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Acta Mathematica Scientia 2024年第2期44卷 484-514页

作者：林孝盛杨大春杨四辈袁文 Laboratory of Mathematics and Complex Systems(Ministry of Education of China) School of Mathematical SciencesBeijing Normal UniversityBeijing100875China School of Mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhou730000China

Assume that L is a non-negative self-adjoint operator on L^(2)(ℝ^(n))with its heat kernels satisfying the so-called Gaussian upper bound estimate and that X is a ball quasi-Banach function space onℝ^(n) satisfying some mild *** HX,L(ℝ^(n))be the Hardy space associated with both X and L,which is defined by the Lusin area function related to the semigroup generated by *** this article,the authors establish various maximal function characterizations of the Hardy space HX,L(ℝ^(n))and then apply these characterizations to obtain the solvability of the related Cauchy *** results have a wide range of generality and,in particular,the specific spaces X to which these results can be applied include the weighted space,the variable space,the mixed-norm space,the Orlicz space,the Orlicz-slice space,and the Morrey ***,the obtained maximal function characterizations of the mixed-norm Hardy space,the Orlicz-slice Hardy space,and the Morrey-Hardy space associated with L are completely new.

关键词： Hardy space ball quasi-Banach function space Gaussian upper bound estimate non-negative self-adjoint operator maximal function

maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schriidinger operators

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Frontiers of Mathematics in China 2015年第5期10卷 1203-1232页

作者： Dachun YANG Dongyong YANG School of Mathematical Sciences Beijing Normal University Laboratory of Mathematics and Complex Systems Ministry of Education Beijing 100875 China School of Mathematical Sciences Xiamen University Xiamen 361005 China

Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 1oc（Rn, Rn） and 0 ≤ V ∈Lloc（Rn） tions of the Musielak-Orlicz-Hardy space HA,^（IRn）, defined by the Lusin area function associated with {e-t2A}t〉0, in terms of the Lusin area function associated with {e-t√A}t〉0, the radial maximal functions and the non- tangential maximal functions associated with {e-t2A}t〉o and {e-t√A}t〉0, respectively. The boundedness of the Riesz transforms LkA-U1/2, k ∈ {1, 2... n}, from HA,φ（Rn） to Lφ（Rn） is also presented, where Lk is the closure of δ/δxk iak in L2（Rn）. These results are new even when φ（x,t）：= w（x）tp for all x ∈Rn and t∈ （0, ＋∞） with p ∈ （0, 1] and ω∈ A∞（Rn）（the class of Muckenhoupt weights on Rn）.

关键词： Magnetic SchrSdinger operator Musielak-Orlicz-Hardy space Lusinarea function growth function maximal function Riesz transform

maximal function characterizations of Hardy spaces on RD-spaces and their applications

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Science China Mathematics 2008年第12期51卷 2253-2284页

作者： Loukas GRAFAKOS Department of Mathematics University of Missouri

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a “dimension” n. For α ∈ (0, ∞) denote by H α p (X), H d p (X), and H *,p (X) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calderón reproducing formula, it is shown that all these Hardy spaces coincide with L p (X) when p ∈ (1,∞] and with each other when p ∈ (n/(n + 1), 1]. An atomic characterization for H ?,p (X) with p ∈ (n/(n + 1), 1] is also established; moreover, in the range p ∈ (n/(n + 1),1], it is proved that the space H *,p (X), the Hardy space H p (X) defined via the Littlewood-Paley function, and the atomic Hardy space of Coifman andWeiss coincide. Furthermore, it is proved that a sublinear operator T uniquely extends to a bounded sublinear operator from H p (X) to some quasi-Banach space B if and only if T maps all (p, q)-atoms when q ∈ (p, ∞)∩[1, ∞) or continuous (p, ∞)-atoms into uniformly bounded elements of B.

