CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES LANZHE LIU Received 4 February 2006; Revised 20 September 2006; Accepted 28 September 2006 The continuity of some multilinear operators related to certain convolution operators on the Triebel-Lizorkin space is obtained. The operators include Littlewood-Paley operator and Marcinkiewi cz operator. Copyright © 2006 Lanzhe Liu. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let T be the Calder ´ on-Zygmund singular integral operator, a well-known result of Coif- man et al. (see [6]) states that the commutator [b,T]( f ) = T(bf) −bT(f )(whereb ∈ BMO) is bounded on L p (R n )(1<p<∞); Chanillo (see [1]) proves a similar result when T is replaced by the fractional integr al operator; in [8, 9], these results on the Triebel- Lizorkin spaces and the case b ∈ Lipβ (where Lipβ is the homogeneous Lipschitz space) are obtained. The main purpose of this paper is to study the continuity of some multi- linear operators related to certain convolution operators on the Triebel-Lizorkin spaces. In fact, we will obtain the continuity on the Triebel-Lizorkin spaces for the multilinear operators only under certain conditions on the size of the operators. As the applications, the continuity of the multilinear operators related to the Littlewood-Paley oper ator and Marcinkiewicz operator on the Triebel-Lizorkin spaces are obtained. 2. Notations and results Throughout this paper, Q will denote a cube of R n with side parallel to the axes, and for a cube Q,let f Q =|Q| −1  Q f (x)dx and f # (x) = sup x∈Q |Q| −1  Q |f (y) − f Q |dy.For 1 ≤ r<∞ and 0 ≤δ<n,let M δ,r ( f )(x) =sup x∈Q  1 |Q| 1−δr/n  Q   f (y)   r dy  1/r , (2.1) Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 58473, Pages 1–11 DOI 10.1155/JIA/2006/58473 2 Continuity of multilinear operators we denote M δ,r ( f ) = M r ( f )ifδ = 0, which is the Hardy-Littlewood maximal function when r = 1 (see [10]). For β>0andp>1, let ˙ F β,∞ p be the homogeneous Triebel-Lizorkin space, and let the Lipschitz space ˙ ∧ β be the space of functions f such that f  ˙ ∧ β = sup x, h∈R n , h=0   Δ [β]+1 h f (x)   |h| β < ∞ , (2.2) where Δ k h denotes the kth difference operator (see [9]). We are going to study the multilinear operator as follows. Let m be a positive integer and let A be a function on R n . We denote R m+1 (A;x, y) =A(x) −  |α|≤m 1 α! D α A(y)(x − y) α . (2.3) Definit ion 2.1. Let F(x,t)defineonR n ×[0,+∞), denote F t ( f )(x) =  R n F(x − y,t) f (y)dy, F A t ( f )(x) =  R n R m+1 (A;x, y) |x − y| m F(x − y,t) f (y)dy. (2.4) Let H be the Hilbert space H ={h : h< ∞}such that, for each fixed x ∈ R n , F t ( f )(x) and F A t ( f )(x) may be viewed as a mapping from [0,+∞)toH. Then, the multilinear operators related to F t is defined by T A ( f )(x) =   F A t ( f )(x)   ; (2.5) and also define T(f )(x) =F t ( f )(x). In particular, consider the following two sublinear operators. Definit ion 2.2. Fix ε>0, n>δ ≥ 0. Let ψ be a fixed function which satisfies the following properties: (1)  ψ(x)dx =0; (2) |ψ(x)|≤C(1 + |x|) −(n+1−δ) ; (3) |ψ(x + y) −ψ(x)|≤C|y| ε (1 + |x|) −(n+1+ε−δ) when 2|y| < |x|. The multilinear Littlewood-Paley operator is defined by g A δ ( f )(x) =   ∞ 0   F A t ( f )(x)   2 dt t  1/2 , (2.6) where F A t ( f )(x) =  R n R m+1 (A;x, y) |x − y| m ψ t (x − y) f (y)dy (2.7) Lanzhe Liu 3 and ψ t (x) =t −n+δ ψ(x/t)fort>0. Denote that F t ( f ) =ψ t ∗ f , and also define that g δ ( f )(x) =   ∞ 0   F t ( f )(x)   2 dt t  1/2 , (2.8) which is the Littlewood-Paley g function w hen δ = 0 (see [11]). Let H be the space H ={h : h=(  ∞ 0 |h(t)| 2 dt/t) 1/2 < ∞},then,foreachfixedx ∈ R n , F A t ( f )(x) may be viewed as a mapping from [0,+∞)toH, and it is clear that g δ ( f )(x) =   F t ( f )(x)   , g A δ ( f )(x) =   F A t ( f )(x)   . (2.9) Definit ion 2.3. Let 0 ≤ δ<n,0<γ≤ 1andΩ be homogeneous of degree zero on R n such that  S n−1 Ω(x  )dσ(x  ) = 0. Assume that Ω ∈ Lip γ (S n−1 ), that is, there exists a con- stant M>0 such that for any x, y ∈ S n−1 , |Ω(x) −Ω(y)|≤M|x − y| γ . The multilinear Marcinkiewicz operator is defined by μ A δ ( f )(x) =   ∞ 0   F A t ( f )(x)   2 dt t 3  1/2 , (2.10) where F A t ( f )(x) =  |x−y|≤t Ω(x − y) |x − y| n−1−δ R m+1 (A;x, y) |x − y| m f (y)dy; (2.11) denote F t ( f )(x) =  |x−y|≤t Ω(x − y) |x − y| n−1−δ f (y)dy, (2.12) and also define that μ δ ( f )(x) =   ∞ 0   F t ( f )(x)   2 dt t 3  1/2 , (2.13) which is the Marcinkiewicz operator when δ = 0 (see [12]). Let H be the space H ={h : h=(  ∞ 0 |h(t)| 2 dt/t 3 ) 1/2 < ∞}. Then, it is clear that μ δ ( f )(x) =   F t ( f )(x)   , μ A δ ( f )(x) =   F A t ( f )(x)   . (2.14) It is clear that Definitions 2.2 and 2.3 are the particular examples of Definition 2.1. Note that when m = 0, T A is just the commutator of F t and A, while when m>0, it is nontrivial generalizations of the commutators. It is well known that multilinear oper- ators are of great interest in harmonic analysis and have been widely studied by many authors (see [2–5, 7]). The main purpose of this paper is to study the continuity for the multilinear opera tors on the Triebel-Lizorkin spaces. We will prove the following theo- rems in Section 3. Theorem 2.4. Let g A δ be the mult ilinear Littlewood-Paley operator as in Definition 2.2.If 0 <β<min(1,ε) and D α A ∈ ˙ ∧ β for |α|=m, then 4 Continuity of multilinear operators (a) g A δ maps L p (R n ) continuously into ˙ F β,∞ q (R n ),for1 <p<n/δand 1/q = 1/p−δ/n; (b) g A δ maps L p (R n ) continuously into L q (R n ) for 1 <p<n/(δ + β) and 1/p− 1/q = (δ +β)/n. Theorem 2.5. Let μ A δ be the multilinear Marcinkiewiz operator as in Definition 2.3.If0 < β<min(1/2, γ) and D α A ∈ ˙ ∧ β for |α|=m, then (a) μ A δ maps L p (R n ) continuously into ˙ F β,∞ q (R n ) for 1 <p<n/δand 1/q =1/p−δ/n, (b) μ A δ maps L p (R n ) continuously into L q (R n ) for 1 <p<n/(δ + β) and 1/p−1/q = (δ +β)/n. 3. Main theorem and proof We first prove a general theorem. Theorem 3.1 (main theorem). Let 0 ≤ δ<n, 0 <β<1,andD α A ∈ ˙ ∧ β for |α|=m.Sup- pose F t , T,andT A are the same as in Definition 2.1,ifT is bounded from L p (R n ) to L q (R n ) for 1 <p<n/δand 1/q = 1/p−δ/n,andT satisfies the following size condition:   F A t ( f )(x) −F A t ( f )  x 0    ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 f (x) (3.1) for any cube Q with supp f ⊂ (2Q) c and x ∈Q, then (a) T A is bounded from L p (R n ) to ˙ F β,∞ q (R n ) for 1 <p<n/δand 1/q =1/p−δ/n, (b) T A is bounded from L p (R n ) to L q (R n ) for 1 <p<n/(δ + β) and 1/q = 1/p−(δ + β)/n. To prove the theorem, we need the following lemmas. Lemma 3.2 (see [9]). For 0 <β<1, 1 <p< ∞, f  ˙ F β,∞ p ≈     sup Q 1 |Q| 1+β/n  Q   f (x) − f Q   dx     L p ≈     sup ·∈Q inf c 1 |Q| 1+β/n  Q   f (x) −c   dx     L p . (3.2) Lemma 3.3 (see [9]). For 0 <β<1, 1 ≤ p ≤∞, f  ˙ ∧ β ≈ sup Q 1 |Q| 1+β/n  Q   f (x) − f Q   dx ≈ sup Q 1 |Q| β/n  1 |Q|  Q   f (x) − f Q   p dx  1/p . (3.3) Lemma 3.4 (see [1, 2]). Suppose that 1 ≤ r<p<n/δand 1/q = 1/p−δ/n. Then   M δ,r ( f )   L q ≤ Cf  L p . (3.4) Lanzhe Liu 5 Lemma 3.5 (see [5]). Let A be a function on R n and D α A ∈L q (R n ) for |α|=m and some q>n. Then   R m (A;x, y)   ≤ C|x − y| m  |α|=m  1    Q(x, y)     Q(x,y)   D α A(z)   q dz  1/q , (3.5) where  Q(x, y) isthecubecenteredatx and has side length 5 √ n|x − y|. Proof of Theorem 3.1 (main theorem). Fix a cube Q = Q(x 0 ,l)andx ∈ Q.Let  Q = 5 √ nQ and  A(x) = A(x) −  |α|=m (1/α!)(D α A)  Q x α ,thenR m (A;x, y) = R m (  A;x, y)andD α  A = D α A −(D α A)  Q for |α|=m.Wewrite,for f 1 = fχ  Q and f 2 = fχ R n \  Q , F A t ( f )(x) =  R n R m+1   A;x, y  |x − y| m F(x − y,t) f (y)dy =  R n R m+1   A;x, y  |x − y| m F(x − y,t) f 2 (y)dy +  R n R m   A;x, y  |x − y| m F(x − y,t) f 1 (y)dy −  |α|=m 1 α!  R n F(x − y,t)(x − y) α |x − y| m D α  A(y) f 1 (y)dy, (3.6) then   T A ( f )(x) −T  A  f 2  x 0    =     F A t ( f )(x)   −   F  A t  f 2  x 0      ≤     F t  R m   A;x,·  |x −·| m f 1  (x)     +  |α|=m 1 α!     F t  (x −·) α |x −·| m D α  Af 1  (x)     +   F  A t  f 2  (x) −F  A t  f 2  x 0    = A(x)+B(x)+C(x), (3.7) thus, 1 |Q| 1+β/n  Q   T A ( f )(x) −T  A ( f )  x 0    dx ≤ 1 |Q| 1+β/n  Q A(x)dx + 1 |Q| 1+β/n  Q B(x)dx + 1 |Q| 1+β/n  Q C(x)dx :=I + II +III. (3.8) 6 Continuity of multilinear operators Now, let us estimate I, II,andIII, respectively. First, for x ∈ Q and y ∈  Q, using Lemmas 3.3 and 3.5,weget   R m   A;x, y    ≤ C|x − y| m  |α|=m sup x∈  Q   D α A(x) −  D α A   Q   ≤ C|x − y| m |Q| β/n  |α|=m   D α A   ˙ ∧ β , (3.9) thus, taking r, s such that 1 ≤ r<pand 1/s =1/r −δ/n,bythe(L r ,L s ) boundedness of T and Holder’ inequality, we obtain I ≤ C  |α|=m   D α A   ˙ ∧ β 1 |Q|  Q   T  f 1  (x)   dx ≤ C  |α|=m   D α A   ˙ ∧ β   T  f 1    L s |Q| −1/s ≤ C  |α|=m   D α A   ˙ ∧ β   f 1   L r |Q| −1/s ≤ C  |α|=m   D α A   ˙ ∧ β M δ,r ( f )(x). (3.10) Secondly, using the following inequality (see [9]):    D α A −  D α A   Q  fχ  Q   L r ≤ C|Q| 1/s+β/n   D α A   ˙ ∧ β M δ,r ( f )(x), (3.11) and similar to the proof of I,wegain II ≤ C  |α|=m   D α A   ˙ ∧ β M δ,r ( f )(x). (3.12) For III, using the size condition of T,wehave III ≤ C  |α|=m   D α A   ˙ ∧ β M δ,1 ( f )(x). (3.13) We now put these estimates together; and taking the supremum over all Q such that x ∈Q, and using Lemmas 3.2 and 3.4,weobtain   T A ( f )   ˙ F β,∞ q ≤ C  |α|=m   D α A   ˙ ∧ β f  L p . (3.14) This completes the proof of (a). (b) By same argument as in proof of (a), we have 1 |Q|  Q   T A ( f )(x) −T  A  f 2  x 0    dx ≤ C  |α|=m   D α A   ˙ ∧ β  M δ+β,r ( f )+M δ+β,1 ( f )  , (3.15) thus,  T A ( f )  # ≤ C  |α|=m   D α A   ˙ ∧ β  M δ+β,r ( f )+M δ+β,1 ( f )  . (3.16) Lanzhe Liu 7 Now, using Lemma 3.4,wegain   T A ( f )   L q ≤ C    T A ( f )  #   L q ≤ C  |α|=m   D α A   ˙ ∧ β    M δ+β,r ( f )   L q +   M δ+β,1 ( f )   L q  ≤ Cf  L p . (3.17) This completes the proof of (b) and the theorem.  