RIMS Kôkyûroku Bessatsu
B42 (2013), 109–136
Hardy spaces with variable exponent
By
Mitsuo Izuki∗, Eiichi Nakai∗∗ and Yoshihiro Sawano∗∗∗
Abstract
In this paper we first make a view of Lebesgue spaces with variable exponent. After
reviewing fundamental properties such as completeness, duality and associate spaces, we reconsider Hardy spaces with variable exponent. We supplement what we obtained in our earlier
paper. In Part I we collect some known basic properties toghther with their proofs. In Part II
we summarize and reinforce what we obtained in [30, 36].
Contents
I
Basic theory on function spaces with variable exponents 112
§ 1. Introduction
§ 2. The usual Lebesgue spaces-Elementary properties
§ 3. Lebesgue spaces with variable exponents
§ 3.1. Elementary properties
§ 3.2. The associate space
§ 3.3. Norm convergence, modular convergence and convergence in measure
§ 3.4. Duality (The generalized F. Riesz representation theorem)
§ 3.5. Some estimates of the norms
Received September 30, 2012. Revised December 13, 2012.
2000 Mathematics Subject Classification(s): 46E30
Mitsuo Izuki was supported by Grant-in-Aid for Scientific Research (C), No. 24540185, Japan
Society for the Promotion of Science. Eiichi Nakai was supported by Grant-in-Aid for Scientific
Research (C), No. 24540159, Japan Society for the Promotion of Science. Yoshihiro Sawano was
supported by Grant-in-Aid for Young Scientists (B) No. 24740085 Japan Society for the Promotion
of Science.
∗ Department of Mathematics, Tokyo Denki University, Adachi-ku, Tokyo 120-8551, Japan
e-mail: [email protected]
∗∗ Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan
e-mail: [email protected]
∗∗∗ Department of Mathematics and Information Science, Tokyo Metropolitan University, 1-1 MinamiOhsawa, Hachioji, Tokyo 192-0397, Japan.
e-mail: [email protected]
c 2013 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
⃝
110
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
§ 4. Banach function spaces
II
Hardy spaces with variable exponent
120
§ 5. Fundamental properties
§ 5.1. Definition of Hardy spaces
§ 5.2. Poisson integral characterization
§ 5.3. Atomic decomposition
§ 6. Atomic decompositions
§ 7. Applications of atomic decomposition
§ 7.1. Molecular decomposition
§ 7.2. Boundedness of singular integral operators
§ 7.3. Littlewood-Paley characterization
§ 8. Campanato spaces with variable growth conditions
§ 8.1. Definition of Campanato spaces with variable growth conditions
§ 9. Duality H p( · ) (Rn )-Lq,ϕ,d (Rn )
§ 9.1. Dual of H p0 (Rn ) ∩ H 1 (Rn ) with 0 < p0 ≤ 1
§ 9.2. Dual spaces of H p( · ) (Rn )
§ 9.3. An open problem
§ 10. Hölder-Zygmund spaces with variable exponents
§ 10.1. Definition of Hölder-Zygmund spaces with variable exponents
§ 11. Local Hardy spaces with variable exponents
References
Notation
In the whole paper we will use the following notation:
(1) Given a measurable set S ⊂ Rn , we denote the Lebesgue measure by |S| and the
characteristic function by χS .
(2) Given a measurable set S ⊂ Rn and a function f on Rn , we denote the mean value
∫
∫
∫
1
of f on S by fS or −S f (x) dx, namely, fS = −S f (x) dx := |S|
f (x) dx．
S
(3) The set N0 consits of all non-negative integers.
(4) Given a malti-index α = (α1 , . . . , αn ) ∈ N0 n , we write
|α| :=
n
∑
αν .
ν=1
Hardy spaces with variable exponent
111
In addition the derivative of f is denoted by
∂ |α| f
D f :=
αn .
1
∂xα
1 . . . ∂xn
α
(5) A symbol C always stands for a positive constant independent of the main parameters.
(6) An open cube Q ⊂ Rn is always asssumed to have sides parallel to the coordinate
∏n
axes. Namely we can write Q = Q(x, r) := ν=1 (xν − r/2 , xν + r/2) using x =
(x1 , . . . , xn ) ∈ Rn and r > 0.
(7) We define an open ball by
B(x, r) := {y ∈ Rn : |x − y| < r},
where x ∈ Rn and r > 0.
(8) Given a positive number s, a cube Q = Q(x, r) and an open ball B = B(x, r), we
define sQ := Q(x, sr) and sB := B(x, sr).
(9) The set Ω ⊂ Rn is measurable and satisfies |Ω| > 0.
∞
(10) The set Ccomp
(Ω) consists of all compactly supported and infinitely differentiable
functions f defined on Ω.
(11) The uncentered Hardy–Littlewood maximal operator M is given by
∫
1
M f (x) := sup
|f (y)| dy,
B∋x |B| B
where the supremum is taken over all open ballls B containing x. We can replace
the open balls {B} by the open cubes {Q}.
(12) By “a variable exponent”, we mean a measurable function p( · ) : Ω → (0, ∞).
The symbol “( · )” emphasizes that the function p does not always mean a constant
exponent p ∈ (0, ∞). Given a variable exponent p( · ) we define the following:
(a) p− := ess.inf x∈Ω p(x) = sup{a : p(x) ≥ a a.e. x ∈ Ω}.
(b) p+ := ess.supx∈Ω p(x) = inf{a : p(x) ≤ a a.e. x ∈ Ω}.
(c) Ω0 := {x ∈ Ω : 1 < p(x) < ∞} = p−1 ((1, ∞)).
(d) Ω1 := {x ∈ Ω : p(x) = 1} = p−1 (1).
(e) Ω∞ := {x ∈ Ω : p(x) = ∞} = p−1 (∞).
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Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
(f ) the conjugate exponent p′ ( · ):



