Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
25(1) (2009), 85–104
www.emis.de/journals
ISSN 1786-0091
LIPSCHITZ ESTIMATES FOR MULTILINEAR
COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR
WANG MEILING AND LIU LANZHE
Abstract. In this paper, we will study the continuity of multilinear commutator generated by Littlewood-Paley operator and Lipschitz functions on
Triebel-Lizorkin space, Hardy space and Herz-Hardy space.
1. Introduction
Let T be the Calderón-Zygmund operator, Coifman, Rochberg and Weiss
(see [4]) proves that the commutator [b, T ](f ) = bT (f ) − T (bf )(where b ∈
BM O(Rn )) is bounded on Lp (Rn ) for 1 < p < ∞. Chanillo (see [2]) proves a
similar result when T is replaced by the fractional operators. In [8, 16], Janson and Paluszynski study these results for the Triebel-Lizorkin spaces and the
case b ∈ Lipβ (Rn ), where Lipβ (Rn ) is the homogeneous Lipschitz space. The
main purpose of this paper is to discuss the boundedness of multilinear commutator generated by Littlewood-Paley operator and bj on Triebel-Lizorkin
space, Hardy space and Herz-Hardy space, where bj ∈ Lipβ (Rn ).
2. Preliminaries and Definitions
Throughout this paper, M (f ) will denote the Hardy-Littlewood maximal
function of f , and write Mp (f ) = (M (f p ))1/p for 0 < p < ∞, RQ will denote
a cube of Rn with side parallel to the axes. Let fQ = |Q|−1 Q f (x)dx and
R
f # (x) = supx∈Q |Q|−1 Q |f (y) − fQ |dy. Denote the Hardy spaces by H p (Rn ).
It is well known that H p (Rn )(0 < p ≤ 1) has the atomic decomposition characterization(see [12][17][18]). For β > 0 and p > 1, let Ḟpβ,∞ (Rn ) be the homogeneous Tribel-Lizorkin space. The Lipschitz space Lipβ (Rn ) is the space
2000 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Littlewood-Paley operator; Multilinear commutator; TriebelLizorkin space; Herz-Hardy space; Herz space; Lipschitz space.
85
86
WANG MEILING AND LIU LANZHE
of functions f such that
||f ||Lipβ = sup
x,y∈Rn
|f (x) − f (y)|
< ∞.
|x − y|β
x6=y
Lemma 1 ([16]). For 0 < β < 1, 1 < p < ∞, we have
¯¯
¯¯
Z
¯¯
¯¯
1
¯¯
¯¯
||f ||Ḟpβ,∞ ≈ ¯¯sup
|f
(x)
−
f
|dx
¯¯
Q
¯¯ Q |Q|1+ nβ Q
¯¯ p
L
¯¯
¯¯
Z
¯¯
¯¯
1
¯¯
¯¯
≈ ¯¯sup inf
|f (x) − c|dx¯¯ .
β
1+
c
¯¯ ·∈Q
¯¯ p
|Q| n Q
L
Lemma 2 ([16]). For 0 < β < 1, 1 ≤ p ≤ ∞, we have
Z
1
||f ||Lipβ ≈ sup
|f (x) − fQ |dx
β
Q |Q|1+ n Q
¶1/p
µ
Z
1
1
p
≈ sup
|f (x) − fQ | dx
.
β
|Q| Q
Q |Q| n
Lemma 3 ([2]). For 1 ≤ r < ∞ and β > 0, let
!1/r
Ã
Z
1
|f (y)|r dy
,
Mβ,r (f )(x) = sup
βr
1−
x∈Q
|Q| n Q
suppose that r < p < n/β, and 1/q = 1/p − β/n, then
||Mβ,r (f )||Lq ≤ C||f ||Lp .
Lemma 4 ([5]). Let Q1 ⊂ Q2 , then
|fQ1 − fQ2 | ≤ C||f ||Lipβ |Q2 |β/n .
Definition 1. Let 0 < p ≤ 1. A function a(x) on Rn is called a H p -atom , if
1) supp a ⊂ B(x0 , r) for some x0 and for some r > 0 (or for some r ≥ 1),
−1/p
2) ||a||
,
R L∞ ≤ γ|B(x0 , r)|
3) Rn a(x)x dx = 0 for all γ with 0 ≤ |γ| ≤ [n(1/p − 1)].
Lemma 5. (see [14][15]) Let 0 < p ≤ P
1. A distribution f on Rn is in H p (Rn )
the distributional sense,
if and only if f can be written as f = ∞
j=−∞ λj aj in
P∞
p
where each aj is H -atom and λj are constants with j=−∞ |λj |p < ∞. Moreover,
à ∞
!1/p
X
||f ||H p = inf
|λj |p
,
j=−∞
where the infimum take over all decompositions of f as above.
Definition 2. Let 0 < p, q < ∞, α ∈ R, Bk = {x ∈ Rn , |x| ≤ 2k }, Ak =
Bk \Bk−1 and χk = χAk for k ∈ Z.
LIPSCHITZ ESTIMATES . . .
87
1) The homogeneous Herz space is defined
K̇qα,p (Rn ) = {f ∈ LqLoc (Rn \{0}) : ||f ||K̇qα,p < ∞},
where
"
||f ||K̇qα,p =
∞
X
#1/p
2kαp ||f χk ||pLq
;
k=−∞
2) The nonhomogeneous Herz space is defined by
Kqα,p (Rn ) = {f ∈ LqLoc (Rn ) : ||f ||Kqα,p < ∞},
where
||f ||Kqα,p =
"∞
X
#1/p
2kαp ||f χk ||pLq + ||f χB0 ||pLq
.
k=1
Definition 3. Let α ∈ R, 0 < p, q < ∞.
