keywords Atomic decomposition, Triebel-Lizorkin spaces, Marcinkiewicz integral.

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TRIEBEL-LIZORKIN SPACES WITH APPLICATIONS TO THE MARCINKIEWICZ INTEGRAL(なめらかでない斉次トリーベルリゾルキンの分解とマ

ルチンキーヴィッツ積分への応用)(英文)

KEISUKE ASAMI

keywords Atomic decomposition, Triebel-Lizorkin spaces, Marcinkiewicz integral.

2010 classification 41A17, 42B35

Abstract. The aim of this paper is to develop a theory of non-smooth decomposition in homogeneous Triebel-Lizorkin spaces. As a byproduct, we can recover the decomposition results for Hardy spaces as a special case. The result extends what Frazier and Jawerth obtained in 1990. The result by Frazier and Jawerth covers only the limited range of the parameters but the result in this paper is valid for all admissible parameters for Triebel- Lizorkin spaces. As an application of the main results, we prove that the Marcinkiewicz operator is bounded. What is new in this paper is to reconstruct sequence spaces other than classical ℓ

^p

spaces.

1. Introduction

It is well known that the Triebel-Lizorkin spaces ˙ F _p,q ^s (R ⁿ ) for 0 < p ≤ 1, 0 < p ≤ q < ∞ and s ∈ R admit the non-smooth atomic decomposition (see [2, Theorem 7.4], [6]). The aim in this paper is to remove this restriction and to study the non-smooth decomposition of ˙ F _p,q ^s (R ⁿ ) for 0 < p < ∞ , 0 < q ≤ ∞ and s ∈ R.

Definition 1.1. Let 0 < p < ∞ , 0 < q ≤ ∞ and s ∈ R. Let φ ∈ C _c ^∞ (R ⁿ ) satisfy χ B(4) \ B(2) ≤ φ ≤ χ _B(8) _\ _B(1) . The homogeneous Triebel-Lizorkin space ˙ F _p,q ^s (R ⁿ ) is defined to be the set of all f ∈ S ^′ (R ⁿ )/ P (R ⁿ ) for which the quantity

∥ f ∥ F ˙

_p,q^s

≡ ∥{ 2 ^js φ j (D)f } j ∈

Z

∥ L

^p

(l

^q

)

is finite, where φ j (x) ≡ φ(2 ⁻ ^j x), P (R ⁿ ) denotes the set of all polynomials on R ⁿ , and ψ(D)f (x) ≡ F ⁻ ¹ ψ ∗ f(x) (x ∈ R ⁿ )

for ψ ∈ S (R ⁿ ) and f ∈ S ^′ (R ⁿ ) and ∥{ f j } j ∈

Z

∥ L

^p

(l

^q

) stands for the vector-norm of a sequence { f j } ^∞ j= −∞ of mesurable functions:

(1.1) ∥{ f j } j ∈

Z

∥ L

^p

(l

^q

) ≡



  ˆ

Rⁿ



 ∑ ^∞

j= −∞

| f j (x) | ^q





p q

dx



 

1 p

, 0 < p, q ≤ ∞ .

The space ˙ F _p,q ^s (R ⁿ ) realizes many function spaces: Indeed,

F ˙ _p,2 ⁰ (R ⁿ ) = L ^p (R ⁿ ) (1 < p < ∞ ), F ˙ _p,2 ⁰ (R ⁿ ) = H ^p (R ⁿ ) (0 < p ≤ 1)

1

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with equivalence of quasi-norms, where H ^p (R ⁿ ) stands for the Hardy space. See [3, Theorem 6.1.2] for the first equivalence and [4, Theorem 2.2.9] for the second equivalence. Thus, our result will cover the ones for Hardy spaces as well as Lebesgue spaces.

To handle ˙ F _p,q ^s (R ⁿ ), it may be convenient to work on the corresponding sequence space f ˙ _p,q ^s (R ⁿ ): it is simpler to handle sequences than to handle distributions.

Definition 1.2. For ν ∈ Z and m = (m 1 , m 2 , . . . , m n ) ∈ Z ⁿ , we define Q ν,m ≡

∏ n j=1

[ m _j

2 ^ν , m _j + 1 2 ^ν

) .

Denote by D = D (R ⁿ ) the set of such cubes. The elements in D (R ⁿ ) are called dyadic cubes.

We adopt the definition by Grafakos; see [4, Definition 2.3.5].

Definition 1.3. Let 0 < q ≤ ∞ and s ∈ R. We consider the set of sequences { r Q } Q ∈D ⊂ C such that the function

g ^s _q ( { r Q } Q ∈D ; x) ≡



 ∑

Q ∈D

( | Q | ⁻

ⁿ^s

| r Q | χ Q (x)) ^q





1 q

(x ∈ R ⁿ ) is in L ^p (R ⁿ ). Let 0 < p < ∞ . For such sequences r = { r _Q } Q ∈D we set

∥ r ∥

f

˙

_p,q^s

≡ ∥ g ^s _q (r) ∥ L

^p

.

