论文标题
一阶差异操作员和与逆量度相关的多项式的新家族的分数力量
Fractional powers of first order differential operators and new families of polynomials associated to inverse measures
论文作者
论文摘要
首先，我们建立了具有零订单项的一阶差分运算符的分数幂理论，获得了PDE属性并分析相应的分数Sobolev空间。特别是，我们的研究表明，具有反度量的Lebesgue和Sobolev空间（如逆高斯度量）在一阶操作员的分数力量理论中起着基本作用。其次，并部分由这种理论动机，为开发\ emph {Inverse}测度的谐波分析的基础奠定了基础。我们发现了与逆高斯，拉瓜和雅各比的多项式家庭的新家族，并使用生成和rodrigues公式以及三届复发关系来表征它们。此外，在这些反度设置中，我们证明了几个基本奇异积分运算符的界限。
First, we establish the theory of fractional powers of first order differential operators with zero order terms, obtaining PDE properties and analyzing the corresponding fractional Sobolev spaces. In particular, our study shows that Lebesgue and Sobolev spaces with inverse measures (like the inverse Gaussian measure) play a fundamental role in the theory of fractional powers of the first order operators. Second, and motivated in part by such a theory, we lay out the foundations for the development of the harmonic analysis for \emph{inverse} measures. We discover new families of polynomials related to the inverse Gaussian, Laguerre, and Jacobi measures, and characterize them using generating and Rodrigues formulas, and three-term recurrence relations. Moreover, we prove boundedness of several fundamental singular integral operators in these inverse measure settings.
