Besov and Triebel-Lizorkin Spaces Associated with Flag Kernel Singular Integrals

Besov and Triebel-Lizorkin Spaces Related to Singular Integrals with Flag Kernels

Dachun Y

School of Mathematical Sciences Beijing Normal University

Laboratory of Mathematics and Complex Systems Ministry of Education

Beijing 100875 — People’s Republic of China [email protected]

Received: August 11, 2008 Accepted: January 8, 2009

ABSTRACT

Let s1, s2 ∈ (−1,1) and s = (s1, s2). In this paper, the author introduces the Besov space B˙pq^s (R²) with p, q ∈ [1,∞] and the Triebel-Lizorkin space F˙pq^s(R²) with p ∈ (1,∞) and q ∈ (1,∞] associated to singular integrals with flag kernels. Some basic properties, including their dual spaces, some equiv- alent norm characterizations via Littlewood-Paley functions, lifting properties and some embedding theorems, on these spaces are given. Moreover, the au- thor obtains the boundedness of flag singular integrals and fractional integrals on these spaces.

Key words: Besov space, Triebel-Lizorkin space, flag singular integral, flag fractional integral, Littlewood-Paley operator, dual space, lifting, embedding.

2000 Mathematics Subject Classification: 42B35, 42B20, 42B25, 46E35.

1. Introduction

In order to study the ^b-complex on certain quadratic CR submanifolds of Cⁿ, Nagel, Ricci, and Stein [6] introduced the notion of singular integrals with flag kernels onRⁿ. Since the flag singular integral is a special case of product singular integrals, the boundedness of flag singular integrals onL^p(Rⁿ) withp∈(1,∞) is a simple corol- lary of the boundedness of the corresponding product singular integrals. Recently,

The author is supported by National Science Foundation for Distinguished Young Scholars (No. 10425106) of China.

Han and Lu in [3] developed a corresponding Hardy space theory for flag singular integrals onRⁿ.

Motivated by [3, 6, 7], letting s₁, s₂ ∈ (−1,1) and s = (s₁, s₂), in this paper, we introduce the Besov space ˙B_pq^s (R²) with p, q ∈ [1,∞] and the Triebel-Lizorkin space ˙F_pq^s(R²) with p∈ (1,∞) andq ∈(1,∞] associated to singular integrals with flag kernels. Some basic properties, including their dual spaces, some equivalent norm characterizations via Littlewood-Paley functions, lifting properties, and some embedding theorems on these spaces are given. Moreover, we obtain the boundedness of flag singular integrals and fractional integrals on these spaces.

For the simplicity of presentation, we work on flag integrals on R². However, our method works for more general product Euclidean spaces.

It was proved in [5, 6] that flag kernels onR² are closely connected with product kernels onR²×R. We denote any element ofR²×Rby the 3-tuplex= (x1, x2, x3), where (x1, x2)∈R²andx3∈R. We endowR²with the following dilation that for any δ >0 andx= (x1, x2)∈R²,δx= (δx1, δ²x2) and the norm thatkxk= (x²₁+|x2|)^1/2, which is equivalent to |x1|+|x2|^1/2. Obviously, the homogeneous dimension of R² is 3.

In order to express the cancellation conditions introduced by Nagel, Ricci, and Stein in [6], we introduce the following terminology. Ak-normalized bump func- tion onRⁿis aC^k-function supported on the unit ball withC^k-norm bounded by 1. It was proved by Nagel, Ricci, and Stein in [6, Remark 2.1.7] that Definitions 1.1 and 1.2 given below are essentially independent of the choice ofk∈N. Hence we usually speak of normalized bump functions rather thank-normalized bump functions.

In what follows, we denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. Constants with sub- scripts, such as C1, do not change in different occurrences. We also use subscripts to indicate which parameters the constant depends on. Moreover, letN≡ {1,2, . . .} andZ+≡N∪ {0}.