A sequence λ = { λ _Q } Q ∈D is said to belong to ˙ f _p,q ^s (R ⁿ ) if ∥ λ ∥

f

^˙

_p,q^s

< ∞ . Sometimes, we identity λ = { λ _ν,m } ν ∈

Z,m

∈

Zⁿ

with λ = { λ _Q } Q ∈D via λ _ν,m = λ _Q when Q = Q _ν,m .

To obtain our result, we follow the book [4] by Grafakos.

Definition 1.4. Let 0 < p < ∞ , 0 < q ≤ ∞ and s ∈ R. A sequence r = { r Q } Q ∈D is called an

∞ -atom for ˙ f _p,q ^s (R ⁿ ) with cube Q 0 if there exists a dyadic cube Q 0 such that

(1.2) g ^s _q ( { r _Q } Q ∈D ; · ) ≡



 ∑

Q ∈D

( | Q | ⁻

ⁿ^s

| r _Q | χ _Q ) ^q





1 q

≤ χ _Q

₀

.

Our first theorem is as follows:

Theorem 1.5. Suppose that we are given parameters p, q, s, u satisfying 0 < p < ∞ , 0 < q ≤ ∞ , s ∈ R, 0 < u ≤ min(1, q).

(1) For any t ∈ f ˙ _p,q ^s (R ⁿ ), there exists a decomposition

(1.3) t =

∑ ∞ j=1

λ _j r _j ,

where each r j is an ∞ -atom for f ˙ _p,q ^s with cube Q j and { λ j } ^∞ j=1 satisfies

(1.4)



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

≤ C ∥ t ∥

f

˙

_p,q^s

.

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(2) If a sequence { Q _j } ^∞ j=1 of cubes and a sequence { λ _j } ^∞ j=1 of complex numbers satisfy

(1.5)



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

< ∞ ,

then for any ∞ -atoms r j for f ˙ _p,q ^s (R ⁿ ) with cube Q j , the series t given by (1.3) belongs to f ˙ _p,q ^s (R ⁿ ).

In Theorem 1.5 the case of s ∈ R, 0 < p = u ≤ 1 and p ≤ q ≤ ∞ is proved in [2, Theorem 7.2]. In this case there is no condition on the position of the cubes since

(1.6)



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

=



 ∑ ^∞

j=1

| λ j | ^p | Q j |





1 p

.

We can refine our Theorem 1.5.

Definition 1.6. Let 0 < p < ∞ , 0 < q ≤ ∞ , v ∈ (0, ∞ ) and s ∈ R. One says that a sequence r = { r Q } Q ∈D is called a v-atom for ˙ f _p,q ^s (R ⁿ ) with cube Q 0 if there exists a dyadic cube Q 0 such that

supp(g _q ^s ( { r Q } Q ∈D ; · )) ⊂ Q 0 , ∥ g ^s _q ( { r Q } Q ∈D ; · ) ∥ L

^v

≤ | Q 0 |

^v¹

. We can refine the latter half of Theorem 1.5 as follows:

Theorem 1.7. In addition to the assumption in Theorem 1.5, let v ∈ (max(1, p), ∞ ). If a sequence { Q j } ^∞ j=1 of cubes and a sequence { λ j } ^∞ j=1 of complex numbers satisfy (1.5), then for any v-atoms r j with cube Q j , the series t given by (1.3) belongs to f ˙ _p,q ^s (R ⁿ ).

The above results cover the ones in [2, Section 7]. What is new about this paper is the case where p > min(q, 1). The case when p > 1 and q = 2 is especially interesting because this yields the decomposition for L ^p (R ⁿ ) = ˙ F _p,2 ⁰ (R ⁿ ).

We now transform the results to the one of the sequences.

Definition 1.8 (Atoms for Triebel-Lizorkin spaces). Let 0 < p < ∞ , 0 < q ≤ ∞ , s ∈ R. Let ν ∈ Z and m ∈ Z ⁿ . Suppose that the integers K, L ∈ Z satisfy K ≥ 0 and L ≥ − 1. A function a ∈ C ^K (R ⁿ ) is said to be a smooth (K, L)-atom centered at Q _0,m for ˙ f _p,q ^s (R ⁿ ), if it is supported on 3 Q _0,m and if it satisfies the diﬀerential inequality and the moment condition:

∥ ∂ ^α a ∥ L

^∞

≤ 2 ^ν ^| ^α ^| , | α | ≤ K, (1.7)

ˆ

Rⁿ

x ^β a(x) dx = 0, | β | ≤ L.

(1.8)

The case L = − 1 is excluded in (1.8).

To state our main result, we present the following definition:

Definition 1.9. Let 0 < p < ∞ , 0 < q ≤ ∞ , s ∈ R. We say that A is a non-smooth atom for F ˙ _p,q ^s (R ⁿ ) with cube ˜ Q if there exists a cube ˜ Q such that A = ∑

Q ⊂ Q ˜

r Q a Q where r = { r Q } Q ∈D is

an ∞ -atom for ˙ f _p,q ^s (R ⁿ ) and each a Q is a smooth (K, L) − atom centered at Q.

