Mapping properties of the Marcinkiewicz integrals on homogeneous groups

MAPPING PROPERTIES OF THE MARCINKIEWICZ INTEGRALS ON HOMOGENEOUS GROUPS

Youngwoo Choi and Kyung Soo Rim

Abstract. Under the cancellation property and the Lipschitz con- dition on kernels, we prove that the Marcinkiewicz integrals de- fined on a homogeneous group H are bounded from H

(H) to L

(H), from L

(H) to BM O (H), and from L

(H) to L

(H) for 1 < p < ∞ assuming the L

-boundedness for some q > 1.

1. Introduction

Stein [8] defined a higher dimensional analogue of the Marcinkiewicz integral by

(1.1) µ _Ω f (x) =

Z ∞ 0

|F t (x)| ² dt t ³



, where

F t (x) = Z

|y−x|<t

Ω (x − y)

|x − y| ⁿ⁻¹ f (y) dy,

Ω is a homogeneous function of degree zero whose restriction to S ⁿ⁻¹ belongs to Λ ^α S ⁿ⁻¹
and satisfies the cancellation property,

Z

S

Ω(x ^′ ) dσ x ^′

= 0.

Here, S ⁿ⁻¹ denotes the unit sphere in R ⁿ and Λ ^α S ⁿ⁻¹
denotes the Lipschitz space of order α on S ⁿ⁻¹ . The continuity of Marcinkiewicz

Received June 26, 2000. Revised April 19, 2001.

2000 Mathematics Subject Classification: Primary 42B20; Secondary 47B38.

Key words and phrases: Marcinkiewicz integrals, homogeneous groups.

The first author was supported by grant No. 1999-2-112-004-3 from the Interdis-

ciplinary Research program of the KOSEF.

integrals is very useful in harmonic analysis [9, 10, 15]. Stein [8] proved that if Ω is in Λ ^α S ⁿ⁻¹
with 0 < α ≤ 1, then

(1.2)

 {x ∈ R ⁿ : µ Ω f (x) > λ}

   ≤ C

λ kf k _L

and

(1.3) kµ Ω f k _L

≤ C p kf k _L

,

where 1 < p ≤ 2, and if Ω is an integrable odd function, then (1.4) kµ Ω f k _L

≤ C p kf k _L

for 2 < p < ∞.

The problem most immediately suggested by Marcinkiewicz [7], who conjectured the L ^p -boundedness of (1.1) for n = 1 and for Ω(t) = sign t until Zygmund [14] proved that the conjecture holds for 1 < p < ∞, has been extensively studied beginning with the 1958’s article of Stein [8].

Benedek, Calderon and Panzone [1] proved that if Ω ∈ C ¹ (R ⁿ \ {0}) is a homogeneous function of degree zero satisfying the cancellation property, then µ Ω is bounded on L ^p (H) for 1 < p < ∞. Torchinsky and Wang considered the weighted L ^p -boundedness of µ _Ω , and showed that if Ω is in Λ ^α S ^Q−1
and µ Ω is bounded on L ^p (R ⁿ ) for 1 < p < ∞, then for ω satisfying an A p condition, µ _Ω is bounded on L ^p (ω) [13]. Further results on (1.1) were obtained when Ω satisfies some smoothness conditions [1], [8] and [13].

In this paper, we prove the H ¹ -L ¹ , L ^∞ _c -BM O and L ^p -L ^p (1 < p < ∞) boundedness of Marcinkiewicz integrals defined on homogeneous groups under the cancellation property and the Lipschitz condition on Ω and under the L ^q -boundedness of µ _Ω .

This paper is organized as follows: in the next section, some prelim- inary materials are introduced. The main theorem is stated in Section 3. End-point results appear in Sections 4 and 5. Combined with an in- terpolation argument, the L ^p boundedness for 1 < p < ∞ will be shown in Section 6.

2. Preliminaries and notations

In this section, we introduce notations related to homogeneous groups

along with some preliminary materials. Mainly, we follow [6].

2.1. Homogeneous groups

A nilpotent Lie group H with a dilation group {δ r } _r>0 is said to be a homogeneous group. The dilation group is given by

δ _r = exp (A ln r)

with a suitable matrix A having positive eigenvalues.

