Báo cáo toán học: "Weighted Estimates of Multilinear Singular Integral Operators with Variable Calder´n-Zygmund Kernel for the Extreme Cases" ppt

9LHWQD P -RXUQDORI 0$ 7+ 0$ 7, &6 ‹ 9$67 Weighted Estimates of Multilinear Singular Integral Operators with Variable Liu Lanzhe College of Math.. The weighted endpoint estimates for the

9LHWQD P -RXUQDO

RI 0$ 7+ (0$ 7, &6

‹ 9$67 

Weighted Estimates of Multilinear

Singular Integral Operators with Variable

Liu Lanzhe

College of Math and Compt., Changsha Univ of Sci and Tech.

Changsha 410077, China

Received February 28, 2005

Abstract. The weighted endpoint estimates for the multilinear singular integral operators with variable Calder´on-Zygmund kernel on some Hardy and Herz type Hardy spaces are obtained

1 Introduction

Let b ∈ BMO(R n ) and T be the Calder´on-Zygmund operator The commutator

[b, T ] generated by b and T is defined by [b, T ]f (x) = b(x)T f (x) − T (bf)(x) By

a classical result of Coifman, Rochberg and Weiss(see [9]), we know that the

commutator [b, T ] is bounded on L p (R n ) for 1 < p < ∞ In [13], the

bound-edness properties of the commutators for the extreme values of p are proved, and in [3], the weak (H1, L1)-boundedness of the multilinear operator related

to some singular integral operator are obtained In [2], Calder´on and Zygmund introduce some singular integral operators with variable kernel and discuss their boundedness In [10], the authors obtain the boundedness for the

commuta-tors generated by the singular integral operacommuta-tors with variable kernel and BM O

functions In [16], the authors prove the boundedness for the multilinear

oscilla-tory singular integral operators generated by the operators and BM O functions.

In recent years, the theory of Herz space and Herz type Hardy space, as a lo-cal version of Lebesgue space and Hardy space, have been developed (see[11,

∗Supported by the NNSF (Grant: 10271071).

14, 15]) The main purpose of this paper is to establish the weighted endpoint continuity properties of the multilinear singular integral operators with variable Calder´on-Zygmund kernel on Hardy and Herz type Hardy spaces

2 Notations and Theorems

Throughout this paper, we denote the Muckenhoupt weights by A p for 1 

p < ∞ (see [12]) Q will denote a cube of R n with sides parallel to the axes

For a cube Q and a locally integrable function f , let f (Q) = 

Q f (x)dx, f Q =

|Q| −1

Q

f (x)dx and f#(x) = sup

x∈Q |Q| −1

Q |f(y) − f Q |dy Moreover, f is said to

belong to BM O(R n ) if f# ∈ L ∞ (R n) and define that ||f|| BMO = ||f#|| L ∞

Also, we give the concepts of the atom and weighted H1 space A function a

is called a H1(w) atom if there exists a cube Q such that a is supported on Q,

||a|| L ∞ (w)  w(Q) −1 and 

R n

a(x)dx = 0 It is well known that the weighted

Hardy space H1(w) has the atomic decomposition characterization (see [1, 12]) For k ∈ Z, define B k={x ∈ R n:|x|  2 k } and C k = B k \ B k−1 Denote by

χ k the characteristic function of C k and ˜χ k the characteristic function of C k for

k ≥ 1 and ˜χ0the characteristic function of B0

Definition 1 Let 1 < p < ∞ and w1, w2 be two non-negative weight functions

on R n

(1) The homogeneous weighted Herz space is defined by

˙

K p (w1, w2; R n) ={f ∈ L p loc (R n \ {0}) : f K˙p (w1,w2)< ∞},

where

f K˙p (w1,w2)=

∞



k=−∞

[w1(B k)]1−1/p fχ k  L p (w2 );

(2) The nonhomogeneous weighted Herz space is defined by

K p (w1, w2; R n) ={f ∈ L p loc (R n) :f K p (w1,w2)< ∞},

where

f K p=

∞



k=0

[w1(B k)]1−1/p f ˜χ k  L p (w2 );

(3) The homogeneous weighted Herz type Hardy space is defined by

H ˙ K p (w1, w2; R n) ={f ∈ S  (R n

) : G(f ) ∈ ˙ K p (w1, w2; R n)}, where

||f|| H ˙ K p (w1,w2)=||G(f)|| K˙p (w1,w2);

