Báo cáo hóa học: " Research Article Uniform Boundedness for Approximations of the Identity with Nondoubling Measures Dachun Yang and Dongyong Yang" pptx

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 19574, 25 pages doi:10.1155/2007/19574 Research Article Uniform Boundedness for Approximations of the Identity with Nondoubling Measures Dachun Yang and Dongyong Yang Received 15 May 2007; Accepted 19 August 2007 Recommended by Shusen Ding Let μ be a nonnegative Radon measure on R d which satisfies the growth condition that there exist constants C 0 > 0andn ∈ (0,d] such that for all x ∈R d and r>0, μ(B(x,r)) ≤ C 0 r n ,whereB(x,r)istheopenballcenteredatx and having radius r. In this paper, the authors establish the uniform boundedness for approximations of the identity introduced by Tolsa in the Hardy space H 1 (μ) and the BLO-type space RBLO (μ). Moreover, the authors also introduce maximal operators . ᏹ s (homogeneous) and ᏹ s (inhomogeneous) associated with a g iven approximation of the identity S,andprovethat . ᏹ s is bounded from H 1 (μ)toL 1 (μ)andᏹ s is bounded from the local atomic Hardy space h 1,∞ atb (μ)to L 1 (μ). These results are proved to play key roles in establishing relations between H 1 (μ) and h 1,∞ atb (μ), BMO-type spaces RBMO (μ)andrbmo(μ)aswellasRBLO(μ)andrblo (μ), and also in character izing rbmo (μ)andrblo(μ). Copyright © 2007 D. Yang and D. Yang. 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 Recall that a nondoubling measure μ on R d means that μ is a nonnegative Radon measure which only satisfies the following growth condition, namely, there exist constants C 0 > 0 and n ∈ (0,d] such that for all x ∈ R d and r>0, μ  B(x,r)  ≤ C 0 r n , (1.1) where B(x,r)istheopenballcenteredatx and having radius r.Suchameasureμ is not necessary to be doubling, which is a key assumption in the classical theory of harmonic analysis. In recent years, it was shown that many results on the Calder ´ on-Zygmund theory 2 Journal of Inequalities and Applications remain valid for nondoubling measures; see, for example, [1–9]. One of the main moti- vations for extending the classical theory to the nondoubling context was the solution of several questions related to analytic capacity, like Vitushkin’s conjecture or Painlev ´ e’s problem; see [10–12] or survey papers [13–16] for more details. In particular, Tolsa [8] constructed a class of approximations of the identity and used it to develop a Littlewood-Paley theory with nondoubling measures in L p (μ)withp ∈ (1,∞) and establish some T(1) theorems. The main purpose of this paper is to investi- gate behaviors of approximations of the identity and some kind of maximal operators associated with it at the extremal cases, namely, when p = 1orp =∞.Tobeprecise,in this paper, we first establish the uniform boundedness for approximations of the identity in the Hardy space H 1 (μ)ofTolsa[7, 9] and the BLO-type space RBLO(μ)ofJiang[1], respectively. We then introduce the homogeneous maximal operator ˙ ᏹ S and inhomoge- neous maximal operator ᏹ S and prove that ˙ ᏹ S is bounded from H 1 (μ)toL 1 (μ)andᏹ S is bounded from the local atomic Hardy space h 1,∞ atb (μ)toL 1 (μ). These results are proved in [17] to play key roles in establishing relations between H 1 (μ)andh 1,∞ atb (μ), BMO-ty pe spaces RBMO(μ)andrbmo(μ)aswellasBLO-typespacesRBLO(μ)andrblo(μ), and also