A note on end properties of Marcinkiewicz integral

A NOTE ON END PROPERTIES OF MARCINKIEWICZ INTEGRAL

Yong Ding

Abstract. In this note we give the mapping properties of the Marcinkiewicz integral µ

Ω

at some end spaces. More precisely, we first prove that µ

Ω

is a bounded operator from H

^1,∞

(R

ⁿ

) to L

^1,∞

(R

ⁿ

). As a corollary of the results above, we obtain again the weak type (1,1) boundedness of µ

Ω

, but the condition assumed on Ω is weaker than Stein’s condition. Finally, we show that µ

Ω

is bounded from BMO(R

ⁿ

) to BMO(R

ⁿ

). The results in this note are the ex- tensions of the results obtained by Lee and Rim recently.

1. Introduction

Suppose that S ⁿ⁻¹ is the unit sphere in R ⁿ (n ≥ 2) equipped with the Lebesgue measure dσ. Let Ω be a homogeneous function of degree zero on R ⁿ satisfying Ω ∈ L ¹ (S ⁿ⁻¹ ) and

(1.1)

Z

S

ⁿ⁻¹

Ω(x ⁰ )dσ(x ⁰ ) = 0,

where x ⁰ = x/|x| for any x 6= 0. Then the Marcinkiewicz integral oper- ator µ _Ω of higher dimension is defined by

µ _Ω (f )(x) = µ Z _∞

0

¯ ¯F _Ω,t (x) ¯

¯ ² dt t ³

¶ _1/2 , where

F Ω,t (x) = Z

|x−y|≤t

Ω(x − y)

|x − y| ⁿ⁻¹ f (y)dy.

Received August 9, 2004.

2000 Mathematics Subject Classification: 42B25, 42B30.

Key words and phrases: Marcinkiewicz integral, weak Hardy space, BMO.

The research was supported partly by NSF of China(Grant No. 10271016).

It is well known that the operator µ Ω was first defined by Stein in [9].

He proved that if Ω satisfies a Lip _α (0 < α ≤ 1) condition on S ⁿ⁻¹ , then µ _Ω is of type (p,p) for 1 < p ≤ 2 and of weak type (1,1). Subsequently Benedek, Calder´on and Panzone showed in [1] that if Ω ∈ C ¹ (S ⁿ⁻¹ ) then µ _Ω is of type (p,p) for 1 < p < ∞. In 2000, Ding, Fan and Pan[4] proved the following result:

Theorem A. Suppose that Ω ∈ H ¹ (S ⁿ⁻¹ ) and satisfies (1.1). Then for 1 < p < ∞, there exists a constant C > 0, independent of f, such that kµ Ω (f )k L

^p

≤ Ckf k L

^p

.

Here H ¹ (S ⁿ⁻¹ ) denotes the Hardy space on the unit sphere (see [2]

and [3] for the definition and properties of H ¹ (S ⁿ⁻¹ )). Note that the following relationship between H ¹ (S ⁿ⁻¹ ) and the other function spaces on S ⁿ⁻¹ :

(1.2) Lip _α (S ⁿ⁻¹ ) ⊂ L ^q (S ⁿ⁻¹ ) ⊂ L log ⁺ L(S ⁿ⁻¹ )

⊂ H ¹ (S ⁿ⁻¹ ) ⊂ L ¹ (S ⁿ⁻¹ ),

where all inclusions are proper for any 0 < α ≤ 1 and q > 1. Hence, Theorem A improved the results in [9] (for 1 < p ≤ 2)) and [1] mentioned above.

Recently, Lee and Rim[8] discussed the (H ¹ , L ¹ ), (L ² ∩ L ^∞ , BM O) and (L ^p , L ^p ) (1 < p < ∞) boundedness of µ Ω when there exist constants C > 0 and ρ > 1 such that

(1.3) |Ω(x ⁰ ) − Ω(y ⁰ )| ≤ C (log _|x

0

−y ¹

⁰

| ) ^ρ , holds uniformly in x ⁰ , y ⁰ ∈ S ⁿ⁻¹ .

It is easy to see that the condition (1.3) is weaker than the Lip α

(0 < α ≤ 1) condition on S ⁿ⁻¹ . In addition, it is also obvious if Ω satisfies (1.3) for some ρ > 1, then by (1.2)

(1.4) Ω ∈ L ^∞ (S ⁿ⁻¹ ) ⊂ L ^q (S ⁿ⁻¹ ) (1 < q < ∞) ⊂ H ¹ (S ⁿ⁻¹ ).

