The Journal of Fourier Analysis and Applications Volume 8, Issue 3, 2002

Embedding Theorem on Spaces of Homogeneous Type Youngshen Han and Chin-Cheng Lin Communicated by Guido Weiss α,q ABSTRACT. In [5] the embedding theorem for the Besov spaces B˙ p with −ε < α < ε and α,q 1 ≤ p, q ≤ ∞, and Triebel–Lizorkin spaces F˙p with −ε < α < ε and 1 < p, q < ∞, on spaces of homogeneous type was obtained. In this article the embedding theorem is generalized to α,q the Besov spaces B˙ p with p0 < p ≤ ∞ and 0 < q ≤ ∞ for p0 < 1, and the Triebel–Lizorkin α,q spaces F˙p with p1 < p < ∞ and p1 < q < ∞ for p1 < 1. The proofs are new even for Rn .

1.

Introduction

We begin by recalling the definitions necessary for introducing the Besov and Triebel– Lizorkin spaces on spaces of homogeneous type. A quasi-metric d on a set X is a function d : X × X  → [0, ∞] satisfying: (i) d(x, y) = 0 if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) there exists a constant A < ∞ such that for all x, y, and z ∈ X, d(x, y) ≤ A[d(x, z) + d(z, y)]. Any quasi-metric defines a topology, for which the balls B(x, r) = {y ∈ X : d(y, x) < r} form a base. However, the balls themselves need not to be open when A > 1.

Definition 1 ([2]). A space of homogeneous type (X, d, µ) is a set X together with a quasi-metric d and a nonnegative measure µ on X such that µ(B(x, r)) < ∞ for all x ∈ X and all r > 0, and there exists A < ∞ such that for all x ∈ X and all r > 0, µ(B(x, 2r)) ≤ A µ(B(x, r)) .

Math Subject Classifications. 42B25, 46F05. Keywords and Phrases. Besov and Triebel–Lizorkin spaces, discrete Calderón formula, embedding theorem, spaces of homogeneous type. Acknowledgements and Notes. This article was written while the authors were visiting Washington University in St. Louis.

c 2002 Birkh¨auser Boston. All rights reserved
ISSN 1069-5869

292

Youngshen Han and Chin-Cheng Lin

Here µ is assumed to be defined on a σ -algebra which contains all Borel sets and all balls B(x, r). Macias and Segovia [10] have shown that one can replace d by another quasi metric ρ such that there exist c < ∞ and some θ, 0 < θ < 1, ρ(x, y) ≈ inf{µ(B) : B is a ball containing x and y} ,
  
    
ρ(x, y) − ρ x , y
≤ cρ x, x  θ ρ(x, y) + ρ x  , y 1−θ for all x, x  , and y ∈ X , where the expression a ≈ b means, as usual, that there are constants c1 and c2 (independent of the main parameters involved) such that c1 ≤ a/b ≤ c2 . We will suppose that µ(X) = ∞ and µ({x}) = 0 for all x ∈ X. These hypotheses allow us to construct an approximation to the identity (see [8]).

Definition 2. A sequence {Sk }k∈Z of operators is called to be an approximation to the

identity if Sk (x, y), the kernels of Sk , are functions from X × X into C such that there exist a constant C, some 0 < ε ≤ θ, and some c < ∞, for all k ∈ Z and all x, x  , y, and y  ∈ X, (i) Sk (x, y) = 0 if ρ(x, y) ≥ c2−k and Sk ∞ ≤ C2k , (ii) |Sk (x, y) − Sk (x  , y)| ≤ C2k(1+ε) ρ(x, x  )ε , (iii) |Sk (x, y) − Sk (x, y  )| ≤ C2k(1+ε) ρ(y, y  )ε , (iv) |[Sk (x, y) − Sk (x, y  )] − [Sk (x  , y) − Sk (x  , y  )]| ≤ Cρ(x, x  )ε ρ(y, y  )ε 2k(1+2ε) ,  (v) Sk (x, y)dµ(y) = 1, X (vi) Sk (x, y)dµ(x) = 1. X

See [3] for the existence of such a sequence of operators, there all conditions are introduced and checked except the condition (iv) in Definition 2. It is easy to see that the same construction in [3] satisfies the condition (iv). To define the Besov and Triebel– Lizorkin spaces on spaces of homogeneous type we need the following definition (see [8]).

Definition 3. Fix two exponents 0 < β ≤ θ and γ > 0. A function f defined on X is said to be a strong smooth molecule (or test function) of type (β, γ ) centered at x0 ∈ X with width d > 0, if f satisfies the following conditions:

dγ , (d + ρ(x, x0 ))1+γ  ρ(x, x  ) β dγ 1 (ii) |f (x) − f (x  )| ≤ c (d + for ρ(x, x  ) ≤ d + ρ(x, x0 ) (d + ρ(x, x0 ))1+γ 2A ρ(x, x0 )),  (i) |f (x)| ≤ c

f (x)dµ(x) = 0.