(4) The nonhomogeneous weighted Herz type Hardy space is defined by

HK p (w1, w2; R n) ={f ∈ S  (R n

) : G(f ) ∈ K p (w1, w2; R n)}, where

f HK p (w1,w2)=G(f) K p (w1,w2)

and G(f ) is the grand maximal function of f

The Herz type Hardy spaces have the atomic decomposition characteriza-tion

Definition 2 Let 1 < p < ∞ and w1, w2∈ A1 A function a(x) on R n is called

a central (n(1 − 1/p), p; w1, w2)-atom (or a central (n(1 − 1/p), p; w1, w2)-atom

of restrict type), if

(1) Supp a ⊂ B(0, r) for some r > 0 (or for some r ≥ 1);

(2) ||a|| L p (w2 ) [w1(B(0, r))] 1/p−1 ,

(3) 

R n a(x)dx = 0.

Lemma 1 (see [11, 15]) Let w1, w2 ∈ A1 and 1 < p < ∞ A temperate distri-bution f belongs to H ˙ K p (w1, w2; R n )(or HK p (w1, w2; R n )) if and only if there

exist central (n(1−1/p), p; w1, w2)-atoms (or central (n(1 −1/p), p; w1, w2)-atoms

of restrict type) a j supported on B j = B(0, 2 j ) and constants λ j , 

j |λ j | < ∞ such that f =∞

j=−∞ λ j a j (or f =∞

j=0 λ j a j ) in the S  (R n ) sense, and

f H ˙ K p (w1,w2)( or f HK p (w1,w2))≈

j

|λ j |.

In this paper, we will study a class of multilinear operators related to the singular integral operators with variable kernel, whose definitions are following

Definition 3 Let k(x) = Ω(x)/ |x| n : R n \ {0} −→ R k is said to be a Calder´ on-Zygmund kernel if

(a) Ω∈ C ∞ (R n \ {0});

(b) Ω is homogeneous of degree zero;

(c) 

Σ

Ω(x)x α dσ(x) = 0 for all multi-indices α ∈ (N{0}) n with |α| = N, where Σ = {x ∈ R n:|x| = 1} is the unit sphere of R n

Definition 4 Let k(x, y) = Ω(x, y)/ |y| n : R n × (R n \ {0}) −→ R k is said to

be a variable Calder´ on-Zygmund kernel if

(d) k(x, ·) is a Calder´on-Zygmund kernel for a.e x ∈ R n ;

(e) max|γ|2n∂ |γ|

∂ γ y Ω(x, y)

L ∞ (R n ×Σ) = M < ∞.

Let m be a positive integer and A be a function on R n Set

R m+1 (A; x, y) = A(x) − 

|α|m

1

α! D

α

A(y)(x − y) α

and

Q m+1 (A; x, y) = R m (A; x, y) − 

|α|=m

1

α! D

α

A(x)(x − y) α

.

The multilinear singular integral operators with variable Calder´ on-Zygmund kernel are defined by

˜

T A (f )(x) =



R n

Ω(x, x − y)

|x − y| n+m Q m+1 (A; x, y)f (y)dy

and

T A (f )(x) =



R n

Ω(x, x − y)

|x − y| n+m R m+1 (A; x, y)f (y)dy,

where Ω(x, y)/|y| n is a variable Calder´ on-Zygmund kernel We also define

T (f )(x) =



R n

Ω(x, x − y)

|x − y| n f (y)dy, which is the singular integral operator with variable Calder´ on-Zygmund kernel

(see [2]).

Note that when m = 0, T A is just the commutator of T and A (see [10]) While when m > 0, T A is the non-trivial generalizations of the commutator

From [16], we know that T A is bounded on L p (w) for 1 < p  ∞ and w ∈ A1

In this paper, we will study the weighted endpoint continuity properties of the multilinear operators ˜T A on Hardy and Herz type Hardy spaces

We shall prove the following theorems in Sec 3

Theorem 1 Let w ∈ A1 and D α A ∈ BMO(R n ) for all α with |α| = m Then

˜

T A is bounded from H1(w) to L1(w).

