I have an oscillatory integral:

$$ \int u(x,y) e^{i\lambda f(x,y)} dx $$

with $f(x,y)\in \mathbb{C}^{\infty}$ a complex-valued function in a neighborhood of $(0,0)$ satisfying:

$$ \text{Im} f \geq 0 \quad \text{Im} f(0,0) =0 \quad f'_x(0,0) = 0 \quad \text{det}f'_{xx}(0,0) \neq 0$$

I also have $u(0,0)=0$. However, $u$ is not differentiable at $(0,0)$. This does not allow me to apply stationary phase theorems from Hormander (*The analysis of linear partial differential operators, v.1*). Is there an asymptotic ($\lambda \rightarrow +\infty$) bound on this integral?

**Edit**
I also checked that $u$ has the form of inner product on Hilbert space:

$$ u(x,y) = \langle g(y), P_{(0,\infty)}(\text{sgn}(x) A) h(y) \rangle $$

with $P_{(0,\infty)}$ spectral projector onto positive eigenvalues, $\text{sgn}(x)$ sign function, $A$ linear self-adjoint operator. The spectrum of $\text{sgn}(x) A$ is discrete.