To prove Th e or e m s 2.4 and 2.5, since g δ and μ δ are all bounded from L p (R n )toL q (R n ) for 1 <p<n/δand 1/q = 1/p−δ/n (see [11, 12]), it suffices to verify that g A δ and μ A δ satisfy the size condition in Theorem 3.1 (main theorem). Suppose supp f ⊂ (2Q) c and x ∈ Q = Q(x 0 ,l). Note that |x 0 − y|≈|x − y| for y ∈ (2Q) c . For g A δ ,wewrite F  A t ( f )(x) −F  A t ( f )  x 0  =  R n \  Q  ψ t (x − y) |x − y| m − ψ t  x 0 − y    x 0 − y   m  R m   A;x, y  f (y)dy +  R n \  Q ψ t  x 0 − y  f (y)   x 0 − y   m  R m   A;x, y  − R m   A;x 0 , y  dy −  |α|=m 1 α!  R n \  Q  ψ t (x − y)(x − y) α |x − y| m − ψ t  x 0 − y  x 0 − y  α   x 0 − y   m  D α  A(y) f (y)dy = I 1 + I 2 + I 3 . (3.18) By the condition on ψ,weobtain   I 1   ≤ C  R n \  Q   x −x 0     x 0 − y   m+1   R m   A;x, y      f (y)     ∞ 0 tdt  t +   x 0 − y    2(n+1−δ)  1/2 dy + C  R n \  Q   x −x 0   ε   x 0 − y   m   R m   A;x, y      f (y)     ∞ 0 tdt  t +   x 0 − y    2(n+1+ε−δ)  1/2 dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=0  2 k+1  Q\2 k+1  Q    x −x 0     x 0 − y   n+1−δ +   x −x 0   ε   x 0 − y   n+ε−δ    f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=1  2 −k +2 −kε   1   2 k  Q   1−δ/n  2 k  Q   f (y)   dy  ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x). (3.19) 8 Continuity of multilinear operators For I 2 , by the formula (see [5]): R m   A;x, y  − R m   A;x 0 , y  =  |η|<m 1 η! R m−|η|  D η  A;x,x 0  (x − y) η (3.20) and Lemma 3.5,weget   R m   A;x, y  − R m   A;x 0 , y    ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n   x −x 0     x 0 − y   m−1 , (3.21) thus, similar to the proof of I 1 ,   I 2   ≤ C  R n \  Q   R m   A;x, y  − R m   A;x 0 , y      x 0 − y   m+n−δ   f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=0  2 k+1  Q\2 k  Q   x −x 0     x 0 − y   n+1−δ   f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x). (3.22) For I 3 , similar to the proof of I 1 ,weobtain   I 3   ≤ C  |α|=m  R n \  Q    x −x 0     x 0 − y   n+1−δ +   x −x 0   ε   x 0 − y   n+ε−δ    f (y)     D α  A(y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=1  2 k(β−1) +2 k(β−ε)  M δ,1 ( f )(x) ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x) (3.23) so that   F  A t ( f )(x) −F  A t ( f )  x 0    ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x). (3.24) Lanzhe Liu 9 For μ A δ ,wewrite   F  A t ( f )(x) −F  A t ( f )  x 0    ≤   ∞ 0   |x−y|≤t, |x 0 −y|>t   Ω(x − y)     R m   A;x, y    |x − y| m+n−1−δ   f (y)   dy  2 dt t 3  1/2 +   ∞ 0   |x−y|>t, |x 0 −y|≤t   Ω  x 0 − y      R m   A;x 0 , y      x 0 − y   m+n−1−δ   f (y)   dy  2 dt t 3  1/2 +   ∞ 0   |x−y|≤t,|x 0 −y|≤t      Ω(x − y)R m (  A;x, y) |x − y| m+n−1−δ − Ω  x 0 − y  R m   A;x 0 , y    x 0 − y   m+n−1−δ        f (y)   dy  2 dt t 3  1/2 + C  |α|=m   ∞ 0      |x−y|≤t  Ω(x − y)(x − y) α |x − y| m+n−1−δ −  |x 0 −y|≤t Ω  x 0 − y  x 0 − y  α   x 0 − y   m+n−1−δ  × D α  A(y) f (y)dy      2 dt t 3  1/2 := J 1 + J 2 + J 3 + J 4 . (3.25) Then