31 Kragujevac J. Math. 27 (2005) 31–46.
LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITLLEWOOD-PALEY OPERATOR Zhang Mingjun and Liu Lanzhe Department of Mathematics Hunan University Changsha, 410082, P.R.of China (e-mails:
[email protected],
[email protected] )
(Received March 11, 2005)
Abstract. In this paper, we will study the continuity of multilinear commutator generated by Littlewood-Paley operator and the of function b, wich belongs to Lipschiz space, in the Triebel-Lizorkin, Hardy and Herz-Hardy space.
1. INTRODUCTION Let T be a Calder´on-Zygmund operator. Coifman, Rochberg and Weiss proved [4] that the commutator [b, T ](f )(x) = b(x)T (f )(x) − T (bf )(x) is bounded on Lp (Rn ) for 1 < p < ∞ and b ∈ BM O. Chanillo proved a similiar result [2] when T is replaced by the fractional operators. Janson and Paluszynski study these result [7, 15] for the Triebel-Lizorkin spaces and when b ∈ Lipβ , where Lipβ 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 continuity of b ∈ Lipβ in the Triebel-Lizorkin, Hardy and Herz-Hardy space.
32 2. PRELIMINARIES AND DEFINITIONS M (f ) denotes the Hardy-Littlewood maximal function of f and Mp (f ) = (M (f p ))1/p for 0 < p < ∞. Q denotes a cube of Rn with side parallel to the axes. Let fQ = |Q|−1
R Q
f (x)dx and f # (x) = supx∈Q |Q|−1
R Q
|f (y) − fQ |dy. Mark Hardy spaces
by H p (Rn ). It is well known that H p (Rn )(0 < p ≤ 1) has the atomic decomposition [11, 16, 17]. For β > 0 and p > 1, let F˙ β,∞ be the homogeneous Tribel-Lizorkin space. p
n
The Lipschitz space Lipβ (R ) is the space of functions f such that ||f ||Lipβ = sup
x,y∈Rn x6=y
|f (x) − f (y)| < ∞. |x − y|β
Lemma 1. [15] For 0 < β < 1 and 1 < p < ∞, ||f ||F˙pβ,∞ ≈ ≈
¯¯ ¯¯ ¯¯ ¯¯ Z ¯¯ ¯¯ 1 ¯¯sup |f (x) − fQ |dx¯¯¯¯ β ¯¯ ¯¯ Q |Q|1+ n Q ¯¯ p L ¯¯ ¯¯ ¯¯ ¯¯ Z ¯¯ ¯¯ 1 ¯¯ ¯¯sup inf |f (x) − c|dx β ¯¯ ¯¯ ¯¯ ¯¯ ·∈Q c |Q|1+ n Q
.
Lp
Lemma 2. [15] For 0 < β < 1 and 1 ≤ p ≤ ∞, ||f ||Lipβ ≈ sup Q
≈ sup Q
Z
1 β
|Q|1+ n
Ã
1 β
|Q| n
Q
|f (x) − fQ |dx
1 Z |f (x) − fQ |p dx |Q| Q
!1/p
.
Lemma 3. [2] For 1 ≤ r < ∞ and β > 0, let
Mβ,r (f )(x) = sup x∈Q
1/r
Z
1
r
βr
|Q|1− n
Q
|f (y)| dy
if we suppose that r < p < β/n, and 1/q = 1/p − β/n, then ||Mβ,r (f )||Lq ≤ C||f ||Lp .
.
33
Lemma 4. [5] If Q1 ⊂ Q2 then |fQ1 − fQ2 | ≤ C||f ||Λ˙ β |Q2 |β/n .
Lemma 5. [10] Let 0 < β ≤ 1, 1 < p < n/β, 1/q = 1/p − β/n and b ∈ Lipβ (Rn ). Then gψb is bounded from Lp (Rn ) to Lq (Rn ). Definition 1. Let 0 < p, q < ∞, α ∈ R, Bk = {x ∈ Rn , |x| ≤ 2k }, Ak = Bk \Bk−1 and χk = χAk for k ∈ Z. 1) The homogeneous Herz space is defined by K˙ qα,p = {f ∈ LqLoc (Rn \{0}), ||f ||K˙ qα,p < ∞}, where
||f ||K˙ qα,p =
1/p
∞ X
2kαp ||f χk ||pLq
.
k=−∞
2) The nonhomogeneous Herz space is defined by Kqα,p (Rn ) = {f ∈ LqLoc (Rn ), ||f ||Kqα,q (Rn ) < ∞}, where ||f ||Kqα,p (Rn ) =
"∞ X
#1/p kαp
2
||f χk ||pLq
+
||f χB0 ||pLq
k=1
Definition 2. Let α ∈ R and 0 < p, q < ∞. (1) The homogeneous Herz type Hardy space is defined by H K˙ qα,p (Rn ) = {f ∈ S 0 (Rn ) : G(f ) ∈ K˙ qα,p (Rn )}, and ||f ||H K˙ qα,p = ||G(f )||K˙ qα,p ;
.
