Fourier Integral Operator

1. Oct 30, 2014

super_al57

Hi everybody! I'm studying the Fourier integral operators but I can't resolve a pass. I'm considering the following operator:
$$Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_\theta^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$$ where $$Au\in C^0 (\mathbb{R}^m)$$. I know that $$Au\in C^0 (\mathbb{R}^m)$$ is well defined as oscillating integral if I use the pseudodifferential operator $$L=\frac{1}{1+\mid\nabla_{y,\theta}\Psi\mid^2}(1+h\nabla_y\bar{\Psi}D_y+h\nabla_{\theta}\bar{\Psi}D_{\theta})$$. I have to demonstrate, using integration by parts, that $$L=\mathcal{O}(<\theta>^{-k})$$.
Could anyone help me? Thanks

2. Nov 4, 2014

Greg Bernhardt

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 8, 2014

super_al57

:L I have no idea about how to begin. And I have no further information.