This paper introduces localized John--Nirenberg--Campanato spaces and their preduals, establishing new relationships and invariance properties, thereby extending the theory of localized function spaces.
Contribution
The authors define new localized function spaces, connect them via duality, and prove invariance properties, advancing the understanding of localized harmonic analysis.
Findings
01
Defined localized John--Nirenberg--Campanato spaces.
02
Established the predual relationship with localized Hardy-kind spaces.
03
Proved invariance of the Hardy-kind space under certain parameter conditions.
Abstract
Let p∈(1,∞), q∈[1,∞), s∈Z+, α∈[0,∞) and X be Rn or a cube Q0⫋Rn. In this article, the authors first introduce the localized John--Nirenberg--Campanato space jn(p,q,s)α(X) and show that the localized Campanato space is the limit case of jn(p,q,s)α(X) as p→∞. By means of local atoms and the weak-∗ topology, the authors then introduce the localized Hardy-kind space hk(p′,q′,s)α(X) which proves the predual space of jn(p,q,s)α(X). Moreover, the authors prove that hk(p′,q′,s)α(X) is invariant when 1<q<p, where p′ or q′ denotes the conjugate number of p or q, respectively. All these results are new even for the localized John--Nirenberg space.
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Harmonic Analysis Research · Advanced Mathematical Modeling in Engineering · Advanced Banach Space Theory
Full text
Localized John–Nirenberg–Campanato Spaces
00footnotetext: 2010 Mathematics Subject Classification. Primary 42B35; Secondary 42B30, 42B25, 46E35. Key words and phrases. cube, Euclidean space,
localized John–Nirenberg–Campanato space, Hardy-kind space, local atom, duality.
This project is supported by the National
Natural Science Foundation of China (Grant Nos. 11571039, 11761131002 and 11671185).
Jingsong Sun, Guangheng Xie and Dachun Yang 111Corresponding author/June 03, 2019/Final version.
Abstract Let p∈(1,∞), q∈[1,∞), s∈Z+, α∈[0,∞)
and X be Rn or a cube Q0⫋Rn.
In this article, the authors first introduce the localized John–Nirenberg–Campanato
space jn(p,q,s)α(X)
and show that the localized Campanato space is the limit case of jn(p,q,s)α(X) as p→∞.
By means of local atoms and the weak-∗ topology,
the authors then introduce the localized Hardy-kind space
hk(p′,q′,s)α(X) which proves the predual space of
jn(p,q,s)α(X). Moreover, the authors prove that hk(p′,q′,s)α(X)
is invariant when 1<q<p, where p′ or q′ denotes the conjugate number of p or q, respectively.
All these results are new even for the localized John–Nirenberg space.
1 Introduction
Apart from the classical missingBMO space (the space of functions with bounded mean oscillation),
John and Nirenberg [14] also introduced a class of larger spaces,
which are now called the John–Nirenberg spaces JNp with p∈(1,∞).
The missingBMO space is closely related to the JNp spaces.
Particularly, for any cube Q0⫋Rn, missingBMO(Q0) is just the
limit case of JNp(Q0) as p→∞; see, for instance, [5, 3, 22].
Although JNp spaces have not been studied as systematically as the
missingBMO space, JNp spaces and their variants still attract much attention.
For instance, Campanato [5] used the embedding of JNp into weak Lp
to prove the Stampacchia interpolation theorem;
Aalton et al. [1] introduced the notion of JNp on the doubling metric space and showed
the corresponding John–Nirenberg inequality;
Hurri-Syrjänen et al. [13] and Marola and Saari [18]
established Reimann–Rychener local-to-global results for
JNp in the setting of Rn or metric measure spaces, respectively;
Berkovits et al. showed in [2] that JNp embeds into weak Lp
both in Euclidean spaces with dyadic cubes and in spaces of homogeneous type with metric balls;
Dafni et al. [9] proved Lp⫋JNp and introduced
a Hardy-kind space which further proves the predual space of JNp.
It is well known that Fefferman and Stein [11] showed
that the dual of the Hardy space H1(Rn) is the space missingBMO(Rn).
Later, Coifman and Weiss [8]
gave a more generalized result via proving that, for any given p∈(0,1],
the dual of the Hardy space Hp(Rn) is the Campanato space
Cp1−1,1,⌊n(p1−1)⌋(Rn) introduced in [4],
where ⌊n(p1−1)⌋ denotes the largest integer not greater than
n(p1−1). Notice that C0,1,0(Rn) coincides with missingBMO(Rn).
Very recently,
Tao et al. [22] introduced the John–Nirenberg–Campanato space, which is a generalization of
the classical John–Nirenberg space and is also closely related to the Campanato space.
In the same article, Tao et al. also found the predual space of
the John–Nirenberg–Campanato space and showed the corresponding John–Nirenberg type inequality.
On the other hand, the localized missingBMO(Rn) space, denoted by bmo(Rn),
was originally introduced by Goldberg [12].
In the same article, Goldberg also introduced the localized Campanato space
Λα(Rn) with α∈(0,∞),
which proves the dual space of the local Hardy space. Later,
Jonsson et al. [15] constructed the local Hardy space and the localized
Campanato space on the subset of Rn;
Chang [6] studied the localized Campanato space on bounded Lipschitz domains;
Chang et al. [7]
studied the local Hardy space and its dual space on smooth domains as well as their applications to
boundary value problems.
For more articles concerning localized missingBMO or Campanato spaces or their variants,
we refer the reader to, for instance, [19, 24, 25, 23, 10].
However, a theory on localized John–Nirenberg–Campanato spaces, even on localized
John–Nirenberg spaces, is still missing.
Let p∈(1,∞), q∈[1,∞), s∈Z+, α∈[0,∞)
and X be Rn or a cube Q0⫋Rn.
In this article, we first introduce the localized John–Nirenberg–Campanato
space jn(p,q,s)α(X)
and show that the localized Campanato space is the limit case of jn(p,q,s)α(X) as p→∞.
By means of local atoms and the weak-∗ topology,
we then introduce the localized Hardy-kind space
hk(p′,q′,s)α(X) which proves the predual space of
jn(p,q,s)α(X). Moreover, we prove that hk(p′,q′,s)α(X)
is invariant when 1<q<p, where p′ or q′ denotes the conjugate number of p or q, respectively.
All these results are new even for the localized John–Nirenberg space.
To be precise, this article is organized as follows.
In Section 2, we first introduce the notion of
the localized John–Nirenberg–Campanato space jn(p,q,s)α(X) with admissible (p,q,s,α),
which is a class of newly-defined spaces even for the special case, the localized John–Nirenberg spaces;
see Definition 2.3 below. Then we establish the relationships between jn(p,q,s)α(X)
and the John–Nirenberg–Campanato space JN(p,q,s)α(X) from [22]
(see Propositions 2.10 and 2.11 below).
Concretely, via the dyadic subcubes and some ideas from the proofs of [15, Theorem 4.1],
we prove that jn(p,q,s)α(X)=JN(p,q,s)α(X)∩Lp(X) with equivalent norms,
where p∈(1,∞), q∈[1,p], s∈Z+ and α∈(0,∞).
Moreover, we also show that the localized Campanato space is the
limit case of jn(p,q,s)α(X) as p→∞;
see Propositions 2.13 and 2.14 below.
In Section 3,
by the John–Nirenberg lemma for JN(p,q,s)α(X) in [22, Proposition 1.19]
(or, see Lemma 3.2 below) and
the continuous embedding jn(p,q,s)α(X)⊂JN(p,q,s)α(X)
(see Proposition 2.10 below),
we first show that jn(p,q,s)α(X) is invariant
on q∈[1,p) with admissible (p,q,s,α); see Proposition 3.1 below.
Via selecting appropriated cubes,
we then establish the relationship between jn(p,q,s)α(X) and Lebesgue spaces;
see Proposition 3.4 below.
Section 4 is aimed at constructing the predual space of jn(p,q,s)α(X)
with p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞).
For this purpose, using the local atoms and the weak-∗ topology,
we introduce the localized Hardy-kind space hk(p′,q′,s)α(X);
see Definition 4.4 below.
Then, via making full use of “local” property and
borrowing some ideas from the proofs of [9, Theorem 6.6] and [22, Theorem 1.16],
we prove that hk(p′,q′,s)α(X) is the
predual space of jn(p,q,s)α(X);
see Theorem 4.11 below.
Remarkably, differently from the Lp-convergence which was used by
Dafni et al. [9] to introduce the predual space of the John–Nirenberg space,
we use the weak-∗ convergence on (jn(p,q,s)α(X))∗ to introduce hk(p′,q′,s)α(X).
This allows us to exchange the order of the integration and the sum of the sequence of constant multiples
of local atoms in the proof of the duality theorem; see Remarks
4.3 and 4.6 below.
We point out that, for any given p∈(1,∞), q∈[1,p) and cube Q0⫋Rn,
hk(p′,q′,0)0(Q0) is equivalent to a new localized Hardy-kind space
hkp′,q′(Q0) which is defined by the same way as that used in
[9, Definition 6.1]; see Proposition 4.14 below.
In Section 5,
via decomposing the local w-atom, with w∈(1,∞), into the sum of the sequence of
scalar multiples of local ∞-atoms and a polynomial
in the sense of weak-∗ topology, and
some arguments similar to those used in the proof of [9, Proposition 6.4]
(see also [22, Proposition 1.23]),
we show that, for appropriate indices v, s and α,
hk(v,w,s)α(X)
is invariant on w∈(v,∞];
see Proposition 5.1 below.
As a counterpart of Proposition 3.4,
we establish the relation between
localized Hardy-kind spaces and Lebesgue spaces;
see Proposition 5.6 below.