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The following theorem, which is a conclusion of this note, extends [4, Corollary 2.3.9]. Define σ p ≡ n

( 1 min(1,p) − 1

)

and σ p,q ≡ max(σ p , σ q ).

Theorem 1.10. Let 0 < p < ∞ , 0 < q ≤ ∞ , s ∈ R, 0 < u ≤ min(1, q) , and let L ≥ max( − 1, [σ p,q − s])

where [ · ] denotes the Gauss sign. Then we have the following.

(1) Let f ∈ F ˙ _p,q ^s (R ⁿ ). Then we can write f =

∑ ∞ j=1

λ _j A _j

in S ^′ (R ⁿ )/ P (R ⁿ ), where { A _j } ^∞ j=1 is a sequence of non-smooth atoms and { λ _j } ^∞ j=1 and { Q j } ^∞ j=1 satisfy suppA j ⊂ 3Q j and

(1.9)



 ∑ ^∞

j=1

| λ j | ^u χ Q

j





1 u

L

^p

≤ C ∥ f ∥ F ˙

_p,q^s

.

(2) Suppose that each A j is a non-smooth atom with cube Q j and the complex sequence { λ j } ^∞ j=1 satisfies



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

< ∞ .

Then by letting f ≡ ∑ _∞

j=1 λ j A j , the sum converges in S ^′ (R ⁿ )/ P (R ⁿ ) and satisfies

∥ f ∥ F ˙

_p,q^s

≤ C



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

.

In Theorem 1.10 the case of s ∈ R, 0 < p = u ≤ 1 and p ≤ q ≤ ∞ is [2, Theorem 7.4 (ii)].

To conclude this section, we recall the following definition to compare our atoms with the ones in Hardy spaces.

Definition 1.11 (Atoms in Hardy spaces). Let 0 < p ≤ 1 < v ≤ ∞ . Fix L ≥ L 0 ≡ [σ p ]. A (non-smooth) (p, v)-atom centered at a cube Q is an L ^v (R ⁿ )-function A which is supported on Q and satisfies the moment condition of order L, that is,

ˆ

Rⁿ

x ^α A(x) dx = 0 for all multi-indexes α with | α | ≤ L and ∥ A ∥ L

^v

≤ | Q |

¹^v

.

Let s = 0, 0 < p < ∞ , q = 2 and 1 < v < ∞ . In Theorem 1.10, the function A _j is a (p, v)-atom modulo a multiplicative constant since

∥ A j ∥ L

^v

∼ ∥ A j ∥ F ˙

_v,2⁰

∼ ∥ g ₂ ⁰ (r j ) ∥ L

^v

≤ | Q j |

¹^v

.

The second equivalence follows from the Littlewood–Paley theory, which indicates ˙ F _v2 ⁰ (R ⁿ ) ∼ L ^v ( R ⁿ ).

We organize the remaining part of this paper as follows: Sections 2-4 are developed to the

proof of the theorems above. As an application, we prove the boundedness of the Marcinkiewicz

operators. Basically, the key idea is to investigate closely the behavior of these operators for

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non-smooth atoms. In [5, Theorem 2.1], Liu and Yang proposed a criterion for the case of 0 < p ≤ 1 and q ≥ p. Here, we will remove the restiction 0 < p ≤ 1. Our results will be valid for 1 ≤ p < ∞ and 1 < q < ∞ as well as for some extra parameters. Unfortunately, we can not present a general criterion for the operators to be bounded from homogeneous Triebel–Lizokin spaces to Banach spaces. This disadvantage comes from the fact that we need to take care of the position of the support of the atoms.

2. Proof of Theorem 1.5

We recall the following facts in [4, p. 115–116]. Let t = { t Q } Q ∈D be a sequence, and let s ∈ R and 0 < q ≤ ∞ .

Lemma 2.1. Let R ∈ D . Define g _q,R ^s ( { t Q } Q ∈D ; x) ≡



 ∑

Q ∈D , R ⊂ Q

( | Q | ⁻

ⁿ^s

| t Q | χ Q (x)) ^q





1 q

(x ∈ R ⁿ ) and

D ν ≡ { Q ∈ D ; l(Q) = 2 ⁻ ^ν } . (1) If R 1 ⊂ R 2 and x ∈ R ⁿ , then g _q,R ^s

2

(t; x) ≤ g _q,R ^s

1

(t; x).

(2) For any x ∈ R ⁿ ,

(2.1) lim

ν →∞

∑

Q ∈D

ν

χ _Q (x)g _q,Q ^s (t; x) = 0.

(3) For any x ∈ R ⁿ ,

(2.2) lim

ν →−∞

∑

Q ∈D

ν

χ Q (x)g _q,Q ^s (t; x) = g _q ^s (t; x).

