I'm trying to find asymptotics for an oscillatory integral on $\mathbb{S}^{n-1}$, for which my advisor said I should use stationary phase arguments. The particular, he claims that:

If $\lambda\gg 1$, then $$I(\lambda,x) = \int_{\mathbb{S}^{n-1}}(x\cdot y)e^{i\lambda(x\cdot y)}\,d\sigma(y),$$ is $\mathcal{O}(|x|(\lambda |x|)^{-\frac{n-1}{2}})$ when $|x|\geq \lambda^{-1}$.

When I hear stationary phase, I think of working with operators of the form
$$
L = \frac{\nabla_y(x\cdot y)}{i\lambda |\nabla_y (x\cdot y)|^2} \cdot \nabla_y,
$$
since then $L^N[e^{i\lambda (x\cdot y)}] = e^{i\lambda (x\cdot y)}$, for any $N \geq 1$, and I can use integration by parts to move these operators over to the $(x\cdot y)$ term. ** However**, wouldn't that require that the integral be defined over an $n$-dimensional region, rather than an $(n-1)$-dimensional surface?

I wouldn't be struggling so much if the gradient $\nabla_y$ could be taken in Cartesian coordinates. But we have a surface integral in $d\sigma(y)$, meaning that we would need to parametrize our surface with $n-1$ parameters, say $\omega = (\omega_1, \ldots, \omega_{n-1}) \in \Omega \subset \mathbb{R}^{n-1}$, with any derivates now being taken in these new variables. Specifically $\nabla_y$ would generate an $n$-vector, while $\nabla_{\omega}$ would generate an $n-1$-vector. So how should I go about applying the same kind of stationary phase arguments to this new integral? $$ I(\lambda, x) = \int_{\Omega} (x \cdot y(\omega)) e^{i\lambda (x\cdot y(\omega))} \,dV(\omega) $$ Now that $y$ depends on $\omega$, the gradients $\nabla_\omega (x\cdot y(\omega))$ become much trickier to get a grasp on. I'm particularly struggling trying to argue how we can find regions where $|\nabla_\omega (x\cdot y(\omega))| > 0$, so that our $L$ type operators are properly defined.

Am I going about this all wrong? If I can get the big-oh asymptotics I mentioned above in some other way, it doesn't really matter. I just need to prove these results as a lemma to something bigger. Any help is much appreciated!