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Non-symmetric differentially subordinate martingales and sharp weak-type bounds for fourier multipliers

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Science China Mathematics 2024年第8期67卷 1807-1826页

作者： Meryem Akboudj Yong Jiao Adam Osękowski School of Mathematics and Statistics Hunan Key Laboratory of Analytical Mathematics and ApplicationsCentral South UniversityChangsha 410075China Faculty of Mathematics Informatics and MechanicsUniversity of WarsawWarsaw 02-097Poland

Let p>2 be a given *** this paper,we prove,with the best constant,the weak-type(p,p)inequality■for a large class of non-symmetric fourier multipliers T_(m) obtained via modulation of jumps of certain Lévy *** particular,the estimate holds for appropriate linear combinations of second-order Riesz transforms and skew versions of the Beurling-Ahlfors operator on the complex *** proof rests on a novel probabilistic bound for Hilbert-space-valued martingales satisfying a certain non-symmetric subordination *** applications to harmonic functions and Riesz systems on Euclidean domains are indicated.

关键词： weak-type bound fourier multiplier martingale differential subordination laminate

Operator-Valued fourier multipliers on Multi-dimensional Hardy Spaces

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Chinese Annals of Mathematics,Series B 2011年第2期32卷 293-302页

作者： Shangquan BU Department of Mathematical Sciences Tsinghua University Beijing 100084 China

The author establishes operator-valued fourier multiplier theorems on multi-dimensional Hardy spaces Hp(Td;X),where 1 ≤ p < ∞,d ∈ N,and X is an AUMD Banach space having the property (α).The suffcient condition on ... 详细信息

The author establishes operator-valued fourier multiplier theorems on multi-dimensional Hardy spaces Hp(Td;X),where 1 ≤ p multiplier is a Marcinkiewicz type condition of order 2 using Rademacher boundedness of sets of bounded linear *** is also shown that the assumption that X has the property (α) is necessary when d ≥ 2 even for scalar-valued *** the underlying Banach space does not have the property (α),a suffcient condition on the multiplier of Marcinkiewicz type of order 2 using a notion of d-Rademacher boundedness is also given.

关键词： H~p-Spaces fourier multiplier Rademacher boundedness d-Rademacher boundedness

A class of singular integrals on the n-complex unit sphere

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Science China Mathematics 1999年第12期42卷 1233-1245页

作者： Michael Cowling 钱涛 Department of Pure Mathematics University of New South Walse

The operaton on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic fourier multipliers, and the Cauchy-Dunford bounded and holomorphic functional calculus of the radial Dirac operator $D = \sum\nolimits_{k = 1}^n {z_k \frac{\partial }{{\partial _{z_k } }}} $ . The equivalence between the three fom and the strong-type (p,p), 1

关键词： singular integral fourier multiplier the unit sphere in C n lunetional calculus

Solutions of second order degenerate integro-differential equations in vector-valued function spaces

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Science China Mathematics 2013年第5期56卷 1059-1072页

作者： BU ShangQuan CAI Gang Department of Mathematical Sciences Tsinghua University

We study the well-posedness of the second order degenerate integro-differential equations （P2）：（Mu）t＇（t）＋ a（Mu）＇（t） = Au（t）＋ ft_c~ a（t - s）Au（s）ds ＋ f（t）, 0 ≤ t ≤ 27r, with periodic boundary conditions Mu（O） = Mu（27r）, （Mu）＇（O） = （Mu）＇（2π）, in periodic Lebesgue-Bochner spaces LP（T,X）, periodic Besov spaces BBp,q（T, X） and periodic Triebel-Lizorkin spaces F~,q（＇F, X）, where A and M are closed linear operators on a Banach space X satisfying D（A） C D（M）, a C LI（R＋） and a is a scalar number. Using known operator- valued fourier multiplier theorems, we completely characterize the well-posedness of （P2） in the above three function spaces.

关键词： fourier multiplier degenerate integro-differential equation well-posedness Besov spaces Triebel- Lizorkin spaces

SOME multiplier THEOREMS FOR ANISOTROPIC HARDY SPACES——In Memory of Professor Yongsheng Sun

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Analysis in Theory and Applications 2006年第4期22卷 339-352页

作者： Yong Ding Senhua Lan Beijing Normal University China School of Mathematical SciencesBeijing Normal University Beijing 100875 P. R. China

Let A be a symmetric expansive matrix and H^p（R^n） be the anisotropic Hardy space associated with A. For a function m in L∞（R^n）, an appropriately chosen function η in Cc^∞（R^n） and j ∈ Z define mj（ξ） = m（A^jξ）η（ξ）. The authors show that if 0 〈 p 〈 1 and mj belongs to the anisotropic nonhomogeneous Herz space K1^1/P^-1,p（R^n）, then m is a fourier multiplier from H^p（R^n） to L^V（R^n）. For p = 1, a similar result is obtained if the space K1^0.1（R^n） is replaced by a slightly smaller space K（w）. Moreover, the authors show that if 0 〈 p 〈 1 and if the sequence {（mj）^v} belongs to a certain mixednorm space, depending on p, then m is also a fourier multiplier from H^p（R^n） to L^v（R^n）.

关键词： anisotropic Hardy space anisotropic Herz space fourier multiplier

HLDER CONTINUOUS SOLUTIONS FOR SECOND ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

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Acta Mathematica Scientia 2011年第3期31卷 765-777页

作者：步尚全 Department of Mathematical Science University of Tsinghua

We study Hlder continuous solutions for the second order integro-differential equations with infinite delay （P1）： u′′（t）＋cu′（t）＋∫t-∞β（t-s）u′（s）ds＋∫t-∞γ（t-s）u（s）ds = Au（t）-∫t-∞δ（t-s）Au（s）ds ＋ f（t）on the line R, where 0 〈 α 〈 1, A is a closed operator in a complex Banach space X, c ∈ C is a constant, f ∈ Cα（R,X） and β,γ,δ∈L1（R＋）.Under suitable assumptions on the kernels β, γ and δ, we completely characterize the Cα- well-posedness of （P_1） by using operator-valued Cα-fourier multipliers.

关键词： fourier multiplier C~α-well-posedness integro-differential equations

REGULARIZED SEMIGROUPS GENERATED BY PSEUDODIFFERENTIAL OPERATORS ON C~α

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Annals of Differential Equations 1998年第2期14卷 329-335页

作者：张寄洲武汉大学

Let Op(f) be a pseudodifferential operator with symbol f∈ S m ρ,0 having constant coefficients. We prove that Op(f) generates a regularized semigroup or cosine function on C α (R n) (0<α<1... 详细信息

关键词： pseudodfferential operator regularized semigroup and cosine function fourier multiplier

Characterization of Robustness with respect to Small Delay for Exponential Stability of Linear Dynamical Systems

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Characterization of Robustness with respect to Small Delay f...