In this note we will consider the mapping properties of the Marcinkie-

wicz integral µ _Ω on some end function spaces when Ω satisfies (1.3)

for ρ > 2. More precise, we will first prove that µ _Ω is bounded from

H ^1,∞ (R ⁿ ) to L ^1,∞ (R ⁿ ). Here H ^1,∞ (R ⁿ ) denotes the weak Hardy space,

which was introduced first by Fefferman and Soria in [6]. To state our

results, let us recall the definition of H ^1,∞ (R ⁿ ).

Definition 1. Suppose that φ ∈ C ₀ ^∞ (R ⁿ ) with R

φ 6= 0. Denote f ₊ ^∗ (x) = sup _t>0 |(φ _t ∗ f )(x)|, where φ _t (x) = t ⁻ⁿ φ(x /t ). A function f is said to belong to the weak Hardy space H ^1,∞ (R ⁿ ) if f ₊ ^∗ ∈ L ^1,∞ (R ⁿ ), i.e., there exists a constant C > 0 such that for any β > 0

sup

β>0

β|{x ∈ R ⁿ : f ₊ ^∗ (x) > β}| ≤ C.

The smallest constant C satisfying the above inequality is called the H ^1,∞ (R ⁿ ) norm of f , which is denoted by kf k _H

1,∞

.

We have the following conclusions.

Theorem 1. Let Ω satisfy (1.1) and (1.3) for some ρ > 2. Then µ _Ω is bounded operator from H ^1,∞ (R ⁿ ) to L ^1,∞ (R ⁿ ). That is, there exists a constant C > 0 such that for any f ∈ H ^1,∞ (R ⁿ ) and β > 0,

(1.5) |{x : µ Ω f (x) > β}| ≤ Ckf k _H

^1,∞

/β.

Remark 1. Let M be the Hardy-Littlewood maximal operator. It is known that if φ ∈ C ₀ ^∞ (R ⁿ ) and f ∈ L ¹ (R ⁿ ), then f ₊ ^∗ (x) ≤ CM (f )(x) (see [10, pp. 62, Theorem 2]). Hence, by the weak (1, 1) boundedness of M , it is easy to see that the space L ¹ (R ⁿ ) is continuously embedded as a subspace of the space H ^1,∞ (R ⁿ ), and kf k _H

1,∞

≤ Ckf k _L

1

for any f ∈ L ¹ (R ⁿ ). Thus we get immediately the following corollary of Theorem 1.

Corollary. If Ω satisfy (1.1) and (1.3) for some ρ > 2, then µ Ω is of weak type (1, 1). That is, there exists a constant C > 0 such that for any f ∈ L ¹ (R ⁿ ) and β > 0, |{x : µ _Ω f (x) > β}| ≤ Ckf k _L

1

/β.

Remark 2. In [9], Stein proved that µ Ω is of weak (1,1) if Ω ∈ Lip _α (S ⁿ⁻¹ ) for 0 < α ≤ 1. In [5], Fan and Sato showed that if Ω ∈ L log ⁺ L(S ⁿ⁻¹ ), then µ Ω is also of weak (1,1). Hence, the condition (1.3) of Corollary is weaker than Stein’s condition but stronger than Fan-Sato’s condition.

The second result in this note shows that µ Ω is also bounded from BMO(R ⁿ ) to BMO(R ⁿ ).

Definition 2. A locally integrable function f (x) is said to belong to BMO(R ⁿ ) if

kf k BM O := sup

Q

1

|Q|

Z

Q

|f (x) − f Q |dx < ∞,

where the supremum is taken over all cubes in R ⁿ with sides parallel to the coordinate axes, and f _Q denotes the average of f over Q, i.e.

f _Q = _|Q| ¹ R

Q f (x)dx.

A well-known important fact on BMO(R ⁿ ) is that for any 1 ≤ p < ∞,

kf k _{BM O} ∼ sup

Q

µ 1

|Q|

Z

Q

|f (x) − f _Q | ^p dx

¶ _1/p

< ∞.

