Interpolation of uniformly absolutely continuous operators

Math. Nachr., 1 – 21 (2012) / DOI 10.1002/mana.201100205

Interpolation of uniformly absolutely continuous operators

Fernando Cobos∗1, Amiran Gogatishvili∗∗2, Bohumır Opic∗∗∗3, and Lubos Pick†3

1 Departamento de Analisis Matematico, Facultad de Matematicas, Universidad Complutense de Madrid, Plazade Ciencias 3, 28040 Madrid, Spain

2 Mathematical Institute, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, CzechRepublic

3 Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University,Sokolovska 83, 186 75 Praha 8, Czech Republic

Received 10 August 2011, revised 31 May 2012, accepted 1 June 2012Published online 17 August 2012

Key words Uniformly absolutely continuous operators, interpolation, type of an interpolation method, com-pactness, quasiconcave function, weighted inequalities, K-functional, fundamental function of interpolationmethod, dilation operator

MSC (2010) Primary: 46B70, 46E35

This paper is dedicated to Professor Hans Triebel on the occasion of his 75th birthday

We develop a method suitable for interpolation of uniformly absolutely continuous operators. We then apply thismethod to establishing compactness of operators and embeddings especially in the limiting situations, wherethe classical interpolation methods fail. We study types of certain interpolation methods and their sharpness.

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1 Introduction and main results

One of the principal tasks of functional analysis and its applications is to establish a continuous embedding, and,even more important, a compact embedding, between a given pair of function spaces. In particular, this is sowhen the norms of the function spaces in question measure in some way the smoothness of functions, sometimesby means of the size of the (possibly higher-order) gradient. The primary role among such spaces is playedby Sobolev and Besov spaces. Compact embeddings of Sobolev and Besov spaces into other types of functionspaces (whose norms are more relaxed and depend only on the size of the functions involved) are indispensablein applications, namely in the theory of PDE’s, mathematical physics, and many other branches of mathematicsincluding probability, hypercontractive semigroups, etc.

To establish compactness of a given embedding is often a difficult problem, in many cases the key one. Forexample, the reason why the regularity of solutions to the Navier–Stokes equations is so difficult in dimension 3whereas it is very easy in dimension 2 is caused by the fact that an appropriate Sobolev embedding is compact in2D but not in 3D. The task is even more involved when limiting cases of embeddings are studied.

To be more precise, let us give an illustrative example. Let Ω ⊂ Rn be a bounded Lipschitz domain, 1 ≤ p < n,

and let W 1,p(Ω) be the Sobolev space (for detailed definitions see, for example, [1]). It is well known thatW 1,p(Ω) is continuously embedded into the two-parameter Lorentz space L

n pn −p ,p(Ω) (defined, for example,

in [2]). We complement this embedding by the trivial bounded inclusion Lp(Ω) ↪→ Lp(Ω), where (and throughout

∗ e-mail: [email protected], Phone: +34 913 944 453, Fax: +34 913 944 613∗∗ e-mail: [email protected], Phone: +420 222 090 786, Fax: +420 222 090 701∗∗∗ e-mail: [email protected], Phone: +420 221 913 367, Fax: +420 222 323 390† Corresponding author: e-mail: [email protected], Phone: +420 221 913 264, Fax: +420 222 323 390

c© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 F. Cobos et al.: Interpolation of uniformly absolutely continuous operators

the paper) the symbol ↪→ means a continuous embedding. The situation can be depicted on a diagram as

W 1,p(Ω) ↪→ Ln p

n −p ,p(Ω),

Lp(Ω) ↪→ Lp(Ω).

}(1.1)

If we interpolate the pair(W 1,p(Ω), Lp(Ω)

)on the left-hand side of embeddings in (1.1) by some known in-

terpolation method, then we usually obtain some kind of a Besov space. The corresponding range space on theright-hand side will often be either a Lorentz space or some of its generalizations, depending on the interpolationmethod used. If the interpolation method is limiting in some sense, then we obtain an embedding whose domainis a function space that lies “near” one of the “endpoints”, that is, either near W 1,p(Ω) or near Lp(Ω). This cor-responds, respectively, to situations when the smoothness parameter in the resulting Besov space is either closeto one or close to zero.

Suppose now that our goal is to obtain compactness of such an embedding. Unfortunately, all known inter-polation methods which result in a compact embedding require that at least one of the endpoint embeddings iscompact (see [7] and the references given there). But in our case, none of the two embeddings in (1.1) providesus with such a courtesy, hence the key requirement is not satisfied. Let us see what one can do to make thingsbetter. The first embedding in (1.1), W 1,p(Ω) ↪→ L

n pn −p ,p(Ω), can be easily turned to a compact one simply by

replacing the target space Ln p

n −p ,p(Ω) by an appropriately larger one, for example by Lq (Ω) with q < npn−p or by

a Zygmund class of type Ln p

n −p (log L)α (Ω) with α < 0. Yet finer amendments are available when other scales offunction spaces are used. If we do this and then apply a suitable limiting interpolation method, we obtain a certaincompact embedding of Besov-type spaces into Lorentz-type spaces. Of course, there will be an unavoidable lossof information caused by damaging the sharpness of the initial embedding but this can be held under control.

Assume now, however, that we are interested in compact embeddings near the other endpoint, namely Lp(Ω)(in other words, we study spaces of Besov-type with the smoothness close to zero). It turns out that when thisendpoint is at the spotlight, then the above-described method is completely useless. The simple reason is that nosymmetrical argument is available here since the embedding Lp(Ω) ↪→ Lp(Ω) can by no means be turned intoa compact one by any conceivable enlargement of the range space Lp(Ω) within the range of rearrangement-invariant Banach function spaces. To see this, it suffices to note that the embedding L∞(Ω) ↪→ L1(Ω) is notcompact.

The principal goal of this paper is to develop an elementary method that circumvents the above-describedobstacle and enables one to obtain limiting compact embeddings.

The main idea is to replace the compactness of an endpoint embedding by a weaker requirement, namely bythe uniform absolute continuity. This property (sometimes also called in literature almost compactness – see, forexample, [16, Remark 3.17.9], [15] or [17]) is of course a good deal weaker than compactness. There exists anelegant way of obtaining compactness when merely the uniform absolute continuity is known. To this end, wecombine two pieces of classical knowledge. First, a set is compact in a Banach function space if it is uniformlyabsolutely continuous and compact in measure (cf. e.g., [2, Exercise 8, p. 31]). Second, bounded sets in Besov-type spaces are compact in L1(Ω), and therefore in measure.

We shall now describe in detail our method. Suppose we want to prove that a Besov-type space B, with thesmoothness parameter close to zero, embeds compactly into a Banach function space X(Ω). We first slightlyenlarge the range space Lp(Ω) in the second embedding of (1.1). Note that the embeddings of type Lp(Ω) ↪→Lq (Ω), q < p, are uniformly absolutely continuous. Again, finer tuning is possible with more delicate scales offunction spaces. The question “how much” this space should be enlarged depends on the (prescribed) range spaceX . Next, we apply an appropriate interpolation method to the two embeddings in (1.1). To this end, we shallprove general interpolation theorems that guarantee that the resulting embedding is still uniformly absolutelycontinuous. Together with the fact that bounded sets in Besov-type spaces are compact in measure, this yields thecompactness of the resulting embedding.

Throughout the paper, we work without an explicit reference to the well-known basic concepts from the inter-polation theory and theory of Banach function spaces and rearrangement-invariant spaces. For precise definitionsand more detailed study of these objects, the reader is referred, for example, to [2], [3], [5], [18].

We shall now state the main results and define the key notions needed for the understanding of the principaltheorems. We start with the uniform absolute continuity.

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Math. Nachr. (2012) / www.mn-journal.com 3

Definition 1.1 Let A be a Banach space and let B be a Banach function space over a measure space (R, μ).Let T be a bounded linear operator defined on A with values in B (notation T : A → B). Then T is said to beuniformly absolutely continuous (notation T : A

∗→ B) if the image under T of the unit ball in A is uniformlyabsolutely continuous in B. This means that

limn→∞

sup‖f ‖A ≤1

‖χEn· Tf‖B = 0

for every sequence {En} of μ-measurable subsets of R such that χEn↘ 0 μ–a.e.