Lemma 2.2. For k ∈ Z, we set

A k ≡ { R ∈ D : g _q,R ^s (t; x) > 2 ^k , for x ∈ R } . (1) [4, p. 116] If Q ∈ D does not belong to any A k , k ∈ Z, then t Q = 0.

(2) [4, p. 115] For each k ∈ Z, A k+1 ⊂ A k . (3) [4, p. 115 (2.3.16)]

(2.3) { x ∈ R ⁿ : g _q ^s (t; x) > 2 ^k } = ∪

R ∈A

k

R. (4) [4, p. 115 (2.3.17)] For all k ∈ Z, (2.4)



 ∑

Q ∈D\A

k

( | Q | ⁻

ⁿ^s

| t _Q | χ _Q ) ^q





1 q

≤ 2 ^k .

Lemma 2.3. Let A k be as in Lemma 2.2.

Let t = { t _Q } Q ∈D be a sequence indexed by Q ∈ D . Assume g _q ^s (t; x) < ∞ for a.e. x ∈ R ⁿ . We set

B k ≡ { J ∈ D : J is a maximal dyadic cube in A k \ A k+1 } . For J ∈ B k , we define

v(k, J ) ≡ { v(k, J) Q } Q ∈D ≡ { t Q χ _A

_k

_\A

_k+1

(Q)χ _{ _S _∈D _: _S _⊂ _J _} (Q) } Q ∈D ,

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r(k, J ) ≡ 2 ⁻ ^k ⁻ ¹ v(k, J ).

(1) [4, p. 116 (2.3.18)] and [4, p. 116 (2.3.21)] We have t = ∑

k ∈

Z

∑

J ∈B

k

v(k, J ) = ∑

k ∈

Z

∑

J ∈B

k

2 ^k+1 r(k, J ).

(2.5)

(2) [4, p. 116 (2.3.19)] For all k ∈ Z and J ∈ B k , g _q ^s (v(k, J)) ≤ 2 ^k+1 .

Remark that in the definition of t(k, J ), t(k, J ) _Q = v _Q χ _{ _S _∈D _: _S _⊂ _J _} (Q) if Q ∈ A k \ A k+1 is a cube contained in J otherwise t(k, J) _Q = 0.

Now we prove Theorem 1.5.

Let t ∈ f ˙ _p,q ^s (R ⁿ ) be given.

By (2.5), we can write

t = ∑

k ∈

Z

∑

J ∈B

k

2 ^k+1 r(k, J ).

Let ι ≡ (ι 1 , ι 2 ) : N → { (k, J ) : k ∈ Z, J ∈ B k } be a bijection. By letting λ j ≡ 2 ^ι

¹

^(j)+1 and r j ≡ r(ι 1 (j), ι 2 (j)), we can write t = ∑ _∞

j=1 λ j r j . Therefore we get the desired decomposition (1.3). We will check that r _j is an ∞ -atom. Letting k = ι ₁ (j), J = Q _j = ι ₂ (j), we have

g _q ^s (r j ) = g ^s _q (r(k, J )) = g ^s _q (2 ⁻ ^k ⁻ ¹ t(k, J )) = 2 ⁻ ^k ⁻ ¹ g ^s _q (t(k, J)).

Now, suppose that t(k, J ) = { v _Q } Q ∈D . If v _Q ̸ = 0, then g ^s _q (t(k, J)) ≤ 2 ^k+1 . Furthermore if t(k, J ) = 0, then g ^s _q (t(k, J )) = 0. Therefore, since g _r ^s (r j ) ≤ χ J holds, it follows that r j is an

∞ -atom with cube J .

Recall that any J ∈ B k is a cube in A k and that B k is disjoint family. So, we have

(2.6) ∑

J ∈B

k

χ J ≤ χ _∪

_Q∈Ak

Q = χ _{ _g

s q

(t)>2

^k

} .

Using (2.6), we calculate



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

=

( ∑

k ∈

Z

∑

J ∈B

k

(2 ^k+1 χ J ) ^u )

¹_u

L

^p

≤ C

( ∑

k ∈

Z

2 ^ku χ _{ _g

s q

(t)>2

^k

}

)

_u¹

L

^p

≤ C





[1+log

₂

g

_q^s

(t)]

∑

k= −∞

2 ^ku





1 u

L

^p

.

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If we calculate the geometric series, then we obtain



 ∑ ^∞

j=1

| λ _j | ^u χ _Q

_j





1 u

L

^p

≤ C (

2 ^[1+log

²

^g

^s^q

^(t)]u 1 − 2 ⁻ ^u

)

¹_u

L

^p

≤ C (

2 ^(1+log

²

^g

^q^s

^(t))u 1 − 2 ⁻ ^u

)

_u¹

L

^p

= C

( 2 ^u

1 − 2 ⁻ ^u g _q ^s (t) ^u )

¹_u

L

^p

= C g _q ^s (t)

L

^p

= C ∥ t ∥

f

^˙

_p,q^s

keeping in mind that u > 0.

