Global $L^p$ continuity of Fourier integral operators

Abstract:

In this paper we establish global $L^p(\mathbb {R}^{n})$-regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on $L^p(\mathbb {R}^{n})$, $1<p<\infty$, as well as to be bounded from Hardy space $H^1(\mathbb {R}^{n})$ to $L^1(\mathbb {R}^{n})$. This extends local $L^p$-regularity properties of Fourier integral operators, as well as results of global $L^2(\mathbb {R}^{n})$ boundedness, to the global setting of $L^p(\mathbb {R}^{n})$. Global boundedness in weighted Sobolev spaces $W^{\sigma ,p}_s(\mathbb {R}^{n})$ is also established, and applications to hyperbolic partial differential equations are given.