SCHR ¨ ODINGER OPERATORS

WEAK TYPE ESTIMATES FOR COMMUTATORS GENERATED BY THE RIESZ TRANSFORM ASSOCIATED WITH

SCHR ¨ ODINGER OPERATORS

Zhiping Wang, Bolin Ma, Chuanmei Bi and Canqin Tang

(Received November 2007)

1. Introduction

In recent years, many authors have been interested in the problems of har- monic analysis associated with Schr¨ odinger operators. The Schr¨ odinger operator is the operator defined by

A = −∆ + V (x), x ∈ IR ⁿ ,

where ∆ denotes the Laplacian operator. The function V is called a potential function. One topic of interest is the boundedness of Riesz transforms associated with Schr¨ odinger operators on L ^p (IR ⁿ ) for 1 < p < ∞ with nonnegative potential functions, which are defined as

R j = ∂

∂x _j A ⁻

¹²

, j = 1, 2, · · · , n. (1) In [14], J. Zhong proved that if V is a nonnegative polynomial then Riesz trans- forms, defined at (1), are Calderon-Zygmund operators. Later, potential functions which are local integrable and in some inverse H¨ older class were investigated by several authors. For q > 1, a local integrable function V (≥ 0) belongs to B q , the inverse H¨ older class, if

 1

|B|

Z

B

V ^q (x)dx

 ^1/q

≤ C 1

|B|

Z

B

V (x)dx,

holds for every ball B ⊂ IR ⁿ . Z. Shen, [13], showed the relation between the inverse H¨ older index q, n and L ^p spaces on which the Riesz transforms are bounded. In [13], when the index q ≥ ⁿ ₂ we know that the Riesz transforms are bounded on some L ^p spaces. In [1], Auscher et al improve Shen’s results and show that q ≥ ⁿ ₂ is not a necessary restriction for the boundedness of Riesz transforms. For p = 1, the associated Hardy space and its dual BMO _A were introduced and studied by Dziubanski et al in [4, 5, 6, 7].

Another topic in harmonic analysis associated with Schr¨ odinger operators is that of the commutators of Riesz transforms and BMO _A functions. In classical harmonic analysis, commutators of singular integral operators have been extensively studied

2000 Mathematics Subject Classification : 42B20, 42B25.

Key words and phrases : Schr¨ odinger operator, Riesz Transform, Commutator, Hardy spaces.

This work was supported by NNSF 70971014.

The second author was supported by NNSF 10771054.

and it well known that BMO functions are bounded on L ^p for 1 < p < ∞ (see [3, 10, 12]). Let T be a linear operator. The commutator of T and b is defined as

[b, T ]f (x) = b(x)T f (x) − T (bf )(x).

In [3], Coifman, Rochberg and Weiss proved that [b, T ] is a bounded operator on L ^p (IR ⁿ ) (1 < p < ∞) if and only if b is a BM O function. Furthermore, for p = 1, C. Perez obtained the weak type estimate and the boundedness from H _b ¹ (IR ⁿ ) into L ¹ (IR ⁿ ) in [11]. The boundedness properties of commutators of Riesz transforms associated with Schr¨ odinger operators and BMO _A were investigated by Guo etc (in [9]) and Chu and Ma([2]). By giving the estimates of the kernels of Riesz transform associated with Schr¨ odinger operators, Guo etc. obtained the L ^p (IR ⁿ )(1 < p <

p ₀ )-boundedness of commutator generated by Riesz transform associated with the Schr¨ odinger operator and BM O _A type functions. By using the sharp maximal function estimates and establishing the “good-λ inequality”, Chu and Ma proved the same result. Motivated by their work, we study the endpoint estimates when p = 1. In this paper we will give a weak type estimate and discuss the boundedness of the commutators on Hardy spaces.

2. Notation and the Main Results.

Henceforth, Q will always denote a cube with sides parallel to the axes. λQ, where λ > 0, denotes the cube with the same center as Q and dilated by λ. For a locally integrable function f , we denote the average of f on Q by

f Q = 1

|Q|

Z

Q

f (y)dy.

Also B = B(x, r) will denote a ball centered at x with radius r and corresponding notation applies for λB and f B .

In this paper, we assume that V ∈ B q is a nonnegative function.

