- Email: [email protected]
ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
2005,25B(3):545-554
ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 1 Hu Guoen ( tll oo,~
)
Department of Applied Mathematics, University of Information Engineering, P. O. Box 1001-747, Zhengzhou 450002, China
Abstract LP(lRn ) (1 < P < 00) boundedness and a weak type endpoint estimate are considered for the commutators of singular integral operators. A condition on the associated kernel is given under which the L 2(lR n ) boundedness of the singular integral operators implies the U(lR n ) boundedness (1 < P < 00) and the weak type (H1(lR n ) , L1(lRn ) ) boundedness for the corresponding commutators. A new interpolation theorem is also established. Key words Commutator, singular integral operators, BMO(lRn ) , interpolation
2000 MR Subject Classification
1
42B2Q
Introduction
We will work on R", n ?: 1. Let T : CD (lR n ) with associated kernel K in the sense that
T f(x) = (
----t
(CD (lR n ) )' be a linear continuous operator
JRn K(x, y)f(y)dx,
if x
f/. supp f.
(1)
For a function b E BMO(lRn ) (the space of functions of bounded mean value oscillation which was introduced by John and Nirenberg, see [12, Chapter IV] for definition and properties of this space), define the commutator generated by T and b by
nf(x)
=
b(x)Tf(x) - T(bf)(x), f
E
CO'(lRn ) .
(2)
A well-known result of Coifman, Rochberg and Weiss[2) states that if K is a standard CalderonZygmund kernel, that is, for some positive constants C and E with 0 < E ::; 1, IK(x, y)1 ::; Clx - Yl-n, xi- Y, IK(x, y) - K(x ', y)1 1 Received
+ IK(y, x) - K(y, x')1 ::; C I~x_-y~~~€' if Ix - yl ?: 21x - xii,
April 21, 2003. This research was supported by the NNSF of China (10271015)
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and if T is bounded on L 2(lR n ) , then Ti, is bounded on LP(lR n ) for all 1 < p < 00. By wavelet analysis, Deng, Yan and Yang[3],[4] considered the LP(lR n ) boundedness for the operator
n
when K satisfies some minimum regularity condition. For each positive integer k, set
"((k)
=
r
sup
R>O, lul+lvl
(IK(x + u, y + v) - K(x, y)1
+IK(y + v, x + u) - K(y, x)l)dy. Deng, Yan and Yang (see [3] and [4]) showed that if Tl E BMO(lR n), T*l E BMO(lR n), T enjoys the weak bounded property, K satisfies the size condition
r
sup
R>O; xEIRn J R< Ix-yl '5.2R
(IK(x, y)1 + IK(y, x)l)qy <
(3)
00
and the regularity condition 00
I>2"((k) <
(4)
00,
k=1
then Ti; is bounded on L 2 (IR n ) . Furthermore, if the regularity condition (4) is replaced by
L k2+II/p-I/21"((k) < 00
(4')
00,
k=1
for some fixed p with 1 < P < 00, then Ti, is bounded on LP(lR n ) . The main purpose of this paper is to give another condition on the associated kernel K which together with the L 2 (IR n ) boundedness of T implies the LP(IR n ) boundedness of Tb for all
1 < P < 00. We remark that our argument is fairly different from that used in [2], [3] and [4]. One of our observation is that some endpoint estimates for T b which via our new interpolation theorem imply the LP(lR n ) boundedness for can be obtained from the L 2(lR n ) boundedness
n
of T directly. Our result concerning the LP(lR n ) boundedness (1 < P < as follows.
o
00)
for Tb can be stated
o
Theorem 1 Let T : C (lR n ) ~ (C (lR n ) )' be a linear continuous operator with associated kernel K. Suppose that there is a positive constant C such that for any R > 0, sup z , zEIRn
sup
f
Iy-y'l
k
r
Jly-zl
r
(IK(x, y)1 + IK(y, x)I)lx - ylndy::::: CRn,
J 2k R< Ix-yl '5.2 k + 1 R
(IK(x, y) - K(x, y')1 + IK(y, x) - K(y', x)l)dx :::::
If T is bounded on L 2(lR n ) , then for b E BMO(lR n), 1
n
(5)
c.
(6)
is bounded on LP(lR n ) for any p with
00.
To prove Theorem 1, we will use the following endpoint estimate which has independent interest.
Co
Theorem 2 Let T: (IR n ) ~ (Co (IR n ) )' be a linear continuous operator with associated kernel K. Suppose that K satisfies (5) and (6). If T is bounded on L 2 (lR n ) , then for any bE BMO(lR n), n is bounded from HI (IR n ) to weak L I (lR n ) , that is, there is a positive constant C such that for any). > 0 and f E HI (IR n ) ,
Hu: ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
No.3
547
where HI (R") is the standard Hardy space, see [12, Chap. IV] for definition and properties of this space. Throughout this paper, C denotes the constants that are independent of the main parameters involved but whose values may differ from line to line. Let M be the standard Hardy-Littlewood maximal operator. For fixed 0 < r < 00, define the operator M; by
Mrf(x) = [M(lfn(x)] l/r and the sharp maximal operator Mj! by
Mj! f(x)
=
1 sup inf ( IQI
xEQcEC
1 Q
)
If(y) - crdy l/r ,
where the supremum is taken over all cubes whose sides are parallel to the coordinate axes. The operator Mr, which will be denoted by M#, is the Fefferman-Stein sharp maximal operator. It is obvious that for any 0 < r < 1, Mj! f(x) ~ M# f(x).
2
Proof of Theorems We begin with some lemmas. Lemma 1
For any 1 < q <
00
and nonnegative numbers a, band t,
This lemma follows directly from the fact that for any 1
~ q
<
00
and a, b 2: 0,
The following lemma is an extension of the interpolation theorem of Han[7] and will be used in the proof of Theorem 1. Lemma 2 Let 0 < PI ~ 1 < P2 < 00, T l and T 2 be two sublinear operators from Co (lR. n to M (lR. n ) (the set of all measurable functions on lR. n ) . Suppose that
)
\
(1) T l is bounded from HPl(lR. n ) to weak LPl(lR. n ) ; (2) T2 is bounded on LP2 (lR. n ) ; (3) there is a positive constant B such that for bounded function
f
with compact support,
Then we have (a) T, is bounded on LP(lR.n ) for any P with 1 < P < P2; (b) if 0
< PI < 1, then T l is bounded from HP(R") to LP(R") for any P with PI < P < 1.