For any v∈(1,∞), w∈(1,∞] and cube Q0⫋Rn,
we then establish the
relation between hk(v,w,0)0(Q0) and the localized Hardy space h1(Q0);
see Proposition 5.7 below.
Finally, we state some conventions on notation. We always let
N:={1,2,3,…} and Z+:=N∪{0}.
The symbolC always denotes a positive constant independent
of the main parameters but may vary from line to line.
Constants with subscripts, such as c0 and C(s), are invariant in different occurrences.
If f≤Cg, we then write f≲g or g≳f and, if f≲g≲f, we then write
f∼g. We also use the following
convention: If f≤Cg and g=h or g≤h, we then write f≲g∼h
or f≲g≲h, rather thanf≲g=h
or f≲g≤h. For normed spaces X1 and X2,
the symbolX1⊂X2 means that
there exists a positive constant C such that,
for any f∈X1, f∈X2 and ∥f∥X2≤C∥f∥X1.
For any set E⊂Rn, the symbol 1E denotes its characteristic function
and the symbol ∣E∣ its Lebesgue measure. For any cube Q, we use the symbol
ℓ(Q) to denote its side length. We also let ℓ(Rn):=∞.
For any set M, the symbol #M represents its cardinality.
Also, for any p∈[1,∞], let p′ be the conjugate index of p, namely, p1+p′1=1.
For any a∈R, the symbol⌊a⌋ denotes the largest integer not greater than a.
2 Localized John–Nirenberg–Campanato spaces
In this section, we first introduce the localized John–Nirenberg–Campanato
space and then establish the relations among
the localized John–Nirenberg–Campanato space, the John–Nirenberg–Campanato
space and the localized Campanato space.
We first introduce some symbols.
Throughout the article, the symbol X always denotes Rn or a cube Q0⫋Rn.
In what follows, for any given p∈[1,∞), the space Lp(X) is defined to be the set of
all measurable functions f such that
∥f∥Lp(X):=(∫X∣f(x)∣pdx)p1<∞
and the symbol Lmissinglocp(X) denotes the collection of all measurable functions f such that
∥f1E∥Lp(X)<∞ for any bounded set E⊂X.
The symbolL∞(X) denotes the set of all measurable functions f
such that ∥f∥L∞(X)<∞,
where the norm∥f∥L∞(X) denotes the essential supremum of f on X.
Let s∈Z+. In what follows, we use the symbolPs(X) to denote the set of
all polynomials of degree not greater than s on X and the symbolQ a cube of Rn with finite length,
but, not necessary to be closed.
For any integrable function f on a cube Q⊂X, let
[TABLE]
here and hereafter, in all integral representations, if there exists no confusion,
we omit the differential dx.
Moreover, for any s∈Z+, the symbol PQ(s)(f) denotes
a unique polynomial from Ps(Q) such that
[TABLE]
where β:=(β1,…,βn)∈Z+n and ∣β∣:=∑i=1nβi. Furthermore, it holds true that
[TABLE]
where the constant C(s)∈[1,∞) only depends on s. For more details on PQ(s)(f), see, for instance, [16, 17, 21].
Clearly, if s=0, then PQ(s)(f)=fQ. For any c0∈(0,ℓ(X)), let
[TABLE]
Now, we recall the definition of the localized Campanato space, which was first introduced by Goldberg in [12, Theorem 5].
Definition 2.1**.**
Let q∈[1,∞), s∈Z+ and α∈[0,∞). Fix c0∈(0,ℓ(X)).
The localized Campanato spaceΛ(α,q,s)(X) is defined to be the set of all measurable functions
f∈Lmissinglocq(X) such that
[TABLE]
where the supremum is taken over all cubes Q in X.
Remark 2.2**.**
(i)
If X:=Rn, q=1, s=0 , α=0 and c0=1,
then Λ(α,q,s)(X) is just the local version of missingBMO(Rn), bmo(Rn), in
Goldberg [12]. We also write bmo(X):=Λ(0,1,0)(X).
(ii)
In Definition 2.1, if PQ,c0(s)(f) is replaced by PQ(s)(f),
then Λ(α,q,s)(X) becomes the Campanato space
C(α,q,s)(X), which was first introduced in [4].
In what follows, we fix the constant c0∈(0,ℓ(X)).
Now, we introduce the localized John–Nirenberg–Campanato space.
Definition 2.3**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞).
Fix the constant c0∈(0,ℓ(X)).
The localized John–Nirenberg–Campanato spacejn(p,q,s)α,c0(X) is defined to be the set of all functions f∈Lmissinglocq(X) such that
[TABLE]
where the supremum is taken over all collections of interior pairwise disjoint cubes
{Qj}j∈N in X.
Remark 2.4**.**
In Definition 2.3,
if PQj,c0(s)(f) is replaced by PQj(s)(f),
then we obtain the John–Nirenberg–Campanato spaceJN(p,q,s)α(X), which was originally introduced in [22, Definition 1.2].
Let JNp(X):=JN(p,1,0)0(X). If Q0⫋Rn is a cube,
JNp(Q0) is just the classical John–Nirenberg space, which originated from [14].
Now, we show that jn(p,q,s)α,c0(X) in Definition 2.3
is independent of the choice of the positive constant c0.
Proposition 2.5**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+, α∈[0,∞), c1∈(0,ℓ(X))
and c2∈(c1,ℓ(X)). Then
jn(p,q,s)α,c1(X)=jn(p,q,s)α,c2(X) with equivalent norms.
Proof.
Let p,q,s,α,c1 and c2 be as in this proposition.
Let {Qj}j∈N be any interior pairwise disjoint cubes in X and
[TABLE]
We first prove jn(p,q,s)α,c1(X)⊂jn(p,q,s)α,c2(X).
Let f∈jn(p,q,s)α,c1(X). For any j∈J, by the definition of PQj,c0(s)(f), we have
[TABLE]
From this, the Minkowski inequality, (2.1) and the Hölder inequality, it follows that, for any j∈J,
[TABLE]
Moreover, for any j∈N∖J, we have PQj,c2(s)(f)=PQj,c1(s)(f),
which, together with (2.2), implies that, for any j∈N,
[TABLE]
From this, the arbitrariness of {Qj}j∈N and Definition 2.3, it follows that
[TABLE]
This proves jn(p,q,s)α,c1(X)⊂jn(p,q,s)α,c2(X).
Next, we show jn(p,q,s)α,c2(X)⊂jn(p,q,s)α,c1(X).
Let f∈jn(p,q,s)α,c2(X).
By the definition of J, the Minkowski inequality and Definition 2.3, we have
[TABLE]
Now, we estimate I1.
If X=Rn, let l1:=c2 and if X⫋Rn is a cube,
let l1:=ℓ(X)(⌊c2ℓ(X)⌋)−1.
Hence, l1∈[c2,2c2).
Choose interior pairwise disjoint cubes {Ri}i∈N in X such that
ℓ(Ri)=l1 for any i∈N and X=⋃i∈NRi.
For any j∈J, let
Rj:={Ri:Ri∩Qj=∅}.
Then Mj:=#Rj≤2n. Rewrite Rj as {Rj,k}k=1Mj
and let Rj,k:=∅ for any integer k∈(Mj,2n].
For any i∈N, let
[TABLE]
Then #Qi≤(c1l1+2)n≤(c12c2+2)n.
From this and the Minkowski inequality, we deduce that
[TABLE]
Combining this, (2.3) and the arbitrariness of {Qj}j∈N, we have f∈jn(p,q,s)α,c1(X) and
[TABLE]
Thus,
jn(p,q,s)α,c2(X)⊂jn(p,q,s)α,c1(X).
This finishes the proof of Proposition 2.5.
∎
Remark 2.6**.**
Based on Proposition 2.5, in what follows, we write jn(p,q,s)α(X):=jn(p,q,s)α,c0(X).
Especially, if q=1, s=0 and α=0, then jn(p,q,s)α(X) becomes the localized John–Nirenberg spacejnp(X):=jn(p,1,0)0(X), which is also a new space.
The following proposition indicates that the localized John–Nirenberg–Campanato space is a Banach space.
Proposition 2.7**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞).
Then jn(p,q,s)α(X) is a Banach space.
Proof.
Let p, q, s and α be as in this proposition and the constant c0∈(0,ℓ(X)).
It is easy to show that jn(p,q,s)α(X)
is a normed space. Then we only need to prove that jn(p,q,s)α(X) is complete.
Let {fk}k=1∞⊂jn(p,q,s)α(X) and ∑k=1∞∥fk∥jn(p,q,s)α(X)<∞.
Now, we claim that there exists a measurable function f on X such that
[TABLE]
Indeed, if X is a cube Q0⫋Rn, by the Minkowski inequality, we have
[TABLE]
Thus, (∑k=1∞∣fk∣)q is integrable on Q0
and hence ∑k=1∞∣fk∣ is finite almost everywhere on Q0.
Letting f:=∑k=1∞fk,
then (2.4) holds true when X=Q0.
If X=Rn,
choose interior pairwise disjoint cubes {Ri}i∈N
such that Rn=⋃i∈NRi and ℓ(Ri)∈[c0,∞).
For any i∈N, since (2.4) holds true when X=Ri,
we deduce that there exists a function gi on Ri such that
gi=∑k=1∞fk1Ri almost everywhere. Let f:=∑i∈Ngi.
Then f=∑k=1∞fk almost everywhere and hence (2.4) also
holds true when X=Rn. This proves the above claim.
Now, we show that f∈jn(p,q,s)α(X) and ∥f−∑k=1Nfk∥jn(p,q,s)α(X)→0 as N→∞.
To this end, let {Qj}j∈N be interior pairwise disjoint cubes in X. For any Qj,
there exists a cube Qj
such that Qj⊂Qj⊂X and ℓ(Qj)∈[c0,ℓ(X)).
