Sparse Bounds for Rough Fourier Integral Operators

TL;DR

This paper establishes sharp pointwise bounds for rough Fourier integral operators with certain amplitude and phase conditions, and uses these bounds to derive sparse bounds, advancing understanding of their boundedness properties.

Contribution

The paper proves sharp pointwise bounds for rough Fourier integral operators and derives sparse bounds, extending previous results to rough phase and amplitude conditions.

Findings

Established sharp pointwise bounds for rough Fourier integral operators.

Derived sparse form bounds using the pointwise bounds.

Identified conditions under which these bounds hold, including non-degeneracy assumptions.

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.