Theorem 2. Let Ω satisfy (1.1) and (1.3) for some ρ > 2. Suppose that f (x) ∈ BM O(R ⁿ ) and there is a measurable set E ⊂ R ⁿ with

|E| > 0 such that µ _Ω (f )(x) < ∞ for any x ∈ E. Then µ _Ω (f )(x) < ∞ a.e. on R ⁿ and

(1.6) kµ _Ω (f )k _{BM O} ≤ Ckf k _{BM O} , where the constant C is independent of f.

Remark 3. In [7], Han gave an example to show that there exist an Ω and f ₀ ∈ L ^∞ (R ⁿ ) such that µ _Ω (f ₀ )(x) = ∞ for any x ∈ R ⁿ . There- fore, µ _Ω can not map an L ^∞ (R ⁿ ) function into a BMO(R ⁿ ) function in general.

2. Proof of Theorem 1

We need the following Fefferman-Soria’s decomposition theorem of function in H ^1,∞ (R ⁿ ) (See [6]).

Theorem B. Given a function f ∈ H ^1,∞ (R ⁿ ), there exists a se- quence of bounded functions {f _k } ^∞ _k=−∞ with the following properties:

(a) f − P

|k|≤N f k tends to zero in the sense of distributions;

(b) Each f _k may be further decomposed as f _k = P

i h ^k _i in L ¹ and {h ^k _i } satisfies

(i) supp(h ^k _i ) ⊂ B _i ^k := B(x ^k _i , r ^k _i ), where B(x, r) denotes the ball in R ⁿ with the center at x and radius r. Moreover, P

i |B _i ^k | ≤ C 1 2 ^−k and P

i χ _B

k

i

(x) ≤ C, where C 1 ∼ kf k _H

^1,∞

; (ii) kh ^k _i k _∞ ≤ C2 ^k , where C is independent of i and k;

(iii) R

h ^k _i (x)dx = 0, for every i and k.

Now let us turn to the proof of Theorem 1. We need to show that

there exists a constant C > 0 such that (1.5) holds for any f ∈ H ^1,∞ (R ⁿ )

and β > 0. To do this, for any given β > 0, we take k 0 satisfying 2 ^k

⁰

≤ β < 2 ^k

⁰

⁺¹ , then by Theorem B we may write

f =

k

0

X

k=−∞

f _k + X ∞ k=k

0

+1

f _k := F ₁ + F ₂ and f _k = X

i

h ^k _i ,

where h ^k _i satisfies (i), (ii) and (iii). Now denote A _k = suppf _k , then A _k =

∪ _i B _i ^k and |A _k | ≤ P

i |B _i ^k | ≤ C2 ^−k kf k _H

1,∞

. Note that kf _k k _∞ ≤ C2 ^k , we have

kF ₁ k ₂ ≤

k

0

X

k=−∞

kf _k k ₂ ≤ C

k

0

X

k=−∞

2 ^k |A _k | ^1/2

≤ C

k

₀

X

k=−∞

2 ^k/2 kf k ^1/2 _H

1,∞

≤ Ckf k ^1/2 _H

1,∞

β ^1/2 .

Using (1.4) and Theorem A, it is easy to see that

|{x : µ _Ω (F ₁ )(x) > β}| ≤ kµ _Ω (F ₁ )k ² ₂ /β ²

≤ CkF 1 k ² ₂ /β ²

≤ Ckf k _H

1,∞

/β.

(2.1)

On the other hand, let

B ¯ _i ^k = B(x ^k _i , 2(3/2) ^(k−k

⁰

^)/n r _i ^k ) and B k

₀

= [ ∞ k=k

0

+1

[

i

B ¯ ^k _i .

We have

(2.2)

|B _k

₀

| ≤ X

k=k

0

+1

X

i

| ¯ B _i ^k |

= X

k=k

₀

+1

X

i

2 ⁿ (3/2) ^k−k

⁰

|B _i ^k |

≤ C X ∞ k=k

0

+1

(3/2) ^k−k

⁰

2 ^−k kf k _H

1,∞

≤ Ckf k _H

^1,∞

/β.

Thus, to prove (1.5) it suffices to show

(2.3) |{x ∈ (B _k

₀

) ^c : µ _Ω (F ₂ )(x) > β}| ≤ Ckf k _H

1,∞

/β.