For any N∈N, by (2.1), the Hölder inequality and Definition 2.3, we have
[TABLE]
which implies that ∑k=1∞[∣PQj(s)(fk)∣+∣fk∣] is integrable on Qj.
From this and the dominated convergence theorem,
we deduce that, for any N∈N, β∈Z+n and ∣β∣≤s,
[TABLE]
Thus, PQj(s)(∑k=N∞fk)=∑k=N∞PQj(s)(fk).
Combining this, the Minkowski inequality and Definition 2.3, we find that
[TABLE]
Therefore, ∥∑k=N∞fk∥jn(p,q,s)α(X)≤∑k=N∞∥fk∥jn(p,q,s)α(X).
From this, (2.4) and ∑k=1∞∥fk∥jn(p,q,s)α(X)<∞, we deduce that
f∈jn(p,q,s)α(X) and
Let p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞).
Next, we consider the relations between the localized John–Nirenberg–Campanato space jn(p,q,s)α(X) and
the John–Nirenberg–Campanato space JN(p,q,s)α(X).
To do this, we first need to recall the notion of JN(p,q,s)α(X)
from [22, Definition 1.2] as follows.
Definition 2.8**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞).
The John–Nirenberg–Campanato spaceJN(p,q,s)α(X) is defined to be the set of all functions f∈Lmissinglocq(X) such that
[TABLE]
where the supremum is taken over all collections of interior pairwise disjoint cubes
{Qj}j∈N in X.
To achieve our target, we also need the following technical lemma.
Lemma 2.9**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+, α∈[0,∞) and Q0⫋Rn be a cube.
Then there exists a positive constant C such that, for any a∈Ps(Q0),
[TABLE]
Proof.
Let p, q, s and α be as in this lemma and a∈Ps(Q0). From Definition 2.3,
it follows that ∥a∥Lq(Q0)≤∣Q0∣α+q1−p1∥a∥jn(p,q,s)α(Q0).
We then only need to show ∥a∥jn(p,q,s)α(Q0)≲∥a∥Lq(Q0).
Let {Qj}j∈N be any interior pairwise disjoint cubes in Q0 and
J:={j∈N:ℓ(Qj)≥c0}, here and hereafter, c0∈(0,ℓ(Q0)).
Observe that, for any j∈N, PQj(s)(a)=a.
By this and the definitions of PQj,c0(s)(a) and J, we know that
[TABLE]
which, combined with Definition 2.3, implies that ∥a∥jn(p,q,s)α(Q0)≲∥a∥Lq(Q0).
This finishes the proof of Lemma 2.9.
∎
From Lemma 2.9, we deduce that
Ps(Q0) is a subspace of jn(p,q,s)α(Q0).
In what follows, the space jn(p,q,s)α(Q0)/Ps(Q0) is defined by setting
[TABLE]
where ∥f∥jn(p,q,s)α(Q0)/Ps(Q0):=infa∈Ps(Q0)∥f+a∥jn(p,q,s)α(Q0).
Proposition 2.10**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞). Then
(i)
jn(p,q,s)α(X)⊂JN(p,q,s)α(X);
(ii)
if Q0⫋Rn is a cube, then
JN(p,q,s)α(Q0)=jn(p,q,s)α(Q0)/Ps(Q0) with equivalent norms;
(iii)
Lp(R)⫋jnp(R)⫋JNp(R).
Proof.
We first prove (i). Let f∈jn(p,q,s)α(X)
and {Qj}j∈N be interior pairwise disjoint cubes in X.
From (2.1), the definition of PQj,c0(s)(f) and the Hölder inequality, it follows that
[TABLE]
By this and the arbitrariness of {Qj}j∈N, we have
∥f∥JN(p,q,s)α(X)≲∥f∥jn(p,q,s)α(X).
This proves (i).
For (ii), let f∈jn(p,q,s)α(Q0)/Ps(Q0).
For any a∈Ps(Q0),
by Definition 2.8 and (i),
we find that
[TABLE]
which implies that f∈JN(p,q,s)α(Q0) and
∥f∥JN(p,q,s)α(Q0)≲∥f∥jn(p,q,s)α(Q0)/Ps(Q0).
Thus,
[TABLE]
Next, we prove JN(p,q,s)α(Q0)⊂jn(p,q,s)α(Q0)/Ps(Q0).
Let f∈JN(p,q,s)α(Q0), g:=f−PQ0(s)(f) and
{Qj}j∈N be interior mutually disjoint cubes in Q0.
Let J:={j∈N:ℓ(Qj)≥c0}. Then #J≤c0n∣Q0∣.
From this, the Minkowski inequality, it follows that
[TABLE]
Combining this and the arbitrariness of {Qj}j∈N, we conclude that
[TABLE]
Therefore, f∈jn(p,q,s)α(Q0)/Ps(Q0) and
hence JN(p,q,s)α(Q0)⊂jn(p,q,s)α(Q0)/Ps(Q0).
This proves (ii).
Finally, we prove (iii). Let a∈R be any non-zero constant. Clearly, ∥a∥JNp(R)=0.
For any N∈[c0,∞), let IN:=[−N,N]. From the definition of jnp(R), we deduce that
[TABLE]
Thus, a∈JNp(R)∖jnp(R). Combining this and (i), we obtain jnp(R)⫋JNp(R).
Now, we show Lp(R)⫋jnp(R).
Let f∈Lp(R). By the Hölder inequality, we have
[TABLE]
where the supremum is taken over all collections of interior pairwise disjoint intervals {Ij}j∈N in R.
Thus, Lp(R)⊂jnp(R).
Then we only need to find a function which belongs to jnp(R)∖Lp(R).
Recall that Dafni et al. [9, Proposition 3.2] constructed a function g∈JNp(R)∖Lp(R) and
they also showed that g∈L1(R) in [9, Lemma 3.4].
Let {Ij}j∈N be interior mutually disjoint intervals in R
and J:={j∈N:ℓ(Ij)≥c0}. Then we have
[TABLE]
which further implies that ∥g∥jnp(R)≲∥g∥JNp(R)+∥g∥L1(R).
Thus, we have g∈jnp(R)∖Lp(R). This finishes the proof of (iii) and hence of Proposition 2.10.
∎
In what follows, for any normed spaces X1 and X2,
the space X1∩X2 denotes the intersection X1∩X2
equipped with the norm
[TABLE]
Proposition 2.11**.**
Let p∈(1,∞), q∈[1,p], s∈Z+ and α∈(0,∞). Then
jn(p,q,s)α(X)=JN(p,q,s)α(X)∩Lp(X).
To prove this proposition, we need the following lemma which can be found in [15, Theorem 1.1].
Lemma 2.12**.**
Let q∈[1,∞), s∈Z+, Q⫋Rn be a cube and P∈Ps(Q). Then
[TABLE]
where the positive constant C(s,n) depends only on s and the dimension n.
Let p,q,s,α be as in this proposition and c0∈(0,ℓ(X)). We first show
JN(p,q,s)α(X)∩Lp(X)⊂jn(p,q,s)α(X). Let
f∈JN(p,q,s)α(X)∩Lp(X),
{Qj}j∈N be interior pairwise disjoint cubes in X
and J:={j∈N:ℓ(Qj)≥c0}. By this, the definition of PQj,c0(s)(f) and
the Hölder inequality, we have
[TABLE]
which implies that f∈jn(p,q,s)α(X) and
∥f∥jn(p,q,s)α(X)≲max{∥f∥JN(p,q,s)α(X),∥f∥Lp(X)}. This proves
JN(p,q,s)α(X)∩Lp(X)⊂jn(p,q,s)α(X).
Now, we show
[TABLE]
Since jn(p,q,s)α(X)⊂JN(p,q,s)α(X) [see Proposition 2.10(i)],
it follows that
we only need to show
jn(p,q,s)α(X)⊂Lp(X).
Let f∈jn(p,q,s)α(X).
First we assume that X=Rn and c0=1.
For any k∈Z+, let
Dk:={2−k[(0,1]k+l]:l∈Zn}
be the collection of all dyadic subcubes with length 2−k of Rn.
Then rewrite Dk as {Qj(k)}j∈N.
Clearly, for any l,k∈Z+ and l≤k, there exists a map
ϕk,l:N→N such that Qj(k)⊂Qϕk,l(j)(l)
for any j∈N.
From the Hölder inequality, ∣Qj(k)∣=2−nk
and Definition 2.3,
we deduce that, for any k∈N,
[TABLE]
which, combined with
Qj(k)⊂Qϕk,k−1(j)(k−1), implies that
[TABLE]
By the Minkowski inequality, (2.6) and (2.7), we have, for any k∈N,
[TABLE]
From Lemma 2.12, we deduce that,
for any k,l,j∈N, l≤k,
P∈Ps(Rn) and Qϕk,l(j)(l)⊃Qj(k),
[TABLE]
which, together with (2.8) and some arguments similar to those used in the proof of (2.7), implies that
[TABLE]
where C(s,n) denotes a positive constant depending on s and n.
By this, the Minkowski inequality and (2.6),
we conclude that, for any k∈Z+,
[TABLE]
where the first equality holds true because, for any j∈N,
PQϕk,0(j)(0),1(s)(f)=0.
From this, the Lebesgue differential theorem and the Fatou lemma, it follows that
[TABLE]
Combining this and Proposition 2.5, we obtain jn(p,q,s)α(Rn)⊂Lp(Rn).
If X is a cube Q0⫋Rn, the proof of jn(p,q,s)α(Q0)⊂Lp(Q0) is similar to
the proof of jn(p,q,s)α(Rn)⊂Lp(Rn) and the details are omitted. Therefore,
jn(p,q,s)α(X)⊂jn(p,q,s)α(X)∩Lp(X). This finishes the proof of Proposition 2.11.