(1) The homogeneous Herz type Hardy space is defined by
H K̇qα,p (Rn ) = {f ∈ S 0 (Rn ) : G(f ) ∈ K̇qα,p (Rn )},
and
||f ||H K̇qα,p = ||G(f )||K̇qα,p ;
(2) The nonhomogeneous Herz type Hardy space is defined by
HKqα,p (Rn ) = {f ∈ S 0 (Rn ) : G(f ) ∈ Kqα,p (Rn )},
and
||f ||HKqα,p = ||G(f )||Kqα,p ;
where G(f ) is the grand maximal function of f , that is
G(f )(x) = sup
sup |f ∗ ϕt (y)|,
ϕ∈Km |x−y|<t
where Km = {ϕ ∈ S(Rn ) : supx∈Rn ,|α|≤m (1 + |u|)m+n |Dα ϕ(u)| ≤ 1},
ϕt (x) = t−n ϕ(x/t) for t > 0, m is a positive integer and S(Rn ) is the
Schwartz class (see [17, p. 88]).
The Herz type Hardy spaces have the atomic decomposition characterization.
Definition 4. Let α ∈ R, 1 < q < ∞. A function a on Rn is called a central
(α, q)-atom (or a central (a, q)-atom of restrict type), if
1) supp a ⊂ B(0, r) for some r > 0 (or for some r ≥ 1),
2) ||a||
r)|−α/n ,
R Lq ≤ |B(0,
η
3) Rn a(x)x dx = 0 for all η with |η| ≤ [α − n(1 − 1/q)].
88
WANG MEILING AND LIU LANZHE
Lemma 6 ([6, 15]). Let 0 < p < ∞, 1 < q < ∞ and α ≥ n(1 − 1/q). A
temperate distribution f belongs to H K̇qα,p (Rn ) (or HKqα,p (Rn )) if and only
if there exist central (α, q)-atoms (or central (α,Pq)-atoms of restrict type) aj
p
supported on Bj = B(0, 2j ) and constants λj ,
j |λj | < ∞ such that f =
P∞
P∞
0
n
j=−∞ λj aj (or f =
j=0 λj aj ) in the S (R ) sense, and
Ã
!1/p
X
.
||f ||H K̇qα,p ( or ||f ||HKqα,p ) ≈
|λj |p
j
Definition 5 ([7]). Let α ∈ R, 1 < p, q < ∞.
1) A measure function f is said to belong to homogeneous weak Herz
space W K̇qα,p (Rn ), if
||f ||W K̇qα,p = sup λ(
λ>0
+∞
X
2kαp |{x ∈ Ek : |f (x)| > λ}|p/q )1/p < ∞;
−∞
2) A measure function f is said to belong to inhomogeneous weak Herz
space W Kqα,p (Rn ), if
+∞
X
2kαp |{x ∈ Ek : |f (x)| > λ}|p/q
||f ||W Kqα,p = sup λ(
λ>0
k=1
+|{x ∈ B0 : |f (x) > λ|p/q }|)1/p < ∞.
Definition 6. Let µ > 1, n > δ > 0 and ψ be a fixed function which satisfies
the following properties:
R
1) Rn ψ(x)dx = 0,
2) |ψ(x)| ≤ C(1 + |x|)−(n+1−δ) ,
3) |ψ(x + y) − ψ(x)| ≤ C|y|(1 + |x|)−(n+2−δ) when 2|y| < |x|.
Given a positive integer m and the locally integrable function bj (1 ≤ j ≤ m).
The multilinear commutator of Littlewood-Paley operator is defined by
#1/2
"Z Z
µ
¶nµ
dydt
t
~b
~b
|Ft (f )(x, y)|2 n+1
,
gµ,δ (f )(x) =
n+1
t
+
|x
−
y|
t
R+
where
"m
Y
Z
~
Ftb (f )(x, y)
=
Rn
#
(bj (x) − bj (z)) ψt (y − z)f (z)dz.
j=1
When m = 1, set
"Z Z
b
gµ,δ
(f )(x) =
µ
n+1
R+
t
t + |x − y|
¶nµ
dydt
|Ftb (f )(x, y)|2 n+1
t
#1/2
,
LIPSCHITZ ESTIMATES . . .
where
89
Z
Ftb (f )(x, y)
=
(b(x) − b(z))ψt (y − z)f (z)dz
Rn
and ψt (x) = t−n+δ ψ(x/t) for t > 0. Set Ft (f )(x) = f ∗ ψt (x), we also define
that
"Z Z
#1/2
¶nµ
µ
dydt
t
gµ,δ (f )(x) =
|Ft (f )(y)|2 n+1
,
n+1
t
+
|x
−
y|
t
R+
which is the Littlewood-Paley
[18]).
n operator (see
o
RR
Let H be the space H = h : ||h|| = ( Rn+1 |h(y, t)|2 dydt/tn+1 )1/2 < ∞ ,
+
then, for each fixed x ∈ Rn Ft (f )(x) may be viewed as a mapping from [0, +∞)
to H, and it is clear that
°
°µ
¶nµ/2
°
°
t
°
°
gµ,δ (f )(x) = °
Ft (f )(y)°
°
° t + |x − y|
and
°µ
°
¶nµ/2
°
°
t
~b
~b
°
°
gµ,δ (f )(x) = °
Ft (f )(x, y)° .
° t + |x − y|
°
~
b
Note that when b1 = · · · = bm , gµ,δ
is just the m order commutator. It is well
known that commutators are of great interest in harmonic analysis and have
been widely studied by many authors (see [1, 2, 3, 4, 8, 10, 9, 11, 16]). Our
main purpose is to establish the boundedness of the multilinear commutator
on Triebel-Lizorkin space, Hardy space and Herz-Hardy space.
Lemma 7 ([9]). Let 0 < δ < n, 1 < p < n/δ, 1/q = 1/p − δ/n and w ∈ A1 .
Then gµ,δ is bounded from Lp (w) to Lq (w).
Given some functions bj (j = 1, . . . , m) and a positive integer m and 1 ≤
Q
m
j ≤ m, we set ||~b||Lipβ = m
j=1 ||bj ||Lipβ and denote by Cj the family of all
finite subsets σ = {σ(1), . . . , σ(j)} of {1, . . . , m} of j different elements. For
σ ∈ Cjm , set σ c = {1, . . . , m} \ σ. For σ = {σ(1), . . . , σ(j)} ∈ Cjm , set ~bσ =
(bσ(1) , . . . , bσ(j) ), bσ = bσ(1) . . . bσ(j) and ||~bσ ||Lipβ = ||bσ(1) ||Lipβ . . . ||bσ(j) ||Lipβ .
