A One Dimensional Proof of the H\"older Inequality using the Heat Equation

TL;DR

This paper presents a novel proof of the H"older Inequality leveraging supersolutions of the Heat Equation and a monotonicity formula, offering a new perspective on classical analysis tools.

Contribution

It introduces a heat equation-based proof of the H"older Inequality using supersolutions and monotonicity formulas, connecting PDE techniques with fundamental inequalities.

Findings

Proof of H"older Inequality via heat equation supersolutions

Application of monotonicity formulas in inequality proofs

New PDE-based approach to classical analysis results

Abstract

This paper gives a proof of the H\"older Inequality by using supersolutions of the Heat Equation. The proof is based on a monotonicity formula for the heat equation presented in Tobias Colding's lectures at MIT.