  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 H\"ormander-type adapted to the Hermite setting on
spaces with non-positive smoothness; 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.