Theorem 2 Let 1 < p < ∞, w1, w2∈ A1and D α A ∈ BMO(R n ) for all α with

|α| = m Then ˜ T A is bounded from ˙ HK p (w1, w2; R n ) (resp HK p (w1, w2; R n))

to ˙ K p (w1, w2; R n )(resp HK p (w1, w2; R n))

3 Proofs of Theorems

To prove the theorems, we need the following lemma

Lemma 2 (see [7]) Let A be a function on R n and D α A ∈ L q (R n ) for |α| = m and some q > n Then

|R m (A; x, y) |  C|x − y| m 

|α|=m

| ˜ Q(x, y)|



˜

Q(x,y)

|D α A(z)| q

dz

1/q

,

where ˜ Q(x, y) is the cube centered at x and having side length 5 √

n |x − y| Proof of Theorem 1 It suffices to show that there exists a constant C > 0 such

that for every H1(w)-atom a (that is that a satisfies: suppa ⊂ Q = Q(x0, r),

||a|| L ∞ (w)  w(Q) −1 and

a(y)dy = 0 (see [1])), the following holds:

|| ˜ T A (a) || L1(w)  C.

Without loss of generality, we may assume l = 2 Write



R n

˜

T A (a)(x)w(x)dx =

2Q

+



(2Q) c

˜

T A (a)(x)w(x)dx := I1+ I2.

For I1, by the following equality

Q m+1 (A; x, y) = R m+1 (A; x, y) + 

|α|=m

1

α! (x − y) α (D α A(x) − D α A(y)),

we get

| ˜ T A (a)(x) |  |T A (a)(x) | + C 

|α|=m

|[D α A, T ]a(x) |,

thus, ˜T A is L p (w)-bounded for 1 < p  ∞ (see[10, 16]), we see that

I1 C|| ˜ T A (a) || L ∞ (w) w(2Q)  C||a|| L ∞ (w) w(Q)  C.

To obtain the estimate of I2, we need to estimate ˜T A (a)(x) for x ∈ (2Q) c Denote ˜A(x) = A(x) −|α|=m 1

α! (D α A) 2Q x α , then D α A = D˜ α A − (D α A) 2Q

for |α| = m, Q m (A; x, y) = Q m( ˜A; x, y) and Q m+1 (A; x, y) = R m (A; x, y) −



|α|=m α!1D α A(x)(x − y) α By [4, 10], we know that

˜

T A (f )(x) =

∞



k=1

g k



h=1

a hk (x)



R n

Y hk (x − y)

|x − y| n+m Q m+1 (A; x, y)f (y)dy

:=

∞



k=1

g k



h=1

a hk (x)S hk A (f )(x),

where g k  Ck n−2,||a hk || L ∞  Ck −2n,|Y hk (x − y)|  Ck n/2−1and



Y hk |x − y| (x − y) n − Y hk |x − x (x − x0

0| n   Ck n/2 |x

0− y|/|x − x0| n+1

for|x − x0| > 2|x0− y| > 0 we write, by the vanishing moment of a and for

x ∈ (2Q) c,

S hk A (a)(x) =



R n

Y hk (x − y)

|x − y| m+n − |x − x Y hk (x − x0

0| m+n

R m( ˜A; x, y)a(y)dy

+



R n

Y hk (x − x0

|x − x0| m+n [R m( ˜A; x, y) − R m( ˜A; x, x0)]a(y)dy

− C 

|α|=m



R n

Y hk (x − y)(x − y) α

|x − y| m+n − Y hk (x |x − x − x0)(x − x0 α

0| m+n

D α A(x)a(y)dy˜

= I2(1)(x) + I2(2)(x) + I2(3)(x);

For I2(1)(x), by Lemma 1 and the following inequality (see [17])

|b Q1− b Q2|  C log(|Q2|/|Q1|)||b|| BMO for Q1⊂ Q2,

we know that, for x ∈ Q and y ∈ 2 j+1 Q \ 2 j Q(j ≥ 1),

|R m( ˜A; x, y)|  C|x − y| m 

|α|=m

(||D α A|| BMO+|(D α

A) 2Q(x,y) − (D α

A) 2Q |)

 Cj|x − y| m 

|α|=m

||D α A|| BMO ,

note that|x − y| ∼ |x − x0| for y ∈ Q and x ∈ R n \ 2Q, then

|I2(1)(x) |  Ck n/2

R n

|y − x0|

|x − x0| m+n+1 |R m( ˜A; x, y) ||a(y)|dy

 Ck n/2 

|α|=m

||D α A|| BMO j



Q

|y − x0|

|x − x0| n+1 |a(y)|dy

 Ck n/2 

|α|=m

||D α A|| BMO j |Q| 1/n+1

|x − x0| n+1 w(Q) −1 .

