# A supersolutions perspective on hypercontractivity

## TL;DR

This paper presents an algebraic approach using supersolutions to establish hypercontractivity inequalities for Markov semigroups with diffusion generators under curvature conditions.

## Contribution

It introduces a novel algebraic closure property of supersolutions that leads to hypercontractivity, applicable to a broad class of diffusion semigroups.

## Key findings

- Establishes a new algebraic closure property of supersolutions.
- Derives a monotone quantity that yields hypercontractivity inequalities.
- Applicable to Markov semigroups with diffusion generators satisfying curvature conditions.

## Abstract

The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature condition.