On the $L^p$-bound for trigonometric integrals (Q2416492)

This paper deals with a class of oscillatory integrals $J(a,\psi)$ with purely imaginary phase function $iF(x,a)$ and amplitude $\psi(x,a)$ having compact support in $x\in\mathbb R^n$, $a \in\mathbb R^m$ being a parameter. Assume that $n=2$, $m=6$ and $F$ is a cubic polynomial in $x\in\mathbb R^2$, while $\psi(\cdot, a)\in C_0^{\infty}(\mathbb R^2)$, $\Vert \psi(\cdot, a)\Vert_{C^3(\mathbb R^2)} \in L^{\infty}(\mathbb R^6)$. It is proved in Th. 1.3 that $J(\cdot,\psi)\in L^p(\mathbb R^6)$ for any $p>8$, while for some non-negative amplitudes $\psi$, $\psi(0,0)>0$ the integral $J(\cdot,\psi)\notin L^p(\mathbb R^6)$ for all $p\leq 8$. In the special case of the cubic polynomial $F$ in $x$ and discontinuous amplitude $\psi\cdot K_Q(x)$, $Q=[0,1]^2$, $\psi \in C_0^1(\mathbb R^2)$ similar result to the above formulated one holds. The problem of finding $\gamma=\inf\{p\geq 1: J(\cdot,\psi)\in L^p(\mathbb R_a^m)\}$ has important applications in Number theory.