学术报告（Prof. Jan Rozendaal 10.9）

摘要：

It is well known that the wave operators $\cos(t\sqrt{-\Delta})$ and $\sin(t\sqrt{-\Delta})$ are not bounded on $L^{p}(\R^{n})$, for $n\geq 2$ and $1\leq p\leq \infty$, unless $p=2$ or $t=0$. In fact, for $1<p<\infty$ these operators are bounded from $W^{(n-1)|\frac{1}{p}-\frac{1}{2}|,p}(\R^{n})$ to $L^{p}(\R^{n})$, and this exponent cannot be improved. This phenomenon is symptomatic of the behavior of Fourier integral operators on $L^{p}(\Rn)$, a class of operators which includes the solution operators to smooth variable-coefficient wave equations.

In this talk, I will introduce a class of Hardy spaces $\mathcal{H}^{p}_{FIO}(\R^{n})$, for $p\in[1,\infty]$, on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on $L^{p}(\Rn)$, and thereby the optimal fixed-time $L^{p}$ regularity of wave equations.

However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on $L^{p}(\Rn)$. In particular, I will indicate how one can use this invariance to obtain the optimal fixed-time $L^{p}$ regularity for wave equations with rough coefficients. I will also mention the connection of these spaces to the local smoothing conjecture and to nonlinear wave equations.

This talk is based on joint work with Andrew Hassell and Pierre Portal (Australian National University), Zhijie Fan (Wuhan University), Naijia Liu, Liang Song and Lixin Yan (Sun Yat-Sen University), and Robert Schippa (Korea Institute for Advanced Study). The research leading to these results has received funding from the Norwegian Financial Mechanism 2014-2021, grant 2020/37/K/ST1/02765.