Proof By the interpolation theorem of Han[7], it suffices to prove the conclusion (a). Let
0< r < Pl. We claim that for each P with r < P ::; 1, M; is bounded on weak LP(lR. n ) , that is, there is a positive constant C = Cr, n, P such that sup,XPI{x E lR. n A>O
:
Mrf(x) >
'x}1
~ CsupsPI{x E lR. n 8>0
:
If(x)1 > s }I.
(7)
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In fact, for the case of p = 1, this was proved by Nazarov, Treil and Volberg[ll]. On the other hand, if r < p < 1, it is easy to verify that for each ,\ > 0, ,\PI{x E ~n: Mrf(x) >
,\}I = ,\PI{x E ~n: Mrjp(lfIP)(x) > ,\P}I ::; Csupsl{x E ~n: If(x)IP 8>0
= supsPI{x E ~n : 8>0
> s}1
If(x)1 > s}1
and the estimate (7) holds. Note that Mj!(Td)(x) ::; Mr(Td)(x). The weak type (HPl(~n), LPI (~n)) boundedness of T 1 along with (7) shows that
(8) Now our goal is to show that for any 1 < p < P2, there is a positive constant C such that for any ,\ > 0 and bounded function f with compact support, I{x E ~n : Mj!(Td)(x) > ,\}I
< C'\-Pllfll~.
(9)
To show this, note that although Mr(T1) is not sublinear, it is quasilinear, i.e., there is a positive
c,
constant such that Mj!(T1(!I + h))(x) ::; Cr(Mj!(TdI)(x) + Mj!(T1h)(x)). For each fixed ,\ > 0 and bounded function f with compact support, applying the Calder6n-Zygmund decomposition to Ifl P at the level ,\P, we can obtain a sequence of cubes {Qj} such that
(1) ,\P <
I~jl ~j If(y)IPdy ::; 2n,\p.
(2) If(y)1 ::; '\, a.e. x
E ~n\
U, Qj.
Let k be the smallest integer such that k> n(1/Pl - 1) - 1, PQj (f) be the unique polynomial of degree no greater than k and satisfies
Set
h(y) =
L hj(y) = L j
and
g(y)
(J(y) - PQj (f)(y))XQj (y)
j
= f(y)XRn\UjQj (y) + L PQj (f) (Y)XQj (y). j
Han's observation tells us that
(see [7] for details). It is obvious that UjQj constant Co. Therefore,
C
{x
E ~n
: Mpf(x) > Co,\} for some positive
suppg C {x E ~n: Mpf(x) > Co'\} usuppf,
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and so g has compact support. A trivial computation now leads to that
I{x E lRn n
:::; I{x E lR
n
:::; I{x E lR
:
Mj!(Td)(x) > 3BCrA}1
:
Mj!(Tlg)(x) > 2BA}1
:
IT2g(x)1 > BA}I + I{x E lR
n
:::; CA -P21IT2gll~~ :::;
+ I{x E lRn :
:
Mj!(Tlh)(x) > BA}I
Mj!(Tlh)(x) > BA}I
+ CA -Plllhll~p1(lRn)
CA-Pllfll~·
Our desired estimate (9) now follows directly. We can now complete the proof of Lemma 2. Recall that M!(Td is quasilinear, as in the proof of the Marcinkiewicz interpolation theorem, we can deduce from (9) that M!(Td is bounded on LP(lR n ) for all p with 1 < P < P2. Set
C~O;k =
{f E cga(lRn ) ,
hn
f(x)xadx = 0, 0:::; lal <
It is readily to see that cgao. k is a subset of HPl (R"). For each fixed R By estimate (8), we.have ,
1
<
l 1
<
Cllfll~p1(lRn)
R
AP/r-ll {x E lR n
R 1/ r
1
R 1/ r
> 0, and f
E cgao. I
I
k(lR n ) .
M(ITdn(x) > A} IdA
:
AP-rl{x E lRn
k}.
:
Mr(Td)(x) > A}IAr-ldA
AP-P1-ldA < 00.
Repeating the proof of Theorem 5 in [5], we know from the last inequality that for
f
E
n
C~o; k(lR ) ,
hn
IMr(Td)(x)IPdx =
hn
(M(lTdn(x))p/r dx :::;
c
hn
(M#(ITdn(x))p/r dx.
(10)
Note that
(M#(ITdn(x)) l/r < CMj!(ITlfl)(x).
(11)
By combining estimates (10) and (11), it yields
By the atomic decomposition of LP(lR n ) (see [6, page 41]), we see that C~o; k(lR n ) is dense in LP(lR n ) for 1 < P < 00. A familiar density argument then leads to the LP(lRn ) boundedness for
r..
Proof of Theorem 2 For each fixed f E Hl(lR n ) , by the atomic decomposition of Hl(lR n ) (see [12, Chap. IV]), we can write
where rj's are complex numbers such that is,
2:: j Irjl :::; 21IfIIHl(lRn), aj's are
(1,00,0) atoms, that
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(i) aj is supported on some cube Qj;
(ii) IlajIILoo(~n) ~ IQjl-\ (iii)
r
l~n
aj (x )dx = O.