(iii) X

This definition was first introduced in [9] for the case X = Rn with the condition (ii) in Definition 3 replaced by 


 
ρ x, x  β dγ dγ
f (x) − f x 
≤ c +  1+γ . d (d + ρ(x, x0 ))1+γ d + ρ x  , x0 The collection of all strong smooth molecules of type (β, γ ) centered at x0 ∈ X with width d > 0 will be denoted by M(β,γ ) (x0 , d). If f ∈ M(β,γ ) (x0 , d), the norm of

Embedding Theorem on Spaces of Homogeneous Type

293

f ∈ M(β,γ ) (x0 , d) is then defined by  f M(β,γ ) (x0 ,d) = inf c ≥ 0 : (i) and (ii) of Definition 3 hold . Now we fix a point x0 ∈ X and denote the class of all f ∈ M(β,γ ) (x0 , 1) by It is easy to see that M(β,γ ) is a Banach space under the norm f M(β,γ ) < ∞. Just as the space of distributions S  defined on Rn , the dual space (M(β,γ ) ) consists of all linear functionals L from M(β,γ ) to C with the property that there exists a finite constant c such that |L(f )| ≤ c f M(β,γ ) for all f ∈ M(β,γ ) . We denote the natural pairing of elements h ∈ (M(β,γ ) ) and f ∈ M(β,γ ) by h, f . It is also easy to see that M(β,γ ) (x1 , d) = M(β,γ ) with equivalent norms for x1 ∈ X and d > 0. Thus, h, f  is well defined for all h ∈ (M(β,γ ) ) and all f ∈ M(β,γ ) (x1 , d) with x1 ∈ X and d > 0. In [8] the Besov and Triebel–Lizorkin spaces on homogeneous type were introduced by use of the family of operators {Dk }k∈Z , where Dk = Sk − Sk−1 and {Sk }k∈Z is an approximation to the identity defined in Definition 2. More precisely, the Besov space α,q B˙ p for −ε < α < ε and 1 ≤ p, q ≤ ∞ is the collection of all f ∈ (M(β,γ ) ) with 0 < β, γ < ε such that M(β,γ ) .

f B˙ pα,q =





2kα Dk (f ) p

q

1/q <∞.

k∈Z

The Triebel–Lizorkin space F˙p for −ε < α < ε and 1 < p, q < ∞ is the collection of all f ∈ (M(β,γ ) ) with 0 < β, γ < ε such that α,q



  q 1/q   kα  <∞. f F˙pα,q =  |D (f )| 2 k   k∈Z

p

In [5] the first author used the continuous version of the Calderón reproducing formula α,q developed in [8] to get the following embedding theorem for the Besov spaces B˙ p with α,q ˙ 1 ≤ p, q ≤ ∞, and Triebel–Lizorkin spaces Fp with 1 < p, q < ∞ on spaces of homogeneous type.

Theorem 1. Suppose that −ε < s1 < s2 < ε. Then the following two embedding maps are continuous: s ,q s ,q (i) B˙ p22  → B˙ p11 for 1 ≤ q ≤ ∞, 1 ≤ p1 , p2 ≤ ∞, and −ε < s2 − 1/p2 = s1 − 1/p1 < ε; s ,q s ,q (ii) F˙p22 2  → F˙p11 1 for 1 < p1 , p2 , q1 , q2 < ∞ and −ε < s2 − 1/p2 = s1 − 1/p1 < ε.

In [6], using the discrete Calderón type reproducing formula obtained in [7], the Besov α,q 1 1 spaces B˙ p were generalized to the case where −ε < α < ε, max{ 1+ε , 1+ε+α } < p ≤ ∞, α,q and 0 < q ≤ ∞, and the Triebel–Lizorkin spaces F˙p were generalized to the case where 1 1 −ε < α < ε, max{ 1+ε , 1+ε+α } < p, q < ∞. In this article we are going to show the α,q following embedding theorem for the Besov spaces B˙ p and the Triebel–Lizorkin spaces α,q ˙ Fp .

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Youngshen Han and Chin-Cheng Lin

Theorem 2. Suppose that −ε < s1 < s2 < ε. Then the following two embedding maps are continuous: s ,q s ,q 1 1 1 1 , 1+ε+s } < p1 ≤ ∞, max{ 1+ε , 1+ε+s } (i) B˙ p22  → B˙ p11 for 0 < q ≤ ∞, max{ 1+ε 2 1 < p2 ≤ ∞, and −ε < s2 − 1/p2 = s1 − 1/p1 < ε; s ,q s ,q 1 1 1 1 , 1+ε+s } < (ii) F˙p22 2  → F˙p11 1 for max{ 1+ε , 1+ε+s } < p1 , q1 < ∞, max{ 1+ε 1 2 p2 , q2 < ∞, and −ε < s2 − 1/p2 = s1 − 1/p1 < ε.

2.

Preliminaries

The proof of Theorem 1 based on the continuous version of the Calderón type reproducing formula on spaces of homogeneous type obtained in [8]. The new idea to prove this type of theorem is that we use the discrete version of the Calderón type reproducing formula α,q on spaces of homogeneous type obtained in [7] to deal with the Besov spaces B˙ p for the 1 1 case where −ε < α < ε, max{ 1+ε , 1+ε+α } < p ≤ ∞, and 0 < q ≤ ∞, and the Triebel– α,q 1 1 ˙ , 1+ε+α } < p, q < ∞. We Lizorkin spaces Fp for the case where −ε < α < ε, max{ 1+ε need the discrete Calderón type reproducing formula on spaces of homogeneous type. To state the discrete Calderón type reproducing formula on spaces of homogeneous type we first recall a result of Christ [1], which is an analogue of the Euclidean dyadic cubes.

Theorem 3. There exist a collection of open subsets {Qkτ ⊂ X : k ∈ Z, τ ∈ Ik }, where Ik denotes some (possibly finite) index set depending on k, and constants δ ∈ (0, 1), a0 > 0, η > 0, c > 0 such that  (i) µ(X\ Qkτ ) = 0 for all k ∈ Z; τ

(ii)

j

j

if j ≥ k, then either Qτ  ⊂ Qkτ or Qτ  ∩ Qkτ = φ;

(iii) for each (k, τ ) and each j < k, there is a unique τ  such that Qkτ ⊂ Qτ  ; j

(iv) diameter (Qkτ ) ≤ cδ k ; (v) each Qkτ contains some ball B(zτk , a0 δ k ).

