Logarithmic interpolation methods and measure of

non-compactness

Blanca F. Besoy and Fernando Cobos ∗

Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad

Complutense de Madrid. Plaza de Ciencias 3, 28040 Madrid, Spain.

Abstract

We derive interpolation formulae for the measure of non-compactness of operators interpo-
lated by logarithmic methods with θ = 0, 1 between quasi-Banach spaces. Applications are
given to operators between Lorentz-Zygmund spaces.
Keywords: Logarithmic interpolation methods, measure of non-compactness, compact op-
erators, Lorentz-Zygmund spaces.
2010 MSC: Primary 46M35, 47H08; Secondary 46B70, 46E30.

1. Introduction

Interpolation theory is a consolidated branch in functional analysis with important applications
in harmonic analysis, partial differential equations, operator theory and other areas in mathe-
matics. See, for example, the books by Butzer and Berens [9], Bergh and Löfström [4], Triebel
[45, 46], Bennett and Sharpley [2], Conness [21] or Amrein, Boutet de Monvel and Georgescu
[1].

The most familiar interpolation methods are the real method (A0, A1)θ,q and the complex
method [A0, A1]θ, introduced by Lions and Peetre [35] and Calderón [10], respectively. If we
work with a couple of Lebesgue spaces (Lp0(R), Lp1(R)) with 1 ≤ p0 6= p1 ≤ ∞, 0 < θ < 1,
1/p = (1 − θ)/p0 + θ/p1 and 1 ≤ q ≤ ∞, it turns out that [Lp0(R), Lp1(R)]θ = Lp(R) (equal
norms) and (Lp0(R), Lp1(R))θ,q = Lp,q(R) (equivalent norms), where Lp,q(R) is a Lorentz space
(see [4, 45]).

Logarithmic perturbations of the real method (A0, A1)θ,q,A have been also studied in detail
in the literature. See, for example, the papers by Gustavsson [31], Doktorskii [24], Evans and
Opic [29], Evans, Opic and Pick [30], Cobos and Segurado [20], Cobos, Fernández-Cabrera and
Mart́ınez [15] and Besoy and Cobos [6]. The norm in (A0, A1)θ,q,A is

‖a‖(A0,A1)θ,q,A
=

( ∞∑
m=−∞

[
2−θm`A(2m)K(2m, a)

]q)1/q

.

Here K is the Peetre’s K-functional, A = (α0, α∞) ∈ R2, `(t) = 1 + | log t| and `A(t) = `α0(t)
if 0 < t ≤ 1, `A(t) = `α∞(t) if 1 < t < ∞. Now θ can take also the extreme values 0 and

∗Corresponding author.
Emails: cobos@mat.ucm.es (F. Cobos), blanca.f.besoy@ucm.es (B.F.Besoy).

1



1, producing spaces very close to A0 and A1, respectively. These extreme cases are the most
important for us here.

Applying logarithmic methods to a couple of Lebesgue spaces, we obtain Lorentz-Zygmund
spaces (see [3, 38, 25]). In particular, if we choose the parameters p0, p1, θ, p, q as above and
α ∈ R, then we have that (Lp0(R), Lp1(R))θ,q,(α,α) = Lp,q(logL)α(R) (see [39, Proposition
6.2/(c)]).

In 1960 Kranosel’skǐı [34] proved that if T is a linear operator which satisfies the assumptions
of the Riesz-Thorin convexity theorem, i.e. restrictions T : Lpj (R) → Lqj (S) are bounded for
1 ≤ p0, q0, p1, q1 ≤ ∞, and in addition q0 < ∞ and T : Lp0(R) → Lq0(S) is compact, then T :
Lp(R)→ Lq(S) is also compact provided that 1/p = (1−θ)/p0+θ/p1, 1/q = (1−θ)/q0+θ/q1 and
0 < θ < 1. This result led to the investigation of interpolation properties of compact operators
under abstract interpolation methods. The first results already appeared in the seminal papers
by Lions and Peetre [35] and Calderón [10]. See [11] for details and other references. As for
the real method, the final result was achieved in 1992 by Cwikel [23] and Cobos, Kühn and
Schonbek [18]. The problem is still open for the complex method.

In 2014 Edmunds and Opic [28] established a limiting variant of Kranosel’skǐı’s theorem
for finite measure spaces with the outcome that if T : Lp0(R) → Lq0(S) is compact and T :
Lp1(R) → Lq1(S) is bounded, then T is also compact when acting between Lorentz-Zygmund
spaces which are very close to Lp0(R) and Lq0(S). Abstract versions of the result by Edmunds
and Opic in terms of logarithmic methods for θ = 0, 1 have been obtained by Cobos, Fernández-
Cabrera and Mart́ınez [13] and Cobos and Segurado [20]. In the last paper it is shown that
the limit variant of Kranosel’skǐı’s theorem also holds when the function spaces are defined on
σ−finite measure spaces. More recently, Besoy [5] has shown that the theorem is still valid if
the function spaces which are target of T are quasi-Banach spaces, that is, it is allowed that
q0, q1, q take any positive values.

The aim of this paper is to complete the research of [28, 13, 20, 5] by estimating the measure
of non-compactness of the operator T acting between the Lorentz-Zygmund spaces in terms of
the measure of non-compactness of the restriction of T from Lp0(R) into Lq0(S).

The behaviour under interpolation of the measure of non-compactness is a problem that has
been studied by a number of authors. The case of the real method was considered by Edmunds
and Texeira [44] assuming some extra conditions in the target couple, and by Cobos, Fernández-
Mart́ınez and Mart́ınez [17] for arbitrary Banach couples. Results for extensions of the real
method have been obtained by Cordeiro [22], Szwedek [41] and Cobos, Fernández-Cabrera and
Mart́ınez [12]. Other quantitative results are due to Edmunds and Netrusov [26, 27] and Szwedek
[42, 43]. The case of bilinear operators has also been recently investigated by Masty lo and Silva
[36] and by the present authors [7].

As far as logarithmic methods for θ = 1, 0, if the operator acts between Banach couples, then
estimates for the measure of non-compactness of the interpolated operator have been derived by
Cobos, Fernández-Cabrera and Mart́ınez [14, 15] with the help of the connection between the
measure of non-compactness and another ideal measure (see [40]). We work here with quasi-
Banach couples where those connections are not available. Hence, our arguments are more
direct. Following ideas originated in the papers by Cobos and Peetre [19] and Cobos, Kühn
and Schonbek [18], we split the operator with the help of certain projections associated to the
vector-valued sequence spaces that arise with the construction of the logarithmic spaces. Then
we proceed to estimate the measure of non-compactness of each piece. Sometimes it suffices
with the interpolation theorem in combination with some other auxiliary results, but for two of
the pieces we have to construct suitable ε−nets. In those cases we rely on compactness in Rn
of certain subsets defined by using the parameters of the logarithmic method, an idea that has

2



his origin in [17] and our previous paper [7].