For each fixed j, let mQj (b) be the mean value of b on the cube Qj. Set bj(x) and write
Td(x) =
= b(x) - mQj(b)
L rjbj(x)Taj(x) + T( L rjbjaj) (x) = TlJ(x) + Tl f(x). 1
j
j
By the Calder6n-Zygmund theory, we know that if K satisfies (5) and (6) and T is bounded on £2 (JRn), then T is also bounded from £1 (JRn) to weak £1 (JRn) , namely, there is a positive constant C such that for any A > 0 and hE £I(JRn),
Therefore,
j
j
Denote by B j the ball with the same center as Qj and having radius n times the side length of Write
o;
~ A-I L
I{x E JRn : ITlf(x)1 > A}I
Ibj(x)IITaj(x)ldx
J
~ A-I
+A- I
L
Irjl
j
r
Ibj(x)IITaj(x)ldx
l~n\2Bj
Ibj(x)IITaj(x)ldx = E + F.
By Holder's inequality and the £2(JRn) boundedness of T, we have
E
~ A-I ~h"'Taj"2(lBjIb(x) -mQj(bWdxf/2 < CA- I
L
Irjlllajl12lQjlI/2 ~ CllfIIHl(~n).
j
On the other hand, for each fixed z
ITaj(x)1 This in turn implies that
ITaj(x)1
~
~
E
Qj, it follows from the vanishing moment of aj that
r
l~n
r
IK(x, y) - K(x, z)lla(y)ldy.
inf IK(x, y) - K(x, z)lla(y)ldy zEQj l~n
< CIQjl-2
r r IK(x, y) - K(x, z)ldydz.
lQj lQj
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Hu: ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
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Let q > 2. Employing Lemma 1, we see that for each fixed j and x E IR n\2Bj ,
r
Ibj(x)IITaj(x)ldx
}Rn\2Bj
r r r \2Bj Ibj(x)IIK(x, y) - K(x, z)ldxdydz < IQjl-2 I= r r r Ib(x) - m2lBj(b)IIK(x, y) - K(x, z)ld~dydz }Qj }Qj }21+1Bj\2IBj +IQjl-2 I= r r r ImQj (b) - m2lnQj (b)IIK(x, y) - K(x, z)ldxdydz }Qj }Qj }21+1Bj\2IBj < IQjl-2 I= z r r r IK(x, y) - K(x, z)ldxdydz }Qj }Qj }21+1Bj\2IBj +IQjl-2I=z-q+l r r r Ib(x) -m2IBj(b)IIK(x, y)ldxdydz }Qj }Qj }2 1nQj\2 +IQjl-2 I= rrr Ib(x) - m2lBj(b)IIK(x, z)ldxdydz }Qj }Qj }2 1Bj\2IBj )
::; CIQjl-2
}a, }Qj }lR
n
1=1
1=1
1=1
1+
1=1
IBj
z-q+l
1+
1=1
< C.
This completes the proof of Theorem 2. Obviously, Theorem 1 follows from Theorem 2, Lemma 2 and the following endpoint estimate for Ti: Lemma 3 Let T : C8" (IR n ) -+ (C8" (IR n ) )' be a linear continuous operator with associated kernel K satisfying (5) and (6). If T is bounded on L2(lR n ) , then for any s > 1, there is a positive constant such that for any bounded function f with compact support,
Proof Without loss of generality, we may assume that IlbIIBMO(Rn) = 1. Let f be a bounded function with compact support. For each x E IR n and a cube Q containing x, decompose f as f(x) = !(X)x4B(X) + f(X)xRn\4B(X) = hex) + hex), where B is the ball with the same center as Q and having diameter n times the side length of Q. Let mB(b) be the mean valueof bon B. For each Yo E Q such that IT((b-mB(b))h)(Yo)1 write
k I~I k + I~I k I~I
::;
< 00,
ITbf(y) - T((mB(b) - b)h(Yo)ldy
ImB(b) - b(y)IITf(y)ldy +
I~I
k
IT((mB(b) - b)h)(y)ldy
IT((mB(b) - b)h)(y) - T((mB(b) - b)h) (Yo) Idy.
Note that (b - mB(b))h E L 2(lRn), IT((b - mB(b))h)(Yo)1 < 00 for almost every Yo E IR n. It follows that 1I 1 inf IQ Inf(y) - cldy ::; CIQI- inf ITbf(y) - T((mB(b) - b)h(yo)ldy cEC
r
YOEQ}Q
552
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ACTA MATHEMATICA SCIENTIA
:::; I~I
k kk
ImB(b) - b(y)IITf(y)ldy +
+IQI- 2
k
I~I
IT((mB(b) - b)h)(y)ldy
IT((mB(b) - b)h)(y) - T((mB(b) - b)h) (yo)ldydyO
=G+H+J. By the Calder6n-Zygmund theory, we know that T is bounded on LP(lR n ) for any 1 < P < Therefore, by Holder's inequality,
1 { , G:::; ( IQI J ImB(b) - b(yW dy Q
)1/8'( IQI1 J(
Q
ITf(yWdy
)1/8 :::; CM
8(Tf)
(x).
The L 2(lR n ) boundedness of T now tells us that
For each fixed y E Q, a straightforward computation leads to that
IT((mB(b) - b)h)(Y) - T((mB(b) - b)h)(yo)1 :::; (
),R,n\4B
ImB(b) - b(z)IIK(y, z) - K(yo, z)llf(z)ldz
f ( +Cllflloo f
:::; Cllflloo
J2 k+ 1B\2 kB
k=l
k=l
ImB(b) - m2k+ 1B(b)1 ( IK(y, z) - K(yo, J2k+lB\2 kB
< C1lflloo + Cllflloo
+Cllflloo
f
k 1-
k
k (
J2 k+ 1B\2 kB
k=l
(
J2 k+ 1B\2 kB
< Cllflloo + Cllflloo
f
f
q
k=l
+C1lflloo
IK(y, z) - K(yo, z)llb(z) - m2k+ 1B(b)ldz
1 q -
k=l
f
k
1
-
k=l
q
(
IK(y, z) - K(yo,
(
z)ldz
IK(y, z) - K(yo, z)llb(z) - m2k+ 1B(bWdz
J2 k+ 1B\2 kB
J2 k+ 1B\2 kB
z)ldz
IK(y, z)llb(z) - m2k+ 1 B ( b W d z
IK(yo, z)llb(z) - m2k+ 1B(bWdz,
where q > 2 and the second-to-last inequality follows from Lemma 1. Therefore,
J :::;
11
Cllflloo + C1lflloo IQ +Cllflloo
f
k=l
k
1 q -
{
f
k=l
k
1 q -
{
(
J2 k+ 1B\2 kB JQ (
J2 k+ 1B\2 kB J Q
IK(y,
m2k+ 1B(bWdz
IK(yo, z)ldYolb(z) - m2k+ 1B(bWdz
< Cllflloo + Cllflloo ~ lk+lB Ib(z) - m2k+ 1B(bWdz < C1lflloo
z)ldylb(z) -
00.