We fix such a collection of open subsets and call all Qkτ in Theorem 3 the “dyadic cubes” in X. Without loss of generality, we may assume δ = 21 in Theorem 3. Let i be a fixed positive integer, and denote by yτk+i the point in Qk+i τ . The discrete Calderón type reproducing formula on spaces of homogeneous type is the following.

Theorem 4 ([7]). Suppose that {Sk }k∈Z is an approximation to the identity defined in Definition 2. Set k }k∈Z such that for all fixed Dk = Sk − Sk−1 . Then there exists a family of operators {D k+i k+i (β,γ )  ) with 0 < β, γ < ε (ε is the regularity exponent of Sk ), yτ ∈ Qτ and all f ∈ (M f (x) =

k∈Z τ ∈Ik+i

      k x, yτk+i Dk (f ) yτk+i , µ Qk+i D τ

(2.1)

  k (x, y), k ∈ where the series converges in (M(β ,γ ) ) with β  > β and γ  > γ . Moreover, D  Z, satisfy the following estimates: for 0 < ε < ε, there exists a constant c > 0 depending

295

Embedding Theorem on Spaces of Homogeneous Type

only on ε and ε  such that


D k (x, y)
≤ c 

2−kε



1+ε , 2−k + ρ(x, y)   ρ x, x   ε

2−kε
D k (x  , y)
≤ c k (x, y) − D 1+ε  2−k + ρ(x, y) 2−k + ρ(x, y)  1  −k for ρ(x, x  ) ≤ 2 + ρ(x, y) ,  2A k (x, y)dµ(x) = k (x, y)dµ(y) = 0 for all k ∈ Z . D D X

(2.2)

X

Finally, we need the following result.

Theorem 5 ([6]). Suppose that {Sk }k∈Z is an approximations to the identity and Dk = Sk − Sk−1 . For in X, f ∈ (M(β,γ ) ) with 0 < β, γ < ε and dyadic cubes Qk+i τ 1 1 , 1+ε+α } < p ≤ ∞ and 0 < q ≤ ∞, then (i) if −ε < α < ε, max{ 1+ε



kα

≈

   
p q/p 1/q    k+i −α+1/p

Dk (f ) y k+i
; µ Qτ τ

2 Dk (f ) p

k∈Z

k∈Z

(ii)

q

1/q



τ ∈Ik+i

1 1 , 1+ε+α } < p, q < ∞, then if −ε < α < ε and max{ 1+ε



   kα q 1/q    2 |D (f )| k   k∈Z

p

  q 1/q      k+i −α
 k+i 


 .  Dk (f ) yτ µ Qτ χQk+i ≈  τ k∈Z τ ∈Ik+i

3.

p

The Proof of Main Theorem

For f ∈ B˙ p , the discrete version of Calderón type reproducing formula (2.1) gives

    k+i  µ(Qk+i Dk (f ) yτk+i , (3.1) Dj (f )(x) = τ )Dj Dk x, yτ s,q

k∈Z τ ∈Ik+i

k (x, yτk+i ) first. Thus, to calculate Dj f p , we have to estimate Dj D

Lemma 1.

k (x, y), the kernel of Dj D k , For 0 < ε < ε, there exists a constant c such that Dj D satisfies the estimate


Dj D k (x, y)
≤ c2−|j −k|ε 

2−(j ∧k)ε



1+ε ,

2−(j ∧k) + ρ(x, y)

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Youngshen Han and Chin-Cheng Lin

where a ∧ b denotes the minimum of a and b.

Proof.

Notice that the kernel of Dj satisfies the conditions (i)−(iv) of Definition 2 and   Dj (x, y) dµ(y) = Dj (x, y) dµ(x) = 0 . X

X

We show the case j ≥ k only; the proof for the case j < k is similar. For j ≥ k, we consider first 2cA2−j ≤ 2−k or ρ(x, y) ≥ 2cA2−k . Then cancellation property Dj (x, z) dµ(z) = 0 yields






Dj D  k (x, y)
=
Dj (x, z)Dk (z, y) dµ(z)



X


 
  =
Dj (x, z) Dk (z, y) − Dk (x, y) dµ(z)

. X

By the fact that ρ(x, z) ≤ c2−j ≤ k , we have of the kernel of D

D j D k (x, y)| ≤ c

 X

1 −k 2A (2


|Dj (x, z)


+ ρ(x, y)) and the smoothness property (2.2)

ρ(x, z)ε



2−kε

ε  

2−k + ρ(x, y)



1+ε dµ(z) .

2−k + ρ(x, y)

The size condition of the kernel of Dj in Definition 2 (i) gives


Dj D k (x, y)
≤ c 

2−j ε



1+ε

2−k + ρ(x, y)

2−kε



= c2−(j −k)ε 



1+ε

2−k + ρ(x, y)

which shows the case for 2cA2−j ≤ 2−k or ρ(x, y) ≥ 2cA2−k . Consider now the case that 2cA2−j > 2−k and ρ(x, y) < 2cA2−k . The size condik give tions on the kernel of Dj and D






≤ c2k .
Dj D k (x, y)
=
 D (x, z) D (z, y) dµ(z) k j

X

By the facts that 2

(j −k)ε

ε

< (2cA) and ρ(x, y) < 2cA2−k , we get




Dj D k (x, y)
≤ c2(k−j )ε 

2−kε



1+ε ,

2−k + ρ(x, y)

which together with the previous estimate shows the case for j ≥ k.