3. Theorems and Proofs
Theorem 1. Let 0 < β < min(1, 1/2m), µ > 3 + 1/n − 2δ/n,1 < p < ∞,
~b
bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m and gµ,δ
be the multilinear commutator of
Littlewood-Paley operator as in Definition 6. Then
~
b
(a) gµ,δ
is bounded from Lp (Rn ) to Ḟpmβ,∞ (Rn ) for 1 < p < n/δ and 1/p −
1/q = δ/n.
~b
(b) gµ,δ
is bounded from Lp (Rn ) to Lq (Rn ) for 1/p − 1/q = (mβ + δ)/n and
1/p > (mβ + δ)/n.
90
WANG MEILING AND LIU LANZHE
Proof. (a). Fixed a cube
Q = (x0 , l) and x̃ ∈ Q. Set ~bQ = ((b1 )Q , . . . , (bm )Q ),
R
−1
where (bj )Q = |Q|
b (y)dy, 1 ≤ j ≤ m. Write f = f χQ + f2 = f χRn \Q =
Q j
f1 + f2 , we have
Z
~
Ftb (f )(x, y)
=
m X
X
m
Y
=
(−1)
j=0 σ∈Cjm
[(bj (x) − (bj )Q ) − (bj (z) − (bj )Q )] ψt (y − z)f (z)dz
Rn j=1
m−j
Z
(~b(z) − ~bQ )σc ψt (y − z)f (z)dz
(~b(x) − ~bQ )σ
Rn
= (b1 (x) − (b1 )Q ) . . . (bm (x) − (bm )Q )Ft (f )(y)
+(−1)m Ft ((b1 − (b1 )Q ) . . . (bm − (bm )Q )f )(y)
Z
m−1
X X
m−j ~
~
+
(−1)
(b(x) − bQ )σ
(~b(z) − ~bQ )σc ψt (y − z)f (z)dz
j=1 σ∈Cjm
~n
R
= (b1 (x) − (b1 )Q ) . . . (bm (x) − (bm )Q )Ft (f )(y)
+(−1)m Ft ((b1 − (b1 )Q ) . . . (bm − (bm )Q )f1 )(y)
+(−1)m Ft ((b1 − (b1 )Q ) . . . (bm − (bm )Q )f2 )(y)
m−1
X X
cm,j (~b(x) − ~bQ )σ Ft ((~b − ~bQ )σc f )(y),
+
j=1 σ∈Cjm
then
~
b
|gµ,δ
(f )(x) − gµ,δ (((b1 )Q − b1 ) . . . ((bm )Q − bm ))f2 )(x0 )|
°µ
¶nµ/2
°
t
~
°
≤°
Ftb (f )(x, y)
° t + |x − y|
°
¶nµ/2
µ
°
t
°
Ft (((b1 )Q − b1 ) . . . ((bm )Q − bm )f2 )(y)°
−
°
t + |x0 − y|
°µ
°
¶nµ/2
°
°
t
°
°
≤°
(b1 (x) − (b1 )Q ) . . . (bm (x) − (bm )Q )Ft (f )(y)°
° t + |x − y|
°
°
°
µ
¶nµ/2
m−1
°
X X °
t
°
°
+
(~b(x) − ~bQ )σ Ft ((~b − ~bQ )σc f )(y)°
°
°
°
t
+
|x
−
y|
j=1 σ∈Cjm
°µ
°
¶nµ/2
°
°
t
°
°
+°
Ft ((b1 − (b1 )Q ) . . . (bm − (bm )Q )f1 )(y)°
° t + |x − y|
°
LIPSCHITZ ESTIMATES . . .
91
°µ
¶nµ/2
m
°
Y
t
°
+°
Ft ( (bj − (bj )Q )f2 )(y)
° t + |x − y|
j=1
°
¶nµ/2
µ
m
°
Y
t
°
−
Ft ( (bj − (bj )Q )f2 )(y)°
°
t + |x0 − y|
j=1
= I1 (x) + I2 (x) + I3 (x) + I4 (x).
Thus,
Z
1
~
b
|gµ,δ
(f )(x) − gµ,δ ((b1 )Q − b1 ) . . . ((bm )Q − bm )f2 )(x0 )|dx
Q
Z
Z
1
1
≤
I1 (x)dx +
I2 (x)dx
|Q|1+mβ/n Q
|Q|1+mβ/n Q
Z
Z
1
1
I3 (x)dx +
I4 (x)dx
+ 1+mβ/n
|Q|
|Q|1+mβ/n Q
Q
= I + II + III + IV.
|Q|1+mβ/n
For I, by using Lemma 2, we have
Z
1
I≤
sup |b1 (x) − (b1 )Q | . . . |bm (x) − (bm )Q | |gµ,δ (f )(x)|dx
|Q|1+mβ/n x∈Q
Q
Z
1
≤ C||~b||Lipβ
|Q|mβ/n
|gµ,δ (f )(x)|dx
1+mβ/n
|Q|
Q
≤ C||~b||Lip M (gµ,δ (f ))(x̃).
β
For II, taking 1 < r < p < q < n/δ, 1/q 0 + 1/q = 1, 1/s0 + 1/s = 1,
1/q = 1/p − δ/n, ps = r by using the Hölder’s inequality and the boundedness
of gµ,δ from Lp to Lq and Lemma 2, we get
Z
m−1
X X
1
|(~b(x) − ~bQ )σ ||gµ,δ ((~b − ~bQ )σc f )(x)|dx
II ≤
1+mβ/n
|Q|
Q
j=1 σ∈C m
j
≤C
m−1
X
X
j=1
σ∈Cjm
1
|Q|mβ/n
µ
1
|Q|
Z
0
|(~b(x) − ~bQ )σ |q dx
Q
µ
¶1/q
Z
1
q
~
~
×
|gµ,δ ((b − bQ )σc f χQ )(x)| dx
|Q| Rn
m−1
X X
1
~bσ ||Lip |Q||σ|β/n 1 ×
≤C
||
β
|Q|mβ/n
|Q|1/q
j=1 σ∈C m
µZ
j
|(~b(x) − ~bQ )σc f (x)χQ (x)| dx
p
×
Rn
¶1/p
¶1/q0
×
92
WANG MEILING AND LIU LANZHE
≤C
m−1
X
X
j=1 σ∈Cjm
µ
×
≤C
m−1
X
1
|Q|
1
|Q|mβ/n
0
||~bσ ||Lipβ |Q||σ|β/n |Q|(−1/q)+(1/ps )+(1−δps/n)/ps ×
Z
X
j=1 σ∈Cjm
0
|(~b(x) − ~bQ )σc |ps dx
Q
1
|Q|mβ/n
¶1/ps0 µ
1
|Q|1−δps/n
¶1/ps
Z
ps
|f (x)| dx
Q
c
||~bσ ||Lipβ |Q||σ|β/n ||~bσc ||Lipβ |Q||σ |β/n Mδ,r (f )(x̃)
≤ C||~b||Lipβ Mδ,r (f )(x̃).
