微分形式上若干算子的范数不等式

发布时间：2018-07-01 18:59

本文选题：微分形式 + BMO范数　；参考：《哈尔滨工业大学》2017年博士论文

【摘要】：微分形式是函数的自然推广,其相关研究发展了欧式空间中的微积分理论。作为处理流形上微积分理论的有力工具,微分形式在偏微分方程、微分几何以及物理学中的力学、电磁学等研究方向发挥着重要的作用。另一方面算子理论在数学、物理、工程、计算机等众多领域更是扮演着不可或缺的角色。在过去的二十年中,微分形式理论,包括微分形式上的方程理论、微分形式的算子理论、微分形式的L~p理论和区域的刻画等迅速发展,成为了当今科学研究的热点领域之一。微分形式的算子理论在现代科学研究中起着重要的作用,并且在众多领域应用广泛。本文主要研究调和分析和偏微分方程中的经典算子,如极大算子、奇异积分算子、Green算子、Dirac算子和其复合算子在微分形式空间上的范数不等式。特别地,针对非齐次A-调和张量及共轭A-调和张量,进一步研究了相关算子的范数有界性。本文主要研究内容包括以下几个方面:首先,利用微分形式的分解性质和基本不等式等工具,结合极大算子及位势算子的有界性,研究了微分形式上极大算子和位势算子的复合算子Ms?P的有界性,并对复合算子Ms?P的L~p范数、BMO范数和Lipschitz范数进行了比较。其次,定义了微分形式上的多重线性Calderón Zygmund奇异积分算子L和L,在一般的微分形式空间上运用Calderón Zygmund分解等技巧讨论了多重线性Calderón Zygmund奇异积分算子的端点弱有界性,为得到算子的强有界性提供了重要支撑。针对非齐次A-调和张量,通过H?lder不等式和特征函数等方法对多重线性Calderón Zygmund奇异积分算子L在微分形式上的L~p范数进行了估计。然后,研究了微分形式上包含Hodge-Dirac算子的复合算子范数不等式,对复合算子M?s?D?G的L~p范数、BMO范数、Lipschitz范数的上界以及相应的加权BMO范数和加权Lipschitz范数的上界进行了估计。并且对Jacobian行列式的子行列式和K-拟正则映射等,估计了其在复合算子M?s?D?G作用下的上界。推广了BMO范数和Lipschitz范数的概念,对微分形式上的复合算子Dk?Gk和Dk+1?Gk的广义BMO范数和广义Lipschitz范数进行了比较。最后,考虑到Orlicz函数理论在近代分析学中的重要作用,结合Orlicz函数和有界平均震荡空间的概念,给出了L~φ-Lipschitz范数和L~φ-BMO范数的定义。利用一类Young函数,G(p,q,C)-类,讨论了同伦算子T作用于微分形式的L~φ-Lipschitz范数和L~φ-BMO范数不等式。然后推广了共轭A-调和张量的范数比较不等式,对共轭A-调和张量u和v的L~φ-BMO范数进行了估计。
[Abstract]:The differential form is a natural generalization of function, and its related research develops the calculus theory in Euclidean space. As a powerful tool for dealing with the calculus theory on manifolds, differential forms play an important role in the research of partial differential equations, differential geometry, mechanics and electromagnetism in physics and so on. On the other hand, operator theory plays an indispensable role in mathematics, physics, engineering, computer and many other fields. In the past twenty years, the theory of differential form, including the theory of differential equation, the theory of differential operator, the theory of differential form and the characterization of region, have become one of the hot fields of scientific research. Differential operator theory plays an important role in modern scientific research and is widely used in many fields. In this paper, the classical operators in harmonic analysis and partial differential equations, such as maximal operator, singular integral operator, Green operator, Dirac operator, and their composition operators, are studied in the differential form space. In particular, for nonhomogeneous A- harmonic Zhang Liang and conjugate A- harmonic Zhang Liang, we further study the norm boundedness of correlation operators. The main contents of this paper include the following aspects: firstly, by using the decomposition property of differential form and the basic inequality, combined with the boundedness of maximal operator and potential operator, In this paper, the boundedness of the composition operator Msmp of the maximal operator and the potential operator in differential form is studied, and the BMO norm and Lipschitz norm of the composition operator MSG P are compared. Secondly, the multiplex linear Calder 贸 n Zygmund singular integral operator L and L are defined in the differential form. The weak boundedness of the extreme point of the multiplex linear Calder 贸 n Zygmund singular integral operator is discussed by using the techniques of Calder 贸 n and Zygmund decomposition on the general differential form space. It provides important support for the strong boundedness of operators. For nonhomogeneous A- harmonic Zhang Liang, the L ~ p norm of multiplex linear Calder 贸 n Zygmund singular integral operator L in differential form is estimated by means of Hillder inequality and eigenfunction. Then, the norm inequality of composition operator containing Hodge-Dirac operator in differential form is studied. The upper bound of Lipschitz norm and the upper bound of weighted BMO norm and weighted Lipschitz norm are estimated. The upper bound of Jacobian determinant subdeterminant, K-quasi regular mapping and so on under the action of composition operator MVX DG is also estimated. In this paper, the concepts of BMO norm and Lipschitz norm are generalized, and the generalized BMO norm and generalized Lipschitz norm of composition operator DkGk in differential form and generalized Lipschitz norm are compared. Finally, considering the important role of Orlicz function theory in modern analysis, the definitions of L ~ 蠁 -Lipschitz norm and L ~ 蠁 -BMO norm are given by combining the concepts of Orlicz function and bounded mean oscillatory space. In this paper, by using a class of Young functions G (pnqnc) -class, we discuss the inequalities of L ~ 蠁 -Lipschitz norm and L ~ 蠁 -BMO norm of homotopy operator T acting on differential form. Then we generalize the norm comparison inequality of conjugate A- harmonic Zhang Liang, and estimate the L蠁 -BMO norm of conjugate A- harmonic Zhang Liang u and v.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O177

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