TAIWANESE JOURNAL OF MATHEMATICS
Vol. 16, No. 4, pp. 1363-1389, August 2012
This paper is available online at http://journal.taiwanmathsoc.org.tw
THE HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT AND
THEIR APPLICATIONS
Hongbin Wang and Zongguang Liu
Abstract. In this paper, a certain Herz-type Hardy spaces with variable exponent
are introduced, and characterizations of these spaces are established in terms of
atomic and molecular decompositions. Using these decompositions, the authors
obtain the boundedness of some operators on the Herz-type Hardy spaces with
variable exponent.
1. INTRODUCTION
In recent years, the theory of function spaces with variable exponents has developed
since the paper [8] of Kováčik and Rákosník appeared in 1991. Lebesgue and Sobolev
spaces with integrability exponent have been extensively investigated, see [5] and the
references therein. Many applications of these spaces were given, see [6]. Very recently,
Izuki [7] introduced the Herz spaces with variable exponent and proved the boundedness
of some sublinear operators on these spaces.
Inspired by [9, 10], we introduce a certain Herz-type Hardy spaces with variable
exponent which is a generalization of classical Herz-type Hardy spaces, and establish
the atomic and molecular decompositions. Using these decompositions, we obtain the
boundedness of some operators on the Herz-type Hardy spaces with variable exponent.
To be precise, we first briefly recall some standard notations in the remainder of this
section. In Section 2, we will define the Herz-type Hardy spaces with variable exponent
α,p
α,p
(Rn ) and HKq(·)
(Rn ), and give their atomic characterizations. In Section 3,
H K̇q(·)
α,p
α,p
we will present the molecular characterizations of H K̇q(·)(Rn ) and HKq(·)(Rn ).
Given an open set Ω ⊂ Rn , and a measurable function p(·) : Ω → [1, ∞), Lp(·)(Ω)
denotes the set of measurable functions f on Ω such that for some λ > 0,
|f (x)| p(x)
dx < ∞.
λ
Ω
Received May 30, 2011, accepted August 8, 2011.
Communicated by Yongsheng Han.
2010 Mathematics Subject Classification: 42B20, 42B30, 42B35.
Key words and phrases: Herz-type Hardy space, Variable exponent, Atomic decomposition, Molecular
decomposition, Operators.
This work was supported by the National Natural Science Foundation of China (Grant No. 11001266 and
No. 11171345) and the Fundamental Research Funds for the Central Universities (Grant No. 2010YS01).
1363
1364
Hongbin Wang and Zongguang Liu
This set becomes a Banach function space when equipped with the Luxemburg-Nakano
norm
|f (x)| p(x)
dx ≤ 1 .
f Lp(·) (Ω) = inf λ > 0 :
λ
Ω
These spaces are referred to as variable Lebesgue spaces or, more simply, as variable
Lp spaces, since they generalized the standard Lp spaces: if p(x) = p is a constant,
then Lp(·)(Ω) is isometrically isomorphic to Lp (Ω). The variable Lp spaces are a
special case of Musielak-Orlicz spaces. For all compact subsets E ⊂ Ω, the space
p(·)
p(·)
Lloc (Ω) is defined by Lloc (Ω) := {f : f ∈ Lp(·)(E)}. Define P(Ω) to be set of
p(·) : Ω → [1, ∞) such that
1 < p− = ess inf{p(x) : x ∈ Ω} ≤ ess sup{p(x) : x ∈ Ω} = p+ < ∞.
Denote p (x) = p(x)/(p(x) − 1).
Let f ∈ L1loc(Rn ), the Hardy-Littlewood maximal operator is defined by
1
|f (y)|dy,
M f (x) = sup
r>0 |Br (x)| Br (x)
where Br (x) = {y ∈ Rn : |x − y| < r}. There exist some sufficient conditions on
p(·) such that the maximal operator M is bounded on Lp(·)(Rn ), see [1-4, 11, 12]. Let
B(Rn ) be the set of p(·) ∈ P(Rn ) such that M is bounded on L p(·)(Rn ).
Let Bk = {x ∈ Rn : |x| ≤ 2k } and Ak = Bk \ Bk−1 for k ∈ Z. Denote Z+ as
the set of positive integers, χk = χAk for k ∈ Z, χ̃k = χk if k ∈ Z+ and χ̃0 = χB0 ,
where χAk is the characteristic function of A k .
Definition 1.1. ([7]). Let α ∈ R, 0 < p ≤ ∞ and q(·) ∈ P(R n). The homogeα,p
(Rn ) is defined by
neous Herz space with variable exponent K̇q(·)
q(·)
α,p
(Rn ) = {f ∈ Lloc (Rn \ {0}) : f K̇ α,p (Rn ) < ∞},
K̇q(·)
q(·)
where
f K̇ α,p (Rn ) =
q(·)
1/p
∞
2kαpf χk pLq(·) (Rn )
.
k=−∞
α,p
(Rn ) is defined by
The non-homogeneous Herz space with variable exponent Kq(·)
q(·)
α,p
(Rn ) = {f ∈ Lloc (Rn ) : f K α,p (Rn ) < ∞},
Kq(·)
q(·)
where
f K α,p (Rn ) =
q(·)
∞
k=0
1/p
2
kαp
p
f χ̃k Lq(·) (Rn )
.
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The Herz-type Hardy Spaces with Variable Exponent
In the proof of our main result, we will use the following lemmas.
Lemma 1.1. ([8]). Let p(·) ∈ P(Rn ). If f ∈ Lp(·)(Rn ) and g ∈ Lp (·) (Rn ), then
f g is integrable on R n and
|f (x)g(x)|dx ≤ rp f Lp(·) (Rn )gLp(·) (Rn ) ,
Rn
where
rp = 1 + 1/p− − 1/p+ .
This inequality is named the generalized H ölder inequality with respect to the
variable Lp spaces.
Lemma 1.2. ([7]). Let p(·) ∈ B(Rn ). Then there exists a positive constant C
such that for all balls B in R n and all measurable subsets S ⊂ B,
χB Lp(·) (Rn )
χS Lp(·) (Rn )
χS Lp(·) (Rn )
χB Lp(·) (Rn )
and
χS Lp (·) (Rn )
χB Lp (·) (Rn )
|B|
,
|S|
|S| δ1
≤C
|B|
|S| δ2
≤C
,
|B|
≤C
where δ1 , δ2 are constants with 0 < δ 1 , δ2 < 1.
Throughout this paper δ2 is the same as in Lemma 1.2.
Lemma 1.3. ([7]). Suppose p(·) ∈ B(R n ). Then there exists a constant C > 0
such that for all balls B in R n ,
1
χB Lp(·) (Rn ) χB Lp(·) (Rn ) ≤ C.
|B|
In [13], we establish the following boundedness theorem on the Herz spaces with
variable exponent for a class of sublinear operators.
