{T}d_k\Vert _{\ell _{\infty }(F_m)}+\Vert Q^-_n\hat{T}d_k\Vert _{\ell _{\infty }(F_m)}\le 2c_4\sigma _0$.

Finally, using (3.3) and (2.9), we derive that there is $N\in \mathbb {N}$ such that if $n\ge N$ then

$$\begin{aligned} \beta (Q^-_n\hat{T}P^-_n&:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m)) \le c_5\Vert Q^-_n\hat{T}P^-_n\Vert _{{\overline{\ell _{p}(G)}}_{0,q;b} , {\overline{\ell _{\infty }(F)}}_{0,q;b}}\\&\le {\left\{ \begin{array}{ll} c_6\sigma _0&{}\text {if} \ \sigma _1 \le \sigma _0, \\ c_6\sigma _0\overline{b}\left( \frac{\sigma _0}{\sigma _1}\right) &{}\text {if} \ \sigma _0 < \sigma _1. \end{array}\right. } \end{aligned}$$

With similar arguments one can show that there is a constant $c_7>0$ such that

$$\begin{aligned}\beta (Q^+_n\hat{T}P^+_n:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m))\le {\left\{ \begin{array}{ll} c_7\sigma _0&{}\text {if} \ \sigma _1 \le \sigma _0, \\ c_7\sigma _0\overline{b}\left( \frac{\sigma _0}{\sigma _1}\right) &{}\text {if} \ \sigma _0 < \sigma _1. \end{array}\right. } \end{aligned}$$

Step 6. Given any quasi-Banach sequence lattice $\Gamma$, we can define a quasi-norm $\Vert \cdot \Vert _{\widetilde{\Gamma }}$ in ${\mathbb {R}}^{2n+1}$ by $\Vert x\Vert _{\widetilde{\Gamma }}=\Vert \tilde{x} \Vert _{\Gamma }$, where $x=(x_k)_{-n\le k\le n}\in {\mathbb {R}}^{2n+1}$, $\tilde{x}=\sum \nolimits ^n_{k=-n}x_ke_k$, $e_k=(\delta ^k_m)_{m\in \mathbb {Z}}$ and $\delta ^k_m$ is the Kronecker delta. Compactness of the unit ball $U=U_{({\mathbb {R}}^{2n+1},\Vert \cdot \Vert _{\widetilde{\Gamma }})}$ in $({\mathbb {R}}^{2n+1},\Vert \cdot \Vert _{\widetilde{\Gamma }})$ will be useful to estimate the measure of non-compactness of the remaining operators.

Let ${\sigma }_j>\beta (T:A_j\rightarrow B_j)$, $j=0,1$. We can find finite sets $\Sigma _j=\{ h^j_1,\dots ,h^j_{L_j}\} \subseteq B_j$ such that

$$\begin{aligned} T(U_{A_j})\subseteq \bigcup ^{L_j}_{l=1}\{h^j_l+{\sigma }_j U_{B_j}\},\quad j=0,1. \end{aligned}$$

(3.4)

Let $N\in \mathbb {N}$ such that $2^{N-1}\le {{\sigma }_1}/{{\sigma }_0}<2^N$ if ${\sigma }_0\le {\sigma }_1$ and let $N=0$ if ${\sigma }_1<{\sigma }_0$.

As for $\hat{T}P_n$, consider the quasi-norm $\Vert \cdot \Vert _{\widetilde{\Lambda }}$ on ${\mathbb {R}}^{2n+1}$ and let $\eta =\left\| \sum \nolimits ^n_{k=-n}{\frac{e_k}{\left\| e_k\right\| _{\Lambda }}}\right\| ^{-1}_\Lambda$. By compactness of $U=U_{({\mathbb {R}}^{2n+1},{\Vert \cdot \Vert }_{\widetilde{\Lambda }})}$, we can find a finite set $\Upsilon =\{ {\lambda }^1,\dots ,{\lambda }^s\} \subseteq U$ such that

$$\begin{aligned} U \subseteq \bigcup ^s_{d=1}\{{\lambda }^d+\eta U \}. \end{aligned}$$

We associate to each ${\lambda }^d={\left( {\lambda }^d_k\right) }_{-n\le k\le n}$ the numbers

$$\begin{aligned} {\varphi }^j_k={\varphi }^j_{k,{\lambda }^d}=\left( \frac{\eta }{{\left\| e_k\right\| }_\Lambda }+\left| {\lambda }^d_k\right| \right) 2^{-kj},\quad j=0,1. \end{aligned}$$

Next, for $-n\le k\le n$, ${\lambda }^d\in \Upsilon$, $h^0_l\in \Sigma _0$ and $h^1_y\in \Sigma _1$ in (3.4), pick any $g_k$ in the intersection $({\varphi }^0_kh^0_l+{\varphi }^0_k{\sigma }_0 U_{B_0})\cap ({\varphi }^1_kh^1_y+{\varphi }^1_k{\sigma }_1 U_{B_1})$ provided it is non-empty and let $g_k=0$ otherwise. Consider the finite set $\Phi$ formed by all sums $\sum \nolimits ^n_{k=-n}{g_k}$. We look at ${\overline{B}}_{0,q;b}$ as a J-space. We have

$$\begin{aligned}\beta (\hat{T}P_n:\Lambda (G_m)\rightarrow \ell _q(b\left( 2^m\right) F_m))\le c_1\beta (T\pi P_n:\Lambda (G_m)\rightarrow {\overline{B}}_{\Lambda ;J}).\end{aligned}$$

We are going to estimate the last term with the help of $\Phi$.