Conversely suppose we are given a sequence r j = { r j,Q } Q ∈D . Denote by Q j the cube for r j

in the definition of atoms. Then setting t =

∑ ∞ j=1

λ _j r _j ,

we have

∥ t ∥

f

˙

_p,q^s

= g ^s _q



 ∑ ^∞

j=1

λ j r j ; ·





L

^p

=

 





g _q ^s



 ∑ ^∞

j=1

λ j r j ; ·









u

L

^p/u

 



1 u

≤

 





 ∑ ^∞

j=1

| λ j | g ^s _q (r j ; · )





u

L

^p/u

 



1 u

≤

 



∑ ∞ j=1

| λ _j | ^u g _q ^s (r _j ; · ) ^u L

^p/u

 



1 u

.

Here we have used u ≤ q to obtain the penultimate inequality and u ≤ 1 to obtain the last inequality. If we use g ^s _r (r j ; · ) ≤ χ Q

_j

, then we obtain

(2.7) ∥ t ∥

f

˙

_p,q^s

≤

 



∑ ∞ j=1

| λ j | ^u χ Q

_j

L

^p/u

 



1 u

=



 ∑ ^∞

j=1

| λ j | ^u χ Q

_j





1 u

L

^p

< ∞ .

Thus, the proof is complete.

We make a brief remark of the method of the proof. The proof of Theorem 1.5 is essentially made up of two tools. The first tool is a method to decompose sequences and the second tool serves to describe the condition of coeﬃcients. The first tool consists of the 1

8 median and the

stopping time argument. In [2, Section 6], Frazier and Jarwerth used them together with L ⁰ ,

the set of all measurable functions f for which { f ̸ = 0 } has finite measure. This method is

refined in § 6.6.4 by Grafakos [4]. Since our proof heavily hinges on § 6.6.4 in [4], we essentially

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used the technique of the paper in [2] and the textbook [4]. What is diﬀerent from these sources is the second tool. As is described in (7.4) of [2] and (7.7) of [2], we have

(2.8) ∥ λ 1 + λ 2 ∥ ^p

f

˙

_p,q^s

≤ ∥ λ 1 ∥ ^p

f

˙

_p,q^s

+ ∥ λ 2 ∥ ^p

f

˙

_p,q^s

(λ 1 , λ 2 ∈ f ˙ _p,q ^s (R ⁿ )) and

(2.9) ∥ f 1 + f 2 ∥ ^p F ˙

_p,q^s

≤ ∥ f 1 ∥ ^p F ˙

_p,q^s

+ ∥ f 2 ∥ ^p F ˙

_p,q^s

(f 1 , f 2 ∈ F ˙ _p,q ^s (R ⁿ ))

for 0 < p ≤ 1, 0 < p ≤ q ≤ ∞ and s ∈ R. Frazier and Jawerth used (2.8) and (2.9) to decompose the sum into small units. One of the important facts on the decomposition of Frazier and Jawerth is that the condition on the position of the cubes Q j does not appear as is hinted in the right-hand side of (1.6). Since (2.8) and (2.9) are no longer available for general case, we need a trick. To accomodate all admissible parameters, we took into account the position of the cubes Q j .

3. Proof of Theorem 1.7 We use the following lemma:

Lemma 3.1. Let 0 < p < ∞ , max(1, q) < p < ∞ . Then for any sequence { A j } ^∞ j=1 of non- negative measurable functions, each of which is supported on a cube Q j , and any sequence { λ j } ^∞ j=1 of non-negative real numbers, we have

∑ ∞ j=1

λ j A j

L

^p

≤ C

∑ ∞ j=1

λ j χ Q

_j

( 1

| Q _j | ˆ

Q

j

A j (y) ^q dy )

¹_q

L

^p

.

Proof. Lemma 3.1 rephrases [7, Lemma 2.5] with 0 < p ≤ 1 and [8, Theorem 1.3.1] with

1 < p < ∞ . □

The proof of Theorem 1.7 is now easy. Just reexamine the proof of Theorem 1.5. Then we notice that everything remains unchanged up to (2.7). Instead of using g _r ^s (r _j ; · ) ≤ χ _Q

_j

we use Lemma 3.1 to have (2.7).

4. Proof of Theorem 1.10

We use the following decomposition results for ˙ F _p,q ^s (R ⁿ ): We invoke the following result in [10, Theorem 13.8].

Theorem 4.1. Let 0 < p < ∞ , 0 < q ≤ ∞ and s ∈ R , and let K be an integer satisfying K ≥ [1 + s] ₊ ≡ max(0, [1 + s]). Furthermore, suppose that L ∈ Z satisfies

(4.1) L ≥ max( − 1, [σ _p,q − s]).

