Mathematical Sciences: Research in Interpolation Theory and Harmonic Analysis

数学科学：插值理论和调和分析研究

• 批准号: 9005436
• 负责人: Yoram Sagher
• 金额: $ 2.7万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-08-01 至 1992-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9005436&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Interpolation; Theory

Work on this project focuses on two general areas of harmonic analysis. The first concerns stability of Fredholm properties in interpolation scales. It is motivated by a recurring theme inthe theory of partial differential equations defined in domainswith relatively irregular boundaries (Lipschitz domains). Namely, when the operators which associate boundary values with solutions are known to be bounded in a certain Lebesgue p-norm say the quadratic - then they turn out to be bounded across alarger range of values of p. The work to be done involves extensions of 'stability range' to larger classes of operators on Banach spaces. Goals include finding upper bounds for the length of the range and reducing assumptions currently in force regarding the dimension of the null space of the operators. The second line of investigation relates to efforts extending classical results on Rademacher and Walsh series, especially in the context of series with large gaps. It is known that in such circumstances, the quadratic norm the coefficients of a series is bounded by the maximum norm of the sum of the series. If the series involved are trigonometric, then the maximum norm must be replaced by p-norms. Recent results have extended the classical comparisons to include the bounded mean oscillation norm. Work will now be done to obtain more general BMO estimates for the same type of problem where the series are restricted to measurable sets rather than intervals. The first hurdle to be overcome is that of drafting a proper definition of BMO for sets of functions defined on other than intervals. Support for this project will be restricted to graduate student stipends and travel allowance.

该项目的工作着重于谐波分析的两个一般领域。 第一个涉及插值量表中弗雷霍姆性质的稳定性。 它是由在域相对不规则边界（Lipschitz域）定义的部分微分方程理论中反复出现的主题的动机。 也就是说，当已知将边界值与溶液相关联的运算符在某个lebesgue p -norm中界定二次时 - 然后，它们被证明是在p的alarger范围内界定的。 要完成的工作涉及“稳定范围”扩展到Banach空间上较大类的运营商。 目标包括在范围的长度上找到上限，并减少有关操作员无效空间尺寸的当前有效的假设。 第二道调查涉及努力扩展Rademacher和Walsh系列的经典成果，尤其是在较大差距的系列范围内。 众所周知，在这种情况下，二次规范系列的系数受到该系列总和的最大规范的界定。 如果所涉及的系列是三角法，则最大规范必须由p-norms代替。 最近的结果将经典比较扩展到包括有限的平均振荡规范。 现在，将完成对相同类型的问题的更通用的BMO估计，其中该系列仅限于可测量的集合而不是间隔。 要克服的第一个障碍是起草适当的BMO定义，以确定在间隔以外定义的功能集。 对该项目的支持将仅限于研究生津贴和旅行津贴。