PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 12, December 2009, Pages 4219–4225 S 0002-9939(09)10044-8 Article electronically published on August 3, 2009
A WAVELET CHARACTERIZATION FOR THE DUAL OF WEIGHTED HARDY SPACES MING-YI LEE, CHIN-CHENG LIN, AND YING-CHIEH LIN (Communicated by Hart F. Smith) p using Abstract. We define the weighted Carleson measure space CM Ow wavelets, where the weight function w belongs to the Muckenhoupt class. Then p p is the dual space of the weighted Hardy space Hw by we show that CM Ow using sequence spaces. As an application, we give a wavelet characterization of BM Ow .
1. Introduction Meyer [4] described the Hardy space H 1 and BM O via wavelets. He offered several characterizations of H 1 in terms of its decompositions with respect to wavelet bases, and characterized BM O in terms of a Carleson condition on wavelet coefficients. A natural extension is to consider their weighted counterparts. In 2001, Garcia-Cuerva and Martell [2] gave a wavelet characterization of weighted Hardy spaces Hwp (R), 0 < p ≤ 1. In this article, we give a wavelet characterization for the dual of Hwp (R), 0 < p ≤ 1. In order to do this, we define the weighted Carleson p and two sequence spaces spw and cpw . We first show that cpw measure space CM Ow p is the dual of spw and then obtain that CM Ow is the dual of Hwp . As a consequence, 1 is the same as BM Ow , and hence we succeed by an approach different from CM Ow the one in [5] for the wavelet characterization of BM Ow . Let ψ be an orthonormal wavelet; that is, ψ ∈ L2 (R) such that the system ψj,k (x) := 2j/2 ψ(2j x − k),
j, k ∈ Z,
is an orthonormal basis for L2 (R). We define the operator Wψ by Wψ f =
|f, ψj,k |2 |Ij,k |−1 χIj,k
1/2 ,
f ∈ L2 (R),
j,k∈Z
Received by the editors November 26, 2008, and, in revised form, May 4, 2009. 2000 Mathematics Subject Classification. Primary 42B30, 42C40. Key words and phrases. BM Ow , orthonormal wavelets, weighted Carleson measure spaces, weighted Hardy spaces. The first author was supported by NSC of Taiwan under Grant #NSC 97-2115-M-008-005. The second and third authors were supported by NSC of Taiwan under Grant #NSC 97-2115M-008-021-MY3. c 2009 American Mathematical Society Reverts to public domain 28 years from publication
4219
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
4220
MING-YI LEE, CHIN-CHENG LIN, AND YING-CHIEH LIN
where Ij,k = [2−j k, 2−j (k + 1)]. Denoting by D the set of all dyadic intervals Ij,k with j, k ∈ Z, and letting ψIj,k = ψj,k , we can also write 1/2 |f, ψI |2 |I|−1 χI . Wψ f = I∈D
Henceforth, we always use I and J to denote dyadic intervals. In what follows, we shall work exclusively with the one-dimensional case. For α ≥ 1, we say that ψ belongs to the regularity class Rα if ψ ∈ C [α] and there exist positive constants C, r, ε satisfying (i) xn ψ(x) dx = 0 for all 0 ≤ n ≤ [α] − 1, R