(1) Let κ = { κ ν,m } ν ∈

Z, m

∈

Zⁿ

∈ f ˙ _p,q ^s (R ⁿ ) and each a ν,m is a smooth L-atom centered at Q ν,m for each ν, m. Then

f ≡

∑ ∞ ν= −∞

∑

m ∈

Zⁿ

κ ν,m a ν,m

converges in S ^′ (R ⁿ )/ P (R ⁿ ) and

(4.2) ∥ f ∥ F ˙

_p,q^s

≤ C ∥ κ ∥

f

˙

_p,q^s

.

(9)

(2) Any f ∈ F ˙ _p,q ^s (R ⁿ ) admits a decomposition:

(4.3) f =

∑ ∞ ν= −∞

∑

m ∈

Zⁿ

κ ν,m a ν,m .

Here, the convergence takes place in S ^′ (R ⁿ )/ P (R ⁿ ), each a _ν,m is a smooth L-atom centered at Q _ν,m and the coeﬃcient κ = { κ _ν,m } ν ∈

N0

, m ∈

Zⁿ

satisfies

(4.4) ∥ κ ∥

f

˙

_p,q^s

≤ C ∥ f ∥ F ˙

_p,q^s

.

We now turn to the proof of Theorem 1.10. First we prove the latter half of Theorem 1.10.

Let f ∈ S ^′ (R ⁿ )/ P (R ⁿ ) be such that f =

∑ ∞ j=1

λ _j A _j .

Let A j = ∑

µ ∈

Z

∑

Q ∈D

µ

r j,Q a Q as in the definition of non-smooth atoms. We set f ^J ≡

∑ J j=1

λ j A j =

∑ J j=1

λ j



 ∑

µ ∈

Z

∑

Q ∈D

µ

r j,Q a Q



 = ∑

µ ∈

Z

∑

Q ∈D

µ



 ∑ ^J

j=1

λ j r j,Q



 a Q .

We set

κ ^J ≡

 



∑ J j=1

λ j r j,Q

 



Q ∈D

.

Let 0 < u ≤ min(1, q). Then we have

∥ κ ^J ∥

f

^˙

_p,q^s

= g ^s _q





 



∑ J j=1

λ _j r _j,Q

 



Q ∈D





L

^p

=



  g _q ^s





 



∑ J j=1

λ _j r _j,Q

 



Q ∈D





u

L

^p/u



 

1 u

≤





∑ J j=1

g _q ^s

( { λ j r j,Q } _Q _∈D ) u

L

^p/u





1 u

=





∑ J j=1

| λ _j | ^u g _q ^s

( { r _j,Q } _Q _∈D ) u

L

^p/u





1 u

≤





∑ J j=1

| λ j | ^u χ Q

_j

L

^p/u





1 u

.

Thus, by Theorem 4.1, we have

∥ f ^J ∥ F ˙

_p,q^s

≤ C ∥ κ ^J ∥

f

˙

_p,q^s

≤ C





∑ J j=1

| λ _j | ^u χ _Q

_j

L

^p/u





1 u

.

(10)

By the Fatou property of ˙ F _p,q ^s (R ⁿ ) or by the classical Fatou lemma, we conclude

∥ f ∥ F ˙

_p,q^s

≤ C





∑ ∞ j=1

| λ j | ^u χ Q

_j

L

^p/u





1 u

.

Also, by letting J ^′ < J,

lim

J →∞ ,J

^′

→∞ ∥ f ^J − f ^J

^′

∥ F ˙

_p,q^s

≤ C



 ∑ ^J

j=J

^′

| λ j | ^u χ Q

_j





1 u

L

^p

= 0,



 ∑ ^J

j=J

^′

| λ j | ^u χ Q

_j





1 u

∈ L ^p .

Thus { f ^J } ^∞ J=1 is a Cauchy sequence in S ^′ (R ⁿ )/ P (R ⁿ ). Therefore lim

J →∞ f ^J = f ∈ S ^′ (R ⁿ )/ P (R ⁿ ).

Next we prove the first half of Theorem 1.10. Let f ∈ F ˙ _p,q ^s (R ⁿ ). Decompose f according to Theorem 4.1, so that (4.3) and (4.4) hold. If Q = Q ν,m , we write λ Q ≡ κ ν,m and a Q ≡ a ν,m . We let

B ≡ { (k, J ) : k ∈ Z, J ∈ B k } . Let

N : j ∈ N 7→ (k j , J j ) ∈ B be an enumeration. Let λ = { λ Q } Q ∈D . Since

g ^s _q (λ; x) < ∞ for almost all x ∈ R ⁿ , we have a decomposition:

λ = ∑

k ∈

Z

∑

J ∈B

k

2 ^k+1 r(k, J ) =

∑ ∞ j=1

2 ^k

^j

⁺¹ r(k _j , J _j ),

where each r(k, J ) = { r(k, J ) _Q } Q ∈D is an ∞ -atom supported on J . According to the proof of Theorem 1.5,

(4.5)



 ∑ ^∞

j=1

2 ^(k

^j

^+1)u χ J

_j





1 u

L

^p

=

( ∑

k ∈

Z

∑

J ∈B

k

2 ^(k+1)u χ J

)

_u¹

L

^p

≤ C ∥ λ ∥

f

˙

_p,q^s

.

