where $S\subset \mathbf{R}^n$ is the unit sphere centered

Reading around the context of the notes one can see that

It seems that should be a ball in .

How can one make sense of the canonical measure , which is originally defined for the sphere , “evaluating” on the ball in , and thus make sense of the second inequality above?

]]>But the group as in Exercise 16 is not written out explicitly and later the following integral is used:

where is the surface measure for the sphere in .

It seems by the context that should be “a” unit sphere. But where should be the center of the sphere?

]]>(Wolff’s has been available online: http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf)

I was trying to give an argument directly showing that

which is “what the author is trying to do”. But I somehow ended up with needing the argument in Wolff’s notes.

If has compact support, then this is fine. But (as mentioned in the comment) only implies that is Schwartz. On the other hand, the tail

contains with *arbitrarily large* . So I was not convinced that

which would give the desired order in using the Schwartz norm in .

More generally, one does not need to halt one’s reading comprehension every time there is a line that you can’t quite justify. I wrote about these sorts of “compilation errors” and how to avoid them at https://plus.google.com/u/0/114134834346472219368/posts/TGjjJPUdJjk

]]>where the author discussed the nondegenerate critical point case, he made a quantitative argument that

“uniformly” in and and accordingly

How can one make sense of the “uniform” big-O in and ? One can see that the remainder term is dominated by

when is small (and being big). But how could it be uniformly true for *all* (and thus justify the later integral)?

One can use Mathematica to do the job of sketching. I plotted the function where

for various values of and .

For fixed (being large), one can see that the graph of the function looks very much like a summation of functions that have () disjoint supports. The bigger is, the larger distance between the “supports” (the place where the function takes non-negligible values) of the functions in the summand, which matches intuitively that

While one can indeed observe that those “supports” interfered with each other when is small, I don’t see how exactly small could “fail” the proof of

if one calculates directly the norm.

Could you give an example that has “unexpectedly small L^2 norm” you mentioned in the previous comment (https://terrytao.wordpress.com/2009/04/06/the-fourier-transform/#comment-509369)?

(It seems that as and I don’t observe any “cancellation”.)

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