- #1
super_al57
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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
$$Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_\theta^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$$ 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