Lemma 1.4. ([13]). Let 0 < α < nδ2 , 0 < p < ∞ and q(·) ∈ B(Rn ). If a
sublinear operator T satisfies
(1.1)
|T f (x)| ≤ Cf 1 /|x|n,
if
dist(x, suppf ) > |x|/2,
for any integrable function f with a compact support and T is bounded on L q(·) (Rn ),
α,p
α,p
(Rn ) and Kq(·)
(Rn ), respectively.
then T is bounded on K̇q(·)
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Hongbin Wang and Zongguang Liu
2. THE ATOMIC CHARACTERIZATIONS AND THEIR APPLICATIONS
In this section, we will give the definition of Herz-type Hardy spaces with variable
α,p
α,p
(Rn ) and HKq(·)
(Rn ). S(Rn ) denotes the Schwartz space of all
exponent H K̇q(·)
rapidly decreasing infinitely differentiable functions on Rn , and S (Rn ) denotes the
dual space of S(Rn ). Let GN f (x) be the grand maximal function of f (x) defined by
GN f (x) = sup |φ∗∇ (f )(x)|,
φ∈AN
where AN = {φ ∈ S(Rn) :
sup
|α|,|β|≤N
|xα Dβ φ(x)| ≤ 1} and N > n + 1, φ∗∇ is the
nontangential maximal operator defined by
φ∗∇ (f )(x) = sup |φt ∗ f (y)|
|y−x|<t
with φt (x) = t−n φ(x/t).
Definition 2.1. Let α ∈ R, 0 < p < ∞, q(·) ∈ P(R n) and N > n + 1.
α,p
(Rn ) is
(i) The homogeneous Herz-type Hardy space with variable exponent HK̇q(·)
defined by
H K̇q(·)(Rn ) = {f ∈ S (Rn ) : GN f (x) ∈ K̇q(·)(Rn )}
α,p
α,p
and we define f H K̇ α,p (Rn ) = GN f K̇ α,p (Rn ) .
q(·)
q(·)
α,p
(Rn )
(ii) The non-homogeneous Herz-type Hardy space with variable exponent HKq(·)
is defined by
α,p
α,p
(Rn ) = {f ∈ S (Rn ) : GN f (x) ∈ Kq(·)
(Rn )}
HKq(·)
and we define f HK α,p (Rn ) = GN f K α,p (Rn ) .
q(·)
q(·)
It is obvious that GN f satisfies (1.1). Thus, by Lemma 1.4, we can easily prove
that if 0 < α < nδ2 , 0 < p < ∞ and q(·) ∈ B(Rn ), then
q(·)
α,p
α,p
(Rn ) ∩ Lloc (Rn \ {0}) = K̇q(·)
(Rn )
H K̇q(·)
and
q(·)
α,p
α,p
(Rn ) ∩ Lloc (Rn ) = Kq(·)
(Rn ).
HKq(·)
If nδ2 ≤ α < ∞, 0 < p < ∞ and q(·) ∈ B(Rn ), then
q(·)
α,p
α,p
(Rn ) ∩ Lloc (Rn \ {0}) K̇q(·)
(Rn )
H K̇q(·)
and
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The Herz-type Hardy Spaces with Variable Exponent
q(·)
α,p
α,p
(Rn ) ∩ Lloc (Rn ) Kq(·)
(Rn ).
HKq(·)
Thus we are interested in the case α ≥ nδ 2 . In this case, we establish characterizations
α,p
α,p
(Rn ) and HKq(·)
(Rn ) in terms of central atomic decompositions.
of the spaces H K̇q(·)
For x ∈ R we denote by [x] the largest integer less than or equal to x.
Definition 2.2. Let nδ 2 ≤ α < ∞, q(·) ∈ P(Rn), and non-negative integer
s ≥ [α − nδ2 ].
(i) A function a on Rn is said to be a central (α, q(·))-atom, if it satisfies
(1) supp a ⊂ B(0, r) = {x ∈ Rn : |x| < r}.
(2) aLq(·) (Rn ) ≤ |B(0, r)|−α/n.
(3) Rn a(x)xβ dx = 0, |β| ≤ s.
(ii) A function a on Rn is said to be a central (α, q(·))-atom of restricted type, if it
satisfies the conditions (2), (3) above and
(1) supp a ⊂ B(0, r), r ≥ 1.
Remark 2.1.
classical case.
If q(x) = q is a constant, then taking δ2 = 1 − 1/q we can get the
Theorem 2.1. Let nδ2 ≤ α < ∞, 0 < p < ∞ and q(·) ∈ B(Rn ). Then we have
α,p
(Rn ) if and only if
(i) f ∈ H K̇q(·)
(2.1)
∞
f=
λk ak ,
in the sense of S (Rn ),
k=−∞
where each ak is a central (α, q(·))-atom with support contained in B k and
< ∞. Moreover,
f H K̇ α,p (Rn ) ≈ inf
q(·)
|λk |p
|λk |p
k=−∞
1/p
∞
∞
,
k=−∞
where the infimum is taken over all above decompositions of f .
α,p
(Rn ) if and only if
(ii) f ∈ HKq(·)
f=
∞
λk ak ,
in the sense of S (Rn ),
k=0
where each ak is a central (α, q(·))-atom of restricted type with support contained in
∞
|λk |p < ∞. Moreover,
Bk and
k=0
f HK α,p (Rn ) ≈ inf
q(·)
∞
k=0
1/p
|λk |
p
,
1368
Hongbin Wang and Zongguang Liu
where the infimum is taken over all above decompositions of f .
Proof. We only prove (i), and (ii) can be proved
in the similar way. To prove the
necessity, choose φ ∈ C0∞ (Rn ) such that φ ≥ 0, Rn φ(x)dx = 1, supp φ ⊂ {x : |x|
≤ 1}. For j ∈ Z+ , let
φ(j) (x) = 2jn φ(2j x).
For each f ∈ S (Rn ), set
f (j) (x) = f ∗ φ(j) (x).
It is obvious that f (j) ∈ C ∞ (Rn ) and lim f (j) = f in the sense of distribution. Let
j→∞
ψ be a radial smooth function such that supp ψ ⊂ {x : 1/2 − ε ≤ |x| ≤ 1 + ε} with
0 < ε < 1/4, ψ(x) = 1 for 1/2 ≤ |x| ≤ 1. Let ψk (x) = ψ(2−k x) for k ∈ Z and
Ãk,ε = {x : 2k−1 − 2k ε ≤ |x| ≤ 2k + 2k ε}.
Observe that supp ψ k ⊂ Ãk,ε and ψk (x) = 1 for x ∈ Ak = {x : 2k−1 < |x| ≤ 2k }.
∞
ψk (x) ≤ 2, |x| > 0. Let
Obviously, 1 ≤
k=−∞

∞

 ψ (x)
ψl (x),
k
Φk (x) =
l=−∞


0, x = 0,
∞
then
x = 0,
Φk (x) = 1 for x = 0. For some m ∈ N, we denote by Pm the class of all the
k=−∞
(j)
real polynomials with the degree less than m. Let Pk (x) = PÃk,ε (f (j)Φk )(x)χÃk,ε (x)
∈ Pm (Rn ) be the unique polynomial satisfying
(j)
f (j) (x)Φk (x) − Pk (x) xβ dx = 0, |β| ≤ m = [α − nδ2 ].
Ãk,ε
Write
f
(j)
(j)
(j)
∞ ∞
(j)
(j)
(j)
(x) =
Pk (x) =
+
.
f (x)Φk (x) − Pk (x) +
k=−∞
I
k=−∞
II
(j)
(j)
(j)
(j)
(j)
, let gk (x) = f (j) (x)Φk (x) − Pk (x) and ak (x) = gk (x)/λk ,
For the term
I
where λk = b|Bk+1 |
α/n
k+1
(GN f )χl Lq(·) (Rn ) , and b is a constant which will be
l=k−1
chosen later. Note that
(j)
supp ak
⊂ Bk+1 ,
(j)
I
=
∞
k=−∞
(j)
λk ak (x).