Definition 1.1. A product kernel onR²×R, relative to the given dilations, is a dis- tributionK onR²×Rwhich coincides with aC^∞function away from the coordinate subspacexj= 0 for j= 1,2,3 and which satisfies:

(i) (Differential inequalities) For each multi-index α= (α1, α2, α3)∈(Z⁺)³, there exists a positive constant Cα so that

|∂^α_x¹

1∂^α_x²

2∂_x^α3

3K(x₁, x₂, x₃)| ≤C_αk(x₁, x₂)k^{−3−α1−2α2}|x₃|^−1−α3 away from the coordinate subspaces, where∂_x^α_iⁱ= ^∂αi

∂x^αi_i . (ii) (Cancellation conditions)

(a) For each multi-index (α1, α2) ∈ (Z+)² and any given normalized bump function ϕ on R and any δ > 0, there exists a positive constant Cα₁,α₂

so that, for all (x₁, x₂)6= (0,0),

Z

R

∂_x^α₁¹∂_x^α₂²K(x1, x2, x3)ϕ(δx3)dx3

≤Cα₁,α₂k(x1, x2)k^{−3−α1−2α2}. (b) For each α₃ ∈ Z+ and any given normalized bump function ϕ on R²

and anyδ >0, there exists a positive constantC_α3 so that, for allx₃6= 0,

Z

R²

∂_x^α₃³K(x1, x2, x3)ϕ(δx1, δ²x2)dx1dx2

≤Cα₃|x3|^−1−α3. (c) For any given normalized bump functionϕonR²×Rand anyδ₁, δ₂>0,

there exists a positive constantC so that

Z

R²×R

K(x₁, x₂, x₃)ϕ(δ₁x₁, δ²₁x₂, δ₂x₃)dx₁dx₂dx₃

≤C.

The following definition of flag kernels is just a special case of flag kernels in [6].

Definition 1.2. A flag kernel onR², relative to the given dilations, is a distribution K on R² which coincides with a C^∞ function away from the coordinate subspace x₁= 0 and which satisfies:

(i) (Differential inequalities) For eachα= (α₁, α₂)∈(Z+)², there exists a positive constant C_α so that, for allx₁6= 0,

|∂_x^α₁¹∂^α_x2²K(x1, x2)| ≤Cα|x1|^−1−α1k(x1, x2)k^−2−2α2. (ii) (Cancellation conditions)

(a) For any given normalized bump functionϕ onR, any α1 ∈ Z+, and any δ >0, there exists a positive constant Cα1 so that, for allx16= 0,

Z

R

∂_x^α1

1K(x₁, x₂)ϕ(δx₂)dx₂

≤C_α1|x1|^−1−α1.

(b) For any given normalized bump function ϕ onR, any α2 ∈ Z+, and any δ >0, there exists a positive constant Cα2 so that for allx26= 0,

Z

R

∂_x^α2

2K(x₁, x₂)ϕ(δx₁)dx₁

≤C_α2|x2|^−1−α2.

(c) For any given normalized bump functionϕonR², and anyδ1, δ2>0, there exists a positive constant Cso that

Z

R²

K(x₁, x₂)ϕ(δ₁x₁, δ₂x₂)dx₁dx₂

≤C.

Remark 1.3. It was pointed by Nagel, Ricci, and Stein in [6] that the single normalized bump function in Definitions 1.1 and 1.2 (ii)-(c) can be replaced by the tensor product of normalized bump functions onR²and R.

The following proposition is completely similar to Proposition 3.2 and Lemma 4.5 in [5], which reveals the relation between the product kernel and the flag kernel.

Proposition 1.4. Let K^] be an integrable function on R²×R which is a product kernel as in Definition 1.1. Then, for (x₁, x₂)∈R²,

K(x₁, x₂) = Z

R

K^](x₁, x₂−x₃, x₃)dx₃ (1) is a flag singular kernel on R².

Conversely, given K∈L¹(R²)which is a flag kernel as in Definition 1.2, then K^](x1, x2, x3) = 1

|x₁|²χ x2

|x₁|²

K(x1, x2+x3),

where χ is a non-negative smooth function supported on [1/2,1] such that R1

1/2χ(t)dt= 1, is an integrable product kernel on R²×Rsuch that (1)holds.

The organization of this paper is as follows. In section 2, we first establish some Calder´on reproducing formulae (see Lemma 2.3), whose dyadic version (see Lemma 2.4) are essentially included in [3]. However, in this paper, we use a slightly different way from [3] to define the topology of S_∞,F(R²); see Definition 2.1 below and Definition 1.6 in [3]. Let s₁, s₂ ∈ (−1,1) and s = (s₁, s₂). With a special choice of approximations of the identity associated to flag kernels (see (1.3) of [3]), we then introduce the norms ofk·kB˙^s_pq(R²) andk·kF˙_pq^s(R²)in Definition 2.5, and using the Calder´on reproducing formulae, we prove in Theorem 2.6 that these norms are independent of the choice of the approximations of the identity associated to flag kernels. Then we introduce the Besov space ˙B^s_pq(R²) and the Triebel-Lizorkin space F˙_pq^s(R²) in Definition 2.7. Some basic properties including dual spaces of these spaces are presented in Propositions 2.9, 2.10, and 2.11. In Theorems 2.8, 2.14 and Corol- lary 2.22, we establish some equivalent norm characterizations of these spaces in- cluding various Littlewood-Paley functions. We remark that Corollary 2.22 clearly reveals the difference between ˙B_pq^s (R²) and ˙F_pq^s(R²) with the classical product Besov and Triebel-Lizorkin spaces in [7].