∞
′
p (x) :=
p(x)
 p(x)−1


1
(x ∈ Ω1 ),
(x ∈ Ω0 ),
(x ∈ Ω∞ ),
1
namely, p(x)
+ p′ 1(x) = 1 always holds for a.e. x ∈ Ω. In particular, if p( · ) equals
to a constant p, then of course p′ ( · ) = p′ is the usual conjugate exponent.
(13) We adopt the following definition of the Fourier transform and its inverse :
∫
∫
−2πix·ξ
−1
f (ξ)e2πix·ξ dξ
Ff (ξ) :=
f (x)e
dx, F f (x) :=
Rn
Rn
for f ∈ L1 (Rn ).
(14) Using this definition of Fourier transform and its inverse, we also define
(0.1)
φ(D)f (x) := F −1 [φ · Ff ](x) = ⟨f, F −1 φ(x − ·)⟩
for f ∈ S ′ (Rn ) and φ ∈ S(Rn ).
Part I
Basic theory on function spaces with
variable exponents
§ 1.
Introduction
Recently, in harmonic analysis, partial differential equations, potential theory and
applied mathematics, many authors investigate function spaces with variable exponents.
In particular, spaces with variable exponent are necessary in the field of electronic fluid
mechanics and the applications to the recovery of graphics.
The theory of Lebesgue spaces with variable exponent dates back to Orlicz’s paper
[33] and Nakano’s books in 1950 and 1951 [31, 32]. In particular, the definition of
Musielak-Orlicz spaces is clearly written in [31]. Later, Kováčik–Rákosnı́k [19] clarified
fundamental properties of Lebesgue spaces with variable exponents and Sobolev spaces
with variable exponents. This important achievement leads to the present hot discussion
of function spaces with variable exponents.
Here is a table of brief history of function spaces with variable exponents:
Hardy spaces with variable exponent
113
• Orlicz [33] (1931) · · · Lp(·) (Ω) with 1 ≤ p− ≤ p+ < ∞.
• Nakano [32] (1951) · · · Lp(·) (Ω) with 1 ≤ p− ≤ p+ < ∞.
• Sharapudinov [37] (1979) · · · Lp(·) ([0, 1]) with 1 ≤ p− ≤ p+ ≤ ∞.
• Kováčik–Rákosnı́k [19] (1991) · · · Lp(·) (Ω) with 1 ≤ p− ≤ p+ ≤ ∞, basic theory.
One of the important problems is to prove the boundedness of the Hardy-Littlewood
maximal operator M . Once this is established, we can expect that this boundedness
can be applied to many parts of analysis. Actually, many authors tackled this hard
problem. The paper [10] by Diening is a pioneering one. Based upon the paper [10],
Cruz-Uribe, Fiorenza and Neugebauer [5, 6] have given sufficient conditions for M to
be bounded on Lebesgue spaces with variable exponents and the condition is referred
to as the log-Hölder condition.
Due to the extrapolation theorem by Cruz-Uribe–Fiorenza–Martell–Pérez [4] about
Lebesgue spaces with variable exponent, we can prove the boundedness of singular integral operators of Calderón–Zygmund type, the boundedness of commutators generated
by BMO functions and singular integral operators and the Fourier multiplier results.
§ 2.
The usual Lebesgue spaces-Elementary properties
In this section, we review classical Lebesgue spaces.
Definition 2.1.
Let 1 ≤ p < ∞. The Lebesgue space Lp (Ω) is the set of all
complex-valued measurable functions f defined on Ω satisfying ∥f ∥Lp (Ω) < ∞, where
(∫
)1/p