2. Logarithmic interpolation spaces

Let (A, ‖ · ‖A) be a quasi-Banach space. We put cA ≥ 1 for the constant in the quasi-triangle
inequality. According to [33, §15.10] or [32, Proposition 1.c.5], there is another quasi-norm ||| · |||
on A which is equivalent to ‖ · ‖A and such that ||| · |||p satisfies the triangle inequality. Here
0 < p ≤ 1 is the number given by the equality cA = 21/p−1. We say that (A, |||·|||) is a p−Banach
space. If 0 < r < p, it is clear that (A, ||| · |||) is also an r−Banach space. Note also that if A is
a p−Banach space then it satisfies the quasi-triangle inequality with constant 21/p−1.

If B is another quasi-Banach space, we write A = B if A and B coincide in the set-theoretical
sense and if they have equivalent quasi-norms.

By a (p-Banach) quasi-Banach couple A = (A0, A1) we mean two (p-Banach) quasi-Banach
spaces A0, A1 which are continuously embedded in the same Hausdorff topological vector space.
For t > 0 and a ∈ A0 +A1, let

K(t, a) = K(t, a;A0, A1) = inf{‖a0‖A0 + t‖a1‖A1 : a = a0 + a1, aj ∈ Aj},

and if a ∈ A0 ∩A1 put

J(t, a) = J(t, a;A0, A1) = max{‖a‖A0 , t‖a‖A1}.

Then K(1, ·) is the quasi-norm of A0 +A1 and J(1, ·) the quasi-norm of A0 ∩A1. If ‖ · ‖Aj is a
p-norm for j = 0, 1, then we put

Kp(t, a) = Kp(t, a;A0, A1) = inf{(‖a0‖pA0
+ tp‖a1‖pA1

)1/p : a = a0 + a1, aj ∈ Aj}.

The functionals K(t, ·) and Kp(t, ·) are equivalent quasi-norms in A0 +A1 with

K(t, a) ≤ Kp(t, a) ≤ 21/p−1K(t, a), a ∈ A0 +A1.

Moreover, Kp(t, ·) and J(t, ·) are also p-norms.
Put `(t) = 1 + | log t| and for A = (α0, α∞) ∈ R2 write

`A(t) = `(α0,α∞)(t) =

{
`α0(t) if 0 < t ≤ 1,

`α∞(t) if 1 < t <∞.

For 0 ≤ θ ≤ 1, 0 < q ≤ ∞ and A ∈ R2, the logarithmic interpolation space Aθ,q,A =
(A0, A1)θ,q,A is formed by all those a ∈ A0 +A1 which have a finite quasi-norm

‖a‖Aθ,q,A = ‖a‖(A0,A1)θ,q,A
=

( ∞∑
m=−∞

[
2−θm`A(2m)K(2m, a)

]q)1/q

(the sum should be replaced by the supremum when q =∞). See [29, 30, 20, 15, 6]. Note that
if A = (0, 0) and 0 < θ < 1, then we recover the classical real interpolation space (A0, A1)θ,q (see
[4, 45, 2, 8]). If 0 < θ < 1 and A 6= (0, 0), then (A0, A1)θ,q,A is a special case of the real method
with function parameter (see [31, 39]).

Observe that K(t, a;A0, A1) = tK(t−1, a;A1, A0). This yields that

(A0, A1)0,q,(α0,α∞) = (A1, A0)1,q,(α∞,α0) (equal quasi-norms). (2.1)

3



In what follows, we focus on the case θ = 1. We assume that{
α0 + 1/q < 0 if 0 < q <∞,
α0 ≤ 0 if q =∞.

(2.2)

Then (A0, A1)1,q,A 6= {0} and (A0, A1)1,q,A is an intermediate space with respect to A, that is

A0 ∩A1 ↪→ (A0, A1)1,q,A ↪→ A0 +A1

where ↪→ means continuous embedding (see [30, Theorem 2.2]).
Given a quasi-Banach space A intermediate with respect to A we write A◦ for the closure of

A0 ∩A1 in A.
It follows from the fundamental lemma (see [4, Lemma 3.3.2]) that a ∈ (A0 + A1)◦ if and

only if min(1, 1/t)K(t, a)→ 0 as t→ 0 and as t→∞. It is easy to check from this fact that if{
α∞ + 1/q ≥ 0 if 0 < q <∞,
α∞ > 0 if q =∞,

(2.3)

then
(A0, A1)1,q,A ⊆ (A0 +A1)◦. (2.4)

The Gagliardo completion A∼j of Aj is formed by all those a ∈ A0 + A1 having a finite
quasi-norm

‖a‖A∼j = sup{t−jK(t, a) : t > 0}

(see [2, 4]). It turns out that a ∈ A0 + A1 belongs to A∼j if and only if there is a sequence
(an) ⊆ Aj with supn ‖an‖Aj <∞ and limn→∞ ‖a− an‖A0+A1 = 0. Let

|||a|||j = inf{sup
n
‖an‖Aj : lim

n→∞
‖a− an‖A0+A1 = 0}.

It turns out that |||a|||j ≤ ‖a‖A∼j ≤ max{cA0 , cA1}|||a|||j (see [2, Theorem V.1.4]).

Minor modifications in the arguments of [2, Theorem V.1.5] show that

K(t, a;A∼0 , A
∼
1 ) ≤ K(t, a;A0, A1) ≤ max{cA0 , cA1}K(t, a;A∼0 , A

∼
1 ). (2.5)

The quasi-Banach couple A is called mutually closed if Aj = A∼j for j = 0, 1.
Let Γ be a quasi-Banach space of real valued sequences with Z as index set. We assume that

Γ contains all the sequences with only finitely many non-zero coordinates and that whenever
(ηm) ∈ Γ and |ξm| ≤ |ηm| for each m ∈ Z, then (ξm) ∈ Γ and ‖(ξm)‖Γ ≤ ‖(ηm)‖Γ. Moreover, we
assume that Γ ↪→ `p + `p(2

−m). Here 0 < p ≤ 1, `p is the space of p-summable sequences and
`p(2

−m) is the weighted space, with (ξm) ∈ `p(2−m) if and only if (2−mξm) ∈ `p.
Let 0 < q ≤ ∞, A = (α0, α∞) ∈ R2 satisfying (2.2) and 0 < p ≤ 1. Then Λ =

(`p, `p(2
−m))1,q,A is an example of quasi-Banach sequence space as introduced above.

If A = (A0, A1) is a p−Banach couple, the J−space AΓ;J = (A0, A1)Γ;J consists of all sums
a =

∑∞
m=−∞ um (convergence in A0 + A1), where (um) ⊆ A0 ∩ A1 and (J(2m, um)) ∈ Γ. The

quasi-norm on AΓ;J is

‖a‖AΓ;J
= ‖a‖(A0,A1)Γ;J

= inf{‖(J(2m, um))‖Γ : a =
∞∑

m=−∞
um}

(see [37]).
The following result is a consequence of (2.4) and [37, Theorem 3.19]. For completeness we

include a proof.