The boundedness of flag singular integrals on ˙B_pq^s (R²) and ˙F_pq^s(R²) is given in The- orem 3.1 and the lifting properties of these spaces via Riesz potential operators is presented in Theorem 4.6.

Finally, in Theorems 5.1 and 5.2, we establish some embedding theorems on ˙B_pq^s (R²) and ˙F_pq^s(R²). The boundedness of flag fractional integrals is given in The- orem 5.4.

2. Besov and Triebel-Lizorkin spaces on R

We first introduce the Calder´on reproducing formulae. To this end, we need to intro- duce some spaces of distributions; see [3].

Definition 2.1. A Schwartz functionf ∈ S(R²) is said to belong to the space of test functions,S_∞,F(R²), related to flag singular integrals, if there exists a Schwartz func- tionf^]∈ S(R²×R) such that, for all (x1, x2)∈R²,

f(x1, x2) = Z

R

f^](x1, x2−x3, x3)dx3,

wheref^] satisfies the following conditions: for allx3∈Rand (α1, α2)∈(Z+)², Z

R²

f^](x1, x2, x3)x^α₁¹x^α₂²dx1dx2= 0, and for all (x₁, x₂)∈R²andα₃∈Z+,

Z

R

f^](x1, x2, x3)x^α₃³dx3= 0.

We endowS∞,F(R²) with the same topology as S(R²), and we denote its dual space byS∞,F(R²)⁰.

Remark 2.2. It is easy to see that the spaceS∞,F(R²) is a closed subspace ofS(R²), and iff ∈ S∞,F(R²) then, for allα₂∈Z+,

Z

R

f(x1, x2)x^α₂²dx2= 0.

Let Px2(R) be the set of all polynomials on R in variable x₂. Then, one can easily see that iff ∈ S_∞,F(R²)⁰, h∈ Px₂(R), andg ∈ S_∞,F(R²), then hf, gi=hf, g+hi, namely,S_∞(R²)⁰/Px₂(R)⊂ S_∞,F(R²)⁰.

We now establish the following Calder´on reproducing formulae by a method es- sentially similar to that of Theorem 3 in the appendix of [2] and a dual argument;

see also [3].

Lemma 2.3. Let ψ⁽¹⁾ ∈ S(R²) with R

R²ψ⁽¹⁾(x1, x2)dx1dx2 = 0 and ψ⁽²⁾ ∈ S(R) with R

Rψ⁽²⁾(x3)dx3 = 0 satisfy the following admissible conditions: for all (ξ1, ξ2)∈R² and (ξ1, ξ2)6= 0,

Z ∞ 0

|ψˆ⁽¹⁾(tξ1, t²ξ2)|²dt t = 1,

and, for all η ∈Rand η6= 0, Z ∞

0

|ψˆ⁽²⁾(tη)|²dt t = 1.

For t1, t2>0 and x1, x2∈R, let ψ_t⁽¹⁾₁ (x1, x2) = 1

t³₁ψ⁽¹⁾x1

t₁,x2

t²₁

, ψ_t⁽²⁾₂ (x2) = 1

t₂ψ⁽²⁾x2

t₂

,

and

ψ_t1t2(x₁, x₂) = Z

R

ψ⁽¹⁾_t

1 (x₁, x₂−x⁰₂)ψ_t⁽²⁾

2 (x⁰₂)dx⁰₂. Then the identity

f(x1, x2) = Z ∞

0

Z ∞ 0

ψt₁t₂∗ψt₁t₂∗f(x1, x2)dt1

t1

dt2

t2

(2) holds in L²(R²),S∞,F(R²), and S∞,F(R²)⁰.

Proof. From the Plancherel principle and the assumptions of the lemma, it is easy to see that (2) holds inL²(R²). The fact that (2) holds inS_∞,F(R²) and a dual argument show that (2) also holds in S_∞,F(R²)⁰. Thus, to finish the proof of Lemma 2.3, we only need to show that (2) holds inS_∞,F(R²). To do so, forf ∈ S_∞,F(R²),i>0, andδi>0 withi= 1,2 and (x1, x2)∈R², set

f_1,2,δ1,δ2(x₁, x₂) = Z δ1

₁

Z δ1

₂

ψ_t1t2∗ψ_t1t2∗f(x₁, x₂)dt₁ t1

dt₂ t2

.