We now define function space BM O A associated with A. To this end, define an auxiliary function

ρ(x, V ) = ρ(x) = 1

m(x, V ) = sup (

r > 0 : 1 r ⁿ⁻²

Z

B(x,r)

V (y)dy ≤ 1 )

. Definition 1. ([7]) Let f ∈ L loc (IR ⁿ ). We say that f belongs to BM O A if there is a constant C ≥ 0, so that 1

|B s | Z

B

_s

|f − f B

s

| ≤ C and 1

|B r | Z

B

_r

|f | ≤ C for all balls B _s = B(x, s), B _r = B(x, r) such that s ≤ ρ(x) ≤ r.

The infimum of all such C is denoted by kf k _{BM O}

_A

.

Definition 2. ([11]) A function a is a b-atom if there is a cube Q for which:

(1) supp a ⊂ Q (2) kak _L

^∞

≤ 1

|Q|

(3) Z

Q

a(y)dy = 0, (4)

Z

Q

a(y)b(y)dy = 0.

The space H _b ¹ (IR ⁿ ) consists of the subspace of L ¹ (IR ⁿ ) functions, f , which can be written as f = P

j λ j a j where a j are b-atoms and λ j are complex numbers with P

j |λ _j | < ∞. We define its space norm as kf k _H

¹

b

= inf



 X

j

|λ j |



 .

Definition 3. ([2]) For δ > 0 we define the δ-sharp maximal operator M _δ ^] (f ) associated with Schr¨ odinger operator as M _δ ^] (f )=M _A ^] (|f | ^δ ) ^1/δ , where

M _A ^] f (x) =



 



 

 sup

x∈B

1

|B|

Z

B

|f (t) − f _B |dt, s ≤ ρ(x) sup

x∈B

1

|B|

Z

B

|f (t)|dt, s > ρ(x) for any B = B(x, s).

Furthermore, a function A : [0, ∞) → [0, ∞) is a Young function if it is contin- uous, convex, and increasing with A(0) = 0, and A(t) → ∞ as t → ∞. Define the A-average of a function f over a cube Q by the following Luxemburg norm

kf k _A,Q = inf {λ > 0 : 1

|Q|

Z

Q

A  |f (y)|

λ



dy ≤ 1}.

Let T be a Riesz transform associated with Schr¨ odinger operator A as at (1).

We can now state our main results.

Theorem 4. Let V ∈ B q , n

2 < q < n and suppose V (x) satisfies the following inequality

Z

B(x,R)

V (y)

|x − y| ⁿ⁻¹ dy ≤ C R ⁿ⁻¹

Z

B(x,R)

V (y)dy (2)

and b ∈ BM O A . Then there exists a positive constant C such that for every smooth function f with compact support and for all λ > 0,

|{y ∈ IR ⁿ : |[b, T ]f (y)| > λ}| ≤ Ckbk BM O

_A

Z

IR

ⁿ

|f (y)|

λ



1 + log ⁺  |f (y)|

λ



dy.

Theorem 5. Let V and b satisfy the same conditions as in Theorem 4. Then [b, T ] is a bounded operator from H _b ¹ (IR ⁿ ) to L ¹ (IR ⁿ ).

3. Preliminary Results

In this section, we state the preliminary results we shall need.

Lemma 6. ([7]) Let p ∈ [1, ∞). Then there exists a constant C = C(n, p) such that for any f ∈ BM O A we have

 1

|B|

Z

B

|f (x) − f B | ^p dx  ^1/p

≤ Ckf k BM O

_A

, for any ball B and

 1

|B|

Z

B

|f (x)| ^p dx  ^1/p

≤ Ckf k BM O

A

, for B = (x, r), r ≥ ρ(x).

Lemma 7. ([2]) Let V ∈ B q and q ≥ n

2 . Then for any γ ∈ IR, (−∆ + V ) ^iγ is of weak type (1, 1). Furthermore, ∇(−∆ + V ) ^−1/2 is of weak type (1, 1) when V ∈ B n . The proof of Lemma 7 is similar to the proof of the standard Calder´ on-Zygmund operator given in [8].

The proof of the next lemma can be found in [2]. However, for completeness, we give a sketch of the proof.

Lemma 8. ([2]) Let V ∈ B q and n

2 ≤ q < n and suppose V (x) satisfies the inequality (2). Then the kernel of the operator ∇(−∆+V ) ^−1/2 satisfies the required estimates of C − Z operator kernel.