∎
The following two propositions show that the localized Campanato space is the limit of the localized John–Nirenberg–Campanato space.
Proposition 2.13**.**
Let p∈(1,∞), q∈[1,∞), s∈Z+, α∈[0,∞) and Q0⫋Rn be a cube. Then,
for any f∈L1(Q0),
[TABLE]
Moreover,
[TABLE]
Proof.
Let p,q,s,α and Q0 be as in this proposition and
c0∈(0,ℓ(Q0)).
Let f∈L1(Q0).
We prove this proposition by two cases.
Case 1) ∥f∥Λ(α,q,s)(Q0)=∞.
For any N∈(0,∞), by Definition 2.1,
we know that there exists a cube QN⊂Q0 such that
[TABLE]
From this, it follows that
[TABLE]
which implies that limp→∞∥f∥jn(p,q,s)α(Q0)=∞.
Thus, in this case,
[TABLE]
Case 2) ∥f∥Λ(α,q,s)(Q0)<∞.
By Definitions 2.1 and 2.3, we know that
[TABLE]
where the supremum is taken over all collections of interior pairwise disjoint cubes {Qj}j∈N in Q0.
Thus, we have f∈jn(p,q,s)α(Q0),
which further implies that
[TABLE]
and
[TABLE]
On the other hand,
from Definition 2.1, we deduce that,
for any ϵ∈(0,∥f∥Λ(α,q,s)(Q0)),
there exists a cube Qϵ such that
Letting p→∞ and ϵ→∥f∥Λ(α,q,s)(Q0), we have
liminfp→∞∥f∥jn(p,q,s)α(Q0)≥∥f∥Λ(α,q,s)(Q0).
By this and (2.10), we obtain
limp→∞∥f∥jn(p,q,s)α(Q0)=∥f∥Λ(α,q,s)(Q0).
From this and (2.9), we further deduce that
Let p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞). Let f∈jn(p,q,s)α(Rn)∩Λ(α,q,s)(Rn).
Then f∈⋂r∈(p,∞)jn(r,q,s)α(Rn) and
[TABLE]
Proof.
Let p, q, s and α be as in this proposition,
c0∈(0,∞) and
f∈jn(p,q,s)α(Rn)∩Λ(α,q,s)(Rn).
For any r∈(p,∞), by Definitions 2.1 and 2.3,
we have
[TABLE]
where the supremum is taken over all collections of interior mutually disjoint cubes {Qj}j∈N in Rn.
Thus, we obtain f∈⋂r∈(p,∞)jn(r,q,s)α(Rn) and, for any r∈(p,∞),
[TABLE]
Letting r→∞, we obtain
limsupr→∞∥f∥jn(r,q,s)α(Rn)≤∥f∥Λ(α,q,s)(Rn).
On the other hand, from some similar arguments to those used in the proof of Proposition 2.13,
we deduce that
[TABLE]
Therefore, ∥f∥Λ(α,q,s)(Rn)=limr→∞∥f∥jn(r,q,s)α(Rn).
This finishes the proof of Proposition 2.14.
∎
Remark 2.15**.**
By Propositions 2.13 and 2.14, we obtain the relations between the localized John–Nirenberg
spaces and the local missingBMO space. Indeed,
if p∈(1,∞) and Q0⫋Rn is a cube, we then have
[TABLE]
if p∈(1,∞) and f∈jnp(Rn)∩bmo(Rn),
then f∈⋂r∈(p,∞)jnr(Rn) and
[TABLE]
Remark 2.16**.**
Recall that the limit case of the John–Nirenberg–Campanato space JN(p,q,s)α(X) or Lp(X)
is the Campanato space C(α,q,s)(X) [see Remark 2.2(ii) for its definition] or L∞(X), respectively;
see, for instance, [22, Propositon 1.5 and Remark 1.6].
From this, Propositions 2.13, 2.14 and 2.11,
we deduce that, for any α∈(0,∞), q∈[1,∞) and s∈Z+,
In this section, we consider the invariance of jn(p,q,s)α(X)
on its indices in the appropriate range.
We first show that, for any p∈(1,∞), s∈Z+ and α∈[0,∞),
jn(p,q,s)α(X) is invariant on q∈[1,p).
Proposition 3.1**.**
Let p∈(1,∞), q∈[1,p), s∈Z+ and α∈[0,∞). Then
jn(p,q,s)α(X)=jn(p,1,s)α(X) with equivalent norms.
To show Proposition 3.1, we need to
use the following John–Nirenberg lemma on JN(p,q,s)α(X),
which is just [22, Proposition 1.19].
Lemma 3.2**.**
Let p∈(1,∞), q∈[1,p), s∈Z+ and α∈[0,∞). Then
JN(p,q,s)α(X)=JN(p,1,s)α(X) with equivalent norms.
Let 1≤q<p<∞, s∈Z+, α∈[0,∞) and c0∈(0,ℓ(X)).
The continuous embedding jn(p,q,s)α(X)⊂jn(p,1,s)α(X)
follows immediately from the Hölder inequality.
Thus, we only need to prove jn(p,1,s)α(X)⊂jn(p,q,s)α(X).
By Lemma 3.2, we know that JN(p,q,s)α(X)=JN(p,1,s)α(X) with equivalent norms.
Combining this and Proposition 2.10(i), we find that,
for any f∈jn(p,1,s)α(X),
[TABLE]
Let {Qj}j∈N be interior pairwise disjoint cubes in X and J:={j∈N:ℓ(Qj)≥c0}.
From the Minkowski inequality, (2.1) and (3.1),
we deduce that, for any f∈jn(p,1,s)α(X),
[TABLE]
which further implies that f∈jn(p,q,s)α(X) and
∥f∥jn(p,q,s)α(X)≲∥f∥jn(p,1,s)α(X).
Thus, jn(p,1,s)α(X)⊂jn(p,q,s)α(X),
which completes the proof of Proposition 3.1.
∎
Remark 3.3**.**
Let s∈Z+, α∈[0,∞) and Q0⫋Rn be a cube.
(i)
If 1<p1<p2<∞ and q∈[1,∞), then, from the Hölder inequality, it follows that jn(p2,q,s)α(Q0)⊂jn(p1,q,s)α(Q0).
(ii)
Recall that the generalized John–Nirenberg inequality [22, Theorem 1.21] states that, for any p∈(0,∞)
and f∈JN(p,1,s)α(Q0), there exists a positive constant C, depending only on n,p and s, such that
[TABLE]
Using this and Proposition 2.10(i), we conclude that the above
John–Nirenberg inequality remains valid when JN(p,1,s)α(Q0)
is replaced by jn(p,1,s)α(Q0).
Now, we discuss the relationship between jn(p,q,s)α(X) and the Lebesgue space.
In what follows, for any given nonnegative constant λ and normed space (X,∥⋅∥X),
the new normed space(λX,∥⋅∥λX) is defined by setting
λX:=X and ∥⋅∥λX:=λ∥⋅∥X.
Proposition 3.4**.**
Let s∈Z+ and Q0⫋Rn be a cube.
(i)
If 1<p≤q<∞, then
∣Q0∣q1−p1jn(p,q,s)0(Q0)=Lq(Q0)
with equivalent norms.
(ii)
If p∈(1,∞), then jn(p,p,s)0(Rn)=Lp(Rn) with equivalent norms.
(iii)
If 1<p<q<∞, α∈[0,p1−q1) and f∈jn(p,q,s)α(Rn), then f=0 almost everywhere.
Proof.
We first show (i). Let 1<p≤q<∞.
For any f∈jn(p,q,s)0(Q0), by Definition 2.3, we have
∥f∥Lq(Q0)≤∣Q0∣q1−p1∥f∥jn(p,q,s)0(Q0).
Thus, we obtain
[TABLE]
Now, we show Lq(Q0)⊂∣Q0∣q1−p1jn(p,q,s)0(Q0).
Let f∈Lq(Q0) and {Qj}j∈N be interior pairwise disjoint cubes in Q0.
By the Minkowski inequality, (2.1), the Hölder inequality and qp≤1, we conclude that
[TABLE]
which, combined with the arbitrariness of {Qj}j∈N, implies that
f∈∣Q0∣q1−p1∥f∥jn(p,q,s)0(Q0) and
∣Q0∣q1−p1∥f∥jn(p,q,s)0(Q0)≲∥f∥Lq(Q0).
Thus, Lq(Q0)⊂∣Q0∣q1−p1jn(p,q,s)0(Q0),
which completes the proof of (i).
Next, we prove (ii).
Let p∈(1,∞).
Choose interior pairwise disjoint cubes {Ri}i∈N such that ℓ(Ri)≥c0 and ⋃i∈NRi=Rn.
For any f∈jn(p,p,s)0(Rn), it is clear that
[TABLE]
Thus, we have f∈Lp(Rn) and jn(p,p,s)0(Rn)⊂Lp(Rn).
For the converse, let {Qj}j∈N be interior pairwise disjoint cubes in Rn.
By the Minkowski inequality, (2.1) and the Hölder inequality, we have
[TABLE]
Combining this and using the arbitrariness of {Qj}j∈N, we obtain f∈jn(p,p,s)0(Rn)
and Lp(Rn)⊂jn(p,p,s)0(Rn).
Thus, jn(p,p,s)0(Rn)=Lp(Rn) with equivalent norms. This proves (ii).
Finally, we show (iii).
For any N∈[c0,∞), let QN:=[−N,N]n.
For any f∈jn(p,q,s)α(Rn), by Definition 2.3, we have
[TABLE]
From this and α+q1−p1<0, it follows that
[TABLE]
Thus, we have f=0 almost everywhere. This finishes the proof of (iii) and hence of Proposition 3.4.