4



Theorem 2.1. Let A = (A0, A1) be a mutually closed p−Banach couple (0 < p ≤ 1). Let
0 < q ≤ ∞, A = (α0, α∞) ∈ R2 satisfying (2.2) and (2.3) and put Λ = (`p, `p(2

−m))1,q,A. Then
we have with equivalent quasi-norms

(A0, A1)1,q,A = (A0, A1)Λ;J .

Proof. Take any a ∈ (A0, A1)Λ;J and let a =
∑∞

m=−∞ um be a J-representation of a with
‖(J(2m, um))‖Λ ≤ 2‖a‖(A0,A1)Λ;J

. For any m ∈ Z, we have

Kp(2
k, a) ≤

( ∞∑
m=−∞

min(1, 2k−m)pJ(2m, um)p
)1/p

.

Hence

‖a‖(A0,A1)1,q,A
≤
( ∞∑
k=−∞

[
2−k`A(2k)

( ∞∑
m=−∞

min(1, 2k−m)pJ(2m, um)p
)1/p]q)1/q

≤ 21/p−1‖(J(2m, um))‖(`p,`p(2−m))1,q,A ≤ 21/p‖a‖(A0,A1)Λ;J
.

Conversely, let a ∈ (A0, A1)1,q,A. Using (2.4) and [37, Theorem 3.2], we can find (um) ⊆ A0∩A1

with a =
∑∞

m=−∞ um (in A0 +A1) and such that

( ∞∑
m=−∞

min(1, 2k−m)pJ(2m, um)p
)1/p

≤ cK(2k, a), k ∈ Z,

where c = cp is a constant independent of a and k. Therefore

‖a‖Λ;J ≤ ‖(J(2m, um))‖(`p,`p(2−m))1,q,A

≤
( ∞∑
k=−∞

[
2−k`A(2k)

( ∞∑
m=−∞

min(1, 2k−m)pJ(2m, um)p
)1/p]q)1/q

≤ c
( ∞∑
k=−∞

[2−k`A(2k)K(2k, a)]q
)1/q

= c‖a‖(A0,A1)1,q,A
.

We complete this section with some results on the shift operator τk defined by τkξ = (ξm+k)
for ξ = (ξm). Here k ∈ Z.

Subsequently, given λ ∈ R we put λ+ = max{0, λ}.
The following estimate is useful:
Let A = (α0, α∞) ∈ R2 and B = (α+

0 + (−α∞)+, α+
∞+ (−α0)+). Then for any u > 0 we have

sup
s>0

`A(su)

`A(s)
≤ `B(u). (2.6)

Inequality (2.6) was established in [15, Lemma 2.1] for u ≥ 1. If 0 < u < 1 it follows by applying
[15, Lemma 2.1] with 1/u > 1. We obtain

sup
s>0

`A(su)

`A(s)
= sup

s>0

`(α∞,α0)(1
s

1
u)

`(α∞,α0)(1
s )
≤ `α

+
0 +(−α∞)+

(
1

u
) = `α

+
0 +(−α∞)+

(u).

5



Lemma 2.2. Let 0 < q ≤ ∞ and A = (α0, α∞) ∈ R2 satisfying (2.2). Put B = ((−α∞)+, α+
∞−

α0). Then
‖τk‖`q(2−m`A(2m)),`q(2−m`A(2m)) ≤ 2k`B(2−k), k ∈ Z.

Proof. For ξ = (ξm) and k ∈ Z, we have

‖τkξ‖`q(2−m`A(2m)) =
( ∞∑
m=−∞

(2−m`A(2m)|ξm+k|)q
)1/q

≤ 2k sup
m∈Z

{
`A(2m)

`A(2m+k)

}
‖ξ‖`q(2−m`A(2m))

≤ 2k`B(2−k)‖ξ‖`q(2−m`A(2m))

where we have used (2.6) in the last inequality.

Lemma 2.3. Let 0 < q ≤ ∞, A = (α0, α∞) ∈ R2 satisfying (2.2), B = ((−α∞)+, α+
∞ − α0),

0 < p ≤ 1 and Λ = (`p, `p(2
−m))1,q,A. Then

‖τk‖Λ,Λ ≤ 21/p−12k`B(2−k), k ∈ Z.

Proof. Let ξ = (ξm) ∈ Λ. For any r ∈ Z, we have

K(2r, ξ; `p, `p(2
−m)) ≤ Kp(2

r, ξ; `p, `p(2
−m))

=
( ∞∑
m=−∞

(
min(1, 2r−m)|ξm|

)p)1/p
≤ 21/p−1K(2r, ξ; `p, `p(2

−m)).

Hence, given any k ∈ Z, we derive

‖τkξ‖Λ =
( ∞∑
r=−∞

[
2−r`A(2r)K(2r, ξ; `p, `p(2

−m))
]q)1/q

≤
( ∞∑
r=−∞

[
2−r`A(2r)

( ∞∑
m=−∞

min(1, 2r−m)p|ξm+k|p
)1/p]q)1/q

=
( ∞∑
r=−∞

[
2−r`A(2r)

( ∞∑
m=−∞

min(1, 2r+k−m)p|ξm|p
)1/p]q)1/q

≤ 21/p−12k sup
r∈Z

{
`A(2r)

`A(2r+k)

}( ∞∑
r=−∞

[
2−(r+k)`A(2r+k)K(2r+k, ξ; `p, `p(2

−m))
]q)1/q

≤ 21/p−12k`B(2−k)‖ξ‖Λ

where the last inequality follows from (2.6).

3. Interpolation of the measure of non-compactness

Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples. By T ∈ L(A,B) we mean that T is a
bounded linear operator from A0 +A1 into B0 +B1 such that the restrictions T : Aj → Bj are
bounded operators for j = 0, 1.

Given 0 < q ≤ ∞, A = (α0, α∞) ∈ R2 satisfying (2.2) and T ∈ L(A,B), it turns out that
the restriction

T : (A0, A1)1,q,A → (B0, B1)1,q,A

6



is bounded. Moreover, the argument given in [15, Theorem 2.2] for Banach couples can be
adapted to our quasi-Banach setting with the effect that if Mj > ‖T‖Aj ,Bj for j = 0, 1, then

‖T‖A1,q,A,B1,q,A
≤ 2M1

(
1 +

(
log

M0

M1

)+
)α+
∞−α0

. (3.1)

In particular, if ‖T‖A1,B1 = 0 then we derive that ‖T‖A1,q,A,B1,q,A
= 0. Note also that if in

addition to (2.2) we also have that (2.3) holds and ‖T‖A0,B0 = 0, then it follows from Theorem
2.1 that ‖T‖A1,q,A,B1,q,A

= 0.

Let A,B be quasi-Banach spaces and T ∈ L(A,B). The operator T is said to be compact if
it transforms each bounded set of A into a set whose closure is compact in B.