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The Herz-type Hardy Spaces with Variable Exponent
(j)
Now we estimate gk Lq(·) (Rn ). To do this, let {φkd : |d| ≤ m} be the orthogonal
polynomials restricted to Ãk,ε with respect to the weight 1/| Ãk,ε |, which are obtained
from {xβ : |β| ≤ m} by Gram-Schmidt’s method, that is
1
k
k
φkν (x)φkµ(x)dx = δνµ .
φν , φµ =
|Ãk,ε | Ãk,ε
(j)
It is readily to see that Pk (x) =
hand, from
1
|Ãk,ε |
f (j) Φk , φkd φkd (x) for x ∈ Ãk,ε . On the other
|d|≤m
k
k
Ãk,ε φν (x)φµ(x)dx = δνµ we infer that
1
|Ã1,ε |
Ã1,ε
φkν (2k−1 y)φkµ (2k−1 y)dy = δνµ .
We can get φkν (2k−1 y) = φ1ν (y) a.e.. That is φkν (x) = φ1ν (21−k x) a.e. for x ∈ Ãk,ε .
Thus |φkν (x)| ≤ C, and for x ∈ Ãk,ε , by the generalized Hölder inequality we have
C
(j)
|f (j)(x)Φk (x)|dx
|Pk (x)| ≤
|Ãk,ε | Ãk,ε
C
f (j)Φk Lq(·) (Rn ) χÃk,ε Lq (·) (Rn ) .
≤
|Ãk,ε |
Therefore, by Lemma 1.3 we have
(j)
gk Lq(·) (Rn )
(j)
≤ f (j) Φk Lq(·) (Rn ) + Pk Lq(·) (Rn )
C
f (j) Φk Lq(·) (Rn )χÃk,ε Lq (·) (Rn )χÃk,ε Lq(·) (Rn)
|Ãk,ε |
≤ f (j) Φk Lq(·) (Rn ) + Cf (j) Φk Lq(·) (Rn )
≤ f (j) Φk Lq(·) (Rn ) +
≤ C(f ∗ φ(j))Φk Lq(·) (Rn)
≤ C
k+1
(GN f )χl Lq(·) (Rn ) .
l=k−1
(j)
(j)
Choose b = C , then ak Lq(·) (Rn ) ≤ |Bk+1 |−α/n and each ak is a central (α, q(·))atom with support contained in B k+1 . Furthermore,
k+1
p
∞
∞
p
pα/n
|λk | ≤ C
|Bk+1 |
(GN f )χl Lq(·) (Rn ) ≤ CGN f pK̇ α,p (Rn ) ,
k=−∞
k=−∞
l=k−1
q(·)
1370
Hongbin Wang and Zongguang Liu
where C is independent of j and f .
(j)
. Let {ψdk : |d| ≤ m} be the dual basis of {x β : |β| ≤ m}
It remains to estimate
II
with respect to the weight 1/| Ãk,ε| on Ãk,ε , that is
1
k β
xβ ψdk (x)dx = δβd .
ψd , x =
|Ãk,ε | Ãk,ε
Similar to the method of [9], let
k
k
ψd (x)χÃk,ε (x) ψdk+1 (x)χÃk+1,ε (x) (j)
−
f (j)(x)Φl (x)xddx.
hk,d (x) =
n
|Ãk,ε |
|Ãk+1,ε |
R
l=−∞
We can write
(j)
=
II
=
=
∞
f (j)Φk , xd ψdk (x)χÃk,ε (x)
k=−∞ |d|≤m
∞ Rn
|d|≤m k=−∞
k
∞
|d|≤m k=−∞
×
=
f
(j)
|Ãk,ε |
∞
|d|≤m k=−∞
where
αk,d = b̃
Φk x dx
l=−∞
ψdk (x)χÃk,ε (x)
−
d
Rn
Ãk,ε (x)
d
|Ãk,ε |
f (j) (x)Φl (x)xd dx
ψdk+1 (x)χÃk+1,ε (x)
|Ãk+1,ε |
(j)
αk,d hk,d (x)/αk,d
k+2
ψ k (x)χ
=
∞
|d|≤m k=−∞
(j)
αk,d ak,d (x),
(GN f )χl Lq(·) (Rn ) |Bk+2 |α/n
l=k−1
and b̃ is a constant which will be chosen later. Note that
k
k d
|Φl (x)x |dx =
|Φl (x)xd |dx ≤ C2k(n+|d|) .
Rn l=−∞
l=−∞
Ãk,ε
By a computation we have
k
f (j) (y)
Φl (y)y d dy ≤ C2k(n+|d|) GN f (x), x ∈ Bk+2 .
Rn
l=−∞
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The Herz-type Hardy Spaces with Variable Exponent
This together with the inequality that
k
k+1
ψd (x)χÃk,ε (x) ψdk+1 (x)χÃk+1,ε (x)
−k(n+|d|)
−
≤ C2
χl (x),
|Ãk,ε |
|Ãk+1,ε |
l=k−1
shows that
(j)
hk,d Lq(·) (Rn )
≤C
k+1
(GN f )χl Lq(·) (Rn ).
l=k−1
(j)
Take b̃ = C . It is readily to verify that each ak,d is a central (α, q(·))-atom with support
k+2
(Gf )χl Lq(·) (Rn ) |Bk+2 |α/n ,
contained in Ãk,ε ∪ Ãk+1,ε ⊂ Bk+2 , and αk,d = C l=k−1
where C is a constant independent of j, f, k and d. Moreover,
k+1
p
∞
|αk,d |p ≤ C
|Bk+2 |αp/n
(GN f )χl Lq(·) (Rn )
k,d
k=−∞
≤ CGN f p
l=k−1
α,p
K̇q(·) (Rn )
where C is independent of j and f .
Thus we obtain that
f (j) (x) =
< ∞,
∞
(j)
λd ad (x),
d=−∞
(j)
where each ad is a central (α, q(·))-atom with support contained in Ãd,ε ∪ Ãd+1,ε ⊂
Bd+2 , λd is independent of j and
∞
1/p
|λd |
≤ CGN f pK̇ α,p (Rn ) < ∞,
p
q(·)
d=−∞
where C is independent of j and f .
Since
(j)
sup a0 Lq(·) (Rn ) ≤ |B2 |−α/n ,
j∈Z+
(jn )
(j)
by the Banach-Alaoglu theorem we can obtain a subsequence {a0 0 } of {a0 } converging in the weak∗ topology of Lq(·)(Rn ) to some a0 ∈ Lq(·)(Rn ). It is readily to
verify that a0 is a central (α, q(·))-atom supported on B2 . Next, since
(jn0 )
sup a1
jn0 ∈Z+
Lq(·) (Rn ) ≤ |B3 |−α/n ,
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Hongbin Wang and Zongguang Liu
(jn1 )
another application of the Banach-Alaoglu theorem yields a subsequence {a1
(jn )
{a1 0 }
} of
which converges weak∗ in Lq(·)(Rn ) to a central (α, q(·))-atom a1 with support in B3 . Furthermore,
(jn )
sup a−1 1 Lq(·) (Rn ) ≤ |B1 |−α/n.
jn1 ∈Z+
(jn
)
(jn )
Similarly, there exists a subsequence {a−1 −1 } of {a−1 1 } which converges weak∗
in Lq(·) (Rn ) to some a−1 ∈ Lq(·)(Rn ), and a−1 is a central (α, q(·))-atom supported
on B1 . Repeating the above procedure for each d ∈ Z, we can find a subsequence
(jn )
(j)
{ad d } of {ad } converging weak∗ in Lq(·)(Rn ) to some ad ∈ Lq(·)(Rn ) which is
a central (α, q(·))-atom supported on Bd+2 . By the usual diagonal method we obtain
(j )
a subsequence {j ν } of Z+ such that for each d ∈ Z, lim ad ν = ad in the weak∗
ν→∞
topology of Lq(·)(Rn ) and therefore in S (Rn ).