34 (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 . The Herz type Hardy spaces have the characterization of the atomic decomposition. Let α ∈ R and 1 < q < ∞. A function a(x) on Rn is called a
Definition 3.
central (α, q)-atom (or a central (a, q)-atom of restrict type), if 1) Suppa ⊂ B(0, r) for some r > 0 (or for some r ≥ 1), 2) ||a||Lq ≤ |B(0, r)|−α/n , 3)
R
Rn
a(x)xη dx = 0 for |η| ≤ [α − n(1 − 1/q)].
Lemma 6. [6, 14] Let 0 < p < ∞, 1 < q < ∞ and α ≥ n(1 − 1/q). A temperate distribution f belongs to H K˙ qα,p (Rn )(or HKqα,p (Rn )) if and only if there exist central (α, q)-atoms(or central (α, q)-atoms of restrict type) aj supported on Bj = B(0, 2j ) and constants λj ,
P j
|λj |p < ∞ such that f =
P∞
j=−∞
λj aj (or f =
P∞
j=0
λj aj )in the
S 0 (Rn ) sense, and
||f ||H K˙ qα,p (or ||f ||HKqα,p ) ∼
X
1/p
|λj |p
.
j
Definition 4. Let ε > 0 and ψ be a fixed function which satisfies the following properties: 1)
R Rn
ψ(x)dx = 0,
2) |ψ(x)| ≤ C(1 + |x|)−(n+1) , 3) |ψ(x + y) − ψ(x)| ≤ C|y|ε (1 + |x|)−(n+1+ε) when 2|y| < |x|. Let m be a positive integer and bj (1 ≤ j ≤ m) be locally integrable functions and ~b = (b1 , · · · , bm ). The multilinear commutator of Littlewood-Paley operator is defined
35 by ~ gψb (f )(x)
where ~ Ftb (f )(x)
Z
=
ÃZ
=
m Y
∞
0
dt ~ |Ftb (x)|2
!1/2
t
,
(bj (x) − bj (y))ψt (x − y)f (y)dy,
Rn j=1
and ψt (x) = t−n ψ(x/t) for t > 0. Let Ft (f ) = ψt ∗ f . Define Littlewood-Paley g function by [17]
ÃZ
gψ (f )(x) =
0
∞
|Ft (f )(x)|
Let H be the space, H = {h : ||h|| = (
R∞ 0
2 dt
!1/2
t
.
|h(t)|2 dt/t)1/2 < ∞}. For each fixed
x ∈ Rn , Ft (f )(x) may be viewed as a mapping from [0, +∞) to H, and it is clear that ~
~
gψ (f )(x) = ||Ft (f )(x)|| and gψb (f )(x) = ||Ftb (f )(x)||. ˜
Note that when b1 = · · · = bm , gψb 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 ([1]-[4][7]-[10][12][15]). Our main purpose is to establish the boundedness of the multilinear commutator on Triebel-Lizorkin, Hardy and HerzHardy space. Q m Let m be a positive integer, 1 ≤ j ≤ m, ||~b||Lipβ = m j=1 ||bj ||Lipβ and Cj
be the family of all finite subsets σ = {σ(1), . . . , σ(j)} of {1, . . . , m} of j different elements. For σ ∈ C m , let σ c = {1, . . . , m} \ σ. For ~b = (b1 , . . . , bm ) and j
σ = {σ(1), . . . , σ(j)} ∈ Cjm , let ~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, ε/m), 1 < p < ∞, ~b = (b1 , . . . , bm ) where ~
bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m and gψb be the multilinear commutator of LittlewoodPaley operator. Then
36 ~ (a) gψb is bounded from Lp (Rn ) to F˙ pmβ,∞ (Rn ). ~
(b) gψb is bounded from Lp (Rn ) to Lq (Rn ) for 1/p − 1/q = mβ/n and 1/p > mβ/n.
