Intrinsic Square Function Characterizations of Hardy Spaces with Variable Exponents
arXiv:1411.5535v1 [math.CA] 20 Nov 2014
Ciqiang Zhuo, Dachun Yang∗ and Yiyu Liang School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
[email protected],
[email protected],
[email protected]
Abstract Let p(·) : Rn → (0, ∞) be a measurable function satisfying some decay condition and some locally log-H¨ older continuity. In this article, via first establishing characterizations of the variable exponent Hardy space H p(·) (Rn ) in terms of the Littlewood-Paley g-function, the Lusin area function and the gλ∗ -function, the authors then obtain its intrinsic square function characterizations including the intrinsic Littlewood-Paley g-function, the intrinsic Lusin area function and the intrinsic gλ∗ -function. The p(·)-Carleson measure characterization for the dual space of H p(·) (Rn ), the variable exponent Campanato space L1,p(·),s (Rn ), in terms of the intrinsic function is also presented. 2010 Mathematics Subject Classification: Primary: 42B25, Secondary: 42B30, 42B35, 46E30 Keywords and phrases: Hardy space, variable exponent, intrinsic square function, Carlson measure, atom.
1
Introduction
Variable exponent Lebesgue spaces are a generalization of the classical Lp (Rn ) spaces, in which the constant exponent p is replaced by an exponent function p(·) : Rn → (0, ∞), namely, they consist of R all functions f such that Rn |f (x)|p(x) dx < ∞. These spaces were introduced by Birnbaum-Orlicz [3] and Orlicz [34], and widely used in the study of harmonic analysis as well as partial differential equations; see, for example, [1, 2, 6, 7, 8, 11, 12, 13, 14, 20, 30, 43, 48, 50]. For a systematic research about the variable exponent Lebesgue space, we refer the reader to [8, 13] Recently, Nakai and Sawano [32] extended the theory of variable Lebesgue spaces via studying the Hardy spaces with variable exponents on Rn , and Sawano in [35] further gave more applications of these variable exponent Hardy spaces. Independently, Cruz-Uribe and Wang in [9] also investigated the variable exponent Hardy space with some weaker conditions than those used in [32], which also extends the theory of variable exponent Lebesgue spaces. Recall that the classical Hardy spaces H p (Rn ) with p ∈ (0, 1] on the Euclidean space Rn and their duals are well studied (see, for example, [10, 38]) and have been playing an important and fundamental role in various fields of analysis such as harmonic analysis and partial differential equations; see, for example, [4, 31]. Communicated by Rosihan M. Ali, Dato’. Received: January 28, 2014; Revised: March 26, 2014. ∗ Corresponding author
1
2
Ciqiang Zhuo, Dachun Yang and Yiyu Liang
On the other hand, the study of the intrinsic square function on function spaces, including Hardy spaces, has recently attracted many attentions. To be precise, Wilson [44] originally introduced intrinsic square functions, which can be thought of as “grand maximal” square functions of C. Fefferman and E. M. Stein from [10], to settle a conjecture proposed by R. Fefferman and E. M. Stein on the boundedness of the Lusin area function S(f ) from the weighted Lebesgue space L2M(v) (Rn ) to the weighted Lebesgue space L2v (Rn ), where 0 ≤ v ∈ L1loc (Rn ) and M denotes the Hardy-Littlewood maximal function. The boundedness of these intrinsic square functions