The point of departure of our method is contained in two theorems which we shall now state. Necessarydefinitions are given in detail in Section 2.

Theorem 1.2 Let C be some subcategory of Banach spaces and let C1 be the category of all compatible pairsof spaces from C. Let A = (A0 , A1) ∈ C1 and B = (B0 , B1) ∈ C1 . Suppose that B0 and B1 are Banach functionspaces over the same measure space (R, μ). Assume that F is an interpolation method of type ϕ on C1 , whereϕ ∈ Φ satisfies

lims→0+

ϕ(s, t) = 0 for any fixed t ∈ (0,∞). (1.2)

If T is a linear operator such that

T : A0∗−→ B0

and

T : A1 −→ B1 ,

then

T : F(A) ∗−→ F(B).

The following assertion is a symmetric counterpart of Theorem 1.2.

Theorem 1.3 Let C be some subcategory of Banach spaces and let C1 be the category of all compatible pairsof spaces from C. Let A = (A0 , A1) ∈ C1 and B = (B0 , B1) ∈ C1 . Suppose that B0 and B1 are Banach functionspaces over the same measure space (R, μ). Assume that F is an interpolation method of type ϕ on C1 , whereϕ ∈ Φ satisfies

limt→0+

ϕ(s, t) = 0 for any fixed s ∈ (0,∞).

If T is a linear operator such that

T : A0−→B0

and

T : A1∗−→ B1 ,

then

T : F(A) ∗−→ F(B).

As we shall see in Section 3 below, the proofs of Theorems 1.2 and 1.3 are very easy. Applications of thesetheorems in concrete situations involving particular interpolation methods can be however rather deep. To thisend, one has first to show that a given interpolation method is of a required type (rather surprisingly, except forthe classical cases, the literature concerning this matter is not rich) and then that the corresponding function ϕsatisfies the assumptions of Theorems 1.2 and 1.3.

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Let (R, μ) be a totally σ-finite measure space. We put M(R) (respectively, M+(R)) for the set of all scalar-valued (respectively, non-negative) functions which are measurable and almost everywhere finite on R. For1 ≤ q ≤ ∞, we denote by Lq (R) the Lebesgue space over (R, μ), equipped with the standard norm ‖ · ‖q ,R.If (a, b) ⊂ R and R = (a, b), we simply write M(a, b), M+(a, b) and Lq (a, b). By W(0,∞) we mean thecollection of all weights on (0,∞), that is, positive measurable finite functions on (0,∞). For v ∈ W(0,∞) andq ∈ [1,∞], we define the weighted Lebesgue space Lq ((0,∞); v(y), dy) as the set of functions in M(0,∞) withthe finite norm ‖f‖q ,v ,(0,∞) := ‖f(y)v(y)‖q ,(0,∞) .

Let Q be the cone of quasiconcave functions on [0,∞), let Q0 be the subcone of Q containing functions hwith the additional property limy→0+ h(y) = 0 and let Q0,∞ be the subcone of Q0 containing functions h withthe additional property limy→∞ y−1h(y) = 0 (see Definition 2.1 below).

Let Q be a subcone of Q. We say that a Banach pair A = (A0 , A1) is Q-abundant if there exists a constantC ∈ (0,∞) such that for every function h ∈ Q there exists some a ∈ A0 + A1 so that

C−1h(t) ≤ K(a, t; A

)≤ Ch(t) for all t ∈ (0,∞),

where K(a, t; A

)stands for the K-functional.

We note that if Q1 and Q2 are subcones of Q such that Q2 ⊂ Q1 and a Banach pair A is Q1-abundant, then Ais also Q2-abundant.

Theorem 1.4 Consider the following compatible pairs of function spaces:

� A = (A0 , A1) :=(L1(0,∞), L∞(0,∞)

);

� B = (B0 , B1) :=(L1(0,∞), L1

((0,∞); 1

y , dy))

;� C = (C0 , C1) :=

(L∞(0,∞), L∞(

(0,∞); 1y , dy

)).

Then A is Q0-abundant, B is Q0,∞-abundant and C is Q0-abundant.

We say that a Banach space Z, which contains functions defined on a totally σ-finite measure space (R, μ),has the lattice property (or that it is a Banach lattice) over the measure space (R, μ) if 0 ≤ g ≤ f μ-a.e. on Rimplies ‖g‖Z ≤ ‖f‖Z . The symbols Et for the dilation operator and ZQ for a certain subcone of Z are explainedin Definition 2.6 below.

In the next theorem we characterize the type of the abstract K-method (see Definition 2.7).

Theorem 1.5 Let Z be a Banach lattice over the measure space ((0,∞), dy) satisfying

0 < ‖min{1, y}‖Z < ∞. (1.3)

Let C be some subcategory of Banach spaces and let C1 be the category of all compatible pairs of spaces from C.Then the abstract K-method is of type ϕ on C1 , where

ϕ(s, t) =

{s‖E t

s‖ZQ

if s, t ∈ (0,∞),0 if s = 0 or t = 0.

Moreover, if Z has the Fatou property and C is such that C1 contains at least one of the pairs A, B, C fromTheorem 1.4, then the abstract K-method is of sharp type ϕ.

When w ∈ W(0,∞), 1 ≤ q ≤ ∞ and Z = Lq

((0,∞); 1

w (y )y1q, dy

), then condition (1.3) reads as

0 <

∥∥∥∥∥min {1, y}w(y)y

1q

∥∥∥∥∥q ,(0,∞)

< ∞ , (1.4)

and the abstract K-method (A0 , A1)Z,K coincides with the real interpolation method with a function parameterw. Moreover, the next theorem shows that the norm ‖Eτ ‖ZQ

of the dilation operator Eτ can be expressed interms of

ψ∗Z,K (·) := ψZ,K (1, ·)



for any Q ∈ {Q0,∞, Q0 , Q}, where ψZ,K stands for the fundamental function of the abstract K-method(A0 , A1)Z,K (cf. Definitions 2.3 and 2.7).

Theorem 1.6 Let w ∈ W(0,∞), q ∈ [1,∞] and let Z = Lq

((0,∞); 1

w (y )y1q, dy

). Assume that (1.4) holds.

Then

‖Eτ ‖ZQ≈ sup

α∈(0,∞)

ψ∗Z,K (ατ)

ψ∗Z,K (α)

= supα∈(0,∞)

‖min{1, ατ ·}‖Z,K

‖min{1, α ·}‖Z,K

for all τ ∈ (0,∞) and any Q ∈ {Q0,∞, Q0 , Q}.

Corollary 1.7 Let w ∈ W(0,∞) and q ∈ [1,∞] be such that (1.4) holds. Let C be some subcategory ofBanach spaces and let C1 be the category of all compatible pairs of spaces from C. Assume that C is such that C1contains at least one of the pairs A, B, C from Theorem 1.4. Then the real interpolation method with the functionparameter w is of sharp type ϕ on C1 , where

ϕ(s, t) =

{sg

(ts

)if s, t ∈ (0,∞),

0 if s = 0 or t = 0,

and

g(τ) := supα∈(0,∞)

∥∥∥∥min{1,ατ ·}w (y )y

1q

∥∥∥∥q ,(0,∞)∥∥∥∥min{1,α ·}

w (y )y1q

∥∥∥∥q ,(0,∞)

for all τ ∈ (0,∞).

Examples 1.8 Theorem 1.5 and Corollary 1.7 can be applied for example to the following subcategories ofBanach spaces:

� the category of Banach spaces;� the category of Banach function spaces;� the category of rearrangement-invariant Banach function spaces (for definition see e.g. [2]);� the category of weighted Lebesgue spaces.

