I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight Hausdorff-Young's inequality and the Brascamp-Lieb inequality. However I can't find any more detail on this. Anyone has a reference for such applications of the induction on scale argument?
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1Ha! I'd be interested in a proof of Cauchy-Schwarz (no t!) using induction on scales. – mathworker21 Jan 30 '23 at 07:10
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1You probably already saw the page from Terry Tao's blog under the tag "induction on scale"...but if not you should a look: https://terrytao.wordpress.com/tag/induction-on-scales/ – Alessandro Della Corte Jan 30 '23 at 08:26
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Related: Basic examples of induction on scales arguments, and What's a nice argument that shows the volume of the unit ball in $\mathbb{R}^n$ approaches 0? – Timothy Chow Jan 31 '23 at 13:46
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I think Guth's proof of the (non-endpoint) multilinear kakeya inequality is a great example of an easier induction on scales argument (without a lot of other tricks or tools involved) if you're looking for an example that introduces the idea. You might also reach out to J. Bennett and ask if has materials from the course he gave with the abstract you linked to. – Mark Lewko Jan 31 '23 at 22:57