WEIGHTED ESTIMATES FOR COMMUTATORS ON NONHOMOGENEOUS SPACES WENGU CHEN AND BING ZHAO Received 14 November 2005; Revised 8 March 2006; Accepted 11 March 2006 Let μ be a Borel measure on R d which may be nondoubling. The only condition that μ must satisfy is μ(Q) ≤ c 0 l(Q) n for any cube Q ⊂ R d with sides parallel to the coordinate axes and for some fixed n with 0 <n ≤ d. This paper is to establish the weighted norm inequality for commutators of Calder ´ on-Zygmund operators with RBMO( μ) functions by an estimate for a variant of the sharp maximal function in the context of the nonho- mogeneous spaces. Copyright © 2006 W. Chen and B. Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let μ be some nonnegative Borel measure on R d satisfying μ(Q) ≤ c 0 l(Q) n (1.1) for any cube Q ⊂ R d with sides parallel to the coordinate axes, where l(Q) stands for the side length of Q and n is a fixed real number such that 0 <n ≤ d. Throughout this paper, all cubes we will consider will be those with sides parallel to the coordinate axes. For r>0, rQ will denote the cube with the same center as Q and with l(rQ) = rl(Q). Moreover , Q(x, r) will be the cube centered at x with side length r. The classical theory of harmonic analysis for maximal functions and singular inte- grals on ( R d ,μ) has been developed under the assumption that the underlying mea- sure μ satisfies the doubling property, that is, there exists a constant c>0suchthat μ(B(x,2r)) ≤ cμ(B(x,r)) for every x ∈ R d and r>0. But recently, many classical results have been proved still valid without the doubling condition; see [1–18] and their refer- ences. Orobitg and P ´ erez [11] have studied an analogue of the classical theory of A p (μ) weights in R d without assuming that the underlying measure μ is doubling. Then, they Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 89396, Pages 1–14 DOI 10.1155/JIA/2006/89396 2 Commutator on nonhomogeneous space obtained weighted norm inequalities for the (centered) Hardy-Littlewood maximal func- tion and corresponding weighted estimates for nonclassical Calder ´ on-Zygmund oper- ators. They also considered commutators of those Calder ´ on-Zygmund operators with BMO(μ) functions. The purpose of this paper is to establish weighted estimates for com- mutators of those nonclassical Calder ´ on-Zygmund operators with RBMO(μ) in this new setting. Let us introduce some notations and definitions. Given two cubes Q ⊂ R in R d ,weset K Q,R = 1+ N Q,R k=1 μ 2 k Q l 2 k Q n , (1.2) where N Q,R is the first integer k such that l(2 k Q) ≥ l(R). K Q,R was introduced by Tolsa in [15]. Given β d (depending on d) big enough (e.g., β d > 2 n ), we say that some cube Q ⊂ R d is doubling if μ(2Q) ≤ β d μ(Q). Given a cube Q ⊂ R d ,letN be the smallest integer ≥ 0suchthat2 N Q is doubling. We denote this cube by Q. Let η>1 b e some fixed constant. We say that a function b(x)isinRBMO(μ)ifthere exists some constant c 1 such that