For III, we choose 1 < r < p < q < n/δ, 1/q = 1/p − δ/n, r = ps, by
the boundness of gµ,δ from Lp (Rn ) to Lq (Rn ) and Hölder’s inequality with
1/s + 1/s0 = 1, we have
Ã
!1/q
Z
m
Y
1
1
III ≤ C
|gµ,δ ( (bj − (bj )Q )f χQ )(x)|q dx
|Q|mβ/n |Q| Rn
j=1
ÃZ
!1/p
m
Y
1
1
| (bj (x) − (bj )Q )|p |f (x)χQ (x)|p dx
≤C
|Q|mβ/n |Q|1/q
n
R
j=1
Ã
!1/ps0
Z Y
m
(−1/q)+1/ps0 +(1−(δps/n)/ps)
1
|Q|
0
| (bj (x) − (bj )Q )|ps dx
×
≤C
mβ/n
|Q|
|Q| Q j=1
µ
¶1/ps
Z
1
ps
×
|f (x)| dx
|Q|1−δps/n Q
≤ C||~b||Lip Mδ,r (f )(x̃).
β
For IV , by the Minkowski’s inequality and by the inequality a1/2 − b1/2 ≤
(a − b)1/2 for a ≥ b ≥ 0, we have
°µ
¶nµ/2
m
°
Y
t
°
Ft ( (bj − (bj )Q )f2 )(y)
°
° t + |x − y|
j=1
°
µ
¶nµ/2
m
°
Y
t
°
−
Ft ( (bj − (bj )Q )f2 )(y)°
°
t + |x0 − y|
j=1
¯µ
ÃZ
"Z Z
¶nµ/2 µ
¶nµ/2 ¯¯
¯
t
t
¯
¯
≤
−
¯
¯
n+1
¯
t + |x0 − y|
(Q)c ¯ t + |x − y|
R+
1/2
!2
m
Y
dydt 
×
|bj (z) − (bj )Q ||ψt (y − z)||f (z)|dz
tn+1
j=1
LIPSCHITZ ESTIMATES . . .

Z

≤C
tnµ/2 |x − x0 |1/2
n+1
R+
Qc
≤C
Ã
Z Z
dydt
× n+1
t
Z Y
m
93
Qm
|bj (z) − (bj )Q ||ψt (y − z)||f (z)|
(t + |x − y|)(nµ+1)/2
¸1/2
dz
ÃZ Z
|bj (z) − (bj )Q ||f (z)||x − x0 |
Qc j=1
t−n dydt
×
(t + |y − z|)2n+2−2δ
¶1/2
µ
1/2
n+1
R+
t
t + |x − y|
¶nµ+1
×
dz.
Set B = B(x, t), then
¶nµ+1
dy
t
t
t + |x − y|
(t + |y − z|)2n+2−2δ
n
ÃRZ
!
¶nµ+1
µ
dy
t
≤ t−n
t + |x − y|
(t + |y − z|)2n+2−2δ
B(x,t)
̰ Z
!
µ
¶nµ+1
X
t
dy
+t−n
t + |x − y|
(t + |y − z|)2n+2−2δ
k
k−1 B
k=1 2 B\2
µZ
22n+2−2δ dy
−n
≤ Ct
2n+2−2δ
B(x,t) (2t + |y − z|)
!
µ
¶nµ+1
∞ Z
X
t
2(k+1)(2n+2−2δ) dy
+
t + 2k−1 t
(2k+1 t + |y − z|)2n+2−2δ
k
k−1 B
k=1 2 B\2
µZ
dy
−n
≤ Ct
2n+2−2δ
B(x,t) (t + |x − z|)
!
Z
∞
k(2n+2−2δ)
X
2
dy
+
2(1−k)(nµ+1)
2n+2−2δ
2k B (t + |x − z|)
k=1
Ã
!
∞
X
1
≤ Ct−n tn +
2−k(nµ+1) 2k(2n+2−2δ) (2k t)n
(t + |x − z|)2n+2−2δ
k=1
Ã
!
∞
X
1
≤C 1+
2k(3n−nµ+1−2δ)
(t + |x − z|)2n+2−2δ
k=1
Z
µ
−n
≤
!2
j=1
C
(t + |x − z|)2n+2−2δ
×
94
WANG MEILING AND LIU LANZHE
and notice that |x − z| ∼ |x0 − z| for x ∈ Q and z ∈ Rn \Q. We obtain
Z Y
m
|bj (z) − (bj )Q ||f (z)||x − x0 |1/2 ×
Qc j=1
ÃZ Z
!1/2
t−n dydt
×
dz
n+1
(t + |y − z|)2n+2−2δ
R+
µZ ∞
¶1/2
Z Y
m
dt
1/2
|bj (z) − (bj )Q ||f (z)||x − x0 |
≤C
dz
(t + |x − z|)2n+2−2δ
Qc j=1
0
∞ Z
m
X
Y
≤C
|x0 − x|1/2 |x0 − z|−(n+1/2−δ) |f (z)|
|bj (z) − (bj )Q |dz
≤C
2−k/2 
µ
≤C
∞
X
¶nµ+1
j=1

k=1
×
t
t + |x − y|
2k Q\2k−1 Q
k=1
∞
X
µ
1
|2k Q|1−δ/n
1
|2k Q|1−δ/n
¯
¯r0 1/r0
Z
m
¯Y
¯
¯
¯
×
¯ (bj (y) − (bj )Q )¯ dy 
¯
2k Q ¯
j=1
¶1/r
Z
r
|f (y)| dy
2k Q
2k(mβ−1/2) ||~b||Lipβ |Q|mβ/n Mδ,r (f )(x̃)
k=1
≤ C||~b||Lipβ |Q|mβ/n Mδ,r (f )(x̃),
so
IV ≤ C||~b||Lipβ Mδ,r (f )(x̃).
