A Good λ Estimate for Multilinear Commutator of Singular Integral on Spaces of Homogeneous Type

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1 Armenian Journal of Mathematics Volume 3, Number 3, 200, A Good λ Estimate for Multilinear ommutator of Singular Integral on Spaces of Homogeneous Type Zhang Qian* and Liu Lanzhe** * School of Mathematics and Systems Science eijing University of Aeronautics & Astronautics eijing, 009, P.R.of hina zqok 000@tom.com ** Department of Mathematics hangsha University of Science and Technology hangsha, 40077, P.R.of hina lanzheliu@63.com Abstract In this paper, a good λ estimate for the multilinear commutator associated to the singular integral operator on the spaces of homogeneous type is obtained. Under this result, we get the(l p (X), L q (X))-boundedness of the multilinear commutator. Key Words: Singular Integral; Multilinear ommutator; Lipschitz Space; MO; Space of Homogeneous Type; Good λ Inequality. Mathematics Subject lassification 2000: 4220, 4225 Introduction Let T be the alderón-zygmund operator, oifman, Rochberg and Weiss [9] proves that the commutator [b, T ](f) = bt (f) T (bf)(where b MO(R n )) is bounded on L p (R n ) for < p <. hanillo [5] proves a similiar result when T is replaced by the fractional operators. In [5], [8], Janson and Paluszynski study these results for the Triebel-Lizorkin spaces and the case b Lip β (R n ), where Lip β (R n ) is the homogeneous Lipschitz space. The main purpose of this paper is to establish the good λ estimate for the multilinear commutator associated to the singular integral operator on the spaces of homogeneous type, where b Lip β (X) or b MO(X). Under this result, we get (L p (X), L q (X))-boundedness of the multilinear commutator. 05

2 06 ZHANG QIAN AND LIU LANZHE 2 Preliminaries and Theorems Give a set X, a function d : X X R + is called a quasi-distance on X if the following conditions are satisfied: (i) for every x and y in X, d(x, y) 0 and d(x, y) = 0 if and only if x = y, (ii) for every x and y in X, d(x, y) = d(y, x), (iii) there exists a constant k such that for every x, y and z in X. d(x, y) k(d(x, z) + d(z, y)) () Let µ be a positive measure on the σ-algebra of subsets of X which contains the r-balls (x, r) = {y : d(x, y) < r}. We assume that µ satisfies a doubling condition, that is, there exists a constant A such that holds for all x X and r > 0. 0 < µ((x, 2r)) Aµ((x, r)) < (2) A structure (X, d, µ), with d and µ as above, is called a space of homogeneous type. The constants k and A in () and (2) will be called the constants of the space. Then let us introduce some notations [3], [3], [8]. Throughout this paper, will denote a ball of X, and for a ball let f = µ() f(x)dµ(x) and the sharp function of f is defined by f # (x) = sup f(y) f dµ(y). x µ() It is well-known that [3] f # (x) sup inf x c f(y) dµ(y). µ() We say that f belongs to MO(X) if f # belongs to L (X) and define f MO = f # L. It has been known that[3] For p < and 0 γ <, let M γ,p (f)(x) = sup x f f 2 k MO k f MO. ( µ() pγ. If γ = 0, M p,γ (f) = M p (f) which is the Hardy-Littlewood maximal function when p =. For 0 < β <, the Lipschitz space β (X) is the space of functions f such that ( ) f β = sup x, h X h 0 ρ [β]+ h f(x) /ρ(x + h, x) β <, 06