Now our proof is reduced to prove that
(2.2)
f=
∞
λdad , in the sense of S (Rn ).
d=−∞
(j )
For each ϕ ∈ S(Rn ), note that supp ad ν ⊂ (Ãd,ε ∪ Ãd+1,ε ) ⊂ (Ad−1 ∪ Ad ∪ Ad+1 ∪
Ad+2 ). We have
f, ϕ = lim
ν→∞
∞
d=−∞
λd
Rn
(j )
ad ν (x)ϕ(x)dx.
See [9] for the details.
Recall that m = [α − nδ2 ]. If d ≤ 0, then by Lemma 1.2 and the generalized
Hölder inequality we have

 β
D ϕ(0) β  (jν )
(jν )

x
ad (x)ϕ(x)dx = ad (x) ϕ(x) −
dx
β!
Rn
Rn
|β|≤m
(j )
|ad ν (x)| · |x|m+1 dx
≤C
Rn
(j )
d(m+1)
|ad ν (x)|dx
≤ C2
Rn
≤ C2d(m+1−α) χBd+2 Lq (·) (Rn)
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The Herz-type Hardy Spaces with Variable Exponent
|Bd+2 | δ2
χB2 Lq (·) (Rn )
≤ C2
|B2 |
|B2 |
χB0 Lq (·) (Rn )
≤ C2d(m+1−α+nδ2 )
|B0 |
d(m+1−α+nδ2 )
−q (x)
≤ C2
inf λ > 0 :
λ
dx ≤ 1
B
0
+
λ−(q ) dx ≤ 1
≤ C2d(m+1−α+nδ2 ) inf 1 ≥ λ > 0 :
d(m+1−α)
= C2
d(m+1−α+nδ2 )
|B0 |
B0
1/(q )+
= C2d(m+1−α+nδ2 ) ,
where C is independent of d.
If d > 0, let k0 ∈ Z+ such that k 0 + α − n > 0, then by Lemma 1.2 and the
generalized Hölder inequality we have
(jν )
(j )
ad (x)ϕ(x)dx ≤ C
|ad ν (x)||x|−k0 dx
Rn
≤ C2
Rn
−d(k0 +α)
χBd+2 Lq (·) (Rn )
|Bd+2 |
χB0 Lq (·) (Rn )
|B0 |
−d(k0 +α−n)
−q (x)
≤ C2
inf λ > 0 :
λ
dx ≤ 1
B0
−d(k0 +α−n)
−(q )+
inf 1 ≥ λ > 0 :
λ
dx ≤ 1
≤ C2
≤ C2−d(k0 +α)
B0
= C2−d(k0 +α−n) ,
where C is independent of d.
Let
|λd|2d(m+1−α+nδ2 ),
|λd|2−d(k0 +α−n) ,
µd =
Then
∞
∞
|µd | ≤ C
d=−∞
and
1/p
|λd|p
≤ CGN f K̇ α,p (Rn ) < ∞
d=−∞
|λd | Rn
d ≤ 0,
d > 0,
q(·)
(j )
ad ν (x)ϕ(x)dx
≤ C|µd |,
which implies that
f, ϕ =
∞
d=−∞
lim λd
ν→∞
Rn
(j )
ad ν (x)ϕ(x)dx
=
∞
d=−∞
λd
Rn
ad (x)ϕ(x)dx.
1374
Hongbin Wang and Zongguang Liu
This establishes the identity (2.2).
To prove the sufficiency, we consider the two cases 0 < p ≤ 1 and 1 < p < ∞.
If 0 < p ≤ 1, it suffices to show that for each central (α, q(·))-atom a,
GN aK̇ α,p (Rn ) ≤ C,
q(·)
with the constant C > 0 independent of a. For a fixed central (α, q(·))-atom a, with
supp a(x) ⊂ B(0, 2 k0 ) for some k0 ∈ Z. Write
GN apK̇ α,p (Rn) =
q(·)
k
0 +3
|Bk |αp/n(GN a)χk pLq(·) (Rn )
k=−∞
∞
+
k=k0 +4
|Bk |αp/n(GN a)χk pLq(·) (Rn)
= I + II.
By the Lq(·)(Rn ) boundedness of the grand maximal operator GN we have
I≤
GN apLq(·) (Rn )
k
0 +3
|Bk |
αp/n
≤
CapLq(·) (Rn )
k=−∞
k
0 +3
|Bk |αp/n ≤ C.
k=−∞
To estimate II, we need a pointwise estimate for GN a(x) on Ak . Let φ ∈ AN , m ∈ N
such that α − nδ2 < m + 1. Denote by Pm the m-th order Taylor series expansion. If
|x − y| < t, then from the vanishing moment condition of a we have
y y−z
−n − Pm
dz a(z) φ
|a ∗ φt (y)| = t n
t
t
R
z m+1
|a(z)| (1 + |y − θz|/t)−(n+m+1) dz
≤ Ct−n
n
t
R
|a(z)||z|m+1 (t + |y − θz|)−(n+m+1) dz,
≤C
Rn
where 0 < θ < 1. Since x ∈ Ak for k ≥ k0 + 4, we have |x| ≥ 2 · 2k0 +1 . From
|x − y| < t and |z| < 2k0 +1 , we have
t + |y − θz| ≥ |x − y| + |y − θz| ≥ |x| − |z| ≥ |x|/2.
Thus,
|a(z)||z|m+1 (|x − y| + |y − θz|)−(n+m+1) dz
|a(z)|dz
≤ C2k0 (m+1) |x|−(n+m+1)
|a ∗ φt (y)| ≤ C
Rn
Rn
≤ C2k0 (m+1) |x|−(n+m+1) |Bk0 |−α/n χBk0 Lq (·) (Rn ) .
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The Herz-type Hardy Spaces with Variable Exponent
Therefore, we have
GN a(x) ≤ C2k0 (m+1)−k(n+m+1) |Bk0 |−α/n χBk0 Lq (·) (Rn ) , x ∈ Ak , and k ≥ k0 + 4.
So by Lemma 1.2 and Lemma 1.3 we have
∞
II ≤ C
2
p[k0 (m+1)−k(n+m+1)]
k=k0 +4
|Bk |
|Bk0 |
pα/n
×χBk0 p q (·) n χBk pLq(·) (Rn )
(R )
L
∞
|Bk | pα/n
p[k0 (m+1)−k(n+m+1)]
≤C
2
|Bk0 |
k=k0 +4
p
×χBk0 pLq (·) (Rn ) |Bk |χBk −1
Lq (·) (Rn )
p
∞
χBk0 Lq (·) (Rn )
=C
2p(k0 −k)(m+1−α)
χBk Lq (·) (Rn )
k=k0 +4
∞
≤C
2p(k0 −k)(m+1−α+nδ2 ) ≤ C.
k=k0 +4
This proves the desired estimate for the case 0 < p ≤ 1.