for any cube Q, 1 μ(ηQ) Q b − m Q b dμ ≤ c 1 , m Q b − m R b ≤ c 1 K Q,R for any two doubling cubes Q ⊂ R, (1.3) where m Q b = 1/μ(Q) Q bdμ. The minimal constant c 1 is the RBMO(μ)normofb,andit will be denoted by b ∗ . The RBMO(μ) function space was introduced by Tolsa in [15] and shares more properties with the classical BMO function space than BMO(μ)space. We say a kernel k(x, y): R d × R d \{(x, y):x = y}→C is an n-dimensional Calder ´ on- Zygmund kernel in the new setting if (1) |k(x, y)|≤A/|x − y| n if x = y, (2) there exists 0 <γ ≤ 1suchthat k(x, y) − k(x , y) + k(y,x) − k(y,x ) ≤ A|x − x | γ |x − y| n+γ (1.4) if |x − y| > 2|x − x |. A bounded linear operator T from L 2 (μ)toL 2 (μ)issaidtobeaCalder ´ on-Zygmund operator with n-dimensional kernel k if for every compacted supported function f ∈ L 2 (μ), Tf(x) = R d k(x, y) f (y)dμ(y)forx ∈ supp f. (1.5) For r>0, we define the truncated operators by T r f (x) = R d \B(x,r) k(x, y) f (y)dμ(y) (1.6) W. Chen and B. Zhao 3 and define the maximal operator associated with T as follows: T ∗ f (x) = sup r>0 T r f (x) . (1.7) 2. Sharp maximal function estimates for commutators In [15], Tolsa defined a sharp maximal operator M # f (x)suchthat f ∈ RBMO(μ) ⇐⇒ M # f ∈ L ∞ (μ), (2.1) where M # f (x) = sup x∈Q 1 μ (3/2)Q Q f − m Q f dμ+sup x∈Q⊂R Q,R doubling m Q f − m R f K Q,R . (2.2) We also consider the noncentered doubling maximal operator N: Nf(x) = sup x∈Q Q doubling 1 μ(Q) Q | f |dμ. (2.3) By [15,Remark2.3],forμ-almost all x ∈ R d one can find a sequence of doubling cubes {Q k } k centered at x with l(Q k ) → 0ask →∞such that lim k→∞ 1 μ Q k Q k b(y)dμ(y) = b(x). (2.4) So, | f (x)|≤Nf(x)forμ-a.e. x ∈ R d . Moreover, it is easy to show that N is of weak type (1,1) and bounded on L p (μ), p ∈ (1,∞]. In order to obtain the estimate for a variant of the sharp maximal function for the commutators of Calder ´ on-Zygmund operators defined as above with RBMO(μ) func- tions, we need the following definition. A function B :[0, ∞) → [0,∞) is called a Young function if it is continuous, convex, increasing, and satisfying B(0) = 0andB(t) →∞as t →∞. We define the B-average of a function f over a cube Q by means of the following Luxemburg norm: f B,Q,(ρ) = inf λ>0: 1 μ(ρQ) Q B f (y) λ dμ ≤ 1 . (2.5) The generalized H ¨ older’s inequality 1 μ(ρQ) Q f (y)g(y) dμ(y) ≤f B,Q,(ρ) g B,Q,(ρ) (2.6) holds, where B is the complementary Young function associated to B.Foreverylocally integrable function f , define its maximal operator M B,(ρ) by M B,(ρ) f (x) = sup x∈Q f B,Q,(ρ) . (2.7) 4 Commutator on nonhomogeneous space Theorem 2.1. Let b ∈ RBMO(μ),let0 <δ< < 1,thereexistsC = C δ, such that M # δ [b,T] f (x) ≤ Cb ∗ M ,(3/2) (Tf)(x)+M 2 (9/8) f (x)+T ∗ f (x) , (2.8) where M # δ f (x) = M # (| f | δ ) 1/δ , M p,(ρ) f (x) = sup x∈Q ((1/μ(ρQ)) Q | f | p dμ) 1/p , 0 <p<∞. Set M (ρ) f (x) = M 1,(ρ) f (x). Before proving the theorem, another equivalent norm for RBMO(μ) is needed. Sup- pose