Proof. Write

∇(−∆ + V ) ^−1/2 f (x) = Z

IR

ⁿ

K(y, x)f (y)dy, where

K(x, y) = − 1 2π

Z

R

(−iτ ) ^−1/2 ∇ _x Γ(x, y, τ )dτ,

and Γ(x, y, τ ) is the fundamental solution for −∆ + V (x) + iτ , τ ∈ IR. From the proof of Theorem 0.8 in [13], we know that

|K(x, y)| ≤ C k

{1 + m(x, V )|x − y|} ^k · 1

|x − y| ⁿ .

Next, we fix x ₀ , y ₀ ∈ IR ⁿ , h ∈ IR ⁿ and |h| < |x ₀ − y ₀ |/4. Let R = |x ₀ − y ₀ |/4 and u(x) = ∇ _y Γ(x, y ₀ , τ ). Since ∇ _y Γ(x, y, τ ) = ∇ _x Γ(y, x, −τ ) (this can be seen by using a similar to the proof of Theorem 0.4 in [13]) for 1

t = 1 q − 1

n , it follows from the imbedding theorem of Morrey that

|u(x 0 + h) − u(x 0 )| ≤ C|h| ¹⁻

^nt

Z

B(x

0

,R)

|∇u| ^t dx

! 1/t

≤ C  |h|

R

 ²⁻

ⁿ_q

{1 + Rm(x 0 , V )} ^k

⁰

sup

B(x

₀

,2R)

|u|

≤ C  |h|

R

 2−

ⁿ_q

1

(1 + |τ | ^1/2 |x 0 − y 0 |) ³ · 1

|x 0 − y 0 | ⁿ⁻¹ . Thus we have proved that for x, y ∈ IR ⁿ , h ∈ IR ⁿ and |h| < |x − y|/4.

|∇ y Γ(x + h, y, τ ) − ∇ y Γ(x, y, τ )| ≤ C

(1 + |τ | ^1/2 |x − y|) ³ · |h| ^δ

|x − y| ^n−1+δ , where δ = 2 − ⁿ _q > 0. This estimate also holds for |x − y|/4 ≤ |h| < |x − y|/2.

Therefore,

|K(x + h, y) − K(x, y)| ≤ C|h| ^δ

|x − y| ^n+δ .

This concludes the proof.

Lemma 9. ([11]) Let M f (x) be the Hardy-Littlewood maximal function of f . For any function f and for all λ > 0, there exists a positive constant C such that

|{y ∈ IR ⁿ : M ² f (y) > λ}| ≤ C Z

IR

ⁿ

|f (y)|

λ



1 + log ⁺  |f (y)|

λ



dy, (3)

here M ² = M ◦ M .

Lemma 10. ([2]) Let γ > 0, λ > 0. For any f ∈ L loc (IR ⁿ ), the “good-λ inequality”

holds. That is

|{x ∈ IR ⁿ : M f (x) > 2λ, M _A ^] f (x) ≤ γλ}| ≤ 2 ⁿ⁺¹ γ|{x ∈ IR ⁿ : M f (x) > λ}|. (4) Proof. Let Ω λ = {x ∈ IR ⁿ : M f (x) > λ}, suppose the measure of this set is finite, otherwise the inequality is obviously true. Then, for any x ∈ Ω λ , there exist a maximal cube Q ^x contained x such that

1

|Q ^x | Z

Q

^x

|f (y)|dy > λ. (5)

Write Q j = {Q ^x : x ∈ Ω λ }. Since the Q ^x ’s are mutually disjoint, so are the Q j ’s.

Therefore,

Ω λ = ∪ j Q j .

Now, to prove Lemma 10, we only need to prove that for every Q j ,

|{x ∈ Q j : M f (x) > 2λ, M _A ^# (f )(x) ≤ γλ}| ≤ 2 ⁿ γ|Q j |. (6) Fix j, x ∈ Q j . Suppose M f (x) > 2λ. By the definition of maximal function of f , it should be that Q ⊃ Q j or Q ⊂ Q j . If Q ⊃ Q j , then the average of |f | on Q is less than or equal to λ, but this will be contradiction to M f (x) > 2λ. It implies Q ⊂ Q j for some j. For x ∈ Q j , we have

M (f − 1

|Q ^∗ _j | Z

Q

^∗_j

|f (t)|dt)χ Q

_j

!

(x) ≥ M (f χ Q

_j

)(x) − 1

|Q ^∗ _j | Z

Q

^∗_j

|f (t)|dt

> 2λ − λ = λ.

Here Q ^∗ _j denotes the cube with the same center as Q j and with twice the length of its side.