The (ball) measure of non-compactness β(T ) = β(T : A→ B) is defined to be the infimum
of the set of all the numbers σ > 0 for which there exists a finite subset {b1, ..., bs} ⊆ B such
that

T (UA) ⊆
s⋃

k=1

{bk + εUB}.

Here UA (respectively, UB) is the closed unit ball of A (respectively, B).
Clearly, β(T ) ≤ ‖T‖A,B. Moreover, T is compact if and only if β(T ) = 0. The next two

properties are easy to check and will be used freely in what follows:

(3.2) If E is another quasi-Banach space and S ∈ L(B,E), then for ST = S ◦ T we have

β(ST : A→ E) ≤ ‖S‖B,Eβ(T : A→ B).

Moreover, if ‖Sb‖E = ‖b‖B for all b ∈ B, then β(T : A→ B) ≤ 2cEβ(ST : A→ E).

(3.3) If X is another quasi-Banach space and R ∈ L(X,A), then

β(TR : X → B) ≤ ‖R‖X,Aβ(T : A→ B).

Moreover, if for any a ∈ A with ‖a‖A < 1, there is x ∈ X with ‖x‖X < 1 and Rx = a,
then β(T : A→ B) ≤ β(TR : X → B).

Lemma 3.1. Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples and let T ∈ L(A,B).
Then

β(T : A∼j → B∼j ) ≤ max{cB0 , cB1}β(T : Aj → Bj), j = 0, 1.

Proof. Suppose j = 0 and let β = β(T : A0 → B0). Given any ε > 0 there is a finite set
{b1, ..., bs} ⊆ B0 such that

T (UA0) ⊆
s⋃

k=1

{bk + (β + ε)UB0}.

Take any a ∈ UA∼0 . There exists (an) ⊆ A0 such that ‖a − an‖A0+A1 → 0 as n → ∞ and
‖an‖A0 ≤ 1 + ε. Since

(Tan) ⊆
s⋃

k=1

{(1 + ε)bk + (β + ε)(1 + ε)UB0},

we can find k0 ∈ {1, ..., s} and a subsequence (Tan′) of (Tan) such that

(Tan′) ⊆ {(1 + ε)bk0 + (β + ε)(1 + ε)UB0}.

7



It follows that
‖Tan′ − (1 + ε)bk0‖B0 ≤ (β + ε)(1 + ε), n′ ∈ N

and

‖Ta− (1 + ε)bk0 − (Tan′ − (1 + ε)bk0)‖B0+B1

≤ ‖T‖A0+A1,B0+B1‖a− an′‖A0+A1 → 0 as n′ →∞.

Hence,

‖Ta− (1 + ε)bk0‖B∼0 ≤ max{cB0 , cB1} sup
n′
{‖Tan′ − (1 + ε)bk0‖B0}

≤ max{cB0 , cB1}(β + ε)(1 + ε).

This yields that

T (UA∼0 ) ⊆
s⋃

k=1

{(1 + ε)bk + (β + ε)(1 + ε) max{cB0 , cB1}UB∼0 }

and therefore β(T : A∼0 → B∼0 ) ≤ max{cB0 , cB1}β.
The case j = 1 can be treated similarly.

Subsequently we are going to work with vector-valued sequence spaces. Let (Wm) be a
sequence of quasi-Banach spaces with the same constant c ≥ 1 in the quasi-triangle inequality
for any m ∈ Z, which is the case if Wm is p−Banach for any m ∈ Z. Let (λm) be a sequence of
positive numbers and 0 < q ≤ ∞. We write

`q(λmWm) = {w = (wm) : wm ∈Wm and ‖w‖`q(λmWm) = ‖(λm‖wm‖Wm)‖`q <∞}.

If Γ is a quasi-Banach sequence lattice, we define Γ(λmWm) in a similar way.

Lemma 3.2. Let 0 < p ≤ 1 and let (Wm) be a sequence of p−Banach spaces. Assume that
0 < q ≤ ∞ and let A = (α0, α∞) ∈ R2 satisfying (2.2). Let Λ = (`p, `p(2

−m))1,q,A. We have
with equivalence of quasi-norms

(`p(Wm), `p(2
−mWm))1,q,A = Λ(Wm).

Proof. Let w = (wm) ∈ (`p(Wm), `p(2
−mWm))1,q,A. For any k ∈ Z and ε > 0, we can find

wj = (wj,m) ∈ `p(2
−jmWm) such that w = w0 + w1 and ‖w0‖`p(Wm) + 2k‖w1‖`p(2−mWm) ≤

(1 + ε)K(2k, w; `p(Wm), `p(2
−mWm)). Hence,( ∞∑

m=−∞
min(1, 2k−m)p‖wm‖pWm

)1/p
≤
(
‖w0‖p`p(Wm) + 2kp‖w1‖p

`p(2−mWm)

)1/p

≤ 21/p−1(1 + ε)K(2k, w; `p(Wm), `p(2
−mWm)).

It follows that

‖w‖Λ(Wm) =
( ∞∑
k=−∞

[2−k`A(2k)K(2k, (‖wm‖Wm); `p, `p(2
−m))]q

)1/q

≤
( ∞∑
k=−∞

[
2−k`A(2k)

( ∞∑
m=−∞

min(1, 2k−m)p‖wm‖pWm

)1/p]q)1/q

≤ 21/p−1
( ∞∑
k=−∞

[2−k`A(2k)K(2k, w; `p(Wm), `p(2
−mWm))]q

)1/q

= 21/p−1‖w‖(`p(Wm),`p(2−mWm))1,q,A .

8



Conversely, if w = (wm) ∈ Λ(Wm), given k ∈ Z, put wj = (wj,m) with

w0,m =

{
wm if m ≤ k,
0 if m > k,

w1,m =

{
0 if m ≤ k,
wm if m > k.

Then w = w0 + w1 and

K(2k, w; `p(Wm), `p(2
−mWm)) ≤

( k∑
m=−∞

‖wm‖pWm

)1/p
+ 2k

( ∞∑
m=k+1

(2−m‖wm‖Wm)p
)1/p

≤ 2
( ∞∑
m=−∞

min(1, 2k−m)p‖wm‖pWm

)1/p
.

Therefore,

‖w‖(`p(Wm),`p(2−mWm))1,q,A =
( ∞∑
k=−∞

[2−k`A(2k)K(2k, w; `p(Wm), `p(2
−mWm))]q

)1/q

≤ 2
( ∞∑
k=−∞

[
2−k`A(2k)

( ∞∑
m=−∞

min(1, 2k−m)p‖wm‖pWm

)1/p]q)1/q

≤ 21/p‖w‖Λ(Wm).

We also recall the following embedding (see [5, Lemma 3.5]).

Lemma 3.3. Let (Wm) be a sequence of quasi-Banach spaces with the same constant c ≥ 1 in
the quasi-triangle inequality. Let 0 < q ≤ ∞ and A = (α0, α∞) ∈ R2 satisfying (2.2). Then

(`∞(Wm), `∞(2−mWm))1,q,A ↪→ `q(2
−m`A(2m)Wm).