Proof. (a) Fix a cube Q = (x0 , l) and x ∈ Q, when m = 1 [10]. Now consider the R case when m ≥ 2. Let ~bQ = ((b1 )Q , . . . , (bm )Q ), where (bj )Q = |Q|−1 Q bj (y)dy and 1 ≤ j ≤ m. If f = f1 + f2 where f1 = f χ2Q and f2 = f χRn \2Q , then Z
~
Ftb (f )(x) =
Rn
(b1 (x) − b1 (y)) · · · (bm (x) − bm (y))ψt (x − y)f (y)dy
= (b1 (x) − (b1 )Q ) · · · (bm (x) − (bm )Q )Ft (f )(x) + (−1)m Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f )(x) +
m−1 X
X
(−1)m−j (b(x) − ~bQ )σ
j=1 σ∈Cjm
Z Rn
(b(y) − ~bQ )σc ψt (x − y)f (y)dy
= (b1 (x) − (b1 )Q ) · · · (bm (x) − (bm )Q )Ft (f )(x) + (−1)m Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f1 )(x) + (−1)m Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f2 )(x) +
m−1 X
X
j=1
σ∈Cjm
(−1)m−j (b(x) − ~bQ )σ Ft ((b − ~bQ )σc f )(x).
Therefore ~
|gψb (f )(x) − gψ (((b1 )Q − b1 ) · · · ((bm )Q − bm )f2 )(x0 )| ~
≤ ||Ftb (f )(x) − Ft (((b1 )Q − b1 ) · · · ((bm )Q − bm )f2 )(x0 )|| ≤ ||(b1 (x) − (b1 )Q ) · · · (bm (x) − (bm )Q )Ft (f )(x)|| +
m−1 X
X
j=1
σ∈Cjm
||(b(x) − ~bQ )σ Ft ((b − ~bQ )σc f )(x)||
+ ||Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f1 )(x)|| + ||Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f2 )(x) − Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f2 )(x0 )|| = I1 (x) + I2 (x) + I3 (x) + I4 (x).
37 Thus Z
1 |Q|
1+ mβ n
≤
~
Q
|gψb (f )(x) − gψ ((b1 )Q − b1 ) · · · ((bm )Q − bm )f2 )(x0 )|dx Z
1 |Q|1+
mβ n
Z
1
+
I1 (x)dx +
Q
Z
1 |Q|1+
mβ n
Q
I2 (x)dx +
Z
1 |Q|1+
mβ n
Q
I3 (x)dx
I4 (x)dx mβ |Q|1+ n Q = I + II + III + IV. By using Lemma 2, we have for I, Z
1
I ≤
|Q|1+
sup |b1 (x) − (b1 )Q | · · · |bm (x) − (bm )Q |
mβ n
x∈Q
≤ C||~b||Lipβ
1 |Q|1+
|Q|
mβ n
mβ n
Q
|gψ (f )(x)|dx
Z Q
|gψ (f )(x)|dx
≤ C||~b||Lipβ M (gψ (f ))(x). Fix r, such that 1 < r < p. Let µ, µ0 be the integers such that µ + µ0 = m, 0 ≤ µ < m and 0 < µ0 ≤ m. By using H¨older’s inequality, the boundedness of gψ on Lr and Lemma 2, we get II ≤
m−1 X
X
j=1 σ∈Cjm
≤ C
m−1 X
|Q|1+
X
j=1 σ∈Cjm
≤ C
m−1 X
X
j=1 σ∈Cjm
≤ C
m−1 X
Z
1
X
j=1 σ∈Cjm
mβ n
Q
|(~b(x) − ~bQ )σ ||gψ ((~b − ~bQ )σc f )(x)|dx
µZ
1 |Q|1+
mβ n
Q
µZ
1 |Q|1+ 1 |Q|1+
mβ n
Q
0 |(~b(x) − ~bQ )σ |r dx
0 |(~b(x) − ~bQ )σ |r dx
¶1/r0 µZ Q
¶1/r0 µZ Q
|gψ ((~b − ~bQ )σc f )(x)|r dx |(~b(x) − ~bQ )σc f (x)|r dx 0
µβ µ β ||~bσ ||Lipβ |Q| n ||~bσc ||Lipβ |Q| n mβ |Q| 1 r0
n
µZ r
Q
|f (x)| dx
≤ C||~b||Lipβ Mr (f )(x); By H¨older’s inequality, we have for III III =
Z
1 |Q|1+
mβ n
Q
|gψ ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f1 )(x)|dx
