Sparse Bounds for Rough Fourier Integral Operators

Abstract

We prove pointwise bounds for rough Fourier integral operators by the $L^p$ Hardy-Littlewood maximal function. We assume the Fourier integral operators have amplitudes in $L^{\infty }{S}_{\rho }^m$ and phases $\phi$ such that $\phi (x,\xi )-x\cdot \xi \in L^{\infty }{\Phi }^1$, and assume a non-degeneracy condition on the matrix ${\partial }_{\xi }^2\phi (x,\xi )$. The pointwise bound holds when \begin{equation*} m < -\fracρ{2}(n-1) - \fracρ{p} - \frac{n}{p}(1-ρ), \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.