H has a natural homogeneous norm | · |, and the homogeneous di- mension Q. Abusing the notation, the bi-invariant measure on H will be denoted by | · |.

Remark 2.1. Let H, {δ r }, | · | and Q be as above. The eigenvalues of the matrix defining {δ _r } are listed as 1 = d 1 ≤ d 2 ≤ · · · ≤ d n and we let ¯ d = max {d i : i = 1, · · · , n}.

1. |δ r x| = r|x| for each x ∈ H, r > 0.

2. There exist C ₁ , C ₂ > 0 such that

C ₁ kxk ≤ |x| ≤ C 2 kxk ^{d ¯} whenever |x| ≤ 1.

Here, k·k denotes the euclidean norm.

3. There exists a constant γ > 0 such that for all x, y ∈ H,

|x ◦ y| ≤ γ(|x| + |y|) for all x ∈ H (2.1)

 |x ◦ y| − |x| 

 ≤ γ|y| for all x, y ∈ H such that |y| ≤ |x|/2.

(2.2)

4. |δ r E| = r ^Q |E|.

5. We let

S = {x ∈ H : |x| = 1} .

There is a unique Radon measure σ on S such that for all f ∈ L ¹ (H),

Z

H

f (x) dx = Z ∞

0

Z

S

f (δ r y)r ^Q−1 dσ(y) dr.

2.2. The Hardy space H ¹ (H)

For the definition of the Hardy space H ¹ (H), we refer the interested

readers to [6].

2.2.1. H _q,0 ¹ -atoms. Let q ∈ (1, ∞]. A function a(x) on H is said to be an H _q,0 ¹ -atom (associated to a ball B) if it satisfies the following conditions:

(a) a(x) is supported in ¯ B;

(b) |a(x)| ≤ |B|

⁻¹ almost everywhere; and (c) R

H a(x) dx = 0.

Remark 2.2. Let a(x) be an H _∞,0 ¹ -atom. Then we have kak _L

(H) ≤ 1.

(2.3)

2.3. Atomic decomposition

An equivalent way of looking at H ¹ (H) is the decomposition of ele- ments in H ¹ (H) into H _q,0 ¹ -atoms.

Theorem 2.1 (Decomposition Theorem). Let q ∈ (1, ∞]. For f ∈ H ¹ (H) there exist a collection of H _q,0 ¹ atoms {a k } _k∈N and a sequence of nonnegative real numbers {λ k } _k∈N with P _∞

k=1 λ _k < ∞ so that f =

∞

X

k=1

λ _k a _k

in the sense of distributions, and we have kf k _H

≈

∞

X

k=1

λ k .

2.4. BMO

Definition 2.3. A locally integrable function f : H → C is said to be in BM O if there exists a constant C such that for each ball B

1

|B|

Z

B

|f (x) − f B | dx ≤ C, holds, where

f _B = 1

|B|

Z

B

f (x) dx.

3. Marcinkiewicz integrals

Let Ω be a measurable function on a homogeneous group H, which is homogeneous of degree 0 in the sense that

Ω (δ r x) = Ω(x)

holds for a.e. x ∈ H \ {0} and r > 0. We define the Marcinkiewicz integral µ _Ω f as follows:

(3.1) µ _Ω f (x) =

Z _∞

0

|F t (x)| ² dt t ³



, where

F t (x) = Z

B

(x)

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 f (y) dy.

We will study the mapping properties of µ _Ω . To be more specific, we will prove:

Theorem 3.1. Let H, Ω and µ _Ω be as above. We assume the follow- ing:

• Ω 

 S ∈ Λ ^α (S);

• R

S Ω (x ^′ ) dσ (x ^′ ) = 0; and

• µ Ω is bounded in L ^q (H) for some q > 1.

Then the following inequalities hold:

kµ _Ω f k _L

≤ C ₁ kf k _H

, f ∈ H ¹ (H) (3.2)

kµ Ω f k _{BM O} ≤ C ∞ kf k _L

, f ∈ L ^∞ _c (H) (3.3)

and

kµ _Ω f k _L

≤ C p kf k _L

, f ∈ L ^p (H) (3.4)

for 1 < p < ∞.