We put these estimates together, by using Lemma 1 and taking the supremum
over all Q such that x̃ ∈ Q, we obtain
~
b
||gµ,δ
(f )||Ḟqmβ,∞ ≤ C||~b||Lipβ (||M (gµ,δ (f )) + Mδ,r (f )| |Lq )
≤ C||~b||Lipβ (||M (gµ,δ (f ))||Lq + ||Mδ,r (f )| |Lq )
≤ C||~b||Lipβ (||gµ,δ (f )||Lq + ||Mδ,r (f )| |Lq )
≤ C||~b||Lipβ ||f ||Lp .
This completes the proof of (a).
(b). By some argument as in proof of (a), we have
Z
1
~b
|gµ,δ
(f )(x) − gµ,δ (((b1 )Q − b1 ) . . . ((bm )Q − bm )f2 )(x0 )|dx
|Q| Q
Z
Z
Z
Z
1
1
1
1
≤
I1 (x)dx +
I2 (x)dx +
I3 (x)dx +
I4 (x)dx
|Q| Q
|Q| Q
|Q| Q
|Q| Q
= V1 + V2 + V3 + V4 .
LIPSCHITZ ESTIMATES . . .
95
For V1 , taking 1/s = 1/r − δ/n, by the boundedness of gµ,δ from Lr (Rn ) to
Ls (Rn ), so we have
Z
1
sup |b1 (x) − (b1 )Q | . . . |bm (x) − (bm )Q | |gµ,δ (f )(x)|dx
V1 ≤
|Q| x∈Q
Q
µ
¶1/s
Z
1
≤ C||~b||Lipβ |Q|mβ/n
|gµ,δ (f )(x)|s dx
|Q| Q
µZ
¶1/r
mβ/n−1/s
r
~
≤ C||b||Lipβ |Q|
|f (x)| dx
µ
= C||~b||Lipβ
Q
1
|Q|1−(mβ+δ)r/n
¶1/r
Z
r
|f (x)| dx
Q
≤ C||~b||Lipβ Mmβ+δ,r (f )(x̃).
For V2 , taking 1/s0 + 1/s = 1,1/s = 1/r − δ/n, by using the Hölder’s inequality
and the boundedness of gµ,δ from Lr (Rn ) to Ls (Rn ), we get
V2 ≤
m−1
X
j=1
≤C
X 1 Z
|(~b(x) − ~bQ )σ ||gµ,δ ((~b − ~bQ )σc f )(x)|dx
|Q|
Q
σ∈C m
j
m−1
X
j=1
¶1/s0
X µ 1 Z
s0
~
~
|(b(x) − bQ )σ | dx
×
|Q|
Q
m
σ∈C
j
µ
¶1/s
Z
1
s
~
~
×
|gµ,δ ((b − bQ )σc f χQ )(x)| dx
|Q| Rn
¶1/s0
µ
Z
m−1
X X
1
0
−1/s
s
|(~b(x) − ~bQ )σ | dx
×
≤C
|Q|
|Q|
Q
m
j=1 σ∈C
j
µZ
|(~b(x) − ~bQ )σc f (x)χQ (x)| dx
¶1/r
r
×
Rn
≤C
m−1
X
X
j=1 σ∈Cjm
µ
c
|Q|−1/s+1/r ||~bσ ||Lipβ |Q||σ|β/n ||~bσc ||Lipβ |Q||σ |β/n ×
¶1/r
Z
1
r
×
|f (x)| dx
|Q| Q
µ
¶1/r
Z
1
r
~
≤ C||b||Lipβ
|f (x)| dx
|Q|1−(mβ+δ)r/n Q
≤ C||~b||Lip Mmβ+δ,r (f )(x̃).
β
96
WANG MEILING AND LIU LANZHE
For V3 , by the boundness of gµ,δ from Lr (Rn ) to Ls (Rn ) and Hölder’s inequality
we get
Ã
!1/s
Z
m
Y
1
V3 ≤ C
|gµ,δ ( (bj − (bj )Q )f χQ )(x)|s dx
|Q| Rn
j=1
!1/r
ÃZ
m
Y
| (bj (x) − (bj )Q )|r |f (x)χQ (x)|r dx
≤ C|Q|−1/s
Rn
≤ C|Q|
−1/s
|Q|
j=1
mβ/n
|Q|
µ
1/r
||~b||Lipβ
1
|Q|
¶1/r
Z
r
|f (x)| dx
Q
≤ C||~b||Lipβ Mmβ+δ,r (f )(x̃).
For V4 , similar to the proof of IV in (a), we get
Z
m
Y
|bj (z) − (bj )Q ||f (z)||x − x0 |1/2 ×
(Q)c j=1
ÃZ Z
×
µ
n+1
R+
Z
≤C
|x0 − x|
t
t + |x − y|
1/2
|x0 − z|
¶nµ+1
t−n dydt
(t + |y − z|)2n+2−2δ
−(n+1/2−δ)
|f (z)|
(Q)c
≤C
∞
X
k=1
µ
×
µ
≤C
dz
|bj (z) − (bj )Q |dz
j=1

2−k/2 
m
Y
!1/2
Z
1
|2k Q|1−δ/n
1
k
|2 Q|1−δ/n
¯m
¯r0 1/r0
¯Y
¯
¯
¯
×
¯ (bj (y) − (bj )Q )¯ dy 
¯
2k Q ¯
j=1
¶1/r
Z
r
|f (y)| dy
2k Q
1
|2k Q|1−(mβ+δ)r/n
¶1/r
Z
r
|f (y)| dy
2k Q
||~b||Lipβ
≤ C||~b||Lipβ Mmβ+δ,r (f )(x̃).