For any $u=(u_m)\in U_{\Lambda (G_m)}$, we can find ${\lambda }^d\in \Upsilon$ such that

$$\begin{aligned} |J(2^k,u_k)-\lambda ^d_k | \Vert e_k\Vert _\Lambda \le \Vert (J(2^k,u_k)-{\lambda }^d_k)\Vert _{\widetilde{\Lambda }}\le \eta ,\quad -n\le k\le n. \end{aligned}$$

It follows that $|J(2^k,u_k)|\le \frac{\eta }{\Vert e_k\Vert _\Lambda }+|{\lambda }^d_k|=2^{kj}\varphi ^j_k$. This yields that $\Vert u_k\Vert _{A_j}\le \varphi ^j_k$, $-n\le k\le n$, $j=0,1$. By (3.4), there are $h^0_l\in \Sigma _0$ and $h^1_y\in \Sigma _1$ such that

$$\begin{aligned} \Vert Tu_k -{\varphi }^0_kh^0_l\Vert _{B_0}\le \varphi ^0_k{\sigma }_0 \end{aligned}$$

and

$$\begin{aligned} \Vert Tu_k-{\varphi }^1_kh^1_y\Vert _{B_1}\le {\varphi }^1_k{\sigma }_1. \end{aligned}$$

Hence, the intersection $({\varphi }^0_kh^0_l+{\varphi }^0_k{\sigma }_0 U_{B_0})\cap ({\varphi }^1_kh^1_y+{\varphi }^1_k{\sigma }_1 U_{B_1})$ is not empty and for the $g_k$ corresponding to that intersection we have

$$\begin{aligned} J(2^{k-N},Tu_k-g_k)&\le \max \{\Vert Tu_k-{\varphi }^0_kh^0_l\Vert ^p_{B_0} +\Vert {\varphi }^0_kh^0_l-g_k\Vert ^p_{B_0}, \\&\, \quad \quad \quad \, \, 2^{(k-N)p}(\Vert Tu_k-\varphi ^1_kh^1_y\Vert ^p_{B_1}+\Vert {\varphi }^1_kh^1_y-g_k\Vert ^p_{B_1})\}^{1/p}\\&\le 2^{1/p}\max \{\sigma _0,2^{-N}\sigma _1\}\varphi ^0_k. \end{aligned}$$

Then, $g=\sum \nolimits ^n_{k=-n}{g_k}$ belongs to $\Phi$ and

$$\begin{aligned}&\Vert T \pi P_nu-g\Vert _{\overline{B}_{\Lambda ;J}} =\left\| \sum ^n_{k=-n}{(Tu_k-g_k)}\right\| _{\overline{B}_{\Lambda ;J}}\\&\le \Vert \tau _N(\dots 0,0,J(2^{-n-N},Tu_{-n}-g_{-n}),\dots ,J(2^{n-N},Tu_n-g_n),0,0,\dots )\Vert _\Lambda \\&\le 2^{1/p}\Vert \tau _N\Vert _{\Lambda ,\Lambda }\max \{ {\sigma }_0,2^{-N}{\sigma }_1\} \Vert (\dots ,0,0,{\varphi }^0_{-n},\dots ,{\varphi }^0_n,0,0,\dots )\Vert _\Lambda \\&\le c_2\overline{b}(2^{-N})\max \{ {\sigma }_0,2^{-N}{\sigma }_1\} \end{aligned}$$

where we have used Lemma 2.2 and definition of ${\varphi }^0_k$ in the last inequality. Whence, according to the choice of N and (2.3), we obtain that

$$\begin{aligned} \beta (\hat{T}P_n:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m))&\le c_1\beta (T\pi P_n:\Lambda (G_m)\rightarrow {\bar{B}}_{\Lambda ;J})\\&\le c_3\overline{b}(2^{-N})\max \{ {\sigma }_0,2^{-N}{\sigma }_1\}\\&\le {\left\{ \begin{array}{ll} c_4\sigma _0&{}\text {if} \ \sigma _1 \le \sigma _0, \\ c_4\sigma _0\overline{b}\left( \frac{\sigma _0}{\sigma _1}\right) &{}\text {if} \ \sigma _0 < \sigma _1. \end{array}\right. } \end{aligned}$$

Next we consider $Q_n\hat{T}(P^+_n+P^-_n)$. This time we work with $\overline{A}_{0,q;b}$ and ${\overline{B}}_{0,q;b}$ realized as K-spaces. We put $\Delta =\ell _q(b(2^m))$. We have

$$\begin{aligned} \beta (Q_n\hat{T}(P^+_n+P^-_n):\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m)) \le c_5\beta (Q_n\iota T:\overline{A}_{0,q;b}\rightarrow \Delta (F_m)). \end{aligned}$$