Lemma 2. Let {aj k }j,k∈Z and {bk }k∈Z be nonnegative sequences. If

(aj k )q + aj k < ∞ for all 0 < q ≤ 1 , j ∈Z

then

k∈Z

j ∈Z

k∈Z

q aj k bk

≤C

(bk )q k∈Z

for all q > 0 .

297

Embedding Theorem on Spaces of Homogeneous Type

Proof.

For 0 < q ≤ 1, using the inequality (x + y)q ≤ x q + y q , we have

q



q q

q q aj k bk = (aj k bk ) = aj k bk ≤ C bk . j

k

j

k

k

j

k

For q > 1, let 1/q + 1/q  = 1 and apply Hölder’s inequality to get

j

q aj k bk



1/q  1/q  a j k b k aj k

=

k

j

≤

C

k

j

q ≤

q

aj k b k ≤ C

k

j

q aj k b k



k

q/q  aj k

k

q

bk .

k

1 1 , 1+ε+s } < p1 ≤ ∞, and Let −ε < s1 < s2 < ε, 0 < q ≤ ∞, max{ 1+ε 1 s2 ,q 1 1 ˙ max{ 1+ε , 1+ε+s2 } < p2 ≤ ∞. Suppose that f ∈ Bp2 and −ε < s2 −1/p2 = s1 −1/p1 < ε. If p1 > 1, using (3.1) and the estimate in Lemma 1, we have




 k  +i   k+i k  +i    k  +i 

Dk (f ) y k+i
=
   µ Qτ  Dk Dk yτ , yτ  Dk (f ) yτ 

τ
k  ∈Z τ  ∈Ik  +i

≤c



k  ∈Z τ  ∈Ik  +i







2−|k−k |ε µ(Qkτ +i ) 





 2−(k∧k )ε
Dk  (f ) y k +i
,    τ  1+ε  2−(k∧k ) + ρ yτk+i , yτk +i

×

where ε is the regularity exponent in the Calderón type reproducing formula satisfying 1 1 −ε < −ε  < s1 < s2 < ε < ε, −ε  < s2 − 1/p2 = s1 − 1/p1 < ε , max{ 1+ε  , 1+ε  +s } < 1 1 1 k+i ) ≈ 2−k , and Minkowski’s p1 , and max{ 1+ε  , 1+ε  +s } < p2 . By Theorem 5, µ(Qτ 2 p 1 inequality for l , we thus have

f B˙ s1 ,q ≤ c p1

≤c

 k∈Z

τ ∈Ik+i

 k∈Z

×

τ ∈Ik+i

2

q/p1 1/q (−s1 +1/p1 )p1

 
Dk (f ) y k+i
p1 µ Qk+i τ τ (−s1 +1/p1 )p1  µ Qk+i τ −(k∧k  )ε



k  ∈Z τ  ∈Ik  +i

τ ∈Ik+i

     2−|k−k |ε µ Qkτ +i

 


 
p1 q/p1 1/q
Dk  (f ) y k +i


 1+ε   2−(k∧k ) + ρ yτk+i , yτk +i   1/p1  =c 2ks1 q µ Qk+i τ k∈Z

k  ∈Z

τ

τ  ∈Ik  +i

     2−|k−k |ε µ Qkτ +i



  

p1 q/p1 1/q  2−(k∧k )ε
Dk  (f ) y k +i
   τ  1+ε 2−(k∧k ) + ρ yτk+i , yτk +i          ≤c 2ks1 q 2−|k−k |ε µ Qk+i µ Qkτ +i τ ×

k∈Z

k  ∈Z

τ ∈Ik+i

τ  ∈Ik  +i



    k+i k  +i 1+ε
 k  +i 
p1 1/p1 q 1/q 2−(k∧k )ε  −(k∧k  )

2 + ρ yτ , yτ  . Dk  (f ) yτ  × r

(3.2)

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Youngshen Han and Chin-Cheng Lin





It is easy to see that, by ρ(yτk+i , yτk +i ) ≈ ρ(yτk+i , y) for y ∈ Qkτ +i ,

τ  ∈Ik  +i

=



 

Qkτ +i

τ  ∈Ik+i



≤c

τ  ∈Ik+i



X









2−(k∧k )ε  1+ε dµ(y)    2−(k∧k ) + ρ yτk+i , yτk +i



2−(k∧k )ε  1+ε dµ(y)   2−(k∧k ) + ρ yτk+i , y

Qkτ +i



=c



2−(k∧k )ε  1+ε   2−(k∧k ) + ρ yτk+i , yτk +i

   µ Qkτ +i 



2−(k∧k )ε   1+ε dµ(y) ≤ c ,  2−(k∧k ) + ρ yτk+i , y

and similarly

τ ∈Ik+i





2−(k∧k )ε 1+ε    2−(k∧k ) + ρ yτk+i , yτk +i

  µ Qk+i  τ 



≤c X



2−(k∧k )ε 1+ε dµ(y) ≤ c .     2−(k∧k ) + ρ y, yτk +i

For p2 > 1, let p2 be its conjugate index. We then apply Hölder’s inequality to 



τ ∈Ik+i

  µ Qk+i τ

τ  ∈Ik  +i



τ

to get



2−(k∧k )ε  1+ε   2−(k∧k ) + ρ yτk+i , yτk +i

   µ Qkτ +i 

p2 1/p2




  
×
Dk  (f ) yτk +i
≤

 τ ∈Ik+i



  µ Qk+i τ



τ  ∈Ik  +i





2−(k∧k )ε 1+ε    2−(k∧k ) + ρ yτk+i , yτk +i

   µ Qkτ +i 





p2 /p2


 k  +i 
p2 2−(k∧k )ε  k+i k  +i 1+ε
Dk  (f ) yτ 
 2−(k∧k ) + ρ yτ , yτ  τ  ∈Ik  +i

       k+i  2−(k∧k )ε ≤c µ Qkτ +i µ Qτ  1+ε    2−(k∧k ) + ρ yτk+i , yτk +i τ ∈Ik+i τ  ∈Ik  +i 
 k  +i 
p2 1/p2

× Dk  (f ) yτ  ×

≤c

 τ  ∈Ik  +i

   µ Qkτ +i 

  
  
p µ Qkτ +i
Dk  (f ) yτk +i
2

1/p2 .