that for a given function b ∈ L 1 loc (μ) there exist some c 2 and a collection of numbers {b Q } Q (i.e., for each cube Q, there exists b Q ∈ R)suchthat sup Q 1 μ(ηQ) Q b − b Q dμ ≤ c 2 , b Q − b R ≤ c 2 K Q,R for any two cubes Q ⊂ R. (2.9) Then, set b ∗∗ = inf c 2 , where the infimum is taken over all the constants c 2 and all the numbers {b Q } satisfying (2.9). By [15, Lemma 2.8, page 99], for a fixed η>1, the norms · ∗ and · ∗∗ are equivalent. Proof of Theorem 2.1. We follow the argument from [15, proof of Theorem 9.1]. Let Q = Q(x,r) be a cube with center x and side length r.For0<δ<1andα,β ∈ R,wehave ||α| δ −|β| δ |≤|α − β| δ .Let{b Q } Q be a sequence of numbers satisfying Q b − b Q dμ ≤ 2μ(2Q)b ∗∗ , (2.10) for all cubes Q and b Q − b R ≤ 2K Q,R b ∗∗ (2.11) for all cubes Q, R with Q ⊂ R. For any cube Q, we denote h Q :=−m Q (T((b − b Q ) fχ R d \ (4/3)Q)). We will show that for all x, Q with x ∈ Q, 1 μ (3/2)Q Q [b,T] f − h Q δ dμ 1/δ ≤ Cb ∗∗ M ,(3/2) (Tf)(x)+M 2 (9/8) f (x) ; (2.12) and for all cubes Q, R with Q ⊂ R, x ∈ Q, h Q − h R ≤ Cb ∗∗ M 2 (9/8) f (x)+T ∗ f (x) K 2 Q,R . (2.13) To obtain (2.12)forsomefixedcubeQ and x with x ∈ Q,werewrite[b,T] f : [b,T] f = b − b Q Tf− T b − b Q f 1 − T b − b Q f 2 , (2.14) W. Chen and B. Zhao 5 where f 1 = fχ (4/3)Q , f 2 = f − f 1 . Let us estimate the term (b − b Q )Tf first. Take 1 <r< ε/δ.ByH ¨ older’s inequality, we have 1 μ (3/2)Q Q b(y) − b Q Tf(y) δ dμ(y) 1/δ ≤ 1 μ (3/2)Q Q b(y) − b Q δr dμ(y) 1/δr 1 μ (3/2)Q Q Tf(y) δr dμ(y) 1/δr ≤ Cb ∗∗ M δr,(3/2) (Tf)(x) ≤ Cb ∗∗ M ε,(3/2) (Tf)(x). (2.15) Since T : L 1 (μ) → L 1,∞ (μ) (see [9]) and 0 <δ<1, Kolmogorov’s inequality and gener- alized H ¨ older’s inequality yield 1 μ (3/2)Q Q T b − b Q f 1 (y) δ dμ(y) 1/δ ≤ 1 μ (3/2)Q (4/3)Q b(y) − b Q f (y) dμ(y) ≤ C b − b Q expL,(4/3)Q,(9/8) f LLog L,(4/3)Q,(9/8) , (2.16) while John-Nirenberg inequality implies that 1 μ (3/2)Q (4/3)Q exp b(y) − b Q Cb ∗ dμ(y) ≤ C 0 . (2.17) So there exists a positive constant C such that for all cubes Q, b − b Q expL,(4/3)Q,(ρ) ≤ Cb ∗ . (2.18) Therefore 1 μ((3/2)Q) Q T b − b Q f 1 (y) δ dμ(y) 1/δ ≤ Cb ∗ M LLog L,(9/8) f (x). (2.19) In order to p rove (2.12), we only need to estimate |T((b − b Q ) f 2 ) − h Q | δ .Notethat K Q,2 k (4/3)Q = 1+ k+1 j=1 μ 2 j Q l 2 j Q n ≤ 1+(k +1)C 0 ≤ Ck. (2.20) 6 Commutator on nonhomogeneous space For x, y ∈ Q,wehave T b − b Q f 2 (x) − T b − b Q f 2 (y) ≤ C R d \(4/3)Q |y − x| γ |z − x| n+γ b(z) − b Q f (z) dμ(z) ≤ C ∞ k=1 2 k (4/3)Q\2 k−1 (4/3)Q l(Q) γ |z−x| n+γ b(z)−b 2 k (4/3)Q + b Q − b 2 k (4/3)Q f (z) dμ(z) ≤ C ∞ k=1 2 −kγ 1 l 2 k Q n 2 k (4/3)Q b(z) − b 2 k (4/3)Q f (z) dμ(z) + C ∞ k=1 k2 −kγ b ∗ 1 l 2 k Q n 2 k (4/3)Q f (z) dμ(z) ≤ C ∞ k=1 2 −kγ 1 μ (9/8)2 k (4/3)Q 2 k (4/3)Q b(z) − b 2 k (4/3)Q f (z) dμ(z) + C ∞ k=1 k2 −kγ b ∗ M (9/8) f (x) ≤ C ∞ k=1 2 −kγ b − b 2 k (4/3)Q expL,2 k (4/3)Q,(9/8) f LLog L,2 k (4/3)Q,(9/8) + Cb ∗ M (9/8) f (x) ≤ Cb ∗ M LLog L,(9/8) f (x)+Cb ∗ M (9/8) f (x). (2.21) For ρ>1, it is easy to see M (ρ) f (x) ≤ M LLog L,(ρ) f (x). Thus T b − b Q f 2 (x) − T b − b Q f 2 (y) ≤ Cb ∗ M LLog L,(9/8) f (x). (2.22) According to Jensen’s inequality, we obtain 1 μ (3/2)Q Q T b − b Q f 2 (y) − m Q T b − b Q f 2 δ dμ(y) 1/δ ≤ 1 μ (3/2)Q Q T b − b Q f 2 (y) − m Q T b − b Q f 2 dμ(y) ≤ Cb ∗ M LLog L,(9/8) f (x). (2.23) Note that for ρ>1, M 2 (ρ) f (x) ≈ M LLog L,(ρ) f (x). By (2.15), (2.16), and (2.23)weobtain (2.12). W. Chen and B. Zhao 7 For {h Q } Q ,wewanttoprove(2.13). Consider two cubes Q ⊂ R and x ∈ Q. We denote N = N Q,R +1.Wewriteh Q − h R in the following way: m Q T b − b Q fχ R d \(4/3)Q − m R T b − b R fχ R d \(4/3)R ≤ m Q T b − b Q fχ 2Q\(4/3)Q + m Q T b Q − b R fχ R d \2Q + m Q T b − b R fχ 2 N Q\2Q + m Q T b − b R fχ R d \2 N Q − m R T b − b R fχ R d \2 N Q + m R T b − b R fχ 2 N Q\(4/3)R = M 1 + M 2 + M 3 + M 4 + M 5 . (2.24) Let us estimate M 1 .Fory ∈ Q we have T b − b Q fχ 2Q\(4/3)Q (y) ≤ C l(2Q) n 2Q b − b Q | f |dμ ≤ C b − b Q expL,2Q,(9/8) f LLog L,2Q,(9/8) ≤ Cb ∗ M LLog L,(9/8) f (x) ≤ Cb ∗ M 2 (9/8) f (x). (2.25) So we derive M 1 ≤ Cb ∗ M 2 9/8) f (x). Let us consider M 2 .Forx, y ∈ Q, Tf χ R d \2Q (y) = R d \2Q f (z)k(y, z)dμ(z) ≤ R d \2Q f (z) k(y,z) − k(x,z) dμ(z) + R d \2Q k(x,z) f ( z)dμ(z) ≤ R d \2Q |y − z| γ |y − z| n+γ f (z) dμ(z) + T ∗ f (x) ≤ C sup Q 0 x 1 l Q 0 n Q 0 | f |dμ + T ∗ f (x) ≤ CM (9/8) f (x)+T ∗ f (x). (2.26) Thus M 2 = b R − b Q Tf χ R d \2Q ≤ CK Q,R b ∗ T ∗ f (x)+CM 2 (9/8) f (x) . (2.27) For the t erm M 4 , we execute the process as in (2.21). For any y,z ∈ R d ,weget T b − b R fχ R d \2Q (y) − T b − b R fχ R d \2Q (z) ≤ Cb ∗ M LLog L,(9/8) f (x) ≤ Cb ∗ M 2 (9/8) f (x). (2.28) 8 Commutator on nonhomogeneous space The term M 5 can be estimated as M 1 .Wecanobtain M 5 ≤ Cb ∗ M 2 (9/8) f (x). (2.29) FinallywehavetodealwithM 3 .Fory ∈ Q,wehave b 2 k+1 Q − b R ≤ CK 2 k+1 Q,R b ∗ ≤ CK Q,R b ∗ . (2.30) Then, T b − b R fχ 2 N \2Q (y) ≤ C N−1 k=1 1 l 2 k Q n 2 k+1 Q\2 k Q b − b R | f |dμ ≤ C N−1 k=1 1 l 2 k Q n 2 k+1 Q b − b 2 k+1 Q | f |dμ + C N−1 k=1 × 1 l 2 k Q n 2 k+1 Q b 2 k+1 Q − b R | f |dμ ≤ C N−1 k=1 b − b 2 k+1 Q expL,2 k+1 Q,(9/8) f LLog L,2 k+1 Q,(9/8) + C N−1 k=1 K Q,R b ∗ μ 2 k+1 Q l 2 k Q n 1 μ 2 k+1 Q 2 k+1 Q | f |dμ ≤ Cb ∗ M LLog L,(9/8) f (x)+CK Q,R b ∗ N−1 k=1 μ 2 k+1 Q l 2 k Q n M (9/8) f (x) ≤ Cb ∗ M LLog L,(9/8) f (x)+CK 2 Q,R b ∗ M (9/8) f (x) ≤ Cb ∗ M 2 (9/8) f (x)K 2 Q,R . (2.31) Taking the mean over Q,weget M 3 ≤ Cb ∗ M 2 (9/8) f (x)K 2 Q,R . (2.32) By the estimates on M 1 , M 2 , M 3 , M 4 , M 5 ,wecanget(2.13). Let us see how from (2.12)and(2.13) one obtains (2.8). If Q is a doubling cube and x ∈ Q,thenwehaveby(2.12) m Q [b,T] f δ − h δ Q 1/δ ≤ 1 μ(Q) Q [b,T] f δ − h δ Q dμ 1/δ ≤ Cb ∗ M ,(3/2) (Tf)(x)+M 2 (9/8) f (x)+T ∗ f (x) . (2.33) W. Chen and B. Zhao 9 Also, for any cube Q x, K Q, Q ≤ C, and then by (2.12)and(2.13)weget 1 μ((3/2)Q) Q [b,T] f δ − m Q [b,T] f δ dμ 1/δ ≤ 1 μ (3/2)Q Q [b,T] f