If we combine (4.4) and (4.5), then we obtain

(4.6) ∥ f ∥ F ˙

_p,q^s

≥ C



 ∑ ^∞

j=1

2 ^(k

^j

^+1)u χ _J

_j





1 u

L

^p

.

We claim that (4.7)

∑ ∞ j=1

∑ ∞ ν= −∞

∑

Q ∈D

ν

2 ^k

^j

⁺¹ r(k j , J j ) Q a Q = lim

T →∞

∑ ∞ ν= −∞

∑

Q ∈D

ν

∑ T j=1

2 ^k

^j

⁺¹ r(k j , J j ) Q a Q

(11)

in ˙ F _p,q ^s (R ⁿ ). In fact, for T ∈ N, we have

∑ ∞ ν= −∞

∑

Q ∈D

ν

∑ ∞ j=1

2 ^k

^j

⁺¹ r(k _j , J _j ) _Q a _Q −

∑ T j=1

∑ ∞ ν= −∞

∑

Q ∈D

ν

2 ^k

^j

⁺¹ r(k _j , J _j ) _Q a _Q _˙

F

_p,q^s

=

∑ ∞ ν= −∞

∑

Q ∈D

ν

∑ ∞ j=1

2 ^k

^j

⁺¹ r(k _j , J _j ) _Q a _Q − ∑ ^∞

ν= −∞

∑

Q ∈D

ν

∑ T j=1

2 ^k

^j

⁺¹ r(k _j , J _j ) _Q a _Q _˙

F

_p,q^s

=

∑ ∞ ν= −∞

∑

Q ∈D

ν

∑ ∞ j=T+1

2 ^k

^j

⁺¹ r(k _j , J _j ) _Q a _Q _˙

F

_p,q^s

≤ C

∑ ∞ j=T +1

2 ^k

^j

⁺¹ r(k _j , J _j ) _˙

f_p,q^s

thanks to Theorem 4.1. We define λ j ≡ 2 ^k

^j

⁺¹ . Since g ^s _q (r(k j , J j )) ≤ χ J

_j

, we have

∑ ∞ ν= −∞

∑

Q ∈D

ν

∑ ∞ j=1

2 ^k

^j

⁺¹ r(k j , J j ) Q a Q −

∑ T j=1

∑ ∞ ν= −∞

∑

Q ∈D

ν

2 ^k

^j

⁺¹ r(k j , J j ) Q a Q

_˙

F

_p,q^s

≤ C g ^s _q



 ∑ ^∞

j=T+1

2 ^k

^j

⁺¹ r(k j , J j )





L

^p

= C



 g ^s _q



 ∑ ^∞

j=T+1

2 ^k

^j

⁺¹ r(k j , J j )





u

L

^pu





1 u

≤ C





∑ ∞ j=T +1

2 ^(k

^j

^+1)u g _q ^s (r(k j , J j )) ^u L

^pu





1 u

≤ C



 ∑ ^∞

j=T +1

| λ _j | ^u χ _J

_j





1 u

L

^p

.

Letting T → ∞ , we obtain (4.7). Thus, we conclude from (4.7) that f =

∑ ∞ ν= −∞

∑

Q ∈D

ν

λ _Q a _Q

=

∑ ∞ ν= −∞

∑

Q ∈D

ν

∑ ∞ j=1

2 ^k

^j

⁺¹ r(k _j , J _j ) _Q a _Q

=

∑ ∞ j=1

∑ ∞ ν= −∞

∑

Q ∈D

ν

2 ^k

^j

⁺¹ r(k j , J j ) Q a Q . If we denote

A j ≡

∑ ∞ ν= −∞

∑

Q ∈D

ν

r(k j , J j ) Q a Q (j ∈ N),

(12)

then we have

f =

∑ ∞ j=1

λ j A j .

Thus, we obtain the desired decomposition.

5. Applications to the boundedness of the Marcinkiewicz operators Let 0 < ρ < n, 1 < q < ∞ . The Marcinkiewicz operator is defined by

(5.1) µ _Ω,ρ,q f (x) ≡ (ˆ _∞

0

1 t ^ρ

ˆ

B(t)

f (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy ^q dt

t )

¹_q

,

where we write B(r) = {| x | < r } ⊂ R ⁿ for r > 0 here and below.

