On a Microlocal Version of Young's Product Theorem

TL;DR

This paper extends Young's product theorem for H"older distributions by employing microlocal analysis and Sobolev wavefront sets to define products even when the sum of regularities is non-positive.

Contribution

It introduces microlocal techniques to generalize the conditions under which H"older distributions can be multiplied, surpassing classical limitations.

Findings

Established new sufficient conditions for distribution multiplication

Extended the product theorem to cases where +eta0

Applied Sobolev wavefront set techniques to distribution theory

Abstract

A key result in distribution theory is Young's product theorem which states that the product between two H\"older distributions $u\in\mathcal{C}^\alpha(\mathbb{R}^d)$ and $v\in\mathcal{C}^\beta(\mathbb{R}^d)$ can be unambiguously defined if $\alpha+\beta>0$. We revisit the problem of multiplying two H\"older distributions from the viewpoint of microlocal analysis, using techniques proper of Sobolev wavefront set. This allows us to establish sufficient conditions which allow the multiplication of two H\"older distributions even when $\alpha+\beta\leq 0$.