We only need to show that for any fixed N ∈Z+ and all (x₁, x₂)∈R², there exists a positive constantC=C_f,ψ(1),ψ⁽²⁾,N such that

|f(x1, x2)−f₁,₂,δ₁,δ₂(x1, x2)| ≤C 1+ 1

δ₁

2+ 1 δ₂

(1 +k(x1, x2)k)^−N. (3) Obviously, we may assume that 0 < _i < 1 < δ_i for i = 1,2 in (3). To prove (3), we write

f(x1, x2)−f₁,₂,δ₁,δ₂(x1, x2) = Z 1

0

Z ∞ 0

ψt₁t₂∗ψt₁t₂∗f(x1, x2)dt1

t₁ dt2

t₂

+ Z δ₁

₁

Z ₂ 0

· · · + Z δ₁

₁

Z ∞ δ₂

· · · + Z ∞

δ₁

Z ∞ 0

· · ·

=H₁+H₂+H₃.

We only estimate H₁. The same technique also works forH₂ and H₃. To estimate H₁, we further decompose it into

H1= Z ₁

0

Z 1 0

ψt₁t₂∗ψt₁t₂∗f(x1, x2)dt1

t₁ dt2

t₂ + Z ₁

0

Z ∞ 1

· · · =H₁¹+H₁².

Letϕ⁽ⁱ⁾=ψ⁽ⁱ⁾∗ψ⁽ⁱ⁾ fori= 1,2 andϕt₁t₂ =ϕ⁽¹⁾_t1 ∗ϕ⁽²⁾_t2 . Then it is easy to see that ϕt1t2=ψt1t2∗ψt1t2. Sincef ∈ S∞,F(R²), by Definition 2.1, there exists a functionf^] as in Definition 2.1 such that, for all (x₁, x₂)∈R²,

f(x1, x2) = Z

R

f^](x1, x2−x3, x3)dx3.

Using this fact, we have ψt₁t₂∗ψt₁t₂∗f(x1, x2)

= Z

R

Z

R²

Z

R

ϕ⁽¹⁾_t

1 (x₁−y₁, x₂−y₂−z)ϕ⁽²⁾_t

2 (z−y₃)f^](y₁, y₂, y₃)dz dy₁dy₂dy₃. By the vanishing moments ofϕ⁽¹⁾ andϕ⁽²⁾, we further write

ψ_t1t2∗ψ_t1t2∗f(x₁, x₂)

= Z

R

Z

|z−y3|≤|z|/2

Z

k(x1−y1,x2−y2−z)k≤k(x1,x2−z)k/2

ϕ⁽¹⁾_t1 (x1−y1, x2−y2−z)

×ϕ⁽²⁾_t

2 (z−y₃)n

f^](y₁, y₂, y₃)−f^](x₁, x₂−z, y₃)

−

f^](y1, y2, z)−f^](x1, x2−z, z)o

dy1dy2dy3dz

+ Z

R

Z

|z−y3|>|z|/2

Z

k(x1−y1,x2−y2−z)k≤k(x1,x2−z)k/2

· · · +

Z

R

Z

|z−y3|≤|z|/2

Z

k(x1−y1,x₂−y2−z)k>k(x1,x₂−z)k/2

· · · +

Z

R

Z

|z−y₃|>|z|/2

Z

k(x₁−y₁,x₂−y₂−z)k>k(x₁,x₂−z)k/2

· · · =I₁+I₂+I₃+I₄.

We denote the corresponding terms of H₁¹ to Ii, respectively, by H_1,i¹ , where i= 1,2,3,4, and by similarity we only estimateH_1,1¹ and H_1,2¹ . By the mean value

theorem, we have

|I1| ≤C Z

R

Z

R

Z

R²

ϕ⁽¹⁾_t

1 (x₁−y₁, x₂−y₂−z)

×

|x₁−y₁|

(1 +k(x1, x2−z)k)^N+2 + |x₂−y₂−z|^1/2 (1 +k(x1, x2−z)k)^N+2

× ϕ⁽²⁾_t

2 (z−y₃)

|z−y₃| 1

(1 +|z|)^N dy₁dy₂dy₃dz

≤Ct1t2

Z

R

1

(1 +k(x1, x2−z)k)^N+2 1

(1 +|z|)^N dz≤Ct1t2

1

(1 +k(x1, x2)k)^N. From this, we can easily deduce a desired estimate forH_1,1¹ . Similarly, we have

|I2| ≤Ct1

Z

R

Z

|z−y3|>|z|/2

ϕ⁽²⁾_t2 (z−y3)

|y3−z|

× 1

(1 +k(x1, x2−z)k)^N+2 dy3dz

≤Ct1t2

Z

R

1

(1 +k(x1, x2−z)k)^N+2 1 (1 +|z|)^N dz

≤Ct1t2

1

(1 +k(x1, x2)k)^N, which gives a desired estimate of H_1,2¹ .

