Boundedness of certain commutators on Triebel–Lizorkin spaces

J. Math. Anal. Appl. 330 (2007) 1264–1272 www.elsevier.com/locate/jmaa

Boundedness of certain commutators on Triebel–Lizorkin spaces ✩ Liya Jiang a , Pu Zhang b,1 , Houyu Jia c,∗ a Department of Mathematics, Zhejiang University of Technology, Hangzhou 310014, PR China b Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, PR China c Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China

Received 1 March 2006 Available online 14 September 2006 Submitted by L. Grafakos

Abstract In this paper, two types of commutators are considered, and the boundedness of these operators on Triebel–Lizorkin spaces are discussed. © 2006 Elsevier Inc. All rights reserved. Keywords: Commutators; Triebel–Lizorkin spaces

1. Introduction Let Ω be homogeneous of degree zero, integrable on the sphere S n−1 and satisfy the vanishing condition  Ω(x  ) dx  = 0. (1.1) S n−1

✩

Supported by ZJNSF No. Y605149.

* Corresponding author.

E-mail addresses: [email protected] (L. Jiang), [email protected] (P. Zhang), [email protected] (H. Jia). 1 Current address: Department of Mathematics, Mudanjiang Teachers College, Mudanjiang 157012, PR China.

0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.07.101

L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

The classical singular integral operator is defined by  Ω(x − y) Tf (x) = p.v. f (y) dy. |x − y|n

1265

(1.2)

Rn

For b ∈ BMO(R n ), the commutator generated by T and b is defined as follows:   Ω(x − y)  [b, T ](f )(x) = b(x) − b(y) f (y) dy. |x − y|n

(1.3)

Rn

Coifman, Rochberg and Weiss [6] proved if Ω ∈ Lipα (S n−1 ), then the above commutator is bounded on Lp (R n ) if and only if b ∈ BMO(R n ). A celebrated result of Coifman and Meyer [4] states that if Ω ∈ C 1 (S n−1 ), then for 1 < p < ∞, the Lp (R n ) boundedness for [b, T ] could be obtained from the weighted Lp estimate with Ap weights for the operator T , here and in what follows, we denote by Ap (1  p  ∞) the weight function class of Muckenhoupt. Alvarez, Bagby, Kurtz and Pérez [1] developed the idea of Coifman and Meyer and established a generalized boundedness criterion for the commutators of linear operators. We state their result in the following strong form which will be used in the proof of our theorems. Theorem A. [1] Let T be a linear operator. If for all w ∈ Aq (R n ) (1 < q < ∞), the inequality Tf Lp (w)  Cf Lp (w) holds, then for b ∈ BMO(R n ) we have   [b, T ]f  p  CbBMO f Lp (w) . L (w)

On the other hand, in [14] the authors studied the boundedness properties of [b, T ] on the s,q Triebel–Lizorkin space F˙p , for s > 0 and Ω ∈ L log+ L(S n−1 ). In this note, we will consider the case s = 0, and get the following results. Theorem 1.1. Let 1 < p, q < ∞ and [b, T ] be as in (1.3). If Ω ∈ L(log L)2 (S n−1 ), then [b, T ] 0,q is bounded from F˙p into itself. The second operator we consider is a kind of generalized commutators as follows. For x = (x1 , x2 , . . . , xn ) ∈ R n , suppose that Ω satisfies the following vanishing condition:  Ω(x  )xj dx  = 0, for each j with 1  j  n. (1.4) S n−1

Let a be a function on R n with ∇a ∈ BMO(R n ). The generalized commutator Ta is defined by

 Ta f (x) = p.v. Rn

 Ω(x − y)  a(x) − a(y) − ∇a(y) · (x − y) f (y) dy. n+1 |x − y|

(1.5)

The behavior of the operator Ta on Lp (R n ) was studied in [2] for ∇a ∈ L∞ (R n ) (or Lq (R n )) n ) and Ω ∈ Lip (R n ). Later, Hofmann [9] improved the result in [3] and in [3] for ∇a ∈ BMO(R 1  and proved that Ω ∈ q>1 Lq (S n−1 ) is a sufficient condition for the Lp (R n ) boundedness of Ta . Recently, Hu Guoen [11] gave a sufficient condition for the L2 (R n ) boundedness for Ta . s,q In 1990, Hofmann studied the boundedness of Ta on Triebel–Lizorkin spaces F˙p , when  ∇a ∈ L∞ (R n ) and Ω ∈ L∞ (S n−1 ) (or Ω ∈ Lq (S n−1 )). Now, we extend Theorem 4.4 in [10], and get the following result.