3 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL where k h denotes the k-th difference operator [8] and the existence of ρ is guaranteed by the following Lemma. In this paper, we will study some multilinear commutators as follows. Definition. Suppose b j (j =,, m) are the fixed locally integrable functions on X. Let T be the singular integral operator as T (f)(x) = K(x, y)f(y)dµ(y), X where K is a locally integrable function on X X \{(x, y) : x = y} and satisfies the following properties: () K(x, y) µ((x, d(x, y))), (2) K(x, y) K(x, y ) + K(y, x) K(y (d(y, y, x) ) δ µ((y, d(x, y)))(d(x, y)) δ, when d(x, y) 2d(y, y ), with some δ (0, ]. The multilinear commutator of the singular integral operator is defined by T b (f)(x) = m (b j (x) b j (y))k(x, y)f(y)dµ(y). X j= and T b (f)(x) = sup ε>0 T b ε (f)(x), where T b ε (f)(x) = j= m (b j (x) b j (y))k(x, y)f(y)dµ(y) Note that when b = = b m, T b is just the m order commutator. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors[] [6], [2] [4], [9] [2]. Our main purpose is to find the good λ estimate for the multilinear commutator T b, and with this result to find (L p (X), L q (X))-boundedness for the multilinear commutator T b. Given some functions b j (j =,, m) and a positive integer m and j m, we set b Lipβ = m j= b j Lipβ, b MO = m j= b j MO and denote by j m the family of all finite subsets σ = {σ(),, σ(j)} of {,, m} of j different elements, σ = j is the element number of σ. For σ m j, set σ c = {,, m} \ σ. For b = (b,, b m ) and σ = {σ(),, σ(j)} m j, set b σ = (b σ(),, b σ(j) ), b σ = b σ() b σ(j), b σ β = b σ() β b σ(j) β and b σ MO = b σ() MO b σ(j) MO. In what follows, > 0 always denotes a constant that is independent of main parameters involved but whose value may differ from line to line. For any index p [, ], we denote by p its conjugate index, namely, /p + /p =. Now we state our results as following. Theorem.. Let 0 < β < and b j β (X) for j =,, m. 07

4 08 ZHANG QIAN AND LIU LANZHE (a). Suppose < r < p <. Then there exists ξ 0 > 0 such that, for any 0 < ξ < ξ 0 and λ > 0, ({ }) µ x X : T b (f)(x) > 3λ, b β M mβ,p (f)(x) ξλ ξ r µ({x X : T b (f)(x) > λ}) (b). T b is bounded from L p (X) to L q (X) for < p < /mβ and /q = /p mβ. Theorem 2.. Let b j MO(X) for j =,, m. (a). Suppose < r < p <. Then there exists ξ 0 > 0 such that, for any 0 < ξ < ξ 0 and λ > 0, ({ }) µ x X : T b (f)(x) > 3λ, b MO M p (f)(x) ξλ ξ r µ({x X : T b (f)(x) > λ}) (b). T b is bounded on L p (X) for < p <. 3 Proofs of Theorems To prove the theorems, we need the following lemmas. Lemma. [6] Let d be a quasi-distance on a set X. Then there exists a quasi-distance d on X, a finite constant and a number 0 < α <, such that d is equivalent to d and, for every x, y and z in X d (x, y) d (z, y) d (x, z) α (d (x, y) + d (z, y)) α. Lemma 2. if x Ω X, then where f # β (x) = sup Proof. Since f L () µ() f f L ( ) β f(x) f µ() +β dµ(x). inf f # u β (u), f(x) dµ(x), so we have f f L ( ) (f f ) L ( ) f f µ( dµ(x) ) f f dµ(x) µ() β inf µ() f # u β (u). 08

5 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL Lemma 3. [],[7] Let 0 < β <, p, then f β sup f(x) f µ() +β dµ(x) ( sup f(x) f µ() β p dµ(x) µ() sup inf f(x) c dµ(x) c µ() +β ( sup inf f(x) c p dµ(x) c µ() β µ() Proof. Step. We first show f β sup f(x) f µ() +β dµ(x) The difference k h is defined for each x such that x,, x + kh Ω X. Let Ω h be the set of all points x such that there is a ball x Ω with x + ih x, i = 0,,, k. Fix h and set Ω h = {x Ω h : x,, x + kh are Lebesgue points of f}, then Ω h \ ω h has measure zero. If x ω h is fixed, set y i = x + ih with i = 0,,, k. hoose as the smallest ball with {y 0, y,, y k } Ω. Since each y i is a Lebesgue point of f, if we choose balls {y i }, then f (y i ) f(y i ) and so according to Lemma 2, Since k h (f ) = 0, we have Therefore f (y i ) f(y i ) = lim {y i } f (y i ) f (y i ) f # β (y i)µ() β. k h(f, x) = k h(f f, x) max ik f(y i) f (y i ) max ik f # β (y i) h β. Step 2. Then we prove sup k h(f, x) k f # β (x + ih) h β a.e. x Ω h. i=0 f(x) f µ() +β dµ(x) sup y Hölder s inequality, we have µ() +β ( f(x) f dµ(x) µ() +β = µ() β ( f(x) f µ() β p dµ(x) µ() ( µ() f(x) f p dµ(x) µ() /p f(x) f p dµ(x). Taking a sup over x in both sides of the above inequality, we finish the step 2. 09