4. H ¹ -L ¹ boundedness

In this section, we establish H ¹ -L ¹ boundedness of the Marcinkiewicz

integral. In view of Theorem 2.1 and the sublinearity of µ Ω , it suffices to

verify the inequality (3.2) when f is an arbitrary H _∞,0 ¹ -atom. Let a(x)

be an H _∞,0 ¹ -atom supported in B r (x ₀ ). We split the integral into two parts,

Z

H

µ _Ω a(x) dx = Z

B

(x

)

µ _Ω a(x) dx + Z

H \B

(x

)

µ _Ω a(x) dx

≡ I + II.

4.1. Estimation on I By hypothesis, we have (4.1)

Z

B

(x

)

[µ _Ω a(x)] ^q dx . Z

B

(x

)

|a(x)| ^q dx . |B _r (x ₀ )| ^−q+1 . By H¨ older’s inequality and (4.1), we obtain

I ≤ Z

B

(x

)

[µ _Ω a(x)] ^q dx

!

q

|B 2λr (x ₀ )|

. 1.

4.2. Estimation on II

Before we proceed, we introduce some simple facts on balls in H.

Definition 4.1. For E ⊂ H and x 6∈ E, we will use the following notation.

d(x, E) = inf 

 x ◦ y ⁻¹ 

 : y ∈ E . Lemma 4.2. Let x 6∈ B _2λr (x 0 ). Then we have

d (x, B r (x ₀ )) ≥ r.

Proof. Suppose

d (x, B r (x ₀ )) < r.

Then there exists y ∈ B _r (x ₀ ) such that 

y ◦ x ⁻¹ 

 < r. So we get 

 x ◦ x ⁻¹ ₀ 

 ≤ λ 

 x ◦ y ⁻¹   + 

 y ◦ x ⁻¹ ₀  

< 2λr.

A contradiction to x 6∈ B _2λr (x ₀ ).

Lemma 4.3. Let x 6∈ B _2λr (x ₀ ) and y ∈ B r (x ₀ ). Then we have 

 x ◦ y ⁻¹ 

 ≤ 2λ 

 x ◦ x ⁻¹ ₀ 

 ≤ 4λ ² 

 x ◦ y ⁻¹ 

 .

Proof. Observe that  x ◦ y ⁻¹ 

 ≤ λ 

x ◦ x ⁻¹ ₀   + 

x ₀ ◦ y ⁻¹  

≤ 2λ 

 x ◦ x ⁻¹ ₀  

≤ 2λ ² 

 x ◦ y ⁻¹   + 

 y ◦ x ⁻¹ ₀  

≤ 4λ ² 

 x ◦ y ⁻¹   , from

 x ◦ y ⁻¹ 

 ≥ d (x, B r (x ₀ )) ≥ r ≥ 

 y ◦ x ⁻¹ ₀   .

Lemma 4.4. Let x 6∈ B _2λr (x ₀ ) and t < d (x, B r (x ₀ )). Then we have B r (x ₀ ) ∩ B t (x) = ∅.

Proof. If y ∈ B r (x 0 ), then we have 

 y ◦ x ⁻¹ 

 ≥ d (x, B r (x 0 )) ≥ t.

Also, observe the following fact.

Fact 4.5. There exist constants C > 0, ε ∈ (0, 1) and ρ > 0 such that

 δ s x ◦ x ⁻¹ 

 ≤ C|1 − s| ^ρ |x|

uniformly in x ∈ H and |1 − s| < ε.

Fix x ∈ H \ B _2λr (x ₀ ). Then, by Lemma 4.2 we have d (x, B _r (x ₀ )) ≤ 

x ◦ x ⁻¹ ₀ 

 ≤ 2λd (x, B r (x ₀ )) . We have

µ _Ω a(x) ² = Z ∞

d(x,B

(x

))

     Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 a(y) dy     

2 dt t ³

= Z _∞

d(x,B

(x

))

     Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

t |x ◦ y ⁻¹ | ^Q−1 a(y) dy     

· J t a(x) dt t ² , where

J t a(x) =      Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 a(y) dy

.

Lemma 4.6. Let Ω, a, and B r (x ₀ ) be as above. Then we have 

    Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

t |x ◦ y ⁻¹ | ^Q−1 a(y) dy     

. Ma(x), whenever t > d (x, B r (x ₀ )), x ∈ H \ B _2λr (x ₀ ).