¶1/r
¶1/r
¶1/r
38
≤ C
≤ C
1 |Q|
1+ mβ n
1 |Q|
1+ mβ n
Rn
|gψ (
m Y
1/r
(bj − (bj )Q )f1 )(x)|dx
|Q|1−1/r
j=1
Z |Q|1−1/r
1
≤ C
Z
1−1/r
|
m Y
1/r
(bj (x) − (bj )Q )f (x)|r dx
2Q j=1
mβ ||~b||Lipβ |Q| n
µZ r
mβ |Q| |Q|1+ n ≤ C||~b||Lipβ Mr (f )(x);
2Q
¶1/r
|f (x)| dx
Since |x0 − y| ≈ |x − y| for y ∈ (2Q)c , using by Lemma 4 and the condition of ψ, we have for IV , ||Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f2 )(x) − Ft ((b1 − (b1 )Q ) · · · (bm − (bm )Q )f2 )(x0 )|| Z ∞ Z ≤ (
(2Q)c
0
|ψt (x − y) − ψt (x0 − y)||f (y)|
(2Q)c
≤ C ≤ C ≤ C
(2Q)c
|x0 − x|ε |x0 − y|−(n+ε) |f (y)|
1/2
m Y
|bj (y) − (bj )Q |dy
j=1
k+1 Q\2k Q k=1 2 ∞ X −kε k+1
k=1 ∞ X
2
t
t|x − x0 | dt |f (y)| |bj (y) − (bj )Q |dy n+1+ε (t + |x0 − y|) t j=1
∞ Z X
2
|bj (y) − (bj )Q |dy)
m Y
ε
Z
≤ C
1/2
2 dt
j=1
Z ∞ Z ≤ C 0
m Y
|2
|x0 − x|ε |x0 − y|−(n+ε) |f (y)|
Q|
2−kε |2k+1 Q|
|bj (y) − (bj )Q |dy
j=1
Z −1 2k+1 Q mβ n
m Y
|f (y)|
m Y
(|bj (y) − (bj )2k+1 Q | + |(bj )2k+1 − (bj )Q |)dy
j=1
||~b||Lipβ M (f )
k=1 mβ ≤ C||~b||Lipβ |Q| n M (f )
∞ X
2(mβ−ε)k
k=1
≤ C||~b||Lipβ |Q|
mβ n
M (f ).
The following holds IV ≤ C||~b||Lipβ M (f ). Putting these estimates together, taking the supremum over all Q such that x ∈ Q
39 and by using Lemma 1, we obtain ~ ||gψb (f )(x)||F˙pmβ,∞ ≤ C||~b||Lipβ ||f ||Lp .
This complete the proof of (a). (b) Like in the proof of (a), we have 1 Z ~b |g (f )(x) − gψ (((b1 )Q − b1 ) · · · ((bm )Q − bm )f2 )(x0 )|dx |Q| Q ψ 1 Z 1 Z 1 Z 1 Z ≤ I1 (x)dx + I2 (x)dx + I3 (x)dx + I4 (x)dx |Q| Q |Q| Q |Q| Q |Q| Q ≤ C||~b||Lipβ (Mmβ,1 (gψ (f )) + Mmβ,r (f ) + Mmβ,r (f ) + Mmβ,1 (f )). Thus ~ (gψb (f ))# ≤ C||~b||Lipβ (Mmβ,1 (gψ (f )) + Mmβ,r (f ) + Mmβ,1 (f )).
By using Lemma 3 and the boundedness of gψ , we have ~
~
||gψb (f )||Lq ≤ C||(gψb (f ))# ||Lq ≤ C||~b||Lipβ (||Mmβ,1 (gψ (f ))||Lq + ||Mmβ,r (f )||Lq + ||Mmβ,1 (f )||Lq ) ≤ C||f ||Lp . This completes the proof of (b). Theorem 2. Let 0 < β ≤ 1, max(n/(n + mβ), n/(n + mε)) < p ≤ 1, 1/q = ~ 1/p − mβ/n and ~b = (b1 , . . . , bm ) where bj ∈ Lipβ (Rn ) for 1 ≤ j ≤ m. Then gψb is bounded from H p (Rn ) to Lq (Rn ). Proof. It is enough to show that there exists a constant C > 0 such that for every H p -atom a, ~
||gψb (a)||Lq ≤ C. Let a be a H p -atom supported on a cube Q = Q(x0 , r), ||a||L∞ ≤ |Q|−1/p and
R
Rn
a(x)xγ dx = 0 for |γ| ≤ [n(1/p − 1)]. When m = 1 see [10]. Now consider the case m ≥ 2. ~ ||gψb (a)(x)||Lq
ÃZ
≤
|x−x0 |≤2r
= I + II.