6 0 ZHANG QIAN AND LIU LANZHE Step 3. Last we get sup ( /p f(x) f µ() β dµ(x)) p f µ(). β Set ω k (f, t) = sup k h(f) L (Ω h ), then f β h t = sup t β ω k (f, t). We claim that t>0 f f L () µ() β f β. (3) It is enough to verify (3) for the unit ball 0 since the case of arbitrary then follows from a linear change of variables. Now suppose (3) does not hold for 0. In this case, there is a sequence of functions (f m ) such that inf f m f L ( 0 ) m f m β. If we let (f ) m denote best L ( 0 ) approximate to f m, m =, 2,, then by rescaling if necessary, we find functions g m = λ m (f m (f ) m ) such that = inf g m f L ( 0 ) = g m L ( 0 ) m g m β. Thus {g m } is precompact in L ( 0 ) and for an appropriate subsequence, g mj g with g L ( 0 ). It follows that g β ( 0 ) = lim j g mj β ( 0 ) = 0. On the other hand, inf g f m L ( 0 ) = and so we have a contradiction. For the case of general ball, we note that if f is defined on and A is the linear transformation which maps 0 onto, then the function f = f A has a modulus of smoothness which satisfies ω k ( f, t) = l /p ω k (f, lt) with l the radius of. Thus f β ( 0 ) = l β /p f β (). For any ball x, we have therefore sup ( /p f f dµ(x)) p f f L µ() (), ( /p b(x) b µ() β dµ(x)) p b µ(). β The proofs of the third and fourth parts of Lemma 3 are similar, so we omit the details. And this completes the Lemma 3. 0

7 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL... Lemma 4. [0] Let f L (X) and α > 0, and assume that µ(x) > α f L. Then f may be decomposed as f = g + b where () g 2 L 2 α f L, (2)b = b j, where each b j is supported on some ball (x j, r j ), (3) b j dµ = 0, (4) b j L αµ((x j, r j )), (5) j µ((x j, r j )) α f L. Lemma 5. [5] Let 0 ν <, r < p < /ν and /q = /p ν, then Proof. We first show that Let us consider the set E, where M ν,r (f) L q f L p. µ({x : M ν,r (x) > λ}) E = {x : M ν,r (x) > λ}. ( ) r/( νr) λ f L r. y the Lemma 4, it follows that there exists a sequence of balls j, with bounded overlap so that E j= j and so that on each j, we have f r dµ(y) λ r. µ( j ) νr j ( ) r/q Now µ(e) r/q µ(j ) µ( j ) r/q, q = r/( νr); the last inequality is true because r/q. Hence Now µ( j ) νr λ r j f r dµ(y) and r/q = νr, so µ(j ) r/q ( f r χj) dµ(y). λ r j µ(j ) r/q λ r f r L r and µ(e) λ q f q L r. Note now that if r < p < /ν, then using Hölder s inequality M ν,r (x) M ν,p (x). Therefore, by the preceeding arguments, we have ( ) p/( νp) µ(e) λ f L p. The Lemma 5 follows by the Marcinkiewicz interpolation theorem in homogeneous spaces.

8 2 ZHANG QIAN AND LIU LANZHE Proof. Theorem (a). When m =, by the Whitney decomposition, {x X : T b (f)(x) > λ} may be written as a union of balls { k } with mutually disjoint interiors and with distance from each to X \ k k comparable to the diameter of k. It suffices to prove the good λ estimate for each k. There exists a constant = (n) such that for each k, the cube k intersects X \ k k, where k denotes the ball with the same center as k and with the diam k = diam k. Then, for each k, there exists a point x 0 = x 0 (k) k such that T b (f)(x 0 ) λ. Now, we fix a ball k. Without loss of generality, we may assume there exists a point z = z(k) with b β M β,p (f)(z) ξλ. Set k = k and write f = f +f 2 for f = fχ k and f 2 = fχ X\k. We turn to the estimates on f and f 2. The estimates on f. For x k, /r = /p + /q < ( ) /r T b (f ) L r (b (x) b (y))k(x, y)f (y) r dµ(y) ) /q ( µ( k ) ( b (x) b (y) q dµ(y) k k ( µ( k ) µ( k ) β+/q b β µ( k β µ( k ) βp k Let η > 0, we have µ( k ) /r b β M β,p (f)(z). µ({x X : T b (f )(x) > ηλ}) (ηλ) r T b (f ) r L r (ηλ) r [ b β M β,p (f)(z)] r µ( k ) /r (ηλ) r (ξλ) r µ( k ) (ξ/η) r µ( k ). The estimates on f 2. Let H = H(X) be a large positive integer depending only on X. We consider the following two cases: ase. diam( k ) ε Hdiam( k ). Set V (x, y) = (b (x) b (y))k(x, y)f(y). hoose x 0 k such that x 0 X \ k k. For x k, following [8], we have T b ε (f 2 )(x) [V (x, y) V (x 0, y)]f 2 (y)dµ(y) + V (x 0, y)f(y) dµ(y) R(x) + V (x 0, y)f(y) dµ(y) + T b ε (f)(x 0 ) R(x 0 ) = I + II + III + IV, 2