Given 0 < q ≤ ∞, A ∈ R2 satisfying (2.2) and any quasi-Banach couple B = (B0, B1),
there are natural vector valued sequence spaces that arise with the construction of (B0, B1)1,q,A.
Indeed, let Fm = (B0 + B1,K(2m, ·)), m ∈ Z. The sequence (Fm) is formed by quasi-Banach
spaces with the same constant c = max{cB0 , cB1} in the quasi-triangle inequality. Consider
the spaces `q(2

−m`A(2m)Fm), `∞(Fm) and `∞(2−mFm). The operator ιb = (..., b, b, b, ...) is a
metric injection from (B0, B1)1,q,A into `q(2

−m`A(2m)Fm). Note also that ι : Bj → `∞(2−mjFm),
j = 0, 1, is bounded with quasi-norm less than or equal to 1.

We use this notation to state the next lemma which follows from a bilinear result of the
present authors [7, Lemma 3.2].

Lemma 3.4. Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples and let T ∈ L(A,B).
Fix j ∈ {0, 1} and put βj = β(T : Aj → Bj). Assume that there is a quasi-Banach space X and
bounded linear operators Rn ∈ L(X,Aj) such that ‖Rn‖X,Aj ≤ 1 and limn→∞ ‖TRn‖X,B0+B1 =
0. Then the following holds.

a) If βj = 0, then there is a subsequence (n′) such that limn′→∞ ‖ιTRn′‖X,`∞(2−mjFm) = 0.

b) If βj > 0, then there is a constant C > 0 independent of T and a subsequence (n′) such that
limn′→∞ ‖ιTRn′‖X,`∞(2−mjFm) ≤ Cβj.

9



We shall also need later the following consequence of [7, Lemma 3.1].

Lemma 3.5. Let A,B,Z be quasi-Banach spaces and let D be a dense subspace of A. As-
sume that T ∈ L(A,B) and let (Sn) ⊆ L(B,Z) such that sup{‖Sn‖B,Z : n ∈ N} < ∞ and
limn→∞ ‖SnTu‖Z = 0 for all u ∈ D. Let β = β(T : A→ B). Then the following holds.

a) If β = 0 then limn→∞ ‖SnT‖A,Z = 0.

b) If β > 0 then there is a constant C > 0 independent of T and there is L ∈ N such that
‖SnT‖A,Z ≤ Cβ for all n ≥ L.

To complete the preparation for the central result of the paper, note that if Γ is a p−Banach
sequence lattice then it induces a p-norm ‖·‖Γ̃ in R2n+1, n ∈ N. Indeed, if x = (xk)

n
k=−n ∈ R2n+1,

put ‖x‖Γ̃ = ‖x̃‖Γ where x̃ =
∑n

k=−n xkek and ek = (δkm) with δkm being the Kronecker delta.
The unit ball U = U(R2n+1,‖·‖Γ̃) is compact in (R2n+1, ‖ · ‖Γ̃) and therefore for any ε > 0 we can
find an ε−net for U .

Subsequently, in order to make the arguments more transparent, constants are denoted by
c1, c2,... In this way we emphasize the new constants that show up and a change in an existing
constant.

Theorem 3.6. Let A = (A0, A1), B = (B0, B1) be mutually closed quasi-Banach couples and
let T ∈ L(A,B). Let 0 < q ≤ ∞ and A = (α0, α∞) ∈ R2 satisfying that{

α0 + 1/q < 0 ≤ α∞ + 1/q if 0 < q <∞,
α0 ≤ 0 < α∞ if q =∞.

Then we have:

a) β(T : (A0, A1)1,q,A → (B0, B1)1,q,A) = 0 if β(T : A1 → B1) = 0,

b) β(T : (A0, A1)1,q,A → (B0, B1)1,q,A) ≤ Cβ(T : A1 → B1) if β(T : A0 → B0) = 0,

c) β(T : (A0, A1)1,q,A → (B0, B1)1,q,A) ≤ Cβ(T : A1 → B1)

(
1 +

(
log β(T :A0→B0)

β(T :A1→B1)

)+
)α+
∞−α0

if

β(T : Aj → Bj) > 0 for j = 0, 1.

Here C > 0 is a constant independent of A, B and T .

Proof. Without loss of generality we can suppose that the spaces A0, A1, B0, B1 are p−Banach
for some 0 < p ≤ 1. Furthermore, since A is mutually closed, by Theorem 2.1, we have that

(A0, A1)1,q,A = (A0, A1)Λ;J with Λ = (`p, `p(2
−m))1,q,A.

For each m ∈ Z let

Gm = (A0 ∩A1, J(2m, ·;A0, A1)), Fm = (B0 +B1,K(2m, ·;B0, B1)).

The operator π(um) =
∑∞

m=−∞ um (convergence in A0 + A1) is surjective from Λ(Gm) into
(A0, A1)Λ;J and it induces the quasi-norm of (A0, A1)Λ;J . On the other hand, as we pointed out

before Lemma 3.4, if ∆ = `q(2
−m`A(2m)) and ιb = (..., b, b, b, b, ...), then ι is a metric injection

from (B0, B1)1,q,A into ∆(Fm). Let T̂ = ιTπ : Λ(Gm) → ∆(Fm). According to (3.2) and (3.3),
we get

β(T : A1,q,A → B1,q,A) ≤ c1β(ιT : A1,q,A → ∆(Fm)) ≤ c2β(T̂ : Λ(Gm)→ ∆(Fm)). (3.4)

10



With the aim of estimating the last measure of non-compactness, consider the couples

`p(G) = (`p(Gm), `p(2
−mGm)) and `∞(F ) = (`∞(Fm), `∞(2−mFm))

and note that π ∈ L
(
`p(2

−mjGm), Aj
)
, j = 0, 1, with quasi-norm less than or equal to 1. On

the other hand, ι ∈ L
(
Bj , `∞(2−mjFm)

)
, j = 0, 1, also with quasi-norm less than or equal to 1.

The relevant picture to keep in mind is

`p(Gm)
π−−→ A0

T−−−→ B0
ι−−→ `∞(Fm)

`p(2
−mGm)

π−−→ A1
T−−−→ B1

ι−−→ `∞(2−mFm)

`p(G)1,q,A
π−−→ A1,q,A

T−−−→ B1,q,A
ι−−→ `∞(F )1,q,A.

Moreover, by Lemmata 3.2 and 3.3, we have

Λ(Gm) ↪→ `p(G)1,q,A and `∞(F )1,q,A ↪→ ∆(Fm).

On `p(G) we can consider the following projections. For n ∈ N and u = (um), let

Pnu = (..., 0, 0, u−n, u−n+1, ..., un−1, un, 0, 0, ...),

P+
n u = (..., 0, 0, un+1, un+2, un+3, ...),

P−n u = (..., u−n−3, u−n−2, u−n−1, 0, 0, ...).

