Asymptotic behavior of oscillatory Hilbert transform

Question

Does anyone know what is the leading term in the asymptotics of $$ P.V. \int\limits_{ -\infty }^{ +\infty } \frac{e^{i \lambda x^3 } f( x ) dx }{ x }, $$ as $ \lambda \to +\infty $? Assume $ f \in C_{0}^{\infty}(\mathbb{R}) $. In literature I have only found results for $ P.V. \int\limits_{ -\infty }^{ +\infty } \frac{e^{i \lambda x } f( x ) dx }{ x } $ and $ P.V. \int\limits_{ -\infty }^{ +\infty } \frac{e^{i \lambda x^{n} } f( x ) dx }{ x } $ with $ n $ even.