∎
Remark 3.5**.**
If 1<p≤q<∞, s∈Z+ and
α∈(0,∞)∩[p1−q1,∞), the relation between
jn(p,q,s)α(Rn) and Lq(Rn) is still unknown.
4 Localized Hardy-kind spaces and duality
In this section, using the local atom,
we introduce the localized Hardy-kind space and show that this space is
the predual of the localized John–Nirenberg–Campanato space.
Definition 4.1**.**
Let v∈[1,∞), w∈(1,∞], s∈Z+ and α∈[0,∞). Fix c0∈(0,ℓ(X)) and
let Q denote a cube in Rn. Then a function
a on Rn is called a local (v,w,s)α,c0-atom supported in Q if
(i)
missingsupp(a):={x∈Rn:a(x)=0}⊂Q;
(ii)
∥a∥Lw(Q)≤∣Q∣w1−v1−α;
(iii)
when ℓ(Q)<c0,
∫Qa(x)xβdx=0 for any β∈Z+n and ∣β∣≤s.
Let p∈(1,∞) and Q0⫋Rn be a cube.
Dafni et al. [9] introduced the Hardy-kind space HKp′(Q0) and proved in [9, Theorem 6.6] that
HKp′(Q0) is the predual space of JNp(Q0). Here the symbol HK might mean Hardy-kind.
Later, Tao et al. [22] introduced the generalized Hardy-kind space, which is the predual space of
the John–Nirenberg–Campanato space. Motivated by this, we introduce the localized Hardy-kind space.
To this end, we first introduce a new polymer.
In what follows, the symbol(jn(p,q,s)α,c0(X))∗ denotes the
dual space of jn(p,q,s)α,c0(X) equipped with the weak-∗ topology.
Definition 4.2**.**
Let v∈(1,∞), w∈(1,∞], s∈Z+, α∈[0,∞) and c0∈(0,ℓ(X)). The spacehk(v,w,s)α,c0(X)
is defined to be the set of all g∈(jn(v′,w′,s)α,c0(X))∗ such that
[TABLE]
where 1/v+1/v′=1=1/w+1/w′,
{aj}j∈N are local (v,w,s)α,c0-atoms supported, respectively,
in interior pairwise disjoint subcubes {Qj}j∈N of X,
{λj}j∈N⊂C and
∑j∈N∣λj∣v<∞.
Any g∈hk(v,w,s)α,c0(X)
is called a local(v,w,s)α,c0-polymer on X and let
[TABLE]
where the infimum is taken over all such decompositions of g as above.
Remark 4.3**.**
For any given v, w, s, α and c0 as in Definition 4.2,
let {aj}j∈N be local (v,w,s)α,c0-atoms supported, respectively, in interior pairwise disjoint subcubes
{Qj}j∈N of X,
{λj}j∈N⊂C and
∑j∈N∣λj∣v<∞.
We claim that ∑j∈Nλjaj converges in (jn(v′,w′,s)α,c0(X))∗,
where 1/v+1/v′=1=1/w+1/w′.
Indeed, for any given f∈jn(v′,w′,s)α,c0(X) and any l∈N, m∈Z+,
by Definition 4.1(iii) and the Hölder inequality, we have
[TABLE]
From this and ∑j∈N∣λj∣v<∞,
it follows that the claim holds true.
By the same argument as used in the estimation of (4.1), we also obtain
[TABLE]
which, together with Definition 4.2, further implies that, for
any g∈hk(v,w,s)α,c0(X) and f∈jn(v′,w′,s)α,c0(X),
[TABLE]
This means that we indeed have g∈(jn(v′,w′,s)α,c0(X))∗.
Now, we introduce the localized Hardy-kind space.
Definition 4.4**.**
Let v∈(1,∞), w∈(1,∞], s∈Z+, α∈[0,∞) and c0∈(0,ℓ(X)).
The localized Hardy-kind space hk(v,w,s)α,c0(X) is defined to be the set of all
g∈(jn(v′,w′,s)α,c0(X))∗ such that
there exists a sequence {gi}i∈N⊂hk(v,w,s)α,c0(X) such that
∑i∈N∥gi∥hk(v,w,s)α,c0(X)<∞ and
[TABLE]
For any g∈hk(v,w,s)α,c0(X), let
[TABLE]
where the infimum is taken over all decompositions of g as in (4.3).
Remark 4.5**.**
For any given v, w, s, α and c0 as in Definition 4.4,
let g∈hk(v,w,s)α,c0(X) and {gi}i∈N⊂hk(v,w,s)α,c0(X).
If g=∑i∈Ngi in (jn(v′,w′,s)α,c0(X))∗,
we then claim that
[TABLE]
Indeed, by Definition 4.4, we know that, for any ϵ∈(0,∞) and i∈N,
there exists a sequence {gi,j}j∈N⊂hk(v,w,s)α,c0(X)
such that
∑j∈N∥gi,j∥hk(v,w,s)α,c0(X)≤∥gi∥hk(v,w,s)α,c0(X)+2−iϵ and
gi=∑j∈Ngi,j in (jn(v′,w′,s)α,c0(X))∗.
From this and g=∑i∈Ngi=∑i∈N∑j∈Ngi,j
in (jn(v′,w′,s)α,c0(X))∗, we deduce that
[TABLE]
which, combined with the arbitrariness of ϵ, implies that the above claim holds true.
Remark 4.6**.**
Let v, w, s, α and c0 be as in Definition 4.4.
If {gi}i∈N⊂hk(v,w,s)α,c0(X) and
∑i∈N∥gi∥hk(v,w,s)α,c0(X)<∞,
we then claim that ∑i∈Ngi convergences in (jn(v′,w′,s)α,c0(X))∗.
Indeed, by Remark 4.3, we have,
for any given f∈jn(v′,w′,s)α,c0(X) and any l∈N, m∈Z+,
[TABLE]
By this and ∑i∈N∥gi∥hk(v,w,s)α,c0(X)<∞,
we conclude that the above claim holds true.
Clearly, if letting g:=∑i∈Ngi in (jn(v′,w′,s)α,c0(X))∗, then
[TABLE]
From this and Definition 4.4, it follows that,
for any g∈hk(v,w,s)α,c0(X),
[TABLE]
The following proposition indicates that hk(v,w,s)α,c0(X) is independent of the choice of
the positive constant c0.
Proposition 4.7**.**
Let v∈(1,∞), w∈(1,∞], s∈Z+, α∈[0,∞) and 0<c1<c2<ℓ(X).
Then hk(v,w,s)α,c1(X)=hk(v,w,s)α,c2(X) with equivalent norms.
Proof.
Let v, w, s, α, c1 and c2 be as in this proposition.
Clearly, any local (v,w,s)α,c2-atom is also a local (v,w,s)α,c1-atom.
By this and Proposition 2.5, we know that, for any G∈hk(v,w,s)α,c2(X),
[TABLE]
Thus, we have G∈hk(v,w,s)α,c1(X) and hence hk(v,w,s)α,c2(X)⊂hk(v,w,s)α,c1(X).
Next, we prove hk(v,w,s)α,c1(X)⊂hk(v,w,s)α,c2(X).
For any g∈hk(v,w,s)α,c1(X), by Definition 4.2, we know that
there exist a sequence {aj}j∈N of local (v,w,s)α,c1-atoms supported,
respectively, in interior pairwise disjoint cubes {Qj}j∈N
and {λj}j∈N⊂C such that
(∑j∈N∣λj∣v)v1≤2∥g∥hk(v,w,s)α,c1(X)
and g:=∑j∈Nλjaj in (jn(v′,w′,s)α,c1(X))∗.
Let J:={j∈N:c1≤ℓ(Qj)<c2}.
Observe that, for any j∈N∖J, aj is a local (v,w,s)α,c2-atom.
By Remark 4.3, we know that ∑j∈N∖Jλjaj converges in (jn(v′,w′,s)α,c2(X))∗.
Let g0:=∑j∈N∖Jλjaj in (jn(v′,w′,s)α,c2(X))∗. Then
[TABLE]
If X=Rn, let l1:=c2 and if X⫋Rn is a cube,
let l1:=ℓ(X)(⌊c2ℓ(X)⌋)−1.
It is clear that l1∈[c2,2c2).
Choose interior pairwise disjoint cubes {Ri}i∈N such that ℓ(Ri)=l1 and X=⋃i∈NRi.
For any i∈N, let Qi:={Qj:j∈J,Qj∩Ri=∅}.
Then
[TABLE]
Rewrite Qi as {Qi,k}k=1Mi and
let Qi,k:=∅ for any integer k∈(Mi,K].
Besides, for any integer k∈[1,Mi], we rewrite the atom supported in Qi,k as ai,k and its corresponding
coefficient as λi,k; for any integer k∈(Mi,K], let ai,k:=0 and λi,k:=0.
For any j∈J, let
[TABLE]
Then #Rj≤2n.
Let
[TABLE]
For any k∈{1,…,K} and i∈N, let ai,k:=C1ai,k1Ri.
Clearly, ai,k is a local (v,w,s)α,c2-atom supported in Ri.
From the definition of λi,k and
#Rj≤2n, we deduce that, for any k∈{1,…,K},
[TABLE]
Combining this and Remark 4.2, we obtain ∑i∈NC1λi,kai,k
converges in (jn(v′,w′,s)α,c2(X))∗.
For any k∈{1,…,K}, let gk:=∑i∈NC1λi,kai,k
in (jn(v′,w′,s)α,c2(X))∗. Then
[TABLE]
Now, we claim that g=g0+∑k=1Kgk in (jn(v′,w′,s)α,c2(X))∗.