One of the most important operators for which its uniform absolute continuity is intensively studied, is theidentity operator I on various function spaces. If, for two Banach spaces X and Y , one has I : X

∗→ Y , then we

say that X is uniformly absolutely continuously embedded into Y and write X∗

↪→ Y . We shall write X ↪→↪→ Yto denote that X is compactly embedded into Y .

The results stated so far can be applied to a number of particular instances. We shall present one such a resulton a compact embedding of a Besov-type space involving a limiting smoothness. We first recall the definition ofthe suitable Besov space. For p ∈ [1,∞), we shall denote by ω1(f, t)p the value of the first-order modulus ofcontinuity of a function f at t with respect to Lp(Rn ), defined by

ω1(f, t)p := sup|h|≤t

‖f(x + h) − f(x)‖p,Rn ,

where h ∈ Rn and t ∈ [0,∞). A positive function v will be said to be slowly varying (or s.v. for short) on

(0,∞) if for each ε > 0, tεv(t) is almost non-decreasing and t−εv(t) is almost non-increasing on (0,∞). Theset of all such functions is denoted by SV(0,∞). We analogously treat such functions on (0, 1) and denote thecorresponding class by SV(0, 1).

Definition 1.9 Let p ∈ [1,∞) and q ∈ [1,∞]. Let v ∈ SV(0, 1) satisfy∥∥∥t−1q v(t)

∥∥∥q ,(0,1)

= ∞. (1.5)



The Besov space B0,vp,q = B0,v

p,q (Rn ) consists of those functions f ∈ Lp(Rn ) having a finite norm

‖f‖B 0 , vp , q (Rn ) := ‖f‖Lp (Rn ) +

∥∥∥t−1q v(t)ω1(f, t)p

∥∥∥q ,(0,1)

. (1.6)

We will denote �(t) := (1+| log t|), t ∈ (0,∞). We recall that the function v(t) := �β (t) belongs to SV(0,∞)(and also to SV(0, 1)) for every β ∈ R. We write B0,β

p,q when v(t) = �β (t).

Theorem 1.10 Let p ∈ [1,∞), q ∈ [1,∞] and let β ∈ R be such that

β +1q≥ 0 if q < ∞ and β > 0 if q = ∞.

If Ω ⊂ Rn is a bounded domain and δ < 0, then

B0,βp,q (Rn ) ↪→↪→ Y , (1.7)

where

Y = Y (Ω) :={

f ∈ Lp (log L)δ (Ω); ‖f‖Y < ∞}

and

‖f‖Y :=∥∥∥t−

1q �β (t)

∥∥∥�δ (τ)f∗(τ)‖p,(0,t)‖q ,(0,1) .

Remark 1.11 Let us finally note that no handy necessary and sufficient conditions are known in the literaturefor the embedding

X(Ω)∗

↪→ Y (Ω) (1.8)

to hold for a pair of rearrangement-invariant Banach function spaces X(Ω), Y (Ω), where Ω is a Lipschitz domainin R

n (the precise definition of an r.i. space can be found, for example, in [2]). One exception is the conditionexpressed in terms of the product operator P , defined on M(Ω) × M(Ω) by P (f, g) := fg, f, g ∈ M(Ω). Itwas shown in [17, Theorem 6.5] that (1.8) is true if and only if there exists a rearrangement-invariant Banachfunction space V (Ω) �= L1(Ω) such that

P : X(Ω) × Y (Ω) −→ V (Ω).

Some reasonable sufficient conditions for (1.8) are given, for example, in [10] or [17]. Necessary and suffi-cient conditions for uniformly absolutely continuous embeddings are known for some special classes of functionspaces. For example, in the recent work [15], a characterization of (1.8) is given when X(Ω) and Y (Ω) areclassical Lorentz spaces of type Λ and Γ.

The paper is structured as follows. In Section 2 we collect necessary definitions of all the key notions thatappear in the paper, including the introduction of three important cones containing quasiconcave functions on[0,∞). In Section 3 we prove all the assertions stated in Section 1 except for Theorem 1.10, which is proved inSection 4. Aside from the proofs, we also state and prove in Section 3 important Lemma 3.2 in the spirit of [14,Theorem 3.7].

2 Preliminaries

As usual, we denote a characteristic function of a measurable set E by χE and we use the convention a∞ = 0 for

a ∈ [0,∞), 00 = 0 and ∞

∞ = 0.For two non-negative expressions (i.e., functions or functionals) A, B, the symbol A � B (or A � B) means

that A ≤ CB (or CA ≥ B), where C is a positive constant independent of appropriate quantities involved in Aand B. If A � B and A � B, we write A ≈ B and say that A and B are equivalent.

Definition 2.1 A real function f is said to be almost non-decreasing on an interval J ⊂ R if it is equivalentto a non-decreasing function on J . An almost non-increasing function is defined in an analogous way. A non-negative function h on [0,∞) is called quasiconcave (notation h ∈ Q) if h(t) = 0 if and only if t = 0, h



is non-decreasing on (0,∞) and h(t)t is non-increasing on (0,∞). By Q0 we shall denote the subcone of Q

containing those functions h that additionally satisfy

limt→0+

h(t) = 0

and by Q0,∞ the subcone of Q0 containing the functions h satisfying the condition

limt→∞

h(t)t

= 0.

Definition 2.2 We say that a function ϕ : [0,∞) × [0,∞) → [0,∞), ϕ �≡ 0, belongs to the class Φ if it hasthe following properties:

(i) ϕ(0, 0) = 0,

(ii) ϕ(s, t) is almost non-decreasing in each variable separately,

(iii) ϕ(s, t) is positively homogeneous of degree 1, that is, ϕ(λs, λt) = λϕ(s, t) for every λ, s, t ∈ [0,∞).

We say that two normed linear spaces X and Y are equal, written X = Y , if they coincide in the set-theoreticalsense and their norms are equivalent, that is,

‖f‖X ≈ ‖f‖Y for all f ∈ X.

Definition 2.3 Let F be an interpolation method. Then the fundamental function ψF : [0,∞) × [0,∞) → R

of the method F is given by

F (sR, tR) = ψF (s, t)R.

We put ψ∗F (t) := ψF (1, t) for every t ∈ [0,∞).

Remarks 2.4

(i) Let ψ be a fundamental function of some interpolation method. Then, by [5, Proposition 2.3.10, p. 145],ψ ∈ Φ.

(ii) Let ϕ ∈ Φ. Then it is not hard to verify that ϕ is quasiconcave in each variable separately. Moreover, theproperties (ii) and (iii) of Definition 2.2 imply that ϕ(s, t) > 0 if s �= 0 and t �= 0 and if we define thefunction ϕ∗ by ϕ∗(t) := ϕ(1, t) for all t ∈ [0,∞), then ϕ∗ is quasiconcave and

ϕ(s, t) = sϕ

(1,

t

s

)= sϕ∗

(t

s

)for all s ∈ (0,∞).

(iii) On the other hand, for every quasiconcave function ϕ∗ : [0,∞) → [0,∞) the function ϕ, defined by

ϕ(s, t) :=

⎧⎨⎩sϕ∗

(t

s

)if s, t ∈ (0,∞),

0 if s = 0 or t = 0,

satisfies ϕ ∈ Φ.

(iv) Let ϕ ∈ Φ. Then there exists a constant C ∈ (0,∞) such that ϕ(s, t) ≤ C max{s, t} for every s, t ∈[0,∞). Indeed, by properties (ii) and (iii) of Definition 2.2, we get for s, t ∈ (0,∞) that

ϕ(s, t) = max{s, t}ϕ(

s

max{s, t} ,t

max{s, t}

)≤ C max{s, t}ϕ(1, 1),

and the claim follows.

Definition 2.5 Let ϕ ∈ Φ, let C be a subcategory of Banach spaces and let C1 be the category of compatiblepairs of spaces from C.