If 1 < p < ∞, write
∞
p
∞
|Bk |αp/n
|λl |(GN al )χk Lq(·) (Rn )
GN f pK̇ α,p (Rn ) ≤
q(·)
k=−∞
l=−∞
∞
p
∞
αp/n
≤C
|Bk |
|λl |al Lq(·) (Rn )
k=−∞
∞
+C
|Bk |
αp/n
l=k−1
k−2
k=−∞
p
|λl |(GN al )χk Lq(·) (Rn )
l=−∞
= III + IV.
Using the Hölder inequality, we have
∞
p
∞
|Bk |αp/n
|λl ||Bl |−α/n
III ≤ C
≤C
≤C
≤C
k=−∞
∞
k=−∞
∞
k=−∞
∞
l=−∞
|Bk |
|Bk |
αp/n
l=k−1
∞
|λl | |Bl |
l=k−1
∞
αp/(2n)
l=k−1
|λl |p .
p
−αp/(2n)
∞
l=k−1
|λl |p|Bl |−αp/(2n)
p/p
|Bl |
−αp /(2n)
1376
Hongbin Wang and Zongguang Liu
Now suppose α − nδ2 < m + 1. As in the argument for II, we can obtain that
IV
≤C
≤C
≤C
≤C
∞
k=−∞
∞
k−2
|λl |2
l=−∞
k−2
k=−∞ l=−∞
k−2
∞
k=−∞
∞
l(m+1)−k(n+m+1)
|Bk |
|Bl |
p
α/n
p
χBlLq (·) (Rn )χBk Lq(·) (Rn )
|λl |2(l−k)(m+1−α+nδ2 )
|λl |p 2(l−k)(m+1−α+nδ2 )p/2
l=−∞
k−2
p/p
2(l−k)(m+1−α+nδ2
)p /2
l=−∞
|λl |p .
l=−∞
This finishes the proof of Theorem 2.1.
α,p
(Rn ) ∩ S(Rn ), we can replace f (j) by f in the proof
Remark 2.2. If f ∈ H K̇q(·)
of the necessity, and have
f (x) =
∞
λk ak (x) +
k=−∞
∞
µk bk (x),
k=−∞
where
ak Lq(·) (Rn ) ≤ |B(0, 2k+1 )|−α/n, bk Lq(·) (Rn ) ≤ |B(0, 2k+2 )|−α/n,
supp ak ⊂ Ãk,ε , supp bk ⊂ Ãk,ε ∪ Ãk+1,ε ,
and
0 ≤ λk , µk ≤ C2αk
k+1
GN (f )χj Lq(·) (Rn ),
j=k−1
with N > n + α + 1.
Remark 2.3. For the case 0 < p ≤ 1, if we remove the condition supp ak ⊂ Bk ,
then the conclusion of Theorem 2.1 is also true .
As an application of the atomic decomposition theorems, we shall extend Lemma
1.4 to the case of α ≥ nδ2 .
Theorem 2.2. Let nδ2 ≤ α < ∞, 0 < p < ∞, q(·) ∈ B(Rn ) and the integer
s = [α − nδ2 ]. If a sublinear operator T satisfies that
(i) T is bounded on L q(·)(Rn );
1377
The Herz-type Hardy Spaces with Variable Exponent
(ii) there exists a constant δ > 0 such that s + δ > α − nδ 2 , and for any compact
support function f with
f (x)xβ dx = 0, |β| ≤ s,
Rn
T f satisfies the size condition
|T f (x)| ≤ C(diam(supp f ))s+δ |x|−(n+s+δ) f 1 ,
(2.3)
dist(x, suppf ) ≥ |x|/2.
if
α,p
α,p
Then T can be extended to be a bounded operator from H K̇q(·)(Rn ) to K̇q(·)(Rn ) (or
α,p
α,p
(Rn ) to Kq(·)
(Rn )).
bounded from HK q(·)
α,p
(Rn ). By
It suffices to prove homogeneous case. Suppose f ∈ HK̇q(·)
∞
λj bj in the sense of S (Rn ), where each bj is a central
Theorem 2.1, f =
Proof.
j=−∞
(α, q(·))-atom with support contained in Bj and

f H K̇ α,p (Rn ) ≈ inf 
q(·)
1/p
∞
|λj |p
.
j=−∞
Therefore, we get
T f pK̇ α,p (Rn ) =
q(·)
∞
2kαp (T f )χk pLq(·) (Rn )
k=−∞
∞
≤C
2
kαp
k−2
|λj |(T bj )χk Lq(·) (Rn )
p
j=−∞
p ∞
kαp
2
|λj |(T bj )χk Lq(·) (Rn )
k=−∞
+
∞
k=−∞
j=k−1
= C(I1 + I2 ).
Let us first estimate I 1 . By (2.3) and the generalized Hölder inequality, we get
−(n+s+δ) j(s+δ)
2
|bj (y)|dy
|T bj (x)| ≤ C|x|
Bj
≤ C2−k(n+s+δ) 2j(s+δ) bj Lq(·) (Rn ) χBj Lq (·) (Rn )
≤ C2j(s+δ−α)−k(s+δ+n) χBj Lq (·) (Rn ) .
So by Lemma 1.1 and Lemma 1.2, we have
1378
Hongbin Wang and Zongguang Liu
(T bj )χk Lq(·) (Rn )
(2.4)
≤ C2j(s+δ−α)−k(s+δ+n) χBj Lq (·) (Rn ) χBk Lq(·) (Rn )
≤ C2j(s+δ−α)−k(s+δ) 2−kn |Bk |χBk −1q (·) n χBj Lq (·) (Rn )
L
(R )
χBj Lq (·) (Rn )
= C2j(s+δ−α)−k(s+δ)
χBk Lq (·) (Rn )
(s+δ+nδ2 )(j−k)−jα
≤ C2
.
Therefore, when 0 < p ≤ 1, by nδ2 ≤ α < s + δ + nδ2 , we get
I1 =
(2.5)
∞
2
kαp
k=−∞
∞
≤C
=C
2
k=−∞
∞
k−2
|λj |(T bj )χk Lq(·) (Rn )
j=−∞
k−2
kαp
p [(s+δ+nδ2 )(j−k)−jα]p
|λj | 2
j=−∞
∞
∞
2(j−k)(s+δ+nδ2 −α)p ≤ C
|λj |p
j=−∞
p
|λj |p .
j=−∞
k=j+2
When 1 < p < ∞, take 1/p + 1/p = 1. Since nδ2 ≤ α < s + δ + nδ2 , by (2.4) and
the Hölder inequality, we have
∞
I1 ≤ C
≤C
2
kαp
k−2
|λj |2
(s+δ+nδ2 )(j−k)−jα
j=−∞
k=−∞
∞ k−2
p (j−k)(s+δ+nδ2 −α)p/2
|λj | 2
k=−∞ j=−∞
p/p
k−2
(j−k)(s+δ+nδ2 −α)p /2
×
2
j=−∞
(2.6)
≤C
=C
≤C
p
∞ k−2
p (j−k)(s+δ+nδ2 −α)p/2
|λj | 2
j=−∞
∞
p
|λj |
2(j−k)(s+δ+nδ2 −α)p/2
j=−∞
k=j+2
∞
|λj |p .
