I am trying to understand how estimates on sublevel integrals imply estimates on oscillatory integrals. Specifically in [this](http://arxiv.org/abs/0804.1579) 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!



Oscillatory integral decay & sublevel set growth
The author supplies copious references -- Stein's book seems particularly relevant...