δ − h Q δ dμ 1/δ + h Q δ − h Q δ 1/δ + h Q δ − m Q [b,T] f δ 1/δ ≤ 1 μ (3/2)Q Q [b,T] f − h Q δ dμ 1/δ + h Q − h Q + h δ Q − m Q [b,T] f δ 1/δ ≤ Cb ∗ M ,(3/2) (Tf)(x)+M 2 (9/8) f (x)+T ∗ f (x) . (2.34) On the other hand, for all doubling cubes Q ⊂ R with x ∈ Q such that K Q,R ≤ P 0 ,where P 0 is the constant in [15, Lemma 9.3, page 143]. By (2.13)wehave h Q − h R ≤ Cb ∗ M 2 (9/8) f (x)+T ∗ f (x) K Q,R P 0 . (2.35) So by [15, Lemma 9.3, page 143], we get h Q − h R ≤ Cb ∗ M 2 (9/8) f (x)+T ∗ f (x) K Q,R (2.36) for all doubling cubes Q ⊂ R with x ∈ Q, using (2.13)again,weget m Q [b,T] f δ − m R [b,T] f δ ≤ m Q [b,T] f δ − h δ Q + h δ Q − h δ R + h δ R − m R [b,T] f δ ≤ C b ∗ M ,(3/2) (Tf)(x)+M 2 (9/8) f (x)+T ∗ f (x) K Q,R δ . (2.37) From the above estimates, we can obtain M # δ [b,T] f (x) ≤ Cb ∗ M ,(3/2) (Tf)(x)+M 2 (9/8) f (x)+T ∗ f (x) . (2.38) Now we are in the position to give the definition of weights we will consider. Here we will consider the A p (μ) weights introduced by Orobitg and P ´ erez in [11]. So we need the assumption that μ(∂Q) = 0 for any cube Q with sides parallel to the coordinates axes. Let 1 <p< ∞ and let p = p/(p − 1). We say that a weight w satisfies the A p (μ)condi- tion if there exists a constant K such that for all cubes Q 1 μ(Q) Q wdμ 1 μ(Q) Q w 1−p dμ p−1 ≤ K. (2.39) And we define the A ∞ (μ)classasA ∞ (μ) = p>1 A p (μ). 10 Commutator on nonhomogeneous space Theorem 2.2. Let 0 <p< ∞,letρ>1, w(x) ∈ A ∞ (μ) defined above, then R d Tf(x) p w(x)dμ(x) ≤ C R d M (ρ) f (x) p w(x)dμ(x) (2.40) holds for every function f for which the left-hand side is finite. Proof. For each > 0 we define the maximal operator T ∗ f (x) = sup δ> T δ f (x) . (2.41) We only need to prove that for w ∈ A ∞ (μ), there exist suitable constants α, β, ε such that w x : T ∗ f (x) > (1 + β)t, M (ρ) f (x) ≤ εt ≤ αw x : T ∗ f (x) >t , t>0, (2.42) for all α p < (1 + β) −1 .Wemayassume f is nonnegative and locally integrable. Follow the idea of [11], we first consider the special case when w = 1, then (2.42)turnsto μ x : T ∗ f (x) > (1 + β)t, M (ρ) f (x) ≤ εt ≤ αμ x : T ∗ f (x) >t . (2.43) Since Ω ={x ∈ R d : T ∗ f (x) >t} is open, we decompose it into disjoint Whitney cubes Ω = j Q j ,whereQ j are disjoint and 2ρdiam(Q j ) ≤ dist(Q j ,Ω c ) ≤ 8ρdiam(Q j ), and ev- ery point of R d at most lies in 4ρQ j cubes. Obviously 4ρQ j ⊂ Ω. We will show that for given β>0, 0 <α<1, there exists c = c(β,α,n) such that for all j, μ x ∈ Q j : T ∗ f (x) > (1 + β)t, M (ρ) f (x) ≤ εt ≤ αμ 4Q j . (2.44) Summing over all j,wehave μ x ∈ R d : T ∗ f (x) > (1 + β)t, M (ρ) f (x) ≤ εt ≤ α4 n μ(Ω). (2.45) Choose α such that α4 n < 1, then we can obtain (2.42) in the special case. For the general case w,recallthatifw ∈ A ∞ (μ), then by [11, Lemma 2.3, page 2017], there exist positive constants c, δ such that for all cubes Q and all E ⊂ Q, w(E) w(Q) ≤ c μ(E) μ(Q) δ . (2.46) Looking back at (2.44), we get w x ∈ Q j : T ∗ f (x) > (1 + β)t, M (ρ) f (x) ≤ εt ≤ cα δ w 4Q j . (2.47) Summing again over j,weobtain w x : T ∗ f (x) > (1 + β)t, M (ρ) f (x) ≤ εt ≤ cα δ 4 n w(Ω). (2.48) Choosing α such that cα δ 4 n < (1 + β) −1 ,wecanget(2.42). [...]