1/p2

299

Embedding Theorem on Spaces of Homogeneous Type

On the other hand, we also have  
 
2−(k∧k )ε
Dk  (f ) y k +i
 τ  +i 1+ε   k+i k 2−(k∧k ) + ρ yτ , yτ  τ  ∈Ik  +i

p2 1/p2     k  +i  2−(k∧k )ε µ Qτ  ≤ 1+ε     2−(k∧k ) + ρ yτk+i , yτk +i τ  ∈Ik  +i     
  
p 1/p2 × µ Qkτ +i
Dk  (f ) yτk +i
2

   µ Qkτ +i 

τ  ∈Ik  +i

 ≤c X

×





τ  ∈Ik  +i

≤ c2



2−(k∧k )ε  1+ε   2−(k∧k ) + ρ yτk+i , y

p2

1/p2 dµ(y)

  
  
p µ Qkτ +i
Dk  (f ) yτk +i
2

−(k∧k  )(1/p2 −1)



1/p2



µ

τ  ∈Ik  +i

  
p 
 Qkτ +i
Dk  (f ) yτk +i
2

1/p2 ,

where the integral converges due to p2 > 1/(1 + ε ). Together with the above two inequalities, by the Riesz–Thorin interpolation theorem, we obtain     k  +i  2−(k∧k )ε k+i µ(Qτ ) µ Qτ   1+ε    2−(k∧k ) + ρ yτk+i , yτk +i τ ∈Ik+i τ  ∈Ik  +i


 k  +i 
p1 1/p1

(3.3) × Dk  (f ) yτ  

≤ c2−(k∧k )(1/p1 −1/p2 )

 τ  ∈Ik  +i

  
p   
µ Qkτ +i
Dk  (f ) yτk +i
2

1/p2

for p1 > p2 > 1. Hence, (3.2) and (3.3) yield     f B˙ s1 ,q ≤ c 2ks1 q 2−|k−k |ε 2−(k∧k )(1/p1 −1/p2 ) p1

k∈Z

×

 τ  ∈Ik  +i

=c

 k∈Z

×

 k∈Z


p   
 µ Qkτ +i
Dk  (f )(yτk +i )
2 





1/p2 q 1/q



2ks1 2−|k−k |ε 2−(k∧k )(1/p1 −1/p2 ) 2−k s2

k  ∈Z



τ  ∈Ik  +i

≡c

k  ∈Z

k  ∈Z

  (−s2 +1/p2 )p2
 

Dk  (f ) y k +i
p2 µ Qkτ +i τ

q 1/q akk  bk 

,

1/p2 q 1/q

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Youngshen Han and Chin-Cheng Lin

where 







akk  = 2ks1 2−|k−k |ε 2−(k∧k )(1/p1 −1/p2 ) 2−k s2 and 

bk  =

τ  ∈Ik  +i


  (−s2 +1/p2 )p2

Dk  (f ) y k +i
p2 µ Qkτ +i τ

1/p2 .

  (q∧1) Since s1 < s2 < ε  , both k  akk  and k akk  converge for all q > 0. We now use Hölder’s inequality for q > 1, and the inequality (x + y)q ≤ x q + y q for q ≤ 1, so that

q ≤c

akk  bk 

k  ∈Z

k  ∈Z

(q∧1) q bk 

for all q > 0 .

akk 

This together with Theorem 5 give us

f B˙ s1 ,q ≤ c



p1

=c

k∈Z k  ∈Z

1/q (q∧1) q akk  bk 

 k  ∈Z

τ  ∈Ik  +i

≤ c f B˙ s2 ,q

≤c

k  ∈Z

q bk 


  (−s2 +1/p2 )p2

Dk  (f ) y k +i
p2 µ Qkτ +i τ

q/p2 1/q

p 1 > p2 > 1 .

for

p2

1/q



If p1 > 1 ≥ p2 , by inequality (3.2) and the fact (x + y)p2 /p1 ≤ x p2 /p1 + y p2 /p1 , we obtain f B˙ s1 ,q ≤ c p1



2ks1 q





2−|k−k |ε

k  ∈Z

k∈Z



×

2



 τ ∈Ik+i

−(k∧k  )ε



  µ Qk+i τ

τ  ∈Ik  +i

   µ Qkτ +i

p1 1/p1 q 1/q




Dk  (f ) y k +i


τ  1+ε   2−(k∧k + ρ yτk+i , yτk +i   p2 /p1       ≤c 2ks1 q 2−|k−k |ε µ Qk+i µ Qkτ +i τ k  ∈Z

k∈Z

2

×

τ  ∈Ik  +i

τ ∈Ik+i

−(k∧k  )ε

p2 1/p2 q 1/q


 k  +i 


 Dk  (f ) y 

1+ε    2−(k∧k ) + ρ yτk+i , yτk +i     ≤c 2ks1 q 2−|k−k |ε 2−k/p1 k  ∈Z

k∈Z



×

2

τ ∈Ik+i τ  ∈Ik  +i −(k∧k  )ε

 1+ε   2−(k∧k ) + ρ yτk+i , yτk +i



τ

p2

  p µ Qkτ +i 2


 