We suppose ˆ

S

ⁿ−1

Ω(ω) dσ(ω) = 0, Ω ∈ C ¹ (S ⁿ ⁻ ¹ ), where S ⁿ ⁻ ¹ = {| x | = 1 } . According to [9, Theorem 1], we have

∥ µ Ω,ρ,q f ∥ L

^p

≤ C ∥ f ∥ F ˙

_p,q⁰

if 1 < p < ∞ . We remark that if A is a non-smooth atom supported in 3Q 0

∥ A ∥ F ˙

_p,q_˜^s

≤ C ∥ g _q ^s ( { r Q } Q ∈D , · ) ∥ L

^p^˜

(R

ⁿ

) ≤ C ∥ χ 3Q

₀

∥ L

^p^˜

(R

ⁿ

) ≤ C | 3Q 0 |

^p¹^˜

. (5.2)

Theorem 5.1. The estimate ∥ µ Ω,ρ,q f ∥ L

^p

≤ C ∥ f ∥ F ˙

_p,q⁰

for all f ∈ F ˙ _p,q ⁰ if nq

nq + 1 < p < ∞ , 1 < q < ∞ .

The rest of this paper is developed to the proof of Theorem 5.1. Let f ∈ F ˙ _p,q ⁰ (R ⁿ ). Let f =

∑ ∞ j=1

λ j A j

be a decomposition as in Theorem 1.10 (1). Then we have µ Ω,ρ,q f (x) ≤

∑ ∞ j=1

| λ j | µ Ω,ρ,q A j (x)

=

∑ ∞ j=1

| λ j | χ 3nQ

j

(x)µ Ω,ρ,q A j (x) +

∑ ∞ j=1

| λ j | χ

Rⁿ

\ 3nQ

_j

(x)µ Ω,ρ,q A j (x).

Lemma 5.2. Let x ∈ R ⁿ \ 3nQ j .

(1) Let 0 < t < | x − c(Q j ) | − ³ ^√ ₂ ⁿ ℓ(Q j ). Then 1

t ^ρ ˆ

B(t)

A j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy = 0.

(2) Let | x − c(Q _j ) | − ³ ^√ ₂ ⁿ ℓ(Q _j ) ≤ t ≤ | x − c(Q _j ) | + ³ ^√ ₂ ⁿ ℓ(Q _j ). Then 1

t ^ρ ˆ

B(t)

A j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy

≤ C ℓ(Q j ) ⁿ

| x − c(Q j ) | ⁿ .

(13)

(3) Let t > | x − c(Q j ) | + ³ ^√ ₂ ⁿ ℓ(Q j ). Then 1

t ^ρ ˆ

B(t)

A _j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy

≤ C ℓ(Q _j ) ⁿ⁺¹ t ^ρ | x − c(Q j ) | ⁿ ⁻ ^ρ+1 .

Proof. (1) This is clear from the condition on the support.

(2) We observe 1

t ^ρ ˆ

B(t)

A _j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy ≤ C 1

t ^ρ ˆ

supp(A

_j

(x −· ))

A _j (x − y) 1

| y | ⁿ ⁻ ^ρ dy.

Note that | y | ≥ | x − c(Q j ) |− ³ ^√ ₂ ⁿ ℓ(Q j ). Hence, 1

| y | ⁿ ⁻ ^ρ ≤ 1 ( | x − c(Q j ) | − ³ ₂ √

nℓ(Q j )) ⁿ ⁻ ^ρ . Now by x ∈ R ⁿ \ 3nQ _j ,

| x − c(Q j ) | − 3 2

√ nℓ(Q j ) = 1

2 | x − c(Q j ) | + 1

2 | x − c(Q j ) | − 3 2

√ nℓ(Q j ) ≥ 1

2 | x − c(Q j ) | . Hence 1

| y | ⁿ ⁻ ^ρ ≤ 2 ^ρ ⁻ ⁿ

| x − c(Q j ) | ⁿ ⁻ ^ρ . Also, 1

t ^ρ ≤ 1

( | x − c(Q j ) | − ³ ₂ √

nℓ(Q j )) ^ρ ≤ 2 ^ρ

| x − c(Q j ) | ^ρ . Therefore,

1 t ^ρ

ˆ

B(t)

A j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy

≤ C ℓ(Q j ) ⁿ

| x − c(Q _j ) | ⁿ (3) We use the moment condition to have:

1 t ^ρ

ˆ

B(t)

A j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy

= 1 t ^ρ

ˆ

supp(A

_j

(x −· ))

A _j (x − y) Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ dy

= 1 t ^ρ

ˆ

supp(A

_j

(x −· ))

A j (x − y)

( Ω(y/ | y | )

| y | ⁿ ⁻ ^ρ − Ω((x − c(Q _j ))/ | x − c(Q _j ) | )

| x − c(Q j ) | ⁿ ⁻ ^ρ

) dy.

Note that if A j (x − y) ̸ = 0, then | x − y − c(Q j ) | ≤ ³ ^√ ₂ ⁿ ℓ(Q j ). Hence

keywords Atomic decomposition, Triebel-Lizorkin spaces, Marcinkiewicz integral.

TRIEBEL-LIZORKIN SPACES WITH APPLICATIONS TO THE MARCINKIEWICZ INTEGRAL(なめらかでない斉次トリーベルリゾルキンの分解とマ

KEISUKE ASAMI