Thus,

|{x ∈ Q j : M f (x) > 2λ}| ≤     

{x ∈ Q j : M (f − 1

|Q ^∗ _j | Z

Q

^∗_j

|f (t)|dt)χ Q

_j

!

(x) > λ}

     Since M is of weak type (1,1), we obtain

1 λ

Z

Q

_j

    

f (y) − 1

|Q ^∗ _j | Z

Q

^∗_j

f (t)dt     

dy ≤ 2 ⁿ |Q _j | λ

1

|Q ^∗ _j | Z

Q

^∗_j

    

f (y) − 1

|Q ^∗ _j | Z

Q

^∗_j

f (t)dt     

dy

≤ 2 ⁿ |Q j |

λ M ^# (f )(ξ _j )

≤ 2 ⁿ |Q j |

λ M _A ^# (f )(ξ j ), where ξ _j ∈ Q _j .

To prove the inequality (6), we suppose M _A ^# (f )(ξ j ) ≤ γλ for some ξ j ∈ Q j ,

otherwise the set on the left side in inequality (6) is empty, and the inequality (6)

holds. By using the last inequality, we can deduce that inequality (6) holds for above ξ j ∈ Q j . This finish the proof of Lemma 10.

4. Proof of the Main Theorems

In this section we first give the δ-sharp type estimate which is the key estimate for Lemma 12.

Lemma 11. Let T be a Riesz transform associated with the Schr¨ odinger operator, b ∈ BM O A . Then, for 0 < δ < ε, there exists a positive constant C = C δ,ε such that

M _δ ^] ([b, T ]f )(x) ≤ Ckbk BM O

A

(M ε (T f )(x) + M ² f (x)), for all smooth function f .

Proof. We prove the Lemma 11 by separating it into two cases.

Let B = B(x, r 0 ) be an arbitrary ball. We recall that 0 < δ < 1 implies

||α| ^δ − |β| ^δ | ≤ |α − β| ^δ for α, β ∈ IR.

Case 1, r 0 ≤ ρ(x). Then for all c ∈ IR, we have

 1

|B|

Z

B

 |[b, T ]f (y)| ^δ − |c| ^δ  dy  1/δ

≤  1

|B|

Z

B

|[b, T ]f (y) − c| ^δ dy  1/δ

. (7) Let f = f 1 + f 2 = f χ 2B + f χ (2B)

^c

, for an arbitrary constant a we can write

[b, T ]f = (b − a)T f − T ((b − a)f ₁ ) − T ((b − a)f ₂ ),

Choose c = (T ((b − a)f ₂ )) _B , a = b _2B , then we can estimate the left hand side of (7) by a multiple of

 1

|B|

Z

B

 |[b, T ]f (y)| ^δ − |c| ^δ   dy  ^1/δ

≤ C( 1

|B|

Z

B

|(b(y) − b 2B )T f (y)| ^δ dy) ^1/δ +C( 1

|B|

Z

B

|T ((b − b _2B )f ₁ )(y)| ^δ dy) ^1/δ +C( 1

|B|

Z

B

|T ((b − b 2B )f 2 )(y) − (T ((b − b 2B )f 2 )) B | ^δ dy) ^1/δ

= I + II + III.

To estimate I we use H¨ older’s inequality with 1 < r < ε δ , 1

r + 1

r ^‘ = 1 and Lemma 6. We obtain

I ≤ C  1

|B|

Z

B

|b(y) − b 2B | ^δr dy  ^1/δr ( 1

|B|

Z

B

|T f (y)| ^δr dy) ^1/δr

≤ Ckbk BM O

_A

M δr (T f )(x) ≤ Ckbk BM O

_A

M ε (T f )(x).

For part II we apply Lemma 7 and Kolmogorov’s inequality. Then

II ≤ C

|B|

Z

2B

|T ((b(y) − b _2B )f )(y)|dy ≤ C

|2B|

Z

2B

|(b(y) − b _2B )f (y)|dy

≤ Ckb − b _2B k expL,2B kf k LlogL,2B ≤ Ckbk BM O

A

M _LlogL f (x),

here we used the fact kb − b B k expL,B ≤ Ckbk BM O

_A

(see details in the proof of 3.1 in [11]).

