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

Abstract

Let $T_N^{j,k}:L^p(B)\to L^q([0,1])$ be the oscillatory integral operators defined by $T_N^{j,k}f(s):={\int }_{B}f(x)e^{ıN{|x|}^j{s}^k}dx,(j,k)\in \{1,2{\}}^2,$ where $B$ is the unit ball in ${\mathbb{R}}^n$ and $N>>1.$ We compare the asymptotic behaviour as $N\to +\infty$ of the operator norms $\parallel T_N^{j,k}{\parallel }_{L^p(B)\to L^q([0,1])}$ 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 Schrödinger equation.