|f (x)|p dx
(1 ≤ p < ∞),
Ω
∥f ∥Lp (Ω) :=
ess.sup
|f (x)|
(p = ∞).
x∈Ω
Theorem 2.2 (Hölder’s inequality).
Let 1 ≤ p ≤ ∞. We have that for all f ∈
p′
L (Ω) and all g ∈ L (Ω),
∫
|f (x)g(x)| dx ≤ ∥f ∥Lp (Ω) ∥g∥Lp′ (Ω) .
p
Ω
Applying Hölder’s inequality, we obtain the following.
Theorem 2.3 (Minkowski’s inequality).
Let 1 ≤ p ≤ ∞. We have that for all
p
f, g ∈ L (Ω),
∥f + g∥Lp (Ω) ≤ ∥f ∥Lp (Ω) + ∥g∥Lp (Ω) .
Corollary 2.4.
If 1 ≤ p ≤ ∞, then ∥ · ∥Lp (Ω) is a norm.
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Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
§ 3.
Lebesgue spaces with variable exponents
Lebesgue spaces with variable exponent have been studied intenstively for these two
decades right after some basic properties was established by Kováčik–Rákosnı́k [19]. We
refer to the surveys [16, 17, 34] and a new book [7] for recent developments. In this
section we state and recall some known basic properties.
§ 3.1.
Elementary properties
Definition 3.1.
Given a measurable function p( · ) : Ω → [1, ∞], we define the
Lebesgue space with variable exponent
Lp( · ) (Ω) := {f : ρp (f /λ) < ∞ for some λ > 0},
∫
where
ρp (f ) :=
{x∈Ω : p(x)<∞}
|f (x)|p(x) dx + ∥f ∥L∞ ({x∈Ω : p(x)=∞}) .
Moreover, define
∥f ∥Lp( · ) (Ω) := inf {λ > 0 : ρp (f /λ) ≤ 1} .
Remark 1.
We easily see that, if p( · ) equals to a constant p0 , then
Lp( · ) (Ω) = Lp0 (Ω) and
∥f ∥Lp( · ) (Ω) = ∥f ∥Lp0 (Ω)
are true.
Now we review the definition of modular.
Definition 3.2.
Let M(Ω) be the set of all complex-valued measurable functions
defined on Ω and X ⊂ M(Ω). A functional ρ : X → [0, ∞] is said to be a modular if
the following conditions are fulfilled:
(a) ρ(0) = 0.
(b) For all f ∈ X and λ ∈ C with |λ| = 1, we have ρ(λf ) = ρ(f ).
(c) ρ is convex, namely, we have that for all f, g ∈ X and all 0 ≤ t ≤ 1,
ρ(tf + (1 − t)g) ≤ tρ(f ) + (1 − t)ρ(g).
(d) For every f ∈ X such that 0 < ρ(f ) < ∞, the function
(3.1)
(0, ∞) ∋ λ 7→ ρ(λf )
is left-continuous, namely, limλ→1−0 ρ(λf ) = ρ(f ) holds.
Hardy spaces with variable exponent
115
(e) If ρ(f ) = 0, then f = 0.
A modular ρ is said to be a continuous modular if (d)’ is satisfied:
(d)′ For every f ∈ X such that 0 < ρ(f ) < ∞, the function defined by (3.1) is continuous.
Theorem 3.3.
Let p( · ) : Ω → [1, ∞] be a variable exponent. Then ρp ( · ) is a
modular. If p( · ) satisfies ess.supx∈Ω \ Ω∞ p(x) < ∞, then ρp ( · ) is a continuous modular.
Lemma 3.4.
Assume 0 < ∥f ∥Lp( · ) (Ω) < ∞.
(
)
f
(1) ρp ∥f ∥ p( · )
≤ 1.
L
(Ω)
)
(
(2) If ess.supx∈Ω \ Ω∞ p(x) < ∞, then ρp
f
∥f ∥Lp( · ) (Ω)
= 1 holds.
Theorem 3.5.
Let p( · ) : Ω → [1, ∞] be a variable exponent. Then ∥ · ∥Lp( · ) (Ω)
is a norm (often referred to as the Luxemberg–Nakano norm).
Lemma 3.6.
Let p( · ) : Ω → [1, ∞] be a variable exponent.
(1) If ∥f ∥Lp( · ) (Ω) ≤ 1, then we have ρp (f ) ≤ ∥f ∥Lp( · ) (Ω) ≤ 1.
(2) Conversely if ρp (f ) ≤ 1, then ∥f ∥Lp( · ) (Ω) ≤ 1 holds.
(3) Assume that 1 ≤ p̃+ = supx∈Ω\Ω∞ p(x) < ∞. If ρp (f ) ≤ 1, then ∥f ∥Lp( · ) ≤
ρp (f )1/p̃+ ≤ 1.
Finally, we remark that Lp( · ) (Rn ) is a complete space.
Theorem 3.7.
space.
The norm ∥ · ∥Lp( · ) (Ω) is complete, that is, Lp( · ) (Ω) is a Banach
§ 3.2.
The associate space
Given a measurable function p( · ) : Ω → [1, ∞], we defined the Lebesgue space with
variable exponent by Definition 3.1.
For p( · ) : Ω → [1, ∞], we define p′ ( · ) : Ω → [1, ∞] as
1=
1
1
+ ′
.
p(x) p (x)
By no means the function p′ ( · ) stands for the derivative of p( · ).
The aim of this section is to state results related to duality.
116
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
Theorem 3.8 (Generalized Hölder’s inequality).
Let p( · ) : Ω → [1, ∞] be a
′
p( · )
variable exponent. Then, for all f ∈ L
(Ω) and all g ∈ Lp ( · ) (Ω),
∫
|f (x)g(x)| dx ≤ rp ∥f ∥Lp( · ) (Ω) ∥g∥Lp′ ( · ) (Ω) ,
Ω
where
rp = 1 +
1
1
−
.
p−
p+
′
It is well known that Lp (Ω) (1 ≤ p < ∞) has Lp (Ω) as its dual. This is not the
case when p = ∞. The notion of associated spaces is close to dual spaces, which is used
in the theory of function spaces. It is sometimes referred to as the Köthe dual. In the
case of variable Lebesgue spaces the definition is given as follows:
Definition 3.9.
Let p( · ) : Ω → (1, ∞) be a variable exponent. The associate
p′ ( · )
space of L
(Ω) and its norm are defined as follows:
{
}
′
Lp ( · ) (Ω)′ = f is measurable : ∥f ∥Lp′ ( · ) (Ω)′ < ∞ ,
{∫
}
∥f ∥Lp′ ( · ) (Ω)′ := sup f (x)g(x) dx : ∥g∥Lp′ ( · ) (Ω) ≤ 1 .
Ω
Remark 2.
of Lemma 3.6.
The condition ∥g∥Lp′ ( · ) (Ω) ≤ 1 is equivalent to ρp′ (g) ≤ 1 by virtue
Given a variable exponent p( · ) : Ω → [1, ∞], write
Theorem 3.10.
rp := 1 +
1
1
−
.
p−
p+
Then we have that for all f ∈ Lp( · ) (Ω),
(3.2)
∥f ∥Lp( · ) (Ω) ≤ ∥f ∥Lp′ ( · ) (Ω)′ ≤ rp ∥f ∥Lp( · ) (Ω) ,
′
in particular, Lp( · ) (Ω) = Lp ( · ) (Ω)′ holds with norm equivalence.
In order to prove Theorem 3.10 we use the next lemma.
Lemma 3.11.
Let p( · ) : Ω → [1, ∞] be a variable exponent. If ∥f ∥Lp′ ( · ) (Ω)′ ≤ 1,
then ρp (f ) ≤ ∥f ∥Lp′ ( · ) (Ω)′ holds.
In order to prove Lemma 3.11, we use the following Lemmas 3.12 and 3.13.
Lemma 3.12.
Let p( · ) : Ω → [1, ∞] be a variable exponent. If ∥f ∥Lp′ ( · ) (Ω)′ <
∞ and ρp′ (g) < ∞, then we have
∫
f (x)g(x) dx ≤ ∥f ∥ p′ ( · ) ′ max{1, ρp′ (g)}.
L
(Ω)
Ω
117
Hardy spaces with variable exponent
Lemma 3.13.
If 1 < p(x) < ∞ a.e. x ∈ Ω, ρp (f ) < ∞ and ∥f ∥Lp′ ( · ) (Ω)′ ≤ 1,
then ρp (f ) ≤ 1 holds.
Lemma 3.11 is a direct consequence of Lemma 3.13. Indeed, if ∥f ∥Lp′ ( · ) (Ω)′ ≤ 1,
then we have
(
ρp
)
f
∥f ∥Lp′ ( · ) (Ω)′
∫ (
=
Ω
|f (x)|
∥f ∥Lp′ ( · ) (Ω)′
)p(x)
dx ≤ 1
from Lemma 3.13. Note that
(
)p(x)
1
≥
∥f ∥
Lp′ ( · ) (Ω)′
1
∥f ∥Lp′ ( · ) (Ω)′
because ∥f ∥Lp′ ( · ) (Ω)′ ≤ 1. Hence,
ρp (f )
1
=
∥f ∥Lp′ ( · ) (Ω)′
∥f ∥Lp′ ( · ) (Ω)′
∫ (
∫
p(x)
(|f (x)|)
Ω
dx ≤
Ω
|f (x)|
∥f ∥Lp′ ( · ) (Ω)′
)p(x)
dx ≤ 1.
Hence, we have ρp (f ) ≤ ∥f ∥Lp′ ( · ) (Ω)′ .
§ 3.3.
Norm convergence, modular convergence and convergence in
measure
Here we investigate the relations between several types of convergences.
Theorem 3.14.
(j = 1, 2, 3, . . .).
Let p( · ) : Ω → [1, ∞] be a variable exponent and fj ∈ Lp( · ) (Ω)
(1) If limj→∞ ∥fj ∥Lp( · ) (Ω) = 0, then limj→∞ ρp (fj ) = 0 holds.
(2) Assume that |Ω\Ω∞ | > 0. The following two conditions (A) and (B) are equivalent:
(A) ess.supx∈Ω\Ω∞ p(x) < ∞.
(B) If limj→∞ ρp (fj ) = 0, then limj→∞ ∥fj ∥Lp( · ) (Ω) = 0 holds.
p( · )
Theorem 3.15.
If a sequence {fj }∞
(Ω) converges in Lp( · ) (Ω), then
j=1 ∈ L
fj converges to 0 in the sense of the Lebesgue measure, namely,
(3.3)
for all ε > 0.
lim |{x ∈ Ω : |fj (x)| > ε}| = 0
j→∞
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Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
As an example of p( · ) satisfying the requirement of Theorem 3.15, we can list

