Sparse Bounds for Rough Fourier Integral Operators

Abstract

We proof pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^\infty S^m_\rho$ and phases $\varphi$ such that $\varphi(x,\xi) - x\cdot\xi \in L^\infty \Phi^1$, and assume a non-degeneracy condition on the matrix $\partial^2_\xi\varphi(x,\xi)$. The pointwise bound holds when \begin{equation*} m < -\frac{\rho}{2}(n-1) - \frac{\rho}{p} - \frac{n}{p}(1-\rho), \end{equation*} which is known to a be sharp condition on $m$ when $\rho=1$, modulo the end-point. Making use of this pointwise bound and known $L^p$ boundedness results when the phase satisfies an additional non-degeneracy condition, we go on to prove sparse form bounds for these operators.