Commutators of Weighted Lipschitz Functions and Multilinear Singular Integrals with Non-Smooth Kernels

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1 Appl. Math. J. Chinese Univ. 20, 26(3: Commutators of Weighted Lipschitz Functions and Multilinear Singular Integrals with Non-Smooth Kernels LIAN Jia-li MA o-lin 2 WU Huo-xiong 3 Abstract. This paper is devoted to studying the commutators of the multilinear singular integral operators with the non smooth kernels and the weighted Lipschitz functions. Some mapping properties for two types of commutators on the weighted Lebesgue spaces, which extend and generalize some previous results, are obtained. Introduction Let T : S(R n S(R n S (R n be an m linear operator. A locally integrable function K(x, y,, y m defined away from the diagonal x = y = = y m in (R n m+ is called to be an associated kernel of T if T (f,, f m, g = K(x, y,, y m f (y f m (y m g(xdy dy m dx (. R n (R n m holds for all f,, f m, g in S(R n with m j= suppf j suppg =. Throughout this paper, we assume the following size estimate on the kernel K A K(x, y,, y m ( x y + + x y m mn (.2 for some A > 0 and all (x, y,, y m with x y j for some j. The study of the multilinear Calderón Zygmund singular integral operators (i.e., whose kernels satisfy the standard Calderón Zygmund smoothness conditions can go back to the works of Coifman and Meyer [2-4], and has advanced significantly in recent years. For a systematic analysis of such operators, one can refer to the articles by Grafakos and Torres [4-6]. Also see [6, 20, 25, 28, 33] for the variant weighted estimates on the multilinear singular integral and fractional integral operators. Very recently, a kind of multilinear operators whose kernels have regularity significantly weaker than that of the standard Calderón-Zygmund operators Received: MR Subject Classification: 4220, 4225, 4670, 47G30. Keywords: Commutator, multilinear singular integral, non-smooth kernel, weighted Lipschitz function, weights. Digital Object Identifier(DOI: 0.007/s Supported by the National Natural Science Foundation of China (077054, and the NFS of Fujian Province of China (No. 200J003.

2 354 Appl. Math. J. Chinese Univ. Vol. 26, No. 3 has been studied in [9-3] to generalize the classical theory of Calderón Zygmund operator. The assumptions made on this kind of kernels base on a class of integral operators {A t } t>0 which play the role of an approximation to the identity operator, see [9]. The operators A t are assumed to be associated with kernels a t (x, y in the sense that for any f L p (R n, p, A t f(x = a t (x, yf(ydy, R n and the following condition holds ( x y a t (x, y h t (x, y = t n/s s h (.3 t in which s is a positive fixed constant and h is a positive, bounded, decreasing function satisfying for some η > 0. lim r rn+η h(r s = 0 (.4 We shall also recall the transposes of T. Note that (. implies that the m linear operator T : S(R n S(R n S (R n is linear on every entry and consequently it has m formally transposes. The j th transpose T,j of T is defined via T,j (f,, f j, f j, f j+,, f m, g = T (f,, f j, g, f j+,, f m, f j (.5 for all f,, f m, g in S(R n. It is easy to check that the kernel K,j of T,j is related to the kernel K of T via the identity K,j (x, y,, y j, y j, y j+,, y m = K(y j, y,, y j, x, y j+,, y m. (.6 To unify the notations, we may occasionally denote T by T,0 and K by K,0. Now we are in a position to state the assumptions on the non smooth kernel K. Assumption (H. Assume that for each i =,, m, there exist operators {A (i t } t>0 with kernels a (i t (x, y that satisfy conditions (.3 and (.4 with constants s and η and that for every j = 0,,, m, there exist kernels K,j,(i t (x, y,, y m such that T,j (f,, A (i t f i,, f m, g = R n K,j,(i t (R n m (x, y,, y m f (y f m (y m g(xdy dy m dx, for all f,, f m, g in S(R n with m j= suppf j suppg =. There exist a function φ C(R with suppφ [, ] and a constant ɛ > 0 so that for every j = 0,,, m and every i =,, m, we have K,j (x, y,, y m K,j,(i t (x, y,, y m A m ( yi y k ( x y + + x y m mn φ whenever 2t /s x y i. k=,k i t /s + (.7 At ɛ/s ( x y + + x y m mn+ɛ, Assumption (H2. Assume that there exist operators { t } t>0 with kernels b t (x, y that satisfy conditions (.3 and (.4 with constants s and η and also that there exist kernels K (0 t (x, y,, y m such that the representation is valid K (0 t (x, y,, y m = K(z, y,, y m b t (x, zdx; (.9 R n (.8

