The Hypercontractivity theorem (or Bonami Beckner inequality) is a very useful tool. Unfortunately, it isn't easy to carry over to other spaces than the uniform boolean cube.

In Ryan O'Donnel's Analysis of Boolean Functions, he mentions the following theorem which considers the more general problem of general product spaces:

enter image description here The "normal" Bonami Beckner however also has a two-function version:

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Do you know of anything similar for the case of general product spaces, or $p$-biased functions in particular?


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