新变指标Besov-Triebel-Lizorkin型函数空间及算子有界性

项目名称： 新变指标Besov-Triebel-Lizorkin型函数空间及算子有界性

项目编号： No.11201043

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 汤灿琴

作者单位： 大连海事大学

项目金额： 22万元

中文摘要： 随着非线性弹性力学、电流变学及图像恢复等实际问题的发展,具有变指数增长性条件的非线性问题成为一个新兴的研究课题. 对这些非线性问题比较合适的函数空间是变指数函数空间. 本课题拟研究变指数的Besov-Triebel-Lizorkin型空间的特征刻画. 如原子分解、分子分解、小波、差分及多项式局部逼近等多种途径进行等价刻画，然后探讨拟微分算子在这些新空间上的有界性，建立Sobelev嵌入性质及乘子的点态估计，迹的性质刻画等. 本项目还将探讨变指数空间上的某些经典算子及交换子的有界性. 建立加权Hardy算子在变指数Lebesgue空间上的最优估计，讨论变指数的分数次积分算子、Hardy算子等调和分析中经典算子与Lip函数生成的交换子在变指数的Morrey空间、Herz空间、Besov-Triebel-Lizorkin空间等函数空间上的相关性质，并将其推广至齐性空间.

中文关键词： Besov-Triebel-Lizorkin型函数空间；变指标分数次积分算子；交换子；Hardy算子；周期解

英文摘要： With the development of the practical problems of nonlinear elasticity, electrorheologiacl fluids and image restoration, with a variable exponent growth conditions of nonlinear problems become an emerging research topic. In the study of such nonlinear problems, variable exponent function space is more appropriate function space. This project firstly intends to study the characterizations of variable exponent Besov-Triebel-Lizorkin space. The smooth atomic and molecular decomposition will be given, and several equivalent forms characterized by wavelets, differences and oscillations (local approxiamtion by polynomials) will also be discussed. Then apply these features to prove the boundedness of pseudo-differential operators on this new kind of spaces. And some crucial problems including Sobelev embedding, pointwise estimate for multipliers and nature of traces will be focused. The project will also explore the nature of some classical operators and commutators on variable exponent spaces. We'll attempt to establish the optimal estimation of weighted Hardy operator from the variable source space to target space, find the condition on weight function. The boundedness of commutator of some classcial operators in harmonic analysis (such as variable fractional integral operator, weigted Hardy operator) and Lip

英文关键词： Besov-Triebel-Lizorkin-type spaces with variable e；Variable fractional integral operator；Commutator；Hardy operator；Periodic Solution