A note on $H^p_w$-boundedness of Riesz transforms and $θ$-Calderón-Zygmund operators through molecular characterization

0.2 MB·English

by  Luong Dang Ky

A NOTE ON Hp-BOUNDEDNESS OF RIESZ TRANSFORMS
w
AND θ-CALDERO´N-ZYGMUND OPERATORS THROUGH
MOLECULAR CHARACTERIZATION
LUONG DANG KY
2
1 Abstract. Let 0 <p 1 and w in the Muckenhoupt class A1. Recently,
≤
0 by using the weighted atomic decomposition and molecular characteriza-
2 tion; Lee, Lin and Yang [11] (J. Math. Anal. Appl. 301 (2005), 394–400)
n established that the Riesz transforms Rj,j = 1,2,...,n, are bounded on
a Hp(Rn). In this note we extend this to the generalcase of weight w in the
w
J Muckenhoupt class A∞ through molecular characterization. One difficulty,

6 which has not been taken care in [11], consists in passing from atoms to
1 allfunctions in Hp(Rn). Furthermore,the Hp-boundedness ofθ-Calder´on-
w w
Zygmund operators are also given through molecular characterization and
]
A atomic decomposition.
C
.
h
t
a 1. Introduction
m
[ Caldero´n-Zygmund operators and their generalizations on Euclidean space
Rn have been extensively studied, see for example [7, 14, 18, 15]. In particular,
3
v Yabuta [18] introduced certain θ-Calder´on-Zygmund operators to facilitate his
4
study of certain classes of pseudo-differential operator.
2
7
Definition 1.1. Let θ be a nonnegative nondecreasingfunction on (0, ) satis-
4
∞
. fying 1 θ(t)dt < . A continuous function K : Rn Rn (x,x) : x Rn C
6 0 t ∞ × \{ ∈ } →
0 is saiRd to be a θ-Calder´on-Zygmund singular integral kernel if there exists a
1
constant C > 0 such that
1
: C
v K(x,y)
i | | ≤ x y n
X | − |
r for all x = y,
a 6
1 x x
′
K(x,y) K(x,y) + K(y,x) K(y,x) C θ | − |
′ ′
| − | | − | ≤ x y n x y
(cid:16) (cid:17)
| − | | − |
for all 2 x x x y .
′
A line|ar−oper|a≤to|r T−: | (Rn) (Rn) is said to be a θ-Calder´on-Zygmund
′
operator if T can be extenSded to→a bSounded operator on L2(Rn) and there exists
2010 Mathematics Subject Classification. 42B20,42B25, 42B30.
Key words and phrases. Muckenhouptweights,weightedHardyspaces,atomicdecompo-
sition, molecular characterization,Riesz transforms, Caldero´n-Zygmundoperators.
1
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 2
w
a θ-Calderon-Zygmund singular integral kernel K such that for all f C (Rn)
∈ c∞
and all x /supp f, we have
∈
Tf(x) = K(x,y)f(y)dy.
Z
Rn
When Kj(x,y) = π−(n+1)/2Γ n+21 xxjy−ny+j1,j = 1,2,...,n, then they are the
(cid:16) (cid:17)| − |
classical Riesz transforms denoted by R .
j
It is well-known that the Riesz transforms R ,j = 1,2,...,n, are bounded
j
on unweighted Hardy spaces Hp(Rn). There are many different approaches to
prove this classical result (see [11, 9]). Recently, by using the weighted molecu-
lar theory (see [10]) and combined with Garc´ıa-Cuerva’s atomic decomposition
[5] for weighted Hardy spaces Hp(Rn), the authors in [11] established that the
w
Riesz transforms R ,j = 1,2,...,n, are bounded on Hp(Rn). More precisely,
j w
they proved that kRjfkHwp ≤ C for every w-(p,∞,ts−1)-atom where s,t ∈ N
satisfyn/(n+s) < p n/(n+s 1) and((s 1)r +n)/(s(r 1))withr isthe
w w w
≤ − − −
critical index of w for the reverse H¨older condition. Remark that this leaves a
gap in the proof. Similar gaps exist in some litteratures, for instance in [10, 15]
when the authors establish Hp-boundedness of Caldero´n-Zygmund type oper-
w
ators. Indeed, it is now well-known that (see [1]) the argument ”the operator
T is uniformly bounded in Hp(Rn) on w-(p, ,r)-atoms, and hence it ex-
tends to a bounded operator onwHp(Rn)” is wro∞ng in general. However, Meda,
w
Sj¨ogren and Vallarino [13] establishes that (in the setting of unweighted Hardy
spaces) this is correct if one replaces L -atoms by Lq-atoms with 1 < q < .