3 LIAN Jia-li, et al. Commutators of weighted Lipschitz functions and multilinear singular integrals 355 and assume also that there exist a function φ C(R with suppφ [, ] and a constant ɛ > 0 such that K(x, y,, y m K (0 t (x, y,, y m m A ( x y + + x y m mn k=,k j ( x yk φ for some A > 0, whenever 2t /s max j m x y j. t /s + At ɛ/s ( x y + + x y m mn+ɛ (.0 Assumption (H3. Assume that there exist operators { t } t>0 with kernels b t (x, y that satisfy conditions (.3 and (.4 with constants s and η and also that there exist kernels K (0 t (x, y,, y m such that (.9 holds. Also assume that there exist positive constants A and ɛ such that K (0 A t (x, y,, y m ( x y + + x y m mn (. whenever 2t /s min j m x y j, and t (x, y,, y m K (0 t (x At ɛ/s, y,, y m ( x y + + x y m mn+ɛ (.2 K (0 whenever 2 x x t /s and 2t /s min j m x y j. One can refer to [, 2] that the assumptions (H, (H2 and (H3 are significantly weaker than the standard Calderón Zygmund kernel conditions for multilinear operators and there does exist examples of singular multilinear operators whose kernels satisfy these weak regularity conditions but do not satisfy the standard Calderón Zygmund kernel regularity. In [] Duong, Grafakos and Yan proved the weak type endpoint estimate and the strong type L p boundedness for the multilinear operators, the analogous results to the classical multilinear Calderón-Zygmund operator can be found in [5, 6]. The weighted norm estimates for such m- th linear operators are obtained in [2, 3] by two different but somehow standard approaches motivated by [6, 32], namely basing on Coltar s inequality and sharp maximal function estimates, respectively. Recently, Hu and Lu [2] improved the results in [2, 3]. On the other hand, in [26], we proved the boundedness on weighted L p spaces for the multilinear commutators of MO functions and multilinear singular integrals with non smooth kernels satisfying (H, (H2 and (H3 (see [3] for the corresponding results of the linear commutators. In this paper, we will consider these commutators with MO functions in place of the weighted Lipschitz functions. We denote f = (f,, f m and let b = (b,, b m be a family of locally integrable functions. We mainly consider the following two types of commutators generated by the multilinear operator T in (. with b: m T Πb (f(x = K(x, y,, y m (b i (x b i (y i f i (y i dy dy m, (.3 (R n m and T Σb (f(x = i= m T j b j (f(x, (.4 j=

4 356 Appl. Math. J. Chinese Univ. Vol. 26, No. 3 where T j b j (f(x = b j (xt (f,, f j,, f m (x T (f,, b j f j,, f m (x, j =,, m. (.5 Here the notations of our commutators are taken from [3], see also [6, 25-28] for the definitions and notations of these commutators. Clearly, for m =, T Πb (f = T Σb (f = [b, T ]f = bt (f T (bf, which is the wellknown the commutator of Coifman-Rochberg-Weiss type. For the operator [b, T ], when T is the Calderón-Zygmund singular integral operator and b Λ β (R n (the homogeneous Lipschitz spaces, Paluszyński [30] established the boundedness of (p, q with < p < n/β and /q = /p β/n. In 2008, Hu and Gu [22] extended this result to the case: b Lip β,µ with µ A. The main purpose of this paper is to extend these results to the multilinear commutators (.3 (.5. Our results can be formulated as follows: Theorem.. Assume that the kernel K satisfies (.2 and the assumptions (H, (H2 and (H3. Let < q,, q m, q < be given numbers satisfying /q + +/q m = /q. Assume that T maps L q (R n L qm (R n to L q (R n. Fix j {,, m} and let /r = /p β/n, < p < r <, 0 < β < and /p + + /p m = /p with < p i <, i =,..., m. Given µ such that µ A (R n and b j Lip β,µ (R n (j =,, m, then we have Furthermore, T j b j (f Lr (µ r b j Lipβ,µ m i= TΣb (f m L b r (µ r j Lipβ,µ j= f i L p i (µ, j =,, m. (.6 m i= f i L p i (µ. (.7 Here, we use the notation I II to denote I CII for some constant C > 0 and the same latter. With the help of Theorem., for the m-linear commutator (.3 we have: Theorem.2. Assume that the kernel K satisfies (.2 and the assumptions (H, (H2 and (H3. Let < q,, q m, q < be given numbers satisfying /q + +/q m = /q. Assume that T maps L q (R n L qm (R n to L q (R n. Let /r i = /p i β i /n, < p i < r i <, 0 < β i <, i =,..., m with /p + +/p m = /p, /r + +/r m = /r, β + +β m = β, and 0 < β <. Given µ such that µ A (R n and b i Lip βi,µ(r n, i =,, m, then we have TΠb (f m Lr b (µ mr i f Lipβi,µ i L p i (µ. (.8 i= Remark.. Observe that for m =, Theorem. and Theorem.2 coincide, which is an improvement of the result in [22] and an extension of the result in [30]. Compared to the results in [22, 30], our results give two aspects generalization: (i weakening the smoothness of the kernel, (ii extending to the multilinear case. We also remark that for b (MO m and the multilinear Calderon-Zygmund singular integrals, Lerner et al.[25] (resp., Perez et al.[3] established the weighted boundedness in