An interpolation method F is said to be of type ϕ on C1 if, for all s, t ∈ [0,∞),

sup{‖T‖F(A)→F(B ) ; ‖T‖A 0 →B0 ≤ s, ‖T‖A 1 →B1 ≤ t, A, B ∈ C1 , T ∈ L

(A, B

)}� ϕ(s, t). (2.1)

An interpolation method F is said to be of sharp type ϕ on C1 if, for all s, t ∈ [0,∞),

sup{‖T‖F(A)→F(B ) ; ‖T‖A 0 →B0 ≤ s, ‖T‖A 1 →B1 ≤ t, A, B ∈ C1 , T ∈ L

(A, B

)}≈ ϕ(s, t).

Definition 2.6 Given a Banach lattice Z over the measure space ((0,∞), dy) and t ∈ (0,∞), we denote byEt the dilation operator on Z defined by

(Etf) (y) := f(yt), y ∈ (0,∞).

If Q is some subcone of Q, we denote by ZQ the cone Z ∩ Q. In such a case if T : Z → Z is an operator defined

on Z ∩ Q, then we put

‖T‖ZQ:= sup

h∈Z∩Q

‖Th‖Z

‖h‖Z.

Our aim now is to define the abstract K-method of interpolation, based on a Banach lattice, and its importantparticular instance called the real method of interpolation with a function parameter.

Definition 2.7 Let Z be a Banach lattice over the measure space ((0,∞), dy) such that

0 < ‖min{1, y}‖Z < ∞. (2.2)

Let A = (A0 , A1) be a compatible pair of Banach spaces. The abstract K-method of interpolation (·, ·)Z ;K isdefined by

‖a‖Z ;K = ‖a‖(A 0 ,A 1 )Z ;K :=∥∥K

(a, y; A

)∥∥Z ((0,∞);dy ) , a ∈ A0 + A1 .

It can be shown that (A0 , A1)Z,K is an interpolation space with respect to the pair A. In the particular case when

Z = Lq

((0,∞); 1

w (y )y1q, dy

), where w ∈ W(0,∞), q ∈ [1,∞] and

0 <

∥∥∥∥∥min {1, y}w(y)y

1q

∥∥∥∥∥q ,(0,∞)

< ∞,

we have

(A0 , A1)Z ;K = (A0 , A1)q ,w ;K ,

where

‖f‖q ,w ;K = ‖f‖(A 0 ,A 1 )q , w ;K:=

∥∥∥∥∥K(f, y; A

)w(y)y

1q

∥∥∥∥∥q ,(0,∞)

.

We shall call this method the real interpolation method with a function parameter.

We refer to [5] for details on the abstract K-method. The method (A0 , A1)q ,w ;K was first introduced byGustavsson in [13] in a less general setting; the function w was assumed to be continuous, non-decreasing, andto satisfy the condition∫ ∞

0w(t)min

{1,

1t

}dt

t< ∞,

where w(t) := sups∈(0,∞)w (st)w (s) .



Remark 2.8 If Z is a Banach lattice over the measure space ((0,∞), dy) such that (2.2) holds, then thefundamental function ψZ,K of the abstract K-method satisfies

ψZ,K (s, t) = ‖min{s, · t}‖Z , s, t ∈ [0,∞)

(cf. [5, p. 438]). In particular, when Z = Lq

((0,∞); 1

w (y )y1q, dy

)with w ∈ W(0,∞) and q ∈ [1,∞], we get

ψLq

((0,∞); 1

w ( y ) y1q

,dy

);K

(s, t) =

∥∥∥∥∥min{s, yt}w(y)y

1q

∥∥∥∥∥q ,(0,∞)

, s, t ∈ [0,∞).

Consequently,

ψ∗

Lq

((0,∞); 1

w ( y ) y1q

,dy

);K

(t) =

∥∥∥∥∥min{1, yt}w(y)y

1q

∥∥∥∥∥q ,(0,∞)

, t ∈ [0,∞). (2.3)

3 Proofs of Theorems 1.2–1.6

P r o o f o f T h e o r e m 1.2. Let {En} be a sequence of subsets of R such that χEn↘ 0 μ-a.e. Applying (2.1)

to the operators TEn, defined by

TEnf := χEn

Tf,

we obtain

‖TEn‖F(A)→F(B ) � ϕ (‖TEn

‖A 0 →B0 , ‖TEn‖A 1 →B1 ) .

Since ϕ is almost non-decreasing in the second variable and ‖TEn‖A 1 →B1 ≤ ‖T‖A 1 →B1 , the lattice property of

the Banach function space B1 yields

‖TEn‖F(A)→F(B ) � ϕ (‖TEn

‖A 0 →B0 , ‖T‖A 1 →B1 ) for all n ∈ N.

Now, the second variable is fixed and, by the uniform absolute continuity of T from A0 to B0 ,

limn→∞

‖TEn‖A 0 →B0 = 0.

Thus, the result follows from (1.2).

P r o o f o f T h e o r e m 1.3. The argument is analogous to that in the proof of Theorem 1.2. We omit thedetails.

P r o o f o f T h e o r e m 1.4. (i) First, let h ∈ Q0 . Then (cf. [2, Chapter 2, Proposition 5.10]) there exists a non-negative concave function h on [0,∞), called the least concave majorant of h, such that

12h ≤ h ≤ h. (3.1)

By [2, remarks on p. 72], for a non-decreasing concave function h, there exists an α ≥ 0 and a non-negativenon-increasing function φ on (0,∞) such that

h(y) = α +∫ y

0φ(s) ds for all y ∈ (0,∞).

Since h ∈ Q0 implies that h ∈ Q0 , we get α = 0. Therefore,

h(y) =∫ y

0φ(s) ds = K(φ, y;A0 , A1), y ∈ (0,∞).

Consequently, A is Q0-abundant.



(ii) Let h ∈ Q0,∞ and let h be the least concave majorant of h. Then (3.1) holds and h ∈ Q0,∞. More-over, h(y) =

∫ y

0 Φ(s) ds, where Φ is a non-negative non-increasing function on (0,∞) satisfying Φ(∞) :=lims→∞ Φ(s) = 0 since

0 = limy→∞

h(y)y

≥ limy→∞

yΦ(y)y

= Φ(∞) ≥ 0.

Put

h1(t) :=

⎧⎨⎩t−1

∫ t

0h(y) dy, t ∈ (0,∞),

0, t = 0.

Then

h(t) ≥ h1(t) ≥12h

(t

2

)≥ 1

4h(t) for all t ∈ [0,∞). (3.2)

Indeed,

h1(t) = t−1∫ t

0h(y) dy ≤ h(t), t ∈ (0,∞),

and

h1(t) ≥ t−1∫ t

t2

h(y) dy ≥ 12h

(t

2

), t ∈ (0,∞).

Since h is concave on (0,∞) and h(0) = 0,

h

(t

2

)≥ 1

2h(t) for all t ≥ 0.

Consequently, (3.2) holds, which implies that

h1 ≈ h ≈ h on [0,∞). (3.3)

Moreover,

h′1(t) = − 1

t2

∫ t

0

(∫ y

0Φ(s) ds

)dy +

1t

∫ t

0Φ(s) ds

= −h1(t)t

+h(t)

tfor all t ∈ (0,∞). (3.4)

Hence,

h′1 is an absolutely continuous function on (0,∞) and lim

t→∞h′

1(t) = 0. (3.5)

Together with (3.2), identity (3.4) implies that h′1 ≥ 0 on (0,∞).

We also claim that

h′1 is non-increasing on (0,∞). (3.6)

Indeed, since h′1 is an absolutely continuous function on (0,∞), it suffices to show that h′′

1 ≤ 0 a.e. on (0,∞).Using (3.4), we obtain that

h′′1 (t) = − 2

t2

[h(t) − h1(t) −

t

2Φ(t)

]for a.e. t ∈ (0,∞). (3.7)



Moreover, making use of the definitions of h and h1 and Fubini’s theorem, we arrive at

h(t) − h1(t) −t

2Φ(t) =

1t

∫ t

0Φ(s)s ds − t

2Φ(t). (3.8)

Since Φ is non-increasing on (0,∞),

1t

∫ t

0Φ(s)s ds ≥ 1

tΦ(t)

∫ t

0s ds =

t

2Φ(t).

