From discrete flow of Beckner to continuous flow of Janson in complex hypercontractivity

TL;DR

This paper demonstrates how Beckner's monotonicity result on the Hamming cube can be used to establish the monotonicity of a flow introduced by Janson in the context of the Hausdorff--Young inequality, linking discrete and continuous hypercontractivity flows.

Contribution

It provides a new proof connecting Beckner's discrete flow to Janson's continuous flow, enriching the understanding of hypercontractivity.

Findings

Beckner's monotonicity implies Janson's flow monotonicity

Establishes a link between discrete and continuous hypercontractivity flows

Simplifies proof of monotonicity in Hausdorff--Young inequality

Abstract

We show how Beckner's montonicity result on Hamming cube easily implies the monotonicity of a flow introduced by Janson in Hausdorff--Young inequality