j=−∞
k=−∞
∞
Let us now estimate I2 . When 0 < p ≤ 1, by Lq(·)(Rn ) boundedness of T , we
1379
The Herz-type Hardy Spaces with Variable Exponent
have
I2 =
∞
2
kαp
k=−∞
∞
≤C
(2.7)
≤C
=C
k=−∞
∞
k=−∞
∞
2
2
∞
|λj |(T bj )χk Lq(·) (Rn )
j=k−1
∞
kαp
kαp
j=k−1
∞
|λj |
p
bj pLq(·) (Rn )
|λj | |Bj |
p
−αp/n
j=k−1
j+1
|λj |p
j=−∞
p
∞
2(k−j)αp ≤ C
|λj |p .
j=−∞
k=−∞
When 1 < p < ∞, by Lq(·)(Rn ) boundedness of T and the Hölder inequality, we have
∞
∞
p/2
kαp
p
2
|λj | (T bj )χk Lq(·) (Rn )
I2 ≤ C
k=−∞
∞
×
j=k−1
p /2
(T bj )χk Lq(·) (Rn )
j=k−1
≤C
(2.8)
≤C
≤C
=C
∞
k=−∞
∞
k=−∞
∞
k=−∞
∞
j=−∞
2
2
kαp
kαp
∞
j=k−1
∞
|λj |
p
p/p
p/2
bj Lq(·) (Rn )
∞
|λj | |Bj |
∞
p
−αp/(2n)
|λj |p |Bj |−αp/(2n)
j=k−1
|λj |p
j+1
2(k−j)αp/2 ≤ C
k=−∞
p/p
j=k−1
∞
j=k−1
2kαp/2
p/2
bj Lq(·) (Rn )
∞
|Bj |
−αp /(2n)
p/p
j=k−1
|λj |p.
j=−∞
Combining (2.5)-(2.8), we have
T f K̇ α,p (Rn ) ≤ Cf H K̇ α,p (Rn ) .
q(·)
q(·)
Thus, the proof of Theorem 2.2 is completed.
Furthermore, we will consider the Calderón-Zygmund operator T of Coifman and
Meyer with associated standard kernel K in the sense of
K(x, y)f (y)dy, x ∈ supp f,
T f (x) =
Rn
which is a L q(·)(Rn )-bounded operator. See [2] for more details.
1380
Hongbin Wang and Zongguang Liu
Theorem 2.3. Suppose that T is an above Calder ón-Zygmund operator, and
that 0 < δ ≤ 1 is the constant associated with the standard kernel K, then for
α,p
α,p
(Rn ) to K̇q(·)
(Rn ).
nδ2 ≤ α < nδ2 + δ and 0 < p < ∞, T is bounded from H K̇q(·)
Proof. Note that nδ 2 ≤ α < nδ2 + δ implies that s = [α − nδ 2 ] = 0. Thus, the
operator T considered here satisfies (2.3) with s = 0. The desired conclusion follows
from Theorem 2.2 directly.
3. THE MOLECULAR CHARACTERIZATIONS AND THEIR APPLICATIONS
In this section, we will consider the molecular decomposition of the Herz-type
Hardy spaces with variable exponent. We first give the notation of molecule.
Definition 3.1. Let nδ 2 ≤ α < ∞, 0 < p < ∞, q(·) ∈ P(Rn), and s ≥ [α−nδ2 ]
be a non-negative integer. Set ε > max{s/n, α/n − δ2 }, a = δ2 − α/n + ε and
b = δ2 + ε. A function Ml ∈ Lq(·)(Rn ) with l ∈ Z (or l ∈ N) is said to be a dyadic
central (α, q(·); s, ε)l-molecule (or dyadic central (α, q(·); s, ε)l)-molecule of restricted
type) if it satisfies
(1) Ml Lq(·) (Rn ) ≤ 2−lα .
1−a/b
a/b
(2) Rq(·)(Ml ) = Ml Lq(·) (Rn ) | · |nb Ml (·)Lq(·) (Rn ) < ∞.
(3) Rn Ml (x)xβ dx = 0, for any β with |β| ≤ s.
Definition 3.2. Let nδ 2 ≤ α < ∞, 0 < p < ∞, q(·) ∈ P(Rn), and s ≥ [α−nδ2 ]
be a non-negative integer. Set ε > max{s/n, α/n−δ2 }, a = δ2 −α/n+ε and b = δ2 +ε.
(i) A function M ∈ Lq(·)(Rn ) is said to be a central (α, q(·); s, ε)-molecule if it
satisfies
1−a/b
a/b
(1) Rq(·)(M ) = M Lq(·) (Rn ) | · |nb M (·)Lq(·) (Rn ) < ∞.
(2) Rn M (x)xβ dx = 0, for any β with |β| ≤ s.
(ii) A function M ∈ L q(·)(Rn ) is said to be a central (α, q(·); s, ε)-molecule of
restricted type if it satisfies (1), (2) in (i) and
(3) M Lq(·) (Rn ) ≤ 1.
The following lemma proves that molecule is a generalization of atom.
Lemma 3.1. Let nδ2 ≤ α < ∞, 0 < p < ∞, q(·) ∈ P(Rn ), s ≥ [α − nδ2 ] be a
non-negative integer, ε > max{s/n, α/n − δ 2 }, a = δ2 − α/n + ε and b = δ2 + ε.
If M is a central (α, q(·))-atom (or (α, q(·))-atom of restricted type), M is also a
central (α, q(·); s, ε)-molecule (or (α, q(·); s, ε)-molecule of restricted type) such that
Rq(·)(M ) ≤ C with C independent of M .
1381
The Herz-type Hardy Spaces with Variable Exponent
Proof. We only need consider the case that a is a (α, q(·))-atom with support on
a ball B(0, r). A straightforward computation leads to that
1−a/b
a/b
M Lq(·) (Rn ) | · |nb M (·) q(·) n ≤ r nb(1−a/b)M Lq(·) (Rn ) ≤ Cr α r −α ≤ C.
L
(R )
Now we give the molecular decomposition of the Herz-type Hardy spaces with
variable exponent.
Theorem 3.1. Let nδ2 ≤ α < ∞, 0 < p < ∞, q(·) ∈ B(Rn ), and s ≥ [α − nδ2 ]
be a non-negative integer. Set ε > max{s/n, α/n − δ 2 }, a = δ2 − α/n + ε and
b = δ2 + ε. Then we have
α,p
(Rn ) if and only if f can be represented as
(i) f ∈ H K̇q(·)
∞
f=
λk Mk ,
in the sense of S (Rn ),
k=−∞
where each Mk is a dyadic central (α, q(·); s, ε) k-molecule, and
Moreover,
f H K̇ α,p (Rn ) ≈ inf
q(·)
|λk |p
|λk |p < ∞.
k=−∞
1/p
∞
∞
,
k=−∞
where the infimum is taken over all above decompositions of f .