∞
∞
See also [19] for L2-atoms with a different method from [13]. Later, the au-
thors in [2] extended these results to the weighted anisotropic Hardy spaces.
More precisely, it is claimed in [2] that the operator T can be extended to a
bounded operator on Hp(Rn) if it is uniformly bounded on w-(p,q,r)-atoms
w
for q < q < ,r [n(q /p 1)] where q is the critical index of w.
w w w
∞ ≥ −
Motivated by [11, 10, 15, 1, 2], in this paper, we extend Theorem 1 in [11] to
A weights (see Theorem 1.1); Theorem 4 in [10] (see Theorem 1.2), Theorem
∞
3 in [15] (see Theorem 3.1) to θ-Calder´on-Zygmund operators; and fill the gaps
oftheproofsbyusingtheatomicdecompositionandmolecularcharacterization
of Hp(Rn) as in [11].
w
Throughout the whole paper, C denotes a positive geometric constant which
is independent of the main parameters, but may change from line to line. In
Rn, we denote by B = B(x,r) an open ball with center x and radius r > 0.
For any measurable set E, we denote by E its Lebesgue measure, and by Ec
the set Rn E. | |
\
Let us first recall some notations, definitions and well-known results.
Let 1 p < . A nonnegative locally integrable function w belongs to the
≤ ∞
Muckenhoupt class A , say w A , if there exists a positive constant C so
p p
∈
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 3
w
that
1 1 p 1
w(x)dx (w(x)) 1/(p 1)dx − C, if 1 < p < ,
− −
B Z (cid:16) B Z (cid:17) ≤ ∞
| | | |
B B
and
1
w(x)dx Cess-infw(x), if p = 1,
B Z ≤ x B
| | ∈
B
for all balls B in Rn. We say that w A if w A for some p [1, ).
p
∈ ∞ ∈ ∈ ∞
It is well known that w A , 1 p < , implies w A for all q > p.
p q
∈ ≤ ∞ ∈
Also, if w A , 1 < p < , then w A for some q [1,p). We thus write
p q
∈ ∞ ∈ ∈
q := inf p 1 : w A to denote the critical index of w. For a measurable
w p
{ ≥ ∈ }
set E, we note w(E) = w(x)dx its weighted measure.
E
The following lemmaRgives a characterization of the class A , 1 p < .
p
≤ ∞
It can be found in [6].
Lemma A. The function w A , 1 p < , if and only if, for all non-
p
∈ ≤ ∞
negative functions and all balls B,
1 p 1
f(x)dx C f(x)pw(x)dx.
(cid:16) B Z (cid:17) ≤ w(B) Z
| |
B B
A close relation to A is the reverse H¨older condition. If there exist r > 1
p
and a fixed constant C > 0 such that
1 1/r 1
wr(x)dx C w(x)dx for every ball B Rn,
(cid:16) B Z (cid:17) ≤ (cid:16) B Z (cid:17) ⊂
| | | |
B B
we say that w satisfies reverse H¨older condition of order r and write w RH .
r
∈
It is known that if w RH , r > 1, then w RH for some ε > 0. We thus
r r+ε
∈ ∈
write r := sup r > 1 : w RH to denote the critical index of w for the
w r
{ ∈ }
reverse H¨older condition.
The following result provides us the comparison between the Lebesgue mea-
sure of a set E and its weighted measure w(E). It also can be found in [6].
