A martingale proof of L2 boundedness of Clifford-valued singular integrals

Summary This paper presents a theory of Clifford algebra-valued martingales on a -finte measure space, with respect to a pseudoaccretive weight. A novel dual pair system of Haar functions associated with the Clifford martingale is constructed, and Littlewood-Paley estimates are established. The dual pair system of Clifford Haar functions is used to give a new proof of the boundedness of the Cauchy principal value integral on Lipschitz surfaces, and of the Clifford-valued T(b) theorem.Research supported by the Australian Research Council.Research supported by the National Science Foundation of China.Research carried out as a National Research Fellow.     

Three-line Theorem on the Octonions

The three-line theorem on the octonions is obtained, which generalizes the result of J.Peetre and P. Sjoelin from the associative Clifford algebra to non-associative octonion algebra.     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

Blocking analysis andT(1) theorem

TheT(1) theorem with a weak condition on the distribution kernel is proved by using a new method—blocking analysis. It improves a result of Meyer’s. Project supported by the National Natural Science Foundation of China (Grant No. 19631080) and the Foundation of Advanced Research Center, Zhongshan University.     

The Semigroup Structure of Left Clifford Semirings

In this paper,we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings.We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.     

Classification of monoids by injectivities II. CC-injectivity

We characterize monoids over which all S-acts are CC-injective and find conditions under which CC-injectivity of all acts implies that all acts are C-injective. Research of X. Zhang supported by the China Scholarship Council No. 2006101056. Research of Y. Chen supported by the NNSF of China (No. 10771077) and the NSF of Guangdong Province (No. 06025062).     

Index of Lagrangian submanifolds of ℂℙ n and the Laplacian of 1-forms

Inequalities, involving the two first eigenvalues of the Laplacian acting on 1-forms of minimal Lagrangian submanifolds of the complex projective space, are obtained. The Clifford torus in the complex projective plane is characterized by its index.Research partially supported by a DGICYT grant No. PB90-0014-C03-02.     

The Mehler Formula for the Generalized Clifford-Hermite Polynomials

The Mehler formula for the Hermite polynomials allows for an integral representation of the one-dimensional Fractional Fourier transform. In this paper, we introduce a multi-dimensional Fractional Fourier transform in the framework of Clifford analysis. By showing that it coincides with the classical tensorial approach we are able to prove Mehler＇s formula for the generalized Clifford-Hermite polynomials of Clifford analysis.     

General spectral flow formula for fixed maximal domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory. This work was supported in part by The Danish Science Research Council, SNF grant 21-02-0446. The second author is partially supported by FANEDD 200215, 973, Program of MOST, Fok Ying Tung Edu. Funds 91002, LPMC of MOE of China, and Nankai University.     

Reflection of transversal progressing waves for semilinear strictly hyperbolic systems

We discuss the reflection of weak singularities of solutions to semilinear hyperbolic systems by using the method of energy estimates in the space of conormal distributions, and consequently obtain the general results on the reflection of weak singularities of the solution to a high order semilinear strictly hyperbolic equation. Research partially supported by the National Natural Science Foundation of China     

Classification of monoids by injectivities I. C-injectivity

A new kind of injectivity, namely, C-injectivity is investigated. Classifications of a monoid by properties of C-injective acts and right ideals are presented. It is shown that the class of monoids over which all acts are C-injective generalizes the class of self-injective monoids. On the way we complete an early result by Johnson and McMorris (Proc. Am. Math. Soc. 36(2), 385–388, 1972). Research of X. Zhang supported by the China Scholarship Council No. 2006101056. Research of Y. Chen supported by the NNSF of China (No. 10771077) and the NSF of Guangdong Province (No. 06025062).     

A Remark on Multilinear Oscillatory Singular Integrals

Abstract In this paper, we study multilinear oscillatory singular integrals with rough kernel, and give a criterion of certain boundedness for this kind of operator Research supported by NNSF of China (No.19131080) and NEDF of China     

Weak monotonicity inequality and partial regularity for harmonic maps

The notion of locally weak monotonicity inequality for weakly harmonic maps is introduced and various results on this class of maps are obtained. For example, the locally weak monotonicity inequality is nearly equivalent to theε-regularity. Project supported by the National Natural Science Foundation of China (Grant No. 19571028) and the Guangdong Provincial Natural Science Foundation of China.     

On weak neighborhood systems and spaces

We introduce and study the concepts of weak neighborhood systems, weak neighborhood spaces, ω(ψ, ψ′)-continuity, ω-continuity and ω*-continuity on WNS’s. This work was supported by a grant from Research Institute for Basic Science at Kangwon National University.     

Continuity in weak topology:higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity in weak topologies.From this,we will prove that the fundamental matrix solutions of k-th, k≥2,order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies.Consequently,Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies.Moreover,for the scalar Hill's equations,Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues,and rotation numbers are all continuous in potentials with weak topologies.These results will lead to many interesting variational problems.     

EQUIFOCAL HYPERSURFACES IN SYMMETRIC SPACES

This note investigates the multiplicity problem of principal curvatures of equifocal hypersurfaces in simply connected rank 1 symmetric spaces. Using Clifford representation theory, and the author also constructs infinitely many equifocal hypersurfaces in the symmetric spaces. Project supported by the National Natural Science Foundation of China (No. 19925104), RFDP and the Qiu-Shi Science and Technology Foundation.     

Some relations on completely regular semigroups

A semigroup S is called a Clifford semigroup if it is completely regular and inverse. In this paper, some relations related to the least Clifford semigroup congruences on completely regular semigroups are characterized. We give the relation between Y and ξ on completely regular semigroups and get that Y ^* is contained in the least Clifford congruence on completely regular semigroups generally. Further, we consider the relation Y ^*, Y, ν and ε on completely simple semigroups and completely regular semigroups. This work is supported by Leading Academic Discipline Project of Shanghai Normal University, Project Number: DZL803 and General Scientific Research Project of Shanghai Normal University, No. SK200707.     

Iterative Method for Set-Valued Mixed Quasi-variational Inequalities in a Banach Space

This paper introduces an iterative method for finding approximate solutions of a set-valued mixed quasivariational inequality in the setting of a Banach space. Existence of a solution of this rather general problem and the convergence of the proposed iterative method to a solution are established.The first two authors were partially supported by the National Science Council of the Republic of China. The third author was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in ShanghaiCommunicated by     

A new proof of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the parabolic polyharmonic equations

In this paper we obtain local L ^p estimates for the parabolic polyharmonic equations by a straightforward approach. Yao was supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905), Shanghai Leading Academic Discipline Project (Grant No. J50101) and Key Disciplines of Shanghai Municipality (Grant No. S30104). Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700), National Natural Science Foundation of China (Grant No. 60532080), and the Key Project of Chinese Ministry of Education (Grant No. 306017)     

Essentially normal Hilbert modules and Khomology Ⅲ: Homogenous quotient modules of Hardy modules on the bidisk

In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K-homology invariant defined in the case of the homogenous quotient modules on the bidisk. This work is partially supported by the National Natural Science Foundation of China (Grant No. 10525106), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program