α,p
(Rn ) if and only if
(ii) f ∈ HKq(·)
∞
λk Mk , in the sense of S (Rn ),
f=
k=0
where each Mk is a dyadic central (α, q(·); s, ε) k-molecule of restricted type, and
∞
|λk |p < ∞. Moreover,
k=0
f HK α,p (Rn ) ≈ inf
q(·)
∞
1/p
|λk |
p
,
k=0
where the infimum is taken over all above decompositions of f .
Theorem 3.2. Let nδ2 ≤ α < ∞, 0 < p ≤ 1, q(·) ∈ B(Rn ), and s ≥ [α − nδ2 ]
be a non-negative integer. Set ε > max{s/n, α/n − δ 2 }, a = δ2 − α/n + ε and
b = δ2 + ε. Then we have
α,p
(Rn ) if and only if f can be represented as
(i) f ∈ H K̇q(·)
f=
∞
k=1
λk Mk ,
in the sense of S (Rn ),
1382
Hongbin Wang and Zongguang Liu
where each Mk is a central (α, q(·); s, ε)-molecule, and
f H K̇ α,p (Rn ) ≈ inf
q(·)
∞
∞
|λk |p < ∞. Moreover,
k=1
1/p
|λk |p
,
k=1
where the infimum is taken over all above decompositions of f .
α,p
(Rn ) if and only if
(ii) f ∈ HKq(·)
f=
∞
λk Mk ,
in the sense of S (Rn ),
k=1
where each Mk is a central (α, q(·); s, ε)-molecule of restricted type, and
∞. Moreover,
f HK α,p (Rn ) ≈ inf
q(·)
∞
|λk |p <
k=1
1/p
|λk |p
∞
,
k=1
where the infimum is taken over all above decompositions of f .
By Theorem 2.1, Remark 2.2 and Lemma 3.1, we know that Theorem 3.1 and
Theorem 3.2 can be obtained from the following lemma.
Lemma 3.2. Let nδ2 ≤ α < ∞, 0 < p < ∞, q(·) ∈ B(Rn ), s ≥ [α − nδ2 ] be a
non-negative integer, ε > max{s/n, α/n − δ 2 }, a = δ2 − α/n + ε and b = δ2 + ε.
(i) If 0 < p ≤ 1, there exists a constant C such that for any central (α, q(·); s, ε)molecule (or (α, q(·); s, ε)-molecule of restricted type) M ,
M H K̇ α,p (Rn ) ≤ C (or M HK α,p (Rn ) ≤ C).
q(·)
q(·)
(ii) There exists a constant C such that for any l ∈ Z(or l ∈ N) and dyadic central
(α, q(·); s, ε)l-molecule (or dyadic central (α, q(·); s, ε) l-molecule of restricted type)
Ml ,
Ml H K̇ α,p (Rn ) ≤ C (or Ml HK α,p (Rn ) ≤ C).
q(·)
q(·)
Proof. We only prove (i) for homogeneous case, the proof for non-homogeneous
case and the proof of (ii) are similar. Let M be a central (α, q(·); s, ε)-molecule. Let
−1/α
σ = M Lq(·) (Rn ) , E0 = {x : |x| ≤ σ}
and
Ek,σ = {x : 2k−1 σ < |x| ≤ 2k σ}, k ∈ Z+ .
The Herz-type Hardy Spaces with Variable Exponent
1383
Set Bk,σ = {x : |x| ≤ 2k σ}. Denote by χk,σ the characteristic function of Ek,σ . It
follows that
∞
M (x)χk,σ (x).
M (x) =
k=0
Let Mk (x) = M (x)χk,σ (x). We denote by Pm the class of all real polynomials of
degree m. Let PEk,σ Mk ∈ Pm be the unique polynomial satisfying
(3.1)
Ek,σ
Mk (x) − PEk,σ Mk (x) xβ dx = 0, |β| ≤ s.
Set Qk (x) = (PEk,σ Mk )(x)χk,σ (x). If we can prove that
(a) there is a constant C > 0 and a sequences of numbers {λ k }k such that
∞
|λk |p < ∞, Mk − Qk = λk ak ,
k=0
where each ak is a (α, q(·))-atom;
∞
Qk has a (α, q(·))-atom decomposition,
(b)
k=0
then our desired conclusion can be deduced directly.
We first show (a). Without loss of generality, we can suppose that Rq(·)(M ) = 1,
which implies that
nb
| · | M (·)
−a/(b−a)
= M Lq(·) (Rn ) = σ na.
Lq(·) (Rn)
Set {ϕkl : |l| ≤ s} ⊂ Ps (Rn ) such that
ϕkµ , ϕkν Ek,σ
1
=
|Ek,σ |
Ek,σ
ϕkµ (x)ϕkν (x)dx = δµν .
It is easy to see that
(3.2)
Qk (x) =
Mk , ϕkl Ek,σ ϕkl (x), if x ∈ Ek,σ
|l|≤s
and
C
|Qk (x)| ≤
|Ek,σ |
Ek,σ
|Mk (x)|dx.
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Hongbin Wang and Zongguang Liu
Thus for any k ∈ Z+ , by Lemma 1.3 and b − a = α/n we have
Mk − Qk Lq(·) (Rn )
≤ Mk Lq(·) (Rn) + Qk Lq(·) (Rn )
C
Mk Lq(·) (Rn )χEk,σ Lq (·) (Rn ) χEk,σ Lq(·) (Rn )
|Ek,σ |
≤ Mk Lq(·) (Rn) + CMk Lq(·) (Rn )
≤ Mk Lq(·) (Rn) +
≤ CMk Lq(·) (Rn )
≤ C | · |nb M (·)
= C|2k σ|
|2k σ|−nb
Lq(·) (Rn )
−nb na
−kna
σ
= C2
|Bk,σ |−α .
We see that Mk − Qk = λk ak , with λk = C2−knα and ak a central (α, q(·))-atom
∞
|λk |p < ∞.
supported in Bk,σ . Obviously,
k=0
Next we will show (b). Let {ψ lk : |l| ≤ s} ⊂ Ps (Rn ) be the dual basis of
α
{x : |α| ≤ s} with respect to the weight 1/|E k,σ | on Ek,σ , that is
1
k α
ψ k (x)xαdx = δlα .
ψl , x =
|Ek,σ | Ek,σ l
If set ϕkl (x) =
k ν
βlν
x and ψlk (x) =
|ν|≤s
k
βνγ
ψlk , xγ =
|γ|≤s
k k
τνl
ϕν (x), then we have
|ν|≤s
k
= ψlk , ϕkν =
τνl
So ψlk (x) =
k
k
βνγ
δlγ = βνl
.
|γ|≤s
k k
βνl
ϕν (x). By a computation we deduce that for x ∈ Ek,σ ,
|ν|≤s
Mk , ϕkl Ek,σ ϕkl (x) = Mk ,
k ν
βνl
x Ek,σ ϕkl (x) =
|ν|≤s
k k
Mk , xν Ek,σ βνl
ϕl (x),
|ν|≤s
which together with (3.2) implies that
(3.3)
Qk (x) =
Mk , xl Ek,σ ψlk (x), if x ∈ Ek,σ .
|l|≤s
Now we assert that there is a constant C > 0 such that
(3.4)
|ψlk (x)| ≤ C(2k−1 σ)−|l|.
