Mathematical Sciences: Variable Coefficient Fourier Analysis

这一奖项为一位年轻的总统研究员的研究提供了额外的支持，他致力于与偏微分算子的微局部分析和傅里叶积分算子理论有关的傅里叶分析问题。将在开发Bourain极大值定理的微局部版本方面做工作。结果涉及到与Radon变换有关的极大算子(球面上)，它被证明是对大于2的幂保持勒贝格空间的。平均可以看作是负半阶的傅里叶积分算子。将努力确定是否相同的结果适用于任何相同阶的非去误差算子族，这取决于线性参数的平滑。由于已知完全泛化是不可能的，因此必须确定一些额外的条件。Carleson和Sjlin研究的一种相关类型的算子是已知的保持四次勒贝格空间的。在这个项目中，我们将把算符扩展为包含Laplace-Beltrami算符的指数的积分变换。如果这是可能的，那么平面上Bochner-Riesz求和的Carleson-Sjolin定理可以推广到紧致流形上。其他工作涉及薛定谔算子的嵌入本征值问题。如果势被变量的倒数所限制，那么霍曼德证明了无穷小的本征函数实际上处处消失。分析这一现象的下一步是确定保证相同结果的潜力的确切条件。一个直接的目标将是找到正确的函数空间(局部勒贝格空间之一)来嵌入势。

This award provides additional support for research by a Presidential Young Investigator working on problems of Fourier analysis related to the microlocal analysis of partial differential operators and the theory of Fourier integral operators. Work will be done in developing a microlocal version of the maximal theorem of Bourgain. The result concerns the maximal operator associated with the Radon transform (over spheres) which was shown to preserve the Lebesgue spaces for powers greater than two. The averaging may be viewed as a Fourier integral operator of negative order one-half. Efforts will be made to determine if the same result holds for any family of nondegererate operators of the same order, depending smoothly on a linear parameter. Some additional conditions have to be established since a total generalization is known to be impossible. A related type of operator studied by Carleson and Sjolin is known to preserve the fourth-power Lebesgue spaces. In this project, work will be done in expanding the operators to integral transforms involving exponential of the Laplace-Beltrami operator. If this is possible then the Carleson-Sjolin theorem for Bochner-Riesz summation in the plane can be extended to compact manifolds. Other work involves the question of embedded eigenvalues for Schrodinger operators. If the potential is bounded by the reciprocal of the variable, then Hormander showed that eigenfunctions which are small at infinity actually vanish everywhere. The next step in analyzing this phenomenon is to determine exact conditions on the potential to guarantee the same result. An immediate goal will be to find the correct function space (one of the local Lebesgue spaces) for embedding the potential.