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I am trying to understand how estimates on sublevel integrals imply estimates on oscillatory integrals. Specifically in this article by M. Greenblatt it says on page 7:

By well-known methods relating sublevel integrals to oscillatory integrals, the above results about [some sublevel integral] have direct implications for [some oscillatory integral].

These methods are not known to me. Can anyone point me to a book or paper, which treats this in detail?

Thanks!

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The author supplies copious references -- Stein's book seems particularly relevant...

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