Dk  (f ) y k +i
p2 τ

1/p2 q 1/q

301

Embedding Theorem on Spaces of Homogeneous Type

≤c



2ks1 q

k  ∈Z

k∈Z

×





2−|k−k |ε 2−k/p1 2k/p2

 τ  ∈Ik  +i



  µ Qk+i  τ

2

  p
  
p µ Qkτ +i 2
Dk  (f ) yτk +i
2

p2 1/p2 q 1/q

−(k∧k  )ε

1+ε    2−(k∧k ) + ρ yτk+i , yτk +i     p
  
p   ≤c 2ks1 q 2−|k−k |ε 2−k/p1 2k/p2 µ Qkτ +i 2
Dk  (f ) yτk +i
2 τ ∈Ik+i

k  ∈Z

k∈Z

τ  ∈Ik  +i

1/p2 q 1/q   2−(k∧k )ε dµ(y) × 1+ε     X 2−(k∧k ) + ρ y, yτk +i     ≤c 2ks1 q 2−|k−k |ε 2−k/p1 2k/p2 2−(k∧k )(1/p2 −1) 

k  ∈Z

k∈Z

×

 τ  ∈Ik  +i

≤c
















  (−s2 +1/p2 )p2

Dk  (f ) y k +i
p2 µ Qkτ +i τ

1/p2 q 1/q

q 1/q dkk  bk 

,

k  ∈Z

k∈Z

where

1/p2 q 1/q

2ks1 2−|k−k |ε 2−k/p1 2k/p2 2−(k∧k )(1/p2 −1) 2−k (1+s2 −1/p2 )

τ  ∈Ik  +i

≡c

  
p   p
µ Qkτ +i 2
Dk  (f ) yτk +i
2

k  ∈Z

k∈Z

×

p2









dkk  = 2ks1 2−|k−k |ε 2−k/p1 2k/p2 2−(k∧k )(1/p2 −1) 2−k (1+s2 −1/p2 )

and bk  =

 τ  ∈Ik  +i


  (−s2 +1/p2 )p2

Dk  (f ) y k +i
p2 µ Qkτ +i τ

1/p2

is the same as before. It is easy to check that

(q∧1)

dkk 

<∞



and

q ≤c

dkk  bk 

k  ∈Z

k

k  ∈Z

(q∧1) q bk 

dkk 

for all q > 0. Use Theorem 5 again to get f B˙ s1 ,q ≤ c



p1

=c

k∈Z k  ∈Z

1/q (q∧1) q bk 

 k  ∈Z

≤ c f B˙ s2 ,q p2

≤c

dkk 

τ  ∈Ik  +i

 k  ∈Z

1/q q

bk 

  (−s2 +1/p2 )p2
 

Dk  (f ) y k +i
p2 µ Qkτ +i τ for

p1 > 1 ≥ p2 .

q/p2 1/q

302

Youngshen Han and Chin-Cheng Lin

If p1 ≤ 1, by (3.1) and the estimate in Lemma 1, we find Dj (f )(x) p1

p1   
  k+i 
p1 1/p1 k+i p1

Dj D  x, y ≤ µ Qk+i (f ) y D k k τ τ τ p 1

k∈Z τ ∈Ik+i

≤c

k∈Z τ ∈Ik+i

p1

  
Dk (f ) y k+i
p1 2−|j −k|ε p1 2(j ∧k)(p1 −1) µ Qk+i τ τ

1/p1 .

Thus, =

f

s ,q B˙ p11





2

j s1

Dj (f ) p1

q

1/q

j ∈Z

≤c





2j s1 p1 2−|j −k|ε p1 2(j ∧k)(p1 −1)

k∈Z τ ∈Ik+i

j ∈Z


 p1

Dk (f ) y k+i
p1 × µ Qk+i τ τ ≤c

 j ∈Z

×

q/p1 1/q



2j s1 p1 2−|j −k|ε p1 2(j ∧k)(p1 −1) 2−k(s2 −1/p2 +1)p1

k∈Z



τ ∈Ik+i

(−s2 +1/p2 )p2

 
Dk (f ) y k+i
p2 µ Qk+i τ τ

p1 /p2 q/p1 1/q

−k and p < p . Using the inequality since µ(Qk+i 1 2 τ )≈2

 q  2j s1 p1 2−|j −k|ε p1 2(j ∧k)(p1 −1) 2−k(s2 −1/p2 +1)p1 j ∈Z

+



2j s1 p1 2−|j −k|ε p1 2(j ∧k)(p1 −1) 2−k(s2 −1/p2 +1)p1 < ∞

for all q > 0 ,

k∈Z

we apply Lemma 2 to get f B˙ s1 ,q ≤ c



p1

k∈Z

τ ∈Ik+i

(−s2 +1/p2 )p2


Dk (f ) y k+i
p2 Qk+i τ τ

µ



q/p2 1/q

for all q > 0. This, by Theorem 5, proves Theorem 2 (i). To prove Theorem 2 (ii), the homogeneity, it suffices to take f F˙ s2 ,q2 = 1. By p2 the discrete version of the Calderón type reproducing formula (2.1) and the estimate in Lemma 1, we have


  k+i 
   k+i
 |Dj (f )(x)| =

µ Qk+i (f ) y D D x, y D j k k τ τ τ
k∈Z τ ∈Ik+i

≤c

k∈Z τ ∈Ik+i

2

−|j −k|ε




 k+i 
2−(j ∧k)ε

.    Dk (f ) yτ 1+ε 2−(j ∧k) + ρ x, yτk+i

  µ Qk+i  τ

(3.4)