Indeed, for any f∈jn(v′,w′,s)α,c2(X),
by (4.5) and an argument similar to that used in the estimation of (4.2), we obtain
[TABLE]
From this, the definitions of ai,k, ai,k and
λi,k, ⋃iRi=Rn and Proposition 2.5, we deduce that
[TABLE]
This proves the above claim. By this claim, (4.4), (4.6) and
K≤(c1l1+2)n, we further conclude that
[TABLE]
Now, for any G∈hk(v,w,s)α,c1(X), by Definition 4.4,
we know that there exists a sequence
{gi}i∈N⊂hk(v,w,s)α,c1(X) such that
[TABLE]
and
G:=∑i∈Ngi in (jn(v′,w′,s)α,c1(X))∗.
From this, Proposition 2.5, Remark 4.5 and (4.7), we deduce that
[TABLE]
Therefore, we have G∈hk(v,w,s)α,c2(X) and hence
hk(v,w,s)α,c1(X)⊂hk(v,w,s)α,c2(X).
This finishes the proof of Proposition 4.7.
∎
Remark 4.8**.**
Based on Proposition 4.7, henceforth, we simply write the local
(v,w,s)α,c0-atom, the spaces hk(v,w,s)α,c0(X) and hk(v,w,s)α,c0(X), respectively,
as the local (v,w,s)α-atom, the spaces hk(v,w,s)α(X) and hk(v,w,s)α(X).
As is well known, a bounded linear functional
on a dense subspace in hk(v,w,s)α(X) can be continuously extended to
the whole space hk(v,w,s)α(X).
To show the duality theorem, we first introduce a dense subspace of hk(v,w,s)α(X).
Definition 4.9**.**
Let v∈(1,∞), w∈(1,∞], s∈Z+ and α∈[0,∞).
The space hk(v,w,s)αfin(X) is defined to be
the set of all finite linear combinations of local
(v,w,s)α-atoms supported, respectively, in cubes in X.
Remark 4.10**.**
Let v, w, s and α be as in Definition 4.9. We claim that
hk(v,w,s)αfin(X) is dense in hk(v,w,s)α(X).
Indeed, for any g∈hk(v,w,s)α(X), by Definitions 4.2 and 4.4,
we know that there exists a representation
[TABLE]
where {ai,j}i,j∈N are local
(v,w,s)α-atoms supported, respectively, in cubes {Qi,j}i,j∈N,
{Qi,j}j∈N for any given i∈N have pairwise disjoint interiors,
and ∑i∈N(∑j∈N∣λi,j∣v)v1<∞. It is easy to see
that, for any l,m∈N, ∑i=1l∑j=1mλi,jai,j∈hk(v,w,s)αfin(X) and
[TABLE]
This proves the above claim.
In what follows, for any given normed space X, we use
the symbolX∗ to denote its dual space.
Theorem 4.11**.**
Let v∈(1,∞), 1/v+1/v′=1, w∈(1,∞),
1/w+1/w′=1, s∈Z+ and α∈[0,∞).
Then jn(v′,w′,s)α(X)=(hk(v,w,s)α(X))∗ in the following sense:
(i)
For any given f∈jn(v′,w′,s)α(X), then the linear functional
[TABLE]
can be extended to a bounded linear functional on hk(v,w,s)α(X).
Moreover, it holds true that ∥Lf∥(hk(v,w,s)α(X))∗≤∥f∥jn(v′,w′,s)α(X).
(ii)
Any bounded linear functional L on hk(v,w,s)α(X) can be represented by a function f∈jn(v′,w′,s)α(X)
in the following sense:
[TABLE]
Moreover, there exists a positive constant C, depending only on s, such that
∥f∥jn(v′,w′,s)α(X)≤C∥L∥(hk(v,w,s)α(X))∗.
Proof.
Let v, w, s and α be the same as in this theorem and
c0∈(0,ℓ(X)).
Let f∈jn(v′,w′,s)α(X). For any g∈hk(v,w,s)αfin(X),
let
[TABLE]
By Remarks 4.3 and 4.6, we have
∣⟨Lf,g⟩∣≤∥f∥jn(v′,w′,s)α(X)∥g∥hk(v,w,s)α(X).
Combining this and Remark 4.10, we then complete the proof of (i).
Now, we show (ii).
Let L represent a bounded linear functional on hk(v,w,s)α(X).
We now claim that there exists a function f on X such that (4.8) holds true.
Indeed, if X is a cube Q0⫋Rn, by Definition 4.4, we know that,
for any h∈Lw(Q0),
[TABLE]
Write LQ0 to be the restriction of L
to Lw(Q0). Thus, LQ0 is bounded on Lw(Q0). By the well-known duality
(Lw(Q0))∗=Lw′(Q0), we find that there exists a unique function f∈Lw′(Q0) such that
[TABLE]
here and hereafter, 1/w+1/w′=1.
Since hk(v,w,s)αfin(Q0) is contained in Lw(Q0) as sets,
this proves (4.8) when X is a cube Q0⫋Rn.
If X=Rn, for any i∈N, let Ri:=[−c0−i,c0+i]n.
Let LRi denote the restriction of L to Lw(Ri).
Using the same argument as that used in the estimation of (4.9),
we find a unique function fi∈Lw′(Ri) such that
[TABLE]
From this, it follows that, for any i∈N and h∈Lw(Ri),
[TABLE]
Hence,
fi+1=fi almost everywhere on Ri. Let
[TABLE]
For any g∈hk(v,w,s)αfin(X), then g has a compact support in X and hence
there exists an i0∈N such that
missingsupp(g)⊂Ri0. Since g∈Lw(Ri0), it follows that
⟨L,g⟩=∫Ri0fi0g=∫Rnfg.
This proves (4.8) when X=Rn. Thus, the above claim holds true.
Now, we still need to show ∥f∥jn(v′,w′,s)α(X)≲∥L∥(hk(v,w,s)α(X))∗.
Suppose {Qi}i∈N are interior mutually disjoint cubes in X.
Then we know that, for any i∈N,
[TABLE]
For any i∈N, choose ai such that ∥ai∥Lw(Qi)≤∣Qi∣w1 and
[TABLE]
and let Ai:=∣Qi∣−α[\fintQi∣f−PQi,c0(s)(f)∣w′]w′1.
For any N∈N, by the fact that (ℓv)∗=ℓv′, where 1/v+1/v′=1, we
choose {λi}i=1N⊂[0,∞) such that
(∑i=1N∣Qi∣λiv)v1≤1 and
[TABLE]
For any N∈N, let
[TABLE]
From (2.1) and the Hölder inequality, we deduce that
[TABLE]
where C(s) is the same positive constant as in (2.1).
For any i∈{1,…,N}, let
[TABLE]
Clearly, {ai}i=1N
are local (v,w,s)α-atoms supported, respectively, in {Qi}i=1N.
By this, we obtain gN∈hk(v,w,s)αfin(X).
Moreover, from the choice of {λi}i=1N, we deduce that
which, together with the arbitrariness of N and {Qi}i∈N, further implies that
[TABLE]
This finishes the proof of (ii) and hence of Theorem 4.11.
∎
For any given cube Q0, by the way similar to that used in [9, Definition 6.1], we can construct
the localized Hardy-kind space hkv,w(Q0) with 1<v<w≤∞, which proves to
be equivalent with hk(v,w,0)0(Q0) in Proposition 4.14 below.
Definition 4.12**.**
Let v∈(1,∞), w∈(v,∞] and Q0⫋Rn be a cube.
The localized Hardy-kind spacehkv,w(Q0) is defined to be the set of all g∈Lv(Q0) such that
[TABLE]
where {ai,j}i,j∈N are local (v,w,0)0-atoms
supported, respectively, in subcubes
{Qi,j}i,j∈N of Q0,
{Qi,j}j∈N for any given i∈N have pairwise disjoint interiors,
{λi,j}i,j∈N⊂C and
[TABLE]
For any g∈hkv,w(Q0), define
[TABLE]
where the infimum is taken over all such decompositions of g as above.
Remark 4.13**.**
Let 1<v<w≤∞ and Q0⫋Rn be a cube.
(i)
Let {ai,j}i,j∈N be local (v,w,0)0-atoms supported, respectively, in subcubes
{Qi,j}i,j∈N of Q0,
{Qi,j}j∈N for any given i∈N have pairwise disjoint interiors,
{λi,j}i,j∈N⊂C
and ∑i∈N(∑j∈N∣λi,j∣v)v1<∞.
We claim that ∑i∈N∑j∈Nλi,jai,j
converges in Lv(Q0). Indeed, by the Hölder inequality,
we know that, for any l∈N and m∈Z+,
[TABLE]
which, together with (∑j∈N∣λi,j∣v)v1<∞, implies that
∑j∈Nλi,jai,j converges in Lv(Q0).
Combining this and ∑i∈N(∑j∈N∣λi,j∣v)v1<∞,
we then complete the proof of the above claim.
Moreover, we also have
[TABLE]
2. (ii)
We claim that hkv,w(Q0)⊂Lv(Q0) with a continuous embedding.
Indeed, let g∈hkv,w(Q0). By (i) of this remark and Definition 4.12, we know that
g∈Lv(Q0) and ∥g∥Lv(Q0)≤∥g∥hkv,w(Q0).
Proposition 4.14**.**
Let v∈(1,∞), w∈(v,∞] and Q0⫋Rn be a cube.
Then hkv,w(Q0)=hk(v,w,0)0(Q0) with equivalent norms.
Proof.
Let v, w and Q0 be as in Proposition 4.14.
We first show hkv,w(Q0)⊂hk(v,w,0)0(Q0).