Lemma B. Let w A RH , p 1 and r > 1. Then there exist con-
p r
∈ ∩ ≥
stants C ,C > 0 such that
1 2
E p w(E) E (r 1)/r
C | | C | | − ,
1 2
B ≤ w(B) ≤ B
(cid:16) (cid:17) (cid:16) (cid:17)
| | | |
for all cubes B and measurable subsets E B.
Given a weight function w on Rn, as usua⊂l we denote by Lq(Rn) the space of
w
aLspl∞wlaf(cuRensn,c)ttihiosenLsw∞fei(sgRahtnti)esdfayniHndagrkkdffykkLsL∞wpqwa=c:=esk(fHRkRpLn(∞|Rf.n()xA,)np|qaw>lo(g0xo,)udcsxal)ny1/btqoe<tdh∞eefin.cleWadsshiinecnatleqrHm=asr∞doyf,
w
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 4
w
maximal functions. Namely, let φ be a function in (Rn), the Schwartz space
S
of rapidly decreasing smooth functions, satisfying φ(x)dx = 1. Define
Rn
φ (x) = t nφ(x/t), t > 0,xR Rn,
t −
∈
and the maximal function f by
∗
f (x) = sup f φ (x) , x Rn.
∗ t
| ∗ | ∈
t>0
Then Hp(Rn) consists of those tempered distributions f (Rn) for which
f Lpw(Rn) with the (quasi-)norm ∈ S′
∗ ∈ w
kfkHwp = kf∗kLpw.
In order to show the Hp-boundedness of Riesz transforms, we characterize
w
weighted Hardy spaces in terms of atoms and molecules in the following way.
Definition of a weighted atom. Let 0 < p 1 q and p = q
≤ ≤ ≤ ∞ 6
such that w A . Let q be the critical index of w. Set [ ] the integer
q w
function. For∈s N satisfying s [n(q /p 1)], a function a· Lq (Rn) is
∈ ≥ w − ∈ w
called w-(p,q,s)-atom centered at x if
0
(i) supp a B for some ball B centered at x ,
0
⊂
(ii) kakLqw ≤ w(B)1/q−1/p,
(iii) a(x)xαdx = 0 for every multi-index α with α s.
Let HpR,qR,sn(Rn) denote the space consisting of tempered|di|s≤tributions admitting
w
a decomposition f = ∞j=1λjaj in S′(Rn), where aj’s are w-(p,q,s)-atomsand
∞j=1|λj|p < ∞. AndPfor every f ∈ Hwp,q,s(Rn), we consider the (quasi-)norm
P
∞ 1/p ′ ∞
kfkHwp,q,s = inf |aj|p : f =S λjaj, {aj}∞j=1 are w-(p,q,s)-atoms .
n(cid:16)Xj=1 (cid:17) Xj=1 o
Denote by Hp,q,s(Rn) the vector space of all finite linear combinations of w-
w,fin
(p,q,s)-atoms, and the (quasi-)norm of f in Hp,q,s(Rn) is defined by
w,fin
k k
1/p
kfkHwp,,qfi,ns := infn(cid:16)Xj=1 |λj|p(cid:17) : f = Xj=1 λjaj,k ∈ N,{aj}kj=1 are w-(p,q,s)-atomso.
We have the following atomic decomposition for Hp(Rn). It can be found
w
in [5] (see also [2, 8]).
Theorem A. If the triplet (p,q,s) satisfies the conditions of w-(p,q,s)-atoms,
then Hp(Rn) = Hp,q,s(Rn) with equivalent norms.
w w
The molecules corresponding to the atoms mentioned above can be defined
as follows.
Definition of a weighted molecule. For 0 < p 1 q and p = q,
≤ ≤ ≤ ∞ 6
let w A with critical index q and critical index r for the reverse H¨older
q w w
∈
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 5
w
condition. Set s [n(q /p 1)], ε > max sr (r 1) 1n 1+(r 1) 1,1/p
w w w − − w −
≥ − { − − −
1 , a = 1 1/p+ε, and b = 1 1/q+ε. A w-(p,q,s,ε)-molecule centered at
x} is a fun−ction M Lq (Rn) sa−tisfying
0(i) M.w(B(x , ∈x )w)b Lq (Rn),
0 ·− 0 ∈ w
(ii) kMkaL/qwbkM.w(B(x0,·−x0))bk1L−qwa/b ≡ Nw(M) < ∞,
(iii) M(x)xαdx = 0 for every multi-index α with α s.