Hu: ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
No.3
553
(recall that IlbIIBMO(jRn) = 1), where we have invoked the John-Nirenberg inequality which states that for two positive constants Co and C 1 ,
This gives the desired estimate for J and then completes the proof of Theorem 1.
3
Some Applications
This section is devoted to some applications of the theorems established in Section 2. At first, we consider the commutators of the homogeneous singular integral operator defined by
-
nf(x) =
1 jRn
O(x - y) n), (b(x) - b(y)) I In f(y)dy, f E Cg"(lR X -
Y
(12)
where 0 is homogeneous of degree zero, integrable on the unit sphere and has mean value zero. This operator was first considered by Coifman, Rochberg and Weiss[2]. It was proved in [2] that
(1 < ex ~ 1) is a sufficient condition for the operator 'h is bounded on LP(lR n ) P < 00. We proved in [8] that if 0 E L(logL)2(sn-1), then Tb is bounded on on
o E Lipo(sn-1) for any 1
<
LP(lR n ) for any 1 < P <
00
and in [9] that if sup
r
(ES n- 1Jsn-l
IO(())llog{3(I()~rl)d(), .,
for some (3 > k + 1, then Tb is bounded on L 2(lR n ) . Now set
w(8) = sup
Ipl'5,8
r
Jsn-l
IO(px) - O(x)ldx,
where the supremum is taken over all rotations on the unit sphere, and Ipl denotes the distance of p from the identity rotation. We have Corollary 1 Let 0 be homogeneous of degree zero, integrable on the unit sphere and have mean value zero. Suppose that
. Jt' w(8)log(2 + J)Jd8 1 1 < o
(13)
00,
Then for any bE BMO(lRn), (a) Tb is bounded on LP(lR n ) for all 1 < P < 00; (b) Tb is bounded from H 1(lRn ) to weak L 1(lRn ) . Proof By the well-known result of Calderon, Weiss and Zygmundl'", we know that the condition (13) implies 0 E LlogL(sn-1) and so the homogeneous singular integral operator
-
Tf(x) = p. v. is bounded on LP(lR n ) for any 1
00.
1I
O(x-y) I f(y)dy jRn x - y n
Let K(x, y)
= ~j~;~).
By the same argument as
that used in [1] , we see that 0 satisfies (13) also implies that K(x, y) satisfies (5). Note that for any R > 0,
1
sup IO(x - y)ldy YI,xEjRn IY-YII
~ CR- n
rIO(y)ldy.
Jsn-l
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ACTA MATHEMATICA SCIENTIA
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Corollary 1 now follows from Theorem 1 and Theorem 2 directly. Remark We do not know whether the condition (13) implies n E L(logL)2(sn-l). Even if this is true, the conclusion (b) of Corollary 1 is new. Now we return our attention to the commutator defined by (2). Let T be a singular integral operator with associated kernel K in the sense (1). A well-known result of Meyer[lO] states that if T1 E BMO(ffin ) , T*1 E BMO(JRn), T enjoys the weak boundedness property, K satisfies the size condition (3) and
L k"((k) < 00
00,
(14)
k=l
then T is bounded on LP(JRn) for any 1 < p < 00. Note that the size condition (5) implies (3). This together with Theorem 1 gives that Corollary 2 Let T : CO' (JR n) -> (Co (JRn))' be a linear continuous operator with associated kernel K satisfies (5) and (14). If T1 E BMO(JRn), T*1 E BMO(JRn), T enjoys the weak boundedness property, then for any b E BMO(JRn), the corresponding commutator Ti, is bounded on LP(lR n) for any 1 < p < 00 and is bounded from HI (IR n) to weak L 1(lRn). References 1 Calderon A P, Weiss M, Zygmund A. On the existence of singular integrals. Proc Syrnp Pure Math, 1967, 10: 56-73 2 Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103: 611-636 3 Deng D, Yan L, Yang Q. L 2 boundedness of commutators of Calderon-Zygrnund singular integral operators. Progr Natur Sci (English Ed), 1998, 8: 416-427 4 Deng D, Yan L, Yang Q. LP boundedness of commutators of Calderon-Zygrnund singular integral operators. Adv in Math (China), 1998, 27: 259-269 5 Fefferman C, Stein E M. HP spaces of several variables. Acta Math, 1972, 129: 137-193 6 Frazier M, Jawerth B, Weiss G. Littlewood-Paley Theory and the Study of Function Spaces, CBMS. Amer Math Soc Providence, Rhode Island, 1991 7 Han Y. A version of Calderon-Zygmund decomposition and its applications. Sci Sinica, 1985, 28: 134-146 8 Hu G. LP(lRn ) boundedness for the commutator of a homogeneous singular integral operator. Studia Math, 2003, 154: 13-27 9 Hu G. L2(lR n ) boundedness for the commutators of convolution operators. Nagoya Math J, 2001, 163: 55-70 10 Meyer Y. La minimalite de l'espace Besev i3~' 1 et la continuite des operateurs definis par des integrals singuliers. Monografias de Matematicas, Vol 4, Univ Autonoma de Madrid, 1986 11 Nazarov F, Treil S, Volberg A. Weak type estimate and Cotlar inequalities for Calderon-Zygmund operaators on nonhomogeneous spaces. International Math Res Notices 1998, 9: 463-487 12 Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton, NJ: Princeton Univ Press, 1993
ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS 1 Hu Guoen ( tll oo,~
)
Department of Applied Mathematics, University of Information Engineering, P. O. Box 1001-747, Zhengzhou 450002, China
Abstract LP(lRn ) (1 < P < 00) boundedness and a weak type endpoint estimate are considered for the commutators of singular integral operators. A condition on the associated kernel is given under which the L 2(lR n ) boundedness of the singular integral operators implies the U(lR n ) boundedness (1 < P < 00) and the weak type (H1(lR n ) , L1(lRn ) ) boundedness for the corresponding commutators. A new interpolation theorem is also established. Key words Commutator, singular integral operators, BMO(lRn ) , interpolation
2000 MR Subject Classification
1
42B2Q
Introduction
We will work on R", n ?: 1. Let T : CD (lR n ) with associated kernel K in the sense that
T f(x) = (
----t
(CD (lR n ) )' be a linear continuous operator
JRn K(x, y)f(y)dx,
if x
f/. supp f.