5 LIAN Jia-li, et al. Commutators of weighted Lipschitz functions and multilinear singular integrals 357 product Lebesgue spaces of the commutators T Σb (resp., T Πb for the more general weights. Here we aren t clear whether our results hold for the more general weights as in [25, 3], which are interesting. The rest of this paper is organized as follows. In Section 2, we recall some standard definitions and lemmas, and also prove some useful estimates. Section 3 is devoted to the proofs of our theorems. Throughout this paper, we use the letter C to denote a positive constant that may vary at each occurrence is dependent of various indexes but independent of the essential variable. For any p, the p is always used to denote the dual index such that = /p + /p. 2 Preliminary A non-negative function µ defined on R n is called weight if it is locally integral. A weight µ is said to belong to the Muckenhoupt class A p (R n, < p <, if there exists a constant C such that ( µ( for every ball R n, where µ( = µ(xdx. The class A (R n is defined by replacing (2. with µ( (µ(x p dx p C (2. Cµ(x, a.e. x R n (2.2 for every ball x. The class A (R n can be characterized as A = p< A p. Many properties of the weights can be found in the books [8, 7, 9], we only collect some of them in the following lemma which will be used below: Lemma 2.. (i A p A q for p < q ; (ii If µ A, then µ θ A for 0 θ ; (iii For < p <, µ A p if and only if µ p A p. A locally integrable function f belongs to the weighted Lipschitz space Lip p β,µ (Rn for p, 0 < β < and µ A if ( /p f(x f µ( β/n p µ(x dx p C <. (2.3 µ( sup x The smallest bound C satisfying (2.3 is then taken to be the norm of f denoted by f Lip p. β,µ Put Lip β,µ = Lip β,µ. Obviously, for the case µ =, then Lip β,µ is the homogeneous Lipschitz space Λ β. Let µ A (R n, García-Cuerva [8] proved that the spaces Lip p β,µ coincide, and the norms of Lip p are equivalent with respect to different values of p provided that p. β,µ The important properties of the weights are the weighted estimates for the maximal function, the sharp maximal function and their variants. We first recall the maximal function defined by M(f(x = sup f(y dy. (2.4 x

6 358 Appl. Math. J. Chinese Univ. Vol. 26, No. 3 It is well known that for < p <, M maps L p (µ into itself if and only if µ A p, see [7, p.690]. The sharp maximal function is defined by M (f(x = sup f(y f dy sup inf f(y c dy. (2.5 x x c We also recall the variants M δ (f(x = M( f δ /δ (x and M δ f(x = M ( f δ /δ (x. We denote the weighted fractional maximal operators by ( M α,µ,s (f(x = sup f(y s s µ(ydy. (2.6 x µ( sα/n Recall that M α := M α,, is the standard fractional maximal operator M α (f(x = sup f(y dy. (2.7 x α/n For maybe confusion, M δ is always for the variants of (2.4, while M α, M β etc. are for those of (2.7. Our main tools in the proofs of Theorem..2 are the following weighted estimates: Lemma 2.2. (Kolmogorov s inequality[9, p.485] Let (X, ω be a probability measure space and let 0 < p < q <, then there is a constant C = C p,q such that for any measurable function f f L p (ω C f L q, (ω. (2.8 Lemma 2.3. ([7, p.323] Let 0 < p, δ <, and µ A (R n, then M δ (f Lp (µ M δ (f L p (µ. (2.9 Lemma 2.4. Suppose that 0 < α < n, 0 < s < p < n/α, /q = /p α/n. If µ A (R n, then M α,µ,s (f L q (µ f L p (µ. (2.0 Proof. For s =, this lemma is a special case of Theorem (.6 in []. Indeed, since µ A (R n implies that µ(x satisfying doubling size condition, (2.0 follows by taking v = w =, γ = α/n in [, Theorem (.6]. Now for s, we observe from (2.6 that M α,µ,s (f = [M sα,µ, ( f s ] /s. Then by using (2.0 for s =, we obtain M α,µ,s (f L q (µ = M sα,µ, ( f s /s L q/s (µ f s /s L p/s (µ = f L p (µ, (2. since 0 < sα < n, /(q/s = /(p/s sα/n. In the proof of Theorem.2, we also need the following weighted estimates for M α : Lemma 2.5. Suppose that 0 < α < n, < p < n/α and /q = /p α/n. If µ A +q/p (R n, then M α (f L q (µ f Lp (µ p/q. (2.2 Proof. y [29, Theorem 4], (2.2 holds for µ /q A(p, q (see [7, 29] for the definition of A(p, q weights.. ut by (2.7 on [7, p.32], we know that this condition is equivalent to µ A +q/p. We build some auxiliary estimates below which will be used in the proof of Theorem..2.