Proof. There are two cases.

Case 1. t ≤ 2d (x, B r (x ₀ )).

     Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

t |x ◦ y ⁻¹ | ^Q−1 a(y) dy     

. kΩk _∞

t · d (x, B _r (x ₀ )) ^Q−1 Z

B

(x)

|a(y)| dy . kΩk _∞

t ^Q Z

B

(x)

|a(y)| dy . Ma(x).

Case 2. t ≥ 2 (d (x, B _r (x ₀ ))). From B _r (x ₀ ) ∩ B _t (x) ⊂ B _d(x,B

_(x

₎₎ (x), we have

     Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

t |x ◦ y ⁻¹ | ^Q−1 a(y) dy     

. kΩk _∞

d (x, B _r (x ₀ )) ^Q Z

B

(x

)

|a(y)| dy . Ma(x),

which completes the proof.

For J t a(x) we have the following:

Lemma 4.7. With Ω, a, and B _r (x ₀ ) as above,

J t a(x) .



 

 

 

 

tMa(x)

if d (x, B r (x 0 )) ≤ t ≤ λ (d (x, B r (x 0 )) + 2λr) r ^ν 

 x ◦ x ⁻¹ ₀  

−Q+1−ν

if t ≥ λ (d (x, B r (x ₀ )) + 2λr)

for any x ∈ H \ B _2λr (x ₀ ), where ν = min {α, ρα, 1}.

Proof. We have two cases.

Case 1. t ≥ λ (d (x, B _r (x ₀ )) + 2λr).

From B r (x ₀ ) ⊂ B t (x), R

B

(x

) a(y) dy = 0 and the Lipschitz condition on Ω| _S , we get

J _t a(x) =      Z

B

(x

)∩B

(x)

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 a(y) dy     

=      Z

B

(x

)

"

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 − Ω x ◦ x ⁻¹ ₀


 x ◦ x ⁻¹ ₀  

Q−1

#

a(y) dy      . Notice the following:

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 − Ω x ◦ x ⁻¹ ₀
 x ◦ x ⁻¹ ₀ 

Q−1

≤ 

 Ω x ◦ y ⁻¹
− Ω x ◦ x ⁻¹ ₀
 

|x ◦ y ⁻¹ | ^Q−1 + 

Ω x ◦ x ⁻¹ ₀
  ·

1

|x ◦ y ⁻¹ | ^Q−1 − 1  x ◦ x ⁻¹ ₀ 

Q−1

.    

δ _|x◦y

|

x ◦ y ⁻¹
◦ δ | ^x◦x

|

x ◦ x ⁻¹ ₀
−1    

α

 x ◦ x ⁻¹ ₀  

Q−1 +

 y ◦ x ⁻¹ ₀   

 x ◦ x ⁻¹ ₀  

Q . From (2.1), we obtain

δ

|

| x ◦ y ⁻¹
◦ δ

|

| x ◦ x ⁻¹ ₀
−1

     .

δ

|

|

 x ◦ y ⁻¹
◦ x ◦ x ⁻¹ ₀
−1      +

δ

|

| x ◦ x ⁻¹ ₀
◦ δ

|

| x ◦ x ⁻¹ ₀
−1

. r

|x ◦ y ⁻¹ | +



1 − | ^x◦x

|

|x◦y

|

 ρ

 x ◦ x ⁻¹ ₀   

 x ◦ x ⁻¹ ₀  

. r

 x ◦ x ⁻¹ ₀  

+ r ^ρ

 x ◦ x ⁻¹ ₀  

ρ ,

and so,

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 − Ω x ◦ x ⁻¹ ₀
 x ◦ x ⁻¹ ₀ 

Q−1

. r ^α

 x ◦ x ⁻¹ ₀  

Q+α−1 + r ^ρα

 x ◦ x ⁻¹ ₀  

Q+ρα−1 + r 

 x ◦ x ⁻¹ ₀  

Q

. r ^ν

 x ◦ x ⁻¹ ₀  

Q+ν−1 , since 

 x ◦ x ⁻¹ ₀ 

 ≥ 2λr. Therefore we obtain J t a(x) . r ^ν  x ◦ x ⁻¹ ₀ 

Q−1+ν . Case 2. t ≤ λ (d (x, B r (x 0 )) + 2λr).