2 (x ∈
/ B(0, 1)),
p(x) = 2 + ∞ · χB(0,1) (x) =
∞ (x ∈ B(0, 1)).
Here we assumed B(0, 1) ⊂ Ω.
§ 3.4.
Duality (The generalized F. Riesz representation theorem)
′
Here we show that a counterpart of the Lp (Ω)-Lp (Ω) duality is available in the
variable setting.
Definition 3.16.
Let p( · ) : Ω → [1, ∞] be a variable exponent. The dual space
of L
(Ω) and its norm are defined by
{
}
Lp( · ) (Ω)∗ := T : Lp( · ) (Ω) → C : T is linear and bounded ,
{
}
∥T ∥Lp( · ) (Ω)∗ := sup |T (u)| : ∥u∥Lp( · ) (Ω) ≤ 1 .
p( · )
′
It is natural to ask ourselves whether Lp (·) (Ω) is naturally identified with the dual
of Lp(·) (Ω). Half of the answer is given by the next theorem.
Theorem 3.17.
Let p( · ) : Ω → [1, ∞] be a variable exponent. Given a function
p′ ( · )
f ∈L
(Ω) we define the functional
∫
Tf (u) :=
f (x)u(x) dx (u ∈ Lp( · ) (Ω)).
Ω
Then, the integral defining Tf u converges absolutely. Also, the functional Tf belongs to
Lp( · ) (Ω)∗ and the estimate
∥f ∥Lp′ ( · ) (Ω) ≤ ∥Tf ∥Lp( · ) (Ω)∗ ≤ (1 + 1/p− − 1/p+ )∥f ∥Lp′ ( · ) (Ω) .
′
In particular Lp ( · ) (Ω) ⊂ Lp( · ) (Ω)∗ is true.
When p+ < ∞, then we can give an affirmative answer to the above question.
Theorem 3.18.
Let p( · ) : Ω → [1, ∞) be a variable exponent such that
p+ < ∞.
′
For all linear functionals F ∈ Lp( · ) (Ω)∗ there uniquely exists a function f ∈ Lp ( · ) (Ω)
such that
∫
F (u) =
f (x)u(x) dx (u ∈ Lp( · ) (Ω)).
Ω
Moreover, we have the norm estimate
(3.4)
∥f ∥Lp′ ( · ) (Ω) ≤ ∥F ∥Lp( · ) (Ω)∗ ≤ (1 + 1/p− − 1/p+ )∥f ∥Lp′ ( · ) (Ω) .
′
In particular Lp( · ) (Ω)∗ ⊂ Lp ( · ) (Ω) is true.
Hardy spaces with variable exponent
§ 3.5.
119
Some estimates of the norms
The following is a crucial inequality and very useful, because it is by no means easy
to measure the Lp( ·) (Rn )-norm of the characteristic functions.
Lemma 3.19 ([30]).
0 < p− ≤ p+ < ∞.
Suppose that p( · ) is a function satisfying (5.1), (5.2) and
1. For all cubes Q = Q(z, r) with z ∈ Rn and r ≤ 1, we have |Q|1/p− (Q) ≲ |Q|1/p+ (Q) .
In particular, we have
(3.5)
|Q|1/p− (Q) ∼ |Q|1/p+ (Q) ∼ |Q|1/p(z) ∼ ∥χQ ∥Lp( · ) ,
where p+ (Q) = ess.supx∈Q p(x) and p− (Q) = ess.inf x∈Q p(x).
2. For all cubes Q = Q(z, r) with z ∈ Rn and r ≥ 1, we have
∥χQ ∥Lp( · ) ∼ |Q|1/p∞ .
Here the implicit constants in ∼ do not depend on z and r > 0.
Remark 3.
The equivalence (3.5) can be implicitly found in [12, Lemma 2.5].
§ 4.
Banach function spaces
In this subsection we outline the definition of Banach function spaces and the Fatou
lemma. For further information we refer to Bennet–Sharpley [3].
Definition 4.1.
Let M(Ω) be the set of all measurable and complex-valued
functions on Ω. A linear space X ⊂ M(Ω) is said to be a Banach function space if there
exists a functional ∥ · ∥X : M(Ω) → [0, ∞] with the following conditions:
Let f, g, fj ∈ M(Ω) (j = 1, 2, . . .).
(1) f ∈ X holds if and only if ∥f ∥X < ∞.
(2) (Norm property):
(A1) (Positivity): ∥f ∥X ≥ 0.
(A2) (strict Positivity) ∥f ∥X = 0 if and only if f = 0 a.e..
(B) (Homogeneity): ∥λf ∥X = |λ| · ∥f ∥X .
(C) (The triangle inequality): ∥f + g∥X ≤ ∥f ∥X + ∥g∥X .
(3) (Symmetry): ∥f ∥X = ∥ |f | ∥X ．
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Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
(4) (Lattice property): If 0 ≤ g ≤ f a.e., then ∥g∥X ≤ ∥f ∥X .
(5) (Fatou property): If 0 ≤ f1 ≤ f2 ≤ . . . and lim fj = f , then lim ∥fj ∥X = ∥f ∥X .
j→∞
j→∞
(6) For all measurable sets F with |F | < ∞, we have ∥χF ∥X < ∞．
(7) For
∫ all measurable sets F with |F | < ∞, there exists a constant CF > 0 such that
|f (x)| dx ≤ CF ∥f ∥X ．
F
Both the usual Lebesgue spaces Lp (Ω) with constant exponent
Example 4.2.
1 ≤ p ≤ ∞ and the Lebesgue spaces Lp( · ) (Ω) with variable exponent p( · ) : Ω → [1, ∞]
are Banach function spaces.
Lemma 4.3 (The Fatou lemma).
Let X be a Banach function space and fj ∈ X
(j = 1, 2, . . .). If fj converges to a function f a.e. Ω and lim inf j→∞ ∥fj ∥X < ∞, then
we have f ∈ X and ∥f ∥X ≤ lim inf j→∞ ∥fj ∥X .
Remark 4.
In the proof of Lemma 3.4 we use the Fatou lemma with X =
L ({p(x) < ∞}), L∞ (Ω∞ ).
1
Part II
Hardy spaces with variable exponent
The role of this part is to survey Hardy spaces with variable exponent. In this part we
summarize what we obtained in [30, 36].
§ 5.
Fundamental properties
Let p(·) : Rn → (0, ∞) be an exponent such that 0 < p− = infn p(x) ≤ p+ =
x∈R
sup p(x) < ∞. Here and below, for the sake of simplicity, we shall postulate the fol-
x∈Rn
lowing conditions on p( · ).
(5.1)
(5.2)
1
1
for |x − y| ≤ ,
log(1/|x − y|)
2
1
for |y| ≥ |x|.
|p(x) − p(y)| ≲
log(e + |x|)
(log-Hölder continuity) |p(x) − p(y)| ≲
(decay condition)
Remark that (5.1) and (5.2) are necessary when we consider the property of maximal
operators.
121
Hardy spaces with variable exponent
§ 5.1.
Definition of Hardy spaces
Recall that the space Lp( · ) (Rn ), the Lebesgue space with variable ∫exponent p( · ), is
defined as the set of all measurable functions f for which the quantity
is finite for some ε > 0. The quasi-norm is given by
{
}
)p(x)
∫ (
|f (x)|
∥f ∥Lp( · ) := inf λ > 0 :
dx ≤ 1
λ
Rn
Rn
|εf (x)|p(x) dx
for such a function f .
In the celebrated paper [13], by using a suitable family FN , C. Fefferman and
E. Stein defined the Hardy space H p (Rn ) with the norm given by
−n
−1
′
n
∥f ∥H p := sup sup |t φ(t ·) ∗ f | , f ∈ S (R )
t>0 φ∈FN
Lp
for 0 < p < ∞. Here, in this part, we aim to replace Lp (Rn ) with Lp( · ) (Rn ) and
investigate the function space obtained in this way.
The aim of the present paper is to review the definition of Hardy spaces with
variable exponents and then to consider and apply the atomic decomposition. As is
the case with the classical theory, we choose a suitable subset FN ⊂ S(Rn ), which we
describe.
Definition 5.1.
1. Topologize S(Rn ) by the collection of semi-norms {pN }N ∈N given by
∑
sup (1 + |x|)N |∂ α φ(x)|
pN (φ) :=
|α|≤N
x∈Rn
for each N ∈ N. Define
(5.3)
FN := {φ ∈ S(Rn ) : pN (φ) ≤ 1}.
2. Let f ∈ S ′ (Rn ). Denote by Mf the grand maximal operator given by
Mf (x) := sup{|t−n ψ(t−1 ·) ∗ f (x)| : t > 0,
ψ ∈ FN },
where we choose and fix a large integer N .
3. The Hardy space H p( · ) (Rn ) is the set of all f ∈ S ′ (Rn ) for which the quantity
∥f ∥H p( · ) := ∥Mf ∥Lp( · )
is finite.
122
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
The definition of FN dates back to the original work [38].
The following theorem about the definition of H p( · ) (Rn ) is obtained in [30].
Let φ ∈ S(Rn ) be a function such
Theorem
5.2 ([30, Theorem 1.2 and 3.3]).
∫
that
φ(x) dx ̸= 0. We define
Rn
(5.4)
∥f ∥H p( · )
φ,∗
−n
−1
:= sup
|t
φ(t
·)
∗
f
|
t>0
f ∈ S ′ (Rn ).
,
Lp( · )
Then the norms ∥ · ∥H p( · ) and ∥f ∥H p( · ) are equivalent.
φ,∗
Note that it can happen that 0 < p− < 1 < p+ < ∞ in our setting.
§ 5.2.
Poisson integral characterization
Now we consider the Poisson integral characterization. Recall that f ∈ S ′ (Rn ) is a
bounded
distribution in terms of Stein, if f ∗ φ ∈ L∞ (Rn ) for all φ ∈ S(Rn ), and that
√
e−t −∆ f = F −1 (e−t|ξ| Ff ) (f ∈ S ′ (Rn )) denotes the Poisson semi-group for bounded
distributions f . We refer to [38, p.89] for more details. Let ψ ∈ S(Rn ) be chosen to
satisfy
χQ(0,1) ≤ Fψ ≤ χQ(0,2) .
(5.5)
With this preparation in mind, we can define
e−t
√
−∆
f := [e−t
√
−∆
(1 − ψ)] ∗ f + e−t
√
−∆
[ψ ∗ f ],
if f is a bounded distribution.
We have the following characterization.
Theorem 5.3 ([30, Theorem 3.4]).
Suppose that p( · ) satisfies (5.1), (5.2) and
′
n
0 < p− ≤ p+ < ∞. Let f ∈ S (R ). Then the following are equivalent.
1. f ∈ H p( · ) (Rn ),
2. f is a bounded distribution and sup |e−t
t>0
§ 5.3.
√
−∆
f | ∈ Lp( · ) (Rn ).
Atomic decomposition
Here is another key result which we shall highlight. To formulate we adopt the
following definition of the atomic decomposition.
Definition 5.4 ((p( · ), q)-atom).
Let p( · ) : Rn → (0, ∞), 0 < p− ≤ p+ < q ≤
∞ and q ≥ 1. Fix an integer d ≥ dp( · ) := min {d ∈ N ∪ {0} : p− (n + d + 1) > n}. A
function a on Rn is called a (p( · ), q)-atom if there exists a cube Q such that
123
Hardy spaces with variable exponent
(a1) supp(a) ⊂ Q,
(a2) ∥a∥
Lq
|Q|1/q
≤
,
∥χQ ∥Lp( · )
∫
(a3)
Rn
a(x)xα dx = 0 for |α| ≤ d.
The set of all such pairs (a, Q) will be denoted by A(p( · ), q).
Under this definition, we define the atomic Hardy spaces with variable exponents.
Here and below we denote
(5.6)
p := min(p− , 1).
p( · ),q
∞
n
Definition 5.5 (Sequence norm A({κj }∞
j=1 , {Qj }j=1 ) and Hatom (R )).
∞
sequences of nonnegative numbers {κj }∞
j=1 and cubes {Qj }j=1 , define
Given
(5.7)



