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

Let $T_N^{j,k}:L^p(B)\to L^q([0,1])$ 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$,B,$istheunitballin${$\mathbb{R}$}^{n},$and$,N,>>1.$ We compare the asymptotic behaviour as $N\to +\mathrm{\infty }$ of the operator norms $\parallel T_N^{j,k}{\parallel }_{L^p(B)\to L^q([0,1])}$ for all $p,q\in [1,+\mathrm{\infty }]\mathrm{.}$ 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.