in characterizing rbmo(μ)andrblo(μ). An interesting open problem is if H 1 (μ)and h 1,∞ atb (μ) can be characterized by ˙ ᏹ S and ᏹ S , respectively. The organization of this paper is as follows. In Section 2, we recall some necessary definitions and notation, including the definitions and characterizations of the spaces H 1 (μ), RBLO(μ), h 1,∞ atb (μ), and approximations of the identity. Section 3 is devoted to prove that approximations of the identity are uniformly bounded on H 1 (μ)andRBLO(μ). In Section 4, we introduce the homogeneous maximal operator ˙ ᏹ S and the inhomoge- neous maximal operator ᏹ S associated with a given approximation of the identity S, and prove that ˙ ᏹ S is bounded from H 1 (μ)toL 1 (μ)andᏹ S is bounded from h 1,∞ atb (μ) to L 1 (μ). Since the approximation of the identity in [8] strongly depends on “dyadic” cubes constructed by Tolsa in [ 8, 9], it is expectable that properties of these “dyadic” cubes will play a key role in the proofs of all these results in this paper. In [17], we introduce a quantity on these “dyadic” cubes, which further clarifies the geometric properties of “dyadic” cubes of Tolsa in [8, 9]; see Lemma 2.18 below. These properties together with some known properties of “dyadic” cubes (see, e.g., [8, Lemmas 3.4 and 4.2]) indeed play key roles in the whole paper. We finally make some convention. Throughout the paper, we always denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. Constant with subscript such as C 1 does not change in di fferent occurrences. The notation Y  Z means that there exists a constant C>0suchthatY ≤CZ, while Y  Z means that there exists a constant C>0suchthatY ≥ CZ. The symbol A ∼ B means that A  B  A.Moreover,foranyD ⊂R d , we denote by χ D the characteristic function of D. We also set N ={ 1,2, }. 2. Preliminaries Throughout this paper, by a cube Q ⊂ R d , we mean a closed cube whose sides are parallel to the axes and centered at some point of supp(μ), and we denote its side length by l(Q) D. Yang and D. Yang 3 and its center by x Q .Ifμ(R d ) < ∞,wealsoregardR d as a cube. Let α, β be two positive constants, α ∈ (1,∞)andβ ∈ (α n ,∞). We say that a cube Q is an (α,β)-doubling cube if it satisfies μ(αQ) ≤ βμ(Q), where and in what follows, given λ>0 and any cube Q, λQ denotes the cube concentric with Q and having side length λl(Q). It was pointed out by Tolsa (see [7, pages 95-96] or [8, Remark 3.1]) that if β>α n ,thenforanyx ∈ supp(μ) and any R>0,thereexistssome(α,β)-doubling cube Q centered at x with l(Q) ≥ R,and that if β>α d ,thenforμ-almost everywhere x ∈ R d , there exists a sequence of (α,β)- doubling cubes {Q k } k∈N centered at x with l(Q k ) → 0ask →∞. Throughout this paper, by a doubling cube Q,wealwaysmeana(2,2 d+1 )-doubling cube. For any cube Q,let  Q be the smallest doubling cube which has the form 2 k Q with k ∈N ∪{0}. Given two cubes Q, R ⊂ R d ,letx Q be the center of Q,andQ R be the smallest cube concentric with Q containing Q and R. The following coefficients were first introduced by Tolsa in [7]; see also [8, 9]. Definit ion 2.1. Given two cubes Q,R ⊂ R d ,wedefine δ(Q,R) = max   Q R \Q 1   x −x Q   n dμ(x),  R Q \R 1   x −x R   n dμ(x)  . (2.1) We may t reat points x ∈ R d as if they were cubes (with side length l(x) = 0). So, for any x, y ∈ R d and cube Q ⊂ R d , the notation δ(x,Q)andδ(x, y)makesense. We now recall