TheRaRbnove quantity N (M) is called the w-molecular| n|o≤rm of M.
w
In [10], Lee and Lin proved that every weighted molecule belongs to the
weighted Hardy space Hp(Rn), and the embedding is continuous.
w
Theorem B. Let 0 < p 1 q and p = q, w A , and (p,q,s,ε) be
q
≤ ≤ ≤ ∞ 6 ∈
the quadruple in the definition of molecule. Then, every w-(p,q,s,ε)-molecule
M centered at any point in Rn is in Hwp(Rn), and kMkHwp ≤ CNw(M) where
the constant C is independent of the molecule.
Although, in general, one cannot conclude that an operator T is bounded
on Hp(Rn) by checking that their norms have uniform bound on all of the cor-
w
responding w-(p, ,s)-atoms (cf. [1]). However, this is correct when dealing
∞
with w-(p,q,s)-atoms with q < q < . Indeed, we have the following result
w
∞
(see [2, Theorem 7.2]).
Theorem C. Let 0 < p 1, w A , q (q , ) and s Z satisfying
w
s [n(q /p 1)]. Suppose≤that T :∈Hp,∞q,s(Rn∈) H∞p(Rn) is a l∈inear operator
≥ w − w,fin → w
satisfying
sup{kTakHwp : a is any w−(p,q,s)−atom} < ∞.
Then T can be extended to a bounded linear operator on Hp(Rn).
w
Our first main result, which generalizes Theorem 1 in [11], is as follows:
Theorem 1.1. Let 0 < p 1 and w A . Then, the Riesz transforms are
bounded on Hp(Rn). ≤ ∈ ∞
w
For the next result, we need the notion T 1 = 0.
∗
Definition 1.2. Let T be a θ-Calder´on-Zygmund operator. We say that T 1 =
∗
0 if Tf(x)dx = 0 for all f Lq(Rn),1 < q , with compact support
Rn ∈ ≤ ∞
and R f(x)dx = 0.
Rn
WRe now can give the Hp-boundedness of θ-Calder´on-Zygmund type opera-
w
tors, which generalizes Theorem 4 in [10] by taking q = 1 and θ(t) = tδ, as
follows:
Theorem 1.2. Given δ (0,1], n/(n + δ) < p 1, and w A RH
q r
∈ ≤ ∈ ∩
with 1 q < p(n + δ)/n,(n + δ)/(n + δ nq) < r. Let θ be a nonnegative
≤ −
nondecreasing function on (0, ) with 1 θ(t)dt < , and T be a θ-Calder´on-
Zygmund operator satisfying T∞∗1 = 0. RT0het1n+δT is b∞ounded on Hwp(Rn).
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 6
w
2. Proof of Theorem 1.1
In order to prove the main theorems, we need the following lemma (see [6,
page 412]).
Lemma C. Let w A ,r > 1. Then there exists a constant C > 0 such
r
∈
that
1 1
w(x)dx C w(B)
Z x x nr ≤ σnr
0
Bc | − |
for all balls B = B(x ,σ) in Rn.
0
Proof of Theorem 1.1. For q = 2(q +1) (q , ), then s := [n(q/p 1)]
w w
∈ ∞ − ≥
[n(q /p 1)]. We now choose (and fix) a positive number ε satisfying
w
−
(2.1)
max sr (r 1) 1n 1+(r 1) 1,q/p 1 < ε < t(s+1)(nq) 1+q 1 1,
w w − − w − − −
{ − − − } −
for some t N,t 1 and max sr (r 1) 1n 1 + (r 1) 1,q/p 1 <
w w − − w −
∈ ≥ { − − − }
t(s+1)(nq) 1 +q 1 1.