In what follows, we denote bybacthe integer no more thana∈R. To estimateH₁², by the vanishing moments ofϕ⁽¹⁾ andf^], we write

ψt₁t₂∗ψt₁t₂∗f(x1, x2)

= Z

R

Z

|y3|≤|z|/2

Z

k(x1−y1,x₂−y2−z)k≤k(x1,x₂−z)k/2

ϕ⁽¹⁾_t1 (x1−y1, x2−y2−z)

×

ϕ⁽²⁾_t

2 (z−y₃)−

bN/2c

X

γ=0

(−y₃)^γd^γ dz^γϕ⁽²⁾_t

2 (z)

×[f^](y1, y2, y3)−f^](x1, x2−z, y3)]dy1dy2dy3dz +

Z

R

Z

|y₃|>|z|/2

Z

k(x₁−y₁,x₂−y₂−z)k≤k(x₁,x₂−z)k/2

· · · +

Z

R

Z

|y3|≤|z|/2

Z

k(x1−y1,x2−y2−z)k>k(x1,x2−z)k/2

· · · +

Z

R

Z

|y3|>|z|/2

Z

k(x1−y1,x₂−y2−z)k>k(x1,x₂−z)k/2

· · · =J₁+J₂+J₃+J₄,

and we also denote the corresponding terms ofH₁²to J_i, respectively, byH_1,i² , where i = 1,2,3,4. By similarity, we only estimate H_1,1² . The Taylor expansion theorem yields

|J₁| ≤C Z

R

t₁

(1 +k(x1, x2−z)k)^N+2 1 t2

1

(t2+|z|)^bN/2c+1dz

≤C t1

(1 +k(x1, x₂)k)^N+2 1 t^bN/2c+1₂

+C t1

(1 +|x1|)^N 1 t₂

1

(t2+|x2|)^bN/2c+1, which yields a desired estimate forH_1,1² . This finishes the proof of Lemma 2.3.

Similarly, we have a ‘discrete’ version of Lemma 2.3 and we omit the details of its proof; see [3].

Lemma 2.4. Let ψ⁽¹⁾ ∈ S(R²) with R

R²ψ⁽¹⁾(x1, x2)dx1dx2 = 0 and ψ⁽²⁾ ∈ S(R) with R

Rψ⁽²⁾(x3)dx3 = 0 satisfy the following admissible conditions that for all (ξ1, ξ2)∈R² and (ξ1, ξ2)6= 0,

∞

X

k₁=−∞

|ψˆ⁽¹⁾(2^−k1ξ₁,2^−2k1ξ₂)|² = 1,

and for all η ∈ R and η 6= 0, P∞

k₂=−∞|ψˆ⁽²⁾(2^−k2η)|² = 1. For all k1, k2 ∈ Z and x1, x2∈R, let ψ_k⁽¹⁾

1(x1, x2) = 2^3k1ψ⁽¹⁾(2^k1x1,2^2k1x2),ψ_k⁽²⁾

2(x2) = 2^k2ψ⁽²⁾(2^k2x2), and ψk1k2(x1, x2) =

Z

R

ψ⁽¹⁾_k

1(x1, x2−x⁰₂)ψ_k⁽²⁾

2(x⁰₂)dx⁰₂. Then the identity

f(x1, x2) =

∞

X

k1=−∞

∞

X

k2=−∞

ψk₁k₂∗ψk₁k₂∗f(x1, x2) (4)

holds in L²(R²),S_∞,F(R²), and S_∞,F(R²)⁰.

We now introduce the normsk·kB˙^s_pq(R²)andk·kF˙_pq^s(R²)forf ∈ S_∞,F(R²)⁰and using Lemma 2.3, we prove that they are independent of choices ofψ⁽¹⁾ andψ⁽²⁾.