关键词： space of homogeneous type Calderón reproducing formula space of test function maximal function Hardy space atom Littlewood-Paley function sublinear operator quasi-Banach space 42B25 42B30 47B38 47A30

Note on Hardy-Littlewood maximal function

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Journal of Mathematical Research with Applications 1995年第1期 40-40页

作者： Chen Liyuan(Dept.of Economic. Hangzou University, Hangzou 310028)

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关键词： maximal function boundedness

Commutators of maximal functions on spaces of homogeneous type and their weighted,local versions

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Frontiers of Mathematics in China 2022年第4期17卷 625-652页

作者： Zunwei FU Elodie POZZI Qingyan WU School of Mathematics and Statistics Linyi UniversityLinyi276005China School of Mathematical Sciences Qufu Normal UniversityQufu273100China Department of Mathematics and Statistics Saint Louis UniversitySt.LouisMO63103USA

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous *** addition,with the aid of interpolation theory,we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO ***,a concrete example shows that the local version of commutators also has an independent interest.

关键词： maximal function space of homogeneous type BMO space weight commutator

WEIGHTED INEQUALITIES FOR CERTAIN maximal functionS IN ORLICZ SPACES

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Analysis in Theory and Applications 2001年第4期17卷 65-76页

作者： Zhu Xuexian (Peking University, China) 1. Peking University China

Let Mg be the maximal operator defined by $$M_g f\left( x \right) = \sup \frac{{\int_a^b {f\left( y \right)g\left( y \right){\text{d}}y} }}{{\int_a^b {g\left( y \right){\text{d}}y} }}$$ , where g is a positive locally integrable function on R and the supremum is taken over all intervals [a,b] such that 0≤a≤x≤b/η(b?a), here η is a non-increasing function such that η (0) = 1 and $\mathop {{\text{lim}}}\limits_{t \to {\text{ + }}\infty } \eta \left( t \right) = 0$ η (t) = 0. This maximal function was introduced by H. Aimar and L. L. Forzani [AF]. Let Φ be an N - function such that Φ and its complementary N - function satisfy Δ2. It gives an A′Φ(g) type characterization for the pairs of weights (u,v) such that the weak type inequality $$u\left( {\left\{ {x \in {\text{R}}\left| {M_g f\left( x \right) >\lambda } \right.} \right\}} \right) \leqslant \frac{C}{{\Phi \left( \lambda \right)}}\int_{\text{R}} {\Phi \left( {\left| f \right|v} \right)} $$ holds for every f in the Orlicz space LΦ(v). And, there are no (nontrivial) weights w for which (w,w) satisfies the condition A′Φ(g).

关键词： ORLICZ SPACES maximal operator maximal function Orlicz space

ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

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Acta Mathematica Scientia 2011年第5期31卷 2081-2098页

作者：邓浏睿李中凯马柏林伍火熊 College of Mathematics and Econometrics Hunan University School of Mathematical Sciences Capital Normal University College of Science and Information Engineering Jiaxing University School of Mathematical Sciences Xiamen University

In this paper, the authors study the mapping properties of singular integrals on product domains with kernels in L（log＋L）ε（Sm-1 × Sn-1）（ε = 1 or 2） supported by hyper-surfaces. The Lp bounds for such singular integral operators as well as the related Marcinkiewicz integral operators are established, provided that the lower dimensional maximal function is bounded on Lq（R3） for all q 1. The condition on the integral kernels is known to be optimal.

关键词： multiple singular integral Marcinkiewicz integral maximal function hypersurface rough kernelε

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

Variable integral and smooth exponent Besov spaces associated to non-negative self-adjoint operators

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Frontiers of Mathematics in China 2020年第6期15卷 1245-1263页

作者： Jingshi XU School of Mathematics and Computing Science Guilin University of Electronic TechnologyGuilin 541004China

We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint *** we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these ... 详细信息

关键词： Besov space variable exponent maximal function non negative self-adjoint operators atomic decomposition

Characterizations of product Hardy space associated to Schrodinger operators

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Applied Mathematics(A Journal of Chinese Universities) 2019年第4期34卷 379-392页

作者： ZHAO Kai LIU Su-ying JIANG Xiu-tian School of Mathematics and Statistics Qingdao University

Let L1 and L2 be the Schrodinger operators on Rn and Rm, respectively. By using different maximal functions and Littlewood-Paley g function on distinct variables, in this paper,some characterizations for functions in ... 详细信息

关键词： Product Hardy space Schrodinger operator characterization Littlewood-Paley function maximal function