Together with (3.8) and (3.7), this yields (3.6).The properties of h′

1 imply that

h′1(t) =

∫ ∞

t

(−h′′1 (s)) ds for all t ∈ (0,∞).

Thus, putting f(s) := −sh′′1 (s) a.e. in (0,∞), we obtain that f ≥ 0 a.e. in (0,∞) and that

h1(t) =∫ t

0h′

1(y) dy =∫ t

0

(∫ ∞

y

f(s)s

ds

)dy, t ∈ (0,∞).

Hence, by Fubini’s theorem,

h1(t) =∫ ∞

0|f(s)|min

{1,

t

s

}ds for all t ∈ [0,∞).

On the other hand (cf. [3, (3), p. 116]),

K

(g, t;L1(0,∞), L1

((0,∞);

1y, dy

))≈

∫ ∞

0|g(y)|min

{1,

t

y

}dy for all t ∈ [0,∞).

Consequently,

h1(t) ≈ K

(f, t;L1(0,∞), L1

((0,∞);

1y, dy

))for all t ∈ [0,∞).

Together with (3.3), this shows that B is Q0,∞-abundant.

(iii) By [2, Chapter 5, Exercise 5(a)], for every function f ∈ L∞(0,∞) + L∞((0,∞); 1

y , dy)

, one has

K

(f, y;L∞(0,∞), L∞

((0,∞);

1y, dy

))≈ f(y) for all y ∈ (0,∞), (3.9)

where f is the least concave majorant of f . Let h ∈ Q0 . Then h(0) = h(0) = 0. Thus, applying (3.9) to f = hand using (3.1), we obtain that C is Q0 abundant.

Remark 3.1 It is easy to verify that the least quasi-concave majorant f of a function f ∈ L∞(0,∞) +L∞

((0,∞); 1

y , dy)

is given by

f(y) := sups∈(0,y ]

s supx∈[s,∞)

|f(x)|x

, y ∈ (0,∞).

On exchanging the suprema, we can rewrite this as

f(y) = supx∈(0,∞)

min{

1,y

x

}|f(x)|, y ∈ (0,∞). (3.10)

On the other hand, the least concave majorant f of f satisfies 12 f ≤ f ≤ f and f ∈ Q. Consequently, f ≈ f .

Together with (3.9) and (3.10), this implies that

K

(f, t;L∞(0,∞), L∞

((0,∞);

1y, dy

))≈ sup

x∈(0,∞)min

{1,

t

x

}|f(x)| for all t ∈ (0,∞). (3.11)



P r o o f o f T h e o r e m 1.5. Fix s, t ∈ (0,∞). Let us denote

GZ (s, t) := sup

{sup

x∈X 0 +X 1 ,K (x,·;X )∈Z

∥∥K(Tx, α; Y

)∥∥Z∥∥K

(x, α; X

)∥∥Z

;

T ∈ L(X, Y

), X, Y ∈ C1 , ‖T‖X 0 →Y0 ≤ s, ‖T‖X 1 →Y1 ≤ t

}.

First, let X, Y ∈ C1 and let T ∈ C be a linear operator defined on X0 + X1 and with values in Y0 + Y1 .Assume that ‖T‖X 0 →Y0 ≤ s and ‖T‖X 1 →Y1 ≤ t. If x ∈ X0 + X1 , and x = x0 + x1 , where xi ∈ Xi , i = 0, 1,then Tx = Tx0 + Tx1 and Txi ∈ Yi , i = 0, 1. Thus, for any α ∈ (0,∞), we have

K(Tx, α; Y

)≤ ‖Tx0‖Y0 + α‖Tx1‖Y1

≤ s‖x0‖X 0 + tα‖x1‖X 1 = s

(‖x0‖X 0 +

tα

s‖x1‖X 1

).

Taking the infimum over all such decompositions, we arrive at the inequality

K(Tx, α; Y

)≤ sK

(x,

tα

s; X

).

Consequently,

GZ (s, t) ≤ sup

⎧⎨⎩ sup

x∈X 0 +X 1 ,K(x,·;X

)∈Z

s∥∥K

(x, tα

s ; X)∥∥

Z∥∥K(x, α; X

)∥∥Z

; X ∈ C1

⎫⎬⎭

≤ sup

⎧⎨⎩ sup

x∈X 0 +X 1 ,K(x,·;X

)∈Z

s∥∥E t

sK

(x, α; X

)∥∥Z∥∥K

(x, α; X

)∥∥Z

; X ∈ C1

⎫⎬⎭ .

Since, for every x ∈ X0 + X1 , the function K(x, α; X

)(of the variable α ∈ [0,∞)) belongs to Q, we get

GZ (s, t) ≤ suph∈Z∩Q

s∥∥E t

sh∥∥

Z

‖h‖Z= s

∥∥E ts

∥∥ZQ

. (3.12)

Hence, the abstract K-method is of type s∥∥E t

s

∥∥ZQ

.We shall now prove the sharpness of this type under the additional assumptions that Z has the Fatou property

and C is such that C1 contains at least one of the pairs A, B, C from Theorem 1.4. We recall that s, t ∈ (0,∞)remain fixed. We define the operators Ts,t and T s,t on M+(0,∞) by

(Ts,th)(y) := th

(t

sy

)and

(T s,th

)(y) := sh

(t

sy

), y ∈ (0,∞).

Using a change of variables, we obtain, for any h ∈ M+(0,∞),

∥∥Ts,th∥∥

1,(0,∞) =∫ ∞

0th

(t

sx

)dx = s

∫ ∞

0h(y) dy = s‖h‖1,(0,∞)

and

∥∥Ts,th∥∥

L1 ((0,∞); 1x ,dx) =

∫ ∞

0th

(t

sx

)dx

x= t

∫ ∞

0h(y)

dy

y= t‖h‖L1 ((0,∞); 1

y ,dy),

hence

‖Ts,t‖L1 (0,∞)→L1 (0,∞) = s and ‖Ts,t‖L1 ((0,∞); 1y ,dy)→L1 ((0,∞); 1

y ,dy) = t.



Moreover, obviously,

‖Ts,t‖L∞(0,∞)→L∞(0,∞) = t and ‖T s,t‖L∞(0,∞)→L∞(0,∞) = s. (3.13)

Also, for any h ∈ L∞((0,∞); 1

y , dy)

,

∥∥T s,th∥∥

L∞((0,∞); 1y ,dy) =

∥∥∥∥sh

(ty

s

)∥∥∥∥L∞((0,∞); 1

y ,dy)= t

∥∥∥∥∥h(

tys

)tys

∥∥∥∥∥L∞(0,∞)

= t

∥∥∥∥h(z)z

∥∥∥∥L∞(0,∞)

= t‖h‖L∞((0,∞); 1y ,dy),

whence ∥∥T s,t∥∥

L∞((0,∞); 1y ,dy)→L∞((0,∞); 1

y ,dy) = t.

Let A, B and C be from Theorem 1.4. Then, for any f ∈ A0 + A1 and every y ∈ (0,∞), we have

K(Ts,tf, y; A

)=

∫ y

0tf∗

(tx

s

)dx = s

∫ t ys

0f∗ (z) dz = sK

(f,

ty

s; A

). (3.14)

Similarly, for any f ∈ B0 + B1 and every y ∈ (0,∞), we have

K(Ts,tf, y; B

)≈

∫ ∞

0min

{1,

y

x

}tf

(tx

s

)dx

= s

∫ ∞

0min

{1,

ty

sz

}f(z) dz ≈ sK

(f,

ty

s; B

),

and, for any f ∈ C0 + C1 and every y ∈ (0,∞), we have, by (3.11),

K(T s,tf, y; C

)≈ sup

x∈(0,∞)min

{1,

y

x

}sf

(tx

s

)

= s supz∈(0,∞)

min{

1,ty

sz

}f (z) ≈ sK

(f,

ty

s; C

),

Assume first that A ∈ C1 . By Theorem 1.4, A is Q0-abundant. Thus, for any h ∈ Q0 , there exists fh ∈ A0 +A1such that

c−1h(y) ≤ K(fh , y; A

)≤ ch(y), y ∈ (0,∞), (3.15)

where c is some positive constant independent of h and y. Thus, using (3.13), (3.14) and (3.15), we obtain

GZ (s, t) ≥ ‖Ts,t‖AZ ;K →AZ ;K= sup

0 =f∈AZ ;K

‖Ts,t(f)‖AZ ;K

‖f‖AZ ;K

≥‖Ts,t(fh)‖AZ ;K

‖fh‖AZ ;K

=

∥∥K(Ts,t(fh), y; A

)∥∥Z∥∥K

(fh , y; A

)∥∥Z

=s∥∥K

(fh , t

s y; A)∥∥

Z∥∥K(fh , y; A

)∥∥Z

≥ c−2s∥∥E t

sh∥∥

Z

‖h‖Z.

