Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

Abstract

We obtain new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel–Lizorkin spaces and use such tools to prove boundedness properties of Hermite pseudo-multipliers on those spaces. The notion of molecule we develop leads to boundedness of pseudo-multipliers associated to symbols of Hormander-type adapted to the Hermite setting on spaces for which the smoothness allowed includes non-positive values; in particular, we obtain continuity results for such operators on Lebesgue and Hermite local Hardy spaces. As a byproduct of our results on boundedness properties of pseudo-multipliers, we show that Hermite Besov spaces and Hermite Triebel–Lizorkin spaces are closed under non-linearities.