15 December 2005, Volume 21 Issue 6

    

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  An Endpoint Estimate for the Commutator of Singular Integrals

  Yong Zhong SUN, Wei Yi SU

  Acta Mathematica Sinica. 2005, 21(6): 1249-1258. https://doi.org/10.1007/s10114-004-0468-2

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  It is well known that the commutator T_b of the singular integral operator T with a BMO function b is bounded on L^p(Rⁿ),1<p<∞.In this paper,we consider the endpoint estimates for a kind of commutator of singular integrals.A BMO-type estimate for T_b is obtained under the assumption b∈LMO.
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  Unitary Equivalence, Similarity and Calculation of K₀-Group

  Hua HE, Chun Lan JIANG

  Acta Mathematica Sinica. 2005, 21(6): 1259-1268. https://doi.org/10.1007/s10114-004-0461-9

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  In this paper,we are concerned with the classi.cation of operators on complex separable Hilbert spaces,in the unitary equivalence sense and the similarity sense,respectively.We show that two strongly irreducible operators A and B are unitary equivalent if and only if W^*(A⊕B)≈M₂(C),and two operators A and B in B₁(Ω) are similar if and only if Moreover,we obtain V (H^∞(Ω,μ)) ≈N and K₀(H^∞(Ω,μ))≈Z by the technique of complex geometry,where Ω is a bounded connected open set in C,and μ is a completely non-reducing measure on Ω.
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  A Strong Approximation Theorem for Quasi-associated Sequences

  Wen Sheng WANG

  Acta Mathematica Sinica. 2005, 21(6): 1269-1276. https://doi.org/10.1007/s10114-004-0471-7

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  By combining the Csörgő-Révész quantile transform methods and the Skorohod-Strassen martingale embedding theorem,we prove a strong approximation theorem for quasi-associated random variables with mean zero and finite (2+δ)th moment under polynomial decay rate.As a consequence,the decay rate for a strong approximation theorem of associated sequences of Yu (1996) is weakened.
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  Computing Degree of Maps Between Manifolds

  Yan Hong DING, Jian Zhong PAN

  Acta Mathematica Sinica. 2005, 21(6): 1277-1284. https://doi.org/10.1007/s10114-005-0639-9

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  We determine the degrees of maps between two given closed (n-1)-connected 2n-manifolds when n≡1(mod 8).
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  Nonlinear Biharmonic Equations with Critical Potential

  Hui XIONG, Yao Tian SHEN

  Acta Mathematica Sinica. 2005, 21(6): 1285-1294. https://doi.org/10.1007/s10114-004-0502-4

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  In this paper,we study two semilinear singular biharmonic equations:one with sub-critical exponent and critical potential,another with sub-critical potential and critical exponent.By Pohozaev identity for singular solution,we prove there is no nontrivial solution for equations with critical exponent and critical potential.And by using the concentrate compactness principle and Mountain Pass theorem,respectively,we get two existence results for the two problems.Meanwhile,we have compared the changes of the critical dimensions in singular and non-singular cases,and we get an interesting result.
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  Asymptotics for Certain Harmonic Functions and the Martin Compactificationon on the Quaternionic Heisenberg Group

  Jing Wen LUAN, Fu Liu ZHU

  Acta Mathematica Sinica. 2005, 21(6): 1295-1308. https://doi.org/10.1007/s10114-005-0554-0

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  In this paper,we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases.We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group.Moreover a Martin compactification of the quaternionic Heisenberg group is constructed,and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.
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  A Priori Bounds for Periodic Solutions of a Kind of Second Order Neutral Functional Differential Equation with Multiple Deviating Arguments

  Shi Ping LU, Wei Gao GE

  Acta Mathematica Sinica. 2005, 21(6): 1309-1314. https://doi.org/10.1007/s10114-005-0567-8

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  The main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows:
  
  In order to do so,we analyze the structure of the linear difference operator A:C_2π→C_2π, to determine some fundamental properties first,which we are going to use throughout this paper.Meanwhile,we also prove some new inequalities which are useful for estimating a priori bounds of periodic solutions.
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  3-Manifold Containing Separating Incompressible Surfaces of All Positive Genera

  Ji Ming MA, Rui Feng QIU

  Acta Mathematica Sinica. 2005, 21(6): 1315-1318. https://doi.org/10.1007/s10114-005-0586-5

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  In this paper,we shall prove that for any orientable 3-manifold M,there is a link L=K∪K₁∪K₂∪K₃ with four components in M,such that the complement of L,say M_L,contains separating essential closed surfaces of all positive genera.
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  Positive Periodic Solutions of a Class of Non-autonomous Single Species Population Model with Delays and Feedback Control