Passing to the supremum over all h ∈ ZQ 0 , we get, for any subcategory of Banach spaces C such that C1 con-tains A,

GZ (s, t) � s∥∥E t

s

∥∥ZQ 0

. (3.16)

When C ∈ C1 , the same argument with Ts,t replaced by T s,t works just as well, and the final lower estimate ofϕ(s, t) is the same.



When B ∈ C1 , the same proof (with the operator Ts,t) works again, but because the pair B is merely Q0,∞-abundant, we obtain in this case only

GZ (s, t) � s∥∥E t

s

∥∥ZQ 0 ,∞

.

Since the inequality ‖T‖ZQ 0≥ ‖T‖ZQ 0 ,∞

automatically holds for any operator T , we get

GZ (s, t) � s∥∥E t

s

∥∥ZQ 0 ,∞

(3.17)

in any of the three cases.We now claim that

‖Eτ ‖ZQ≈ ‖Eτ ‖ZQ 0

≈ ‖Eτ ‖ZQ 0 ,∞for any τ ∈ (0,∞). (3.18)

First, the inequalities

‖Eτ ‖ZQ≥ ‖Eτ ‖ZQ 0

≥ ‖Eτ ‖ZQ 0 ,∞(3.19)

are trivial. Next, let h ∈ Q. Then there exist α, β ∈ [0,∞) and a positive non-increasing function φ on (0,∞)such that limy→∞ φ(y) = 0 and

h(t) = α + βt +∫ t

0φ(y) dy, t ∈ (0,∞).

Then

‖h (τy)‖Z

‖h‖Z≤ ‖α‖Z

‖α‖Z+

β‖τt‖Z

β‖t‖Z+

∥∥ ∫ τ t

0 φ(y) dy∥∥

Z∥∥ ∫ t

0 φ(y) dy∥∥

Z

≤ 1 + τ + ‖Eτ ‖ZQ 0 ,∞,

since the function∫ t

0 φ(y) dy belongs to Q0,∞. Hence,

‖Eτ ‖ZQ= sup

h∈Q

‖h (τy)‖Z

‖h‖Z≤ 1 + τ + ‖Eτ ‖ZQ 0 ,∞

. (3.20)

Now, we define

hn (y) := y1− 1n for y ∈ (0,∞) and n ∈ N.

Then hn ∈ Q0,∞ for every n ∈ N, n ≥ 2, and, moreover, hn (y) ↗ y for every y ∈ (0,∞) as n → ∞. Thus,using the Fatou property of Z, we have

‖Eτ ‖ZQ 0 ,∞= sup

h∈Q 0 ,∞

‖h (τy)‖Z

‖h‖Z≥ sup

n∈N

‖hn (τy)‖Z

‖hn‖Z.

Since

‖hn (τy)‖Z

‖hn‖Z−→ ‖τy‖Z

‖y‖Z= τ,

we obtain

‖Eτ ‖ZQ 0 ,∞≥ τ. (3.21)

Similarly, if gn (y) := min{1, ny} for y ∈ (0,∞) and n ∈ N, then gn ∈ Q0,∞ for every n ∈ N and, moreover,gn ↗ χ(0,∞) on (0,∞) as n → ∞. Hence,

‖gn (τy)‖Z

‖gn‖Z−→

∥∥χ(0,∞)∥∥

Z

‖χ(0,∞)‖Z= 1,



so that

‖Eτ ‖ZQ 0 ,∞≥ 1. (3.22)

Combining (3.19), (3.20), (3.21) and (3.22), we arrive at

‖Eτ ‖ZQ≤ 3‖Eτ ‖ZQ 0 ,∞

≤ 3‖Eτ ‖ZQ 0≤ 3‖Eτ ‖ZQ

,

and (3.18) follows.Inserting (3.18) in (3.17) and (3.16), we get

GZ (s, t) � s∥∥E t

s

∥∥ZQ

, (3.23)

which, together with (3.12), gives the desired sharpness of the type of the abstract K-method.

In order to prove Theorem 1.6 and Corollary 1.7, we need the following auxiliary result (cf. [14] and [12]).

Lemma 3.2 Let u, v be weights on (0,∞) and let q ∈ [1,∞]. Then

suph∈Q

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)= sup

h∈Q 0

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)

= suph∈Q 0 ,∞

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)≈ sup

α∈(0,∞)

‖u(x)min {x, α}‖q ,(0,∞)

‖v(x)min {x, α}‖q ,(0,∞). (3.24)

(The constants implicitly involved in the symbol ≈ in (3.24) do not depend on u and v.)

P r o o f. We denote

M := supα∈(0,∞)

‖u(y)min {y, α}‖q ,(0,∞)

‖v(y)min {y, α}‖q ,(0,∞).

Assume first that q = ∞. We recall that, by exchanging suprema, whenever F,G are non-negative functionson (0,∞) and F is non-increasing, then

supt∈(0,∞)

F (t)G(t) = supt∈(0,∞)

F (t) sups∈(0,t]

G(s), t ∈ (0,∞), (3.25)

while, when F is non-decreasing, then

supt∈(0,∞)

F (t)G(t) = supt∈(0,∞)

F (t) sups∈[t,∞)

G(s), t ∈ (0,∞). (3.26)

Define

u(t) :=1t

sups∈(0,t]

s supy∈[s,∞)

u(y), t ∈ (0,∞), (3.27)

and v analogously. Then tu(t) is clearly non-decreasing on (0,∞). On the other hand, by the change of variablesσ = s

t , we have

u(t) := sups∈(0,1]

σ supσt≤y<∞

u(y), t ∈ (0,∞),

hence u is non-increasing on (0,∞). Now let h ∈ Q. Then h is non-decreasing and h(t)t is non-increasing on

(0,∞). Therefore, using (3.27), (3.25) and (3.26), we obtain

‖uh‖∞,(0,∞) = sup0<t<∞

h(t)t

sup0<s≤t

s sups≤y<∞

u(y) = sup0<t<∞

h(t)t

t supt≤y<∞

u(y)

= sup0<t<∞

h(t) supt≤y<∞

u(y) = sup0<t<∞

h(t)u(t) = ‖uh‖∞,(0,∞) . (3.28)



Moreover, if α ∈ (0,∞), then

‖u(y)min {y, α}‖∞,(0,∞) = max{

sup0<y≤α

yu(y), α supα≤y<∞

u(y)}

= αu(α). (3.29)

Assume that M < ∞. Applying (3.29) and its analogue with u replaced by v, we get

M = supα∈(0,∞)

‖u(y)min {y, α}‖∞,(0,∞)

‖v(y)min {y, α}‖∞,(0,∞)= sup

α∈(0,∞)

αu(α)αv(α)

= supα∈(0,∞)

u(α)v(α)

,

and, consequently, u(α) ≤ Mv(α), α ∈ (0,∞). Together with (3.28), this implies that

suph∈Q

‖uh‖∞,(0,∞)

‖vh‖∞,(0,∞)= sup

h∈Q

‖uh‖∞,(0,∞)

‖vh‖∞,(0,∞)≤ sup

h∈Q

‖Mvh‖∞,(0,∞)

‖vh‖∞,(0,∞)= M.