Using the Minkowski inequality, we get Z

(B

_k0

)

^c

µ _Ω (F ₂ )(x)dx

= Z

(B

_k0

)

^c

µ Z _∞

0

¯ ¯

¯ ¯ X ∞ k=k

₀

+1

X

i

Z

|x−y|≤t

Ω(x − y)

|x − y| ⁿ⁻¹ h ^k _i (y)dy

¯ ¯

¯ ¯

2 dt t ³

¶ _1/2 dx

≤ Z

(B

_k0

)

^c

X ∞ k=k

0

+1

X

i

µ Z _∞

0

¯ ¯

¯ ¯ Z

|x−y|≤t

Ω(x − y)

|x − y| ⁿ⁻¹ h ^k _i (y)dy

¯ ¯

¯ ¯

2 dt t ³

¶ _1/2 dx

≤ C X ∞ k=k

₀

+1

X

i

(I 1 + I 2 ), where

I 1 = Z

(B

_k0

)

^c

µ Z _|x−x

^k

i

|+2r

^k_i

0

¯ ¯

¯ ¯ Z

|x−y|≤t

Ω(x − y)

|x − y| ⁿ⁻¹ h ^k _i (y)dy

¯ ¯

¯ ¯

2 dt t ³

¶ _1/2 dx and

I ₂ = Z

(B

_k0

)

^c

µ Z _∞

|x−x

^k_i

|+2r

^k_i

¯ ¯

¯ ¯ Z

|x−y|≤t

Ω(x − y)

|x − y| ⁿ⁻¹ h ^k _i (y)dy

¯ ¯

¯ ¯

2 dt t ³

¶ _1/2 dx.

It is easy to check that when y ∈ B _i ^k , for any x ∈ (B k

₀

) ^c , |x − x ^k _i | ∼

|x − y| ∼ |x − x ^k _i | + 2r _i ^k , and (2.4)

¯ ¯

¯ ¯ 1

(|x − x ^k _i | + 2r _i ^k ) ² − 1

|x − y| ²

¯ ¯

¯ ¯ ≤ C r _i ^k

|x − y| ³ . Hence by Ω ∈ L ^∞ (S ⁿ⁻¹ ) and (2.4), we have

I 1 ≤ Z

(B

_k0

)

^c

Z

B

_i^k

|Ω(x − y)|

|x − y| ⁿ⁻¹ |h ^k _i (y)|

µ Z

|x−y|≤t≤|x−x

^k_i

|+2r

_i^k

1 t ³ dt

¶ _1/2 dydx

≤ C2 ^k kΩk _L

∞

(S

ⁿ⁻¹

)

× Z

(B

_k0

)

^c

Z

B

^k_i

1

|x − y| ⁿ⁻¹

¯ ¯

¯ ¯ 1

(|x − x ^k _i | + 2r ^k _i ) ² − 1

|x − y| ²

¯ ¯

¯ ¯

1/2

dydx

≤ C2 ^k (r ^k _i ) ^1/2 Z

B

^k_i

Z _∞

2(3/2)

(k−k0)/n

r

^k_i

1

s ^1+1/2 dsdy

≤ C2 ^k (r ^k _i ) ^1/2 |B _i ^k | ¡ 2 3

¢ _(k−k

₀

_)/(2n)

(r ^k _i ) ^−1/2 .

Thus (2.5)

X ∞ k=k

0

+1

X

i

I ₁ ≤ C X ∞ k=k

0

+1

( 2

3 ) ^(k−k

⁰

^)/(2n) kf k _H

1,∞

≤ Ckf k _H

1,∞

. Now let us consider I 2 . As above we know that if x ∈ (B k

₀

) ^c , y ∈ B _i ^k , then |x − x ^k _i | ∼ |x − y| ∼ |x − x ^k _i | + 2r _i ^k , and

(2.6)

¯ ¯

¯ ¯ x − y

|x − y| − x − x ^k _i

|x − x ^k _i |

¯ ¯

¯ ¯ ≤ C r ^k _i

|x − x ^k _i | . Thus by (1.3) and (2.6), we get

(2.7)

¯ ¯Ω(x − y) − Ω(x − x ^k _i ) ¯

¯ =

¯ ¯

¯ ¯Ω

µ x − y

|x − y|

¶

− Ω

µ x − x ^k _i

|x − x ^k _i |

¶¯ ¯

¯ ¯

≤ C

¡ log ^|x−x _r

k^ki

^|

i

¢ _ρ .