... Lebesgue spaces [3] for non-doubling measures, The Journal of Fourier Analysis and Applications 7 (2001), no 5, 469–487 , Two-weight norm inequalities for maximal operators and fractional integrals on non[4] homogeneous spaces, Indiana University Mathematics Journal 50 (2001), no 3, 1241–1280 [5] Y C Han, Weighted estimates for higher-order commutators of singular integral operators on nonhomogeneous spaces,... 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Marcinkiewicz interpolation theorem, we can get the desired result Theorem 2.4 Let 0 < p < ∞, let ρ > 1, w ∈ A∞ (μ), b ∈ RBMO(μ) Then there exists constant C such that Rd p [b,T] f w(x)dμ(x) ≤ C p M(ρ) f (x) w(x)dμ(x) Rd (2.57) holds for every function f for which the left-hand side is finite Proof For w ∈ A∞ (μ) and b ∈ RBMO(μ), by the estimate for the variant of the sharp maximal function, we get Rd [b,T]... result with Corollary 2.5 for higher-order commutators But Theorems 2.1, 2.2, and 2.4 in our paper are new and are of independent interest in themselves Acknowledgments The authors would like to express their deep thanks to the referee for his several valuable remarks and suggestions This work was partly supported by NNSF of China (no 10371080) and Scientific Research Foundation for Returned Overseas Chinese... nondoubling measures in terms of a grand maximal operator, Transac[17] tions of the American Mathematical Society 355 (2003), no 1, 315–348 [18] J Verdera, The fall of the doubling condition in Calder´n-Zygmund theory, Publicacions o Matem` tiques 2002 (2002), Vol Extra, 275–292 a [16] Wengu Chen: Institute of Applied Physics and Computational Mathematics, P.O Box 8009, Beijing 100088, China E-mail address:... (otherwise the left-hand set of (2.44) would be empty) Set z ∈ Ωc , that is, T ∗ f (z) ≤ t such that dist(z,Q) = dist(Q,Ωc ) By a simple computation, we get 5 Q ⊂ P ≡ Q b, r ⊂ 4Q ⊂ B ≡ Q(z,18r) 2 (2.49) Set f1 = f χB , f2 = f − f1 Then for x ∈ Q, γ > , by the growth condition (1.1), Tγ f1 (x) ≤ Tγ f χP (x) + f χB\P Rd |x − y |n dμ(y) ≤ T ∗ f χP (x) + c rn B f (y)dμ(y) ≤ T ∗ f χP (x) + cM(ρ) f (x)(b) ≤ T ∗ f . func- tion and corresponding weighted estimates for nonclassical Calder ´ on- Zygmund oper- ators. They also considered commutators of those Calder ´ on- Zygmund operators with BMO(μ) functions. The. weighted norm inequality for commutators of Calder ´ on- Zygmund operators with RBMO( μ) functions by an estimate for a variant of the sharp maximal function in the context of the nonho- mogeneous spaces. Copyright. Calder ´ on- Zygmund operators on non-homogeneous spaces, Publicacions Matem ` atiques 44 (2000), no. 2, 613–640. [3] , OntheexistenceofprincipalvaluesfortheCauchyintegralonweightedLebesguespaces for