on the weighted Lebesgue spaces Lpω (Rn ), when p ∈ (1, ∞) and ω belongs to Muckenhoupt weights Ap (Rn ), was proved by Wilson [45]. The intrinsic square functions dominate all square functions of the form S(f ) (and the classical ones as well), but are not essentially bigger than any one of them. Similar to the Fefferman-Stein and the Hardy-Littlewood maximal functions, their generic natures make them pointwise equivalent to each other and extremely easy to work with. Moreover, the intrinsic Lusin area function has the distinct advantage of being pointwise comparable at different cone openings, which is a property long known not to hold true for the classical Lusin area function; see Wilson [44, 45, 46, 47] and also Lerner [24, 25]. Later, Huang and Liu in [19] obtain the intrinsic square function characterizations of the weighted Hardy space Hω1 (Rn ) under the additional assumption that f ∈ L1ω (Rn ), which was further generalized to the weighted Hardy space Hωp (Rn ) with p ∈ (n/(n + α), 1) and α ∈ (0, 1) by Wang and Liu in [42], under another additional assumption. Very recently, Liang and Yang in [28] established the s-order intrinsic square function characterizations of the Musielak-Orlicz Hardy space H ϕ (Rn ), which was introduced by Ky [23] and generalized both the Orlicz-Hardy space (see, for example, [21, 41]) and the weighted Hardy space (see, for example, [16, 36]), in terms of the intrinsic Lusin area function, the intrinsic g-function and the intrinsic gλ∗ -function with the best known range λ ∈ (2 + 2(α + s)/n, ∞). More applications of such intrinsic square functions were also given by Wilson [46, 47] and Lerner [24, 25]. Motivated by [28], in this article, we establish intrinsic square function characterizations of the variable exponent Hardy space H p(·) (Rn ) introduced by Nakai and Sawano in [32], including the intrinsic Littlewood-Paley g-function, the intrinsic Lusin area function and the intrinsic gλ∗ -function by first obtaining characterizations of H p(·) (Rn ) via the Littlewood-Paley g-function, the Lusin area function and the gλ∗ -function. We also establish the p(·)-Carleson measure characterization for the dual space of H p(·) (Rn ), the variable exponent Campanato space L1,p(·),s (Rn ) in [32], in terms of the intrinsic square function. To state the results, we begin with some notation. In what follows, for a measurable function p(·) : Rn → (0, ∞) and a measurable set E of Rn , let p− (E) := ess inf p(x) and p+ (E) := ess sup p(x). x∈E
n
x∈E
n
∗
For simplicity, we let p− := p− (R ), p+ := p+ (R ) and p := min{p− , 1}. Denote by P(Rn ) the collection of all measurable functions p(·) : Rn → (0, ∞) satisfying 0 < p− ≤ p+ < ∞. For p(·) ∈ P(Rn ), the space Lp(·) (Rn ) is defined to be the set of all measurable functions such that ) ( p(x) Z |f (x)| dx ≤ 1 < ∞. kf kLp(·) (Rn ) := inf λ ∈ (0, ∞) : λ Rn Remark 1.1. It was pointed out in [32, p. 3671] (see also [8, Theorem 2.17]) that the follows hold true: (i) kf kLp(·) (Rn ) ≥ 0, and kf kLp(·) (Rn ) = 0 if and only if f (x) = 0 for almost every x ∈ Rn ; (ii) kλf kLp(·) (Rn ) = |λ|kf kLp(·) (Rn ) for any λ ∈ C; (iii) kf + gkℓLp(·) (Rn ) ≤ kf kℓLp(·) (Rn ) + kgkℓLp(·) (Rn ) for all ℓ ∈ (0, p∗ ];
3
Intrinsic Square Function Characterizations
(iv) for all measurable functions f with kf kLp(·) (Rn ) 6= 0,
R
Rn
[|f (x)|/kf kLp(·)(Rn ) ]p(x) dx = 1.