For part III, we use the properties of kernel function K(x, y) and Fubini’s theo- rem. For any y, z ∈ B, w ∈ (2B) ^c , we have following estimate

III ≤ 1

|B|

Z

B

|T ((b − b 2B )f ₂ )(y) − (T ((b − b _2B )f ₂ )) _B |dy

= 1

|B|

Z

B

   Z

IR

ⁿ

K(y, w)(b(w) − b 2B )f 2 (w)dw

− 1

|B|

Z

B

Z

IR

ⁿ

K(z, w)(b(w) − b 2B )f 2 (w)dwdz    dy

≤ 1

|B| ² Z

B

Z

B

Z

IR

ⁿ

\2B

|K(y, w) − K(z, w)||(b(w) − b 2B )f (w)|dwdzdy

≤ 1

|B| ² Z

B

Z

B

∞

X

j=1

Z

2

^j+1

B\2

^j

B

|y − z| ^δ

|z − w| ^n+δ |b(w) − b _2B ||f (x)|dwdzdy

≤ C

∞

X

j=1

r ₀ ^δ (2 ^j r ₀ ) ^n+δ

Z

2

^j+1

B

|b(w) − b 2B ||f (w)|dw

≤ C

∞

X

j=1

2 ^−jδ (2 ^j r 0 ) ⁿ

Z

2

^j+1

B

|b(w) − b ₂

j+1

B ||f (w)|dw

+C

∞

X

j=1

2 ^−jδ |b ₂

j+1

B − b 2B | 1 (2 ^j r 0 ) ⁿ

Z

2

^j+1

B

|f (w)|dw

≤ C

∞

X

j=1

2 ^−jδ kb(w) − b 2

^j+1

B k exp L,2

^j+1

B kf k L log L,2

^j+1

B + Ckbk BM O

A

∞

X

j=1

j

2 ^jδ M f (x)

≤ Ckbk BM O

_A

M LlogL f (x) + Ckbk BM O

_A

M f (x) ≤ Ckbk BM O

_A

M LlogL f (x).

Here we have used the fact that

|b ₂

j+1

B − b 2B | ≤ |b ₂

j+1

B − b ₂

j

B + b ₂

j

B − . . . + b ₂

2

B − b 2B |

≤ |b ₂

j+1

B − b ₂

j

B | + . . . + |b ₂

2

B − b _2B | ≤ Cjkbk _{BM O}

_A

.

Case 2, r ₀ > ρ(x). Write [b, T ]f = bT (f ) − T (bf ₁ ) − T (bf ₂ ). Then

 1

|B|

Z

B

|[b, T ]f (y)| ^δ dy

 1/δ

≤

 1

|B|

Z

B

|bT f (y)| ^δ dy

 1/δ

+

 1

|B|

Z

B

|T (bf 1 )(y)| ^δ dy

 1/δ

+

 1

|B|

Z

B

|T (bf 2 )(y)| ^δ dy

 1/δ

= I

⁰

+ II

⁰

+ III

⁰

.

To estimate I

⁰

, we again use H¨ older’s inequality to deduce that I

⁰

≤

 1

|B|

Z

B

|b(y)| ^δr

0

dy

 ^1/δr

0

 1

|B|

Z

B

|T (f )(y)| ^δr dy

 ^1/δr

≤ kbk BM O

_A

M δr (T (f ))(x) ≤ kbk BM O

_A

M ε (T f )(x).

Following the argument giving the estimate for II we obtain the desired estimate for II

⁰

.

Now we turn to estimate III

⁰

. Making use of the estimation of the kernel, we have

III

⁰

≤ 1

|B|

Z

B

   Z

IR

ⁿ

\2B

K(y, w)b(w)f (w)dw    dy

≤ C 1

|B|

Z

B

Z

IR

ⁿ

\2B

  

C _k

{1 + |y − w|m(y, V )} ^k 1

|y − w| ⁿ b(w)f (w)    dwdy

≤ C 1

|B|

Z

B

∞

X

j=0

Z

2

^j+2

B\2

^j+1

B

  

C _k

{1 + |y − w|m(y, V )} ^k    · 1

|y − w| ⁿ |b(w)f (w)|

≤ C 1

|B|

Z

B

∞

X

j=1

Z

2

^j+1

B\2

^j

B

  

C k

{|y − w|m(y, V )} ^k 1

|y − w| ⁿ b(w)f (w)    dwdy

≤ C 1

|B|

Z

B

∞

X

j=1

1 2 ^kj (2 ^j r ₀ ) ⁿ

Z

2

^j+1

B\2

^j

B

|b(w)f (w)|dwdy

≤ C

∞

X

j=1

1 2 ^kj |2 ^j B|

Z

2

^j+1

B

|b(w)f (w)|dw

≤ C

∞

X

j=1

1

2 ^kj kbk _expL,2

j+1

B kf k _LlogL,2

j+1

B

≤ Ckbk BM O

_A

M LlogL f (x) ≤ Ckbk BM O

_A

M ² f (x),

where we have used the fact M f (x) ≤ M LlogL f (x) ≈ M ² f (x) (see [11] p.170).

