Sharp \(L^p\) decay of oscillatory integral operators with certain homogeneous polynomial phases in several variables (Q2423851)

In this paper, the authors establish the sharp $L^p(\mathbb{R}^n)$ decay of the following oscillatory operator \begin{align*} T_{\lambda}(f)(x):=\int_{\mathbb{R}^n}e^{i\lambda S(x,y)}\psi(x,y)f(y)\,\text{d}y, \quad \forall\, x\in\mathbb{R}^n, \end{align*} where $n\geq2$, $\psi(x,y)$ is a smooth function supported in a compact neighborhood of the origin, $S(x,y):=\sum_{|\alpha|+|\beta|=d}a_{\alpha,\beta}x^\alpha y^\beta$ is a real-valued homogeneous polynomial in higher dimensions with degree $d$, and $\alpha$, $\beta$ are multi-indices. We are glad to see higher-dimensional results for these kinds of operators. A key method to obtain these results is the interpolation of analytic families of operators. Similar beautiful results are were obtained by \textit{A. Greenleaf} et al. [J. Funct. Anal. 244, No. 2, 444--487 (2007; Zbl 1127.35090)], \textit{Z. Shi} et al. [Math. Z. 286, No. 3--4, 1277--1302 (2017; Zbl 06780328)] and \textit{C. Yang} [Illinois J. Math. 48, No. 4, 1093--1103 (2004; Zbl 1073.45010)].