  Feng De CHEN, Xiao Xin CHEN, Jin De CAO, An Ping CHEN

  Acta Mathematica Sinica. 2005, 21(6): 1319-1336. https://doi.org/10.1007/s10114-005-0585-6

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  With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree,several verifiable criteria are established for the global existence of positive periodic solutions of a class of non-autonomous single species population model with delays (both state-dependent delays and continuous delays) and feedback control.After that,by constructing a suitable Lyapunov functional,sufficient conditions which guarantee the existence of a unique globally asymptotic stable positive periodic solution of a kind of nonlinear feedback control ecosystem are obtained.Our results extend and improve the existing results,and have further applications in population dynamics.
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  The Self-similar Solution to Some Nonlinear Integro-differential Equations Corresponding to Fractional Order Time Derivative

  Chang Xing MIAO, Han YANG

  Acta Mathematica Sinica. 2005, 21(6): 1337-1350. https://doi.org/10.1007/s10114-005-0546-0

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  In this paper we study the self-similar solution to a class of nonlinear integro-differential equations which correspond to fractional order time derivative and interpolate nonlinear heat and wave equation.Using the space-time estimates which were established by Hirata and Miao in [11] we prove the global existence of self-similar solution of Cauchy problem for the nonlinear integro-differential equation in .
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  The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

  Bin XU, Rong YUAN

  Acta Mathematica Sinica. 2005, 21(6): 1351-1360. https://doi.org/10.1007/s10114-005-0590-9

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  This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations,by using a fixed point theorem in the mixed monotone operators (Ma,Y.:On a class of mixed monotone operators and a kind of two-point bounded value problem.Indian J.Math.,41(2),211-220 (1999)).Some known results are extended.
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  The Iyengar Type Inequalities with Exact Estimations and the Chebyshev Central Algorithms of Integrals

  Xing Hua WANG, Shi Jun YANG

  Acta Mathematica Sinica. 2005, 21(6): 1361-1376. https://doi.org/10.1007/s10114-005-0578-5

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  In this paper,both low order and high order extensions of the Iyengar type inequality are obtained.Such extensions are the best possible in the same sense as that of the Iyengar inequality.Furthermore,the Chebyshev central algorithms of integrals for some function classes and some related problems are also considered and investigated.
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  Large Deviations for Empirical Measures of Not Necessarily Irreducible Countable Markov Chains with Arbitrary Initial Measures

  Yi Wen JIANG, Li Ming WU

  Acta Mathematica Sinica. 2005, 21(6): 1377-1390. https://doi.org/10.1007/s10114-005-0596-3

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  All known results on large deviations of occupation measures of Markov processes are based on the assumption of (essential) irreducibility.In this paper we establish the weak* large deviation principle of occupation measures for any countable Markov chain with arbitrary initial measures.The new rate function that we obtain is not convex and depends on the initial measure,contrary to the (essentially) irreducible case.
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  Homomorphisms between JC*-algebras and Lie C*-algebras

  Chun Gil PARK, Jin Chuan HOU, Sei Qwon OH

  Acta Mathematica Sinica. 2005, 21(6): 1391-1398. https://doi.org/10.1007/s10114-005-0629-y

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  It is shown that every almost *-homomorphism h:A→B of a unital JC^*-algebra A to a unital JC^*-algebra B is a *-homomorphism when h(rx)=rh(x)(r>1) for all x∈A,and that every almost linear mapping h:A→B is a *-homomorphism when h(2ⁿu oy)=h(2ⁿu)oh(y),h(3ⁿuoy)=h(3ⁿu)oh(y) or h(qⁿuoy)=h(qⁿu)oh(y) for all unitaries u∈A,all y∈A,and n=0,1,....Here the numbers 2,3,q depend on the functional equations given in the almost linear mappings.class="a-plus-plus">We prove that every almost *-homomorphism h:A→B of a unital Lie C^*-algebra A to a unital Lie C^*-algebra B is a *-homomorphism when h(rx)=rh(x)(r>1) for all x∈A.
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  Additive Mappings Preserving Rank-one Idempotents

  B. KUZMA

  Acta Mathematica Sinica. 2005, 21(6): 1399-1406. https://doi.org/10.1007/s10114-005-0598-1

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  A complete characterization of additive mappings Φ:B(X)→B(Y),which map rank-one idempotents to themselves or to zero,is given.
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  Chaos for Subshifts of Finite Type

  Huo Yun WANG, Jin Cheng XIONG

  Acta Mathematica Sinica. 2005, 21(6): 1407-1414. https://doi.org/10.1007/s10114-005-0540-6

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  This paper deals with chaos for subshifts of finite type.We show that for any subshift of finite type determined by an irreducible and aperiodic matrix,there is a finitely chaotic set with full Hausdorff dimension.Moreover,for any subshift of finite type determined by a matrix,we point out that the cases including positive topological entropy,distributional chaos,chaos and Devaney chaos are mutually equivalent.
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  Compactness of Composition Operators from the Bloch Space B to Q_K Spaces