Since t ∼ d (x, B r (x ₀ )), we get

J t a(x) ≤ kΩk _∞      Z

B

(x)

1

|x ◦ y ⁻¹ | ^Q−1 |a(y)| dy     

. 1

t ^Q−1 Z

B

(x)

|a(y)| dy . tMa(x).

The proof of Lemma 4.7 is completed.

Thus, for x ∈ H \ B _2λr (x ₀ ),

µ _Ω a(x) ² .

Z ∞ d(x,B

(x

))

J _t a(x) dt t ²

!

· Ma(x)

=

Z _λ(d(x,B

_(x

_))+2λr)

d(x,B

(x

))

J t a(x) dt t ² +

Z _∞

λ(d(x,B

(x

))+2λr)

J _t a(x) dt t ²

!

· Ma(x)

.

Z λ(d(x,B

(x

))+2λr) d(x,B

(x

))

dt t

!

Ma(x) ²

+ Z _∞

λ(d(x,B

(x

))+2λr)

r ^ν  x ◦ x ⁻¹ ₀  

Q−1+ν

dt t ²

!

· Ma(x)

. r

d (x, B r (x ₀ )) · Ma(x) ² + r ^ν 

 x ◦ x ⁻¹ ₀  

Q+ν · Ma(x), and so we obtain

µ _Ω a(x) . r

· Ma(x) 

 x ◦ x ⁻¹ ₀  

1 2

+ r

· Ma(x)

 x ◦ x ⁻¹ ₀ 

Q+ν 2

. Pick p ₁ , p ₂ , q ₁ , and q ₂ with the following properties:

• 2Q < p 1 < ∞;

• _Q+ν ^2Q < p ₂ < 2;

• _p ¹

1

+ _q ¹

1

= 1; and

• _p ¹

+ _q ¹

2

= 1.

From H¨ older’s inequality, the Maximal theorem, and (2.3), we obtain r

Z

H \B

(x

)

Ma(x) 

 x ◦ x ⁻¹ ₀  

1 2

dx

. r

Z

H \B

(x

)

 x ◦ x ⁻¹ ₀  

−

dx

!

p1

kMak _q

. r

Z _∞

2r

ρ ⁻

^+Q−1 dρ



p1

kak _q

. 1

and

r

Z

H \B

(x

)

Ma(x)

 x ◦ x ⁻¹ ₀  

Q+ν 2

dx

. r

Z

H \B

(x

)

|x − x 0 | ⁻

dx

!

p2

kMak

. r

Z _∞

2r

ρ ⁻

^+Q−1 dρ



p2

kak

1 2q2

2

. 1.

This shows

II . 1.

Altogether, we obtain Z

H

µ Ω a(x) dx . 1 and the proof is complete.

5. L ^∞ -BM O boundedness

In this section, we study the L ^∞ -BM O boundedness of the Mar- cinkiewicz integrals. Let f ∈ L ^∞ (H) be compactly supported and let B r (x ₀ ) be any ball. We write

f = f χ _B

_(x

₎ + f χ H \B

(x

)

≡ f 1 + f ₂ .

Then f ₁ ∈ L ² (H). By H¨ older’s inequality and by hypothesis, Z

B

(x

)

µ _Ω f ₁ (x) dx ≤ |B r (x ₀ )|

Z

B

(x

)

[µ _Ω f ₁ (x)] ^q dx

!

. |B r (x ₀ )|

kf 1 k _L

(H)

. |B r (x ₀ )|

|B 2λr (x ₀ )|

kf 1 k _L

. |B r (x ₀ )| · kf k _L

.

Thus we obtain

(5.1) 1

|B r (x ₀ )|

Z

B

(x

)

µ _Ω f ₁ (x) dx . kf k _L

.

For x ∈ B r (x ₀ ) we have 

 x ◦ y ⁻¹ 

 > r whenever y ∈ H \ B _2λr (x ₀ ). Let x ∈ B _r (x ₀ ) and let

III =

Z _∞

0

|F t (x) − F _t (x ₀ )| ² dt t ³



.