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L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

Theorem 1.2. Suppose 1 < p, q < ∞ and q˜ = max{q, q  }. Let Ta be as in (1.5). If Ω ∈ Lq˜ (S n−1 ) and ∇a ∈ BMO(R n ), then there exists a positive constant 0 = 0 (n, p, q) such that for 0 < α,q α < 0 , Ta is bounded from F˙p into itself. Let φ(x) be a radial C0∞ function supported in the unit ball with integral zero and ∞ 2 dt −n φ( x ) and denote by Q be the convolution operator with ˆ t 0 (φ(t)) t = 1. Write φt (x) = t t ∞ 2 dt kernel φt . Then 0 Qt t = I . For s ∈ R, 1  p < ∞, 1  q  ∞, the homogeneous Triebel– s,q Lizorkin spaces F˙p is defined by  ∞ 1   dt q    −sq q f F˙ps,q =  t |Qt f | (1.6)  ,   t p

0

with the usual modification if q = ∞. s,q Replaced φ by φ˜ in our definitions, we get the same spaces as F˙p with equivalent norms, where φ˜ satisfies the same properties as φ announced above. Set ∞   dt , (1.7) p(x) = φt ∗ φt |x| t 1

the observation of Han and Sawyer [13] shows that p(|x|) is a radial C0∞ function with support in a ball of radius C and mean value zero. Thus, if we denote by Pt the convolution operator with kernel pt (x), then  ∞ 1  q  dt   t −sq |Pt f |q (1.8)  ∼ f F˙ps,q .    t 0

p

2. Proof of Theorem 1.1 In this section, we denote the operator [b, T ] by Tb . It suffices to prove that, for f, g ∈ S(R n ),

∞∞

dt ds



2 2 Qs Tb Qt (f )(x)g(x) dx (2.1)

 Cf F˙ 0,q gF˙ 0,q  .

p

t s

p 0 0 Rn

Since the operator Tb∗ , the adjoint operator of Tb , is similar to the operator Tb , the proof of this theorem can be reduced to prove the following inequality:

∞ t 

dt ds



2 2 Qs Tb Qt (f )(x)g(x) dx (2.2)

 Cf F˙ 0,q gF˙ 0,q  .

p

t s

p 0 0 Rn

∞ Noting that Ps = s Q2t dtt , we only need to prove

∞



ds



2 Qs Tb Ps (f )(x)g(x) dx  Cf F˙ 0,q g ˙ 0,q  .

Fp  p

s

0 Rn

Write

  Qs Tb Ps (f )(x) = (Qs T )b Ps (f ) (x) − (Qs )b T Ps (f )(x) = I + II.

(2.3)

L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

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By the Hölder’s inequality and (1.8), we have

∞



ds



2 Qs Tb Ps (f )(x)g(x) dx



s

0 Rn

 ∞





Qs Tb Ps (f )(x)

Qs (g)(x) ds dx  s Rn 0

 ∞  ∞ 1  1  q q   ds ds     |I |q |II|q   +  gF˙ 0,q  .     s s p p

0

0

p

It is easy to check that  ∞  ∞ 1  1  q q  

 

ds ds q    

T Ps (f )

|II|q    C   CTf F˙ 0,q  Cf F˙ 0,q , p p     s s 0

p

(2.4)

p

0

where we use Theorem A and following lemma. Lemma 2.1. [8] Let 1 < q < ∞ and Ts be a class of linear operators. If for all w ∈ A1 and s  0 there holds Ts f Lp (w)  Cf Lp (w) and Ts f Lp (w)  Cf Lp (w) , then for all 1 < p < ∞,  ∞  ∞ 1  1  q q   ds ds     |Ts fs |q |fs |q    C      s s 0

p

0

  f s ∈ Lp .

p

According to [5], we decompose Ω(x) as follows. Let



 θ0 = x  ∈ S n−1 : Ω(x  )  1 ,



 θd = x  ∈ S n−1 : 2d−1  Ω(x  )  2d (d  1), Ω˜ d (x) = Ω(x)χθd (x),  ˜ n−1 Ωd (x) dx ˜ Ωd (x) = Ωd (x) − S . ωn Then

Ωd (x) = Ω(x),

d0



Ωd (x) dx = 0, S n−1

Ωd ∞  C2d , Ωd L1  C2d |θd |,

d 2 2d |θd |  CΩL(log L)2 . d0

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L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