Let now $\eta =\left\| \sum \nolimits ^n_{k=-n}{\frac{e_k}{{\left\| e_k\right\| }_{\Delta }}}\right\| ^{-1}_{\Delta }$ and consider on ${\mathbb {R}}^{2n+1}$ the quasi-norm $\Vert \cdot \Vert _{\widetilde{\Delta }}$. There is a finite set $\Psi =\{ \mu ^1,\dots ,\mu ^s\}\subseteq U =U_{({\mathbb {R}}^{2n+1},\Vert \cdot \Vert _{\widetilde{\Delta }})}$ such that

$$\begin{aligned} U \subseteq \bigcup ^s_{d=1}{\left\{ {\mu }^d+\eta U \right\} }. \end{aligned}$$

Starting from ${\mu }^d=({\mu }^d_k)_{-n\le k\le n}$ we define the numbers

$$\begin{aligned} {\psi }^j_k={\psi }^j_{k,{\mu }^d}=\overline{b}(2^{-N})\left( \frac{\eta }{{\Vert e_k\Vert }_{{\Delta }}}+|{\mu }^d_k|\right) 2^{-{(k+N)}j},\quad j=0,1, \end{aligned}$$

where N was defined in the following line to (3.4). Let $\Omega$ be the finite subset of $\Delta (F_m)$ formed by all vectors $z^{d,l,y}=(z^{d,l,y}_m)_{m\in \mathbb {Z}}$ where

$$\begin{aligned}z^{d,l,y}_m={\left\{ \begin{array}{ll} 0&{}\text {if} \ m\notin [-n,n] \\ \psi ^0_mh^0_l+{\psi }^1_mh^1_y&{}\text {if} \ -n\le m\le n, \end{array}\right. } \end{aligned}$$

where $h^0_\ell \in \Sigma _0$ and $h^1_y\in \Sigma _1$ are the vectors of (3.4). We refer to $z^{d,l,y}$ as the element of $\Omega$ associated to ${\mu }^d$, $h^0_l$ and $h^1_y$.

Given any $a\in U_{\overline{A}_{0,q;b}}$, using the shift operator $\tau _N$ and (2.4), we have

$$\begin{aligned} \Vert (K(2^{m+N},a))\Vert _{\Delta }\le \Vert \tau _N\Vert _{\Delta }\Vert a\Vert _{\overline{A}_{0,q;b}}\le \overline{b}(2^{-N}). \end{aligned}$$

Therefore, there is ${\mu }^d\in \Psi$ such that

$$\begin{aligned} \Vert (K(2^{m+N},a)-\overline{b}(2^{-N}){{\mu }^d_m)}_{-n\le m\le n}\Vert _{\widetilde{\Delta }}<\eta \overline{b}(2^{-N}). \end{aligned}$$

Hence

$$\begin{aligned} |K(2^{m+N},a)-\overline{b}(2^{-N}){\mu }^d_m|\Vert e_m\Vert _{\Delta }<\eta \overline{b}(2^{-N}),\quad -n\le m\le n, \end{aligned}$$

and so $K(2^{m+N},a)<{\psi }^0_m$ for $-n\le m\le n$. It follows that we can decompose $a=a_{0,m}+a_{1,m}$ with $a_{j,m}$ belonging to $A_j$ and such that $\Vert a_{0,m}\Vert _{A_0}+2^{m+N}\Vert a_{1,m}\Vert _{A_1}<{\psi }^0_m$. Therefore, there are $h^0_l\in \Sigma _0$ and $h^1_y\in \Sigma _1$ such that

$$\begin{aligned} \Vert Ta_{0,m}-{\psi }^0_mh^0_l\Vert _{B_0}\le {\psi }^0_m{\sigma }_0 \end{aligned}$$

and

$$\begin{aligned} \Vert Ta_{1,m}-{\psi }^1_mh^1_y\Vert _{B_1}\le {\psi }^1_m{\sigma }_1,\quad -n\le m\le n. \end{aligned}$$

If we take $z=z^{d,l,y}$ the element of $\Omega$ associated to ${\mu }^d$, $h^0_l$, and $h_y^1,$ then we have

$$\begin{aligned} \Vert Q_n\iota Ta-z\Vert _{\Delta (F_m)}&=\Vert (K(2^m,Ta-z_m^{d,l,y}))_{-n\le m\le n}\Vert _{\widetilde{\Delta }}\\&\le \big \Vert (\Vert Ta_{0,m}-{\psi }^0_mh^0_l\Vert _{B_0}+2^m\Vert Ta_{1,m}-{\psi }^1_mh^1_y\Vert _{B_1})_{-n\le m\le n} \big \Vert _{\widetilde{\Delta }}\\&\le \Vert ({\psi }^0_m{\sigma }_0+2^m{\psi }^1_m{\sigma }_1)_{-n\le m\le n}\Vert _{\widetilde{\Delta }}\\&\le c_6\overline{b}(2^{-N})({\sigma }_0+2^{-N}{\sigma }_1). \end{aligned}$$