J 1 ≤ C  R n \  Q   f (y)     R m   A;x, y    |x − y| m+n−1−δ   |x−y|≤t<|x 0 −y| dt t 3  1/2 dy ≤ C  R n \  Q   f (y)     R m   A;x, y    |x − y| m+n−1−δ   x 0 −x   1/2 |x − y| 3/2 dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=1 2 −k/2 1   2 k  Q   1−δ/n  2 k  Q   f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x), (3.26) similarly, we have J 2 ≤ C  |α|=m D α A ˙ ∧ β |Q| β/n M δ,1 ( f )(x). For J 3 , by the following inequality (see [12]):      Ω(x − y) |x − y| m+n−1−δ − Ω  x 0 − y    x 0 − y   m+n−1−δ      ≤ C    x −x 0     x 0 − y   m+n−δ +   x −x 0   γ   x 0 − y   m+n−1−δ+γ  , (3.27) 10 Continuity of multilinear operators we gain J 3 ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n  R n \  Q    x −x 0     x 0 − y   n−δ +   x −x 0   γ   x 0 − y   n−1−δ+γ  ×   |x 0 −y|≤t, |x−y|≤t dt t 3  1/2   f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=1  2 −k +2 −γk  M δ,1 ( f )(x) ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x). (3.28) For J 4 , similar to the proof of J 1 , J 2 ,andJ 3 ,weobtain J 4 ≤ C  |α|=m  R n \  Q    x −x 0     x 0 − y   n+1−δ +   x −x 0   1/2   x 0 − y   n+1/2−δ +   x −x 0   γ   x 0 − y   n+γ−δ  ×   D α  A(y)   f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n ∞  k=1  2 k(β−1) +2 k(β−1/2) +2 k(β−γ)  1   2 k  Q    2 k  Q   f (y)   dy ≤ C  |α|=m   D α A   ˙ ∧ β |Q| β/n M δ,1 ( f )(x). (3.29) These yield the desired results. Acknowledgment The author would like to express his gratitude to the referee for his comments and sug- gestions. References [1] S. Chanillo, Anoteoncommutators, Indiana University Mathematics Journal 31 (1982), no. 1, 7–16. [2] W. Chen, A Besov estimate for multilinear singular integrals, Acta Mathematica Sinica. English Series 16 (2000), no. 4, 613–626. [3] J. Cohen, A sharp est i mate for a multilinear singular integral in R n , Indiana University Mathe- matics Journal 30 (1981), no. 5, 693–702. 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(1995), no 1, 1–17 [10] E M Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol 43, Princeton University Press, New Jersey, 1993 [11] A Torchinsky, Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, vol 123, Academic Press, Florida, 1986 [12] A Torchinsky and S L Wang, A note on the Marcinkiewicz integral, Colloquium... vol 123, Academic Press, Florida, 1986 [12] A Torchinsky and S L Wang, A note on the Marcinkiewicz integral, Colloquium Mathematicum 60/61 (1990), no 1, 235–243 Lanzhe Liu: Department of Mathematics, Changsha University of Science and Technology, Changsha 410077, China E-mail address: lanzheliu@163.com . Triebel-Lizorkin spaces for the multilinear operators only under certain conditions on the size of the operators. As the applications, the continuity of the multilinear operators related to the Littlewood-Paley. is to study the continuity of some multi- linear operators related to certain convolution operators on the Triebel-Lizorkin spaces. In fact, we will obtain the continuity on the Triebel-Lizorkin. CONTINUITY OF MULTILINEAR OPERATORS ON TRIEBEL-LIZORKIN SPACES LANZHE LIU Received 4 February 2006; Revised 20 September 2006; Accepted 28 September 2006 The continuity of some multilinear operators