For j ∈ R, denote by Tjd the convolution operator with kernel  j −1  2 < |x|  2j , kjd = Ωd (x)|x|−(n−1) χ then Tb f (x) =

∞∞

Q2s Tjd Q2t (f )(x)

0 0 d0 j ∈R

Hence I=



dt ds . t s

    (χ1 + χ2 + χ3 + χ4 ) Qs Tjd b Ps (f ) (x),

d0 j

where χ1 = {s  2j  2dN s}, χ2 = {s  2dN s  2j }, χ3 = {2−dN s  2j  s}, χ4 = {2j  2−dN s  s}, and N will be chosen late. Next, we will prove that, for i = 1, 2, 3, 4,  ∞

1 

q q 



      ds   d



Q P χ T (f ) (x)    CbBMO 1 + ΩL(log L)2 f F˙ps,q . i s s j



b   s d0 j

0

p

(2.5)

First, we check (2.5) for i = 2, 4. To this end we need the following lemma. Lemma 2.2. Let Tjd be defined as above, then there exists a positive number ε such that        Qs T d f   CΩd ∞ min 1, 2−j s ε , 2−j s −ε f 2 . j 2

(2.6)

Proof. It is enough to consider j = 0. Noting the following obvious inequalities:



  

φ(sξ ˆ )  C min 1, s|ξ |, s|ξ | −ε and

d



kˆ (ξ )  CΩd ∞ min 1, |ξ |, |ξ |−ε , 0

we have





φ(sξ ˆ )kˆ0d (ξ )  CΩd ∞ min 1, s ε , s −ε . Then Lemma 2.2 follows from ∧  ˆ )kˆ d (ξ )fˆ(ξ ). Qs T d f (ξ ) = φ(sξ 0

0

2

On the other hand, for w ∈ Ap , it is easy to check   Qs T d f   CΩd ∞ f p,w . j p,w Interpolating (2.6) and (2.7), we get    ε  −ε Qs T d f  f p,w .  CΩd ∞ min 2−j s , 2−j s j p,w By Theorem A, we have   ε  −ε    Qs T d f   CbBMO Ωd ∞ min 2−j s , 2−j s f p,w . j b p,w

(2.7)

L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

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Choose N such that N ε > 1, it follows from Lemma 2.1 that  ∞

1 

q q 



   ds  

χi Qs Tjd b Ps (f ) (x)

 

  s d0 j

0

 bBMO

p

Ωd ∞ 2−dN ε f F˙ 0,q  CbBMO f F˙ 0,q . p

p

d0

Now, let us check (2.5) for i = 1, 3. By Lemma 4 in [12], and noting that ΩL(log L)2 , we get 



     sup sup Qs Tjd b (fs )  s



d0 dλΩd



p

j

 





        sup sup (Qs )b Tjd (fs )  + sup sup Qs Tjd b (fs )  s

p

j

s

j

p

         CbBMO Ωd 1  sup |fs | + CbBMO λΩd  sup |fs | . p

s

p

s

Thus, by the duality, we have  ∞        Qs Tjd b (fs )    j

0

p

  ∞   ∞     

    d d



Qs T  (Qs )b Tj (fs ) +  j b (fs )      0

j

p

0

j

p

         CbBMO Ωd 1  sup |fs | + CbBMO λΩd  sup |fs | . p

s

s

p

It follows from the interpolation theorem that  ∞

1 

q q 



   ds   d



Q P χ T (f ) (x)   i s s j b

s   0

d0 j

  Ωd 1 + λΩd N df F˙ 0,q  CbBMO

p

p

d0

 CbBMO ΩL(log L)2 f F˙ 0,q . p

Hence the inequality (2.5) is proved and then the proof of Theorem 1.1 is finished. 3. Proof of Theorem 1.2 Let Φ ∈ C0∞ ( 14 , 4) with k(x, y)Φ(2−j |x − y|), where k(x, y) =

+∞

j =−∞ Φ(2

−j r)

= 1, for all r > 0. Denote by kj (x, y) =

 Ω(x − y)  a(x) − a(y) − ∇a(y) · (x − y) . n+1 |x − y|

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L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

Let T j be the convolution operator with kernel kj . Then Ta (f ) =

∞ ∞

  j

Q2s T j Q2t f

dt ds , t s

(3.1)

0 0

∈ S(R n ).

for all f From the proof of Theorem 4.4 in [10] for s  t, to prove Theorem 1.2, it suffices to verify that, for all w ∈ A1 ,   Qs T j f  q  C∇aBMO f Lq (w) for s  2j , (3.2) L (w) and