Definition 2.5. Let ψt₁t₂ be the same as in Lemma 2.3 and s1, s2 ∈ R. For f ∈ S∞,F(R²)⁰, define

kfkB˙_pq^s (R²)= Z ∞

0

Z ∞ 0

t^−s₁ ^1qt^−s₂ ^2qkψt₁t₂∗fk^q_Lp(R²)

dt1

t₁ dt2

t₂ ^1/q

forp, q∈[1,∞], and kfkF˙_pq^s(R²)=

Z ∞ 0

Z ∞ 0

t^−s₁ ^1qt^−s₂ ^2q|ψt₁t₂∗f|^q dt1

t1

dt2

t2

1/q _Lp(

R²)

forp∈(1,∞) andq∈(1,∞], where the usual modifications are made when p=∞ orq=∞.

We recall the definition of the strong maximal function: for anyf ∈L¹_loc(R²) and all (x₁, x₂)∈R²,

M_s(f)(x₁, x₂) = sup

(x1,x2)∈R Rrectangle

1

|R|

Z

R

|f(y₁, y₂)|dy₁dy₂.

Theorem 2.6. Let s1, s2 ∈ (−1,1) and s = (s1, s2). The norm k·kB˙^s_pq(R²) with p, q∈[1,∞]and the norm k·kF˙_pq^s(R²) with p∈(1,∞) and q∈(1,∞] are independent of the choices of ψ⁽ⁱ⁾ for i= 1,2.

Proof. Let ˜ψ⁽ⁱ⁾ be some other functions satisfying the same conditions as ψ⁽ⁱ⁾ for i = 1,2. We denote the corresponding norms, respectively, by k·k

0B˙^s_pq(R²)

and k·k

0F˙_pq^s(R²) and now prove that there exists a positive constant C such that, for allf ∈ S_∞,F(R²)⁰,

kfk

0B˙_pq^s (R²)≤CkfkB˙^s_pq(R²) (5) and

kfk

0F˙_pq^s(R²)≤CkfkF˙_pq^s(R²). (6) To prove (5) and (6), by Lemma 2.3, we first need to establish a desired estimate for ˜ψu1u2∗ψt1t2. By its definition, it is easy to show that, for all (x1, x2)∈R²,

ψ˜u₁u₂∗ψt₁t₂(x1, x2) = ψ˜⁽¹⁾_u1 ∗ψ_t⁽¹⁾₁

∗2 ψ˜_u⁽²⁾₂ ∗ψ⁽²⁾_t2

(x1, x2),

where, and in what follows, ∗₂ denotes the convolution in the second variable. We also seta∧b= min{a, b}anda∨b= max{a, b} fora, b∈R. We claim that

(i) for allt1, u1>0 and all (x1, x2)∈R²,

ψ˜_u⁽¹⁾₁ ∗ψ⁽¹⁾_t1

(x1, x2)

≤Cu1

t₁ ∧ t1

u₁

u1∨t1

(u₁∨t₁+k(x₁, x₂)k)⁴; (7) (ii) for allt2, u2>0 and allz∈R,

ψ˜⁽²⁾_u2 ∗ψ_t⁽²⁾₂ (z)

≤Cu2

t2

∧ t2

u2

u2∨t2

(u2∨t2+|z|)². (8)

By similarity, we only show (7). To this end, by the mean value theorem and some trivial computation, we can easily prove that, for allu₁>0 and (x₁, x₂)∈R²,

ψ˜_u⁽¹⁾

1(x₁, x₂)

≤C u₁

(u1+k(x1, x2)k)⁴, (9) and, for allu₁>0 and (x₁, x₂), (y₁, y₂)∈R² withk(y₁, y₂)k ≤(u₁+k(x₁, x₂)k)/2,

ψ˜_u⁽¹⁾

1(x₁+y₁, x₂+y₂)−ψ˜_u⁽¹⁾

1(x₁, x₂)

≤C k(y1, y2)k u₁+k(x1, x₂)k

u1

(u₁+k(x1, x₂)k)⁴. (10) The estimates (9) and (10), with ˜ψ⁽¹⁾u₁ and u1 replaced respectively by ψ⁽¹⁾_t1 and t1, also hold. We now prove (7) in the caseu1≥t1. In this case, as

Z

R²

ψ_t⁽¹⁾

1 (y₁, y₂)dy₁dy₂= 0, we can write

ψ˜⁽¹⁾_u1 ∗ψ_t⁽¹⁾₁ (x1, x2)

= Z

k(y1,y₂)k≤(u1+k(x1,x₂)k)/2

ψ˜_u⁽¹⁾

1(x₁−y₁, x₂−y₂)−ψ˜_u⁽¹⁾

1(x₁, x₂)