These mappings have quasi-norm 1 when acting from `p(2
−jmGm) into `p(2

−jmGm), j = 0, 1,
or from Λ(Gm) into Λ(Gm). The identity operator I on `p(Gm) + `p(2

−mGm) can be split as
I = Pn + P+

n + P−n , n ∈ N. Moreover

‖P+
n ‖`p(Gm),`p(2−mGm) = 2−(n+1) = ‖P−n ‖`p(2−mGm),`p(Gm). (3.5)

On the couple `∞(F ) we can consider similar sequences of projections with analogous properties.
We denote them by (Qn), (Q+

n ), (Q−n ).
Since

T̂ = T̂Pn +QnT̂ (P+
n + P−n ) +Q+

n T̂P
−
n +Q−n T̂P

+
n +Q−n T̂P

−
n +Q+

n T̂P
+
n , (3.6)

we proceed to estimate the measure of non-compactness of each one of these six operators
acting from Λ(Gm) into ∆(Fm). Take any σj > β(T : Aj → Bj) and let N ∈ N ∪ {0} such that
2N ≤ σ0/σ1 < 2N+1 if σ1 ≤ σ0 and N = 0 if σ0 < σ1.

For T̂Pn we have

β(T̂Pn : Λ(Gm)→ ∆(Fm)) ≤ cβ(TπPn : Λ(Gm)→ (B0, B1)Λ;J).

Let η = ‖
∑n

k=−n
ek
‖ek‖Λ ‖

−1
Λ and consider the quasi-norm ‖ · ‖Λ̃ in R2n+1. We can find Υ =

{λ1, ..., λs} ⊆ U(R2n+1,‖·‖Λ̃) such that

U(R2n+1,‖·‖Λ̃) ⊆
s⋃

d=1

{λd + ηU(R2n+1,‖·‖Λ̃)}.

For each λd = (λdk)
n
k=−n, let

ϕjk = ϕj
k,λd

=

(
η

‖ek‖Λ
+ |λdk|

)
2−kj , j = 0, 1.

11



Besides, there are finite sets Σj = {hj1, ..., h
j
Lj
} ⊆ Bj such that

T (UAj ) ⊆
Lj⋃
`=1

{hj` + σjUBj}, j = 0, 1.

Given any λd ∈ Υ, h0
` ∈ Σ0 and h1

y ∈ Σ1, for any −n ≤ k ≤ n, let gk be an element of
(ϕ0

kh
0
` + ϕ0

kσ0UB0) ∩ (ϕ1
kh

1
y + ϕ1

kσ1UB1) provided the intersection is non-empty and let gk = 0
otherwise. Denote by Φ the collection of all sums

∑n
k=−n gk of the elements gk. Note that Φ is

a finite set. Now we show that Φ is a suitable net for TπPn.
Given any u = (um) ∈ UΛ(Gm), there is λd ∈ Υ such that

|J(2k, uk)− λdk|‖ek‖Λ ≤ ‖(J(2k, uk)− λdk)‖Λ̃ ≤ η, −n ≤ k ≤ n.

Hence, |J(2k, uk)| ≤ η
‖ek‖Λ + |λdk| = ϕ0

k and so ‖uk‖Aj ≤ ϕjk, j = 0, 1, −n ≤ k ≤ n. We can find

h0
` ∈ Σ0 and h1

y ∈ Σ1 such that

‖Tuk − ϕ0
kh

0
`‖B0 ≤ ϕ0

kσ0, ‖Tuk − ϕ1
kh

1
y‖B1 ≤ ϕ1

kσ1.

Thus,
(ϕ0

kh
0
` + ϕ0

kσ0UB0) ∩ (ϕ1
kh

1
y + ϕ1

kσ1UB1) 6= ∅

and for the corresponding gk we obtain that

J(2k+N , Tuk − gk) ≤ max{‖Tuk − ϕ0
kh

0
`‖
p
B0

+ ‖ϕ0
kh

0
` − gk‖

p
B0
,

2(k+N)p(‖Tuk − ϕ1
kh

1
y‖
p
B1

+ ‖ϕ1
kh

1
y − gk‖

p
B1

)}1/p

≤ 21/p−1 max{σ0, 2
Nσ1}ϕ0

k.

For g =
∑n

k=−n gk ∈ Φ, it follows that

‖TπPnu− g‖(B0,B1)Λ;J
= ‖

n∑
k=−n

(Tuk − gk)‖(B0,B1)Λ;J

≤ ‖τ−N (..., 0, 0, J(2−n+N , Tu−n − g−n), ..., J(2n+N , Tun − gn), 0, 0, ...)‖Λ
≤ 21/p−1‖τ−N‖Λ,Λ max{σ0, 2

Nσ1}‖(..., 0, 0, ϕ0
−n, ..., ϕ

0
n, 0, 0, ...)‖Λ

≤ c3‖τ−N‖Λ,Λ max{σ0, 2
Nσ1}.

By our choice of N and Lemma 2.3 we conclude that

β(T̂Pn : Λ(Gm)→ ∆(Fm)) ≤ 21/pc3σ1

(
1 +

(
log

σ0

σ1

)+)α+
∞−α0

.

As for QnT̂ (P+
n + P−n ) we get

β(QnT̂ (P+
n + P−n ) : Λ(Gm)→ ∆(Fm)) ≤ c4β(QnιT : A1,q,A → ∆(Fm)).

To estimate the last measure of non-compactness let σ0, σ1 and N be as before. Put η =
‖
∑n

k=−n
ek
‖ek‖∆ ‖

−1
∆ and choose Ψ = {µ1, ..., µs} ⊆ U(R2n+1,‖·‖∆̃) such that for any x ∈ U(R2n+1,‖·‖∆̃)

there is µd such that ‖x− µd‖∆̃ < η. Let associate to each µd = (µdk)
n
k=−n the numbers

ψjk = ψj
k,µd

= ‖τ−N‖∆,∆
(

η

‖ek‖∆
+ |µdk|

)
2−(k−N)j , j = 0, 1.

12



Choose a σj−net Σj = {hj1, ..., h
j
Lj
} for T (UAj ) in Bj , j = 0, 1, and given any µd ∈ Ψ, h0

` ∈ Σ0

and h1
y ∈ Σ1, let zd,`,y = (zd,`,ym )m∈Z the sequence of vectors defined by

zd,`,ym =

{
0 if m /∈ [−n, n],

ψ0
mh

0
` + ψ1

mh
1
y if − n ≤ m ≤ n.

Clearly, the collection of these zd,`,y forms a finite subset Ω of ∆(Fm). We are going to show
that Ω is a suitable net for QnιT .

Take any a ∈ UA1,q,A
. Since

‖(K(2m−N , a))‖∆ ≤ ‖τ−N‖∆,∆‖a‖A1,q,A
≤ ‖τ−N‖∆,∆,

there is µd ∈ Ψ such that

‖(K(2m−N , a)− ‖τ−N‖∆,∆µdm)nm=−n‖∆̃ < η‖τ−N‖∆,∆.