9 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL... 3 where R(u) = {y X : diam( k ) < ρ(u, y) Hdiam( k )}. Now let us treat I, II and III respectively. For I, we write V (x, y) V (x 0, y) = (b (x) b (y))k(x, y)f 2 (y) (b (x 0 ) b (y))k(x 0, y)f 2 (y) = b (x)k(x, y)f 2 (y) b (y)k(x, y)f 2 (y) b (x 0 )K(x 0, y)f 2 (y) +b (y)k(x 0, y)f 2 (y) b (x 0 )K(x, y)f 2 (y) + b (x 0 )K(x, y)f 2 (y) (b (x) b (x 0 ))K(x, y)f 2 (y) + (b (x 0 ) b (y))(k(x, y) K(x 0, y))f 2 (y) = I + I 2. For I, by Lemma 3, Hölder s inequality and the following inequality, for b β b(x) b b µ() β x y β dµ(y) b β µ() β, we have I dµ(y) 2 v+ ε (x 0 )\ 2 v ε (x 0 ) b (x) b (x 0 ) K(x, y) f(y) dµ(y) ( b β µ( k ) β µ( 2 v ε(x 0 )) µ( 2v+ε(x 0 )) /p 2 v+ (x ε 0) b β µ( k ) β µ( 2 v+ ε(x 0 )) + /p+/p β ( µ( 2 v+ ε(x 0 )) βp 2 v+ ε (x 0 ) b β µ( k ) β µ( 2 v+ ε(x 0 )) β M β,p (f)(z) b β 2 vβ M β,p (f)(z) b β M β,p (f)(z) ξλ. For I 2, by K s properties, µ s doubling condition, Hölder s inequality and Lemma 3, we 3

10 4 ZHANG QIAN AND LIU LANZHE obtain I 2 dµ(y) Therefore I ξλ. ( 2 v+ ε (x 0 )\ 2 v ε (x 0 ) 2 v+ ε (x 0 ) 2 v+ ε (x 0 ) µ( k ) δ µ( 2 v+ ε(x 0 ) +δ b (x 0 ) b (y) (b (x 0 ) b (y))(k(x, y) K(x 0, y))f(y) dµ(y) [ ] δ d(x0, x) d(x, y) µ((x 0, d(x 0, y))) f(y) dµ(y) b (x 0 ) b (y) d(x 0, x) δ f(y) dµ(y) d(x 0, y) +δ ( b (x 0 ) b (y) p dµ(y) 2 v+ ε (x 0 ) 2 v+ ε (x 0 ) µ( k ) δ µ( 2 v+ ε)(x 0 ) δ+β+/p +/p β b β M β,p (f)(z) 2 vδ b β M β,p (f)(z) b β M β,p (f)(z) ξλ. For II and III, note that, for y R(x), ρ(x, y) Hdiam( k ), we get, by Lemma 3 and Hölder s inequality, II b (x 0 ) b (y) K(x, y) f(y) dµ(y) H k ( ( µ(h k ) b (x 0 ) b (y) p dµ(y) H k H k ( µ(h k ) +β+/p +/p β b β µ(h k ) βp f(y) p dµ(y) b β M β,p (f)(z) ξλ. Similar III ξλ. Thus I + II + III ξλ. For IV, since x / k k, then T b ε (f)(x 0 ) λ. For x k, sup T b ε diam( ε (f 2 )(x) ξλ + λ. k ) 4