Applying (2.7) we have (2.8) ¯

¯ ¯

¯ Ω(x − y)

|x − y| ⁿ⁻¹ − Ω(x − x ^k _i )

|x − x ^k _i | ⁿ⁻¹

¯ ¯

¯ ¯

≤

¯ ¯Ω(x − y) − Ω(x − x ^k _i ) ¯

¯

|x − x ^k _i | ⁿ⁻¹ + |Ω(x − y)|

¯ ¯

¯ ¯ 1

|x − y| ⁿ⁻¹ − 1

|x − x ^k _i | ⁿ⁻¹

¯ ¯

¯ ¯

≤ C

|x − x ^k _i | ⁿ⁻¹ ¡

log ^|x−x _r

_k^ki

^|

i

¢ _ρ + r _i ^k

|x − x ^k _i | ⁿ

≤ C

|x − x ^k _i | ⁿ⁻¹ ¡

log ^|x−x _r

_k^ki

^|

i

¢ _ρ .

Now let us return to the estimate of I 2 . Note that when y ∈ B _i ^k , for any x ∈ (B _k

₀

) ^c and t > |x − x ^k _i | + 2r ^k _i , we have B _i ^k ⊂ {y ∈ R ⁿ : |x − y| ≤ t}.

By the cancellation property of h ^k _i (y) and (2.8), we get I ₂ =

Z

(B

_k0

)

^c

µ Z _∞

|x−x

^k_i

|+2r

^k_i

¯ ¯

¯ ¯ Z

|x−y|≤t

µ Ω(x − y)

|x − y| ⁿ⁻¹ − Ω(x − x ^k _i )

|x − x ^k _i | ⁿ⁻¹

¶

× h ^k _i (y)dy

¯ ¯

¯ ¯

2 dt t ³

¶ _1/2

dx

≤ C2 ^k Z

(B

_k0

)

^c

Z

B

^k_i

¯ ¯

¯ ¯ Ω(x − y)

|x − y| ⁿ⁻¹ − Ω(x − x ^k _i )

|x − x ^k _i | ⁿ⁻¹

¯ ¯

¯ ¯

× µ Z

|x−y|≤t,|x−x

^k_i

|<t

1 t ³ dt

¶ _1/2 dydx

≤ C2 ^k Z

B

^k_i

Z

|x−x

^k_i

|>(3/2)

^(k−k0)/n

r

^k_i

1

|x − x ^k _i | ⁿ⁻¹ ¡

log ^|x−x _r

_k^ki

^|

i

¢ _ρ

· 1

|x − x ^k _i | dxdy

≤ C2 ^k |B _i ^k | Z _∞

(3/2)

^(k−k0)/n

r

^k_i

ds s ¡

log _r ^s

k i

¢ _ρ ds.

Note that ρ > 2

(2.9)

Z _∞

(3/2)

^(k−k0)/n

r

^k_i

ds s ¡

log _r ^s

_k

i

¢ _ρ ds

= X ∞ j=1

Z _(3/2)

(j+1)(k−k0)/n

(3/2)

^j(k−k0)/n

ds s(log s) ^ρ ds

≤ X ∞ j=1

¡ 1

log(3/2) ^j(k−k

⁰

^)/n ¢ _ρ

Z _(3/2)

(j+1)(k−k0)/n

(3/2)

^j(k−k0)/n

ds s

= X ∞ j=1

log(3/2) ^(k−k

⁰

^)/n

¡ log(3/2) ^j(k−k

⁰

^)/n ¢ _ρ

≤ C(k − k ₀ ) ^1−ρ . We therefore get by (2.9)

(2.10) I 2 ≤ C2 ^k |B _i ^k |(k − k 0 ) ^1−ρ . Thus

(2.11)

X ∞ k=k

₀

+1

X

i

I ₂ ≤ C X ∞ k=k

₀

+1

(k − k ₀ ) ^1−ρ kf k _H

1,∞

≤ Ckf k _H

1,∞

.

By (2.5) and (2.11), we have (2.12)

Z

(B

_k0

)

^c

µ Ω (F 2 )(x)dx ≤ Ckf k _H

^1,∞

.