So,
|K(2m−N , a)− ‖τ−N‖∆,∆µdm|‖em‖∆ < η‖τ−N‖∆,∆, −n ≤ m ≤ n.

This yields that K(2m−N , a) < ψ0
m, −n ≤ m ≤ n. We can decompose a = a0,m + a1,m with

aj,m ∈ Aj , j = 0, 1, and
‖a0,m‖A0 + 2m−N‖a1,m‖A1 < ψ0

m.

It follows that for some h0
` ∈ Σ0, h1

y ∈ Σ1

‖Ta0,m − ψ0
mh

0
`‖B0 ≤ ψ0

mσ0 and ‖Ta1,m − ψ1
mh

1
y‖B1 ≤ ψ1

mσ1, −n ≤ m ≤ n.

Let z = zd,`,y be the element of Ω associated to µd, h0
` and h1

y. Then,

‖QnιTa− z‖∆(Fm) = ‖(K(2m, Ta− zm))nm=−n‖∆̃
≤ ‖(‖Ta0,m − ψ0

mh
0
`‖B0 + 2m‖Ta1,m − ψ1

mh
1
y‖B1)nm=−n‖∆̃

≤ ‖(ψ0
mσ0 + 2mψ1

mσ1)nm=−n‖∆̃ ≤ 2c∆‖τ−N‖∆,∆(σ0 + 2Nσ1).

By the choice of N and Lemma 2.2, we derive that

β(QnιT : A1,q,A → ∆(Fm)) ≤ 8c∆σ1

(
1 +

(
log

σ0

σ1

)+)α+
∞−α0

.

As for Q+
n T̂P

−
n we are going to estimate the measure of non-compactness by the quasi-norm

of the operator and then to show that it tends to 0 as n→∞. We have ‖Q+
n T̂P

−
n ‖`p(Gm),`∞(Fm) ≤

‖T‖A0,B0 . On the other hand, the factorization

`p(2
−mGm) `∞(2−mFm)

`p(Gm) `∞(Fm)

Q+
n T̂P

−
n

P−n

T̂

Q+
n

13



and (3.5) give that

‖Q+
n T̂P

−
n ‖`p(2−mGm),`∞(2−mGm) ≤ 2−(n+1)‖T̂‖`p(Gm),`∞(Fm)2

−(n+1) ≤ 2−2(n+1)‖T‖A0,B0 .

Hence, according to Lemmata 3.2, 3.3 and estimate (3.1), we get

β(Q+
n T̂P

−
n : Λ(Gm)→ ∆(Fm)) ≤ c‖Q+

n T̂P
−
n ‖`p(G)1,q,A,`∞(F )1,q,A

≤ c2−2(n+1)‖T‖A0,B0(1 + log 22(n+1))α
+
∞−α0 → 0 as n→∞.

As for Q−n T̂P
+
n , we observe that

‖P+
n ‖`p(Gm),`p(2−mGm) = 2−(n+1) and ‖P+

n ‖`p(2−mGm),`p(2−mGm) = 1,

hence ‖P+
n ‖Λ(Gm),`p(2−mGm) ≤ c5. Similarly, ‖Q−n ‖`∞(2−mFm),∆(Fm) ≤ c6. Using the diagram

Λ(Gm) ∆(Fm)

`p(2
−mGm) `∞(2−mFm)

Q−n T̂P
+
n

P+
n

T̂

Q−n

we derive that

β(Q−n T̂P
+
n : Λ(Gm)→ ∆(Fm)) ≤ c5c6β(T̂ : `p(2

−mGm)→ `∞(2−mFm))

≤ c5c6σ1 ≤ c5c6σ1

(
1 +

(
log

σ0

σ1

)+)α+
∞−α0

.

The remaining two operators can be estimated with the help of Lemmata 3.4 and 3.5. Using
the factorization

`p(2
−mGm) B0 +B1

`p(Gm) A0

TπP−n

P−n

π

T

we get
‖TπP−n ‖`p(2−mGm),B0+B1

≤ 2−(n+1)‖T‖A0,B0+B1 → 0 as n→∞.

Whence, Lemma 3.4 yields that there is c7 > 0 and a subsequence (n′) such that

lim
n′→∞

‖T̂P−n′‖`p(2−mGm),`∞(2−mFm) ≤ c7β(Tπ : `p(2
−mGm)→ B1) ≤ c7σ1.

On the other hand, let D be a subset of `p(Gm) formed by all sequences having only a finite
number of coordinates different from 0. Clearly, D is dense in `p(Gm) and if u ∈ D then

‖Q−n T̂ u‖`∞(Fm) ≤ 2−(n+1)‖T̂ u‖`∞(2−mFm) → 0 as n→∞.

According to Lemma 3.5, there is c8 > 0 and L ∈ N such that for all n ≥ L

‖Q−n T̂‖`p(Gm),`∞(Fm) ≤ c8β(T̂ : `p(Gm)→ `∞(Fm)) ≤ c8σ0.

14



Using now Lemmata 3.2, 3.3 and (3.1), if n′ > L we derive that

β(Q−n′ T̂P
−
n′ : Λ(Gm)→ ∆(Fm)) ≤ c9‖Q−n′ T̂P

−
n′‖`p(G)1,q,A,`∞(F )1,q,A

≤ cσ1

(
1 +

(
log

σ0

σ1

)+)α+
∞−α0

.

With a similar argument one can show that there is another subsequence that we also denote
by (n′) and another positive integer that we also call L such that if n′ > L we have that

β(Q+
n′ T̂P

+
n′ : Λ(Gm)→ ∆(Fm)) ≤ cσ1

(
1 +

(
log

σ0

σ1

)+)α+
∞−α0

.

Collecting all these estimates we derive that there is a constant C > 0 independent of A, B
and T such that if we split the operator as in (3.6) with a suitable n then

β(T̂ : Λ(Gm)→ ∆(Fm)) ≤ Cσ1

(
1 +

(
log

σ0

σ1

)+)α+
∞−α0

.

Now take σj = (1 + ε)β(T : Aj → Bj) if β(T : Aj → Bj) > 0 and σj = ε otherwise. Letting
ε→ 0 and using (3.4) the result follows.

Next we get rid of the assumption on the couples of being mutually closed.

Theorem 3.7. Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples and let T ∈ L(A,B).
If q and A are as in Theorem 3.6, then the conclusion of that theorem is still valid.

Proof. The quasi-Banach couples A∼ = (A∼0 , A
∼
1 ), B∼ = (B∼0 , B

∼
1 ) are mutually closed and,

by (2.5), A1,q,A = A∼1,q,A and B1,q,A = B∼1,q,A. Furthermore, T belongs to L(A∼, B∼) and,
according to Lemma 3.1,

β(T : A∼j → B∼j ) ≤ max{cB0 , cB1}β(T : Aj → Bj), j = 0, 1.

Consequently, Theorem 3.6 yields the result.

As a direct consequence of (2.1) and Theorem 3.7 we get the following result for logarithmic
methods with θ = 0.

