Mathematical Sciences: Singular Integrals and Fourier Integrals

这项研究的数学主题是将调和分析与现代微分算子理论相结合，应用于某些类型的积分变换问题。这种分析的物理对应物出现在X射线重建或断层摄影术中，在这种情况下，人们试图通过从有限数量的角度拍摄的低维照片中获得的信息来可视化未知对象。在数学环境中，假设一类曲面平均(Radon变换)是已知的，这里的曲面可以是抽象流形，并且形成所谓的傅立叶积分算子。在所谓的特殊情况下，将努力获得此类平均值或变换的估计值。当人们考虑与d-bar Neumann问题有关的曲线或算子的Hilbert变换时，这种特殊情况下的例子自然地发生在多个复变量的弱伪凸域上。当变换不退化时，均方估计可以推广到所有的幂范数。这项工作将集中在退化情况下，估计更难获得，事实上，Radon变换的范数估计可能会失败，除非是均方情况。我们还将尝试构造一类足以处理欧氏空间中受限k平面变换的可容许性和可逆性的傅立叶积分算子。相关正则关系的构造应与(k平面的)Grassman几何有关。

The mathematical theme of this research is a combination of harmonic analysis and the modern theory of differential operators applied to questions about integral transforms of certain type. A physical counterpart to this analysis occurs in x-ray reconstruction or tomography where one seeks to visualize an unknown object by information derived from lower dimensional pictures taken from a limited number of angles. In the mathematical setting, it is assumed that a class of surface averages (Radon transformations) are known, where the surface here may be an abstract manifold, and form what is called a Fourier integral operator. Efforts will be made to obtain estimates for such averages or transforms in what is called the singular case. Examples in this particular context occur naturally when one considers Hilbert transforms along curves or operators associated with the d-bar Neumann problem on weakly pseudoconvex domains in several complex variables. When the transforms are not degenerate, mean-square estimates can be extended to all power norms. This work will concentrate on the degenerate case where estimates are harder to obtain and, in fact, the norm estimates for Radon transforms may fail, except in the mean-square case. Work will also be done in an attempt to formulate classes of Fourier integral operators sufficient to deal with the problem of admissibility and invertibility for the restricted k-plane transform in Euclidean space. The construction of the relevant canonical relations should be related to the Grassmannian geometry (of the k-planes).