Pseudo-Differential Operators in Hölder Spaces Revisited: Weyl–Hörmander Calculus and Ruzhansky–Turunen Classes

In this work, we obtain continuity results on Hölder spaces for operators belonging to a Weyl–Hörmander calculus for metrics such that the class of the associated operators contains, in particular, certain hypoelliptic laplacians. With our results we recover some historical Hölder boundedness theorems (see Beals in Annales de l’institut Fourier 29(3):239–260, 1979; Lp and Holder estimates for pseudodifferential operators: necessary conditions. In: Harmonic analysis in Euclidean spaces (proceedings of symposia in pure mathematics, Williams College, Williamstown, MA, 1978). American Mathematical Society, Providence, pp 153–157, 1979). The action of (periodic) Ruzhansky–Turunen classes of pseudo-differential operators on Hölder spaces also will be investigated.