 p(x)
p


(
)
∫
∞
p
∑


κj χQj (x)
∞
∞
A({κj }j=1 , {Qj }j=1 ) := inf λ > 0 :
dx ≤ 1 .


λ∥χQj ∥Lp( · ) 
Rn  j=1


p( · ),q
The atomic Hardy space Hatom (Rn ) is the set of all functions f ∈ S ′ (Rn ) such that it
can be written as
(5.8)
f=
∞
∑
κj aj in S ′ (Rn ),
j=1
∞
where {κj }∞
j=1 is a sequence of nonnegative numbers, {(aj , Qj )}j=1 ⊂ A(p(·), q) and
∞
A({κj }∞
j=1 , {Qj }j=1 ) is finite. One defines
∞
∥f ∥H p( · ),q := inf A({κj }∞
j=1 , {Qj }j=1 ),
atom
where the infimum is taken over all admissible expressions as in (5.8).
Suppose that 0 < p− ≤ p+ < ∞. Under these definitions, in Section 6 we formulate
the following.
Theorem 5.6.
The variable Hardy norms given in Theorem 5.2 and the ones
given by means of atoms are isomorphic as long as
q > p+ ≥ 1, or q = 1 > p+ .
124
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
Remark that we could not specify the condition of q precisely in [30] but as the
calculation in [36] shows q > p+ ≥ 1 or q = 1 > p+ suffices.
§ 6.
Atomic decompositions
In this section we consider decomposition.
Here, we define an index dp( · ) ∈ N ∪ {0} by
(6.1)
dp( · ) := min {d ∈ N ∪ {0} : p− (n + d + 1) > n} .
For a nonnegative integer d, let Pd (Rn ) denote the set of all polynomials having
degree at most d.
Let p( · ) : Rn → (0, ∞), 0 < p− ≤ p+ < q ≤ ∞ and q ≥ 1. Recall that we have
defined (p( · ), q)-atoms in Definition 5.4.
In the variable setting as well, we have that atoms have Lp( · ) -norm less than 1.
We denote by A(p( · ), q) the set of all pairs (a, Q) such that a is a (p( · ), q)-atom and
that Q is the corresponding cube.
Remark 5.
1. Define another variable exponent q̃(·) by
(6.2)
1
1
1
= +
p(x)
q
q̃(x)
(x ∈ Rn ).
Then we have
(6.3)
∥f · g∥Lp( · ) ≲ ∥g∥Lq ∥f ∥Lq̃(·)
for all measurable functions f and g [21].
2. A direct consequence of Lemma 3.19 and (6.3) is that ∥a∥Lp( · ) ≲ 1 whenever
(a, Q) ∈ A(p( · ), q).
Of course, as is the case when p( · ) is a constant, Remark 5 can be extended as
follows:
Proposition 6.1 (cf. [30, Proposition 4.2]).
1. Let q > max(1, p+ ). If p( · ) satisfies 0 < p− ≤ p+ < ∞ as well as (5.1) and (5.2),
then we have
∥a∥H p( · ) ≲ 1
for any (a, Q) ∈ A(p( · ), q).
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Hardy spaces with variable exponent
2. If p( · ) satisfies 0 < p− ≤ p+ < 1 as well as (5.1) and (5.2), then we have
∥a∥H p( · ) ≲ 1
for any (a, Q) ∈ A(p( · ), 1).
p( · ),q
The function space Hatom (Rn ) was defined to be the set of all functions f such that
∞
∑
∞
it can be written in the form f =
κj aj in S ′ (Rn ), where A({κj }∞
j=1 , {Qj }j=1 ) < ∞
j=1
and {(aj , Qj )}j∈N ⊂ A(p( · ), q). One defines
∞
∥f ∥H p( · ),q := inf A({κj }∞
j=1 , {Qj }j=1 ),
atom
where the infimum is taken over all expressions as above.
Observe that if p(·) ≡ p+ = p− , that is, p( · ) is a constant function, then we can
recover classical Hardy spaces. Unlike the classical case, (p(·), ∞)-atoms are not dealt
separately. Consequently we have two types of results for (p(·), ∞)-atoms.
p( · ),∞
Definition 6.2 (Hatom,∗ (Rn ), [30, Definition 4.3]).
Let p( · ) : Rn → (0, ∞),
p( · ),∞
0 < p− ≤ p+ < q ≤ ∞ and q ≥ 1. Then f ∈ S ′ (Rn ) is in Hatom,∗ (Rn ) if and only if
∞
there exist sequences of nonnegative numbers {κj }∞
j=1 and {(aj , Qj )}j=1 ⊂ A(p( · ), ∞)
such that
)p(x)
∞
∑
∑∫ (
κj
′
n
(6.4)
f=
dx < ∞.
κj aj in S (R ), and that
∥χ
∥
p( · )
Q
Q
L
j
j
j=1
j
∞
For sequences of nonnegative numbers {κj }∞
j=1 and cubes {Qj }j=1 , define