the notion of cubes of generations in [8, 9]. Definit ion 2.2. We say that x ∈ R d is a stopping point (or stopping cube) if δ(x,Q) < ∞ for some cube Q  x with 0 <l(Q) < ∞.WesaythatR d is an initial cube if δ(Q,R d ) < ∞ for some cube Q with 0 <l(Q) < ∞.ThecubesQ such that 0 <l(Q) < ∞ are called transit cubes. Remark 2.3. In [8, page 67], it was pointed out that if δ(x,Q) < ∞ for some transit cube Q containing x,thenδ(x,Q  ) < ∞ for any other transit cube Q  containing x.Also,if δ(Q, R d ) < ∞ for some transit cube Q,thenδ(Q  ,R d ) < ∞ for any transit cube Q  . Let A be some big positive constant. In particular, we assume that A is much bigger than the constants  0 ,  1 ,andγ 0 , which appear, respectively, in [8,Lemmas3.1,3.2,and 3.3]. Moreover, the constants A,  0 ,  1 ,andγ 0 depend only on C 0 , n,andd.Inwhat follows, for  > 0anda,b ∈ R, the notation a = b ±  does not mean any precise equality but the estimate |a −b|≤  . Definit ion 2.4. As sume that R d is not an initial cube. We fix some doubling cube R 0 ⊂ R d . This will be our “reference” cube. For each j ∈ N,letR −j be some doubling cube concen- tric with R 0 , containing R 0 , and such that δ(R 0 ,R −j ) = jA±  1 (which exists because of [8, Lemma 3.3]). If Q is a transit cube, we say that Q is a cube of generation k ∈ Z if it is a doubling cube, and for some cube R −j containing Q we have δ(Q,R −j ) = ( j + k)A ±  1 . If Q ≡{x}is a stopping cube, we say that Q is a cube of generation k ∈Z if for some cube R −j containing x we ha v e δ(Q,R −j ) ≤ ( j + k)A +  1 . We remark that the definition of cubes of generations is proved in [8, page 68] to be independent of the chosen reference {R −j } j∈N∪{0} in the sense modulo some small errors. 4 Journal of Inequalities and Applications Definit ion 2.5. Assume that R d is an initial cube. Then we choose R d as our “reference” cube. If Q is a transit cube, we say that Q is a cube of generation k ≥ 1, if Q is doubling and δ(Q, R d ) = kA ±  1 .IfQ ≡{x} is a stopping cube, we say that Q is a cube of gen- eration k ≥ 1ifδ(x,R d ) ≤ kA +  1 .Moreover,forallk ≤ 0, we say that R d is a cube of generation k. In what follows, we also regard that R d is a cube centered at all the points x ∈supp(μ). Using [8, Lemma 3.2], it is easy to verify that for any x ∈ supp(μ)andk ∈ Z, there exists adoublingcubeofgenerationk;see[8, page 68]. Throughout this paper, for any x ∈ supp(μ)andk ∈ Z, we denote by Q x,k afixeddoublingcubecenteredatx of generation k. By [18, Proposition 2.1] and Definition 2.5, it follows that for any x ∈ supp(μ), l(Q x,k ) → ∞ as k →−∞. Remark 2.6. We should point out that when R d is an initial cube, cubes of generations in [8] were not assumed to be doubling. However, by using [8, Lemma 3.2], it is easy to check that doubling cubes of generations exist even in this case. Moreover, it is not so difficult to verify that (2,2 d+1 )-doubling cubes in [8]canbereplacedby(ρ,ρ d+1 )-doubling cubes for any ρ ∈ (1,∞). In [8], Tolsa constructed an approximation of the identity S ≡{S k } ∞ k=−∞ related to doubling cubes {Q x,k } x∈R d ,k∈Z , which consists of integral operators given by kernels {S k (x, y)} k∈Z on R d ×R d satisfying the following properties: (A-1) S k (x, y) = S k (y,x)forallx, y ∈ R d ; (A-2) for any k ∈ Z and any x ∈supp(μ), if Q x,k is a transit cube, then  R d S k (x, y)dμ(y) = 1; (2.2) (A-3) if