A function p(·) ∈ P(Rn ) is said to satisfy the locally log-H¨ older continuous condition if there exists a positive constant C such that, for all x, y ∈ Rn and |x − y| ≤ 1/2, |p(x) − p(y)| ≤
(1.1)
C , log(1/|x − y|)
and p(·) is said to satisfy the decay condition if there exist positive constants C∞ and p∞ such that, for all x ∈ Rn , |p(x) − p∞ | ≤
(1.2)
C∞ . log(e + |x|)
In the whole article, we denote by S(Rn ) the space of all Schwartz functions and by S ′ (Rn ) its dual space. Let S∞ (Rn ) denote the space of all Schwartz functions ϕ satisfying R topological β ′ ϕ(x)x dx = 0 for all multi-indices β ∈ Zn+ := ({0, 1, . . .})n and S∞ (Rn ) its topological dual Rn space. For N ∈ N := {1, 2, . . . }, let X sup (1 + |x|)N |Dβ ψ(x)| ≤ 1 , (1.3) FN (Rn ) := ψ ∈ S(Rn ) : x∈Rn n β∈Z+ , |β|≤N
∂ β1 where, for β := (β1 , . . . , βn ) ∈ Zn+ , |β| := β1 + · · · + βn and Dβ := ( ∂x ) · · · ( ∂x∂ n )βn . Then, for 1 ′ n ∗ all f ∈ S (R ), the grand maximal function fN,+ of f is defined by setting, for all x ∈ Rn , ∗ fN,+ (x) := sup {|f ∗ ψt (x)| : t ∈ (0, ∞) and ψ ∈ FN (Rn )} ,
where, for all t ∈ (0, ∞) and ξ ∈ Rn , ψt (ξ) := t−n ψ(ξ/t). For any measurable set E ⊂ Rn and r ∈ (0, ∞), let Lr (E) be the set of all measurable functions 1/r R < ∞. For r ∈ (0, ∞), denote by Lrloc (Rn ) the set of f such that kf kLr (E) := E |f (x)|r dx n all r-locally integrable functions on R . Recall that the Hardy-Littlewood maximal operator M is defined by setting, for all f ∈ L1loc (Rn ) and x ∈ Rn , Z 1 M(f )(x) := sup |f (y)| dy, B∋x |B| B where the supremum is taken over all balls B of Rn containing x. Now we recall the notion of the Hardy space with variable exponent, H p(·) (Rn ), introduced by Nakai and Sawano in [32]. For simplicity, we also call H p(·) (Rn ) the variable exponent Hardy space. Definition 1.2. Let p(·) ∈ P(Rn ) satisfy (1.1) and (1.2), and n (1.4) N∈ + n + 1, ∞ ∩ N. p− The Hardy space with variable exponent p(·), denoted by H p(·) (Rn ), is defined to be the set of all ∗ ∗ f ∈ S ′ (Rn ) such that fN,+ ∈ Lp(·) (Rn ) with the quasi-norm kf kH p(·) (Rn ) := kfN,+ kLp(·) (Rn ) . Remark 1.3. (i) Independently, Cruz-Uribe and Wang in [9] introduced the variable exponent e p(·) , in the following way: Let p(·) ∈ P(Rn ) satisfy that there Hardy space, denoted by H exist p0 ∈ (0, p− ) and a positive constant C, only depending on n, p(·) and p0 , such that (1.5)
kM(f )kLp(·)/p0 (Rn ) ≤ Ckf kLp(·)/p0 (Rn ) .
4
Ciqiang Zhuo, Dachun Yang and Yiyu Liang
e p(·) is defined to be the If N ∈ (n/p0 + n + 1, ∞), then the variable exponent Hardy space H ′ n ∗ p(·) n set of all f ∈ S (R ) such that fN,+ ∈ L (R ). In [9, Theorem 3.1], it was shown that the e p(·) is independent of the choice of N ∈ (n/p0 + n + 1, ∞). space H
(ii) We point out that, in [32, Theorem 3.3], it was proved that the space H p(·) (Rn ) is independent of N as long as N is sufficiently large. Although the range of N is not presented explicitly in [32, Theorem 3.3], by the proof of [32, Theorem 3.3], we see that N as in (1.4) does the work. Let φ ∈ S(Rn ) be a radial real-valued function satisfying supp φb ⊂ {ξ ∈ Rn : 1/2 ≤ |ξ| ≤ 2}
(1.6) and
b |φ(ξ)| ≥ C if 3/5 ≤ |ξ| ≤ 5/3,
(1.7)
where C denotes a positive constant independent of ξ and, for all φ ∈ S(Rn ), φb denotes its Fourier ′ transform. Obviously, φ ∈ S∞ (Rn ). Then, for all f ∈ S∞ (Rn ), the Littlewood-Paley g-function, ∗ the Lusin area function and the gλ -function with λ ∈ (0, ∞) of f are, respectively, defined by setting, for all x ∈ Rn , 1/2 Z ∞ dt , g(f )(x) := |f ∗ φt (x)|2 t 0 )1/2 (Z Z ∞ dy dt S(f )(x) := |φt ∗ f (y)|2 n+1 t 0 {y∈Rn : |y−x| α} and q ∈ [1, ∞) is the uniformly Muckenhoupt weight index of ϕ. To see this, following [32, Example 1.3], for all x ∈ R, let n o p(x) := max 1 − e3−|x|, min 6/5, max{1/2, 3/2 − x2 } .