7 LIAN Jia-li, et al. Commutators of weighted Lipschitz functions and multilinear singular integrals 359 Lemma 2.6. Let µ A (R n, 0 < β < and b Lip β,µ. (i For any k, we have then b (x,2r b (x,2 k+ R kµ(xµ((x, 2 k+ R β/n b Lipβ,µ. (2.3 (ii For any s < and any ball x, we have µ( β/n f(y dy M β,µ,s (f(x. (2.4 (iii For any < s < and any ball x, we have (b(y b f(y dy µ(x b Lipβ,µ M β,µ,s (f(x. (2.5 Proof. The assertion (i is an easy consequence of the definition (2.3. For the assertion (ii, by Hölder s inequality and the definitions (2., (2.6, since µ A A s, we have µ( β/n ( /s ( /s f(y dy µ(β/n f(y s µ(ydy µ(y s s dy ( s µ(β/n µ( /s β/n M β,µ,s (f(x µ(y s s dy µ(/s M β,µ,s (f(x µ( /s = M β,µ,s (f(x. Now for (iii, first notice that since µ A, Lip p β,µ coincide for all p. Then by Hölder s inequality and the definitions (2.2, (2.3, (2.6, for < s <, we have (b(y b f(y dy ( /s ( /s b(y b s µ(y s dy f(y s µ(ydy b Lipβ,µ µ( /s β/n M β,µ,s (T (f, f 2 (x µ(β/n+/s µ(x b Lipβ,µ M β,µ,s (f(x. Finally, we recall the following L p boundedness of T, which can be found in [-3]: Lemma 2.7. Let T be a m linear operator with kernel satisfying the assumptions (H, (H3. Let < q,... q m, q < be given numbers satisfying /q + + /q m = /q. Assume that T maps L q (R n L qm (R n to L q (R n, then the following statements are valid: (i T can be extended to be a bounded operator from L (R n L (R n to L m, (R n ; (ii Let < p, p, p m < be any numbers satisfying /p + + /p m = /p and w A p. Then T can be extended to be a bounded operator from L p (w L pm (w to L p (w. We will prove our theorems in this section. 3 Proofs of Theorems

8 360 Appl. Math. J. Chinese Univ. Vol. 26, No Proof of Theorem. For simplicity, we will only prove for the case m = 2. The arguments for the case m > 2 are similar. We first establish the following crucial lemma: Lemma 3.. Let µ A (R n and b j Lip β,µ with 0 < β <, j =, 2. Let 0 < δ < /2 < < s < n/β. Then we have M δ [T j b j (f, f 2 (x] for j =, 2. µ(x b j Lipβ,µ [M β,µ,s (T (f, f 2 (x + M β,µ,s (f (xm(f 2 (x +M(f (xm β,µ,s (f 2 (x] Proof. Without loss of generality, we only consider the case: j = and denote b by b for convenience. Fix x R n and let = (x, R with R > 0. Taking λ = b, the average of b on, where = (x, 2R. To proceed with, we decompose f i = f 0 i + f i, where f 0 i = fχ, i =, 2. Let c be a constant to be fixed along the proof. Since 0 < δ <, we have ( T b (f, f 2 (y δ c δ dy /δ ( /δ Tb (f, f 2 (y c δ dy ( /δ ( (b(y λt (f, f 2 (y dy δ + ( /δ ( + T ((b λf 0, f2 (y dy δ + C ( /δ + T ((b λf, f2 (y c δ dy := I + I 2 + I 3 + I 4 + I 5. /δ T ((b λf 0, f2 0 (y δ dy T ((b λf, f 0 2 (y δ dy Since 0 < δ <, µ A and b Lip β,µ, by Lemma 2.6 (iii, we can bound the first term I by I (b(y b T (f, f 2 (y dy µ(x b Lipβ,µ M β,µ,s (T (f, f 2 (x. (3.3 To estimate the second term I 2, since 0 < δ < /2, we use the Kolmogorov inequality with p = δ, q = /2, X =, ω = dx, also by Lemma 2.7 (i and Lemma 2.6 (iii, to obtain I 2 T ((b λf 0, f2 0 L 2, ( ( dx ( (b(y b f (y dy f 2 (y 2 dy 2 µ(x b Lipβ,µ M β,µ,s (f (xm(f 2 (x. For the term I 3, using the fact y y y x for any y ( c, y and the size estimate on K, namely, (.2, by Lemma 2.6 (i (iii, we obtain I 3 T ((b λf, f2 0 (y dy (3. /δ (3.2 (3.4