Q x,k is a transit cube, then supp(S k (x, ·)) ⊂ Q x,k−1 ; (A-4) if Q x,k and Q y,k are transit cubes, then there exists a constant C>0suchthat 0 ≤ S k (x, y) ≤ C  l  Q x,k  + l  Q y,k  + |x − y|  n ; (2.3) (A-5) if Q x,k , Q x  ,k ,andQ y,k are transit cubes, and x,x  ∈ Q x 0 ,k for some x 0 ∈ supp(μ), then there exists a constant C>0suchthat   S k (x, y) −S k (x  , y)   ≤ C |x −x  | l  Q x 0 ,k  1  l  Q x,k  + l  Q y,k  + |x − y|  n . (2.4) Moreover, Tolsa also pointed out that (A-1) through (A-5) also hold if any of Q x,k , Q x  ,k , and Q y,k is a stopping cube, and that (A-1), (A-3) through (A-5) also hold if any of Q x,k , Q x  ,k ,andQ y,k coincides with R d , except that (A-2) is replaced by (A-2’). If Q x,k = R d for some x ∈ supp(μ), then S k = 0. In what follows, without loss of generality, for any x ∈ supp(μ), we always assume that Q x,k is not a stopping cube, since the proofs for stopping cubes are similar. We next recall the notions of the spaces H 1 (μ)andRBMO(μ)in[9] and the space RBLO(μ)in[1]. D. Yang and D. Yang 5 Definit ion 2.7. Given f ∈ L 1 loc (μ), we set ᏹ Φ ( f )(x) = sup ϕ∼x      R d fϕdμ     , (2.5) where t he notation ϕ ∼ x means that ϕ ∈ L 1 (μ) ∩C 1 (R d ) and satisfies (i) ϕ L 1 (μ) ≤ 1; (ii) 0 ≤ ϕ(y) ≤ 1/|y −x| n for all y ∈ R d ; (iii) |∇ϕ(y)|≤1/|y −x| n+1 for all y ∈ R d ,where∇=(∂/∂x 1 , ,∂/∂x d ). Definit ion 2.8. The Hardy space H 1 (μ) is the set of all functions f ∈L 1 (μ) satisfying that  R d fdμ=0andᏹ Φ f ∈ L 1 (μ). Moreover, we define the norm of f ∈ H 1 (μ)by f  H 1 (μ) =f  L 1 (μ) +   ᏹ Φ ( f )   L 1 (μ) . (2.6) On the Hardy space, Tolsa established the following atomic characterization (see [7, 9]). Definit ion 2.9. Let η>1and1<p ≤∞. A function b ∈ L 1 loc (μ)iscalledap-atomic block if (i) there exists some cube R such that supp(b) ⊂ R; (ii)  R d b(x)dμ(x) = 0; (iii) for j = 1, 2, there exist functions a j supported on cubes Q j ⊂ R and numbers λ j ∈ R such that b = λ 1 a 1 + λ 2 a 2 ,and   a j   L p (μ) ≤  μ  ηQ j  1/p−1  1+δ  Q j ,R  −1 . (2.7) We th en le t |b| H 1,p atb (μ) =|λ 1 |+ |λ 2 |. A function f ∈ L 1 (μ)issaidtobelongtothespaceH 1,p atb (μ) if there exist p-atomic blocks {b i } i∈N such that f =  ∞ i=1 b i with  ∞ i=1 |b i | H 1,p atb (μ) < ∞.TheH 1,p atb (μ)normof f is defined by f  H 1,p atb (μ) = inf{  ∞ i=1 |b i | H 1,p atb (μ) }, where the infimum is taken over all the possible decompositions of f in p-atomic blocks as above. Remark 2.10. It was proved in [7 , 9] that the definition of H 1,p atb (μ)in[7] is independent of the chosen constant η>1, and for any 1 <p< ∞, all the atomic Hardy spaces H 1,p atb (μ) coincide with H 1,∞ atb (μ) with equivalent norms. Moreover, Tolsa proved that H 1,∞ atb (μ)co- incides with H 1 (μ) with equivalent norms (see [9, Theorem 1.2]). Thus, in the rest of this paper, we identify the atomic Hardy space H 1,p atb (μ)withH 1 (μ), and when we use the atomic characterization of H 1 (μ), we always assume η = 2andp =∞in Definition 2.9. Definit ion 2.11. Let η ∈ (1,∞). A function f ∈ L 1 loc (μ) is said to be in the space RBMO(μ) if there exists some constant  C ≥ 0 such that for any cube Q centered at some point of supp(μ), 1 μ(ηQ)  Q   f (y) −m  Q ( f )   dμ(y) ≤  C, (2.8) 6 Journal of Inequalities and Applications and for any two doubling cubes Q ⊂ R,   m Q ( f ) −m R ( f )   ≤  C  1+δ(Q,R)  , (2.9) where m Q ( f ) denotes