This completes the proof of the lemma 11.

Lemma 12. Let Φ(t) = t(1 + log ⁺ t) where T is a Riesz transform associated with the Schr¨ odinger operator and b ∈ BM O _A . Suppose V ∈ B _q , n

2 < q < n and V still satisfies the inequality (2). Then there exists a constant C such that

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : |[b, T ]f (y)| > t}|

≤ Ckbk BM O

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|, for all smooth functions with compact support.

Proof. Let f be a smooth function with compact support. We have to prove the

inequality above with a constant C is independent of f . Instead of working with

the functional on the left hand side of the inequality, we consider the following functional. For some δ > 0, L Φ,δ (f ) = L δ (f ) is defined by

L _δ (f ) = sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M _δ ([b, T ]f )(y) > t}|.

We claim that the operator [b, T ] satisfies the following inequality. For any 0 <

δ < 1, ε > 0,

L δ (f ) ≤ εCL δ (f ) + C(ε)kbk BM O

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}| (8)

To prove (8), for any t > 0, δ > 0, by Lemma 10, we have

|{y ∈ IR ⁿ : M δ ([b, T ]f )(y) > t}|

= |{y ∈ IR ⁿ : M (|[b, T ]f | ^δ )(y) > t ^δ }|

≤ εC _n |{y ∈ IR ⁿ : M (|[b, T ]f | ^δ ) > t ^δ

2 }| + |{y ∈ IR ⁿ : M _A ^] (|[b, T ]f | ^δ )(y) > εt ^δ }|

= I + II.

To estimate II, let ε = rδ, 1 < r < 1

δ , applying Lemma 11 we can write

II = |{y ∈ IR ⁿ : M _δ ^] ([b, T ]f )(y) > ε ^1/δ t}|

≤ |{y ∈ IR ⁿ : M δr (T f )(y) + M ² f (y) > ε ^1/δ t Ckbk BM O

_A

}|

≤ |{y ∈ IR ⁿ : M δr (T f )(y) > ε ^1/δ t 2Ckbk _{BM O}

_A

}|

+|{y ∈ IR ⁿ : M ² f (y) > ε ^1/ t 2Ckbk BM O

_A

}|.

Choose a = ε ^1/δ 2Ckbk BM O

_A

. Then, dividing the above inequality by Φ(1/t) and using that Φ(t) is doubling, we have

1

Φ(1/t) |{y ∈ IR ⁿ : M _δ ([b, T ]f )(y) > t}|

≤ εC

Φ(1/t) |{y ∈ IR ⁿ : M _δ ([b, T ]f )(y) > t 2 ^1/δ }|

+ 1

Φ(1/t) |{y ∈ IR ⁿ : M _δr (T f )(y) > at}| + 1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > at}|

≤ εC

Φ(2 ^1/δ /t) |{y ∈ IR ⁿ : M _δ ([b, T ]f )(y) > t 2 ^1/δ }|

+ Ckbk BM O

_A

Φ(1/at) |{y ∈ IR ⁿ : M δr (T f )(y) > at}|

+ Ckbk _{BM O}

_A

Φ(1/at) |{y ∈ IR ⁿ : M ² f (y) > at}|

≤ εCL δ (f ) + Ckbk BM O

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M δr (T f )(y) > t}|

+Ckbk _{BM O}

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|.

Now, since 0 < rδ < 1, we can use the estimate M _α ^] (T f )(y) ≤ CM f (y), which holds for all 0 < α < 1. Then,

L δ (f ) ≤ εCL δ (f ) + Ckbk BM O

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M _δr ^] (T f )(y) > t}|

+Ckbk _{BM O}

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|

≤ εCL δ (f ) + Ckbk _{BM O}

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M f (y) > t}|

+Ckbk BM O

A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|

≤ εCL δ (f ) + Ckbk BM O

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|.

