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It is well known that the solution operators $\cos(t\sqrt{-\Delta})$ and $\sin(t\sqrt{-\Delta})$ to the wave equation $\partial_{t}^{2}u=\Delta u$ are not bounded on $L^{p}(\mathbb{R}^{n})$, for $n\geq 2$ and $1\leq p\leq \infty$, unless $p=2$ or $t=0$. In fact, for $1<p<\infty$ these wave operators are bounded from $W^{s_{p},p}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ for $s_{p}:=(n-1)|\frac{1}{p}-\frac{1}{2}|$, and this exponent cannot be improved.

In this talk, I will introduce a class of Hardy spaces $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, for $p\in[1,\infty]$, on which the wave operators are bounded. These spaces also satisfy Sobolev embeddings that allow one to recover the optimal boundedness results on the $L^{p}$-scale. In fact, the spaces are invariant under a wider class of oscillatory integral operators known as Fourier integral operators, as follows from the fact that such operators satisfy a notion of off-singularity decay which is a modification of classical heat kernel bounds.

This talk is based on joint work with Andrew Hassell and Pierre Portal (Australian National University).