the mean of f over cube Q,namely,m Q ( f )=(1/μ(Q))  Q f (y)dμ(y). Moreover, we define the RBMO(μ)normof f by the minimal constant  C as above and denote it by f  RBMO(μ) . Remark 2.12. It was proved by Tolsa [7] that the definition of RBMO(μ)isindepen- dent of the choices of η. As a result, throughout this paper, we always assume η = 2in Definition 2.11. The following space RBLO(μ) was introduced in [1]. It is obvious that L ∞ (μ) ⊂ RBLO(μ) ⊂ RBMO(μ). Definit ion 2.13. A function f ∈ L 1 loc (μ)issaidtobelongtothespaceRBLO(μ)ifthere exists some constant  C ≥ 0 such that for any doubling cube Q, 1 μ(Q)  Q  f (x) −essinf Q f (y)  dμ(x) ≤  C, (2.10) and for any two doubling cubes Q ⊂ R, m Q ( f ) −m R ( f ) ≤  C  1+δ(Q,R)  . (2.11) The minimal constant  C as above is defined to be the norm of f in the space RBLO(μ) and denote it by f  RBLO(μ) . Remark 2.14. Let η ∈ (1,∞). It was proved in [17] that we obtain an equivalent norm of RBLO(μ)if(2.10)and(2.11)inDefinition 2.13 are, respectively, replaced by that there ex- ists a nonnegative constant  C such that for any cube Q centered at some point of supp(μ), 1 μ(ηQ)  Q  f (x) −essinf  Q f (y)  dμ(x) ≤  C, (2.12) and for any two doubling cubes Q ⊂ R, essinf Q f (y) −essinf R f (y) ≤  C  1+δ(Q,R)  . (2.13) If R d is not an initial cube, letting {R −j } ∞ j=0 be as in Definition 2.4,wethendefine the set Ᏸ ={Q ⊂R d : there exists a cube P ⊂ Q and j ∈ N ∪{0} such that P ⊂ R −j with δ(P,R −j ) ≤ ( j +1)A +  1 }.IfR d is an initial cube, we define the set Ᏸ ={Q ⊂ R d :there exists a cube P ⊂ Q such that δ(P, R d ) ≤ A +  1 }. Remark 2.15. In [17], it was pointed out that if Q ∈ Ᏸ,thenanyR containing Q is also in Ᏸ and the definition of the set Ᏸ is independent of the chosen reference {R −j } j∈N∪{0} in the sense modulo some small error (the error is no more than 2  1 +  0 ); see also [8,page 68]. M oreover , it was also proved in [17]thatifμ is the d-dimensional Lebesgue measure on R d , then for any cube Q ⊂ R d , Q ∈ Ᏸ if and only if l(Q)  1. D. Yang and D. Yang 7 In [17], we used the set Ᏸ to introduce the local Hardy spaces h 1,p atb,η (μ), p ∈(1,∞], in the sense of Goldberg [ 19]. Definit ion 2.16. For a fixed η ∈ (1,∞)andp ∈ (1,∞], a function b ∈ L 1 loc (μ)iscalleda p-atomic block if it satisfies (i), (ii), and (iii) of Definition 2.9. A function b ∈ L 1 loc (μ) is called a p-block if it only satisfies (i) and (iii) of Definition 2.9. In both cases, we let |b| h 1,p atb,η (μ) =  2 j =1 |λ j |. Moreover, a function f ∈ L 1 (μ)issaidtobelongtothespaceh 1,p atb,η (μ) if there exist p-atomic blocks or p-blocks {b i } i such that f =  i b i and  i |b i | h 1,p atb,η (μ) < ∞,whereb i is a p-atomic block if supp(b i ) ⊂ R i with R i /∈Ᏸ, while b i is a p-block if supp(b i ) ⊂ R i and R i ∈ Ᏸ. We define the h 1,p atb,η (μ)normof f by letting f  h 1,p atb,η (μ) = inf{  i |b i | h 1,p atb,η (μ) }, where the infimum is taken over all possible decompositions of f in p-atomic blocks or p-blocks as above. Remark 2.17. It was proved in [17] that the definition of h 1,p atb,η (μ) is independent of the chosen constant η>1, and for any 1 <p< ∞, all the atomic Hardy spaces h 1,p atb,η (μ)co- incide with h 1,∞ atb,η (μ) with equivalent norms. Thus, in the rest of this paper, we always assume η = 2andp =∞in Definition 2.16. In what follows, for any cube R and x ∈ R ∩supp(μ), let H x R be the