Consequently,

$$\begin{aligned} \beta (Q_n\hat{T}(P^+_n+P^-_n)&:\Lambda (G_m)\rightarrow \ell _q(b(2^m)F_m)) \le c_5\beta (Q_n\iota T:\overline{A}_{0,q;b}\rightarrow \Delta (F_m))\\&\le c_7\overline{b}(2^{-N})({\sigma }_0+2^{-N}{\sigma }_1)\\&\le {\left\{ \begin{array}{ll} c_8\sigma _0&{}\text {if} \ \sigma _1 \le \sigma _0, \\ c_8\sigma _0\overline{b}\left( \frac{\sigma _0}{\sigma _1}\right) &{}\text {if} \ \sigma _0 < \sigma _1, \end{array}\right. }, \end{aligned}$$

where we have used the value of N and (2.3) in the last inequality.

Step 7. Collecting the estimates of the Steps 3 to 6, we conclude that there is a constant $C>0$ independent of T such that if we split the operator as in the Step 2 and we take a suitable n, then for ${\sigma }_j>\beta (T:A_j\rightarrow B_j)$, we have

$$\begin{aligned}\beta (\hat{T}:\Lambda (G_m)\rightarrow \ell _q(2^mF_m)) \le {\left\{ \begin{array}{ll} C{\sigma }_0&{}\text {if} \ \sigma _1 \le \sigma _0, \\ C\max \left\{ \sigma _0, {\sigma }_0\overline{b}\left( \frac{{\sigma }_0}{{\sigma }_1\ }\right) \right\} &{}\text {if} \ \sigma _0 < \sigma _1. \end{array}\right. } \end{aligned}$$

Then, if $\beta (T:A_0 \rightarrow B_0)=0$, letting $\sigma _0 \rightarrow 0$ and using (2.2) we obtain case (i) of the statement. If $0\le \beta (T:A_1 \rightarrow B_1) < \beta (T:A_0 \rightarrow B_0)$, letting $\sigma _0 \rightarrow \beta (T:A_0 \rightarrow B_0)$ we get the case (ii). Finally, if $0< \beta (T:A_0 \rightarrow B_0) \le \beta (T:A_1 \rightarrow B_1)$, taking ${\sigma }_j=(1+\varepsilon )\beta (T:A_j\rightarrow B_j)$ and letting $\varepsilon$ goes to 0 we derive the case (iii). This finishes the proof. $\square$

Remark 3.2

On the contrary to the case of the real method (see [17, 22]), if $T\in \mathcal {L}(\bar{A},\bar{B})$ and $T:A_1\rightarrow B_1$ is compact, then $T:\bar{A}_{0,q;b}\rightarrow \bar{B}_{0,q;b}$ might not be compact. A counterexample can be found in [13, Remark 2.4].

For limiting methods with $\theta =1$ we have the following direct consequence of (2.7) and Theorem 3.1.

Theorem 3.3

Let $\overline{A}=(A_0,A_1)$, $\overline{B}={(B}_0,B_1)$ be quasi-Banach couples and let $T\in \mathcal {L} (\overline{A},\overline{B})$. Let $0<q\le \infty$ and $v\in SV(0,\infty )$ satisfying

$$\begin{aligned} \left( \int \limits _0^1v(t)^q\frac{\textrm{d}t}{t}\right) ^{1/q}<\infty ,\,\textit{ and also that}\end{aligned}$$

$$\begin{aligned} \left( \int \limits _1^{\infty }v(t)^q\frac{\textrm{d}t}{t}\right) ^{1/q}=\infty \textit{ if }q<\infty \textit{ and } \lim \limits _{t\rightarrow \infty } {v}(t)=\infty \textit{ if } q=\infty . \end{aligned}$$

Then we have

• (i) $\beta (T:\overline{A}_{1,q;v}\rightarrow {\overline{B}}_{1,q;v})=0$ if $\beta (T: A_1\rightarrow B_1)=0$,

• (ii) $\beta (T:\overline{A}_{1,q;v}\rightarrow {\overline{B}}_{1,q;v})\le C\beta (T:A_1\rightarrow B_1)$ if $0 \le \beta (T:A_0\rightarrow B_0) < \beta (T:A_1\rightarrow B_1)$,

• (iii)
  $$\begin{aligned} \beta (T:\overline{A}_{1,q;v}\rightarrow {\overline{B}}_{1,q;v})&\le C \max \Big \{\beta (T:A_1\rightarrow B_1), \\&\quad \beta (T:A_1\rightarrow B_1)\overline{v}\left( \frac{\beta (T:A_1\rightarrow B_1)}{\beta (T:A_0\rightarrow B_0)}\right) \Big \} \end{aligned}$$

  if $0< \beta (T:A_1\rightarrow B_1) \le \beta (T:A_0\rightarrow B_0)$.

Here C is a constant independent of T.