Embedding Theorem on Spaces of Homogeneous Type

303

1 1 To estimate the last expression in (3.4), we claim that, for max{ 1+ε  , 1+ε  +s } < r ≤ 1, 2




 k+i 
2−(j ∧k)ε

   Dk (f ) yτ 1+ε k+i −(j ∧k) + ρ x, yτ 2 τ ∈Ik+i

r 1/r 


Dk (f ) y k+i
χ k+i (y) ≤ c2−k(1−1/r) 2(j ∧k)(1−1/r) inf M , τ Qτ   µ Qk+i  τ

y∈B

(3.5)

τ ∈Ik+i

where B = B(x, 2−(j ∧k) ) and M is the Hardy–Littlewood maximal function. To prove −k (3.5), we have, by the fact r ≤ 1 and µ(Qk+i τ )≈2 , 

 2−(j ∧k)ε
Dk (f ) y k+i
   τ k+i 1+ε −(j ∧k) 2 + ρ x, yτ τ ∈Ik+i 

r  
 k+i 
r 1/r   k+i r 2−(j ∧k)ε

≤ Dk (f ) yτ µ Qτ  1+ε  2−(j ∧k) + ρ x, yτk+i τ

r      k+i r−1 2−(j ∧k)ε µ Qτ =  1+ε  Qk+i 2−(j ∧k) + ρ x, yτk+i τ τ 1/r
 k+i 
r

× Dk (f ) yτ dµ(y)

  µ Qk+i  τ

≤c





r−1



µ(Qk+i τ )

2−(j ∧k)ε



r



1+ε

2−(j ∧k) + ρ(x, y) 1/r
 k+i 
r

× Dk (f ) yτ χQk+i (y) dµ(y) τ X

τ

≤ c2−k(1−1/r)



2−(j ∧k)ε






Dk (f ) y k+i
r χ × τ

= c2

−k(1−1/r)

1+ε

2−(j ∧k) + ρ(x, y)

X

τ

r





1/r

(y) dµ(y) Qk+i τ 2−(j ∧k)ε

r



1+ε 2−(j ∧k) + ρ(x, y)


 

Dk (f ) y k+i
r χ k+i (y) dµ(y) × τ Qτ B

τ

+





∞ 

2−(j ∧k)ε




 

Dk (f ) y k+i
r χ × τ

≤ c2

−k(1−1/r)

 2

1+ε

2−(j ∧k) + ρ(x, y)

m+1 B\2m B m=0 2

τ

r



(j ∧k)r

µ(B) µ(B)

1/r

(y) dµ(y) Qk+i τ 
 

Dk (f ) y k+i
r χ B

τ

τ

(y) dµ(y) Qk+i τ

304

Youngshen Han and Chin-Cheng Lin ∞

+





2((j ∧k)−m)(1+ε )r 2−(j ∧k)ε r

m=0

1/r 


 
µ(2m+1 B)
Dk (f ) y k+i
r χ k+i (y) dµ(y) τ Qτ µ(2m+1 B) 2m+1 B τ  1/r ∞

−k(1−1/r) (j ∧k)r −(j ∧k) (j ∧k)r −(j ∧k) −m[r(1+ε )−1] 2 ≤ c2 2 + 2 2 2 ×

m=0

r  1/r


Dk (f ) y k+i
χ k+i (x  ) × M τ Qτ τ

≤ c2

−k(1−1/r) (j ∧k)(1−1/r)

2



r 
 k+i 
   1/r

M , Dk (f ) yτ χQk+i x τ τ

where 2m B = B(x, 2m 2−(j ∧k) ) and x  is any point in B. Taking infimum over B shows the claim (3.5). Inequalities (3.4) and (3.5) yield

|Dj (f )(x)| ≤ c



2−|j −k|ε 2−k(1−1/r) 2(j ∧k)(1−1/r) 2−ks2

k∈Z

 r 1/r  k+i −s2
 k+i 


Dk (f ) yτ χQk+i × inf M µ Qτ (y) τ y∈B

τ ∈Ik+i

1 1 1 1 for max{ 1+ε Choose r satisfying max{ 1+ε  , 1+ε  +s } < r ≤ 1.  , 1+ε  +s } < r < 2 2 min{p2 , q2 } and r ≤ 1, and denote

r 1/r   k+i −s2
 k+i 


µ Qτ (x) . Dk (f ) yτ χQk+i Fk (x) = M τ τ ∈Ik+i

We now have N 



 q 2j s1
Dj (f )(x)
1

j =−∞

≤c

N j =−∞

2j s1 q1

 k∈Z

1/q1 (3.6) q1 1/q1



2−|j −k|ε 2−k(1−1/r) 2(j ∧k)(1−1/r) 2−ks2 inf Fk (y) y∈B

Since r < min{p2 , q2 }, we use the Fefferman–Stein vector-valued maximal theorem [4]

305

Embedding Theorem on Spaces of Homogeneous Type

and get 

inf Fk (y) ≤ inf

y∈B

q Fk  (y) 2



y∈B

k  ∈Z

≤ µ(B)

−1/p2

  

q2

p2 /q2

Fk  (x)

B

≤ µ(B)

1/q2

1/p2 dµ(x)

k  ∈Z

     q2 1/q2    Fk (x)  