×ψ_t⁽¹⁾

1 (y₁, y₂)dy₁dy₂ +

Z

k(y1,y2)k>(u1+k(x1,x2)k)/2

· · ·

=D1+D2. The estimate (10) yields that

|D₁| ≤C t₁

(u1+k(x1, x2)k)⁴ Z

R²

y₁ t1

,y₂ t²₁

ψ_t⁽¹⁾

1 (y₁, y₂) dy₁dy₂

≤C t1

(u1+k(x1, x2)k)⁴, and the estimates (9) for ˜ψu⁽¹⁾₁ andψ⁽¹⁾_t

1 imply that

|D2| ≤C Z

k(y1,y₂)k>(u1+k(x1,x₂)k)/2

ψ˜⁽¹⁾_u1(x1−y1, x2−y2)

t1

k(y1, y2)k⁴dy1dy2

+C u1

(u₁+k(x1, x₂)k)⁴ Z

k(y1,y2)k>u1/2

t1

k(y1, y₂)k⁴dy₁dy₂

≤C t1

(u1+k(x1, x2)k)⁴,

which proves (7).

Lett, s >0 and (x₁, x₂)∈R². We now estimate Z

R

t

(t+k(x1, x2)−(0, y)k)⁴ s (s+|y|)²dy

≤ Z

|y|≤|x2|/2

t

(t²+|x1|²+|x2−y|)² s (s+|y|)²dy +

Z

|x2|/2<|y|≤2|x2|

· · ·+ Z

|y|>2|x2|

· · ·=E₁+E₂+E₃. ForE1, in this case, we have that|x2−y| ≥ |x2|/2 and

E₁≤ t

(t²+|x1|²+|x2|/2)² Z

R

s

(s+|y|)²dy≤C t

(t+k(x1, x2)k)⁴; forE3, by the fact that

t²+|x1|²+|x2−y| ≥t²+|x1|²+|x2| ≥(t+k(x1, x2)k)²/2, we also obtain an estimate similar toE1. ForE2, we have

E2≤C t t²+|x1|²

s (s+|x2|)²

Z

R

1

(1 +|y|)²dy≤C t (t+|x1|)²

s (s+|x2|)². Thus, for allt, s >0 and (x1, x2)∈R²,

Z

R

t

(t+k(x1, x2)−(0, y)k)⁴ s (s+|y|)²dy

≤C

t

(t+k(x₁, x₂)k)⁴ + t (t+|x₁|)²

s (s+|x₂|)²

. (11) Let M denote the Hardy-Littlewood maximal funnction on R². Now, the esti- mates (7), (8), and (11) and Lemma 2.3 yield that

ψ˜u₁u₂∗f(x1, x2)

≤C Z ∞

0

Z ∞ 0

u1

t1

∧ t1

u1

u2

t2

∧ t2

u2

× Z

R²

u1∨t1

(u1∨t1+k(z1, z2)k)⁴ + u1∨t1

(u1∨t1+|z1|)²

u2∨t2

(u2∨t2+|z2|)²

× |ψt1t2∗f(x₁−z₁, x₂−z₂)|dz₁dz₂dt₁ t1

dt₂ t2

≤C Z ∞

0

Z ∞ 0

u₁ t1

∧ t₁ u1

u₂ t2

∧ t₂ u2

× {M(ψ_t1t2∗f)(x₁, x₂) +M_s(ψ_t1t2∗f)(x₁, x₂)}dt1

t₁ dt2

t₂ , (12)

which together with the Minkowski inequality and the L^p(R²)-boundedness of M andM_syields that, forp∈(1,∞),

ψ˜u₁u₂∗f _Lp(

R²)

≤C Z ∞

0

Z ∞ 0

u₁ t1

∧ t₁ u1

u₂ t2

∧ t₂ u2

kψt1t2∗fkL^p(R²)

dt₁ t1

dt₂ t2

. (13) The estimate (13) combined with the Minkowski inequality shows that

kfk

0B˙^s_pq(R²)

≤C Z ∞

0

Z ∞ 0

Z u1

0

Z u2

0

t₁ u1

s1+1t₂ u2

s2+1

×t^−s₁ ¹t^−s₂ ²kψt1t2∗fkL^p(R²)

dt₁ t1

dt₂ t2

q

du₁ u1

du₂ u2

1/q

+C Z ∞

0

Z ∞ 0

Z u1

0

Z ∞ u₂

t₁ u1

s1+1t₂ u2

s2−1

· · · dt₁ t1

dt₂ t2

q

du₁ u1

du₂ u2

1/q

+C Z ∞

0

Z ∞ 0

Z ∞ u₁

Z u2

0

t₁ u1

^s1−1t₂ u2

^s2+1

· · · dt₁ t1

dt₂ t2

^q du₁ u1

du₂ u2

^1/q

+C Z ∞

0

Z ∞ 0

Z ∞ u₁

Z ∞ u₂

t1

u₁

^s1−1t2

u₂ ^s2−1

· · · dt1

t₁ dt2

t₂ ^q du1

u₁ du2

u₂ ^1/q

=F1+F2+F3+F4.