11 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL... 5 ase 2. ε > Hdiam( k ). Let ε k denote the ball with the same center as k and with the diam k ε = ε. Similar to the proof of ase, we get Thus, we have shown that for x k, sup T b ε>diam( ε (f 2 )(x) ξλ + λ. k ) T b (f 2 )(x) ξλ + λ. Now, choose ξ 0 such that ξ 0 <, let η = and combine the estimates on f with f 2, we get ({ }) µ x k : T b (f)(x) > 3λ, b β M β,p (f)(x) ξλ µ({x k : T b (f )(x) > 2λ ξλ}) + µ({x X : T b (f 2 )(x) > λ + ξλ}) µ({x k : T b (f )(x) > λ}) ξ r µ( k ). When m >, similar to the case m =, there exists a point x 0 = x 0 (k) k such that T b (f)(x 0 ) λ. Now, we fix a ball k. Without loss of generality, we may assume there exists a point z = z(k) with b β M mβ,p (f)(z) ξλ. Set k = k and write f = f +f 2 for f = fχ k and f 2 = fχ X\k. We turn to the estimates on f and f 2. The estimates on f. For x k, /r = /p + /q < ( r ) T b m /r (f ) L r (b j (x) b j (y))k(x, y)f (y) dµ(y) j= ( ) /q m ( µ( k ) b j (x) b j (y) q dµ(y) k k j= µ( k ) µ( k ) mβ+/q b β µ( k mβ ( Let η > 0, we have µ( k ) /r b β M mβ,p (f)(z). µ({x X : T b (f )(x) > ηλ}) (ηλ) r T b (f ) r L r µ( k ) mβp (ηλ) r [ b β M mβ,p (f)(z)] r µ( k ) /r (ηλ) r (ξλ) r µ( k ) (ξ/η) r µ( k ). 5 k

12 6 ZHANG QIAN AND LIU LANZHE The estimates on f 2. Let H = H(X) be a large positive integer depending only on X. We consider the following two cases: ase. diam( k ) ε Hdiam( k ). Set U(x, y) = m j= (b j(x) b j (y))k(x, y)f(y). hoose x 0 k such that x 0 X \ k k. For x k, following [8], we have T b ε (f 2 )(x) + R(x 0 ) [U(x, y) U(x 0, y)]f 2 (y)dµ(y) + U(x 0, y)f(y) dµ(y) + T b ε (f)(x 0 ) = J + JJ + JJJ + JJJJ, R(x) U(x 0, y)f(y) dµ(y) where R(u) = {y X : diam( k ) < ρ(u, y) Hdiam( k )}. Now let us treat J, JJ and JJJ respectively. For J, we write J (b (x) b (x 0 )) (b m (x) b m (x 0 )) + m j= σ j m = J + J 2. ( b(x) b(x 0 )) σ c K(x, y)f 2 (y)dµ(y) ( b(y) b(x 0 )) σ (K(x, y) K(x 0, y)f 2 (y)dµ(y) For J, we have J m b j (x) b j (x 0 ) j= 2 v+ ε (x 0 )\ 2 v ε (x 0 ) K(x, y) f(y) dµ(y) ( b β µ( k ) mβ µ( 2 v ε(x 0 )) µ( 2v+ε(x 0 )) /p 2 v+ (x ε 0) b β µ( k ) mβ µ( 2 v+ ε(x 0 )) + /p+/p mβ ( µ( 2 v+ ε(x 0 )) mβp 2 v+ ε (x 0 ) b β µ( k ) mβ µ( 2 v+ ε(x 0 )) mβ M mβ,p (f)(z) b β 2 vmβ M mβ,p (f)(z) b β M mβ,p (f)(z) ξλ. 6

13 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL... 7 For J 2, let τ, τ N such that τ + τ = m, and τ 0, we get m ( b(x) b(x 0 )) σ c ( b(y) b(x 0 )) σ (K(x, y) K(x 0, y)f(y)dµ(y) J 2 j= σ j m m j= σ j m m j= σ j m τ+τ =m ( b(x) b(x 0 )) σ c ( b(x) b(x 0 )) σ c 2 v+ ε (x 0 )\ 2 v ε (x 0 ) µ( k ) δ µ( 2 v ε(x 0 )) +δ 2 v+ ε (x 0 ) ( b(y) b(x 0 )) σ f(y)dµ(y) µ( ( k ) δ ( µ( 2 v ε(x 0 )) b(y) b(x +δ 0 )) σ p dµ(y) 2 v+ ε (x 0 ) ( 2 v+ ε (x 0 ) b σ c β µ( k ) τβ+δ µ( 2 v+ ε(x 0 )) δ+τ β+/p +/p mβ b σ β M mβ,p (f)(z) τ+τ =m b β µ( k ) τβ+δ µ( 2 v+ ε(x 0 )) δ τβ M mβ,p (f)(z) b β 2 v(δ+τβ) M mβ,p (f)(z) b β M mβ,p (f)(z) ξλ. Therefore J ξλ. For JJ and JJJ, note that, for y R(x), ρ(x, y) Hdiam( k ), we get, by Lemma 3 and Hölder s inequality, m JJ b j (x 0 ) b j (y) K(x, y) f(y) dµ(y) H k j= ( µ(h m ( k ) b j (x 0 ) b j (y) p dµ(y) H k j= H k ( µ(h k ) +mβ+/p +/p mβ b β µ(h k ) mβp f(y) p dµ(y) b β M mβ,p (f)(z) ξλ. Similar JJJ ξλ. Thus J + JJ + JJJ ξλ. For JJJJ, since x / k k, then T b ε (f)(x 0 ) λ. For x k, sup T b ε (f 2 )(x) ξλ + λ. ε diam( k ) 7