9 LIAN Jia-li, et al. Commutators of weighted Lipschitz functions and multilinear singular integrals 36 A (R n \ ( y y + y y 2 2n b(y λ f (y f 2 (y 2 dy dy 2 dy b(y b f (y R n \ y x 2n dy f 2 (y 2 dy 2 b(y b f (y k= 2 k+ \2 k y x 2n dy f 2 (y 2 dy 2 M(f 2 (x k 2 b(y b f (y dy k= 2 k+ ( M(f 2 (x k 2 b(y b (x,2 R f k+ (y dy k= 2 k+ + b b (x,2 k+ R f (y dy 2 k+ M(f 2 (x 2 k[ µ(x b Lipβ,µ M β,µ,s (f (x k= +kµ(x µ((x, 2k+ R β/n ] k 2 b Lipβ,µ f (y dy [ 2 k+ M(f 2 (x 2 k µ(x b Lipβ,µ M β,µ,s (f (x k= + µ((x, 2 k+ R β/n k= k2 k µ(x b Lipβ,µ 2 k+ µ(x b Lipβ,µ M(f 2 (xm β,µ,s (f (x. 2 k+ f (y dy ] (3.5 While for the term I 4, using (.2 and Lemma 2.6 (iii, we bound it by I 4 T ((b λf 0, f2 (y dy A (R n \ ( y y + y y 2 2n b(y λ f (y f 2 (y 2 dy dy 2 f 2 (y 2 b(y b f (y dy R n \ y 2 x 2n dy 2 µ(x b Lipβ,µ M β,µ,s (f (x f 2 (y 2 k= 2 k+ \2 k y 2 x 2n dy 2 2 µ(x b Lipβ,µ M β,µ,s (f (x k 2 f 2 (y 2 dy 2 k= µ(x b Lipβ,µ M β,µ,s (f (xm(f 2 (x. 2 k+ (3.6 For the last term I 5, we will make use of the assumptions (H2 and (H3. Fixing now the value of c by taking c = A t (T ((b λf, f2 (x, where t = R, then we have I 5 T ((b λf, f2 (y A t (T ((b λf, f2 (x dy K(y, y, y 2 Kt 0 (y, y, y 2 (b(y b f (y f 2 (y 2 dy dy 2 dy (R n \ 2 ε/2 ( y y + y y 2 2n+ε (b(y b f (y f 2 (y 2 dy dy 2 dy (R n \ 2 t