~ |gψb (a)(x)|q dx
!1/q
ÃZ
+
|x−x0 |>2r
~ |gψb (a)(x)dx|q
!1/q
40 ~
Choose 1 < p1 < 1/β and q1 such that 1/q1 = 1/p1 − β/n. By the boundednss of gψb from Lp1 (Rn ) to Lq1 (Rn )(see Lemma 5), we get for I ~
I ≤ C||gψb (a)||Lq1 r
n( 1q − q1 ) 1
≤ C||a||Lq1 r
n( 1q − q1 ) 1
≤ C.
Let τ, τ 0 ∈ N such that τ + τ 0 = m, and τ 0 6= 0. We get for II Z
~
|Ftb (a)(x)| ≤ |(b1 (x) − b1 (x0 )) · · · (bm (x) − bm (x0 )) +
m X X
Z
|(b(x) − b(x0 ))σc
j=1 σ∈Cjm
≤ C||~b||Lipβ |x − x0 |mβ · + C||~b||Lipβ
X
B
B
|ψt (x − y) − ψt (x − x0 )||a(y)|dy Z
mβ
|x − x0 | t (t + |x − x0 |)n+1+ε
+ C||~b||Lipβ
(b(y) − b(x0 ))σ ψt (x − y)a(y)dy|
|x − x0 |τ β
X
(ψt (x − y) − ψt (x − x0 ))a(y)dy|
Z
τ +τ 0 =m
≤ C||~b||Lipβ
B
|x − x0 |τ β
τ +τ 0 =m
B
0
|y − x0 |τ β |ψt (x − y)||a(y)|dy
Z B
|x0 − y|ε |a(y)|dy
Z t 0 |y − x0 |τ β |a(y)|dy n+1 (t + |x − x0 |) B
t mβ+ε+n(1− p1 ) · r (t + |x − x0 |)n+1+ε 1 t + C||~b||Lipβ · rmβ+n(1− p ) . n+1 (t + |x − x0 |)
≤ C||~b||Lipβ
Thus ~ |gψb (a)(x)| ≤ C||~b||Lipβ
Z ∞à 0
+ C||~b||Lipβ
t (t + |x − x0 |)n+1+ε
Z ∞à 0
!2
t (t + |x − x0 |)n+1
!2
1/2
dt t
1
· rmβ+ε+n(1− p )
1/2
dt t
1
· rmβ+n(1− p )
1 ≤ C||~b||Lipβ |x − x0 |−n · rmβ+n(1− p ) ,
so II ≤ C||~b||Lipβ · r
mβ+n(1− p1 )
≤ C||~b||Lipβ . This completes the proof of Theorem 2.
ÃZ
!1/q |x−x0 |>2r
|x − x0 |
−nq
dx
41 Theorem 3. Let 0 < β ≤ 1, 0 < p < ∞, 1 < q1 , q2 < ∞, 1/q1 − 1/q2 = mβ/n, n(1 − 1/q1 ) ≤ α < n(1 − 1/q1 ) + mβ and ~b = (b1 , . . . , bm ) where bj ∈ Lipβ (Rn ) for ~ 1 ≤ j ≤ m. Then gψb is bounded from H K˙ qα,p (Rn ) to K˙ qα,p . 1 2
P j Proof. Let f ∈ H K˙ qα,p (Rn ) and f = ∞ j=−∞ λj aj , suppaj ⊂ Bj = B(0, 2 ), aj be 1
a central (α, q)−atom, and
P∞
j=−∞
|λj |p < ∞ (Lemma 6).
When m = 1, we have ||gψb1 ||pK˙ α,p ≤ C q2
∞ X
2kαp
|λj | · ||gψb1 (aj )χk ||Lq2
j=−∞
k=−∞
∞ X
+C
p
k−2 X
2kαp
k=−∞
∞ X
p
|λj | · ||gψb1 (aj )χk ||Lq2
j=k−1
= I1 + II2 . By the boundedness of gψb1 on (Lq1 , Lq2 ), we have for II2 II2 ≤ C||b1 ||pLipβ ≤ C||b1 ||pLipβ
∞ X
2kαp
k=−∞ ∞ X
2kαp
k=−∞
p
∞ X
|λj | · ||aj ||Lq1
j=k−1 ∞ X
p
|λj |p · 2−jα
j=k−1
P P∞ p (k−j)αp ∞ , 0