)p(x)
∫ ∑(


κj
∞
A∗ ({κj }∞
,
{Q
}
)
:=
inf
λ
>
0
:
dx
≤
1
.
j j=1
j=1


λ∥χQj ∥Lp( · )
Qj
j
Now we formulate our atomic decomposition theorem. Let us begin with the space
with q = ∞.
p( · ),q
Hatom,∗ (Rn )
If p( · ) satisfies 0 < p− ≤ p+ < ∞, (5.1)
Theorem 6.3 ([30, Theorem 4.5]).
and (5.2), then, for all f ∈ S ′ (Rn ),
∥f ∥H p( · ) ∼ ∥f ∥H p( · ),∞ ∼ ∥f ∥H p( · ),∞ .
atom
atom,∗
The atomic decomposition for A(p(·), q) can be also obtained.
Theorem 6.4 (cf. [30, Theorem 4.6]).
(i) 0 < p− ≤ p+ < q ≤ ∞ and p+ ≥ 1;
Suppose either (i) or (ii) holds;
126
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
(ii) 0 < p− ≤ p+ < 1 ≤ q ≤ ∞.
Assume p( · ) satisfies (5.1) and (5.2). Then, for all f ∈ S ′ (Rn ), ∥f ∥H p( · ) ∼ ∥f ∥H p( · ),q .
atom
§ 7.
Applications of atomic decomposition
This section is a small modification of [30, Section 5]. We first state Theorem 7.2
based on Theorem 6.4, which refines [30, Theorem 5.2]. And then we recall what we
obtained in [30].
§ 7.1.
Molecular decomposition
Now we investigate molecular decomposition as an application of Theorems 6.3 and
6.4. Here we present a definition of molecules.
Definition 7.1 (Molecules [30, Definition 5.1]).
Let 0 < p− ≤ p+ < q ≤ ∞,
q ≥ 1 and d ∈ [dp( · ) , ∞) ∩ Z be fixed. One says that M is a (p( · ), q)-molecule centered
at a cube Q if it satisfies the following conditions.
√
1. On 2 nQ, M satisfies the estimate ∥M∥Lq (2√nQ) ≤
√
2. Outside 2 nQ, we have |M(x)| ≤
1
∥χQ ∥Lp(·)
1
|Q| q
.
∥χQ ∥Lp(·)
(
)−2n−2d−3
|x − z|
1+
. This condiℓ(Q)
tion is called the decay condition.
3. If α is a multiindex with length less than d, then we have
∫
xα M(x) dx = 0.
Rn
This condition is called the moment condition.
By definition (p( · ), q)-atoms are (p( · ), q)-molecules modulo a multiplicative constant.
As we did in [30], we are able to prove the following result.
Theorem 7.2 (cf. [30, Theorem 5.2]).
Z ∪ [dp( · ) , ∞). Assume either
Let 0 < p− ≤ p+ < q ≤ ∞ and d ∈
p+ < 1 = q or q > p+ = 1.
Assume in addition that p( · ) satisfies (5.1) and (5.2).
127
Hardy spaces with variable exponent
∞
Suppose that {Qj }∞
j=1 = {Q(zj , ℓj )}j=1 is a sequence of cubes and, for each j ∈ N,
that we are given a (p( · ), q)-molecule Mj centered at Qj . If a sequence of positive
numbers {κj }∞
j=1 satisfies
(
)
∞
A {κj }∞
j=1 , {Qj }j=1 = 1, that is,
then we have
∫
 p(x)
p
p
κj χQj 

dx ≤ 1,
∥χQ ∥ p(·) Rn
L
j
j=1
∑
∞
κj Mj j=1
(7.1)
§ 7.2.

∞ ∑
≲ 1.
H p(·)
Boundedness of singular integral operators
If we combine Theorems 6.4 and 7.2, then we obtain the following theorem.
Theorem 7.3 ([30, Theorem 5.5]).
Assume that p( · ) satisfies 0 < p− ≤ p+ <
n
∞, (5.1) and (5.2). Let k ∈ S(R ) and write
Am := sup |x|n+m |∇m k(x)|
x∈Rn
(m ∈ N ∪ {0}).
Define a convolution operator T by
T f (x) := k ∗ f (x)
(f ∈ L2 (Rn )).
Then, T can be extended also to an H p( · ) (Rn )-H p( · ) (Rn ) operator and the norm depends
only on ∥F k∥L∞ and a finite number of collections A1 , A2 , . . . , AN with N depending
only on p( · ).
§ 7.3.
Littlewood-Paley characterization
Now we consider the Littlewood-Paley characterization of the function spaces.
We are going to characterize H p( · ) (Rn ) by means of the Littlewood-Paley decomposition.
The following lemma is a natural extension with |·| in the definition of Mf replaced
2
by ℓ (Z).
We introduce the ℓ2 (Z)-valued function space H p( · ) (Rn ; ℓ2 (Z)). Suppose that we
′
n
are given a sequence {fj }∞
j=−∞ ⊂ S (R ).
Let ψ ∈ S(Rn ) be such that χQ(0,1) ≤ ψ ≤ χQ(0,2) . We set ψk (ξ) := ψ(2−k ξ). With
this in mind, we define

 21 ∞
∑
2 
∥{fj }∞
∥
:=
sup
|ψ
(D)f
|
.
k
j
j=−∞ H p( · ) (ℓ2 )
k∈Z
j=−∞
p( · )
L
128
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
Observe that this is a natural vector-valued extension of
kn −1
k
∥f ∥H p( · ) ∼ sup |2 F ψ(2 ·) ∗ f |
k∈Z
.
Lp( · )
We characterize Hardy spaces with variable exponents. Let us set φj (x) := φ(2−j x),
φj (D)f := F −1 [φ(2−j ·)Ff ] for f ∈ S ′ (Rn ).
Theorem 7.4 ([30, Theorem 5.7]).
Q(0, 4) \ Q(0, 1/4) such that
∞
∑
Let φ ∈ S(Rn ) be a function supported on
|φj (ξ)|2 > 0
j=−∞
for ξ ∈ Rn \ {0}. Then the following norm is an equivalent norm of H p( · ) (Rn ):

