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.

该项目的工作重点是两个一般领域， 谐波分析 第一个是关于Fredholm的稳定性 插值比例的属性。 它的动机是 偏微分方程理论中反复出现的主题 定义在具有相对不规则边界的域中（Lipschitz 域）。 也就是说，当与边界相关联的算子 已知有解的值在某个 Lebesgue p-范数表示二次型，那么它们是 在p值的较大范围内有界。要做的工作 涉及“稳定范围”的扩展， Banach空间上的算子 目标包括找到上界 关于幅度的长度，并减少目前 力关于操作符的零空间的维度。 第二条调查线涉及努力 推广了Rademacher和沃尔什级数的经典结果， 特别是在具有大间隙的系列的背景下。 已知 在这种情况下，系数的二次范数 一个系列的最大范数的总和的限制， 系列. 如果所涉及的级数是三角函数，则 最大范数必须用p范数代替。 最近的结果 扩展了经典的比较，包括有界均值 振荡范数 现在将开展工作，以获得更普遍的 BMO估计相同类型的问题，其中系列是 仅限于可测量的集合而不是区间。 第一 需要克服障碍是起草一个适当的定义， 定义在区间以外的函数集的BMO。 对该项目的支持将仅限于毕业 学生津贴和旅行津贴。