14 8 ZHANG QIAN AND LIU LANZHE ase 2. ε > Hdiam( k ). Let ε k denote the ball with the same center as k and with the diam k ε = ε. Similar to the proof of ase, we get Thus, we have shown that for x k, sup T b ε (f 2 )(x) ξλ + λ. ε>diam( k ) T b (f 2 )(x) ξλ + λ. Now, choose ξ 0 such that ξ 0 <, let η = and combine the estimates on f with f 2, we get ({ }) µ x k : T b (f)(x) > 3λ, b β M mβ,p (f)(x) ξλ µ({x k : T b (f )(x) > 2λ ξλ}) + µ({x X : T b (f 2 )(x) > λ + ξλ}) µ({x k : T b (f )(x) > λ}) ξ r µ( k ). This completes the proof of Theorem (a). (b) follows from (a) and Lemma 5. Proof. Theorem 2(a). When m =, by the Whitney decomposition, {x X : T b (f)(x) > λ} may be written as a union of balls { k } with mutually disjoint interiors and with distance from each to X \ k k comparable to the diameter of k. It suffices to prove the good λ estimate for each k. There exists a constant = (n) such that for each k, the cube k intersects X \ k k, where k denotes the ball with the same center as k and with the diam k = diam k. Then, for each k, there exists a point x 0 = x 0 (k) k such that T b (f)(x 0 ) λ. Now, we fix a ball k. Without loss of generality, we may assume there exists a point z = z(k) with b MO M p (f)(z) ξλ. Set k = k and write f = f +f 2 for f = fχ k and f 2 = fχ X\k. We turn to the estimates on f and f 2. The estimates on f. For x k, /r = /p + /q <, we have ( ) /r T b (f ) L r (b (x) b (y))k(x, y)f (y) r dµ(y) ) /q ( µ( k ) ( b (x) b (y) q dµ(y) k k µ( k ) µ( k ) /q b MO µ( k ( µ( k ) µ( k ) /r b MO M p (f)(z). 8 k

15 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL... 9 Let η > 0, we have µ({x X : T b (f )(x) > ηλ}) (ηλ) r T b (f ) r L r (ηλ) r [ b MO M p (f)(z)] r µ( k ) /r (ηλ) r (ξλ) r µ( k ) (ξ/η) r µ( k ). The estimates on f 2. Let H = H(X) be a large positive integer depending only on X. We consider the following two cases: ase. diam( k ) ε Hdiam( k ). Set V (x, y) = (b (x) b (y))k(x, y)f(y). hoose x 0 k such that x 0 X \ k k. For x k, following [8], we have T b ε (f 2 )(x) + R(x 0 ) [V (x, y) V (x 0, y)]f 2 (y)dµ(y) + V (x 0, y)f(y) dµ(y) + T b ε (f)(x 0 ) = I + II + III + IV, R(x) V (x 0, y)f(y) dµ(y) where R(u) = {y X : diam( k ) < ρ(u, y) Hdiam( k )}. Now let us treat I, II and III respectively. For I, we write V (x, y) V (x 0, y) = (b (x) b (y))k(x, y)f 2 (y) (b (x 0 ) b (y))k(x 0, y)f 2 (y) = b (x)k(x, y)f 2 (y) b (y)k(x, y)f 2 (y) b (x 0 )K(x 0, y)f 2 (y) +b (y)k(x 0, y)f 2 (y) b (x 0 )K(x, y)f 2 (y) + b (x 0 )K(x, y)f 2 (y) (b (x) b (x 0 ))K(x, y)f 2 (y) + (b (x 0 ) b (y))(k(x, y) K(x 0, y))f 2 (y) = I + I 2. For I, by Hölder s inequality and the following inequality, for b MO(X) b(x) b b(x) b dµ(x) b MO, µ() 9