  Hasi WULAN

  Acta Mathematica Sinica. 2005, 21(6): 1415-1424. https://doi.org/10.1007/s10114-005-0584-7

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  Suppose that φ is an analytic self-map of the unit disk Δ.We consider compactness of the composition operator C_φ from the Bloch space B into the spaces Q_K defined by a nonnegative,nondecreasing function K(r) for 0≤r<∞.Our compactness condition depends only on Φ which can be considered as a slight improvement of the known results.The compactness of C_ψ from the Dirichlet space D into the spaces Q_K is also investigated.
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  Gauss Sum of Index 4:(1) Cyclic Case

  Ke Qin FENG, Jing YANG, Shi Xin LUO

  Acta Mathematica Sinica. 2005, 21(6): 1425-1434. https://doi.org/10.1007/s10114-005-0610-9

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  Let p be a prime,m ≥ 2,and (m,p(p-1))=1.In this paper,we will calculate explicitly the Gauss sum in the case of [(Z/mZ)^*:<p>]=4,and-1∈<p>,where q=p^f,f=φ(m)/4,χ is a multiplicative character of F_q with order m,and T is the trace map for F_q/Fp.Under the assumptions [(Z/mZ)^*:〈p〉]=4 and-1∈<p>,the decomposition field of p in the cyclotomic field Q(ζ_m) is an imaginary quartic (abelian) field.And G_(χ) is an integer in K.We deal with the case where K is cyclic in this paper and leave the non-cyclic case to the next paper.
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  Product of Uniform Distribution and Stirling Numbers of the First Kind

  Ping SUN

  Acta Mathematica Sinica. 2005, 21(6): 1435-1442. https://doi.org/10.1007/s10114-005-0631-4

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  Let V_k=u₁u₂...u_k,u_i's be i.i.d～U(0,1),the p.d.f of 1-V_k+1 be the GF of the unsigned Stirling numbers of the first kind s(n,k).This paper discusses the applications of uniform distribution to combinatorial analysis and Riemann zeta function;several identities of Stirling series are established,and the Euler's result for ∑H_n/n^k-1,k≥3 is given a new probabilistic proof.
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  Solving the Multi-discrete Logarithm Problems over a Group of Elliptic Curves with Prime Order

  Jun Quan LI, Mu Lan LIU, Liang Liang XIAO

  Acta Mathematica Sinica. 2005, 21(6): 1443-1450. https://doi.org/10.1007/s10114-005-0632-3

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  In this paper,we discuss the expected number of steps in solving multi-discrete logarithm problems over a group of elliptic curves with prime order by using Pollard's rho method and parallel collision search algorithm.We prove that when using these algorithms to compute discrete logarithms,the knowledge gained through computing many logarithms does not make it easier for finding other logarithms.Hence in an elliptic cryptosystem,it is safe for many users to share the same curve,with different private keys.
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  On Maximal Injectivity

  Ming Yi WANG, Guo ZHAO

  Acta Mathematica Sinica. 2005, 21(6): 1451-1458. https://doi.org/10.1007/s10114-005-0599-0

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  A right R-module E over a ring R is said to be maximally injective in case for any maximal right ideal m of R,every R-homomorphism ƒ:m→E can be extended to an R-homomorphism ƒ':R→E.In this paper,we first construct an example to show that maximal injectivity is a proper generalization of injectivity.Then we prove that any right R-module over a left perfect ring R is maximally injective if and only if it is injective.We also give a partial affirmative answer to Faith's conjecture by further investigating the property of maximally injective rings.Finally,we get an approximation to Faith's conjecture,which asserts that every injective right R-module over any left perfect right self-injective ring R is the injective hull of a projective submodule.
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  Integrability Condition on an Almost Complex Structure and Its Application

  Chia Kuei PENG, Zi Zhou TANG(Zizhou TANG)

  Acta Mathematica Sinica. 2005, 21(6): 1459-1464. https://doi.org/10.1007/s10114-005-0605-6

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  We obtain and apply a simple form of the integrability condition of an almost complex structure to the Hopf manifolds
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  The Martingale Hardy Type Inequalities for Dyadic Derivative and Integral

  Jian Ying NIE, Xing Guo LI, Guo Wei LOU

  Acta Mathematica Sinica. 2005, 21(6): 1465-1474. https://doi.org/10.1007/s10114-005-0627-0