The H¨older inequality and the assumption that si > −1 for i = 1,2 further imply that

F1≤C Z ∞

0

Z ∞ 0

t^−s₁ ^1qt^−s₂ ^2q

ψt₁t₂∗f

q

L^p(R²)

× Z ∞

t₁

Z ∞ t₂

t₁ u1

s1+1t₂ u2

s2+1

du₁ u1

du₂ u2

dt₁ t1

dt₂ t2

1/q

≤C Z ∞

0

Z ∞ 0

t^−s₁ ^1qt^−s₂ ^2q

ψ_t1t2∗f

q

L^p(R²)

dt₁ t1

dt₂ t2

^1/q

=CkfkB˙^s_pq(R²), where, and in what follows, we denote by q⁰ the conjugate index of q, namely, 1/q+ 1/q⁰= 1.

The same argument as for F₁ also yields the desired estimates for F_i with i = 2,3,4. This proves (5) and hence, by symmetry, the independence of the normk·kB˙_pq^s (R²)with respect to the choice ofψ⁽ⁱ⁾, fori= 1,2.

We now turn to the proof of (6). From the estimate (12), it follows that, for all u₁, u₂>0 and (x₁, x₂)∈R²,

u^−s₁ ¹u^−s₂ ²

ψ˜_u1u2∗f(x₁, x₂)

≤C Z ∞

0

Z ∞ 0

u1

t₁ ∧ t1

u₁ u2

t₂ ∧ t2

u₂ t1

u₁

^s1t2

u₂ ^s2

×

M

t^−s₁ ¹t^−s₂ ²ψt₁t₂∗f

(x1, x2) +Ms

t^−s₁ ¹t^−s₂ ²ψt₁t₂∗f

(x1, x2) ^q dt1

t₁ dt2

t₂ ^1/q

,

which combined with Lemma 2.3 and (10) yields that kfk

0F˙_pq^s(R²)≤C

Z ∞ 0

Z ∞ 0

M

t^−s₁ ¹t^−s₂ ²ψ_t1t2∗f

+M_s

t^−s₁ ¹t^−s₂ ²ψ_t1t2∗f

^qdt₁ t1

dt₂ t2

^1/q _Lp(

R²)

≤C

Z ∞ 0

Z ∞ 0

t^−s₁ ^1qt^−s₂ ^2q|ψt₁t₂∗f|^q dt1

t1

dt2

t2

1/q _Lp(

R²)

=CkfkF˙_pq^s(R²),

where we have used the vector-valued inequality of Fefferman-Stein in [1]. This proves (6) and, by symmetry, the independence of the norm k·kF˙_pq^s(R²) with respect to the choice ofψ⁽ⁱ⁾fori= 1,2 . This finishes the proof of Theorem 2.6.

Based on Theorem 2.6, we now introduce the Besov space ˙B^s_pq(R²) and the Triebel- Lizorkin space ˙F_pq^s(R²) as follows.

Definition 2.7. Lets₁, s₂∈(−1,1) ands= (s₁, s₂). The Besov space ˙B_pq^s (R²) with p, q∈[1,∞] is defined by

B˙_pq^s (R²) ={f ∈ S∞,F(R²)⁰:kfkB˙_pq^s (R²)<∞ };

and the Triebel-Lizorkin space ˙F_pq^s(R²) withp∈(1,∞) andq∈(1,∞] is defined by F˙_pq^s(R²) ={f ∈ S∞,F(R²)⁰ :kfkF˙_pq^s(R²)<∞ }.

Theorem 2.6 shows that the definitions of the spaces ˙B_pq^s (R²) and ˙F_pq^s(R²) are independent of the choices ofψ⁽ⁱ⁾withi= 1,2.

From Lemma 2.3 and Lemma 2.4, we deduce the ‘discrete’ characterization of Besov spaces ˙B_pq^s (R²) and Triebel-Lizorkin spaces ˙F_pq^s(R²) as below.