From (2.12) we get (2.3). Hence we finish the proof of Theorem 1.

3. Proof of Theorem 2

Let us begin by recalling a known result.

Lemma 1. Suppose that 1 ≤ p < ∞, η > 0 and f ∈ BM O(R ⁿ ).

Then there exists a C = C(n, p, η) > 0 such that for any cube Q with its center at x ₀ and side length d,

Z

R

ⁿ

d ^η |f (x) − f _Q | ^p

d ^n+η + |x − x 0 | ^n+η dx ≤ Ckf k ^p _{BM O} .

Now let us turn to the proof of Theorem 2. Take any density point ¯ x of E and any cube Q with center at ¯ x, and denote by d the side length of Q. First we show that µ _Ω (f )(x) < ∞ a.e. on Q. To do this, we denote Q ^∗ = 16Q to be the sixteen times extension of Q with its center at ¯ x.

Decompose f (x) as

(3.1) f (x) = f _Q

^∗

+ (f (x) − f _Q

^∗

)χ _Q

^∗

+ (f (x) − f _Q

^∗

)(1 − χ _Q

^∗

) := f ₁ (x) + f ₂ (x) + f ₃ (x).

By (1.1) we get µ Ω (f 1 )(x) ≡ 0. From (1.4) and Theorem A, we obtain Z

Q

|µ Ω (f 2 )(x)| ² dx ≤ Z

R

ⁿ

|µ Ω (f 2 )(x)| ² dx

≤ C Z

R

ⁿ

|f 2 (x)| ² dx

≤ C|Q| kf k ² _{BM O} . Hence

Z

Q

|µ _Ω (f ₂ )(x)|dx ≤ |Q| ^1/2 µ Z

Q

|µ _Ω (f ₂ )(x)| ² dx

¶ _1/2

≤ C|Q| kf k _{BM O} . (3.2)

This shows that µ _Ω (f ₂ )(x) < ∞ a.e. on Q. Since |E| > 0, we have

|Q ∩ E| > 0. Hence there exists an x ₀ ∈ Q ∩ E such that µ _Ω (f )(x ₀ ) < ∞ and µ Ω (f 2 )(x 0 ) < ∞ hold at the same time, and we may chose x 0 to close to ¯ x enough, say, |x ₀ − ¯ x| < d/4. Thus

(3.3) µ _Ω (f ₃ )(x ₀ ) ≤ µ _Ω (f )(x ₀ ) + µ _Ω (f ₂ )(x ₀ ) < ∞.

On the other hand, for any x ∈ Q we have (3.4)

|µ _Ω (f ₃ )(x) − µ _Ω (f ₃ )(x ₀ )|

≤ µ Z _∞

0

¯ ¯

¯ ¯ Z

|x−y|≤t

Ω(x − y)f ₃ (y)

|x − y| ⁿ⁻¹ dy

− Z

|x

0

−y|≤t

Ω(x 0 − y)f 3 (y)

|x ₀ − y| ⁿ⁻¹ dy

¯ ¯

¯ ¯

2 dt t ³

¶ _1/2

≤ µ Z _∞

0

µ Z

|x−y|≤t

|x

0

−y|>t

¯ ¯

¯ ¯ Ω(x − y)

|x − y| ⁿ⁻¹ f ₃ (y)

¯ ¯

¯ ¯dy

¶ ₂ dt t ³

¶ _1/2

+ µ Z _∞

0

µ Z

|x−y|>t

|x

₀

−y|≤t

¯ ¯

¯ ¯ Ω(x ₀ − y)

|x ₀ − y| ⁿ⁻¹ f ₃ (y)

¯ ¯

¯ ¯dy

¶ ₂ dt t ³

¶ _1/2

+ µ Z _∞

0

µ Z

|x−y|≤t

|x

₀

−y|≤t

¯ ¯

¯ ¯ Ω(x − y)

|x − y| ⁿ⁻¹ − Ω(x 0 − y)

|x ₀ − y| ⁿ⁻¹

¯ ¯

¯ ¯|f ³ (y)|dy

¶ ₂ dt t ³

¶ _1/2

:= I 1 (x) + I 2 (x) + I 3 (x).

