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Elementary characterizations of generalized weighted Morrey-Campanato spaces

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Applied Mathematics(A Journal of Chinese Universities) 2010年第2期25卷 162-176页

作者： yang da-chun yang Si-bei School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 China

Let α∈（0,∞ ）, p, q ∈[1, ∞）, s be a nonnegative integer, and ω∈ A1（Rn）（tne class of Muckenhoupt＇s weights）. In this paper, we introduce the generalized weighted Morrey- Campanato space L（α, p, q, s, ω; Rn） and obtain its equivalence on different p∈ [1, β） and 1 integers s ≥[nα] （the integer part of nα）, whereβ = （1/q - α）-1 when α 〈 1/qorβ= ∞ when α ≥1/q We then introduce the generalized weighted Lipschitz space A（α, q, w; Rn） and prove that L（a, p, q, s, w; Rn）С ∧（α, q, w; Rn） when α ∈ （0, ∞）, s ≥[nα], and p∈ [1,β）.

关键词： Morrey-Campanato space Lipschitz space weight.

Endpoint Estimates for Multilinear Fractional Integrals with Non-doubling Measures

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Acta Mathematicae Applicatae Sinica 2014年第3期30卷 755-764页

作者： Hai-bo LIN da-chun yang School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex SystemsMinistry of education College of Science China Agricultural University

Under the assumption that μ is a non-doubling measure on R^d which only satisfies the polynomial growth condition, the authors obtain the boundedness of the multilinear fractional integrals on Morrey spaces, weak-Morrey spaces and Lipschitz spaces associated with it, which, in the case when μ is the d-dimensional Lebesgue measure, also improve the known results.

关键词： multilinear fractional integral Morrey space weak-Morrey space Lipschitz space non-doubling measure

Herz-type Triebel-Lizorkin Spaces, I

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Acta Mathematica Sinica,English Series 2005年第3期21卷 643-654页

作者： Jing Shi XU da chun yang Department of Mathematics Beijing Normal University Beijing 100875 P. R. China Department of Mathematics Hunan Normal University Changsha 410081 P. R. China

Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s（R^n）andKq^α,pFβ^s（R^n）which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given.

关键词： Herz space Triebel-Lizorkin space Maximal function Embedding Multiplier Lifting property

Musielak–Orlicz BMO-type Spaces Associated with Generalized Approximations to the Identity

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Acta Mathematica Sinica,English Series 2014年第11期30卷 1917-1962页

作者： Shao Xiong HOU da chun yang Si Bei yang School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex SystemsMinistry of Education School of Mathematics and Statistics Lanzhou University

Let X be a space of homogenous type and ψ： X × [0,∞） → [0,∞） be a growth function such that ψ（·,t） is a Muckenhoupt weight uniformly in t and ψ（x,·） an Orlicz function of uniformly upper type 1 and lower type p ∈（0,1].In this article,the authors introduce a new Musielak–Orlicz BMO-type space BMOψA（X） associated with the generalized approximation to the identity,give out its basic properties and establish its two equivalent characterizations,respectively,in terms of the spaces BMOψA,max（X） and BMOψA（X）.Moreover,two variants of the John–Nirenberg inequality on BMOψA（X） are *** an application,the authors further prove that the space BMOψΔ1/2（Rn）,associated with the Poisson semigroup of the Laplace operator Δ on Rn,coincides with the space BMOψ（Rn） introduced by Ky.

关键词： Growth function BMO-type space approximation to the identity John–Nirenberg inequality Carleson measure

Atomic Hardy-type Spaces between H1 and L1 on Metric Spaces with Non-doubling Measures

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Acta Mathematica Sinica,English Series 2011年第12期27卷 2445-2468页

作者： Li Guang LIU da chun yang Dong Yong yang School of Mathematical Sciences Beijing Normal UniversityLaboratory. of Mathematics and Complex Systems Ministry of EducationBeijing 100875 P. R. China