For I2(2)(x), by the formula (see [7]):

R m( ˜A; x, y) − R m( ˜A; x, x0) = 

|β|<m

1

β! R m−|β| (D

β A; x, x˜ 0)(x − y) β

and Lemma 1, we have

|R m( ˜A; x, y) − R m( ˜A; x, x0 |  C 

|β|<m



|α|=m

|x − x0| m−|β| |x − y| |β| D α A BMO ,

then

|I2(2)(x) |  Ck n/2 

|α|=m

D α A  BMO j



Q

|y − x0|

|x − x0| n+1 |a(y)|dy

 Ck n/2 

|α|=m

D α A BMO j |Q| 1/n+1

|x − x0| n+1 w(Q) −1 .

Similarly,

|I2(3)(x) |  Ck n/2 

|α|=m

|Q| 1/n+1

|x − x0| n+1 w(Q) −1 |D α A(x)|;˜

Thus, for x ∈ 2 j+1 Q \ 2 j Q(j ≥ 1),

| ˜ T A (a)(x) |  C

∞



k=1

g k



h=1

|a hk (x) |k n/2 

|α|=m

D α A  BMO j |Q| 1/n+1

|x − x0| n+1 w(Q) −1

+ C

∞



k=1

g k



h=1

|a hk (x) |k n/2 

|α|=m

|Q| 1/n+1

|x − x0| n+1 w(Q) −1 |D α A(x)˜ |

 C∞

k=1

k −2n+n/2+n−2 

|α|=m

D α A BMO j |Q| 1/n+1

|x − x0| n+1 w(Q) −1

+ C

∞



k=1

k −2n+n/2+n−2 

|α|=m

|Q| 1/n+1

|x − x0| n+1 w(Q) −1 |D α A(x)|˜

|α|=m

||D α A|| BMO j |Q| 1/n+1

|x − x0| n+1 w(Q) −1

|α|=m

|Q| 1/n+1

|x − x0| n+1 w(Q) −1 |D α A(x)|;˜

Notice that if w ∈ A1, then

w(Q2

|Q2|

|Q1| w(Q1  C for all cubes Q1, Q2 with Q1 ⊂ Q2, and w satisfies the reverse of H¨older’ inequality:

 1

|Q|



Q

w(x) q dx

1/q

|Q| C



Q

w(x)dx

for all cube Q and some 1 < q < ∞(see[12]) Thus, by H¨older’s inequality, we

obtain

∞



j=1



2j+1 Q\2 j Q

| ˜ T A (a)(x) |dx

|α|=m

D α A BMO

∞



j=1

j2 −j |Q|

w(Q)

w(2 j+1 Q)

|2 j+1 Q|

|α|=m

∞



j=1

2−j |Q|

w(Q)

|2 j+1 Q |



2j+1 Q

|D α A(x)|˜ q 

dx

1/q 

×|2 j+11Q|



2j+1 Q

w(x) q dx

1/q

|α|=m

D α A BMO

∞



j=1

j2 −j w(2

j+1 Q)

|2 j+1 Q|

|Q|

w(Q)

|α|=m

D α A BMO

Proof of Theorem 2 We only give the proof for case of homogeneous Herz

type Hardy space Without loss of generality, we may assume l = 2 Let

f ∈ H ˙ K p (w1, w2; R n ), by Lemma 1, f =∞

j=−∞ λ j a j , where a  j s are the central

(n(1 −1/p), p; w1, w2)-atom with suppa j ⊂ B j = B(0, 2 j) and||f|| H ˙ K p (w1,w2)≈



j |λ j | Write

|| ˜ T A (f ) || K˙p (w1,w2)=

∞



k=−∞

[w1(B k)]1−1/p ||χ k T˜A (f ) || L p (w2 )



∞



k=−∞

[w1(B k)]1−1/p

k−1



j=−∞

|λ j |||χ k T˜A (a j)|| L p (w2 ) +

∞



k=−∞

[w1(B k)]1−1/p

∞



j=k

|λ j |||χ k T˜A (a j)|| L p (w2 )

= J1+ J2.

For J2, by the L p (w)-boundedness of ˜ T A for 1 < p < ∞ and w ∈ A1, we get

J2 C

∞



k=−∞

[w1(B k)]1−1/p

∞



j=k

|λ j |||a j || L p (w2 )

 C ∞

k=−∞

[w1(B k)]1−1/p

∞



j=k

|λ j |[w1(B j)]−(1−1/p)

 C ∞

j=−∞

|λ j |

j



k=−∞

w1(B k)

w1(B j)

1−1/p

 C ∞

j=−∞

|λ j |  C||f|| H ˙ K p (w1,w2).

