$L^p$-$L^q$ boundedness of integral operators with oscillatory kernels: Linear versus quadratic phases

TL;DR

The paper investigates how the asymptotic behavior of certain oscillatory integral operators depends on whether their phase functions are linear or quadratic, revealing a dimension-dependent distinction.

Contribution

It provides a detailed comparison of the asymptotic norms of oscillatory integral operators with linear and quadratic phases, highlighting the dimension-dependent nature of their behavior.

Findings

Asymptotic behavior depends on phase linearity or quadraticity for dimensions greater than one.

In dimension one, the behavior differs and does not solely depend on phase type.

Results are motivated by inhomogeneous Strichartz estimates for Schrödinger equations.

Abstract

Let $\,T^{j,k}_{N}:L^{p}(B)\, \rightarrow\,L^{q}([0,1])\,$ be the oscillatory integral operators defined by $\;\displaystyle T^{j,k}_{N}f(s):=\int_{B} \,f(x)\,e^{\imath N{|x|}^{j}s^{k}}\,dx, \quad (j,k)\in\{1,2\}^{2},\,$ where $\,B\,$ is the unit ball in ${\mathbb{R}}^{n}\,$ and $\,N\,>>1.$ We compare the asymptotic behaviour as $\,N\rightarrow +\infty\,$ of the operator norms $\,\parallel T^{j,k}_{N} \parallel_ {L^{p}(B)\rightarrow L^{q}([0,1])}\,$ for all $\,p,\,q\in [1,+\infty].\,$ We prove that, except for the dimension $n=1,\,$ this asymptotic behaviour depends on the linearity or quadraticity of the phase in $s$ only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schr\"{o}dinger equation.