10 362 Appl. Math. J. Chinese Univ. Vol. 26, No. 3 k= k+\k R ε/2 y x (b(y b n+ε/2 f (y dy R j= j+\j ε/2 y 2 x f 2(y n+ε/2 2 dy 2 2 kε/2 (b(y b f (y dy k+ k+ 2 jε/2 f 2 (y 2 dy 2 j+ j+ k= j= µ(x b Lipβ,µ M(f 2 (xm β,µ,s (f (x, where in the last inequality, we use the same computational technique in (3.5. Consequently, substituting (3.3 (3.7 into (3.2, by the definition of M δ, we obtain (3. and conclude Lemma 3.. Now we are ready to return to prove Theorem.. Proof of Theorem.. First, by Lemma 2. we have that µ A A r and hence µ r A r A. Then by Lemma 2.3, we obtain [T j b j (f, f 2 (x] Mδ [T j M Lr (µ r b j (f, f 2 (x] Lr (µ r δ [T j b j (f, f 2 (x] (3.8 Lr (µ r for j =, 2. Hence, by Lemma 3., we reduce to bound the L r (µ r norm of the right-hand side of (3.. For the first term, since /r = /p β/n and taking s such that < s < p < n/β, by Lemma 2.4 and Lemma 2.7 (ii, we have µm β,µ,s (T (f, f 2 Lr (µ r = M β,µ,s (T (f, f 2 Lr (µ T (f, f 2 Lp (µ f L p (µ f 2 L p 2 (µ. For the second term, we let /r = /p 2 + /l and hence /l = /p β/n. Then by Lemma 2.4 again, together with Hölder s inequality, we obtain (3.7 (3.9 µm β,µ,s (f M(f 2 L r (µ r = M β,µ,s (f M(f 2 L r (µ M β,µ,s (f Ll (µ M(f 2 L p 2 (µ f L p (µ f 2 L p 2 (µ. Consequently, by (3., (3.8 (3.0, we conclude the proof of Theorem.. ( Proof of Theorem.2 As before, we only consider the case m = 2 and the proof of Theorem.2 is based on the following estimate of the sharp maximal function: Lemma 3.2. Let µ A (R n and b Lip β,µ, b 2 Lip β2,µ, β + β 2 = β, and 0 < β <. Let 0 < δ < /3 < < s i < n/β i, i =, 2. Then we have M δ [T Πb(f, f 2 ](x µ(x 2 b Lipβ,µ b 2 Lipβ2,µ [M β,µ,s(t (f, f 2 (x + M β,µ,s(f (xm β2,µ,s(f 2 (x] + b Lipβ,µ µ(x+β/n M β (T 2 b 2 (f, f 2 (x + b 2 Lipβ2,µ µ(x+β2/n M β2 (T b (f, f 2 (x, where T i b i (i =, 2 are defined in (.5 for the case: m = 2. (3.

11 LIAN Jia-li, et al. Commutators of weighted Lipschitz functions and multilinear singular integrals 363 Proof. We fix x R n and let = (x, R with R > 0. Taking λ i = b i,, the average of b i on, i =, 2, where = (x, 2R. Let c be a constant to be fixed along the proof. Since 0 < δ < /3, we have ( T Πb (f, f 2 (y δ c δ dy /δ ( ( (b (y λ (b 2 (y λ 2 T (f, f 2 (y δ dy ( /δ + (b (y λ Tb 2 2 (f, f 2 (y δ dy ( /δ + (b 2 (y λ 2 Tb (f, f 2 (y δ dy ( + T ((b λ f, (b 2 λ 2 f 2 (y c δ dy := W + W 2 + W 3 + W 4. /δ T Πb (f, f 2 (y c δ dy /δ /δ (3.2 For the first term W, we can not estimate like (3.3 to use Lemma 2.6 (iii due to that the integral contains three terms. We overcome it by restricting that 0 < δ < /3. Then by Hölder s inequality, the definition (2.3 for p = and Lemma 2.6 (ii, we can bound W as follows: W ( ( /3δ ( b (y λ dy 3δ ( /3δ b 2 (y λ 2 3δ dy /3δ T (f, f 2 (y 3δ dy ( b (y b, dy b 2 (y b 2, dy ( T (f, f 2 (y dy µ( β2/n+ b 2 Lipβ2,µ µ(x 2 b b µ( β/n Lipβ,µ 2 Lipβ2,µ T (f, f 2 (y dy µ(x 2 b b Lipβ,µ 2 M Lipβ2,µ β,µ,s(t (f, f 2 (x. µ( β/n+ b Lipβ,µ ( T (f, f 2 (y dy (3.3 For the terms W 2, W 3, we also can not use Lemma 2.6 (iii. Noticing that 0 < δ < /3, we use the facts /δ = + ( δ/δ and 0 < δ δ < /2. Then by Hölder s inequality and (2.6, we have ( /δ W 2 (b (y λ Tb 2 2 (f, f 2 (y δ dy ( ( ( δ/δ b (y b, dy T b2 (f, f 2 δ/( δ dy µ( +β/n ( (3.4 b Lipβ,µ Tb 2 2 (f, f 2 dy ( b Lipβ,µ µ(x+β/n T 2 β/n b 2 (f, f 2 dy = b Lipβ,µ µ(x+β/n M β (Tb 2 2 (f, f 2 (x.