16 20 ZHANG QIAN AND LIU LANZHE we have I dµ(y) 2 v+ ε (x 0 )\ 2 v ε (x 0 ) b (x) b (x 0 ) K(x, y) f(y) dµ(y) ( b MO µ( 2 v ε(x 0 )) µ( 2v+ε(x 0 )) /p 2 v+ (x ε 0) b MO µ( 2 v+ ε(x 0 )) + /p+/p ( µ( 2 v+ ε(x 0 )) b MO M p (f)(z) ξλ. 2 v+ ε (x 0 ) For I 2, similar to I 2, we obtain I 2 dµ(y) ( 2 v+ ε (x 0 )\ 2 v ε (x 0 ) 2 v+ ε (x 0 ) 2 v+ ε (x 0 ) µ( k ) δ µ( 2 v+ ε(x 0 ) +δ b (x 0 ) b (y) (b (x 0 ) b (y))(k(x, y) K(x 0, y))f(y) dµ(y) [ ] δ d(x0, x) d(x, y) µ((x 0, d(x 0, y))) f(y) dµ(y) b (x 0 ) b (y) d(x 0, x) δ f(y) dµ(y) d(x 0, y) +δ ( b (x 0 ) b (y) p dµ(y) 2 v+ ε (x 0 ) 2 v+ ε (x 0 ) µ( k ) δ µ( 2 v+ ε)(x 0 ) δ+/p +/p b MO M p (f)(z) 2 vδ b MO M p (f)(z) b MO M p (f)(z) ξλ. Therefore I ξλ. For II and III, note that, for y R(x), ρ(x, y) Hdiam( k ), 20

17 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL... 2 we get, by Hölder s inequality, II b (x 0 ) b (y) K(x, y) f(y) dµ(y) H k ( ( µ(h k ) b (x 0 ) b (y) p dµ(y) H k H k ( µ(h k ) +/p +/p b MO µ(h k ) f(y) p dµ(y) b MO M p (f)(z) ξλ. Similar III ξλ. Thus I + II + III ξλ. For IV, since x / k k, then T b ε (f)(x 0 ) λ. For x k, sup T b ε diam( ε (f 2 )(x) ξλ + λ. k ) ase 2. ε > Hdiam( k ). Let ε k denote the ball with the same center as k and with the diam k ε = ε. Similar to the proof of ase, we get Thus, we have shown that for x k, sup T b ε>diam( ε (f 2 )(x) ξλ + λ. k ) T b (f 2 )(x) ξλ + λ. Now, choose ξ 0 such that ξ 0 <, let η = and combine the estimates on f with f 2, we get ({ }) µ x k : T b (f)(x) > 3λ, b MO M p (f)(x) ξλ µ({x k : T b (f )(x) > 2λ ξλ}) + µ({x X : T b (f 2 )(x) > λ + ξλ}) µ({x k : T b (f )(x) > λ}) ξ r µ( k ). When m >, similar to the case m =, there exists a point x 0 = x 0 (k) k such that T b (f)(x 0 ) λ. Now, we fix a ball k. Without loss of generality, we may assume there exists a point z = z(k) with b MO M p (f)(z) ξλ. Set k = k and write f = f +f 2 for f = fχ k and f 2 = fχ X\k. We turn to the estimates on f and f 2. 2

18 22 ZHANG QIAN AND LIU LANZHE The estimates on f. For x k, /r = /p + /q j <, j =,, m ( r ) T b m /r (f ) L r (b j (x) b j (y))k(x, y)f (y) dµ(y) j= ( ) /qj m ( µ( k ) b j (x) b j (y) q j dµ(y) k k j= µ( k ) µ( k ) m j= /q j b MO µ( k ( µ( k ) µ( k ) /r b MO M p (f)(z). k Let η > 0, we have µ({x X : T b (f )(x) > ηλ}) (ηλ) r T b (f ) r L r (ηλ) r [ b MO M p (f)(z)] r µ( k ) /r (ηλ) r (ξλ) r µ( k ) (ξ/η) r µ( k ). The estimates on f 2. Let H = H(X) be a large positive integer depending only on X. We consider the following two cases: ase. diam( k ) ε Hdiam( k ). Set U(x, y) = m j= (b j(x) b j (y))k(x, y)f(y). hoose x 0 k such that x 0 X \ k k. For x k, following [8], we have T b ε (f 2 )(x) + R(x 0 ) [U(x, y) U(x 0, y)]f 2 (y)dµ(y) + U(x 0, y)f(y) dµ(y) + T b ε (f)(x 0 ) = J + JJ + JJJ + JJJJ, R(x) U(x 0, y)f(y) dµ(y) where R(u) = {y X : diam( k ) < ρ(u, y) Hdiam( k )}. Now let us treat J, JJ and JJJ respectively. For J, we write J (b (x) b (x 0 )) (b m (x) b m (x 0 )) + m j= σ j m = J + J 2. ( b(x) b(x 0 )) σ c K(x, y)f 2 (y)dµ(y) ( b(y) b(x 0 )) σ (K(x, y) K(x 0, y)f 2 (y)dµ(y) 22