Then we have

III =



 Z _∞

0

     Z

B

(x)

"

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 − Ω x ₀ ◦ y ⁻¹

|x 0 ◦ y ⁻¹ | ^Q−1

#

f ₂ (y) dy     

2 dt t ³





1 2

=



 Z ∞

r

     Z

B

(x)

"

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 − Ω x ₀ ◦ y ⁻¹

|x 0 ◦ y ⁻¹ | ^Q−1

#

f 2 (y) dy     

2 dt t ³





1 2

.



 Z ∞

r

Z

B

(x

)\B

(x

)

r ^ν

|y − x 0 | ^Q−1+ν dy

! 2

dt t ³





1 2

kf 2 k _L

. Z _∞

r

Z _2λt

2λr

r ^ν s ^ν ds

 2

dt t ³

!

kf 2 k _L

. kf k _L

.

A triangle inequality provides us

|µ Ω f (x) − µ Ω f 2 (x 0 )| ≤ µ Ω f 1 (x) + III, which verifies

1

|B r (x ₀ )|

Z

B

(x

)

|µ Ω f (x) − µ Ω f 2 (x 0 )| dx . kf k _L

. This proves the L ^∞ _c -BM O boundedness.

6. L ^p -boundedness

From H ¹ -L ¹ boundedness and the L ^q -boundedness, it is clear that µ _Ω is bounded in L ^p for 1 < p ≤ q. To prove the L ^p -boundedness for q < p < ∞, we define Stein’s linearizing function ϕ(x, t) which is a function defined for x ∈ H, 0 < t < ∞, so that it satisfies the conditions:

(a) ϕ vanishes if t is small enough, or if t is large enough and is bounded.

(b) For all x, (6.1)

Z ∞ 0

|ϕ(x, t)| ² dt

t ³ ≤ 1

holds.

Now we define T ϕ f (x) =

Z ∞ 0

Z

B

(x)

Ω x ◦ y ⁻¹

|x ◦ y ⁻¹ | ^Q−1 f (y) dyϕ(x, t) dt t ³ . By (a), (b) and by H¨ older’s inequality,

(6.2) |T ϕ f (x)| ≤ µ _Ω f (x) and µ _Ω f (x) = sup

ϕ |T ϕ f (x)|

for all ϕ satisfying (a) and (b) of the above.

It can be shown that T _ϕ is bounded from L ^∞ _c (H) to BM O (H) with uniform bounds for those ϕ with the above properties. An interpolation yields the L ^p boundedness for p ∈ (q, ∞) of T ϕ uniformly in ϕ with above properties, which implies the L ^p boundedness of µ _Ω .

References

[1] A. Benedek, A. Calderon, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365.

[2] A. P. Calder´ on, On the theorem of Marcinkiewicz and Zygmund, Trans. Amer.

Math. Soc. 68 (1950), 55–61.

[3] A. P. Calder´ on and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139.

[4] J. Duoandikoetxea and J. L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541–561.

[5] C. Fefferman and E. M. Stein, H

spaces of several variables, Acta Math. 129 (1972), 137–193.

[6] G. B. Folland and E. M. Stein, Hardy spaces on homogeneous groups, Princeton Univ. Press, 1982.

[7] J. Marcinkiewicz, Sur quelques integrales de type de Dini, Annales de la Soci´et´e Polonaise 17 (1938), 42–50.

[8] E. M. Stein, On the function of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 430–466.

[9] , Singular integrals and differeniability properties of functions, Princeton Univ. Press, 1970.

[10] , Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, 1993.

[11] E. M. Stein and G. Weiss, On the interpolation of analytic families of operators acting on H

spaces, Tˆ ohoku Math. J. 9 (1957), 318–339.

[12] A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, 1986.

[13] A. Torchinsky and S. Wang, A note on the Marcinkiewicz integral, Colloq. Math.

61-62 (1990), 235–243.

[14] A. Zygmund, On certain integrals, Trans. Amer. Math. Soc. 55 (1944), 170–204.

[15] , Trigonometric Series, 2nd ed. Vol. 2, Cambridge Univ. Press, Cam-

bridge, 1968.

Youngwoo Choi

Department of Mathematics Ajou University

Suwon 442-749, Korea

E-mail : [email protected] Kyung Soo Rim

School of Mathematics

Korea Institute for Advanced Study Seoul 130-012, Korea

E-mail : [email protected]