Sorry, I never mean to put words in your mouth. Since I was asking why people would be interesting in these kinds of estimates, I was thinking you were implying something more. It's one thing to say knowing the spectrum alone isn't enough, but that didn't tell me why projecting onto these restricted Eigenspaces would be of interest. Thank you, again, for your time.

And if this indeed the case, then can you explain briefly/broadly why know this kind of information about the spectrum tells you that kind of information?

Thank you for taking the time to post these insights; they are rather interesting. However I'm not sure if I understand some of the bigger implications. Are you saying that studying the spectrum of the manifold (as a whole) is insufficient? But that by looking at the spectrum's distribution amongst higher frequencies (spread out among compact intervals) we can glean more important topological/geometric information about the manifold?

Thanks, again!!

2) Assuming we have clarified the issue with $\mathbf{J}(x,\lambda)$ above, lets call $J_k(x, \lambda)$ the $k$-th component of the vector. Then I can see how stationary phase around $(x,z) = (0,0)$ gives us the $(|x_n| \lambda)^{(n-1)/2}$ behavior, but this would only be asymptotic behavior in each coordinate. Moreover, every $J_k$ would satisfy the same big-oh bounds involving the absolute value $|x_n|$ only; so how do I put these all together to get a uniform bound involving the norm, $|x|$?

1) By bringing the integral inside of the dot product, you are effectively making $\mathbf{J}(x,\lambda)$ a vector-valued function, correct? But isn't your cutoff function, $a(z)$ a scalar-valued function? Should I just be interpreting $a(z)$ as one scalar component of $\mathbf{J}(x,\lambda)$? If not, then wouldn't this seem to contradict the definition for $\mathbf{J}(x,\lambda)$ you had earlier?

I'm still reading through this carefully, but thank you! I have a couple small clarification questions, though, so if you don't mind I'll just enumerate them here in a series of separate comments:

That's quite vague, and not very enlightening. Can you possibly elaborate?