C for all x ∈ R, (1 + |x|)1+[α]+r C (iii) |ψ (n) (x)| ≤ for all x ∈ R and 0 ≤ n ≤ [α]. (1 + |x|)α+ε Here [α] denotes the greatest integer not greater than α. The weight functions mentioned in this article refer to the Muckenhoupt Aq weights. A weight w ≥ 0 belongs to the class Aq , 1 < q < ∞, if there is a constant C > 0 such that q−1 −1/(q−1) w(x) dx w(x) dx ≤ C|I|q for any interval I ⊂ R. (ii) |ψ(x)| ≤
I
I
The class A1 consists of weights w satisfying for some C > 0 that 1 w(x) dx ≤ C · ess inf w(x) for any interval I ⊂ R, x∈I |I| I and A∞ := 1≤q 1 : w ∈ Aq } the critical index of w. We w(x)dx. E use w(E) to denote the weighted measure Let ϕ ∈ S satisfy R ϕ(x)dx = 1. The maximal function f ∗ is defined by f ∗ (x) = sup |f ∗ ϕr (x)|, r>0
−1
where ϕr (x) = r ϕ(x/r), r > 0. The weighted Hardy spaces Hwp consist of those tempered distributions f ∈ S for which f ∗ ∈ Lpw with f Hwp = f ∗ Lpw . We refer readers to [1, 3] for the details about Aq and Hwp . The following theorem was proved by Garcia-Cuerva and Martell [2]. Theorem A. Let 0 < p ≤ 1 and w ∈ A∞ . If ψ ∈ Rα is an orthonormal wavelet with α ≥ qw /p, then there exist two constants 0 < c ≤ C < ∞ such that c f Hwp ≤ Wψ f Lpw ≤ C f Hwp . Definition. For 0 < p ≤ 1 and w ∈ A∞ , let ψ ∈ Rα be an orthonormal wavelet p is the set of all with α ≥ qw /p. The weighted Carleson measure space CM Ow 1 g ∈ Lloc satisfying 1/2 1 2 |I| p g CM Ow := sup |g, ψI | < ∞. 2 w(I) J∈D w(J) p −1 I⊂J
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
p A WAVELET CHARACTERIZATION FOR (Hw )
4221
Remark 1. If w ≡ constant and p = 1, then the above definition reduces to the Carleson condition that characterizes BM O (cf. [4, p. 154]). Theorem A implies that the wavelet characterization of Hwp is independent of the choice of ψ, and p is independent of the hence, by the following Theorem 1, the definition of CM Ow choice of ψ, too. We now state our main result as follows. Theorem 1. For 0 < p ≤ 1 and w ∈ A∞ , let ψ ∈ Rα be an orthonormal wavelet p with α ≥ qw /p. The dual of Hwp is CM Ow in the following sense. p (a) For each g ∈ CM Ow , there is a linear functional g , initially defined on Hwp ∩ L2 , which has a continuous extension to Hwp and g ≤ C g CM Owp . (b) Conversely, every continuous linear functional of Hwp can be realized as p and g CM Owp ≤ C . = g with some g ∈ CM Ow It is known that the dual space of Hw1 is
1 BM Ow = f ∈ L1loc : sup |f (x) − fQ |dx < ∞ interval Q w(Q) Q for w ∈ A∞ , and the dual space of Hwp , 0 < p < 1, is f (x) − PQ (x) r w(x)dx 1/r f (x) r ∈ Lloc (w(x)dx) : w(Q) w(x) w(x) Q (1) ≤ Cw(Q)
1/p−1
for any bounded interval Q
1 for w ∈ Ar , 1 ≤ r < ∞, where fQ = |Q| f (x)dx and PQ is the unique polynomial Q of degree ≤ [qw /p] − 1 such that Q (f (x) − PQ (x))xk dx = 0 for k = 0, 1, · · · , [qw /p] − 1 (see [1]). Thus, we have a wavelet characterization of BM Ow and a p as follows. continuous characterization of CM Ow Corollary 2. Let 0 < p ≤ 1 and w ∈ Ar , 1 ≤ r ≤ ∞. Also let ψ ∈ Rα be an orthonormal wavelet with α ≥ qw /p. (a) For p = 1 and w ∈ A∞ , f ∈ BM Ow if and only if its wavelet coefficients f, ψI satisfy Carleson’s condition: 1 |I| sup |f, ψI |2 ≤ C. w(I) J∈D w(J) I⊂J