−1/p2 

k  ∈Z

p2



q21/q2    k  +i −s2
 
 .
Dk  (f ) y k  +i
χ k +i µ Q ≤ cµ(B)−1/p2  τ τ   Qτ p2

k  ∈Z τ ∈Ik  +i

Applying Theorem 5, we obtain inf Fk (x) ≤ cµ(B)−1/p2 f F˙ s2 ,q2 ≤ c2(j ∧k)/p2 , p2

x∈B

which together with (3.6) shows N

(2

j s1

|Dj (f )(x)|)

j =−∞

≤c

N

2j s1 q1

j =−∞

q1



1/q1



2−|j −k|ε 2−k(1−1/r) 2(j ∧k)(1−1/r) 2−ks2 2(j ∧k)/p2

q1 1/q1 .

k∈Z

Since ε + s2 − 1/p2 > 0 and r > 1/(1 + ε  + s2 ),



2−|j −k|ε 2−k(1−1/r) 2(j ∧k)(1−1/r) 2−ks2 2(j ∧k)/p2 ≤ c2−j (s2 −1/p2 ) .

k∈Z

Thus, N

(2

j s1

|Dj (f )(x)|)

q1

1/q1

≤c

j =−∞

N

2

j s1 q1 −j (s2 −1/p2 )q1

2

j q1 /p1

2

j =−∞

≤c

N

j =−∞

1/q1

1/q1

(3.7) = c2

N/p1

306

Youngshen Han and Chin-Cheng Lin

since s2 − 1/p2 = s1 − 1/p1 . On the other hand, ∞


q
2j s1
Dj (f )(x)
1



1/q1

j =N

=

∞


q
2j (s1 −s2 ) 2j s2
Dj (f )(x)
1



j =N

≤

∞

2

j (s1 −s2 )q1

j =N

≤ c2N(1/p1 −1/p2 )

1/q1





2

1/q1


q Dj (f )(x)
2




j s2


j



q 2j s2
Dj (f )(x)
2



(3.8)

1/q2

1/q2

j

since s1 < s2 and s1 − 1/p1 = s2 − 1/p2 . We now obtain
 ∞


 ∞




q 1/q1
p f ˙1s1 ,q1 = p1 t p1 −1

> t

dt 2j s1
Dj (f )
1 Fp 0

1

≤ p1 ≤ p1

∞ 

j =−∞

2c2(N+1)/p1

t

N/p1 N=−∞ 2c2

∞



j =−∞

2c2(N+1)/p1

N/p1 N=−∞ 2c2

+

∞





∞

q1 1/q1
 js
1

2 Dj (f ) > t

dt


p1 −1



 N



q1 1/q1
p1 −1
j s1

2 Dj (f ) t
j =−∞



q 2j s1
Dj (f )
1



1/q1 

j =N

≤ p1

∞



N/p1 N=−∞ 2c2
 ∞


×



≤ p1

∞



q 1/q1

> t/2

dt 2j s1
Dj (f )
1



j =N

2c2(N+1)/p1

N/p1 N=−∞ 2c2
 ∞

∞





∞

≤ p1 0



by (3.7)



t p1 −1




q 1/q2 1 N(1/p −1/p )
  1 t
dt 2 2j s2
Dj (f )
2 > c2 by (3.8)
2



j =−∞

2c2(N+1)/p1

N/p1 N=−∞ 2c2
 ∞


×



t p1 −1




×



≤ p1

2c2(N+1)/p1



> t

dt

t p1 −1


q
2j s2
Dj (f )
2



j =−∞

1/q2



> ct p1 /p2

dt



since t ≈ 2N/p1





∞


q 1/q2  js
2 2
Dj (f )
2 t p1 −1

> ct p1 /p2

dt j =−∞



Embedding Theorem on Spaces of Homogeneous Type



∞

≤ p2

t





∞

q2 1/q2
 js
2

> ct

dt 2 Dj (f )


p2 −1


0

≤

307

j =−∞

p c f ˙2s2 ,q2 Fp

=c.

2

This proves of Theorem 2 (ii), and hence the proof of main theorem is completed.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

Christ, M. (1990). A T b theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math., 60/61, 601–628. Coifman, R.R. and Weiss, G. (1971). Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Lecture Notes in Math., vol. 242 Springer-Verlag, Berlin and New York. David, G., Journe, J.-L., and Semmes, S. (1985). Operateurs de Calderón–Zygmund, fonctions paraaccretive et interpolation, Rev. Mat. Iberoamericana, 1, 1–56. Fefferman, C. and Stein, E.M. (1971). Some maximal inequalities, Am. J. Math., 93, 107–115. Han, Y.-S. (1995). The embedding theorem for the Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Proc. Am. Math. Soc., 123, 2181–2189. Han, Y.-S. (1998). Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications, Proc. Am. Math. Soc., 126, 3315–3327. Han, Y.-S. Discrete Calderón reproducing formula on spaces of homogeneous type, Acta Math. Sinica, to appear. Han, Y.-S. and Sawyer, E. (1994). Littlewood–Paley theory on spaces of homogeneous type and classical function spaces, Mem. Am. Math. Soc., 110, 1–136. Meyer, Y. (1985). Les nouveaux operateurs de Calderón–Zygmund, Asterisqué, 131, 237–254. Macias, R.A. and Segovia, C. (1979). Lipschitz functions on spaces of homogeneous type, Adv. in Math., 33, 257–270.

Received March 22, 2000 Revision received May 30, 2001 Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310 e-mail: [email protected] Department of Mathematics, National Central University, Chung-li, Taiwan 32054, Republic of China e-mail: [email protected]