12 364 Appl. Math. J. Chinese Univ. Vol. 26, No. 3 Similarly, we have W 3 b 2 Lipβ2,µ µ(x+β2/n M β2 (T b (f, f 2 (x. (3.5 Now we turn to estimate the most delicate term W 4. To proceed with, we decompose f i = fi 0 + f i, where fi 0 = fχ, i =, 2, then ( /δ W 4 T ((b (y λ f 0, (b 2 (y λ 2 f2 0 (y δ dy ( /δ + T ((b (y λ f 0, (b 2 (y λ 2 f2 (y δ dy ( /δ + T ((b (y λ f, (b 2 (y λ 2 f2 0 (y δ dy (3.6 ( /δ + T ((b (y λ f, (b 2 (y λ 2 f2 (y c δ dy := W 4 + W 42 + W 43 + W 44. To estimate W 4, applying again Lemma 2.2, Lemma 2.7 (i and Lemma 2.6 (iii, we obtain W 4 T ((b λ f 0, (b 2 λ 2 f2 0 L 2, ( ( dx ( (b (y b, f (y dy (b 2 (y 2 b 2, f 2 (y 2 dy 2 µ(x 2 b b Lipβ,µ 2 M Lipβ2,µ β,µ,s(f (xm β2,µ,s(f 2 (x. (3.7 For the second term W 42, by the fact y y 2 y 2 x for any y 2 ( c, y, by (.2 and Lemma 2.6 (i (iii we have W 42 T ((b λ f 0, (b 2 λ 2 f2 (y dy A (R n \ ( y y + y y 2 2n b (y λ f (y b 2 (y 2 λ 2 f 2 (y 2 dy dy 2 b b (y b, f (y dy 2(y 2 b 2, f 2(y 2 R n \ y 2 x dy 2n 2 µ(x b M Lipβ,µ β,µ,s(f (x b 2 (y 2 b 2, f 2 (y 2 y 2 x 2n dy 2 Similarly, µ(x b Lipβ,µ M β,µ,s(f (x k= k= k 2 2 k+ \2 k 2 k+ µ(x 2 b Lipβ,µ b 2 Lipβ2,µ M β,µ,s(f (xm β2,µ,s(f 2 (x. b 2 (y 2 b 2, f 2 (y 2 dy 2 (3.8 W 43 µ(x 2 b Lipβ,µ b 2 Lipβ2,µ M β,µ,s(f (xm β2,µ,s(f 2 (x. (3.9 For the last term W 44, we will use the assumptions (H2 and (H3. Now fixing the value of c by taking c = A t (T ((b λ f, (b 2 λ 2 f2 (x, where t = R, then we have W 44 T ((b λ f, (b 2 λ 2 f2 (y A t (T ((b λ f, (b 2 λ 2 f2 (x dy K(y, y, y 2 Kt 0 (y, y, y 2 (R n \ 2 (b (y b, f (y (b 2 (y 2 b 2, f 2 (y 2 dy dy 2 dy

13 LIAN Jia-li, et al. Commutators of weighted Lipschitz functions and multilinear singular integrals 365 k= 2 k+ \2 k t ε/2 ( y y + y y 2 2n+ε (b (y b, f (y (b 2 (y 2 b 2, f 2 (y 2 dy dy 2 dy R k= k+\k ε/2 y x (b (y n+ε/2 b, f (y dy R j= j+\j ε/2 y 2 x (b 2(y n+ε/2 2 b 2, f 2 (y 2 dy 2 2 kε/2 (b (y b, f (y dy k+ k= k+ 2 jε/2 (b 2 (y 2 b 2, f 2 (y 2 dy 2 j+ j+ j= µ(x 2 b b Lipβ,µ 2 M Lipβ2,µ β,µ,s(f (xm β2,µ,s(f 2 (x. This together with (3.7 (3.9 leads to (3.20 W 4 µ(x 2 b Lipβ,µ b 2 Lipβ2,µ M β,µ,s(f (xm β2,µ,s(f 2 (x. (3.2 Consequently, (3. follows from (3.4 (3.6 and (3.2. Lemma 3.2 is proved. Now we are in a position to prove Theorem.2: Proof of Theorem.2. As in the proof of Theorem., since T Πb (f, f 2 L r (µ 2r M δ (T Πb (f, f 2 L r (µ 2r M δ (T Πb(f, f 2 L r (µ 2r, (3.22 we reduce to estimate the L r (µ 2r norm of the each term in the right-hand side of (3.. We estimate each term as follows. For the first term, since /r = /p β/n and taking s such that < s < p < n/β, by Lemma 2.4 and Lemma 2.7 (ii, since µ A, we obtain µ 2 M β,µ,s (T (f, f 2 Lr (µ 2r = M β,µ,s (T (f, f 2 Lr (µ T (f, f 2 L p (µ f L p (µ f 2 L p 2 (µ. (3.23 For the second term, since /r = /r + /r 2, by Hölder s inequality and Lemma 2.4, we get µ 2 M β,µ,s(f M β2,µ,s(f 2 Lr (µ 2r = M β,µ,s(f M β2,µ,s(f 2 Lr (µ M β,µ,s(f L r (µ M β2,µ,s(f 2 L r 2 (µ f L p (µ f 2 L p 2 (µ. (3.24 Now for the last two terms, we are involved in using Theorem. and Lemma 2.5. For the first one of them, we let /r = /l β /n and hence + r/l = r rβ /n >. Since µ A, by Lemma 2. (i, (iii, we obtain µ r+rβ/n A +r/l. Then by Lemma 2.5, we have µ +β /n M β (T 2 b 2 (f, f 2 Lr (µ 2r Noting that /l = /p β 2 /n, then by Theorem., we have = Mβ (T 2 b 2 (f, f 2 Lr (µ r+rβ /n T 2 b 2 (f, f 2 L l (µ l. T 2 b 2 (f, f 2 L l (µ l b 2 Lipβ2,µ f L p (µ f 2 L p 2 (µ. (3.25 Hence, µ +β /n M (T 2 β b 2 (f, f 2 b 2 f Lipβ2 Lr (µ 2r,µ L p (µ f 2 L p 2 (µ. (3.26