Let g∈hkv,w(Q0). By Definition 4.12,
we have
[TABLE]
where {ai,j}i,j∈N are local (v,w,0)0-atoms supported, respectively, in subcubes
{Qi,j}i,j∈N of Q0, {Qi,j}j∈N for any given i∈N
is a collection of interior pairwise disjoint cubes,
{λi,j}i,j∈N⊂C and
[TABLE]
From Remarks 4.3 and 4.6, it follows that
∑i∈N∑j∈Nλi,jai,j converges in (jn(v′,w′,0)0(Q0))∗,
here and hereafter, 1/v+1/v′=1=1/w+1/w′.
Let
g:=∑i∈N∑j∈Nλi,jai,j in (jn(v′,w′,0)0(Q0))∗.
Then g∈hk(v,w,0)0(Q0) and, for any f∈jn(v′,w′,0)0(Q0),
we have
[TABLE]
Now, we claim that g is independent of
the above decomposition of g and hence well defined. Indeed, for any given f∈jn(v′,w′,0)0(Q0) and any N∈(0,∞),
let
[TABLE]
From g∈Lv(Q0)⊂L1(Q0) and the boundedness of fN, it follows that ∫Q0∣gfN∣<∞.
Notice that g=∑i∈N∑j∈Nλi,jai,j in Lv(Q0) and also in L1(Q0).
By this, we have
[TABLE]
Since ai,j∈Lw(Q0), f∈jn(v′.w′,0)0(Q0)⊂Lw′(Q0) and ∣fN∣≤∣f∣,
from the dominated convergence theorem,
we deduce that
which implies that the above claim holds true.
By Definition 4.12, we know that
[TABLE]
which, together with the above claim and the arbitrariness of
{λi,j}i,j∈N and {ai,j}i,j∈N, implies that
[TABLE]
Thus, we have hkv,w(Q0)⊂hk(v,w,0)0(Q0).
Next, we show hk(v,w,0)0(Q0)⊂hkv,w(Q0).
Let g∈hk(v,w,0)0(Q0).
By Definition 4.12, we have
[TABLE]
where {ai,j}i,j∈N are local (v,w,0)0-atoms supported, respectively, in subcubes
{Qi,j}i,j∈N of Q0, {Qi,j}j∈N for any given i∈N have pairwise disjoint interiors,
{λi,j}i,j∈N⊂C and
[TABLE]
From Remark 4.13, we deduce that
∑i∈N∑j∈Nλi,jai,j converges in Lv(Q0).
Let
[TABLE]
in Lv(Q0).
Then g∈hkv,w(Q0).
Now, we show that g is independent of the above decomposition of g.
Suppose that there exists another representation,
[TABLE]
where {bi,j}i,j∈N are local (v,w,0)0-atoms supported in
subcubes {Ri,j}i,j∈N of Q0,
{Ri,j}j∈N for any given i∈N have pairwise disjoint interiors,
{μi,j}i,j∈N⊂C and ∑i∈N(∑j∈N∣μi,j∣v)v1<∞.
Similarly to the estimation of (2.5), we obtain Lv′(Q0)⊂jn(v′,w′,0)0(Q0).
Notice that both ∑i∈N∑j∈Nμi,jbi,j
and ∑i∈N∑j∈Nλi,jai,j
converge in Lv(Q0).
Thus, for any f∈Lv′(Q0),
[TABLE]
which implies that
[TABLE]
Therefore, g is independent of the choice of {λi,j}i,j∈N
and {ai,j}i,j∈N and hence well defined.
By this, we obtain ∥g∥hkv,w(Q0)≤∥g∥hk(v,w,0)0(Q0).
This proves hk(v,w,0)0(Q0)⊂hkv,w(Q0),
which completes the proof of Proposition 4.14.
∎
5 Equivalent norms on hk(v,w,s)α(X)
In this section, we first consider the equivalent relations on localized Hardy-kind spaces.
We then study the limit case of localized Hardy-kind spaces.
The following proposition indicates that, for admissible (v,s,α), hk(v,w,s)α(X) is invariant on w∈(v,∞].
Proposition 5.1**.**
Let v∈(1,∞), w∈(v,∞], s∈Z+ and α∈[0,∞). Then
hk(v,w,s)α(X)=hk(v,∞,s)α(X) with equivalent norms.
Remark 5.2**.**
By Propositions 3.1, 5.1 and Theorem 4.11, we conclude that,
for any p∈(1,∞), q∈[1,∞), s∈Z+ and α∈[0,∞),
the predual space of jn(p,q,s)α(X) is hk(p′,q′,s)α(X),
where 1/p+1/p′=1=1/q+1/q′.
To prove Proposition 5.1, we need the following two technical lemmas.
The proof of the following lemma can be found in [22, Lemma 4.3].
Lemma 5.3**.**
Let w∈[1,∞), C∈(1,∞), γ∈(0,∞), Q0 be a cube in Rn and f∈Lw(Q0).
For any k∈N, let μk:=Ckγ. Then
[TABLE]
Let s∈Z+ and Q⫋Rn be a cube. In what follows,
the symbolLs∞(Q) denotes the set of all functions f∈L∞(Q)
such that, for any β∈Z+n and ∣β∣≤s, ∫Qf(x)xβdx=0.
We also denote by the symbol
MQ(d) the maximal function related to the dyadic subcubes of Q, namely,
for any f∈L1(Q) and x∈Q,
[TABLE]
where the supremum is taken over all dyadic subcubes Q(x) containing x in Q.
The following decomposition lemma contains a refinement of classical Calderón–Zygmund decompositions;
see [22, Lemma 4.4] and also [9, Lemma 6.5] for its proof.
Lemma 5.4**.**
Let s∈Z+, C∈(2n,∞), Q be a cube in Rn, f∈L1(Q) and γ≥\fintQ∣f∣. Then
[TABLE]
almost everywhere,
where Ak,j∈Ls∞(Qk,j) and ∥Ak,j∥L∞(Qk,j)≤2n+1C(s)Ck+1γ,
{Qk,j}j∈N is a collection of interior pairwise disjoint cubes in Q satisfying
Q0,1=Q, Q0,j=∅ for any j∈N∖{1} and
[TABLE]
where C(s) is the same constant as in (2.1).
Furthermore, if f∈Lw(Q), then (5.1) holds true in (JN(v′,w′,s)α(Y))∗
for any v∈(1,∞), w∈(1,∞] and α∈[0,∞), where Y is Rn or a cube which contains Q,
and 1/v+1/v′=1=1/w+1/w′.
Let v∈(1,∞), 1/v+1/v′=1,
w∈(v,∞), 1/w+1/w′=1, s∈Z+ and α∈[0,∞).
Clearly, a local (v,∞,s)α-atom is also a local (v,w,s)α-atom.
By this and Proposition 3.1,
we have hk(v,∞,s)α(X)⊂hk(v,w,s)α(X).
Now, we show hk(v,w,s)α(X)⊂hk(v,∞,s)α(X).
To this end, we first let g∈hk(v,w,s)α(X).
By Definition 4.2, we know that
there exists a sequence of local (v,w,s)α-atoms {al}l∈N
supported, respectively, in interior pairwise disjoint cubes {Ql}l∈N,
and {λl}l∈N⊂C with
(∑l∈N∣λl∣v)v1≤2∥g∥hk(v,w,s)α(X) such that
g=∑l∈Nλlal in (jn(v′,w′,s)α(X))∗.
Without the loss of generality, we may assume ∥al∥L1(Ql)=0.
Let C0∈(2n,∞) and γl:=(\fintQl∣al∣w)w1.
By Lemma 5.4 and Proposition 2.10(i),
we have
[TABLE]
where Ak,jl∈Ls∞(Qk,jl) and
[TABLE]
{Qk,jl}j∈N is a collection of interior pairwise disjoint cubes in Ql satisfying
Q0,1l=Ql, Q0,jl=∅ for any j∈N∖{1} and
Combining this and the definitions of A0,1l and PQl(s)(al),
we know that, for any l∈N, a0l is a local (v,∞,s)α-atom supported in Ql.
From this, Remark 4.3 and
(∑l∈N∣λl∣v)v1≤2∥g∥hk(v,w,s)α(X),
we deduce that ∑l∈N2n+2C(s)C0λla0l converges in (jn(v′,1,s)α(X))∗.
Let g0:=∑l∈N2n+2C(s)C0λla0l
in (jn(v′,1,s)α(X))∗.
Then
[TABLE]
For any k,j∈N,
let ak,jl:=[2n+1C(s)C0k+1γl]−1∣Qk,jl∣−v1−αAk,jl.
By (5.3), we find that ak,jl is a local (v,∞,s)α-atom supported in Qk,jl.
Since Qk,jℓ⊂Qℓ, from (5.4) and the Hölder inequality, we deduce that
[TABLE]
By this, Lemma 5.3 and
the boundedness of MQl(d) on Lw(Ql), we conclude that
[TABLE]
which, together with the definition of γl, the fact that,
for any l∈N, al is a local (v,w,s)α-atom
and (∑l∈N∣λl∣v)v1≤2∥g∥hk(v,w,s)α(X),
implies that
[TABLE]
From this and Remark 4.2, it follows that, for any k∈N,
∑l,j∈N2n+1C(s)C0k+1γl∣Qk,jl∣v1+αλlak,jl
converges in (jn(v′,1,s)α(X))∗.
For any k∈N, let
gk:=∑l,j∈N2n+1C(s)C0k+1γl∣Qk,jl∣v1+αλlak,jl
in (jn(v′,1,s)α(X))∗.
