Let $\phi(x) = x^1_1 - x_2^2$. I wanted to know if the stationary phase result $$\lim_{\lambda \to \infty} \frac{\lambda}{\pi} \int_{\mathbb{R}^2} e^{i \lambda \phi(x)} f(x) dx = f(0)$$ also holds when $f$ is not smooth. In particular I can only assume that $f \in C^0_0(\mathbb{R}^2)$ and also that $f\in H^1(\mathbb{R}^2)$.
Thank you!