(b) For 0 < p < 1 and w ∈ Ar , 1 ≤ r < ∞, f satisfies (1) if and only if p . f ∈ CM Ow Remark 2. When the wavelet ψ has compact support, the above characterization of BM Ow was given by Wu [5]. Here we offer a different but simpler approach. 2. Sequence spaces In this section, we introduce two sequence spaces spw and cpw , 0 < p ≤ 1. Definition. Let 0 < p ≤ 1 and w ≥ 0 be a weight function. The sequence space spw is defined to be the collection of all complex-valued sequences 1/2 p 2 −1 p sw = {sI } : {sI } sw := |sI | |I| χI w(I) and w(I ∩ Ωk+1 ) ≤ w(I) . 2 2 sI · tI = s I · tI ≤ |sI ||tI |, I
˜ ˜ k I∈B k I⊂I
˜ ˜ k I∈B k I⊂I
I∈Bk
I∈Bk
˜ are the maximal dyadic intervals in Bk . Applying the inequality · 1 ≤ where I’s p · , we get 1/2 1/2 w(I) |I| |sI ||tI | ≤ |sI |2 |tI |2 |I| w(I) ˜ ˜ k I∈B k I⊂I
˜ k I∈B k
I∈Bk
≤
I⊂I˜
I⊂I˜
I∈Bk
I∈Bk
˜ k I∈B k
≤ {tI } cpw
|sI |
I⊂I˜
p/2
2 w(I)
|I|
I⊂I˜
I∈Bk
˜ k I∈B k
p/2 1/p |I| |tI | w(I) 2
I∈Bk
˜ w(I)
1−p/2
I⊂I˜
|sI |
p/2 1/p
2 w(I)
|I|
I∈Bk
Write
k = {x ∈ R : Mw (χΩ )(x) > 1/2}, Ω k
where Mw is the weighted Hardy-Littlewood maximal function defined by 1 Mw f (x) = sup |f (x)|w(x)dx. interval Qx w(Q) Q
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
.
p A WAVELET CHARACTERIZATION FOR (Hw )
4223
k for any I ∈ Bk . Since the I’s ˜ are mutually disjoint dyadic intervals, Then I ⊂ Ω
˜ ≤ w(Ω k ). We then apply H¨ w(I) older’s inequality to obtain ˜ I∈B k p/2 1/p w(I) 1−p/2 2 k) sI · tI ≤ {tI } cpw w(Ω |sI | . |I| I
I∈Bk
k
We claim that
|sI |2
I∈Bk
w(I) k ). ≤ C22k w(Ω |I|
k ) ≤ Cw(Ωk ) and the Mw is of weak type (1, 1) with respect to w(x)dx, so w(Ω claim gives sI · tI
≤ C {tI } cpw
I
k) 2 w(Ω
k
≤ C {tI } cpw
1/p
kp
1/p 2kp w(Ωk )
k
≤ C {tI } S Lpw = C {tI } cpw {sI } spw . cp w
To prove the claim, by the definitions of S(x) and Bk , we have k) S 2 (x)w(x) dx ≤ 22k+2 w(Ω k \Ωk+1 Ω
and
S (x)w(x) dx ≥
2
k \Ωk+1 Ω
k \Ωk+1 Ω I∈Bk
=
|sI |
I∈Bk
2w
|sI |2 |I|−1 χI (x)w(x) dx
k \Ωk+1 ) I ∩ (Ω |I|
w(I) 1 |sI |2 . 2 |I|
≥
I∈Bk
(b) Clearly, every ∈ (spw ) is of the form ({sI }) =
s I tI ,
{sI } ∈ spw ,
I
where {tI } is a certain sequence. Fix a dyadic interval J. Let SJ = {I ∈ D : I ⊂ J} and define a measure ν on SJ by dν(I) =
|I| w(J) p −1 2
for I ∈ SJ .
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
4224
By duality,
(2)
MING-YI LEE, CHIN-CHENG LIN, AND YING-CHIEH LIN
1/2 |I| 1 t = I 2 1 −1 w(I) w(I) 2 2 (SJ ,dν) w(J) p I⊂J |I| = sup s t I I 2 1 −1 p 2 {sI }2 (S ,dν) ≤1 I⊂J w(J) w(I) J |I| . sI ≤ sup 2 1 p −1 {sI }2 (S ,dν) ≤1 w(J) p w(I) 2 sw J
1
|tI |2
For {sI } ∈ 2 (SJ , dν), H¨ older’s inequality yields p/2 1/p |I| 1 2 |I| sI χ = |s | (x) w(x) dx I 2 2 1 p w(I) I J w(J) p −1 w(I) 2 w(J) p −1 sw I⊂J 1/2 1 2 |I| ≤ |s | (x)w(x) dx χ I 2 w(I) I w(J) p −1 J I⊂J = {sI } 2 (SJ ,dν) , and hence sup {sI }2 (S
≤1 J ,dν)
sI
|I| w(J) p −1 w(I) 2
1 2
sp w
≤ 1.