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The Herz-type Hardy Spaces with Variable Exponent
We set E = {x ∈ Rn : 1 ≤ |x| ≤ 2}, F = {x ∈ Rn : |x| ≤ 1}, {el : |l| ≤ s} ⊂
1
α
n
Ps (Rn ) satisfying |E|
E el (x)x dx = δlα , and {ẽl : |l| ≤ s} ⊂ Ps (R ) satisfying
1
α
|F | F ẽl (x)x dx = δlα .
Noting that
1
1
k
α
ψ (x)x dx =
(2k−1 σ)|α|ψlk (2k−1 σy)y αdy,
δlα =
|Ek,σ | Ek,σ l
|E| E
we get el (y) = (2k−1 σ)|l|ψlk (2k−1 σy). This in turn leads to that
x , x ∈ Ek,σ .
ψlk (y) = (2k−1 σ)−|l|el k−1
2 σ
Similarly, we have
ψlk (x) = (2k−1 σ)−|l| ẽl
x
, x ∈ F.
σ
Taking C = sup {el L∞ (Rn ), ẽl L∞ (Rn )}. Then (3.4) follows directly.
l:|l|≤s
Now we can conclude (b). Set
Nlk =
∞
|Ej,σ |Mj , xl Ej,σ , k ∈ N.
j=k
It is readily to see that
Nl0
=
∞
|Ej,σ |Mj , x Ej,σ =
l
j=0
∞ j=0
l
M (x)x dx =
Ej,σ
Rn
M (x)xl dx = 0,
and for k ∈ Z+ , there exists Eσ ⊂ Ej,σ such that |E σ | = min{1, |Ej,σ |}, so we have
|Nlk |
≤
∞ j=k Ej,σ
∞ l
Mj (·)| · | ≤C
≤C
j=k
∞
j=k
≤C
|Mj (x)xl |dx
∞
Lq(·) (Rn )
j
(2 σ)
χEj,σ Lq (·) (Rn)
Lq(·) (Rn )
(2j σ)|l|−nb+nδ2 | · |nb Mj (·)
≤C
j=k
|Ej,σ |
|Eσ |
Lq(·) (Rn )
j=k
∞
| · |nb Mj (·)
|l|−nb σ |l|+na−nb+nδ2 2j(|l|−nb+nδ2 )
≤ Cσ |l|−α+nδ2 2k(|l|−nb+nδ2 ) .
δ2
χEσ Lq (·) (Rn )
+
|Eσ |1/(q )
1386
Hongbin Wang and Zongguang Liu
This via (3.4) shows that
(3.5)
|Ek,σ |−1 |Nlk ψlk (x)χk,σ (x)| ≤ Cσ nδ2 −n−α 2−kn(b+1−δ2 ) → 0, if k → ∞.
By Abel transform and (3.5) we have
∞
Qk (x)


k
∞

|Ej,σ |Mj , xl Ej,σ 
=
k=0
|l|≤s
k=0
× |Ek,σ | ψlk (x)χk,σ (x) − |Ek+1,σ |−1 ψlk+1 (x)χk+1,σ (x)
∞
(−Nlk+1 ) |Ek,σ |−1 ψlk (x)χk,σ (x) − |Ek+1,σ |−1 ψlk+1 (x)χk+1,σ (x) .
=
j=0
−1
|l|≤s k=0
Meanwhile, we also have
k+1
|Ek,σ |−1 ψlk (x)χk,σ (x) − |Ek+1,σ |−1 ψlk+1 (x)χk+1,σ (x) Nl
≤ C|Nlk+1 ||Ek+1,σ |−1 |ψlk+1 (x)|
≤ C2−kna |Ek+1,σ |δ2 −1−α/n .
Set λlk = C2−kna and
k+1
) |Ek,σ |−1 ψlk (x)χk,σ (x) − |Ek+1,σ |−1 ψlk+1 (x)χk+1,σ (x) .
alk = λ−1
lk (−Nl
Then we have
∞
Qk (x) =
λlk alk ,
|l|≤s k=0
k=0
with alk a (α, q(·))-atom, and
∞
∞
|λlk |p < ∞.
|l|≤s k=0
The conclusion (b) then holds.
Theorem 3.3.
For any central (α, q(·))-atom f , let
K(x, y)f (y)dy, x ∈
/ supp f
T f (x) =
Rn
satisfying Rn T f (x)dx = 0 be a bounded operator on L q(·)(Rn ) for some q(·) ∈
B(Rn ), and the kernel K satisfies that there are constants C > 0 and 0 < δ ≤ 1 such
that
|y|δ
, |x| ≥ 2|y|.
|K(x, y) − K(x, 0)| ≤ C |x − y|n+δ
1387
The Herz-type Hardy Spaces with Variable Exponent
Then for any α and p with nδ 2 ≤ α
< nδ2 + δ and 0 < p <∞, there exists a constant
C such that T f H K̇ α,p (Rn ) ≤ C or T f HK α,p (Rn) ≤ C .
q(·)
q(·)
Proof. We only prove homogeneous case. Let f be a central (α, q(·))-atom supporting in B(0, r)(r > 0). It suffices to show T f is a central (α, q(·); 0, ε)-molecule
for some 1 + δ/n − δ2 ≥ ε > α/n − δ2 . To this aim, let a = δ2 − α/n + ε, b = δ2 + ε.
Obviously, we only need to verify the size condition for molecules, that is
1−a/b
a/b
≤ C,
Rq(·)(T f ) = T f Lq(·) (Rn ) | · |nb (T f )(·) q(·)
n
L
(R )
with C independent of f . To do this, we first estimate | · |nb (T f )(·)Lq(·) (Rn ).
In fact, we have
nb
(T
f
)(·)
≤ Cr nb T f Lq(·) (Rn ) ≤ Cr nb−α .
·
|
q(·)
|
L
(|·|≤2r)
On the other hand, the vanishing moment of f and the regularity of K give us that for
x with |x| > 2r,
K(x, y)f (y)dy |T f (x)| = Rn
(K(x, y) − K(x, 0))f (y)dy = Rn
|y|δ
|f (y)|dy
≤C
n+δ
Rn |x − y|
1
|f (y)|dy
≤ Cr n+δ |x|−(n+δ)
|B0,r | B0,r
≤ Cr n+δ |x|−(n+δ)M f (x)
and so by nb − n − δ ≤ 0 we have
nb
≤ Cr n+δ | · |nb−n−δ M f (·)
| · | (T f )(·) q(·)
L
(|·|>2r)
≤ Cr
n+δ+nb−n−δ
Lq(·) (|·|>2r)
M f Lq(·) (Rn )
≤ Cr nb f Lq(·) (Rn ) ≤ Cr nb−α .
Thus, we get
1−a/b
a/b
Rq(·)(T f ) = T f Lq(·) (Rn ) | · |nb (T f )(·) q(·)
≤
a/b
Cf Lq(·) (Rn )r (nb−α)(1−a/b)
−αa/b+(nb−α)(1−a/b)
≤ Cr
= C,
L
(Rn )
1388
Hongbin Wang and Zongguang Liu
where C is independent of f .
This completes the proof of Theorem 3.3.
ACKNOWLEDGMENTS
The authors are very grateful to the referees for their valuable comments.
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The Herz-type Hardy Spaces with Variable Exponent
Hongbin Wang and Zongguang Liu
Department of Mathematics
China University of Mining and Technology (Beijing)
Beijing 100083
P. R. China
E-mail: hbwang [email protected]
[email protected]
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