So we have
Z
1
~b
|gµ,δ
(f )(x) − gµ,δ (((b1 )Q − b1 ) . . . ((bm )Q − bm )f2 )(x0 )|dx
|Q| Q
≤ C||~b||Lip Mmβ+δ,r (f )(x̃).
β
Thus,
~b
(gµ,δ
(f ))# ≤ C||~b||Lipβ Mmβ+δ,r (f ).
By using Lemma 3 and the boundedness of gµ,δ , we have
LIPSCHITZ ESTIMATES . . .
97
~
~
b
b
||gµ,δ
(f )||Lq ≤ C||(gµ,δ
(f ))# ||Lq
≤ C||~b||Lipβ ||Mmβ+δ,r (f )||Lq ≤ C||~b||Lipβ ||f ||Lp .
¤
This completes the proof of (b) and the theorem.
Theorem 2. Let 0 < β ≤ 1, µ > 3 + 1/n − 2δ/n, n/(n + mβ) < p ≤ 1,
~
1/q = 1/p − (mβ + δ)/n, bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m. Then gµb , δ is bounded
from H p (Rn ) to Lq (Rn ).
Proof. It suffices to show that there exists a constant C > 0 such that for every
H p -atom a,
~
b
||gµ,δ
(a)||Lq ≤ C.
Let a be a HRp -atom, that is that a supported on a cube Q = Q(x0 , r), ||a||L∞ ≤
|Q|−1/p and Rn a(x)xγ dx = 0 for |γ| ≤ [n(1/p − 1)].
When m = 1 see [9]. Now, consider the case m ≥ 2. Write
¶1/q
µZ
~b
~b
q
||gµ,δ (a)(x)||Lq ≤
|gµ,δ (a)(x)| dx
|x−x0 |≤2r
µZ
+
|x−x0 |>2r
~b
|gµ,δ
(a)(x)dx|q
¶1/q
= I + II.
For I, choose 1 < p1 < n/(mβ +δ) and q1 such that 1/q1 = 1/p1 −(mβ +δ)/n.
~b
By the boundedness of gµ,δ
from Lp1 (Rn ) to Lq1 (Rn ) (see Theorem 1), the size
condition of a and Hölder’s inequality, we get
~b
I ≤ C||gµ,δ
(a)||Lq1 rn(1/q−1/q1 ) ≤ C||~b||Lipβ ||a||Lp1 rn(1/q−1/q1 )
≤ C||~b||Lipβ rn(−1/p+1/p1 ) rn(1/q−1/q1 ) ≤ C||~b||Lipβ .
For II, let τ, τ 0 ∈ N such that τ + τ 0 = m, and τ 0 6= 0. We get
~
|Ftb (a)(x, y)| ≤
Z
(ψt (y − z) − ψt (y − x0 ))a(z)dz|
≤ |(b1 (x) − b1 (x0 )) . . . (bm (x) − bm (x0 ))
+
m X
X
Z
|(~b(x) − ~b(x0 ))σc
Z
≤ C||~b||Lipβ |x − x0 |mβ ·
|ψt (y − z) − ψt (y − x0 )||a(z)|dz
Z
X
0
τβ
|z − x0 |τ β |ψt (y − z)||a(z)|dz
|x − x0 |
B
τ +τ 0 =m
≤ C||~b||Lipβ
(~b(z) − ~b(x0 ))σ ψt (y − z)a(z)dz|
B
j=1 σ∈Cjm
+C||~b||Lipβ
B
|x − x0 |mβ t
(t + |y − x0 |)n+2−δ
B
Z
|x0 − z||a(z)|dz
B
98
WANG MEILING AND LIU LANZHE
X
t
|x − x0 |
(t + |y − x0 |)n+1−δ
τ +τ 0 =m
+C||~b||Lipβ
Z
0
|z − x0 |τ β |a(z)|dz
τβ
B
t
· r1+n(1−1/p) · |x − x0 |mβ
(t + |y − x0 |)n+2−δ
X
t
0
+C||~b||Lipβ
·
rτ β+n(1−1/p) · |x − x0 |τ β .
n+1−δ
(t + |y − x0 |)
τ +τ 0 =m
≤ C||~b||Lipβ
Thus,
~
b
|gµ,δ
(a)(x)| ≤
ÃZ µ
∞
≤ C||~b||Lipβ
0
ÃZ
+C||~b||Lipβ
×
X
rτ
t
(t + |x − x0 |)n+2−δ
∞
µ
0
¶2
dt
t
t
(t + |x − x0 |)n+1−δ
0 β+n(1−1/p)
!1/2
· r1+n(1−1/p) · |x − x0 |mβ
¶2
dt
t
!1/2
×
· |x − x0 |τ β
τ +τ 0 =m
≤ C||~b||Lipβ |x − x0 |−(n+1−δ) · r1+n(1−1/p) · |x − x0 |mβ
X
0
+C||~b||Lipβ |x − x0 |−(n−δ) ·
rτ β+n(1−1/p) · |x − x0 |τ β ,
τ +τ 0 =m
so
¶1/q
µZ
II ≤ C||~b||Lipβ · r1+n(1−1/p)
+C||~b||Lipβ ·
X
|x − x0 |
|x−x0 |>2r
r
−(n+1−δ−mβ)q
dx
¶1/q
µZ
τ 0 β+n(1−1/p)
|x − x0 |
−(n−δ−τ β)q
dx
|x−x0 |>2r
τ +τ 0 =m
= J1 + J2 ,
we get
J1 ≤ C||~b||Lipβ · r1+n(1−1/p)
̰ Z
X
k=1
≤ C||~b||Lipβ · r1+n(1−1/p)
̰
X
!1/q
(2k r)−(n+1−δ−mβ)q dx
2k+1 Q\2k Q
!1/q
2−kq(n+1−δ−mβ) r−q(n+1−δ−mβ) 2(k+1)n rn dx
k=1
≤ C||~b||Lipβ
∞
X
k=1
≤ C||~b||Lipβ .
2−k(n+1−δ−mβ−n/q) r1+n(1−1/p)−(n+1−δ−mβ)+n/q
LIPSCHITZ ESTIMATES . . .
