On a microlocal version of Young’s product theorem

Abstract

AbstractA key result in distribution theory is Young’s product theorem which states that the product between two Hölder distributions $$u\in \mathcal {C}^\alpha (\mathbb {R}^d)$$ u ∈ C α ( R d ) and $$v\in \mathcal {C}^\beta (\mathbb {R}^d)$$ v ∈ C β ( R d ) can be unambiguously defined if $$\alpha +\beta >0$$ α + β > 0 . We revisit the problem of multiplying two Hölder 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 Hölder distributions even when $$\alpha +\beta \le 0$$ α + β ≤ 0 .