On a Microlocal Version of Young's Product Theorem

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 $\alpha+\beta\leq 0$.