To finish the proof of Lemma 12 we need to show that L δ (f ) is finite so that we can choose ε < 1

C to conclude that L _δ (f ) ≤ Ckbk _{BM O}

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|. (9) For each m = 1, 2, 3 · · · we let b m = inf {b, m}. Since kb m k BM O

_A

≤ Ckbk BM O

_A

with C independent of m, we shall prove that L Φ,δ,b is finite with b replaced by b m . For b m → b as m → ∞, we shall let m → ∞ to conclude the proof of inequality (9).

Now, since f is bounded and has compact support, we may assume that supp f ⊂

B(0, s) for some s > 0. Recalling that b = b _m , kbk _L

r

≤ m, for |x| > 2s, by Lemma

8 we have,

|[b, T ]f (x)| ≤ C Z

B(0,s)

|b(x) − b(y)|

|x − y| ⁿ |f (y)|dy

≤ 2Cm

|x| ⁿ Z

B(0,C|x|)

|f (y)|dy ≤ CmM f (x).

Using the above inequality and 0 < δ < 1, t > 0, we obtain 1

Φ(1/t) |{x ∈ IR ⁿ : M _δ ([b, T ]f (x) > t}|

≤ 1

Φ(1/t) |{x ∈ IR ⁿ : M (χ _B(0,2s) [b, T ]f )(x) > t 2 }|

+ 1

Φ(1/t) |{x ∈ IR ⁿ : M (χ _IR

ⁿ

_\B(0,2s) [b, T ]f )(x) > t 2 }|

≤ 1

Φ(1/t) 1 t

Z

B(0,2s)

|[b, T ]f (x)|dx + 1

Φ(1/t) |{x ∈ IR ⁿ : M ² f (x) > Cmt}|

≤ C|B(0, 2s)| 1

|B(0, 2s)|

Z

B(0,2s)

|[b, T ]f (y)| ² dy

! ^1/2

+ C

Φ(1/t) Z

IR

ⁿ

Φ  |Cmf (y)|

t

 dy

≤ C|B(0, 2s)|kbk _{BM O}

_A

1

|B(0, s)|

Z

B(0,s)

|f (y)| ² dy

! ^1/2

+C Z

B(0,s)

Φ(f (y))dy,

here we used the fact that M is of weak type (1,1) and the analog for M ² given in Lemma 9. Since f is smooth with compact support the last expression is finite, and the Lemma 12 is proved.

Proof of Theorem 4 We prove that for each f and λ > 0, there exists a positive constant C such that

|{y ∈ IR ⁿ : |[b, T ]f (y)| > λ}| ≤ Ckbk _{BM O}

_A

Z

IR

ⁿ

|f (y)|

λ (1 + log ⁺ ( |f (y)|

λ ))dy.

By homogeneity it is sufficient to consider the case of λ = 1. Namely,

|{y ∈ IR ⁿ : |[b, T ]f (y)| > 1}| ≤ Ckbk _{BM O}

_A

Z

IR

ⁿ

|f (y)|(1 + log ⁺ |f (y)|)dy.

Assume that f is smooth with compact support, by Lemma 12 we have sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : |[b, T ]f (y) > t}|

≤ Ckbk BM O

_A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|.

For all t > 0, making use of Lemma 9, we deduce that

|{y ∈ IR ⁿ : M ² f (y) > t}| ≤ C Z

IR

ⁿ

Φ( |f (y)|

t )dy ≤ C Z

IR

ⁿ

Φ (|f (y)|) Φ(1/t)dy, Since Φ is submultiplicative. Hence,

|{y ∈ IR ⁿ : |[b, T ]f (y)| > 1}| ≤ sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : |[b, T ]f (y)| > t}|

≤ Ckbk BM O

A

sup

t>0

1

Φ(1/t) |{y ∈ IR ⁿ : M ² f (y) > t}|

≤ Ckbk _{BM O}

_A

Z

IR

ⁿ

Φ(|f (y)|)dy

= Ckbk _{BM O}

_A

Z

IR

ⁿ

|f (y)|(1 + log ⁺ |f (y)|)dy, and this yields the desired estimate.

Recall that BM O may be defined by

BM O(IR ⁿ ) = {f ∈ L ¹ _loc (IR ⁿ ) : kf k BM O = kM ^] f k _∞ < ∞}.

Two basic facts about BM O will be used in the proof of Theorem 5. First

|f ₂

k

B − f B | ≤ C(k + 1)kf k BM O , k > 0.

and then the John-Nirenberg inequality kf k BM O ∼ sup

B

( 1

|B|

Z

B

|f (y) − f B | ^p dy) ^1/p , ∀p ≥ 1.