 12 ∞
∑

2 |φj (D)f |
, f ∈ S ′ (Rn ).
:= (7.2)
∥f ∥Ḟ 0
p( · )2
j=−∞
p( · )
L
§ 8.
§ 8.1.
Campanato spaces with variable growth conditions
Definition of Campanato spaces with variable growth conditions
Recall that dp( · ) is defined in (6.1) to be
dp( · ) := min {d ∈ N ∪ {0} : p− (n + d + 1) > n} .
Let Lqcomp (Rn ) be the set of all Lq (Rn )-functions having compact support. Given a
nonnegative integer d, let
{
}
∫
n
α
n
q
q,d
f (x)x dx = 0, |α| ≤ d .
Lcomp (R ) := f ∈ Lcomp (R ) :
Rn
Likewise if Q is a cube, then we write
{
}
∫
q,d
q
α
L (Q) := f ∈ L (Q) :
f (x)x dx = 0, |α| ≤ d .
Q
p( · ),q
n
n
If d is as in (6.1), then Lq,d
comp (R ) is dense in Hatom (R ), Indeed, it contains all the
p( · ),q
finite linear combinations of (p( · ), q)-atoms from the definition of Hatom (Rn ).
Recall that Pd (Rn ) is the set of all polynomials having degree at most d. For a
locally integrable function f , a cube Q and a nonnegative integer d, there exists a unique
polynomial P ∈ Pd (Rn ) such that, for all q ∈ Pd (Rn ),
∫
(f (x) − P (x))q(x) dx = 0.
Q
129
Hardy spaces with variable exponent
Denote this unique polynomial P by PQd f . It follows immediately from the definition
that PQd g = g if g ∈ Pd (Rn ).
Definition 8.1 ([30, Definition 6.1], Lq,ϕ,d (Rn )).
(0, ∞) be a function and f ∈ Lqloc (Rn ). One denotes
∥f ∥Lq,ϕ,d
1
= sup
Q∈Q ϕ(Q)
(
1
|Q|
Let 1 ≤ q ≤ ∞. Let ϕ : Q →
)1/q
∫
|f (x) −
PQd f (x)|q
dx
,
Q
when q < ∞ and
1
∥f − PQd f ∥L∞ (Q) .
Q∈Q ϕ(Q)
∥f ∥Lq,ϕ,d = sup
when q = ∞. Then the Campanato space Lq,ϕ,d (Rn ) is defined to be the sets of all f ∈
Lqloc (Rn ) such that ∥f ∥Lq,ϕ,d < ∞. One considers elements in Lq,ϕ,d (Rn ) modulo polynomials of degree d so that Lq,ϕ,d (Rn ) is a Banach space. When one writes f ∈ Lq,ϕ,d (Rn ),
then f stands for the representative of {f + P : P is a polynomial of degree d }.
Here and below we abuse notation slightly. We write ϕ(x, r) := ϕ(Q(x, r)) for
x ∈ Rn and r > 0.
§ 9.
Duality H p( · ) (Rn )-Lq,ϕ,d (Rn )
In this section, we shall prove that the dual spaces of H p( · ) (Rn ) are generalized
Campanato spaces with variable growth conditions when 0 < p− ≤ p+ ≤ 1.
§ 9.1.
Dual of H p0 (Rn ) ∩ H 1 (Rn ) with 0 < p0 ≤ 1
In this subsection, let p0 be a constant with 0 < p0 ≤ 1. This subsection is an
auxiliary step to investigate H p( · ) (Rn )∗ .
If p( · ) is a constant function, then the dual is known to exist [14].
Keeping this in mind, we now seek to investigate the structure of Lq,ϕ,d (Rn ).
Recall that bmo(Rn ), the local BMO, is the set of all locally integrable functions
f such that
∫ ∫
∫
∥f ∥bmo := sup − f (x) − − f (y) dy dx + sup − |f (x)| dx < ∞.
Q∈Q Q
|Q|≤1
Q
Q∈Q Q
|Q|=1
Then from the definition of the norms ∥ · ∥BMO and ∥ · ∥bmo we have ∥f ∥BMO ≲ ∥f ∥bmo .
By the well-known H 1 (Rn )-BMO(Rn ) duality, bmo(Rn ) is canonically embedded into
the dual space of H 1 (Rn ).
130
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
Theorem 9.1 ([30, Theorem 7.3]).
Let 0 < p0 ≤ 1 and 1 ≤ q ≤ ∞. Set
1
1
p0 −1
p0 −1
ϕ1 (Q) := |Q|
and ϕ2 (Q) := |Q|
+ 1 for Q ∈ Q. Then we have Lq,ϕ2 ,d (Rn ) ,→
Lq,ϕ1 ,d (Rn ) + bmo(Rn ) in the sense of continuous embedding. More quantitatively, if
we choose ψ ∈ S(Rn ) so that χQ(0,1) ≤ ψ ≤ χQ(0,2) , then we have
∥ψ(D)g∥Lq,ϕ1 ,d ≲ ∥g∥Lq,ϕ2 ,d ,
∥(1 − ψ(D))g∥bmo ≲ ∥g∥Lq,ϕ2 ,d .
Dual spaces of H p( · ) (Rn )
§ 9.2.
Now we specify the dual of H p( · ) (Rn ) with 0 < p− ≤ p+ ≤ 1. It follows from the
p( · ),q
definition of the dual norm that, for all ℓ ∈ (Hatom (Rn ))∗ ,
{
}
∥ℓ∥(H p( · ),q (Rn ))∗ = sup |ℓ(f )| : ∥f ∥H p( · ),q ≤ 1
atom
atom
p( · ),q
is finite and ∥ℓ∥(H p( · ),q (Rn ))∗ is a norm on (Hatom (Rn ))∗ . We prove the following
atom
theorem.
Let p( · ) : Rn → (0, ∞), 0 < p− ≤ p+ ≤
Theorem 9.2 (cf. [30, Theorem 7.5]).
1, p+ < q ≤ ∞ and 1/q + 1/q ′ = 1. Suppose that the integer d is as in (6.1). Define
(9.1)
ϕ3 (Q) :=
∥χQ ∥Lp( · )
|Q|
(Q ∈ Q).
If p( · ) satisfies (5.1) and (5.2), then
p( · ),q
(Hatom (Rn ))∗ ≃ Lq′ ,ϕ3 ,d (Rn )
with equivalent norms. More precisely, we have the following assertions.
1. Let f ∈ Lq′ ,ϕ3 ,d (Rn ). Then the functional
∫
ℓf : a ∈
n
Lq,d
comp (R )
7→
Rn
a(x)f (x) dx ∈ C
p( · ),q
extends to a bounded linear functional on (Hatom (Rn ))∗ such that
∥ℓf ∥(H p( · ),q )∗ ≲ ∥f ∥Lq′ ,ϕ3 ,d .
atom
p( · ),q
2. Conversely, any linear functional ℓ on (Hatom (Rn ))∗ can be realized as above with
some f ∈ Lq′ ,ϕ3 ,d (Rn ) and we have ∥f ∥Lq′ ,ϕ3 ,d ≲ ∥ℓ∥(H p( · ),q )∗ .
atom
In particular, we have
(H p( · ) (Rn ))∗ ≃ Lq′ ,ϕ3 ,d (Rn ).
Hardy spaces with variable exponent
131
Namely, any f ∈ Lq′ ,ϕ3 ,d (Rn ) defines a continuous linear functional on (H p( · ) (Rn ))∗
such that
∫
Lf (a) =
a(x)f (x) dx
Rn
n
p( · )
for any a ∈ Lq,d
(Rn ))∗ is realized
comp (R ) and any continuous linear functional on (H
with some some f ∈ Lq′ ,ϕ3 ,d (Rn ).
Note that there was no need to assume q ≫ 1 in Theorem 9.2, since we refined
Theorem 6.4. When q ≫ 1, this theorem is recorded as [30, Theorem 7.5].
§ 9.3.
Open Problem 9.3.
cases ?
An open problem
Do we have analogies of Theorems 9.1 and 9.2 for general
A partial answer is;
Proposition 9.4.
When p− > 1, then we have
′