To estimate J1, we denote that ˜A(x) = A(x) −|α|=m 1

α! (D α A) 2Q x α Then

Q m (A; x, y) = Q m( ˜A; x, y) Similar to the proof of Theorem 1, we know

˜

T A (a j )(x) =

∞



k=1

g k



h=1

a hk (x)



R n

Y hk (x − y)

|x − y| n+m Q m+1 (A; x, y)a j (y)dy

=

∞



k=1

g k



h=1

a hk (x)S hk A (a j )(x),

and write, by the vanishing moment of a j , for x ∈ (2Q) c,

S hk A (a j )(x) =



R n

Y hk (x − y)

|x − y| m+n − |x| Y hk m+n (x)R m( ˜A; x, y)a j (y)dy

+



R n

Y hk (x)

|x| m+n [R m( ˜A; x, y) − R m( ˜A; x, 0)]a(y)dy

− C 

|α|=m



R n

Y hk (x − y)(x − y) α

|x − y| m+n − Y |x| hk (x)x m+n αD α A(x)a˜ j (y)dy;

Similar to the proof of Theorem 1, we obtain

| ˜ T A (a j )(x) |  C 

|α|=m

||D α A|| BMO 2j

2k(n+1)



B j

|a j (y) |dy

|α|=m

|D α A(x)|˜ 2j

2k(n+1)



B j

|a j (y) |dy

|α|=m

||D α A|| BMO 2j

2k(n+1) ||a j || L p (w2 )

 

B j

w2(y) −1/(p−1) dy

(p−1)/p

|α|=m

|D α A(x)|˜ 2j

2k(n+1) ||a j || L p (w2 )

 

B j

w2(y) −1/(p−1) dy

(p−1)/p

|α|=m

||D α A|| BMO 2j

2k(n+1) [w1(B j)]

1/p−1 

B j

w2(y) −1/(p−1) dy

(p−1)/p

|α|=m

2j

2k(n+1) |D α A(x)|[w˜ 1(B j)]1/p−1

 

B j

w2(y) −1/(p−1) dy

(p−1)/p

.

Notice that w2∈ A1⊆ A p , w2satisfies

sup

Q

 1

|Q|



Q

w2(x)dx

 1

|Q|



Q

w2(x) −1/(p−1) dx

(p−1)/p

 C

and the reverse of H¨older’ inequality for some 1 < q < ∞ (see [12]), thus

J1 C

∞



k=−∞

[w1(B k)]1−1/p

k−1



j=−∞

|λ j | 2j

2k(n+1) [w1(B j)]

−(1−1/p)

× 

B j

w2(y) −1/(p−1) dy

(p−1)/p

× [w2(B k)]1/p+ 

|α|=m

 

B k

|D α A(x)|˜ p

w2(x)dx

1/p

 C

∞



k=−∞

[w1(B k)]1−1/p

k−1



j=−∞

|λ j | 2j

2k(n+1) [w1(B j)]

−(1−1/p)

× 

B j

w2(y) −1/(p−1) dy

(p−1)/p

[w2(B k)]1/p+ 

|α|=m

 1

|B k |

×



B k

|D α A(x)|˜ pq 

dx

1/pq  1

|B k |



B k

w2(x) q dx

1/pq

|B k | 1/p

 C ∞

k=−∞

k−1



j=−∞

|λ j | 2j

2k(n+1)

w1(B k)

w1(B j) 1−1/p

× 

B j

w2(x) −1/(p−1) dx

(p−1)/p

[w2(B k)]1/p

 C

∞



k=−∞

k−1



j=−∞

|λ j | 2j

2k(n+1)

w1(B k)

w1(B j) 1−1/p

× w2(B k)

w2(B j)

1/p

× |B j ||B1

j |



B j

w2(x)dx

1/p 1

|B j |



B j

w2(y) −1/(p−1) dy

(p−1)/p

 C

∞



j=−∞

|λ j |

∞



k=j+1

2j

2k(n+1)

w1(B k)

w1(B j)

|B j |

|B k |

1−1/p

× w2(B k)

w2(B j)

|B j |

|B k |

1/p

|B k |

 C

∞



j=−∞

|λ j |

∞



k=j+1

2j−k

 C

∞



j=−∞

|λ j |  C||f|| H ˙ K p (w1,w2).

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