    Qs T j ∗ f 



Lq (w)

 C∇aBMO f Lq  (w)

for s  2j ,

(3.3)

where (Qs T j )∗ is the adjoint operator of Qs T j . Since the proof of (3.3) is similar to that of (3.2), we only prove (3.2). To this end, we need the following lemma. Lemma 3.1. [9] Let a be a function on R n with derivatives of order one in Lr (R n ) for some r > n. Then  1  r





∇a(z) r dz ,

a(x) − a(y)  C|x − y| 1 (3.4) y |Ix | y

Ix y

where Ix is the cube centered at x with sides parallel to the axes and having side length 2|x − y|. For any x ∈ R n , let Q be a cube centered at x with sides length 2j . Denote by aQ (z) = a(z) −

n

mQ (∂j a) · zj ,

j =1

where mQ (f ) is the average of f on Q. Then, for any x, y ∈ R n , aQ (x) − aQ (y) − ∇aQ (y) · (x − y) = a(x) − a(y) − ∇a(y) · (x − y). Noting that Ix ⊂ 8Q when 2j −2  |x − y|  2j +2 , where 8Q denotes the cube with the same center as Q and with sides length 8 times the sides length of Q. By Lemma 3.1, we get, for x, y ∈ R n with 2j −2  |x − y|  2j +2 , 1  n 

r





1

aQ (x) − aQ (y)  C|x − y|

∂j a(z) − mQ (∂j a) r dz y |Ix | j =1 y

y

Ix

1  n 

r

r

1



∂  C|x − y| a(z) − m (∂ a) dz j Q j 2j n j =1

8Q

 C2 ∇aBMO , j

where we use the fact |m8Q (∂j a) − mQ (∂j a)|  C∂j aBMO . By the Hölder’s inequality, we have

L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

 j p T f  p

p p  CΩq  ∇aBMO L (w)



  q  p M |f | q w(x) dx

Rn p

p

1271



+ CΩq  ∇aBMO

  qr  p qr w(x) dx, M |f |

Rn

where 1 < r < ∞ and is closed to 1. If we choose p such that p > qr > q, then for w ∈ A1 , there holds  j  T f  p  C∇aBMO f Lp (w) . (3.5) L (w) On the other hand, by Theorem 1.2 in [10], T j is bounded on Lr (R n ) for all r with 1 < r < ∞. So interpolation gives the estimate (3.2). Now, let us treat the case s  t. By duality, it suffices to prove

∞ t 



ds dt



Q2s Ta∗ Q2t f (x)g(x) dx

 Cf F˙pα,q gF˙ −α,q  .

t s

p 0 0 Rn

As in [11], we set T˜a f (x) = p.v. −



Rn

and

 Wj f (x) = p.v. Rn

 Ω(y − x)  a(x) − a(y) − ∇a(y) · (x − y) f (y) dy |x − y|n+1

  Ω(x − y)(xj − yj ) ∂j a(x) − ∂j a(y) f (y) dy, |x − y|n+1

1  j  n.

 ˜ = Ω(−x) is also integrable on Write Ta∗ f (x) = T˜a f (x) + nj=1 Wj f (x). Noting that Ω(x) n−1 S and enjoys the same property, by the above argument with the case s  t, we have

∞ t 



dt ds



2 ˜ 2 Qs Ta Qt f (x)g(x) dx (3.6)

 Cf F˙pα,q gF˙ −α,q  .

t s

p 0 0 Rn

Ω(x)x

Denote q˜ = max{q, q  } , then |x| j ∈ Lq˜ (S n−1 ) is homogeneous of degree zero with integral zero for 1  j  n. In [7], there shows that the operator G(f )(x) = p.v.

Ω(x)xj  f (x) |x| 

is bounded on Lq (w) and on Lq (w) for all w ∈ A1 . So by Theorem A, we can see that Wj is  bounded on Lq (w) and on Lq (w) for all w ∈ A1 . Thus, applying the Schwartz’s inequality twice and using Lemma 2.1, it follows immediately that, for each j with 1  j  n,

∞ t 

 ∞







dt

dt ds





2 2 2





Wj Qt f (x) Pt (g) (x) dx

Qs Wj Qt f (x)g(x) dx



t s

t

0 0 Rn

Rn 0

 Cf F˙pα,q g ˙ −α,q  . Fp 

So, the proof of Theorem 1.2 is finished.

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L. Jiang et al. / J. Math. Anal. Appl. 330 (2007) 1264–1272

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