Global Lp continuity of Fourier integral operators

TL;DR

This paper proves global Lp regularity and boundedness of Fourier integral operators, extending previous local and L2 results to a broader global Lp setting and weighted Sobolev spaces.

Contribution

It establishes new global Lp and weighted Sobolev space boundedness results for Fourier integral operators, expanding upon prior local and L2 regularity findings.

Findings

Operators are bounded on L^p for 1<p<∞ with specific amplitude decay

Operators are bounded from Hardy space H^1 to L^1 under certain conditions

Global boundedness in weighted Sobolev spaces W^{σ,p}_s is achieved

Abstract

In this paper we establish global Lp regularity properties of Fourier integral operators. The orders of decay of the amplitude are determined for operators to be bounded on $L^p(\Rn)$, $1<p<\infty$, as well as to be bounded from Hardy space $H^1(\Rn)$ to $L^1(\Rn)$. The obtained results extend local $L^p$ regularity properties of Fourier integral operators established by Seeger, Sogge and Stein (1991) as well as global $L^2(\Rn)$ results of Asada and Fujiwara (1978) and Ruzhansky and Sugimoto (2006), to the global setting of $L^p(\Rn)$. Global boundedness in weighted Sobolev spaces $W^{\sigma,p}_s(\Rn)$ is also established. The techniques used in the proofs are the space dependent dyadic decomposition and the global calculi developed by Ruzhansky and Sugimoto (2006) and Coriasco (1999).