(1)
For a function b E BMO(lRn ) (the space of functions of bounded mean value oscillation which was introduced by John and Nirenberg, see [12, Chapter IV] for definition and properties of this space), define the commutator generated by T and b by
nf(x)
=
b(x)Tf(x) - T(bf)(x), f
E
CO'(lRn ) .
(2)
A well-known result of Coifman, Rochberg and Weiss[2) states that if K is a standard CalderonZygmund kernel, that is, for some positive constants C and E with 0 < E ::; 1, IK(x, y)1 ::; Clx - Yl-n, xi- Y, IK(x, y) - K(x ', y)1 1 Received
+ IK(y, x) - K(y, x')1 ::; C I~x_-y~~~€' if Ix - yl ?: 21x - xii,
April 21, 2003. This research was supported by the NNSF of China (10271015)
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and if T is bounded on L 2(lR n ) , then Ti, is bounded on LP(lR n ) for all 1 < p < 00. By wavelet analysis, Deng, Yan and Yang[3],[4] considered the LP(lR n ) boundedness for the operator
n
when K satisfies some minimum regularity condition. For each positive integer k, set
"((k)
=
r
sup
R>O, lul+lvl
(IK(x + u, y + v) - K(x, y)1
+IK(y + v, x + u) - K(y, x)l)dy. Deng, Yan and Yang (see [3] and [4]) showed that if Tl E BMO(lR n), T*l E BMO(lR n), T enjoys the weak bounded property, K satisfies the size condition
r
sup
R>O; xEIRn J R< Ix-yl '5.2R
(IK(x, y)1 + IK(y, x)l)qy <
(3)
00
and the regularity condition 00
I>2"((k) <
(4)
00,
k=1
then Ti; is bounded on L 2 (IR n ) . Furthermore, if the regularity condition (4) is replaced by
L k2+II/p-I/21"((k) < 00
(4')
00,
k=1
for some fixed p with 1 < P < 00, then Ti, is bounded on LP(lR n ) . The main purpose of this paper is to give another condition on the associated kernel K which together with the L 2 (IR n ) boundedness of T implies the LP(IR n ) boundedness of Tb for all
1 < P < 00. We remark that our argument is fairly different from that used in [2], [3] and [4]. One of our observation is that some endpoint estimates for T b which via our new interpolation theorem imply the LP(lR n ) boundedness for can be obtained from the L 2(lR n ) boundedness
n
of T directly. Our result concerning the LP(lR n ) boundedness (1 < P < as follows.
o
00)
for Tb can be stated
o
Theorem 1 Let T : C (lR n ) ~ (C (lR n ) )' be a linear continuous operator with associated kernel K. Suppose that there is a positive constant C such that for any R > 0, sup z , zEIRn
sup
f
Iy-y'l
k
r
Jly-zl
r
(IK(x, y)1 + IK(y, x)I)lx - ylndy::::: CRn,
J 2k R< Ix-yl '5.2 k + 1 R
(IK(x, y) - K(x, y')1 + IK(y, x) - K(y', x)l)dx :::::
If T is bounded on L 2(lR n ) , then for b E BMO(lR n), 1
n
(5)
c.
(6)
is bounded on LP(lR n ) for any p with
00.
To prove Theorem 1, we will use the following endpoint estimate which has independent interest.
Co
Theorem 2 Let T: (IR n ) ~ (Co (IR n ) )' be a linear continuous operator with associated kernel K. Suppose that K satisfies (5) and (6). If T is bounded on L 2 (lR n ) , then for any bE BMO(lR n), n is bounded from HI (IR n ) to weak L I (lR n ) , that is, there is a positive constant C such that for any). > 0 and f E HI (IR n ) ,
Hu: ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
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547
where HI (R") is the standard Hardy space, see [12, Chap. IV] for definition and properties of this space. Throughout this paper, C denotes the constants that are independent of the main parameters involved but whose values may differ from line to line. Let M be the standard Hardy-Littlewood maximal operator. For fixed 0 < r < 00, define the operator M; by
Mrf(x) = [M(lfn(x)] l/r and the sharp maximal operator Mj! by
Mj! f(x)
=
1 sup inf ( IQI
xEQcEC
1 Q
)
If(y) - crdy l/r ,
where the supremum is taken over all cubes whose sides are parallel to the coordinate axes. The operator Mr, which will be denoted by M#, is the Fefferman-Stein sharp maximal operator. It is obvious that for any 0 < r < 1, Mj! f(x) ~ M# f(x).
2
Proof of Theorems We begin with some lemmas. Lemma 1
For any 1 < q <
00
and nonnegative numbers a, band t,
This lemma follows directly from the fact that for any 1
~ q
<
00
and a, b 2: 0,
The following lemma is an extension of the interpolation theorem of Han[7] and will be used in the proof of Theorem 1. Lemma 2 Let 0 < PI ~ 1 < P2 < 00, T l and T 2 be two sublinear operators from Co (lR. n to M (lR. n ) (the set of all measurable functions on lR. n ) . Suppose that
)
\
(1) T l is bounded from HPl(lR. n ) to weak LPl(lR. n ) ; (2) T2 is bounded on LP2 (lR. n ) ; (3) there is a positive constant B such that for bounded function
f
with compact support,
Then we have (a) T, is bounded on LP(lR.n ) for any P with 1 < P < P2; (b) if 0
< PI < 1, then T l is bounded from HP(R") to LP(R") for any P with PI < P < 1.
