Speaker
Alex Iosevich, University of Rochester
Abstract
In the 70’s, Shmuel Agmon and Lars Hormander asked the following basic question. Consider the partial differential equation \(P(D)u=0\). Then the support of the Fourier transform of \(u\), in the sense of distributions, is contained in the variety \(\{\xi: P(\xi)=0\}\). Which degree of \(L^p\) integrability of \(u\) guarantees that \(u\) is identically \(0\)? The problem was revisited in the early 2000s by Agranovsky and Naryanan, who proved that if \(f\) is a locally integrable function, with the Fourier transform supported on \(k\)-dimensional submanifold of \({\mathbb R}^d\), \(1 \leq k<d\), then if \(f \in L^p({\mathbb R}^d)\) for some \(p \leq \frac{2d}{k}\), then \(f\) is identically \(0\). The result was extended to sets of a given packing dimension by Senthil-Raani in 2014. The exponent is sharp if the manifold is, for example, a sphere, but it is far from sharp if the manifold is a hyperplane. Furthermore, Guo, Iosevich, Zhang, and Zorin-Kranich proved that in the case of the moment curve \(\{(t,t^2, \dots, t^d: t \in [0,1]\}\), the sharp spectral synthesis exponent is \(\frac{d^2+d+2}{2}\), far exceeding the Agranovsky-Narayanan exponent \(\frac{2d}{k}=2d\) in higher dimensions. This shows that the underlying geometry, not just dimensionality, plays a key role in the determination of the sharp exponent. Analogous problems, requiring very different techniques, will also be discussed in the setting of compact Riemannian manifolds without a boundary. In this talk, we are going to discuss a variety of approaches to these problems, including the use of the Fourier ratio, which previously came up in the theory of compressed sensing and approximate Kolmogorov complexity. Connections with restriction theory for the Fourier transform will also be prominently featured. The talk is based on joint works with Shantanu Deodhar, Azita Mayeli, and Emmett Wyman.