− −
−
Clearly, ℓ := t(s +1) 1 s [n(q /p 1)]. Hence, by Theorem B and
w
− ≥ ≥ −
Theorem C, it is sufficient to show that for every w-(p,q,ℓ)-atom f centered
at x and supported in ball B = B(x ,σ), the Riesz transforms R f = K f,
0 0 j j
j = 1,2,...,n, are w-(p,q,s,ε)-molecules with the norm N (R f) C. ∗
w j
≤
Indeed, as w A by q = 2(q + 1) (q , ). It follows from Lq -
∈ q w ∈ w ∞ w
boundedness of Riesz transforms that
(2.2) kRjfkLqw ≤ kRjkLqw→LqwkfkLqw ≤ Cw(B)1/q−1/p.
To estimate kRjf.w(B(x0,|·−x0|))bkLqw where b = 1−1/q+ε, we write
R f.w(B(x , x ))b q = R f(x) qw(B(x , x x ))bqw(x)dx+
k j 0 ·− 0 kLqw Z | j | 0 | − 0|
x x0 2√nσ
| − |≤
+ R f(x) qw(B(x , x x ))bqw(x)dx
j 0 0
Z | | | − |
x x0>2√nσ
| − |
= I +II.
By Lemma B, we have the following estimate,
I = R f(x) qw(B(x , x x ))bqw(x)dx
j 0 0
Z | | | − |
x x0 2√nσ
| − |≤
w(B(x ,2√nσ))bq R f(x) qw(x)dx
0 j
≤ Z | |
x x0 2√nσ
| − |≤
Cw(B)bq R q f q Cw(B)(b+1/q 1/p)q.
≤ k jkLqw Lqwk kLqw ≤ −
→
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 7
w
To estimate II, as f is w-(p,q,ℓ)-atom, by the Taylor’s fomular and Lemma A,
we get
1
K f(x) = K (x y) DαK (x x )(x y)α f(y)dy
j j j 0 0
| ∗ | (cid:12) Z (cid:16) − − X α! − − (cid:17) (cid:12)
(cid:12)(cid:12)|y−x0|≤σ |α|≤ℓ (cid:12)(cid:12)
σℓ+1
C f(y) dy
≤ Z x x n+ℓ+1| |
0
| − |
y x0 σ
| − |≤
σn+ℓ+1
≤ C x x n+ℓ+1w(B)−1/qkfkLqw,
0
| − |
for all x (B(x ,2√nσ))c. As b = 1 1/q + ε, it follows from (2.1) that
0
∈ −
(n + ℓ + 1)q q2nb > nq. Therefore, by combining the above inequality,
−
Lemma B and Lemma C, we obtain
II = R f(x) qw(B(x , x x ))bqw(x)dx
j 0 0
Z | | | − |
x x0>2√nσ
| − |
1
Cσ(n+ℓ+1)qw(B) 1 f q w(B(x , x x ))bqw(x)dx
≤ − k kLqw Z x x (n+ℓ+1)q 0 | − 0|
0
| − |
x x0>2√nσ
| − |
1
Cσ(n+ℓ+1)q q2nbw(B)(b 1/p)q w(x)dx
≤ − − Z x x (n+ℓ+1)q q2nb
0 −
| − |
x x0>2√nσ
| − |
Cw(B)(b+1/q 1/p)q.
−
≤
Thus,
(2.3) kRjf.w(B(x0,|·−x0|))bkLqw = (I +II)1/q ≤ Cw(B)b+1/q−1/p.