Proof of Theorem 5 Let b ∈ BM O. By the atomic decomposition of Hardy space, we only need to prove that there exists a constant C such that

Z

IR

ⁿ

|[b, T ]a(y)|dy ≤ Ckbk BM O kf k _H

1 b

(IR

ⁿ

) , for each b-atom a.

To prove this, suppose supp a ⊂ B(x, r) for some ball B. Then Z

IR

ⁿ

|[b, T ]a(y)|dy = Z

2B

|[b, T ]a(y)|dy + Z

IR

ⁿ

\2B

|[b, T ]a(y)|dy = I + II.

The estimate of I follows the boundedness of [b, T ] on L ² (IR ⁿ ) (see [6]) and the size condition of atom a. That is,

I ≤ C|B|

 1

|2B|

Z

2B

|[b, T ]a(y)| ² dy

 1/2

≤ Ckbk BM O |B|( 1

|B|

Z

B

|a(y)| ² dy) ^1/2

≤ Ckbk BM O |B| kak _L

2

(IR

ⁿ

) ≤ Ckbk BM O .

Now, to estimate II, we split [b, T ] as [b, T ]a = (b − b _B )T a − T ((b − b _B )a), then II ≤

Z

IR

ⁿ

\2B

|(b(x) − b _B )T a(x)|dx + Z

IR

ⁿ

\2B

|T ((b− _B )a)(x)|dx = III + IV.

By Lemma 8, we know that the kernel of T satisfies the kernel’s estimate of Calder´ on-Zygmund operator. Let B = B(x B , r) using the cancellation condition R

B a(y)dy = 0, then III ≤

Z

B

|a(y)|

Z

IR

ⁿ

\2B

|K(x, y) − K(x, x B )||b(x) − b _B |dxdy

= Z

B

|a(y)|

∞

X

j=1

Z

2

^j

r≤|x−x

_B

|<2

^j+1

r

|y − x _B | ^δ

|x − x B | ^n+δ |b(x) − b _B |dxdy

≤ Z

B

|a(y)|dy

∞

X

j=1

2 ^−jδ

|2 ^j+1 B|

Z

2

^j+1

B

|b(x) − b B |dx

≤ C

∞

X

j=1

2 ^−jδ [ 1

|2 ^j+1 B|

Z

2

^j+1

B

|b(x) − b ₂

j+1

B |dx + |b ₂

j+1

B − b B |]

≤ C

∞

X

j=1

2 ^−jδ [kbk BM O + (j + 1)kbk BM O ]

≤ Ckbk BM O . By the definition of a, we have

Z

B

(b(y) − b B )a(y)dy = Z

B

a(y)b(y)dy − b B

Z

B

a(y) = 0,

Moreover, similar to Lemma 3.3 from [9, p.413], we obtain the estimate of the final part. That is,

IV = Z

IR

ⁿ

\2B

|T ((b − b B )a)(x)|dx = Z

IR

ⁿ

\2B

    Z

B

k(x, y)(b(y) − b B )a(y)dy    

dx

≤ Z

IR

ⁿ

\2B

    Z

B

(K(x, y) − K(x, x B ))(b(y) − b B )a(y)dy    

dx

≤ Z

B

|b(y) − b B )| |a(y)|

Z

IR

ⁿ

\2B

|K(x, y) − K(x, x B )|dxdy

≤ Z

B

|b(y) − b B ||a(y)|

∞

X

j=1

Z

2

^j

r≤|x−x

_B

|≤2

^j+1

r

|y − x _B | ^δ

|x − x B | ^n+δ dxdy

≤ Z

|b(y) − b _B ||a(y)|

∞

X

j=1

2 ^−jδ

|2 ^j+1 B|

Z

2

^j+1

B

≤

∞

X

j=1

2 ^−jδ Z

B

|b(y) − b _B ||a(y)|dy

≤ C

Z

B

|b(y) − b B |dy ≤ Ckbk BM O , where δ = 2 − n

q > 0. This concludes the proof of Theorem 5.

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Zhiping Wang Universit´ e de Gen` eve Department of Mathematics Dalian Maritime University Dalian, Liaoning, P. R. China 116026

[email protected]

Bolin Ma

College of Mathematics and Information Engineering

Jiaxing , Zhejiang, P. R. China 314001

[email protected]

Chuanmei Bi and Canqin Tang

Department of Mathematics

Dalian Maritime University

[email protected]