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  Since the Leibniz-Newton formula for derivatives cannot be used in local fields,it is important to investigate the new concept of derivatives in Walsh-analysis,or harmonic analysis on local fields.On the basis of idea of derivatives introduced by Butzer,Schipp and Wade,Weisz has proved that the maximal operators of the one-dimensional dyadic derivative and integral are bounded from the dyadic Hardy space H_p,q to L_p,q of weak type (L₁,L₁),and the corresponding maximal operators of the two-dimensional case are of weak type (H₁^#,L₁).In this paper,we show that these maximal operators are bounded both on the dyadic Hardy spaces H_p and the hybrid Hardy spaces H₁^# 0
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  A Generalization of the Mean Size Formula of Wavelet Packets in L_p Song LI Guo Mao WANG Zhi Song LIU

  Song LI, Guo Mao WANG, Zhi Song LIU

  Acta Mathematica Sinica. 2005, 21(6): 1475-1486. https://doi.org/10.1007/s10114-005-0641-2

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  The purpose of this paper is to investigate the mean size formula of wavelet packets in L_p for 0<p≤∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the L_p-norm or quasi-norm on the subdivision trees.All results will be given in the general setting.
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  Dimension of Maximal Attractors for the m-dimensional Cahn-Hilliard System

  Wei Nian ZHANG(Weinian Zhang)

  Acta Mathematica Sinica. 2005, 21(6): 1487-1494. https://doi.org/10.1007/s10114-005-0633-2

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  On the basis of the existence of the maximal attractor of the m-dimensional Cahn-Hilliard system in the product spaces (L²(Ω))^mand (H²(Ω))^m,in this paper,its Hausdorff dimension is estimated by calculating the orthogonal projection of the linear variational operator of the system.
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  The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

  Jun Feng LI

  Acta Mathematica Sinica. 2005, 21(6): 1495-1508. https://doi.org/10.1007/s10114-005-0560-2

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  In this paper,the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some Lp(Rⁿ) to Lq(Rⁿ) when the m-th order derivatives of A belong to Lr(R_n) for r large enough.By this result,the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions.It follows that the multilinear operators are (Lp,Lp)-type operators for 1
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  The Nonexistence of Expansive Z^d Actions on Graphs

  En Hui SHI, Li Zhen ZHOU

  Acta Mathematica Sinica. 2005, 21(6): 1509-1514. https://doi.org/10.1007/s10114-005-0640-3

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  It is well known that if X is an arc or a circle,then there is no expansive homeomorphism on X.In this paper we prove that there is no expansive Z^d action onX,which answers the two questions raised by us before.In 1979,Manéproved that there is no expansive homeomorphism on infinite dimensional spaces.Contrary to this result,we construct an expansive Z² action on an infinite dimensional space.We also construct an expansive Z² action on a zero dimensional space but no element in Z² is expansive.
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  Some New Perturbation Bounds for Subunitary Polar Factors

  Wen LI

  Acta Mathematica Sinica. 2005, 21(6): 1515-1520. https://doi.org/10.1007/s10114-004-0478-0

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  In this paper,we present some new perturbation bounds for subunitary polar factors in a special unitarily invariant norm called a Q-norm.Some recent results in the Frobenius norm and the spectral norm are extended to the Q-norm on one hand.On the other hand we also present some relative perturbation bounds for subunitary polar factors.
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  Representations of a Noncommutative Associative Algebra Related to Quantum Torus of Rank Three

  Shang Yuan LIN, Bin XIN

  Acta Mathematica Sinica. 2005, 21(6): 1521-1524. https://doi.org/10.1007/s10114-005-0630-5

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  In this paper,we present some modules over the rank-three quantized Weyl algebra,which are closely related to modules over some vertex algebras.The isomorphism classes among these modules are also determined.
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  Laguerre Geometry of Surfaces in R³

  Tong Zhu LI

  Acta Mathematica Sinica. 2005, 21(6): 1525-1534. https://doi.org/10.1007/s10114-005-0642-1

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  Let ƒ:M→R³ be an oriented surface with non-degenerate second fundamental form.We denote by H and K its mean curvature and Gauss curvature.Then the Laguerre volume of ƒ,defined by L(ƒ)=∫(H²-K )/KdM,,is an invariant under the Laguerre transformations.The critical surfaces of the functional L are called Laguerre minimal surfaces.In this paper we study the Laguerre minimal surfaces in R³ by using the Laguerre Gauss map.It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map.In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map.We show also that round spheres are the only compact Laguerre minimal surfaces in R³.And we give a classification theorem of surfaces in R³ with vanishing Laguerre form.
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  Morrey Spaces for Non-doubling Measures

  Yoshihiro SAWANO, Hitoshi TANAKA

  Acta Mathematica Sinica. 2005, 21(6): 1535-1544. https://doi.org/10.1007/s10114-005-0660-z

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  The authors give a natural definition of Morrey spaces for Radon measures which may be non-doubling but satisfy certain growth condition,and investigate the boundedness in these spaces of some classical operators in harmonic analysis and their vector-valued extension.