Remark that a = 1 1/p+ε. Combining (2.2) and (2.3), we obtain
−
N (R f) Cw(B)(1/q 1/p)a/bw(B)(b+1/q 1/p)(1 a/b) C.
w j − − −
≤ ≤
The proof will be concluded if we establish the vanishing moment conditions
of R f. One first consider the following lemma.
j
Lemma. For every classical atom (p,2,ℓ)-atom g centered at x , we have
0
R g(x)xαdx = 0 for 0 α s,1 j n.
j
Z ≤ | | ≤ ≤ ≤
Rn
Proof of the Lemma. Since b = 1 1/q + ε < (ℓ + 1)(nq) 1 < (ℓ + 1)n 1,
− −
−
we obtain 2(n+ℓ+1) 2nb > n. It is similar to the previous argument, we
also obtain that R g an−d R g. x nb belong to L2(Rn). Now, we establish
j j 0
that R g.( x )α L1(Rn) f|o·r−ever|y multi-index α with α s. Indeed,
j 0
· − ∈ | | ≤
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 8
w
since ε > q/p 1 by (2.1), implies that 2(s nb) < (s nb)q < n by
′
− − − −
q = 2(q +1) > 2, where 1/q +1/q = 1. We use Schwartz inequality to get
w ′
R g(x)(x x )α dx R g(x) x x sdx
j 0 j 0
Z | − | ≤ Z | || − |
B(x0,1)c B(x0,1)c
1/2 1/2
R g(x) 2 x x 2nbdx x x 2(s nb)dx
j 0 0 −
≤ (cid:16) Z | | | − | (cid:17) (cid:16) Z | − | (cid:17)
B(x0,1)c B(x0,1)c
C R g. x nb < ,
j 0 L2
≤ k |·− | k ∞
and
1/2
R g(x)(x x )α dx B(x ,1) 1/2 R g(x) 2dx < .
j 0 0 j
Z | − | ≤ | | (cid:16) Z | | (cid:17) ∞
B(x0,1) B(x0,1)
Thus, R g.( x )α L1(Rn) for any α s. Deduce that R g(x)xα
j 0 j
L1(Rn) for any· −α s.∈Therefore, | | ≤ ∈
| | ≤
(R g(x)xα)ˆ(ξ) = C .Dα(\R g)(ξ)
j α j
ˆ
is continuous, with C C (C depends only on s) for any α s, where h
α s s
| | ≤ | | ≤
is used to denote the fourier transform of h. Consequently,
\
R g(x)xαdx = C .Dα(R g)(0) = C .Dα(m gˆ)(0),
j α j α j
Z
Rn
where m (x) = ix / x . Moreover, since g is a classical (p,2,ℓ)-atom, it
j j
− | |
follows from [17, Lemma 9.1] that gˆ is ℓth order differentiable and gˆ(ξ) =
O( ξ ℓ+1) as ξ 0. We write e to be the jth standard basis vector of Rn,
j
| | →
α = (α ,...,α ) a multi-index of nonnegative integers α , ∆ φ(x) = φ(x)
1 n j hej −
φ(x−hej), ∆αhejjφ(x) = ∆hαejj−1φ(x)−∆hαejj−1φ(x−hej) for αj ≥ 2, ∆0hejφ(x) =
φ(x), and ∆α = ∆α1 ...∆αn . Then, the boundedness of m , and C C for
h he1 hen j | α| ≤ s
α s, implies
| | ≤
R g(x)xαdx = C lim h α∆α(m gˆ)(0)
(cid:12)Z j (cid:12) | α|(cid:12)h 0| |−| | h j (cid:12)
(cid:12)Rn (cid:12) (cid:12) → (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
C lim h ℓ+1 α = 0,
−| |
≤ h 0| |
→
for α s by s ℓ. Thus, for any j = 1,2,...,n, and α s,
| | ≤ ≤ | | ≤
R g(x)xαdx = 0.
j
Z
Rn
This complete the proof of the lemma.
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 9
w
Let us come back to the proof of Theorem 1.1. As q/2 = q + 1 > q , by
w w
Lemma A,
1 q/2 1
f(x) 2dx C f(x) qw(x)dx.
(cid:16) B Z | | (cid:17) ≤ w(B) Z | |
| |
B B
Therefore, g := C 1/q B 1/pw(B)1/pf is a classical (p,2,ℓ)-atom since f is
− −
| |
w-(p,q,ℓ)-atom associated with ball B. Consequently, by the above lemma,
R f(x)xαdx = C1/q B 1/pw(B) 1/p R g(x)xαdx = 0
j − j
Z | | Z
Rn Rn
for all j = 1,2,...,n and α s. Thus, the theorem is proved. (cid:3)
| | ≤
Following a similar but easier argument, we also have the following Hp-
w
boundedness of Hilbert transform. We leave details to readers.