Proof By the interpolation theorem of Han[7], it suffices to prove the conclusion (a). Let
0< r < Pl. We claim that for each P with r < P ::; 1, M; is bounded on weak LP(lR. n ) , that is, there is a positive constant C = Cr, n, P such that sup,XPI{x E lR. n A>O
:
Mrf(x) >
'x}1
~ CsupsPI{x E lR. n 8>0
:
If(x)1 > s }I.
(7)
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In fact, for the case of p = 1, this was proved by Nazarov, Treil and Volberg[ll]. On the other hand, if r < p < 1, it is easy to verify that for each ,\ > 0, ,\PI{x E ~n: Mrf(x) >
,\}I = ,\PI{x E ~n: Mrjp(lfIP)(x) > ,\P}I ::; Csupsl{x E ~n: If(x)IP 8>0
= supsPI{x E ~n : 8>0
> s}1
If(x)1 > s}1
and the estimate (7) holds. Note that Mj!(Td)(x) ::; Mr(Td)(x). The weak type (HPl(~n), LPI (~n)) boundedness of T 1 along with (7) shows that
(8) Now our goal is to show that for any 1 < p < P2, there is a positive constant C such that for any ,\ > 0 and bounded function f with compact support, I{x E ~n : Mj!(Td)(x) > ,\}I
< C'\-Pllfll~.
(9)
To show this, note that although Mr(T1) is not sublinear, it is quasilinear, i.e., there is a positive
c,
constant such that Mj!(T1(!I + h))(x) ::; Cr(Mj!(TdI)(x) + Mj!(T1h)(x)). For each fixed ,\ > 0 and bounded function f with compact support, applying the Calder6n-Zygmund decomposition to Ifl P at the level ,\P, we can obtain a sequence of cubes {Qj} such that
(1) ,\P <
I~jl ~j If(y)IPdy ::; 2n,\p.
(2) If(y)1 ::; '\, a.e. x
E ~n\
U, Qj.
Let k be the smallest integer such that k> n(1/Pl - 1) - 1, PQj (f) be the unique polynomial of degree no greater than k and satisfies
Set
h(y) =
L hj(y) = L j
and
g(y)
(J(y) - PQj (f)(y))XQj (y)
j
= f(y)XRn\UjQj (y) + L PQj (f) (Y)XQj (y). j
Han's observation tells us that
(see [7] for details). It is obvious that UjQj constant Co. Therefore,
C
{x
E ~n
: Mpf(x) > Co,\} for some positive
suppg C {x E ~n: Mpf(x) > Co'\} usuppf,
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No.3
and so g has compact support. A trivial computation now leads to that
I{x E lRn n
:::; I{x E lR
n
:::; I{x E lR
:
Mj!(Td)(x) > 3BCrA}1
:
Mj!(Tlg)(x) > 2BA}1
:
IT2g(x)1 > BA}I + I{x E lR
n
:::; CA -P21IT2gll~~ :::;
+ I{x E lRn :
:
Mj!(Tlh)(x) > BA}I
Mj!(Tlh)(x) > BA}I
+ CA -Plllhll~p1(lRn)
CA-Pllfll~·
Our desired estimate (9) now follows directly. We can now complete the proof of Lemma 2. Recall that M!(Td is quasilinear, as in the proof of the Marcinkiewicz interpolation theorem, we can deduce from (9) that M!(Td is bounded on LP(lR n ) for all p with 1 < P < P2. Set
C~O;k =
{f E cga(lRn ) ,
hn
f(x)xadx = 0, 0:::; lal <
It is readily to see that cgao. k is a subset of HPl (R"). For each fixed R By estimate (8), we.have ,
1
<
l 1
<
Cllfll~p1(lRn)
R
AP/r-ll {x E lR n
R 1/ r
1
R 1/ r
> 0, and f
E cgao. I
I
k(lR n ) .
M(ITdn(x) > A} IdA
:
AP-rl{x E lRn
k}.
:
Mr(Td)(x) > A}IAr-ldA
AP-P1-ldA < 00.
Repeating the proof of Theorem 5 in [5], we know from the last inequality that for
f
E
n
C~o; k(lR ) ,
hn
IMr(Td)(x)IPdx =
hn
(M(lTdn(x))p/r dx :::;
c
hn
(M#(ITdn(x))p/r dx.
(10)
Note that
(M#(ITdn(x)) l/r < CMj!(ITlfl)(x).
(11)
By combining estimates (10) and (11), it yields
By the atomic decomposition of LP(lR n ) (see [6, page 41]), we see that C~o; k(lR n ) is dense in LP(lR n ) for 1 < P < 00. A familiar density argument then leads to the LP(lRn ) boundedness for
r..
Proof of Theorem 2 For each fixed f E Hl(lR n ) , by the atomic decomposition of Hl(lR n ) (see [12, Chap. IV]), we can write
where rj's are complex numbers such that is,
2:: j Irjl :::; 21IfIIHl(lRn), aj's are
(1,00,0) atoms, that
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(i) aj is supported on some cube Qj;
(ii) IlajIILoo(~n) ~ IQjl-\ (iii)
r
l~n
aj (x )dx = O.
For each fixed j, let mQj (b) be the mean value of b on the cube Qj. Set bj(x) and write
Td(x) =
= b(x) - mQj(b)
L rjbj(x)Taj(x) + T( L rjbjaj) (x) = TlJ(x) + Tl f(x). 1
j
j
By the Calder6n-Zygmund theory, we know that if K satisfies (5) and (6) and T is bounded on £2 (JRn), then T is also bounded from £1 (JRn) to weak £1 (JRn) , namely, there is a positive constant C such that for any A > 0 and hE £I(JRn),
Therefore,
j
j
Denote by B j the ball with the same center as Qj and having radius n times the side length of Write
o;
~ A-I L
I{x E JRn : ITlf(x)1 > A}I
Ibj(x)IITaj(x)ldx
J
~ A-I
+A- I
L
Irjl
j
r
Ibj(x)IITaj(x)ldx
l~n\2Bj
Ibj(x)IITaj(x)ldx = E + F.