14 366 Appl. Math. J. Chinese Univ. Vol. 26, No. 3 Similarly, µ +β2/n M β2 (Tb (f, f 2 b f Lipβ L r (µ 2r,µ L p (µ f 2 L p 2 (µ. (3.27 Therefore, by (3., (3.22 (3.27, we conclude the proof of Theorem.2. References [] A ernardis, O Salinas. Two-weight norm inequalities for the fractional maximal operator on spaces of homogeneous type, Studia Math, 994, 22(3: [2] R Coifman, Y Meyer. On commutators of singular integral and bilinear singular integrals, Trans Amer Math Soc, 975, 22: [3] R Coifman, Y Meyer. Au delá des opérateurs pseudodifférentiels, Astérisque, 978, 57. [4] R Coifman, Y Meyer. Ondelettes ét opérateurs III, Hermann, Paris, 990. [5] S Chanillo. A note on commutators, Indiana Univ Math J, 982, 3: 7-6. [6] X Chen, Q Y Xue. Weighted estimates for a class of multilinear fractional type operators, J Math Anal Appl, 200, 362(2: [7] Y Ding, S Z Lu. Weighted norm inequalities for fractional integral operators with rough kernel, Canad J Math, 998, 50(: [8] J Duoandikoetxea. Fourier analysis, Vol 29 of Graduate Studies in Mathematics, Amer Math Soc, Providence, RI, USA, 995. [9] X T Duong, A McIntosh. Singular integral operators with non smooth kernels on irregular domains, Rev Mat Iberoamericana, 999, 5: [0] X Duong, L X Yan. Commutators of MO functions and singular integral operators with non smooth kerels, ull Austral Math Soc, 2003, 67: [] X Duong, L Grafakos, L X Yan. Multilinear operators with non smooth kernels and commutators of singular integrals, Trans Amer Math Soc, 200, 362(4: [2] X T Duong, R M Gong, L Grafakos, J Li, L X Yan. Maximal operator for multilinear singular integrals with non smooth kernels, Indiana Univ Math J, 2009, 58(6: [3] R M Gong, J Li. Sharp maximal function estimates for multilinear singular integrals with nonsmooth kernels, Anal Theory Appl, 2009, 25(4: [4] L Grafakos, R H Torres. On multilinear singular integrals of Calderón Zygmund type, Proceedings of the 6-th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, Publ Mat, 2002, Extra: [5] L Grafakos, R H Torres. Maximal operators and weighted norm inequalities for multilinear singular integrals, Indiana Univ Math J, 2002, 5: [6] L Grafakos, R H Torres. Multilinear Calderón Zygmund theory, Adv in Math, 2002, 65: [7] L Grafakos. Classical and Modern Fourier Analysis, Pearson Education, Inc, Prentice Hall, [8] J García-Cuerva. Weighted H p space, Dissertation Summaries in Mathematics, 979, Vol 62. [9] J Garía-Cuerva, J L Rubio de Francia. Weighted Norm Inequalities and Related Topics, North- Holland, Amsterdam, 985.

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