Below we shall give the estimates of I ₁ , I ₂ and I ₃ , respectively. For I ₁ , we have

I 1 (x)

≤ Z

R

ⁿ

|Ω(x − y)|

|x − y| ⁿ⁻¹ |f ₃ (y)|

µ Z

|x−y|≤t

|x

₀

−y|>t

dt t ³

¶ _1/2 dy

≤ CkΩk _L

∞

(S

ⁿ⁻¹

)

Z

(Q

^∗

)

^c

1

|x − y| ⁿ⁻¹ |f ₃ (y)|

¯ ¯

¯ ¯ 1

|x − y| ² − 1

|x ₀ − y| ²

¯ ¯

¯ ¯

1/2

dy.

Note that y ∈ (Q ^∗ ) ^c and x, x ₀ ∈ Q, we have |x − y| ∼ |x ₀ − y| ∼ |¯ x − y|

and ¯

¯ ¯

¯ 1

|x − y| ² − 1

|x ₀ − y| ²

¯ ¯

¯ ¯ ≤ Cd

|x − y| ³ . From this and applying H¨older’s inequality, we get

(3.5)

I ₁ (x) ≤ CkΩk _L

∞

(S

ⁿ⁻¹

)

Z

(Q

^∗

)

^c

d ^1/2 |f ₃ (y)|

|x − y| ^n+1/2 dy

≤ CkΩk _L

∞

(S

ⁿ⁻¹

)

µ Z

(Q

^∗

)

^c

d ^1/2

|x − y| ^n+1/2 dy

¶ _1/2

× µ Z

(Q

^∗

)

^c

d ^1/2 |f 3 (y)| ²

|x − y| ^n+1/2 dy

¶ _1/2

.

It is easy to check that

d ^n+1/2 + |x − y| ^n+1/2 ≤ 2|x − y| ^n+1/2 for x ∈ Q and y ∈ (Q ^∗ ) ^c . Hence by (3.5) and Lemma 1

(3.6) I 1 (x) ≤ C µ Z

(Q

^∗

)

^c

d ^1/2 |f 3 (y)| ²

d ^n+1/2 + |x − y| ^n+1/2 dy

¶ _1/2

≤ Ckf k _{BM O} .

Similarly, we have I ₂ (x) ≤ Ckf k _{BM O} in the same way of estimating I 1 (x). Now let us consider I 3 (x). Using H¨older’s inequality again, we get

I ₃ (x) ≤ C Z

(Q

^∗

)

^c

¯ ¯

¯ ¯ Ω(x − y)

|x − y| ⁿ⁻¹ − Ω(x ₀ − y)

|x 0 − y| ⁿ⁻¹

¯ ¯

¯ ¯|f ³ (y)|

× µ Z

|x−y|≤t,|x

₀

−y|≤t

dt t ³

¶ _1/2 dy

≤ C Z

(Q

^∗

)

^c

¯ ¯

¯ ¯ Ω(x − y)

|x − y| ⁿ⁻¹ − Ω(x ₀ − y)

|x ₀ − y| ⁿ⁻¹

¯ ¯

¯ ¯ |f ₃ (y)|

|x ₀ − y| dy.

Similar to the estimates of (2.6)–(2.8), we get I 3 (x) ≤ C

Z

(Q

^∗

)

^c

|f 3 (y)|dy

|y − x ₀ | ⁿ ¡

log ^|y−x _d

⁰

^| ¢ _ρ . Note that y ∈ (Q ^∗ ) ^c and x ₀ , ¯ x ∈ Q, and hence

|y − x 0 | ≥ |y − ¯ x| − |¯ x − x 0 | ≥ 8d − d/4 > 4d.

Thus

I 3 (x) ≤ C X ∞ j=2

Z

2

^j

d≤|y−x

0

|<2

^j+1

d

|f 3 (y)|dy

|y − x ₀ | ⁿ ¡

log ^|y−x _d

⁰

^| ¢ _ρ .

≤ C X ∞ j=2

µ Z

2

^j

d≤|y−x

0

|<2

^j+1

d

dy

|y − x ₀ | ⁿ ¡

log ^|y−x _d

⁰

^| ¢ _2ρ

¶ _1/2

× µ Z

2

^j

d≤|y−x

0

|<2

^j+1

d

|f 3 (y)| ² dy

|y − x ₀ | ⁿ

¶ _1/2

.