Let （y, d, dλ） be （Rn, ｜·｜,μ）, where ｜·｜ is the Euclidean distance, μ is a nonnegative Radon measure on Rn satisfying the polynomial growth condition, or the Gauss measure metric space （Rn, ｜·｜,dγ）, or the space （S, d, p）, where S - Rn×R＋ is the （ax ＋ b）-group, d is the left-invariant Riemannian metric and p is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces {Xs（Y））0〈s≤∞ and the BM0-type spaces {BM0（y, s）}0〈s≤∞. Let Hi（Y） be the known atomic Hardy space and L01（y） the subspace of f ∈ L1（Y） with integral 0. The authors prove that the dual space of Xs（Y） is SM0（Y,s） when s∈ （0, ∞）, Xs（Y） = H1（Y） when s ∈ （0, 1], and X∞（y） = L01（Y）（or L1（Y））. As applications, the authors show that if T is a linear operator bounded from H1 （Y） to L1 （Y） and from L1（y） to L1,∞（Y）, then for all r ∈ （1, ∞） and s ∈ （r, ∞], T is bounded from Xr（y） to the Lorentz space L1,8（y）, which applies to the Calderon-Zygmund operator on （Rn, ｜·｜,μ）, the imaginary powers of the 0rnstein-Uhlenbeck operator on （Rn, ｜·｜,dγ） and the spectral operator associated with the spectral multiplier on （S, d, p）. All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.

关键词： Non-doubling measure polynomial growth Gauss measure (ax + b)-group atomicHardy-type space BMO-type space operator

Pointwise Characterizations of Besov and Triebel–Lizorkin Spaces with Generalized Smoothness and Their Applications

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Acta Mathematica Sinica,English Series 2022年第4期38卷 623-661页

作者： Zi Wei LI da chun yang Wen YUAN Laboratory of Mathematics and Complex Systems(Ministry of Education of China) School of Mathematical SciencesBeijing Normal UniversityBeijing 100875P.R.China

In this article,the authors first establish the point wise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness on R;via the Hajlasz gradient sequences,which serve as a way to extend these spaces to more general metric measure ***,on metric spaces with doubling measures,the authors further prove that the Besov and the Triebel-Lizorkin spaces with generalized smoothness defined via Hajlasz gradient sequences coincide with those defined via hyperbolic *** an application,some trace theorems of these spaces on Ahlfors regular spaces are established.

关键词： Besov space Triebel-Lizorkin space generalized smoothness Hajlasz gradient hyperbolic filling trace

Boundedness of Some Maximal Commutators in Hardy-type Spaces with Non-doubling Measures

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Acta Mathematica Sinica,English Series 2007年第6期23卷 1129-1148页

作者： Guo En HU Yan MENG da chun yang Department of Applied Mathematics University of Information Engineering P. O. Box 1001-747 Zhengzhou 450002 P. R. China School of Information Renmin University of China Beijing 100872 P. R. China School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China

Let μ be a non-negative Radon measure on R^d which satisfies only some growth conditions. Under this assumption, the boundedness in some Hardy-type spaces is established for a class of maximal Calderón-Zygmund operators and maximal commutators which are variants of the usual maximal commutators generated by Calder6ón- Zygmund operators and RBMO（μ） functions, where the Hardytype spaces are some appropriate subspaces, associated with the considered RBMO（μ） functions, of the Hardv soace H^I（μ） of Tolsa.

关键词： maximal commutator Calderon Zygmund operator RBMO(μ) Hardy-type space non-doubling measure

An Off-Diagonal Marcinkiewicz Interpolation Theorem on Lorentz Spaces

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Acta Mathematica Sinica,English Series 2011年第8期27卷 1477-1488页

作者： Yi Yu LIANG Li Guang LIU da chun yang School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Corr~plex Systems Ministry of Education Beijing 100875 P. R. China Department of Mathematics School of Information Renmin University of China Beijing 100872 P. R. China School of Mathematical Sciences Beijing Normal UniversityLaboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 P. R. China

Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡（0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].

关键词： Quasilinear operator Marcinkiewicz interpolation theorem Lorentz space restricted weak type estimate

A Remark on the Boundedness of Calder6n-Zygmund Operators in Non-homogeneous Spaces

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Acta Mathematica Sinica,English Series 2007年第3期23卷 449-456页

作者： Xiao Li FU Guo En HU da chun yang School of Mathematical Sciences Beijing Normal University Beijing 100875 P. R. China Department of Applied Mathematics University of Information Engineering Zhengzhou 450002 P. R. China

Let μ be a Radon measure on R^d which may be non-doubling. The only condition satisfied by μ is that μ（B（x,r）） ≤ Cr^n for all x∈R^d r 〉 0 and some fixed 0 〈 n ≤ d. In this paper, the authors prove that the boundedness from H^1（μ） into L^1,∞ （μ） of a singular integral operator T with Calderón-Zygmund kernel of HSrmander type implies its L^2（μ）-boundedness.

关键词： Caldern-Zygmund operator Hardy space L^1,∞(μ) Non-doubling measure