4 Applications

Let $(R,\mu )$ be a non-atomic $\sigma$-finite measure space. For $0<p$, $q\le \infty$ and $b\in SV(0,\infty )$, the Lorentz–Karamata space $L_{p,q;b}(R)$ is formed by all (equivalent classes of) measurable functions f on R which have a finite quasi-norm

$$\begin{aligned} \Vert f\Vert _{L_{p,q;b}(R)}=\left( \int ^{\infty }_0{{[t^{{1}/{p}}b(t)f^*(t)]}^q\frac{\textrm{d}t}{t}}\right) ^{{1}/{q}} \end{aligned}$$

(the integral should be replaced by the supremum if $q=\infty$). Here $f^*$ stands for the non-increasing rearrangement of f defined by

$$\begin{aligned} f^*(t)=\inf \{s>0:\mu \{x\in R:|f(x)|>s\} \le t\}. \end{aligned}$$

We refer to [25] and [29] for properties of Lorentz–Karamata spaces. Note that if $b(t)=(1+|\log t|)^a$ we get the Lorentz–Zygmund spaces $L_{p,q}(\log L)_a$ (see [2, 3]). If $\mathbb {A}=(\alpha _0,\alpha _\infty )\in {\mathbb {R}}^2$ and

$$\begin{aligned} b(t)=\ell ^{\mathbb {A}}(t)={\left\{ \begin{array}{ll} (1+|\log t|)^{\alpha _0}&{}\text {for}\ 0<t\le 1, \\ (1+|\log t|)^{{\alpha _\infty }}&{}\text {for} \ 1<t<\infty , \end{array}\right. } \end{aligned}$$

then we obtain the generalized Lorentz–Zygmund spaces $L_{p,q}(\log L)_{\mathbb {A}}(R)$ (see [39]). If $b\equiv 1$ then we obtain the Lorentz spaces $L_{p,q}(R)$ (see [4, 8, 43]) and if, in addition, $p=q$ then we get the Lebesgue spaces $L_p(R)$.

In what follows, we work with couples of Lebesgue spaces

$(L_{p_0}(R),L_{p_1}(R)),(L_{q_0}(S),L_{q_1}(S))$ and operators

$T\in \mathcal {L}((L_{p_0}(R),L_{p_1}(R)),(L_{q_0}(S),L_{q_1}(S)))$. We put

$$\begin{aligned} \beta (T_j)=\beta (T:L_{p_j}(R)\rightarrow L_{q_j}(S)),\quad j=0,1. \end{aligned}$$

It is shown in [29, Corollary 5.3] that

$$\begin{aligned} L_{p,q;b}(R)=(L_1(R),L_{\infty }(R))_{\theta ,q;b} \end{aligned}$$

(4.1)

provided that $1<p<\infty$, $0<\theta <1$, ${1}/{p}=1-\theta$, $0<q\le \infty$ and $b\in SV(0,\infty )$.

As a consequence of Theorem 3.1 we can establish the following result for Lorentz–Karamata spaces.

Theorem 4.1

Let $(R,\mu )$ and $(S,\nu )$ be non-atomic $\sigma$-finite measure spaces. Let $1<p_0<p_1<\infty$, $1<q_0<q_1<\infty$, $0<q<\infty$ and let $b\in SV(0,\infty )$ satisfying (2.8) and (2.12). Put

$$\begin{aligned} b_0(t)=b(t^{{1}/{p_0}-{1}/{p_1}}){\left( \frac{1}{b{(t^{{1}/{p_0-{1}/{p_1}}})}^q}\int ^{\infty }_t{b{(s^{{1}/{p_0}-{1}/{p_1}})}^q\frac{\textrm{d}s}{s}}\right) }^{{1}/{\min \{p_0,q\}}} \end{aligned}$$

and

$$\begin{aligned} b_1(t)=b(t^{1/q_0-1/q_1})\left( \frac{1}{b{(t^{{1}/{q_0}-{1}/{q_1}})}^q}\int ^\infty _t{b{(s^{{1}/{q_0}-{1}/{q_1}})}^q\frac{\textrm{d}s}{s}}\right) ^{1/\max \{ q_0,q\}}. \end{aligned}$$

If $T\in \mathcal {L}((L_{p_0}(R),L_{p_1}(R)),(L_{q_0}(S),L_{q_1}(S)))$ then

$$\begin{aligned} T:L_{p_0,q;b_0}(R)\rightarrow L_{q_0,q;b_1}(S) \ \text {boundedly.} \end{aligned}$$

Moreover, for $\beta (T)=\beta (T:L_{p_0,q;b_0}(R)\rightarrow L_{q_0,q;b_1}(S))$ we have

• (a) $\beta (T)=0$ if $\beta (T_0)=0$,

• (b) $\beta (T)\le C\beta (T_0)$ if $0 \le \beta (T_1) < \beta (T_0)$,

• (c) $\beta (T)\le C \max \left\{ \beta (T_0), \beta (T_0)\overline{b}\left( \beta (T_0) /\beta (T_1)\right) \right\}$ if $0 < \beta (T_0) \le \beta (T_1)$.

