Singular oscillatory integrals on \({\mathbb{R}^n}\) (Q993366)

The authors improve a result of \textit{E. M. Stein} [Beijing lectures in harmonic analysis, Ann. Math. Stud. 112, 307--355 (1986; Zbl 0618.42006)] \[ |I_n(P)|\leq c_d\|\Omega\|_{L^\infty(S_{n-1})}, \] where \(\Omega\) is a function on the unit sphere \(S^{n-1}\) having zero mean value, \[ I_n(P)= \text{p.v.}\int_{\mathbb R^n}e^{i\,P(x)}K(x)\,dx, \] \(K(x)=\frac{\Omega(x/|x|)}{|x|^n}\), \(P\in \mathcal P_{d,n}\) (real polynomials on \(\mathbb R^n\) of degree at most \(d\)) and the constant \(c_d\) depends only on \(d\). Namely, they prove a sharp estimate \[ \sup_{P\in \mathcal P_{d,n}}\Big|\text{p.v.}\int_{\mathbb R^n}e^{i\,P(x)}K(x)\,dx\Big| \leq c\log d\,\big(\|\Omega\|_{L\log L(S^{n-1})}+1\big), \] where \(c\) is a positive constant independent of \(P\) and \(d\). The main ingredient of the proof of this result is a certain estimate for the logarithmic measure of the sublevel set of a real polynomial in one dimension.