Theorem 2.1. Let 0 < p 1 and w A . Then, the Hilbert transform is
bounded on Hp(R). ≤ ∈ ∞
w
3. Proof of theorem 1.2
We first consider the following lemma
Lemma 3.1. Let p (0,1],w A ,1 < q < , and T be a θ-Calder´on-
q
∈ ∈ ∞
Zygmund operator satisfying T 1 = 0. Then, Tf(x)dx = 0 for all w-
∗ Rn
(p,q,0)-atoms f. R
Proof of Lemma 3.1. Let f be an arbitrary w-(p,q,0)-atom associated with
ball B. It is well-known that there exists 1 < r < q such that w A .
r
∈
Therefore, it follows from Lemma A that
f(x) q/rdx C B w(B)1/r f q/r < .
Z | | ≤ | | k kLqw ∞
B
We deduce that f is a multiple of classical (p,q/r,0)-atom, and thus the con-
dition T 1 = 0 implies Tf(x)dx = 0. (cid:3)
∗ Rn
R
Proof of Theorem 1.2. Because of the hypothesis, without loss of generality we
can assume q > 1. Futhermore, it is clear that [n(q /p 1)] = 0, and there
w
−
exists a positive constant ε such that
1 1 n+δ
(3.1) max , 1 < ε < 1.
r 1 p − nq −
n w o
−
Similarly to the arguments in Theorem 1.1, it is sufficient to show that, for
every w-(p,q,0)-atom f centered at x and supported in ball B = B(x ,σ),
0 0
ON Hp-BOUNDEDNESS THROUGH MOLECULAR CHARACTERIZATION 10
w
Tf is a w-(p,q,0,ε)-molecule with the norm N (Tf) C. One first observe
w
≤
that Tf(x)dx = 0 by Lemma 3.1, and
Rn
R
∞
θ(2 k)2knbq < ,
−
∞
X
k=0
where b = 1 1/q +ε, by 1 θ(t)dt < and (3.1). We deduce that
− 0 t1+δ ∞
R
∞ q
(3.2) θ(2 k)2knbq < .
−
∞
X(cid:16) (cid:17)
k=0
As w A , 1 < q < , it follows from [18, Theorem 2.4] that
q
⊂ ∞
(3.3) kTfkLqw ≤ CkfkLqw ≤ Cw(B)1/q−1/p.
To estimate kTf.w(B(x0,|·−x0|))bkLqw, we write
Tf.w(B(x , x ))b q = Tf(x) qw(B(x , x x ))bqw(x)dx+
k 0 ·− 0 kLqw Z | | 0 | − 0|
x x0 2σ
| − |≤
+ Tf(x) qw(B(x , x x ))bqw(x)dx = I +II.
0 0
Z | | | − |
x x0>2σ
| − |
By Lemma B, we have the following estimate,
I = Tf(x) qw(B(x , x x ))bqw(x)dx
0 0
Z | | | − |
x x0 2σ
| − |≤
w(B(x ,2σ))bq Tf(x) qw(x)dx
0
≤ Z | |
x x0 2σ
| − |≤
Cw(B)bq f q Cw(B)(b+1/q 1/p)q.
≤ k kLqw ≤ −
To estimate II, since f is of mean zero, by Lemma A, we have
Tf(x) = (K(x,y) K(x,x ))f(y)dy
0
| | (cid:12) Z − (cid:12)
(cid:12)(cid:12)|y−x0|≤σ (cid:12)(cid:12)
1 y x
0
C θ | − | f(y) dy
≤ Z x x0 n (cid:16) x x0 (cid:17)| |
| − | | − |
y x0 σ
| − |≤
σn σ
≤ C x x nθ x x w(B)−1/qkfkLqw,
0 (cid:16) 0 (cid:17)
| − | | − |