19 A GOOD λ ESTIMATE FOR MULTILINEAR OMMUTATOR OF SINGULAR INTEGRAL For J, we have J m b j (x) b j (x 0 ) j= 2 v+ ε (x 0 )\ 2 v ε (x 0 ) K(x, y) f(y) dµ(y) ( b MO µ( 2 v ε(x 0 )) µ( 2v+ε(x 0 )) /p 2 v+ (x ε 0) b MO µ( 2 v+ ε(x 0 )) + /p+/p ( µ( 2 v+ ε(x 0 )) b MO M p (f)(z) ξλ. 2 v+ ε (x 0 ) For J 2, we get J 2 m j= σ j m m j= σ j m m j= σ j m ( ( b(x) b(x 0 )) σ c ( b(x) b(x 0 )) σ c ( b(x) b(x 0 )) σ c 2 v+ ε (x 0 )\ 2 v ε (x 0 ) µ( k ) δ µ( 2 v ε(x 0 )) +δ ( b(y) b(x 0 )) σ (K(x, y) K(x 0, y)f(y)dµ(y) 2 v+ ε (x 0 ) ( b(y) b(x 0 )) σ f(y)dµ(y) µ( ( k ) δ ( µ( 2 v ε(x 0 )) b(y) b(x +δ 0 )) σ p dµ(y) 2 v+ ε (x 0 ) 2 v+ ε (x 0 ) b σ c MO µ( k ) δ µ( 2 v+ ε(x 0 )) δ+/p +/p b σ MO M γ,p (f)(z) b MO µ( k ) δ µ( 2 v+ ε(x 0 )) δ M p (f)(z) b MO 2 vδ M p (f)(z) b MO M p (f)(z) ξλ. Therefore J ξλ. For JJ and JJJ, note that, for y R(x), ρ(x, y) Hdiam( k ), 23

20 24 ZHANG QIAN AND LIU LANZHE we get, by Hölder s inequality, for /p + /p m + /p = JJ H k m b j (x 0 ) b j (y) K(x, y) f(y) dµ(y) j= ( j µ(h m ( k ) b j (x 0 ) b j (y) p j dµ(y) H k j= H k ( µ(h k ) +/p + /p m+/p b MO µ(h k ) f(y) p dµ(y) b MO M p (f)(z) ξλ. Similar JJJ ξλ. Thus J + JJ + JJJ ξλ. For JJJJ, since x / k k, then T b ε (f)(x 0 ) λ. For x k, sup T b ε (f 2 )(x) ξλ + λ. ε diam( k ) ase 2. ε > Hdiam( k ). Let ε k denote the ball with the same center as k and with the diam k ε = ε. Similar to the proof of ase, we get Thus, we have shown that for x k, sup T b ε (f 2 )(x) ξλ + λ. ε>diam( k ) T b (f 2 )(x) ξλ + λ. Now, choose ξ 0 such that ξ 0 <, let η = and combine the estimates on f with f 2, we get ({ }) µ x k : T b (f)(x) > 3λ, b MO M p (f)(x) ξλ µ({x k : T b (f )(x) > 2λ ξλ}) + µ({x X : T b (f 2 )(x) > λ + ξλ}) µ({x k : T b (f )(x) > λ}) ξ r µ( k ). This completes the proof of Theorem 2(a). (b) follows from (a) and Lemma 5. References [] J. Alvarez, R. J. abgy, D. S. Kurtz and. Pérez, Weighted estimates for commutators of linear operators, Studia Math., 04(993),

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22 26 ZHANG QIAN AND LIU LANZHE [7] M. Paluszynski, haracterization of the esov spaces via the commutator operator of oifman, Rochbeg and Weiss, Indiana Univ. Math. J., 44(995), -7. [8]. Pérez, Endpoint estimate for commutators of singular integral operators, J. Func. Anal., 28(995), [9]. Pérez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators, Michigan Math. J., 49(200), [20]. Pérez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 65(2002), [2] G. Pradolini, O. Salinas, ommutators of singular integrals on space of homogeneous type, preprint, available at [22] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on euclidean and homogeneous spaces, Amer. J. Math., 4(992), [23] E. M. Stein, Harmonic analysis: teal variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton NJ,