Then p(·) satisfies (1.1) and (1.2). Now, let E := (1, 2), then, for all x ∈ R, 1 χR\E (x). 1 + 2|x − 3/2|
M(χE )(x) = χE (x) + It is easy to see that, for all λ ∈ (0, ∞), Z
U(E;1/11)
λp(x) dx =
Z
− 13 2
− 27
R
E
λp(x) dx = λ1/2 and
λp(x) dx >
Z
E
λp(x) dx +
Z
1 2
− 12
λp(x) dx = λ1/2 + λ6/5 .
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Ciqiang Zhuo, Dachun Yang and Yiyu Liang
Thus, we find that lim
λ→∞
R
λp(x) dx
U(E;1/11) R λp(x) E
dx
= ∞,
which implies that there does not exist a positive constant C, independent of λ, such that, Z Z p(x) λ dx ≤ C λp(x) dx. U(E;1/11)
E
Thus, the method used in the proof of [26, Theorem 4.8] is not suitable for the present setting. For any s ∈ Z+ , C s (Rn ) denotes the set of all functions having continuous classical derivatives up to order not more than s. For α ∈ (0, 1] and s ∈R Z+ , let Cα,s (Rn ) be the family of functions φ ∈ C s (Rn ) such that supp φ ⊂ {x ∈ Rn : |x| ≤ 1}, Rn φ(x)xγ dx = 0 for all γ ∈ Zn+ and |γ| ≤ s, and, for all x1 , x2 ∈ Rn and ν ∈ Zn+ with |ν| = s, |Dν φ(x1 ) − Dν φ(x2 )| ≤ |x1 − x2 |α .
(1.10)
For all f ∈ L1loc (Rn ) and (y, t) ∈ Rn+1 := Rn × (0, ∞), let + Aα,s (f )(y, t) :=
sup φ∈Cα,s (Rn )
|f ∗ φt (y)|.
Then, the intrinsic g-function, the intrinsic Lusin area integral and the intrinsic gλ∗ -function of f are, respectively, defined by setting, for all x ∈ Rn and λ ∈ (0, ∞), gα,s (f )(x) :=
Z
∞
0
Sα,s (f )(x) :=
(Z
∞
∗ gλ,α,s (f )(x)
:=
(Z
0
∞
Z
Rn
1/2
,
dy dt [Aα,s (f )(y, t)] n+1 t |y−x| |x − x0 |/2. From this and (2.15), we deduce that 1/2 " #2 Z ∞ dt gα,s (a)(x) = sup |a ∗ φt (x)| 0 t φ∈Cα,s (Rn ) .
.
1
kχQ kLp(·) (Rn ) 1 kχQ kLp(·) (Rn )
r
n+α+s
(Z
r |x − x0 |
∞
|x−x0 | 2
t
−2(n+α+s)
n+α+s
dt
)1/2
n+α+s
.
[M(χQ )(x)] n kχQ kLp(·) (Rn )
,
which implies that (2.16) kgα,s (f )kLp(·) (Rn )
X
√
λj gα,s (aj )χ2 nQj .
j
Lp(·) (Rn )
=: I1 + I2 .
n+α+s
X [M(χQj )] n
λj +
kχQj kLp(·) (Rn )
j
Lp(·) (Rn )
For I1 , by taking bj := gα,s (aj )χ2√nQj for each j in Lemma 2.10, (2.14) and Lemma 2.6, we conclude that
1 q
X λ b |Q | j j j
(2.17) I1 .
