Applying Integration by Parts and Eikonal Equation to Fourier Integral Operators

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super_al57
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Hi! I have a question for you. At the end of the post there's a link. There's the homework which I have to do for an exam. I have to study the Fourier Integral Operator that there is at the begin of the paper. I did almost all the homework but I can't do a couple of things. First: at the point 1, using k-times the integration by parts, I have to prove that |eiψ/l(tL)kau|=O(<θ>m-δN).
Second: at the end I have to estimate [itex]\partial^\alpha\varphi[/itex] derivating the eikonal equation but I don't know how to begin.

http://freepdfhosting.com/08103fe65b.pdf
 
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To answer your questions, I suggest you look at the chapter on Fourier Integral Operators in your textbook and review the methods of integration by parts and deriving the eikonal equation. For the first question, you can use integration by parts to find an expression for |eiψ/l(tL)kau| that is equal to or less than O(<θ>m-δN). And for the second question, you can use the method of differentiation to find an expression for \partial^\alpha\varphi.