Taking the supremum over J ∈ D in (2), we obtain {tI } cpw ≤ .
3. Proof of the main theorem In this section we show that Theorem 1 follows as a consequence of Theorem 3. Let ψ ∈ Rα , α ≥ 1, be an orthonormal wavelet. Define a map P from the family of complex sequences into S by P ({sI }) = sI ψI . I
Define another map L from function space into the family of complex sequences by L(f ) = f, ψI such that all f, ψI ’s are well defined. Figure 1 illustrates the relationship among p . Then P ◦ L L2 is the identity on L2 . For 0 < p ≤ 1 and spw , cpw , Hwp , and CM Ow
swp P
cwp L
H wp
P
L
CMOwp
Figure 1. Diagram for spaces and maps w ∈ A∞ with critical index qw , if α ≥ qw /p, then Theorem A yields (3)
{L(f )} spw ≤ C f Hwp
for f ∈ Hwp ∩ L2
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
p A WAVELET CHARACTERIZATION FOR (Hw )
4225
and (4)
P ({sI }) Hwp ≤ C Wψ P ({sI }) Lpw = C {sI } spw
for {sI } ∈ spw .
p , By the definitions of cpw and CM Ow
{L(g)} cpw = g CM Owp
(5)
p for g ∈ CM Ow
and P ({tI }) CM Owp = {tI } cpw
(6)
for {tI } ∈ cpw .
p Proof of Theorem 1. For g ∈ CM Ow , define a linear functional ˜g by ˜g (f ) = L(f ), L(g) for f ∈ Hwp ∩ L2 .
By (3), (5), and Theorem 3, |˜g (f )| ≤ C L(f ) spw L(g) cpw ≤ C f Hwp g CM Owp for f ∈ Hwp ∩ L2 . Since Hwp ∩ L2 is dense in Hwp , the map ˜g can be extended to a continuous linear functional g on Hwp satisfying g ≤ C g CM Owp . Conversely, let ∈ (Hwp ) and set 1 = ◦ P on spw . It follows from (4) that 1 ∈ (spw ) . By Theorem 3, there exists {tI } ∈ cpw such that 1 ({sI }) = s I · tI for {sI } ∈ spw , I
and {tI } cpw ≈ 1 ≤ C . For f ∈ Hwp ∩ L2 , we have f, ψI tI = L(f ), L(g), (f ) = 1 ◦ L(f ) = where g =
I
I tI ψI . This shows that = g , and (6) gives
g CM Owp = {tI } cpw ≤ C .
Hence, the proof is finished. References Hp
1. J. Garcia-Cuerva, Weighted spaces, Dissertationes Math. 162 (1979), 1-63. MR549091 (82a:42018) 2. J. Garcia-Cuerva and J. M. Martell, Wavelet characterization of weighted spaces, J. Geom. Anal. 11 (2001), 241-264. MR1856178 (2002h:42039) 3. J. Garc´ıa-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. MR807149 (87d:42023) 4. Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, Vol. 37, Cambridge University Press, Cambridge, 1992. MR1228209 (94f:42001) 5. S. Wu, A wavelet characterization for weighted Hardy spaces, Rev. Mat. Iberoamericana 8 (1992), 329-349. MR1202414 (94i:42027) Department of Mathematics, National Central University, Chung-Li, Taiwan 320, Republic of China E-mail address:
[email protected] Department of Mathematics, National Central University, Chung-Li, Taiwan 320, Republic of China E-mail address:
[email protected] Department of Mathematics, National Central University, Chung-Li, Taiwan 320, Republic of China E-mail address:
[email protected]
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