√ √
j kbj kLq (2 nQj ) kχ2 nQj kLp(·) (Rn ) p(·) n L (R )
∗
! p∗ 1/p 1
X
λj bj |Qj | q
. .A({λj }j , {Qj }j ).
√ √
kχ k kb k q p(·) n j 2 nQj L (R ) L (2 nQj )
j
p(·) n L
(R )
For I2 , letting θ := (n + α + s)/n, by Lemma 2.5 and p− ∈ (n/(n + α + s), ∞), we find that
1
θ
X
X λ [M(χ )]θ θ
λ χ j Qj j Qj
I2 . .
kχ kχ Q Q j kLp(·) (Rn ) j kLp(·) (Rn )
j
j
θp(·) n Lp(·) (Rn ) L (R )
!p∗ p1∗
X λ χ j Qj
∼ A({λj }j , {Qj }j ). .
kχ Q j kLp(·) (Rn )
p(·) n
j L
(R )
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Ciqiang Zhuo, Dachun Yang and Yiyu Liang
From this, together with (2.13), (2.16) and (2.17), we deduce that kgα,s (f )kLp(·) (Rn ) . kf kH p(·) (Rn ) , which completes the proof of Theorem 1.8. For s ∈ Z+ , α ∈ (0, 1] and ǫ ∈ (0, ∞), let C(α,ǫ),s (y, t), with y ∈ Rn and t ∈ (0, ∞), be the family of functions ψ ∈ C s (Rn ) such that, for all γ ∈ Zn+ , |γ| ≤ s and x ∈ Rn , |Dγ ψ(x)| ≤ R t−n−|γ| (1 + |y − x|/t)−n−ǫ , Rn ψ(x)xγ dx = 0 and, for all x1 , x2 ∈ Rn , ν ∈ Zn+ and |ν| = s, |x1 − x2 |α |D ψ(x1 ) − D ψ(x2 )| ≤ n+γ+α t ν
ν
" −n−ǫ −n−ǫ # |y − x1 | |y − x2 | . 1+ + 1+ t t
The proof of Theorem 1.10 needs the following Lemma 2.15, whose proof is trivial, the details being omitted. Lemma 2.15. Let s ∈ Z+ , α ∈ (0, 1], ǫ ∈ (0, ∞) and f be a measurable function satisfying (1.12). (i) For any y ∈ Rn and t ∈ (0, ∞), it holds true that e(α,ǫ),s (f )(y, t) = A
sup ψ∈C(α,ǫ),s (y,t)
Z
Rn
ψ(x)f (x) dx .
(ii) If t1 , t2 ∈ (0, ∞), t1 < t2 , y ∈ Rn and ψ ∈ C(α,ǫ),s (y, t1 ), then ( tt12 )n+s+α ψ ∈ C(α,ǫ),s (y, t2 ). ∗ Proof of Theorem 1.10. If f ∈ (L1,p(·),s (Rn ))∗ , e gλ,(α,ǫ),s (f ) ∈ Lp(·) (Rn ) and f vanishes weakly at infinity, then, by Lemma 2.8, we see that f ∈ S ′ (Rn ) and, by the fact that, for all x ∈ Rn , ∗ ∗ (f )(x) gλ∗ (f )(x) . gλ,α,s (f )(x) . geλ,(α,ǫ),s
and Theorem 1.4, we further know that f ∈ H p(·) (Rn ) and
∗ ∗ kf kH p(·) (Rn ) . kgλ∗ (f )kLp(·) (Rn ) . kgλ,α,s (f )kLp(·) (Rn ) . ke gλ,(α,ǫ),s (f )kLp(·) (Rn ) .
This finishes the proof of the sufficiency of Theorem 1.10. Next we prove the necessity of Theorem 1.10. Let f ∈ H p(·) (Rn ). Then, as in the proof of Theorem 1.8, we see that f ∈ (L1,p(·),s (Rn ))∗ and f vanishes weakly at infinity. For all x ∈ Rn , we have (2.18)
∗ [e gλ,(α,ǫ),s (f )(x)]2 Z ∞Z =
+
0 |y−x|
