$L^p$-boundedness of multi-parameter Fourier integral operators

TL;DR

This paper extends the theory of multi-parameter Fourier integral operators, establishing their boundedness on various function spaces under certain non-degeneracy conditions and decomposable phase functions.

Contribution

It generalizes the Seeger-Sogge-Stein theorem to multi-parameter settings with decomposable phases, covering cases where each dimension is at least two.

Findings

Extended boundedness results to local Hardy, Lipschitz, and Sobolev spaces.

Generalized the Seeger-Sogge-Stein theorem for multi-parameter operators.

Established conditions under which these operators are bounded.

Abstract

We study a specific class of Fourier integral operators characterized by symbols belonging to the multi-parameter H\"ormander class $\mathbf{S}^m(\R^{ n_1} \times \R^{ n_2} \times \cdots \times \R^{n_d} )$, where $n= n_1 + n_2 +\cdots + n_d$. Our investigation focuses on cases where the phase function $\Phi(x,\xi)$ can be decomposed into a sum of individual components $\Phi_i(x_i,\xi_i)$, with each component satisfying a non-degeneracy condition. We extend the Seeger-Sogge-Stein theorem under the condition that the dimension $ n_i \ge 2$ for each $1\le i \le d$. As a corollary, we obtain the boundedness of multi-parameter Fourier integral operators on local Hardy spaces, Lipschitz spaces, and Sobolev spaces.