度量测度空间上多线性分数次积分算子的加权模不等式

设(X, d,μ)为满足几何双倍条件的度量测度空间.本文建立了(X, d,μ)上多线性分数次积分算子以及多线性分数次极大函数的弱型加权模不等式.作为应用,本文还建立了(X, d,μ)上多线性分数次积分算子在Morrey空间上的弱型加权模不等式.     

度量测度空间上多线性分数次积分算子的有界性

设(X, d,μ)为一个度量测度空间,满足对于任意的x∈X,μ(B(x, r))关于r在(0,∞)上连续,或者设(X, d,μ)是Hyt?nen意义下满足上双倍条件和几何双倍条件的度量测度空间.在此两种背景条件下,本文建立多线性分数次积分算子I_(m,α)在乘积Lebesgue空间上的端点估计、在乘积Morrey空间上的有界性以及弱型端点估计.     

有非负Ricci曲率的度量测度空间上调和函数的边界问题

设(X,d,μ)是满足非负Ricci曲率条件的度量测度空间.本文研究了(开)上半空间X×R+上调和函数的边界问题.我们得到了:若u(x,t)是定义在上半空间X×R+上的调和函数,且满足Carleson测度条件supxB,rB∫rB0fB(xB,rB)|t▽u(x,t)|2dμ(x)dt/t≤C＜∞,其中▽=(▽x,?)...     

Dini型多线性Caldern-Zygmund算子的多线性迭代交换子在非齐性空间上的有界性

设(X,d,μ)为仅具有几何双倍度量d和上双倍测度μ的一类新的非齐性空间,本文考虑Dini型多线性Caldern-Zygmund奇异积分算子与RBMO(μ)中的向量值函数b生成的多线性迭代交换子在这非齐性空间(X,d,μ)上的有界性.     

鞅极大算子的强弱(Φ1,Φ2)-型不等式

研究了鞅Orlicz空间极大算子的双Φ-不等式,得到了相应不等式成立的一些充要条件,给出了Burkholder-Gundy型双Φ-不等式的等价条件,讨论了鞅的Cianchi弱(Φ1,Φ2)-型不等式与Φ-函数的强于关系的联系.     

非齐度量测度空间上的Herz型Hardy空间

设(X, d,μ)是一个同时满足上双倍条件和几何双倍条件的非齐度量测度空间.本文首先引进了非齐度量测度空间上的Herz空间,并利用中心块得到了该空间的分解定理.然后,根据离散系数K_(B,S)~((ρ),p),引入了非齐度量测度空间上的原子Herz型Hardy空间与分子Herz型Hardy空间,并证明了原子Herz型Hardy空间和分子Herz型Hardy空间的等价性.最后作为应用,本文讨论了Calderón-Zygmund算子在这些空间上的有界性.     

非齐度量测度空间上Morrey-Herz空间上的Calderón-Zygmund算子及其交换子（英文）

设(X,d,μ)是一个同时满足上双倍条件和几何双倍条件的非齐度量测度空间,本文中,引进一类非齐度量测度空间上的Morrey-Herz空间,利用非齐度量测度空间的特征,特别是η-弱逆倍条件,证明Calderón-Zygmund算子及其交换子在Morrey-Herz空间上的有界性.     

非倍测度空间上的Calderón-Zygmund算子（英文）

综述回顾了带有非倍测度的欧氏空间R~d上的Calderon-Zygmund理论中的基本结果.在该背景下欧氏空间上所赋予的测度μ不需要满足通常的双倍条件,只需满足如下增长性条件,即存在正常数n∈(0,d]以及C使得对任意的x∈R~d和r∈(0,∞),μ(B(x,r))≤Cr~n.回顾的主要结果包括:Hardy空间H~1(μ)与正则BMO空间RBMO(μ);与H~1(μ)以及RBMO(μ)相关的插值定理;Calderon-Zygmund分解;T(1)定理与Calderon-Zygmund算子在Lebesgue空间和Hardy空间上的有界性;Cotlar不等式与极大Calderon-Zygmund算子的有界性;多线性Calderon-Zygmund算子在乘积Lebesgue空间上的性质;Calderon-Zygmund算子的加权模不等式;由Calderon-Zygmund算子与RBMO(μ)函数所生成的交换子的有界性.此外,作者还介绍了该研究方面的一些最新进展与成果.     

空间L_p(μ,X)(O

李冲  王祖樾 《数学年刊A辑(中文版)》1991,(2)

度量测度空间上双线性θ-型Marcinkiewicz积分交换子

设(X, d,μ)是Hyt?nen意义下满足几何双倍和上双倍条件的非齐型度量测度空间.在假设控制函数λ满足一定的-弱的逆双倍条件下,该文证明了由双线性θ-型Marcinkiewicz积分Mθ与具离散系数的正则有界平均振荡空间■生成的交换子Mθ,b1,b2从Lp1(μ)×Lp2(μ)到Lp(μ)是有界的,其中1 p1, p2∞且满足■.进一步,还得到了交换子Mθ,b1,b2在Morrey空间上的有界性.     

Properties of the normed cone of semi-Lipschitz functions

Summary We obtain several properties of the normed cone of semi-Lipschitz functions defined on a quasi-metric space (X,d) that vanish at a fixed point x₀∈X. For instance, we prove that it is both bicomplete and right K-sequentially complete, and the unit ball is compact with respect to the topology of quasi-uniform convergence. Furthermore, it has a structure of a Banach space if and only if (X,d) is a metric space.     