99
For J2 , similar to J1 , we have J2 ≤ C||~b||Lipβ .
Combining the estimates for I and II, then leads to the desired result.
¤
Theorem 3. Let 0 < β ≤ 1, µ > 3 + 1/n − 2δ/n, 0 < p < ∞, 1 < q1 , q2 < ∞,
1/q1 − 1/q2 = (mβ + δ)/n, n(1 − 1/q1 ) ≤ α < n(1 − 1/q1 ) + ε, ε < min(1, mβ),
~b
is bounded from H K̇qα,p
(Rn ) to
bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m. Then gµ,δ
1
K̇qα,p
(Rn ).
2
P∞
Proof. By Lemma 7, let f ∈ H K̇qα,p
(Rn ) and f =
λ a , supp aj ⊂
1
P∞ j=−∞ p j j
j
Bj = B(0, 2 ), aj be a central (α, q1 )−atom, and j=−∞ |λj | < ∞.
à k−2
!p
∞
X
X
~b
~b
(aj )χk ||Lq2
(f )||pK̇ α,p ≤ C
2kαp
|λj |||gµ,δ
||gµ,δ
q2
j=−∞
k=−∞
∞
X
+C
Ã
2kαp
k=−∞
∞
X
!p
~b
|λj |||gµ,δ
(aj )χk ||Lq2
= I + II.
j=k−1
~
b
For II, by the boundedness of gµ,δ
on (Lq1 , Lq2 ) (see Theorem 1), we have
II ≤ C||~b||pLipβ
≤ C||~b||pLipβ
∞
X
kαp
2
k=−∞
∞
X
(
∞
X
|λj |||aj ||Lq1 )p
j=k−1
(
∞
X
|λj | · 2(k−j)α )p
k=−∞ j=k−1
≤ C||~b||pLipβ

j+1
∞
P
P (k−j)αp


p

|λ
|
2
, 0<p≤1
j

j=−∞
k=−∞
j+1
j+1
∞
P
P (k−j)αp0 /2 p/p0
P

p
(k−j)αp/2


(
|λ
|
·
2
)(
2
) , 1<p<∞
j

j=−∞ k=−∞
≤ C||~b||pLipβ
∞
X
k=−∞
|λj |p .
j=−∞
For I, we have
~
|Ftb (aj ))(x, y)| ≤
Z
≤ |(b1 (x) − b1 (0)) · · · (bm (x) − bm (0))
+
∞ X
X
Z
|(~b(x) − ~b(0))σc
j=1 σ∈Cjm
≤ C||~b||Lipβ |x|mβ
(ψt (y − z) − ψt (y))aj (z)dz|
Bj
(~b(z) − ~b(0))σ ψt (y − z)aj (z)dz|
Bj
Z
|ψt (y − z) − ψt (y)||aj (z)|dz
Bj
100
WANG MEILING AND LIU LANZHE
Z
X
+C||~b||Lipβ
|x|
0
τβ
|z|τ β |ψt (y − z)||aj (z)|dz
Bj
τ +τ 0 =m
mβ
|x| t
· 2j(1+n(1−1/q1 )−α)
n+2−δ
(t + |y|)
X
|x|τ β t
0
+C||~b||Lipβ
· 2j(τ β+n(1−1/q1 )−α) .
n+1−δ
(t + |y|)
τ +τ 0 =m
≤ C||~b||Lipβ
Thus,
~
b
(aj )(x) ≤
gµ,δ
≤ C||~b||Lipβ |x|
ÃZ
mβ
j(1+n(1−1/q1 )−α)
·2
·
|x|τ β · 2j(τ
∞
×
0
µ
t
(t + |x|)n+1−δ
¶2
t
(t + |x|)n+2−δ
0 β+n(1−1/q )−α)
1
τ +τ 0 =m
ÃZ
µ
0
X
+C||~b||Lipβ
∞
dt
t
¶2
dt
t
!1/2
×
!1/2
≤ C||~b||Lipβ |x|mβ |x|−(n+1−δ) · 2j(1+n(1−1/q1 )−α)
X
0
+C||~b||Lipβ
|x|τ β |x|−(n−δ) · 2j(τ β+n(1−1/q1 )−α) ,
τ +τ 0 =m
and
~
b
||gµ,δ
(aj )χk ||Lq2 ≤
ÃZ
≤ C||~b||Lipβ · 2j(1+n(1−1/q1 )−α) ·
+C||~b||Lipβ
X
τ +τ 0 =m
!1/q2
|x|−(n−(mβ+δ)+1)q2 dx
Bk \Bk−1
j(τ 0 β+n(1−1/q
2
1 )−α)
ÃZ
!1/q2
|x|−(n−τ β−δ)q2 dx
·
Bk \Bk−1
≤ C||~b||Lipβ (2j(1+n(1−1/q1 )−α)−k(1+n(1−1/q1 ))
X
0
0
+
2j(τ β+n(1−1/q1 )−α)−k(τ β+n(1−1/q1 )) )
τ +τ 0 =m
¡
¢
≤ C||~b||Lipβ 2−kα 2(j−k)(1+n(1−1/q1 )−α) + 2(j−k)(mβ+n(1−1/q1 )−α) ,
so
LIPSCHITZ ESTIMATES . . .
∞
X
I ≤ C||~b||pLipβ
Ã
!p
k−2
X
|λj | · 2(j−k)(1+n(1−1/q1 )−α)
j=−∞
k=−∞
Ã
∞
X
+ C||~b||pLipβ
When 0 < p ≤ 1,
I≤
Ã
∞
X
Ã
∞
X
∞
X
≤ C||~b||pLipβ
+
|λj |
≤ C||~b||pLipβ
p(j−k)(1+n(1−1/q1 )−α)
|λj | · 2
!
k−2
X
|λj |p · 2p(j−k)(mβ+n(1−1/q1 )−α)
p
∞
X
2p(j−k)(1+n(1−1/q1 )−α)
k=j+2
∞
X
|λj |p
j=−∞
p
j=−∞
j=−∞
∞
X
|λj | · 2(j−k)(mβ+n(1−1/q1 )−α)
!
k−2
X
k=−∞
Ã
!p
j=−∞
k=−∞
+C||~b||pLipβ
k−2
X
j=−∞
k=−∞
C||~b||pLipβ
101
!