Here $C>0$ is a constant independent of T.

Proof

Let $0<\theta _0<\theta _1<1$ such that ${1}/p_j=1-\theta _j$, $j=0,1$. We have $L_{p_j}(R)=(L_1(R),L_{\infty }(R))_{\theta _j,p_j}$. Hence, we can use the reiteration formula of [29, Theorem 3.2] to work with the space $(L_{p_0}(R),{L_{p_1}(R))}_{0,q;b}$. Then, according to [36, Theorem 4.10] and (4.1), we obtain

$$\begin{aligned} L_{p_0,q;b_0}(R)=(L_1{(R)},L_{\infty }{(R)})_{\theta _0,q;b_0}\hookrightarrow {(L_{p_0}(R),L_{p_1}(R))}_{0,q;b}. \end{aligned}$$

Similarly, but using now [36, Theorem 4.8] with $\eta _j=1{-1}/{q_j},\, j=0,1$, we get

$$\begin{aligned} (L_{q_0}(S),L_{q_1}(S))_{0,q;b}\hookrightarrow (L_1(S),L_{\infty }(S))_{{\eta }_0,q;b_1}=L_{q_0,q;b_1}(S). \end{aligned}$$

Therefore, the result follows interpolating with parameters 0, q, b the couples $(L_{p_0}(R),L_{p_1}(R))$, $(L_{q_0}(S),L_{q_1}(S))$, applying Theorem 3.1 and having in mind the embeddings pointed out above. $\square$

Subsequently, for $\tau \in \mathbb {R}$ and $\mathbb {A}={(\alpha }_0,\alpha _{\infty })\in {\mathbb {R}}^2$, we put $\mathbb {A}+\tau =(\alpha _0+\tau ,\alpha _{\infty }+\tau )$. Recall that ${\alpha }^+=\max \{\alpha ,0\}$ for $\alpha \in {\mathbb {R}}$.

Remark 4.2

Let $\mathbb {A}=(\alpha _0,\alpha _\infty )\in \mathbb {R}^2$ such that $\alpha _{\infty }+1/q<0<\alpha _0+1/q$ and let $b(t)=(1+|\log t|)^{\mathbb {A}}$. Then for the function $b_0$ in Theorem 4.1 we obtain

$$\begin{aligned} b_0(t)&\approx b(t)\left( \frac{1}{b{(t)}^q}\int ^{\infty }_t{b{(s)}^q\frac{\textrm{d}s}{s}}\right) ^{{1}/\min \{p_0,q\}}\\&\approx (1+|\log t|)^{\mathbb {A}}(1+|\log t|)^{1/\min \{p_0,q\}}\\&=(1+|\log t|)^{\mathbb {A}+\frac{1}{\min \{p_0,q\}}}. \end{aligned}$$

Similarly,

$$\begin{aligned} b_1(t)\approx (1+|\log t|)^{\mathbb {A}+\frac{1}{\max \{q_0,q\}}}. \end{aligned}$$

Hence, we have that

$$\begin{aligned} L_{p_0,q;b_0}(R)=L_{p_0,q}{(\log L)}_{\mathbb {A}+\frac{1}{\min \{p_0,q\}}}(R) \end{aligned}$$

and

$$\begin{aligned} L_{q_0,q;b_1}(S)=L_{q_0,q}{(\log L)}_{\mathbb {A}+\frac{1}{\max \{q_0,q\}}}(S). \end{aligned}$$

Moreover, by [14, Lemma 2.1] and [5, (2.6)], we have

$$\begin{aligned} \overline{b}(t)\le (1+|\log t|)^{({\alpha }^+_0-\alpha _{\infty },(-\alpha _0)^+)}. \end{aligned}$$

Consequently, writing down Theorem 4.1 for this choice of b we recover a result of Besoy and Cobos (see [5, Corollary 3.13]), which is a quantitative version of a compactness result of Edmunds and Opic (see [26, Corollary 4] and also [20, Corollary 4.5]).

The following result refers to the case $1<p_1<p_0<\infty$.

Theorem 4.3

Let $(R,\mu )$ and $(S,\nu )$ be non-atomic $\sigma$-finite measure spaces. Let $1<p_1<p_0<\infty$, $1<q_0<q_1<\infty$, $0<q<\infty$ and let $b\in SV(0,\infty )$ satisfying (2.8) and (2.12). Put

$$\begin{aligned} \widetilde{b}_0(t)=b(t^{{1}/{p_0}-{1}/{p_1}})\left( \frac{1}{b(t^{{1}/{p_0}-{1}/{p_1}})^q}\int ^t_0b(s^{{1}/{p_0}-{1}/{p_1}})^q\frac{\textrm{d}s}{s}\right) ^{1/\min \{p_0,q\}} \end{aligned}$$

and

$$\begin{aligned} b_1{(t)}=b(t^{{1}/{q_0}-{1}/{q_1}})\left( \frac{1}{b(t^{{1}/{q_0}-{1}/{q_1}})^q}\int ^{\infty }_tb(s^{{1}/{q_0}-{1}/{q_1}})^q\frac{\textrm{d}s}{s}\right) ^{1/\max \{ q_0,q\}}. \end{aligned}$$