H p(·) (Rn )′ ∼ Lp(·) (Rn )′ ∼ H p (·) (Rn ).
How do we characterize the dual of H p(·) (Rn ) for general cases, that is, without
assuming p+ ≤ 1 ?
Besov spaces and Triebel-Lizorkin spaces are useful tools but about the dual we
have the folloing:
Proposition 9.5.
For 0 < p < 1,
0
n/p−n
hp (Rn ) ∼ Fp2
(Rn ) → B∞∞
(Rn ).
For p = 1,
0
0
(Rn ).
(Rn ) → bmo(Rn ) = F∞2
hp (Rn ) ∼ Fp2
For p > 1,
0
(Rn ) → Fp0′ 2 (Rn ).
hp (Rn ) ∼ Fp2
So, starting from the Triebel-Lizorkin scale, the resulting duals can be both Besov
spaces and Triebel-Lizorkin spaces. Once we mix the situation about p, it seems no
longer possible to determine duals.
§ 10.
Hölder-Zygmund spaces with variable exponents
In this section we assume that
(10.1)
0 < p− ≤ p+ < 1.
132
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
We consider the function spaces of Hölder-Zygmund type and we connect them in
n
particular with the function spaces LD
q,ϕ3 (R ) that we are going to define, where again
∥χQ ∥Lp( · )
we let ϕ3 (Q) =
for Q ∈ Q.
|Q|
§ 10.1.
Definition of Hölder-Zygmund spaces with variable exponents
We define ∆kh to be a difference operator, which is defined inductively by
(10.2)
∆1h f = ∆h f := f (· + h) − f,
∆kh := ∆1h ◦ ∆k−1
h ,
k ≥ 2.
Definition 10.1 ([30, Definition 8.1], Λϕ,d (Rn )). Let ϕ : Rn × (0, ∞) → (0, ∞)
and d ∈ N ∪ {0}. Then Λϕ,d (Rn ), the Hölder space with variable exponent p( · ), is
defined to be the set of all continuous functions f such that ∥f ∥Λϕ,d < ∞, where
∥f ∥Λϕ,d :=
d+1
1
∆ f (x) .
h
x∈Rn , h̸=0 ϕ(x, |h|)
sup
One considers elements in Λϕ,d (Rn ) modulo polynomials of degree d so that Λϕ,d (Rn )
is a Banach space. When one writes f ∈ Λϕ,d (Rn ), then f stands for the representative
of {f + P : P is a polynomial of degree d }.
Several helpful remarks may be in order.
Remark 6 ([30, Remark 8.2]).
1. Assume that there exists a constant µ > 0 such that ϕ(Q) ≲ |Q|µ for all Q with
|Q| ≥ 1. If a continuous function f satisfies ∥f ∥Λϕ,d < ∞, then f is of polynomial
order. In particular the representative of such a function f can be regarded as an
element in S ′ (Rn ). Actually, since f is assumed continuous, f is bounded on a
neighborhood Q(0, 1). Using ∥f ∥Λϕ,d < ∞, inductively on k ∈ N ∪ {0} we can show
that |f (x)| ≲ (k + 1)d+µ+1 for all x with k ≤ |x| ≤ k + 1.
2. It is absolutely necessary to assume that f is a continuous function, when d ≥ 1.
We remark that there exists a discontinuous function f such that ∆d+1
h f (x) = 0 for
n
all x, h ∈ R . See [23] for such an example.
3. The function space Λϕ,d (Rn ) is used to measure the Hölder continuity uniformly,
when ϕ does not depend on x. Such an attempt can be found in [22].
As for Λϕ,d (Rn ), we have the following equivalence.
Theorem 10.2 ([30, Theorem 8.4]).
following conditions.
Assume that ϕ : Q → (0, ∞) satisfies the
133
Hardy spaces with variable exponent
(A1) There exists a constant C > 0 such that
C −1 ≤
ϕ(x, r)
≤ C,
ϕ(x, 2r)
(x ∈ Rn , r > 0).
(A2) There exists a constant C > 0 such that
C −1 ≤
ϕ(x, r)
≤ C,
ϕ(y, r)
(x, y ∈ Rn , r > 0, |x − y| ≤ r).
(A3) There exists a constant C > 0 such that
∫
0
r
ϕ(x, t)
dt ≤ Cϕ(x, r),
t
(x ∈ Rn , r > 0).
Then the function spaces Λϕ,d (Rn ) and Lq,ϕ,d (Rn ) are isomorphic. Speaking more precisely, we have the following :
1. For any f ∈ Λϕ,d (Rn ) we have ∥f ∥Lq,ϕ,d ≲ ∥f ∥Λϕ,d .
2. Any element in Lq,ϕ,d (Rn ) has a continuous representative. Moreover, whenever
f ∈ Lq,ϕ,d (Rn ) ∩ C(Rn ), then f ∈ Λϕ,d (Rn ) and we have ∥f ∥Λϕ,d ≲ ∥f ∥Lq,ϕ,d .
§ 11.
Local Hardy spaces with variable exponents
What we have been doing can be transplanted
∫ into the theory of the local Hardy
n
ψ(x) dx ̸= 0, and if we define the
spaces. For example, if ψ ∈ S(R ) is such that
Rn
norm by
(11.1)
∥f ∥hp( · )
−n
−1
= sup sup |t φ(t ·) ∗ f |
0<t<1 φ∈F
N
,
Lp( · )
then we see that
(11.2)
∥f ∥hp( · )
∼ sup |ψj (D)f |
j∈N
,
Lp( · )
where ψj (ξ) = ψ(2−j ξ).
0
n
To conclude this paper, we establish the norms of hp( · ) (Rn ) and Fp(
· )2 (R ) are
equivalent. Let ψ ∈ S(Rn ) be a bump function satisfying χQ(0,1) ≤ ψ ≤ χQ(0,2) and set
φj := ψ(2−j ·) − ψ(2−j+1 ·)
134
Mitsuo Izuki, Eiichi Nakai and Yoshihiro Sawano
0
n
for j ∈ N. In [11] Diening, Hästö and Roudenko defined the function space Fp(
· )2 (R ),
the one of Triebel-Lizorkin type, with the norm

 21 ∞
∑

2 0
∥f ∥Fp(
:=
∥ψ(D)f
∥
+
|φ
(D)f
|
p(
·
)
j
L
· )2
j=1
p( · )
L
for f ∈ S ′ (Rn ).
Let 0 < p− ≤ p+ < ∞. The function
Theorem 11.1 ([30, Theorem 9.2]).
p( · )
n
0
n
spaces h
(R ) and Fp( · )2 (R ) are isomorphic to each other.
Other results of the present paper have counterpart for hp( · ) (Rn ). For example,
when we consider the local Hardy spaces, their duals will be the Besov spaces defined
in [1] by virtue of the counterpart of Theorems 9.2. The proofs being analogous to the
corresponding proofs for H p( · ) (Rn ), we omit the details.
Acknowledgement
Mitsuo Izuki was indebted to Professor Toshio Horiuchi at Ibaraki University and
the students at Ibaraki University for their kind suggestion of Part I. The authors are
thankful to the anonymous referee for his advice about the structure of the present
paper.
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