By (5.6), we have
[TABLE]
Then, by the definition of ak,jl, we obtain
[TABLE]
From (5.6) and the same argument as that used in the estimation of (4.2),
we deduce that, for any f∈jn(v′,1,s)α(X),
[TABLE]
By this, (5.8), the definition of a0l, (5.2) and Proposition 3.1,
we find that, for any f∈jn(v′,1,s)α(X),
[TABLE]
Thus, g=∑k=0∞gk in (jn(v′,1,s)α(X))∗,
which, combined with (5.5) and (5.7), implies that
[TABLE]
Now, for any G∈hk(v.w.s)α(X), by Definition 4.4,
we find a sequence {gi}i∈N⊂hk(v,w,s)α(X) such that
∑i∈N∥gi∥hk(v,w,s)α(X)≤2∥G∥hk(v.w.s)α(X) and
[TABLE]
From Proposition 3.1, we deduce that
∑i∈Ngi converges in (jn(v′,1,s)α(X))∗.
By this, Remark 4.5 and (5.9), we conclude that
[TABLE]
Therefore, G∈hk(v,∞,s)α(X) and hence hk(v,w,s)α(X)⊂hk(v,∞,s)α(X).
This finishes the proof of Proposition 5.1.
∎
Remark 5.5**.**
Let 1<v1<v2<∞, 1/v1+1/v1′=1=1/v2+1/v2′,
w∈(1,∞], 1/w+1/w′=1, α∈[0,∞),
s∈Z+ and
Q0⫋Rn be a cube. Then we claim that
hk(v2,w,s)α(Q0)⊂hk(v1,w,s)α(Q0). Indeed,
let g∈hk(v2,w,s)α(Q0). Assume that
[TABLE]
where {ai,j}i,j∈N are local (v2,w,s)α-atoms supported, respectively,
in subcubes {Qi,j}i,j∈N of Q0,
{Qi,j}j∈N for any given i∈N is a collection of interior pairwise disjoint cubes,
{λi,j}i,j∈N⊂C and
Observe that ∣Qi,j∣v21−v11ai,j is a local (v1,w,s)α-atom supported in Qi,j.
From this, the Hölder inequality and the interior pairwise disjointness
of {Qi,j}j∈N for any given i∈N, it follows that
[TABLE]
which implies that
[TABLE]
This proves the above claim.
The following proposition might be viewed as a counterpart of Proposition 3.4.
Proposition 5.6**.**
Let v∈(1,∞) and s∈Z+.
(i)
If w∈(1,v] and Q0⫋Rn is a cube, then
hk(v,w,s)0(Q0)=∣Q0∣v1−w1Lw(Q0)
with equivalent norms.
(ii)
Lv(Rn)=hk(v,v,s)0(Rn)* with equivalent norms.*
Proof.
Let v∈(1,∞), 1/v+1/v′=1, s∈Z+ and Q0⫋Rn be a cube.
First, we show (i).
Let w∈(1,v] and 1/w+1/w′=1.
Clearly, ∣Q0∣v1−w1Lw(Q0)⊂hk(v,w,s)0(Q0).
We only need to show hk(v,w,s)0(Q0)⊂∣Q0∣v1−w1Lw(Q0).
Let g∈hk(v,w,s)0(Q0). By Definition 4.4, we know that
[TABLE]
where {ai,j}i,j∈N are local (v,w,s)0-atoms supported,
respectively, in subcubes {Qi,j}i,j∈N of Q0,
{Qi,j}j∈N for any given i∈N have pairwise disjoint interiors,
{λi,j}i,j∈N⊂C and
[TABLE]
Now, we claim that ∑i∈N∑j∈Nλi,jai,j converges in Lw(Q0).
Since {Qi,j}j∈N for any given i∈N are interior pairwise disjoint cubes,
for any i∈N, letting gi:=∑j∈Nλi,jai,j,
then gi is well defined pointwisely.
By the Jensen inequality and wv≥1, we obtain
[TABLE]
From this and the interior pairwise disjointness of {Qi,j}j∈N,
it follows that gi=∑j∈Nλi,jai,j in Lw(Q0),
which, together with ∑i∈N(∑j∈N∣λi,j∣v)v1<∞,
proves the above claim. By this claim, (5.10) and Proposition 3.4(i), we conclude that
g=∑i∈N∑j∈Nλi,jai,j in Lw(Q0).
From this and (5.11), it follows that
[TABLE]
which implies that
[TABLE]
Therefore, hk(v,w,s)0(Q0)⊂∣Q0∣v1−w1Lw(Q0). This proves (i).
For (ii), let c0∈(0,∞), g∈Lv(Rn) and {Ri}i∈N⊂Rn
be interior pairwise disjoint cubes such that
ℓ(Ri)∈[c0,∞) and Rn=⋃i∈NRi.
Let
[TABLE]
Observe that {gi}i∈N are local (v,v,s)0-atoms supported, respectively, in {Ri}i∈N and
[TABLE]
in Lv(Q0) and
also in (jn(v′,v′,s)0(Rn))∗ because of Proposition 3.4(ii).
By Definition 4.4, we have
[TABLE]
This proves Lv(Rn)⊂hk(v,v,s)0(Rn).
Now, we show hk(v,v,s)0(Rn)⊂Lv(Rn).
Let g∈hk(v,v,s)0(Rn). By Definition 4.4, we know that
[TABLE]
where {ai,j}i,j∈N are local (v,v,s)0-atoms supported, respectively, in cubes {Qi,j}i,j∈N,
{Qi,j}j∈N for any given i∈N have pairwise disjoint interiors,
{λi,j}i,j∈N⊂C and ∑i∈N(∑j∈N∣λi,j∣v)v1<∞.
Observe that ∑i∈N∑j∈Nλi,jai,j converges in Lv(Rn).
From this, (5.12) and Proposition 3.4(ii),
it follows that g=∑i∈N∑j∈Nλi,jai,j in Lv(Rn).
By this, we have
[TABLE]
which, combined with the arbitrariness of the decomposition of g, implies that g∈Lv(Rn) and
∥g∥Lv(Rn)≤∥g∥hk(v,v,s)0(Rn).
Thus, hk(v,v,s)0(Rn)⊂Lv(Rn).
This finishes the proof of (ii) and hence of Proposition 5.6.
∎
Recall that, for any given q∈(1,∞],
the atomic localized Hardy spacehat1,q(X)
is defined to be the set of all f∈L1(X) such that
f=∑j∈Nλjaj in L1(X), where {aj}j∈N is a sequence of local (1,q,0)0-atoms
supported, respectively, in cubes {Qj}j∈N⊂X, and {λj}j∈N⊂C with
∑j∈N∣λj∣<∞.
Let ∥g∥hat1,q(X):=inf∑j∈N∣λj∣,
where the infimum is taken over all the above decompositions of g.
Finally, we consider the relation between hk(v,w,s)α(X) and the atomic localized Hardy space.
Proposition 5.7**.**
Let v∈(1,∞), w∈(1,∞] and Q0⫋Rn be a cube.
Then
[TABLE]
Moreover, if g∈⋃v∈(1,∞)hk(v,w,0)0(Q0), then
[TABLE]
where v→1+ means that v∈(1,∞) and v→1.
Proof.
Let g∈hk(v,w,0)0(Q0). From Proposition 4.14, it follows that g∈hkv,w(Q0).
By Definition 4.12, we know that
[TABLE]
where {ai,j}i,j∈N are local (v,w,0)0-atoms supported, respectively,
in subcubes {Qi,j}i,j∈N of Q0,
{Qi,j}j∈N for any given i∈N is a collection of interior pairwise disjoint cubes,
{λi,j}i,j∈N⊂C and ∑i∈N(∑j∈N∣λi,j∣v)v1<∞.
By this and the embedding Lv(Q0)⊂L1(Q0), we obtain
[TABLE]
Notice that, for any i,j∈N, ∣Qi,j∣v1−1ai,j is a local (1,w,0)0-atom supported in Qi,j.
From the Hölder inequality and the interior pairwise disjointness of {Qi,j}j∈N for any given i∈N, we deduce that
[TABLE]
which implies that
[TABLE]
Therefore, g∈hat1,w(Q0) and ∥g∥hat1,w(Q0)≤liminfv→1+∥g∥hk(v,w,0)0(Q0).
This finishes the proof of Proposition 5.7.
∎
Remark 5.8**.**
Let v∈(1,∞), w∈(1,∞] and Q0⫋Rn be a cube.
(i)
It is interesting to ask whether or not ⋃v∈(1,∞)hk(v,w,0)0(Q0)=hat1,w(Q0) and
to find the condition on g such that ∥g∥hat1,w(Q0)=limv→1+∥g∥hk(v,w,0)0(Q0).
(ii)
Let α∈(0,∞) and s∈Z+.
As v→1+,
the relation between the atomic localized Hardy space (see [12])
and hk(v,w,s)α(Q0) is still unknown.
Bibliography25
The reference list from the paper itself. Each links out to its DOI / PubMed record.
1[1] D. Aalto, L. Berkovits, O. E. Kansanen and H. Yue, John–Nirenberg lemmas for a doubling measure, Studia Math. 204 (2011), 21-37.
2[2] L. Berkovits, J. Kinnunen and J. M. Martell, Oscillation estimates, self-improving results and good- λ 𝜆 \lambda inequalities, J. Funct. Anal. 270 (2016), 3559-3590.
3[3] A. Brudnyi and Y. Brudnyi, On Banach structure of multivariate BV spaces I, ar Xiv: 1806.08824.
4[4] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137-160.
5[5] S. Campanato, Su un teorema di interpolazione di G. Stampacchia, Ann. Scuola Norm. Sup. Pisa (3) 20 (1966), 649-652.
6[6] D.-C. Chang, The dual of Hardy spaces on a bounded domain in R n superscript R 𝑛 \textbf{R}^{n} , Forum Math. 6 (1994), 65-81.
7[7] D.-C. Chang, G. Dafni and E. M. Stein, Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in R n superscript R 𝑛 \textbf{R}^{n} , Trans. Amer. Math. Soc. 351 (1999), 1605-1661.
8[8] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.