Observe

(3.7)

µ Z

2

^j

d≤|y−x

₀

|<2

^j+1

d

dy

|y − x 0 | ⁿ ¡

log ^|y−x _d

⁰

^| ¢ _2ρ

¶ _1/2

≤ C

µ Z ₂

^j+1

_d

2

^j

d

ds s ¡

log(s/d) ¢ _2ρ

¶ _1/2

≤ C

¡ log 2 ^j ¢ _ρ

µ Z ₂

^j+1

2

^j

ds s

¶ _1/2

≤ C j ^ρ .

On the other hand, denote by Q j the 2 ^j times extension of Q, then we have

(3.8) µ Z

2

^j

d≤|y−x

0

|<2

^j+1

d

|f 3 (y)| ² dy

|y − x ₀ | ⁿ

¶ _1/2

≤

µ 1

(2 ^j d) ⁿ Z

2

^j

d≤|y−x

₀

|<2

^j+1

d

|f ₃ (y)| ² dy

¶ _1/2

≤

µ 1

(2 ^j d) ⁿ Z

2

^j

d≤|y−x

0

|<2

^j+1

d

(|f (y) − f _Q

_j+1

| ² + |f _Q

_j+1

− f _Q

^∗

| ² )dy

¶ _1/2

≤ C

µ 1

|Q j+1 | Z

Q

j+1

|f (y) − f Q

_j+1

| ² dy + (j + 1) ² 2 ²ⁿ kf k ² _{BM O}

¶ _1/2 . Thus by (3.7) and (3.8) and note that ρ > 2

(3.9) I 3 (x) ≤ C X ∞ j=2

1 j ^ρ

£ 1 + (j + 1)2 ⁿ ¤

kf k BM O ≤ Ckf k BM O .

Summing up (3.4), (3.6) and (3.9), we get

(3.10) |µ _Ω (f ₃ )(x) − µ _Ω (f ₃ )(x ₀ )| ≤ Ckf k _{BM O} . Thus we prove that

µ _Ω (f )(x) ≤ µ _Ω (f ₁ )(x) + µ _Ω (f ₂ )(x)

+ |µ _Ω (f ₃ )(x) − µ _Ω (f ₃ )(x ₀ )| + µ _Ω (f ₃ )(x ₀ )

< ∞ a.e. on Q.

Because Q is any cube with center at ¯ x ∈ E, we get actually µ Ω (f )(x) <

∞ a.e. on R ⁿ .

Finally, let us show that (1.6) holds. In fact, from the process of the above proof we know that for any cube Q ⊂ R ⁿ , there exists an x 0 ∈ Q such that µ _Ω (f ₃ )(x ₀ ) < ∞. Let Q ^∗ = 16Q and write f = f ₁ +f ₂ +f ₃ as in (3.1). From the process of the above proof, we know that µ Ω (f 1 )(x) = 0 on Q, and R

Q |µ _Ω (f ₂ )(x)|dx ≤ C|Q| kf k _{BM O} . And by (3.2) and (3.10) we get

(3.11)

Z

Q

|µ _Ω (f )(x) − µ _Ω (f ₃ )(x ₀ )|dx

≤ Z

Q

|µ _Ω (f ₂ )(x)|dx + Z

Q

|µ _Ω (f ₃ )(x) − µ _Ω (f ₃ )(x ₀ )|dx

≤ C|Q| kf k _{BM O} .

Using (3.10) and (3.11) again, we have Z

Q

|µ _Ω (f )(x) − (µ _Ω (f )) _Q |dx

≤ Z

Q

|µ _Ω (f )(x) − µ _Ω (f ₃ )(x ₀ )|dx + Z

Q

|µ _Ω (f ₃ )(x ₀ ) − (µ _Ω (f )) _Q |dx

≤ C|Q|kf k _{BM O} + Z

Q

|µ _Ω (f )(x) − µ _Ω (f ₃ )(x ₀ )|dx

≤ C|Q|kf k _{BM O} .

Thus we obtain (1.6) and complete the proof of Theorem 2.

Acknowledgement. The author is grateful to the referee for his comments and valuable suggestions.

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Department of Mathematics and Computer Science Nanchang Institute of Aeronautical Technology Nanchang, 330034, P.R.China

and

(Corresponding Address)

Department of Mathematics

Beijing Normal University

Beijing 100875, P.R.China

E-mail: [email protected]