Since, for every α ∈ (0,∞), the function h(y) := min{y, α} satisfies h ∈ Q0,∞, we finally obtain

suph∈Q 0 ,∞

‖uh‖∞,(0,∞)

‖vh‖∞,(0,∞)≥ sup

α∈(0,∞)

‖u(y)min {y, α}‖∞,(0,∞)

‖v(y)min {y, α}‖∞,(0,∞).

Consequently,

M ≤ suph∈Q 0 ,∞

‖uh‖∞,(0,∞)

‖vh‖∞,(0,∞)≤ sup

h∈Q 0

‖uh‖∞,(0,∞)

‖vh‖∞,(0,∞)≤ sup

h∈Q

‖uh‖∞,(0,∞)

‖vh‖∞,(0,∞)≤ M,

and the result for q = ∞ follows.Now let q ∈ [1,∞). Since for every α ∈ (0,∞) the function h(x) := min{x, α} satisfies h ∈ Q0,∞, we get


‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞).

Conversely, let h ∈ Q, h �≡ 0. Then there exists a non-negative non-increasing function φ on (0,∞) andβ ∈ [0,∞) such that

h(t) ≈ β +∫ t

0φ(s) ds.

We next note that

M ≥ supn∈N

∥∥u(y)min{y, 1

n

}∥∥q ,(0,∞)∥∥v(y)min

{y, 1

n

}∥∥q ,(0,∞)

≥ supn∈N

∥∥u(y)min{y, 1

n

}∥∥q ,( 1

n ,∞)∥∥v(y)min{y, 1

n

}∥∥q ,(0,∞)

≥ supn∈N

1n ‖u‖q ,( 1

n ,∞)1n ‖v‖q ,(0,∞)

= supn∈N

‖u‖q ,( 1n ,∞)

‖v‖q ,(0,∞)=

‖u‖q ,(0,∞)

‖v‖q ,(0,∞).

Hence, ‖u‖q ,(0,∞) ≤ M ‖v‖q ,(0,∞) .By [14, Theorem 3.7, p. 151],∥∥∥∥u(t)

∫ t

0φ(s) ds

∥∥∥∥q ,(0,∞)

≤ M

∥∥∥∥v(t)∫ t

0φ(s) ds

∥∥∥∥q ,(0,∞)

.

Thus, altogether,

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)≈

∥∥u(t)(β +

∫ t

0 φ(s) ds)∥∥

q ,(0,∞)∥∥v(t)(β +

∫ t

0 φ(s) ds)∥∥

q ,(0,∞)

≤β ‖u(t)‖q ,(0,∞) +

∥∥u(t)∫ t

0 φ(s) ds∥∥

q ,(0,∞)∥∥v(t)(β +

∫ t

0 φ(s) ds)∥∥

q ,(0,∞)

≤β ‖u(t)‖q ,(0,∞) + M

∥∥v(t)∫ t

0 φ(s) ds∥∥

q ,(0,∞)∥∥v(t)(β +

∫ t

0 φ(s) ds)∥∥

q ,(0,∞)

≤‖u(t)‖q ,(0,∞)

‖v(t)‖q ,(0,∞)+ M

≤ 2M.



Passing to the supremum, we get

suph∈Q

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)≤ 2M.

Combining the estimates established so far, we arrive at


‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)≤ sup

h∈Q 0

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)≤ sup

h∈Q

‖uh‖q ,(0,∞)

‖vh‖q ,(0,∞)≤ 2M,

and the result follows.

P r o o f o f T h e o r e m 1.6. Assume that Q ∈ {Q,Q0 , Q0,∞}. Then, by the multiple change of variables,Lemma 3.2 and (2.3),

‖E ts‖ZQ

= suph∈Q

∥∥∥∥ h( ts y)

w (y )y1q

∥∥∥∥q ,(0,∞)∥∥∥∥ h(y )

w (y )y1q

∥∥∥∥q ,(0,∞)

= suph∈Q

∥∥∥∥∥ h(x)

w( st x)x

1q

∥∥∥∥∥q ,(0,∞)∥∥∥∥ h(y )

w (y )y1q

∥∥∥∥q ,(0,∞)

≈ supα∈(0,∞)

∥∥∥∥∥ min{x,α}w( s

t x)x1q

∥∥∥∥∥q ,(0,∞)∥∥∥∥min{y ,α}

w (y )y1q

∥∥∥∥q ,(0,∞)

= supα∈(0,∞)

∥∥∥∥min{ ts y ,α}

w (y )y1q

∥∥∥∥q ,(0,∞)∥∥∥∥min{y ,α}

w (y )y1q

∥∥∥∥q ,(0,∞)

= supα∈(0,∞)

∥∥∥∥min{ t ys α ,1}

w (y )y1q

∥∥∥∥q ,(0,∞)∥∥∥∥min{ y

α ,1}

w (y )y1q

∥∥∥∥q ,(0,∞)

= supβ∈(0,∞)

∥∥min{ ts β ·, 1}

∥∥Z

‖min{β ·, 1}‖Z

= supβ∈(0,∞)

ψ∗Z,K

(ts β

)ψ∗

Z,K (β),

and the assertion follows.

4 Proof of Theorem 1.10

We start introducing another limiting Besov space and establishing several auxiliary results. Let p ∈ [1,∞),q ∈ [1,∞] and let v ∈ SV(0,∞) be such that (1.5) holds and∥∥∥t−

1q v(t)

∥∥∥q ,(1,∞)

< ∞. (4.1)

The Besov space B0,vp,q = B0,v

p,q (Rn ) consists of those functions f ∈ Lp(Rn ) for which the norm

‖f‖B0 , vp , q (Rn ) := ‖f‖Lp (Rn ) +

∥∥∥t−1q v(t)ω1(f, t)p

∥∥∥q ,(0,∞)

(4.2)

is finite. Next we show that the Besov spaces B0,vp,q and B0,v

p,q coincide with Besov spaces introduced inDefinition 1.9.

Lemma 4.1 Let p ∈ [1,∞), q ∈ [1,∞] and let v ∈ SV(0,∞) be such that (1.5) and (4.1) hold. ThenB0,v

p,q = B0,vp,q .

P r o o f. First note that v ∈ SV(0,∞) implies v ∈ SV(0, 1). Denoting B = B0,vp,q and B = B0,v

p,q , we obtain

‖f‖B ≤ ‖f‖B = ‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(0,∞)

� ‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(0,1)

+∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(1,∞)

(4.3)

= ‖f‖B +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(1,∞)



for all f ∈ Lp . Since ω1(f, t)p ≤ 2‖f‖p for all f ∈ Lp and t > 0, and since (4.1) holds, we get∥∥∥t−1q v(t)ω1(f, t)p

∥∥∥q ,(1,∞)

≤ 2‖f‖p

∥∥∥t−1q v(t)

∥∥∥q ,(1,∞)

≈ ‖f‖p ≤ ‖f‖B

for all f ∈ Lp . Together with (4.3), the last estimate shows that ‖f‖B ≤ ‖f‖B � ‖f‖B , and the resultfollows.

Now we characterize these Besov spaces by interpolation.

Lemma 4.2 Let p ∈ [1,∞), q ∈ [1,∞], X0 = Lp(Rn ) and X1 = W 1,p(Rn ).

