Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

Abstract

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending the previous work by Langenbruch. As a consequence, we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.

The Borel map in the mixed Beurling setting

“…In Sect. 4, we identify the duals ( (M) ) and E (N) (R) with certain weighted spaces of entire functions. This allows us to carry out the solution by dualization (2) in Sect.…”

“…In the short appendix, we prove a technical statement needed in Sect. 4, namely that the entire functions are dense in an auxiliary function space. Since the inclusion of the entire functions is continuous, also the polynomials are dense.…”

Global pseudodifferential operators in spaces of ultradifferentiable functions

“…Since s ≥ nC 1 , the series depending on η in (1. 16) is convergent as it is proved in (1.14). Both n and s depend on λ > 0, therefore the estimate (1.15) holds.…”

“…The space S ω (R d ) is nuclear for every weight function ω. See, for instance, Boiti, Jornet, Oliaro, and Schindl [16,17].…”

“…Hence, we have 16), then, by (0.1), e mω(v+tτ w,v−t(1−τ )w,ξ) ≤ e mLω(2(|v|+(|1−τ |+|τ |)|w|)) e mLω(ξ) e mL ≤ e mL 2 ω(|v|+2 k |w|) e mLω(ξ) e mL 2 +mL ≤ e mL 3 ω(v) e mL k+3 ω(w) e mLω(ξ) e mL k+3 +•••+mL .…”