By Holder's inequality and the £2(JRn) boundedness of T, we have
E
~ A-I ~h"'Taj"2(lBjIb(x) -mQj(bWdxf/2 < CA- I
L
Irjlllajl12lQjlI/2 ~ CllfIIHl(~n).
j
On the other hand, for each fixed z
ITaj(x)1 This in turn implies that
ITaj(x)1
~
~
E
Qj, it follows from the vanishing moment of aj that
r
l~n
r
IK(x, y) - K(x, z)lla(y)ldy.
inf IK(x, y) - K(x, z)lla(y)ldy zEQj l~n
< CIQjl-2
r r IK(x, y) - K(x, z)ldydz.
lQj lQj
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Hu: ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
No.3
Let q > 2. Employing Lemma 1, we see that for each fixed j and x E IR n\2Bj ,
r
Ibj(x)IITaj(x)ldx
}Rn\2Bj
r r r \2Bj Ibj(x)IIK(x, y) - K(x, z)ldxdydz < IQjl-2 I= r r r Ib(x) - m2lBj(b)IIK(x, y) - K(x, z)ld~dydz }Qj }Qj }21+1Bj\2IBj +IQjl-2 I= r r r ImQj (b) - m2lnQj (b)IIK(x, y) - K(x, z)ldxdydz }Qj }Qj }21+1Bj\2IBj < IQjl-2 I= z r r r IK(x, y) - K(x, z)ldxdydz }Qj }Qj }21+1Bj\2IBj +IQjl-2I=z-q+l r r r Ib(x) -m2IBj(b)IIK(x, y)ldxdydz }Qj }Qj }2 1nQj\2 +IQjl-2 I= rrr Ib(x) - m2lBj(b)IIK(x, z)ldxdydz }Qj }Qj }2 1Bj\2IBj )
::; CIQjl-2
}a, }Qj }lR
n
1=1
1=1
1=1
1+
1=1
IBj
z-q+l
1+
1=1
< C.
This completes the proof of Theorem 2. Obviously, Theorem 1 follows from Theorem 2, Lemma 2 and the following endpoint estimate for Ti: Lemma 3 Let T : C8" (IR n ) -+ (C8" (IR n ) )' be a linear continuous operator with associated kernel K satisfying (5) and (6). If T is bounded on L2(lR n ) , then for any s > 1, there is a positive constant such that for any bounded function f with compact support,
Proof Without loss of generality, we may assume that IlbIIBMO(Rn) = 1. Let f be a bounded function with compact support. For each x E IR n and a cube Q containing x, decompose f as f(x) = !(X)x4B(X) + f(X)xRn\4B(X) = hex) + hex), where B is the ball with the same center as Q and having diameter n times the side length of Q. Let mB(b) be the mean valueof bon B. For each Yo E Q such that IT((b-mB(b))h)(Yo)1 write
k I~I k + I~I k I~I
::;
< 00,
ITbf(y) - T((mB(b) - b)h(Yo)ldy
ImB(b) - b(y)IITf(y)ldy +
I~I
k
IT((mB(b) - b)h)(y)ldy
IT((mB(b) - b)h)(y) - T((mB(b) - b)h) (Yo) Idy.
Note that (b - mB(b))h E L 2(lRn), IT((b - mB(b))h)(Yo)1 < 00 for almost every Yo E IR n. It follows that 1I 1 inf IQ Inf(y) - cldy ::; CIQI- inf ITbf(y) - T((mB(b) - b)h(yo)ldy cEC
r
YOEQ}Q
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ACTA MATHEMATICA SCIENTIA
:::; I~I
k kk
ImB(b) - b(y)IITf(y)ldy +
+IQI- 2
k
I~I
IT((mB(b) - b)h)(y)ldy
IT((mB(b) - b)h)(y) - T((mB(b) - b)h) (yo)ldydyO
=G+H+J. By the Calder6n-Zygmund theory, we know that T is bounded on LP(lR n ) for any 1 < P < Therefore, by Holder's inequality,
1 { , G:::; ( IQI J ImB(b) - b(yW dy Q
)1/8'( IQI1 J(
Q
ITf(yWdy
)1/8 :::; CM
8(Tf)
(x).
The L 2(lR n ) boundedness of T now tells us that
For each fixed y E Q, a straightforward computation leads to that
IT((mB(b) - b)h)(Y) - T((mB(b) - b)h)(yo)1 :::; (
),R,n\4B
ImB(b) - b(z)IIK(y, z) - K(yo, z)llf(z)ldz
f ( +Cllflloo f
:::; Cllflloo
J2 k+ 1B\2 kB
k=l
k=l
ImB(b) - m2k+ 1B(b)1 ( IK(y, z) - K(yo, J2k+lB\2 kB
< C1lflloo + Cllflloo
+Cllflloo
f
k 1-
k
k (
J2 k+ 1B\2 kB
k=l
(
J2 k+ 1B\2 kB
< Cllflloo + Cllflloo
f
f
q
k=l
+C1lflloo
IK(y, z) - K(yo, z)llb(z) - m2k+ 1B(b)ldz
1 q -
k=l
f
k
1
-
k=l
q
(
IK(y, z) - K(yo,
(
z)ldz
IK(y, z) - K(yo, z)llb(z) - m2k+ 1B(bWdz
J2 k+ 1B\2 kB
J2 k+ 1B\2 kB
z)ldz
IK(y, z)llb(z) - m2k+ 1 B ( b W d z
IK(yo, z)llb(z) - m2k+ 1B(bWdz,
where q > 2 and the second-to-last inequality follows from Lemma 1. Therefore,
J :::;
11
Cllflloo + C1lflloo IQ +Cllflloo
f
k=l
k
1 q -
{
f
k=l
k
1 q -
{
(
J2 k+ 1B\2 kB JQ (
J2 k+ 1B\2 kB J Q
IK(y,
m2k+ 1B(bWdz
IK(yo, z)ldYolb(z) - m2k+ 1B(bWdz
< Cllflloo + Cllflloo ~ lk+lB Ib(z) - m2k+ 1B(bWdz < C1lflloo
z)ldylb(z) -
00.
