Mathematical Sciences: Research in Interpolation Theory and Harmonic Analysis

该项目的工作集中在谐波分析的两个一般领域。第一个问题是关于Fredholm性在插值尺度上的稳定性。它的动机是在定义在具有相对不规则边界的域(Lipschitz域)中定义的偏微分方程组理论中反复出现的主题。也就是说，当将边值与解联系在一起的算子已知在某一Lebeguep-范数上有界时，即二次范数，那么它们被证明是在更大范围的p的值上有界的。所要做的工作包括将“稳定范围”推广到Banach空间上更大的算子类。目标包括找到范围长度的上界，并减少目前有效的关于算子零空间的维度的假设。第二条线涉及推广关于Rademacher和Walsh级数的经典结果，特别是在具有大间隙的级数的背景下。众所周知，在这种情况下，级数系数的二次范数由级数和的最大范数有界。如果所涉及的级数是三角的，那么最大范数必须用p-范数来代替。最近的结果扩展了经典的比较，包括有界平均振荡范数。现在将进行工作，以获得同一类型问题的更一般的BMO估计，其中系列限于可测量的集合，而不是间隔。要克服的第一个障碍是为区间以外的一组函数起草一个适当的BMO定义。对该项目的支持将仅限于研究生津贴和旅行津贴。

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.