Analysis Seminar

Thursday, February 29, 2024 - 3:30pm

Donggeun Ryou

University of Rochester

Suppose that $\mu$ is a Borel probability measure on $\mathbb{R}^d$ such that $\mu(B(x,r)) \lesssim r^{a}$ for all $x \in \mathbb{R}^d$ and all $r > 0$ and $|\widehat{\mu}(\xi)| \lesssim (1+|\xi|)^{-b/2}$ for all $\xi \in \mathbb{R}^d$. The Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem says that for each $p \geq (4d-4a+2b)/b$, $\|\widehat{f\mu}\|_{L^p(\mathbb{R}^d)} \lesssim_p \|f\|_{L^2(\mu)}$ holds for all $f \in L^2(\mu)$. We use a deterministic construction to prove the optimality of range of $p$ in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $d=1$ and parameter range $0 < a,b \leq d$ and $b\leq 2a$. Previous constructions by Hambrook-{\L}aba and Chen required randomness and only covered the range $0 < b \leq a \leq d=1$.