(i) Let w = v−1 , where v ∈ SV(0,∞) satisfies (1.5) and (4.1). Then

X := (X0 ,X1)w,q ;K = B = B0,vp,q (Rn ). (4.4)

(ii) Let

w(z) =

{v−1(z), z ∈ (0, 1),∞, z ∈ [1,∞),

where w ∈ SV(0, 1) satisfies (1.5) (with v replaced by w). Then

X := (X0 ,X1)w ,q ;K = B = B0,vp,q (Rn ). (4.5)

P r o o f. Putting K(t, f) = K(t, f ;X0 ,X1), we get

‖f‖X =

∥∥∥∥∥K(t, f)

w(t)t1q

∥∥∥∥∥q ,(0,∞)

≈∥∥∥∥∥K(t, f)

w(t)t1q

∥∥∥∥∥q ,(0,1)

+

∥∥∥∥∥K(t, f)

w(t)t1q

∥∥∥∥∥q ,(1,∞)

=: N1 + N2 . (4.6)

Since, by [2, Chapter 5, Theorem 4.12, p. 339],

K(t, f) ≈ min{1, t}‖f‖p + ω1(t, f)p (4.7)

for all f ∈ Lp and t ∈ (0,∞), we see that

N1 ≈∥∥∥t1−

1q v(t)

∥∥∥q ,(0,1)

‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(0,1)

≈ ‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(0,1)

(4.8)

for all f ∈ Lp(Rn ). Using (4.7) and (4.1), we arrive at

N2 ≈∥∥∥t−

1q v(t)

∥∥∥q ,(1,∞)

‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(1,∞)

≈ ‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(1,∞)

(4.9)

for all f ∈ Lp(Rn ).Combining (4.6), (4.8), (4.9) and using (4.2), we obtain

‖f‖X ≈ ‖f‖p +∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(0,1)

+∥∥∥t−

1q v(t)ω1(f, t)p

∥∥∥q ,(1,∞)

≈ ‖f‖B

for all f ∈ Lp(Rn ), and (4.4) is proved.To prove (4.5), note that the choice of w implies that, for all f ∈ Lp(Rn ),

‖f‖X =∥∥∥t−

1q v(t)K(t, f)

∥∥∥q ,(0,1)

= N1 .

Thus, the result follows on making use of (4.8) (and (1.6)).



Subsequently, given any N := (β0 , β∞) ∈ R2 we write

�N (t) :=

{�β0 (t), if 0 < t ≤ 1,

�β∞(t), if 1 < t < ∞.

Lemma 4.3 Let p ∈ [1, n), q ∈ [1,∞] and let v(t) = �N (t), t ∈ (0,∞), where N = (β0 , β∞) ∈ R2 is such

that

β0 +1q≥ 0 if q < ∞ and β0 > 0 if q = ∞, (4.10)

and

β∞ +1q

< 0 if q < ∞ and β∞ ≤ 0 if q = ∞. (4.11)

If Ω ⊂ Rn with Lebesgue measure |Ω| < ∞ and δ < 0, then

B0,vp,q (Rn )

∗↪→

(Lp(log L)δ (Ω), Lp∗,p(Ω)

)w,q ;K , (4.12)

where w = v−1 and 1p∗ = 1

p − 1n .

P r o o f. According to Theorems 1.5 and 1.6, the real interpolation method with the function parameter w isof type

ϕ(s, t) =

⎧⎪⎨⎪⎩

0 if s = 0 or t = 0,

s if 0 < t ≤ s,

s�β0 −β∞(t/s) if s < t < ∞.

Hence, condition (1.2) is satisfied and Theorem 1.2 holds for this method.We have Lp(Rn ) ↪→ Lp(log L)δ (Ω), by which we mean ‖u|Ω‖Lp (log L)δ (Ω) � ‖u‖p,Rn for all u ∈ Lp(Rn ).

In fact Lp(Rn )∗

↪→ Lp(log L)δ (Ω) because δ < 0. Moreover W 1,p(Rn ) ↪→ Lp∗,p(Ω). Therefore, interpolating bythe real method with the function parameter w and using Theorem 1.2 and Lemma 4.2, we derive that

B0,vp,q (Rn ) =

(Lp(Rn ),W 1,p(Rn )

)w,q ;K

∗↪→


)w,q ;K .

Next we consider the space Y introduce in the statement of Theorem 1.10. It is not hard to check that the normof Y is equivalent to

‖f‖Y :=∥∥∥t−

1q �N (t)

∥∥∥�δ (τ)f∗(τ)∥∥

p,(0,t)

∥∥q ,(0,∞) .

In what follows, we denote by Y the space Y with the norm ‖ · ‖Y . The following result is a direct consequenceof [9, Theorem 5.9 and Lemma 8.5].

Lemma 4.4 Let p ∈ [1, n), q ∈ [1,∞] and let v(t) = �N (t), t ∈ (0,∞), where N = (β0 , β∞) ∈ R2 is such

that

β0 +1q

> 0 and β∞ +1q

< 0. (4.13)

Put w = v−1 and 1p∗ = 1

p − 1n . If Ω ⊂ R

n , |Ω| < ∞ and δ < 0, then


)w,q ;K =

(Lp(log L)δ (Ω), L∞(Ω)

)w,q ;K = Y.



Remark 4.5 If one replaces condition (4.13) by (4.10) and (4.11) in Lemma 4.4, then it is possible to provethat (

Lp(log L)δ (Ω), Lp∗,p(Ω))w,q ;K ↪→ Y(

Lp(log L)δ (Ω), L∞(Ω))w,q ;K ↪→ Y

}. (4.14)

Indeed, this is a consequence of [9, Lemmas 5.1 and 8.5].

Lemma 4.6 Let p ∈ [1,∞), q ∈ [1,∞], δ < 0 and let v(t) = �N (t), t ∈ (0,∞), where N = (β0 , β∞) ∈ R2

satisfies (4.10) and (4.11). Let Ω ⊂ Rn with |Ω| < ∞. Then

B0,vp,q (Rn )

∗↪→ Y. (4.15)

P r o o f. If p ∈ [1, n), the result is a consequence of Lemmas 4.3 and 4.4 and Remark 4.5. If p ∈ (n,∞), thenW 1,p(Rn ) ↪→ L∞(Ω). Thus, instead of (4.12), we obtain

B0,vp,q (Rn )

∗↪→

(Lp(log L)δ (Ω), L∞(Ω)

)w,q ;K(

with w = v−1). Together with (4.14), this yields the result.

Finally, if p = n, then we have W 1,p(Rn ) ↪→ L∞,p(log L)−1(Ω) (cf. [4]). Thus, instead of (4.12), we get

B0,vp,q (Rn )

∗↪→

(Lp(log L)δ (Ω), L∞,p(log L)−1(Ω)

)w,q ;K(

with w = v−1). Since also (by [9, Lemmas 5.1 and 8.5])

(Lp(log L)δ (Ω), L∞,p(log L)−1(Ω)

)w,q ;K ↪→ Y, the

proof is finished.

P r o o f o f T h e o r e m 1.10. Let β∞ ∈ R be such that (4.11) holds. Put N = (β0 , β∞) and v(t) = �N (t),t ∈ (0,∞). It follows by [6, Lemma 3.1] and Lemma 4.1 that

B0,vp,q (Rn ) = B0,v

p,q (Rn ) ↪→↪→ Lp(Ω) ↪→ L1(Ω).

On the other hand, every sequence {fi} ⊂ L1(Ω), which converges to f in L1(Ω) also converges to f in measure

on Ω. Since, by Lemma 4.6, B0,vp,q (Rn )

∗↪→ Y = Y , the result follows.

Remark 4.7 (i) Note that Theorem 1.10 remains true when Ω is a bounded Lipschitz domain in Rn and (1.7)

is replaced by B0,vp,q (Ω) ↪→↪→ Y .

(ii) Under the assumptions of Theorem 1.10, one can show (by means of [11, Theorem 4.2]) that

Y ↪→ Lp,q (log L)β+δ+ 1max{p , q } (Ω).

Consequently,

B0,vp,q (Ω) ↪→↪→ Lp,q (log L)β+δ+ 1

max{p , q } (Ω). (4.16)

Note that such a result follows also from [6, Theorem 1.3]. On the other hand, (1.7) is a better result than (4.16).

Acknowledgements This research was supported in part by Spanish Ministerio de Economıa y Competitividad (MTM2010-15814), UCM-BSCH GR35/10-A (Grupo de Investigacion 910348) and grants no. 201/07/0388 and 201/08/0383 of the GrantAgency of the Czech Republic. We thank the referees for their valuable comments to the paper.

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