If $T\in \mathcal {L}((L_{p_0}(R),L_{p_1}(R)),(L_{q_0}(S),L_{q_1}(S)))$ then

$$\begin{aligned} T:L_{p_0,q;{\widetilde{b}}_0}(R)\rightarrow L_{q_0,q;b_1}(S) \ \text {boundedly.} \end{aligned}$$

Moreover, for $\beta (T)=\beta (T:L_{p_0,q;{\widetilde{b}}_0}(R)\rightarrow L_{q_0,q;b_1}(S))$ we have

• (a) $\beta (T)=0$ if $\beta (T_0)=0$,

• (b) $\beta (T)\le C \beta (T_0)$ if $0 \le \beta (T_1) < \beta (T_0)$,

• (c) $\beta (T)\le C \max \left\{ \beta (T_0), \beta (T_0)\overline{b}\left( \beta (T_0) /\beta (T_1)\right) \right\}$ if $0 < \beta (T_0) \le \beta (T_1)$.

Here $C>0$ is a constant independent of T.

Proof

Consider the couple $(L_{\infty }(R),L_1(R))$. We have

$$\begin{aligned} L_{p_j}(R)=(L_{\infty }(R),L_1(R))_{\widetilde{\theta }_j,p_j} \ \ \text {where} \ {\widetilde{\theta }}_j=\frac{1}{p_j}, \ j=0,1. \end{aligned}$$

So $0<{\widetilde{\theta }}_0<{\widetilde{\theta }}_1<1$ and we still can use [29, Theorem 3.2] and [36, Theorem 4.10] to get that

$$\begin{aligned} (L_{\infty }(R),L_1(R))_{{\widetilde{\theta }}_0,q;u}\hookrightarrow (L_{p_0}(R),L_{p_1}(R))_{0,q;b} \end{aligned}$$

where

$$\begin{aligned} u(t)=b(t^{{1}/{p_1}-{1}/{p_0}})\left( \frac{1}{{b(t^{1/{p_1}-{1}/{p_0}})}^q}\int ^{\infty }_tb(s^{{1}/{p_1}-{1}/{p_0}})^q\frac{\textrm{d}s}{s}\right) ^{{1}/\min \{ p_0,q\}}. \end{aligned}$$

According to the relationship between the K-functionals of $(L_{\infty }{(R)},L_{1}{(R)})$ and $(L_{1}(R),L_{\infty }(R))$, making a change of variables and using (4.1), we obtain

$$\begin{aligned} \Vert f\Vert _{{(L_{\infty }(R),L_1(R))}_{\widetilde{\theta }_0,q;u}}&=\left( \int ^{\infty }_0[t^{-{\widetilde{\theta }}_0}u(t)K(t,f;L_{\infty }(R),L_1{(R)})]^q\frac{\textrm{d}t}{t} \right) ^{{1}/{q}}\\&=\left( \int ^{\infty }_0[t^{1-{\widetilde{\theta }}_0}{{\widetilde{b}}_0(t^{-1})K(t^{-1},f;L_1(R),L_{\infty }(R))]}^q\frac{\textrm{d}t}{t}\right) ^{{1}/{q}}\\&=\left( \int ^{\infty }_0[t^{{\widetilde{\theta }}_0-1}{{\widetilde{b}}_0(t)K(t,f;L_1(R),L_{\infty }(R))]}^q\frac{\textrm{d}t}{t}\right) ^{{1}/{q}}\\&=\Vert f\Vert _{{(L_1(R),L_{\infty }(R))}_{1-{\widetilde{\theta }}_0,q;{\widetilde{b}}_0}}\approx \Vert f\Vert _{L_{p_0,q;{\widetilde{b}}_0}(R)}. \end{aligned}$$

Therefore

$$\begin{aligned} L_{p_0,q;{\widetilde{b}}_0}(R)\hookrightarrow (L_{p_0}(R),L_{p_1}(R))_{0,q;b}. \end{aligned}$$

Since the embedding

$$\begin{aligned} (L_{q_0}(S),L_{q_1}{(S)})_{0,q;b}\hookrightarrow L_{q_0,q;b_1}(S) \end{aligned}$$

has been established in Theorem 4.1, we can conclude the result by interpolating with parameters 0, q, b and applying Theorem 3.1. $\square$

Remark 4.4

If $b(t)=(1+|\log t|)^{\mathbb {A}}$ with $\mathbb {A}=(\alpha _0,\alpha _{\infty })$ and $\alpha _{\infty }+{1}/{q}<0<\alpha _0+{1}/{q}$, then

$$\begin{aligned} \widetilde{b}_0(t)\approx (1+|\log t|)^{\widetilde{\mathbb {A}}+\frac{1}{\min \{p_0,q\}}} \end{aligned}$$

where $\widetilde{\mathbb {A}}= (\alpha _{\infty },\alpha _0)$ and Theorem 4.3 gives estimates for the measure of non-compactness of

$$\begin{aligned} T:L_{p_{0},q}(\log L)_{\widetilde{\mathbb {A}}+\frac{1}{\min \{p_0,q\}}}(R)\rightarrow L_{q_0,q}(\log L)_{\mathbb {A}+\frac{1}{\max \{q_0,q\}}}(R). \end{aligned}$$

Proceeding similarly, but using [36, Theorem 4.8], we can derive results for $1<q_1<q_0<\infty$.