largest integer k such that R ⊂ Q x,k . The fol lowing properties of H x R play key roles in the proofs of all theorems in this paper, whose proofs can be found in [17]. Lemma 2.18. The following properties hold. (a) For any cube R and x ∈ R ∩supp(μ), Q x,H x R +1 ⊂ 3R and 5R ⊂Q x,H x R −1 . (b) For any cube R, x ∈ R ∩supp(μ) and k ∈ Z with k ≥ H x R +2, Q x,k ⊂ (7/5)R. (c) For any cube R ⊂ R d and x, y ∈ R ∩supp(μ), |H x R −H y R |≤1. (d) If R d is not an initial cube, then for any cube R and x ∈R ∩supp(μ), H x R ≤ 1 when R ∈ Ᏸ and H x R ≥ 0 when R/∈ Ᏸ.IfR d is an initial cube, then 0 ≤ H x R ≤ 1 for any cube R ∈ Ᏸ and x ∈ R ∩supp(μ). (e) For any cube R and x ∈ R ∩ supp(μ), there exists a constant C>0 such that δ(R,Q x,H x R ) ≤ C and δ(Q x,H x R +1 ,R) ≤C. 3. Uniform boundedness in H 1 (μ) and RBLO(μ) This section is devoted to establishing the boundedness for approximations of the identity in the spaces H 1 (μ)andRBLO(μ). Theorem 3.1. For any k ∈ Z,letS k be as in Section 2. Then there exists a constant C>0 independent of k such that for all f ∈ H 1 (μ),   S k ( f )   H 1 (μ) ≤ Cf  H 1 (μ) . (3.1) Proof. We use some ideas from [20]. By the Fatou lemma, to show Theorem 3.1,itsuffices to prove that for any ∞-atomic block b =  2 j =1 λ j a j as in Definition 2.9, ᏹ Φ (S k (b)) ∈ L 1 (μ)andᏹ Φ (S k (b)) L 1 (μ)   2 j =1 |λ j |,whereᏹ Φ is the maximal operator as in 8 Journal of Inequalities and Applications Definition 2.7.Moreover,ifk ≤ 0andR d is an initial cube, then S k = 0, and Theorem 3.1 holds automatically in this case. Therefore, we may assume that R d is not an initial cube when k ≤ 0. Using the notation as in Definition 2.9 and choosing any x 0 ∈ supp(μ) ∩R, we now consider the following two cases: (1) k ≤ H x 0 R ;(2)k ≥ H x 0 R +1. In case (1), write   ᏹ Φ  S k (b)    L 1 (μ) =  8R ᏹ Φ  S k (b)  (x) dμ(x)+  R d \8R ···≡I + II. (3.2) Since ᏹ Φ is sublinear, we have that I ≤ 2  j=1   λ j    8R ᏹ Φ  S k  a j  (x) dμ(x) = 2  j=1   λ j    2Q j ᏹ Φ  S k  a j  (x) dμ(x)+ 2  j=1   λ j    8R\2Q j ···≡I 1 + I 2 . (3.3) By (A-2) and (A-4), we see that for any x ∈ 2Q j , j =1, 2, and ϕ ∼ x,      R d ϕ(y)S k  a j  (y)dμ(y)     ≤  R d ϕ(y)S k (y,z)   a j (z)   dμ(z)dμ(y) ≤   a j   L ∞ (μ) , (3.4) which implies that ᏹ Φ (S k (a j ))(x) ≤a j  L ∞ (μ) . This together with (2.7) further yields I 1 ≤ 2  j=1   λ j     a j   L ∞ (μ) μ  2Q j   2  j=1   λ j   . (3.5) On the other hand, for any x ∈ 8R \2Q j and z ∈ Q j , j =1, 2, |x −z|∼ |x −x j |,where x j denotes the center of Q j . This observation together with the fact that for any x, y, z ∈ R d ,if|y −z| < (1/2)|x −z|,then|x −z| < 2|x − y|. The properties (A-2) and (A-4) imply that for any x ∈ 8R \2Q j , ϕ ∼ x and z ∈ Q j ,  R d ϕ(y)S k (y,z)dμ(y)   |y−z|≥(1/2)|x−z| ϕ(y) |y −z| n dμ(y)+  |y−z|<(1/2)|x−z| S k (y,z) |x − y| n dμ(y)   |y−z|≥(1/2)|x−z| ϕ(y) |x −z| n dμ(y)+  |y−z|<(1/2)|x−z| S k (y,z) |x −z| n dμ(y)  1   x −x j   n . (3.6) D. Yang and D. Yang 9 From this fact and (2.7), it then follows that      R d ϕ(y)S k  a j  (y)dμ(y)     ≤  Q j   a j (z)    R d ϕ(y)S k (y,z)dμ(y)dμ(z)  1   x −x j   n   a j   L ∞ (μ) μ  Q j   1   x −x j   n 1 1+δ  Q j ,R  . (3.7) Thus, for any x ∈ 8R \2Q j , ᏹ Φ  S k  a j  (x)  1   x −x j   n 1 1+δ  Q j ,R  . (3.8) Moreover, by [8, Lemma 3.1 (a) and (d)], we obtain δ  2Q j ,8R  ≤ δ  Q j ,8R   1+δ  Q j ,R  + δ(R,8R)  1+δ  Q j ,R  . (3.9) Therefore, it follows that I 2  2  j=1   λ j   δ  2Q j ,8R  1+δ  Q j ,R   2  j=1   λ j   . (3.10) To es t im a t e II, by the observation that  R d S k (b)(x)dμ(x) = 0, we write II ≤  R d \8R sup ϕ∼x      R d S k (b)(y)  ϕ(y) −ϕ  x 0  dμ(y)     dμ(x) ≤  R d \8R sup ϕ∼x  2R   S k (b)(y)     ϕ(y) −ϕ  x 0    dμ(y)dμ(x) +  R d \8R sup ϕ∼x      R d \2R S k (b)(y)  ϕ(y) −ϕ  x 0  dμ(y)     dμ(x) ≡ II 1 + II 2 . (3.11) Notice that for any y ∈ 2R and x ∈ 2 m+1 R \2 m R with m ≥ 3, |x −x 0 |≥l(2 m−2 R), and |x 0 − y|≤2 √ dl(R), which implies that |y − x 0 |  |x 0 −x|. This fact together with the mean value theorem yields that for any ϕ ∼ x,   ϕ(y) −ϕ  x 0       y −x 0     x 0 −x   n+1 . (3.12) Moreover, let N j be the smallest integer k such that 2R ⊂ 2 k Q j . Because {S k } k are bounded on L 2 (μ) uniformly, (A-4) together with the H ¨ older inequality, [8, Lemma 3.1], (3.12), 10 Journal of Inequalities and Applications and (2.7)leadsto II 1 ≤ 2  j=1   λ j   ∞  m=3  2 m+1 R\2 m R  sup ϕ∼x  2R\2Q j   S k  a j  (y)     ϕ(y) −ϕ  x 0    dμ(y) +sup ϕ∼x  2Q j   S k  a j  (y)     ϕ(y) −ϕ  x 0    dμ(y)  dμ(x)  2  j=1   λ j   ∞  m=3  2 m+1 R\2 m R l(R)  l  2 m R  n+1   2R\2Q j  Q j   a j (z)   |y −z| n dμ(z)dμ(y) +  μ  2Q j  1/2   2Q j   S k  a j  (y)   2 dμ(y)  1/2  dμ(x)  l(R) 2  j=1   λ j   ∞  m=3 μ  2 m+1 R   l  2 m R  n+1  N j −1  i=1  2 i+1 Q j \2 i Q j  Q j   a j   L ∞ (μ) |y −z| n dμ(z)dμ(y) +  μ  2Q j  1/2   Q j   a j (y)   2 dμ(y)  1/2   2  j=1   λ j     a j   L ∞ (μ)  N j −1  i=1 μ  2 i+1 Q j   l  2 i Q j  n μ  Q j  + μ  2Q j    2  j=1   λ j    1+δ  2Q j ,2R  1+δ  Q j ,R  +1   2  j=1   λ j   . (3.13) To es t im a t e II 2 ,wewrite II 2 ≤ ∞  m=3  2 m+1 R\2 m R ᏹ Φ  S k (b)χ 2 m+2 R\2 m−1 R  (x) dμ(x) + ∞  m=3  2 m+1 R\2 m R sup ϕ∼x  2 m+2 R\2 m−1 R   S k (b)(y)   ϕ  x 0  dμ(y)dμ(x) + ∞  m=3  2 m+1 R\2 m R sup ϕ∼x  R d \2 m+2 R   S k (b)(y)     ϕ(y) −ϕ  x 0    dμ(y)dμ(x) + ∞  m=3  2 m+1 R\2 m R sup ϕ∼x  2 m−1 R\2R   S k (b)(y)     ϕ(y) −ϕ  x 0    dμ(y)dμ(x) ≡ E 1 + E 2 + E 3 + E 4 . (3.14) [...]... λj , (3.39) j =1 which completes the proof of Theorem 3.1 For any k ∈ Z, from Theorem 3.1, the linearity of Sk , the fact that (H 1 (μ))∗ = RBMO(μ), and a dual argument, it is easy to deduce the uniform boundedness of Sk in RBMO(μ) We omit the details Corollary 3.2 For any k ∈ Z, let Sk be as in Section 2 Then there exists a constant C > 0 independent of k such that for all f ∈ RBMO(μ), Sk ( f ) RBMO(μ)... Calderon-Zygmund operators for non doubling measures, ” Mathematische Annalen, vol 319, no 1, pp 89–149, 2001 [8] X Tolsa, “Littlewood-Paley theory and the T(1) theorem with non-doubling measures, ” Advances in Mathematics, vol 164, no 1, pp 57–116, 2001 [9] X Tolsa, The space H 1 for nondoubling measures in terms of a grand maximal operator,” Transactions of the American Mathematical Society, vol 355,... completes the proof of Theorem 4.3 Acknowledgments Dachun Yang is supported by National Natural Science Foundation for Distinguished Young Scholars (no 10425106) and NCET (no 04-0142) of Ministry of Education of China References [1] Y Jiang, “Spaces of type BLO for non-doubling measures, ” Proceedings of the American Mathematical Society, vol 133, no 7, pp 2101–2107, 2005 [2] J Mateu, P Mattila, A Nicolau, and. ..D Yang and D Yang 11 Since ᏹΦ is bounded from H 1 (μ) to L1 (μ) (see [9, Lemma 3.1]) and bounded on L∞ (μ), then