On estimates for trigonometric integrals (Q2862885)

The aim of this article is to estimate the trigonometric integral NEWLINE\[NEWLINE\int_{\Omega}e^{2\pi i F(\bar{x})}\,d\bar{x} \tag{1}NEWLINE\]NEWLINE where \(\Omega\) is a bounded closed simply connected \(n\)-dimensional domain with piecewise smooth boundary in \(\mathbb{R}^n\), \(\bar{x}=(x_1,\dots ,x_n)\), \(d\bar{x}=dx_1\dots dx_n\) and \(F(\bar{x})\) is a polynomial of a special type.NEWLINENEWLINEUsing methods similar to the ones in [Proc. Steklov Inst. Math. 207, 77--85 (1995); translation from Tr. Mat. Inst. Steklova 207, 82--92 (1994; Zbl 0899.11041)], the author obtains some upper bounds for the measure \(\mu (\Omega )\) of the domain \(\Omega\). Then these bounds are applied to estimate the integral (1).