非齐型空间上分数型Marcinkiewicz积分算子的加权估计

令（X，d，μ）为满足所谓上倍双倍条件和几何双倍条件的度量测度空间.设M_β，ρ，q为（X，d，μ）上的分数型Marcinkiewicz积分算子.在本文中，作者证明了若β ∈[0，∞），ρ ∈（0，∞），q ∈（1，∞）且M_β，ρ，q在L²（μ）上有界，则M_β，ρ，q是从加权Lebesgue空间L^p（w）到加权弱Lebesgue空间L^p，∞（w）上有界和从加权Morrey空间L^p，κ，η（ω）到加权弱Morrey空间WL^p，κ，η（ω）上有界.     

The quasi-metric of complexity convergence

For any weightable quasi-metric space (X, d) having a maximum with respect to the associated order ≤ _d , the notion of the quasi-metric of complexity convergence on the the function space (equivalently, the space of sequences) X^ω , is introduced and studied. We observe that its induced quasi-uniformity is finer than the quasi-uniformity of pointwise convergence and weaker than the quasi-uniformity of uniform convergence. We show that it coincides with the quasi-uniformity of pointwise convergence if and only if the quasi-metric space (X, d) is bounded and it coincides with the quasi-uniformity of uniform convergence if and only if X is a singleton. We also investigate completeness of the quasi-metric of complexity convergence. Finally, we obtain versions of the celebrated Grothendieck theorem in this context.     

Pointwise behaviour of <Emphasis Type="Italic">M</Emphasis><Superscript>1,1</Superscript> Sobolev functions

Our main objective is to study Haj?asz type Sobolev functions with the exponent one on metric measure spaces equipped with a doubling measure. We show that a discrete maximal function is bounded in the Haj?asz space with the exponent one. This implies that every such function has Lebesgue points outside a set of capacity zero. We also show that every Haj?asz function coincides with a Hölder continuous Haj?asz function outside a set of small Hausdorff content. Our proofs are based on Sobolev space estimates for maximal functions.     

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Assume that(X,d,μ) is a space of homogeneous type in the sense of Coifman and Weiss(1971,1977). In this article, motivated by the breakthrough work of Auscher and Hyt(o|¨)nen(2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay(for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al.(2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderón reproducing formulae on(X, d,μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure,not necessary to satisfy the reverse doubling condition. It is well known that Calderón reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.     

非线性Φ 伪压缩映象不动点的具误差的迭代过程

该文的目的是要引入比重要的 强伪压缩映象更一般的Φ 伪压缩映象，并且在更一般的假设下，用具误差的 Ishikawa和 Mann迭代过程去研究这类映象不动点的迭代逼近问题。研究结果表明：Φ 伪压缩映象T的一致连续性保证了在任意实Banach空间E中，Ishikawa迭代序列强收敛于T的唯一不动点；进一步，如果E是一致光滑的则T的连续性是不必要的     

Cheeger-harmonic functions in metric measure spaces revisited

Let (X,d,μ)$(X,d,\mu )$ be a complete metric measure space, with μ   a locally doubling measure, that supports a local weak L²$L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X,d,μ)$(X,d,\mu )$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.     

Pointwise behaviour of Orlicz–Sobolev functions

We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space $L^{\Psi}(X)We study the regularity of Orlicz–Sobolev functions on metric measure spaces equipped with a doubling measure. We show that each Orlicz–Sobolev function is quasicontinuous and has Lebesgue points outside a set of capacity zero and that the discrete maximal operator is bounded in the Orlicz–Sobolev space. We also show that if the Hardy–Littlewood maximal operator is bounded in the Orlicz space , then each Orlicz–Sobolev function can be approximated by a H?lder continuous function both in the Lusin sense and in norm. The research is supported by the Centre of Excellence Geometric Analysis and Mathematical Physics of the Academy of Finland.     

向量值空间中的几何酉元

研究向量值空间中的几何酉元.通过数值指标理论刻画向量值空间C(Ω,X),L_∞(μ,X)和L(l_1(Γ),X)中几何酉元的特征,其中X是Banach空间,Ω是紧Hausdorff空间,μ是σ有限测度以及Γ是非空指标集.同时,描述了Banach空间的内射张量积和投射张量积中几何酉元的特征.     

Endpoint estimates of generalized homogeneous Littlewood–Paley <Emphasis Type="Italic">g</Emphasis>-functions over non-homogeneous metric measure spaces

Let (χ, d, μ) be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Under the weak reverse doubling condition, the authors prove that the generalized homogeneous Littlewood-Paley g-function_r (r∈[2,∞)) is bounded from Hardy space H¹(μ) into L¹(μ). Moreover, the authors show that, if f ∈ RBMO(μ), then[_r(f)]^r is either infinite everywhere or finite almost everywhere, and in the latter case,[_r(f)]^r belongs to RBLO(μ) with the norm no more than ‖f‖_RBMO(μ) multiplied by a positive constant which is independent of f. As a corollary, the authors obtain the boundedness of_r from RBMO(μ) into RBLO(μ). The vector valued Calderón-Zygmund theory over (χ, d, μ) is also established with details in this paper.