it is bounded on L p (μ) for any p ∈ (1, ∞) by an argument similar to the proof of [7, Theorem 7.2] The only difference is that in the current case, we do not need to invoke the sharp operator ᏹ in [7, equation (6.4)] On the other hand, by (A-3) and (A-1), we have x supp(Sk... We now consider the uniform boundedness of Sk in RBLO(μ) To this end, we first establish the following lemma, which is a version of [18, Lemma 3.1] for RBLO(μ) Lemma 3.3 There exists a constant C > 0 such that for any two cubes Q ⊂ R and f ∈ RBLO(μ), f (y) − essinf y∈Q f (y) R y − xQ + l(Q) n dμ(y) ≤ C 1 + δ(Q,R) 2 f RBLO(μ) (3.41) Proof The proof of this lemma can be conducted as that of [18, Lemma... Capacities and Operators on Nonhomogeneous Spaces, vol 100 of o CBMS Regional Conference Series in Mathematics, American Mathematical Society Providence, RI, USA, 2003 [17] G Hu, D Yang, and D Yang, “h1 , bmo, blo and Littlewood-Paley g-functions with non-doubling measures, ” submitted [18] D Yang and D Yang, “Endpoint estimates for homogeneous Littlewood-Paley g-functions with non-doubling measures, ”... ··· ≡ J1 + J2 + J3 (4.19) Since the argument of estimates for I1 and I2 in the proof of Theorem 4.2 also works in 2 the current situation, we then have that J1 + J2 j =1 |λ j | To estimate J3 , fix any x0 ∈ R ∩ supp(μ) Notice that for any x ∈ Rd \ 4R and any y ∈ Q j , j = 1,2, |x − y | ∼ |x − x0 | From this fact, Definition 2.16, and (A-4), it follows that for j = 1, 2 and any x ∈ Rd \ 4R, sup Sk a j... 3.4 For any k ∈ Z, let Sk be as in Section 2 Then Sk is uniformly bounded on RBLO(μ), namely, there exists a nonnegative constant C independent of k such that for all f ∈ RBLO(μ), Sk ( f ) RBLO(μ) ≤C f RBLO(μ) (3.44) Proof Without loss of generality, we may assume that f RBLO(μ) = 1 We only need to consider the case that Rd is not an initial cube, since if Rd is an initial cube, then for any k ∈ N, the. .. Orobitg, “BMO for nondoubling measures, ” Duke Mathematical Journal, vol 102, no 3, pp 533–565, 2000 D Yang and D Yang 25 ´ [3] F Nazarov, S Treil, and A Volberg, “Cauchy integral and Calderon-Zygmund operators on nonhomogeneous spaces,” International Mathematics Research Notices, vol 1997, no 15, pp 703– 726, 1997 ´ [4] F Nazarov, S Treil, and A Volberg, “Weak type estimates and Cotlar inequalities for CalderonZygmund... verifies (3.45) Now we estimate (3.46) As in the proof of (3.45), we consider the following three cases: (i) there exists some x0 ∈ Q ∩ supp(μ) such that R ⊂ Qx0 ,k−2 ; (ii) for any x ∈ Q ∩ supp(μ), Q ⊂ Qx,k−2 ; (iii) for any x ∈ Q ∩ supp(μ), R ⊂ Qx,k−2 , and there exists some x0 ∈ Q ∩ supp(μ) such that Q ⊂ Qx0 ,k−2 D Yang and D Yang 19 In case (i), (3.49) together with (3.48) leads to mQ Sk ( f ) − mR Sk . Corporation Journal of Inequalities and Applications Volume 2007, Article ID 19574, 25 pages doi:10.1155/2007/19574 Research Article Uniform Boundedness for Approximations of the Identity with Nondoubling Measures Dachun. establish the uniform boundedness for approximations of the identity in the Hardy space H 1 (μ)ofTolsa[7, 9] and the BLO-type space RBLO(μ)ofJiang[1], respectively. We then introduce the homogeneous. ≤ C and δ(Q x,H x R +1 ,R) ≤C. 3. Uniform boundedness in H 1 (μ) and RBLO(μ) This section is devoted to establishing the boundedness for approximations of the identity in the spaces H 1 (μ)andRBLO(μ). Theorem