Variable Coefficient Fourier Analysis

The PI will study several problems in Geometric Harmonic Analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated with a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it omits, as well as the sound volume. Similar phenomena arise for manifolds, and the PI will study precisely how their shapes, such as how they are curved, affect the properties properties of eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the PI will study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity. This project provides research training opportunities for graduate students. Among the specific problems the PI shall study, they wish to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, they will develop what is called ``global harmonic analysis’’, which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates are Lp-estimates for eigenfunctions and quasimodes and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affect the estimates and the kinds of functions that saturate them. The latter issue is closely related to the much-studied (but still not well-understood) questions of concentration, oscillation, and size properties of modes and quasimodes in spectral asymptotics. These questions are also naturally linked to the long-time properties of the solution operator for the wave equation, Schrodinger equation, and resolvent estimates coming from the metric Laplacian. High frequency solutions and obtaining sharp results under geometric assumptions are particularly interesting. They will also study functions that saturate the estimates in different ways depending on the sign of the sectional curvatures of the manifolds. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.