2p(j−k)(mβ+n(1−1/q1 )−α)
k=j+2
∞
X
|λj |p .
j=−∞
When p > 1,
I ≤ C||~b||pLipβ
Ã
×
Ã
∞
X
k=−∞
k−2
X
×
×
!p/p0
0
2p (j−k)(1+n(1−1/q1 )−α)/2
∞
X
+C||~b||pLipβ
Ã
k=−∞
k−2
X
|λj |p 2p(j−k)(1+n(1−1/q1 )−α)/2
j=−∞
j=−∞
Ã
!
k−2
X
k−2
X
!
|λj |p 2p(j−k)(mβ+n(1−1/q1 )−α)/2
j=−∞
!p/p0
0
2p (j−k)(mβ+n(1−1/q1 )−α)/2
j=−∞
≤ C||~b||pLipβ
∞
X
|λj |p .
j=−∞
From I and II, we have
~b
||gµ,δ
(f )||K̇qα,p
2
Ã
≤ C||~b||Lipβ
∞
X
j=−∞
!1/p
|λj |p
≤ C||f ||H K̇qα,p
1
×
.
102
WANG MEILING AND LIU LANZHE
This completes the proof of Theorem 3.
¤
When α = n(1 − 1/q1 ) + ε, ε < min(1, mβ), this kind of boundedness fails.
Now, we give an estimate of weak type.
Theorem 4. Let 0 < β ≤ 1, 0 < p ≤ 1, 1 < q1 , q2 < ∞ 1/q2 = 1/q1 − (mβ +
~b
n(1−1/q1 )+ε,p
δ)/n, bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m. Then gµ,δ
maps H K̇q1
(Rn )
n(1−1/q1 )+ε,p
continuously into W K̇q2
(Rn ), where 0 < ε < min(1, mβ).
P
Proof. We write f = ∞
λk ak , where each ak is a central (n(1−1/q1 )+ε, q1 )
k=−∞P
p
atom supported on Bk and ∞
k=−∞ |λk | < ∞. Write
~
b
||gµ,δ
||W K̇ n(1−1/q1 )+ε,p ≤
q2
∞
X
≤ sup λ{
λ>0
l(n(1−1/q1 )+ε)p
2
∞
X
~b
|gµ,δ (
λk ak )(x)|
k=l−3
|{x ∈ El :
l=−∞
+ sup λ{
λ>0
∞
X
l(n(1−1/q1 )+ε)p
2
|{x ∈ El :
~b
|gµ,δ
(
l=−∞
l−4
X
> λ/2}|p/q2 }1/p
λk ak )(x)| > λ/2}|p/q2 }1/p
k=−∞
= G1 + G2 .
~
b
By the (Lq1 , Lq2 ) boundedness of gµ,δ
and an estimate similar to that for I1 in
Theorem 3, we get
Gp1
≤C
∞
X
lp(n(1−1/q1 )+ε)
2
l=−∞
∞
X
~b
λk ak )(x)χl ||pq2
||gµ,δ (
k=l−3
≤ C||~b||pLipβ
∞
X
|λk |p .
k=−∞
To estimate G2 , let us now use the estimate
~b
gµ,δ
(ak )(x) ≤ C||~b||Lipβ |x|mβ |x|−(n+1−δ) · 2k(1+n(1−1/q1 )−α)
X
0
+C||~b||Lipβ
|x|τ β |x|−(n−δ) · 2k(τ β+n(1−1/q1 )−α) ,
τ +τ 0 =m
which we get in the proof of Theorem 3. Note that when x ∈ El ,
α = n(1 − 1/q1 ) + ε,
λ<2
l−4
X
~
b
|λk ||gµ,δ
(ak )(x)| ≤
k=−∞
≤ C||~b||Lipβ
l−4
X
˙ l )mβ+δ−n−1
|λk |(2
k=−∞
+C||~b||Lipβ
l−4
X
k=−∞
l−4
X
(2k )1+n(1−1/q1 )−α
k=−∞
|λk |
X
l τ β+δ−n
(2 )
τ +τ 0 =m
l−4
X
(2k )τ
k=−∞
0 β+n(1−1/q )−α
1
LIPSCHITZ ESTIMATES . . .
≤ C||~b||Lipβ
l−4
X
103
˙ l )mβ+δ−n−ε
|λk |(2
k=−∞
≤ C||~b||Lipβ 2l(mβ+δ−n−ε) (
∞
X
|λk |p )1/p ,
k=−∞
for λ > 0, let lλ be the maximal positive integer satisfying
∞
X
lλ (n+ε−mβ−δ)
−1
~
2
≤ C||b||Lipβ λ (
|λk |p )1/p ,
k=−∞
then if l > lλ , we have
|{x ∈ El :
l−4
X
~b
|gµ,δ (
λk ak )|
k=−∞
> λ/2}| = 0,
so we obtain
G2 ≤ sup λ{
λ>0
lλ
X
2l(n(1−1/q1 )+ε)p (2l )np/q2 }1/p
l=−∞
≤ sup λ{
λ>0
lλ
X
(2l )(n+ε−mβ−δ) }
l=−∞
lλ ((n+ε−mβ−δ)
≤ sup λ2
λ>0
≤ C||~b||Lipβ (
∞
X
|λk |p )1/p .
k=−∞
Now, combining the above estimates for G1 and G2 , we obtain
∞
X
~b
~
||gµ,δ (f )||W K̇ n(1−1/q1 )+ε,p ≤ C||b||Lipβ (
|λk |p )1/p .
q2
k=−∞
Theorem 4 follows by taking the infimum over all central atomic decompositions.
¤
Acknowledgement
The authors would like to express their gratitude to the referee for his comments.
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Received June 3, 2008.
Wang Meiling
Department of Mathematics,
Changsha Univesity of Science and Technology,
Changsha, 410077,
P.R.of China.
Liu Lanzhe
Department of Mathematics,
Hunan University,
Changsha, 410082,
P.R.of China.
E-mail address: lanzheliu@163.com