Theorem 3.8. Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples and let T ∈ L(A,B).
Let 0 < q ≤ ∞ and A = (α0, α∞) ∈ R2 satisfying that{

α∞ + 1/q < 0 ≤ α0 + 1/q if 0 < q <∞,
α∞ ≤ 0 < α0 if q =∞.

Then we have:

a) β(T : (A0, A1)0,q,A → (B0, B1)0,q,A) = 0 if β(T : A0 → B0) = 0,

b) β(T : (A0, A1)0,q,A → (B0, B1)0,q,A) ≤ Cβ(T : A0 → B0) if β(T : A1 → B1) = 0,

c) β(T : (A0, A1)0,q,A → (B0, B1)0,q,A) ≤ Cβ(T : A0 → B0)
(

1 +
(

log β(T :A1→B1)
β(T :A0→B0)

)+)α+
0 −α∞

if

β(T : Aj → Bj) > 0 for j = 0, 1.

Here C > 0 is a constant independent of A, B and T .

15



Another consequence of Theorem 3.7 is the following compactness result. Its novelty is
that in the source and the target we have quasi-Banach couples. Note that we require weaker
assumptions on A than in Theorem 3.7.

Theorem 3.9. Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples. Assume that 0 < q ≤
∞ and A = (α0, α∞) ∈ R2 satisfy that{

α0 + 1/q < 0 if 0 < q <∞,
α0 ≤ 0 if q =∞.

If T ∈ L(A,B) and T : A1 → B1 is compact, then T : (A0, A1)1,q,A → (B0, B1)1,q,A is also
compact.

Proof. Step 1. If 0 ≤ α∞ + 1/q and 0 < q < ∞, or 0 < α∞ and q = ∞, then the result follows
from Theorem 3.7.

Step 2. Suppose now that α∞ + 1/q < 0 and 0 < q < ∞, or α∞ ≤ 0 and q = ∞. Take any
α > −1/q. The argument in [20, Corollary 2.5] for Banach couples still work in the quasi-Banach
context showing that

(A0, A1)1,q,A = (A0 +A1, A1)1,q,(α0,α) and (B0, B1)1,q,A = (B0 +B1, B1)1,q,(α0,α).

Since T ∈ L((A0 +A1, A1), (B0 +B1, B1)), the result established in the Step 1 yields that

T : (A0, A1)1,q,A → (B0, B1)1,q,A compactly.

The compactness theorem for θ = 0 reads as follows.

Theorem 3.10. Let A = (A0, A1), B = (B0, B1) be quasi-Banach couples. Assume that 0 <
q ≤ ∞ and A = (α0, α∞) ∈ R2 satisfy that{

α∞ + 1/q < 0 if 0 < q <∞,
α∞ ≤ 0 if q =∞.

If T ∈ L(A,B) and T : A0 → B0 is compact, then T : (A0, A1)0,q,A → (B0, B1)0,q,A compactly.

Remark 3.11. The compactness result of Cwikel [23] and Cobos, Kühn and Schonbek [18] shows
that if any restriction of the operator is compact, then the interpolated operator by the classical
real method is also compact. However, this is not the case for logarithmic methods with θ = 0
or 1. Theorem 3.10 does not hold if we move the compactness assumption from T : A0 → B0

to T : A1 → B1. A counterexample can be found in [13, Remark 2.4]. A similar observation
applies to Theorem 3.9 and a counterexample can be found in [16, Counterexample 7.11].

Next we apply the abstract results to derive quantitative versions of results of Edmunds and
Opic (see [28, Corollary 4 and Theorem 6]). We work with function spaces on σ−finite measure
spaces and the range for parameters q0, q1, q is broader than in [28].

Let (R,µ) be a σ-finite measure spaces. For 0 < p, q ≤ ∞ and A = (α0, α∞) ∈ R2,
the generalized Lorentz-Zygmund space Lp,q(logL)A(R) consists of all (classes of) measurable
functions f on R having a finite quasi-norm

‖f‖Lp,q(logL)A(R) =

(∫ µ(R)

0
[t1/p`A(t)f∗(t)]q

dt

t

)1/q

where f∗ is the non-increasing rearrangement of f . See [3, 38, 25].

16



Corollary 3.12. Let (R,µ) and (S, ν) be σ-finite measure spaces. Take 1 ≤ p0 < p1 < ∞,
0 < q0 < q1 <∞, 0 < q <∞ and A = (α0, α∞) ∈ R2 satisfying that α0 + 1/q < 0 < α∞ + 1/q.
Let T ∈ L((Lp0(R), Lp1(R)), (Lq0(S), Lq1(S))), put βj = β(T : Lpj (R)→ Lqj (S)), j = 0, 1, and

β = β(T : Lp1,q(logL)A+ 1
min(p1,q)

(R)→ Lq1,q(logL)A+ 1
max(q1,q)

(S)).

Then there is a constant C > 0 independent of T such that

a) β ≤ Cβ1

(
1 +

(
log β0

β1

)+)α+
∞−α0

if βj > 0 for j = 0, 1;

b) β ≤ Cβ1 if β0 = 0;

c) β = 0 if β1 = 0.

Proof. Reiteration results of Evans and Opic [29, Theorems 4.7, 5.9 and Corollary 8.4] yield
that

Lp1,q(logL)A+ 1
min(p1,q)

(R) ↪→ (Lp0(R), Lp1(R))1,q,A

and
(Lq0(S), Lq1(S))1,q,A ↪→ Lq1,q(logL)A+ 1

max(q1,q)
(S).

Whence, the result follows from Theorem 3.7.

Corollary 3.13. Let (R,µ), (S, ν) be σ-finite measure spaces. Take 1 < p0 < p1 ≤ ∞, 0 <
q0 < q1 <∞, 0 < q <∞ and A = (α0, α∞) ∈ R2 satisfying that α∞ + 1/q < 0 < α0 + 1/q. Let
T ∈ L((Lp0(R), Lp1(R)), (Lq0(S), Lq1(S))), put βj = β(T : Lpj (R)→ Lqj (S)), j = 0, 1, and

β = β(T : Lp0,q(logL)A+ 1
min(p0,q)

(R)→ Lq0,q(logL)A+ 1
max(q0,q)

(S)).

Then there exists a constant C > 0 independent of T such that

a) β ≤ Cβ0

(
1 +

(
log β1

β0

)+)α+
0 −α∞

if βj > 0, for j = 0, 1;

b) β ≤ Cβ0 if β1 = 0;

c) β = 0 if β0 = 0.

Proof. We can proceed as in the previous corollary but using now Theorem 3.8.

Funding.- This work was supported in part by Agencia Estatal de Investigación y Fondo Europeo
de Desarrollo Regional [grant MTM2017-84508-P]; and the Spanish Ministerio de Educación, Cultura y
Deporte [grant FPU16/02420 to B.F.B.].

Acknowledgments.- The authors would like to thank the referee for his/her useful remarks.

17



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