Hu: ON THE COMMUTATORS OF SINGULAR INTEGRAL OPERATORS
No.3
553
(recall that IlbIIBMO(jRn) = 1), where we have invoked the John-Nirenberg inequality which states that for two positive constants Co and C 1 ,
This gives the desired estimate for J and then completes the proof of Theorem 1.
3
Some Applications
This section is devoted to some applications of the theorems established in Section 2. At first, we consider the commutators of the homogeneous singular integral operator defined by
-
nf(x) =
1 jRn
O(x - y) n), (b(x) - b(y)) I In f(y)dy, f E Cg"(lR X -
Y
(12)
where 0 is homogeneous of degree zero, integrable on the unit sphere and has mean value zero. This operator was first considered by Coifman, Rochberg and Weiss[2]. It was proved in [2] that
(1 < ex ~ 1) is a sufficient condition for the operator 'h is bounded on LP(lR n ) P < 00. We proved in [8] that if 0 E L(logL)2(sn-1), then Tb is bounded on on
o E Lipo(sn-1) for any 1
<
LP(lR n ) for any 1 < P <
00
and in [9] that if sup
r
(ES n- 1Jsn-l
IO(())llog{3(I()~rl)d(), .,
for some (3 > k + 1, then Tb is bounded on L 2(lR n ) . Now set
w(8) = sup
Ipl'5,8
r
Jsn-l
IO(px) - O(x)ldx,
where the supremum is taken over all rotations on the unit sphere, and Ipl denotes the distance of p from the identity rotation. We have Corollary 1 Let 0 be homogeneous of degree zero, integrable on the unit sphere and have mean value zero. Suppose that
. Jt' w(8)log(2 + J)Jd8 1 1 < o
(13)
00,
Then for any bE BMO(lRn), (a) Tb is bounded on LP(lR n ) for all 1 < P < 00; (b) Tb is bounded from H 1(lRn ) to weak L 1(lRn ) . Proof By the well-known result of Calderon, Weiss and Zygmundl'", we know that the condition (13) implies 0 E LlogL(sn-1) and so the homogeneous singular integral operator
-
Tf(x) = p. v. is bounded on LP(lR n ) for any 1
00.
1I
O(x-y) I f(y)dy jRn x - y n
Let K(x, y)
= ~j~;~).
By the same argument as
that used in [1] , we see that 0 satisfies (13) also implies that K(x, y) satisfies (5). Note that for any R > 0,
1
sup IO(x - y)ldy YI,xEjRn IY-YII
~ CR- n
rIO(y)ldy.
Jsn-l
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ACTA MATHEMATICA SCIENTIA
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Corollary 1 now follows from Theorem 1 and Theorem 2 directly. Remark We do not know whether the condition (13) implies n E L(logL)2(sn-l). Even if this is true, the conclusion (b) of Corollary 1 is new. Now we return our attention to the commutator defined by (2). Let T be a singular integral operator with associated kernel K in the sense (1). A well-known result of Meyer[lO] states that if T1 E BMO(ffin ) , T*1 E BMO(JRn), T enjoys the weak boundedness property, K satisfies the size condition (3) and
L k"((k) < 00
00,
(14)
k=l
then T is bounded on LP(JRn) for any 1 < p < 00. Note that the size condition (5) implies (3). This together with Theorem 1 gives that Corollary 2 Let T : CO' (JR n) -> (Co (JRn))' be a linear continuous operator with associated kernel K satisfies (5) and (14). If T1 E BMO(JRn), T*1 E BMO(JRn), T enjoys the weak boundedness property, then for any b E BMO(JRn), the corresponding commutator Ti, is bounded on LP(lR n) for any 1 < p < 00 and is bounded from HI (IR n) to weak L 1(lRn). References 1 Calderon A P, Weiss M, Zygmund A. On the existence of singular integrals. Proc Syrnp Pure Math, 1967, 10: 56-73 2 Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann of Math, 1976, 103: 611-636 3 Deng D, Yan L, Yang Q. L 2 boundedness of commutators of Calderon-Zygrnund singular integral operators. Progr Natur Sci (English Ed), 1998, 8: 416-427 4 Deng D, Yan L, Yang Q. LP boundedness of commutators of Calderon-Zygrnund singular integral operators. Adv in Math (China), 1998, 27: 259-269 5 Fefferman C, Stein E M. HP spaces of several variables. Acta Math, 1972, 129: 137-193 6 Frazier M, Jawerth B, Weiss G. Littlewood-Paley Theory and the Study of Function Spaces, CBMS. Amer Math Soc Providence, Rhode Island, 1991 7 Han Y. A version of Calderon-Zygmund decomposition and its applications. Sci Sinica, 1985, 28: 134-146 8 Hu G. LP(lRn ) boundedness for the commutator of a homogeneous singular integral operator. Studia Math, 2003, 154: 13-27 9 Hu G. L2(lR n ) boundedness for the commutators of convolution operators. Nagoya Math J, 2001, 163: 55-70 10 Meyer Y. La minimalite de l'espace Besev i3~' 1 et la continuite des operateurs definis par des integrals singuliers. Monografias de Matematicas, Vol 4, Univ Autonoma de Madrid, 1986 11 Nazarov F, Treil S, Volberg A. Weak type estimate and Cotlar inequalities for Calderon-Zygmund operaators on nonhomogeneous spaces. International Math Res Notices 1998, 9: 463-487 12 Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton, NJ: Princeton Univ Press, 1993