We finish the paper with some results when the main information on T refers to the restriction from $L_{p_1}(R)$ into $L_{q_1}(S)$.

Theorem 4.5

Let $(R,\mu )$ and $(S,\nu )$ be non-atomic $\sigma$-finite measure spaces. Let $1<p_1<p_0<\infty$, $1<q_1<q_0<\infty$, $0<q<\infty$ and let $v\in SV(0,\infty )$ satisfying that

$$\begin{aligned} \left( \int ^1_0{v{(t)}^q\frac{\textrm{d}t}{t}}\right) ^{{1}/{q}}<\infty \text { and } \ \left( \int ^{\infty }_1{v{(t)}^q\frac{\textrm{d}t}{t}}\right) ^{{1}/{q}}=\infty . \end{aligned}$$

Put

$$\begin{aligned} v_0(t)=v(t^{{1}/{p_0}-{1}/{p_1}})\left( \frac{1}{v(t^{{1}/{p_0}-{1}/{p_1}})^q}\int ^{\infty }_tv(s^{{{1}/{p_0}-{1}/{p_1}}})^q\frac{\textrm{d}s}{s}\right) ^{1/\min \{ p_1,q\}} \end{aligned}$$

and

$$\begin{aligned} v_1(t)=v(t^{{1}/{q_0}-{1}/{q_1}})\left( \frac{1}{v(t^{{1}/{q_0}-{1}/{q_1}})^q}\int ^{\infty }_tv(s^{{{1}/{q_0}-{1}/{q_1}}})^q\frac{\textrm{d}s}{s}\right) ^{1/\max \{ q_1,q\}}. \end{aligned}$$

If $T\in \mathcal {L}((L_{p_0}(R),L_{p_1}(R)),(L_{q_0}(S),L_{q_1}(S)))$ then

$$\begin{aligned} T:{L_{{p_1},q;v_0}}(R)\rightarrow L_{q_1,q;v_1}(S) \ \text {boundedly.} \end{aligned}$$

Furthermore, for $\beta (T)=\beta (T:L_{p_1,q;v_0}(R)\rightarrow L_{q_1,q;v_1}(S))$ we have

• (a) $\beta (T)=0$ if $\beta (T_1)=0$,

• (b) $\beta (T)\le C\beta (T_1)$ if $0 \le \beta (T_0) < \beta (T_1)$,

• (c) $\beta (T)\le C \max \left\{ \beta (T_1), \beta (T_1)\overline{v}\left( \beta (T_1) /\beta (T_0) \right) \right\}$ if $0 < \beta (T_1) \le \beta (T_0)$.

Here $C>0$ is a constant independent of T.

Proof

According to (2.7), for any quasi-Banach couple $(A_0,A_1)$ we have $(A_0,A_1)_{1,q;v}=(A_1,A_0)_{0,q;b}$ where $b(t)=v({1}/{t})$. We also have that

$$\begin{aligned} T\in \mathcal {L}((L_{p_1}(R),L_{p_0}(R)),(L_{q_1}(S),L_{q_0}(S))). \end{aligned}$$

Hence, the wanted result follows by interpolating with parameters 0, q, b and applying Theorem 4.1. $\square$

If $1<p_0<p_1<\infty$ and/or $1<q_0<q_1<\infty$ we can obtain similar results.

Remark 4.6

Let $\mathbb {A}=(\alpha _0,\alpha _{\infty })$ with $\alpha _0+{1}/{q}<0<\alpha _{\infty }+{1}/{q}$ and let $v(t)=(1+|\log t|)^{\mathbb {A}}$. So, v satisfies the assumptions of Theorem 4.5. We have

$$\begin{aligned} v_0(t)&\approx (1+|\log t|)^{\widetilde{\mathbb {A}}+\frac{1}{\min \{p_1,q\}}},\\ v_1(t)&\approx (1+|\log t|)^{\widetilde{\mathbb {A}}+\frac{1}{\max \{q_1,q\}}}. \end{aligned}$$

Moreover, by [5, (2.6)] we know that $\overline{v}(t)\le (1+|\log t|)^{((-\alpha _{\infty })^+,{\alpha _{\infty } ^+ }-\alpha _0)}$. Writing down Theorem 4.5 for this choice of the parameters we obtain estimates for the measure of non-compactness of

$$\begin{aligned} T:L_{p_1,q}(\log L)_{\tilde{\mathbb {A}}+\frac{1}{\min \{p_1,q\}}}(R)\rightarrow L_{q_1,q}{(\log L)}_{\widetilde{\mathbb {A}}+\frac{1}{\max \{q_1,q\}}}(R). \end{aligned}$$