Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type

Quasianalytic functionals and projective descriptions

“…The cases (a) and (b) of Lemma 3.8 were treated in Meyer [15], Theorem 2.8, for a larger class of functions; see also Meise [12], Lemma 2.5. Since the case (c) does not appear there and since the proof is simpler in the special case needed here, we decided to prove the lemma for the convenience of the reader.…”

“…Since Proj 1 E {ω} (G) = {0} (this is proved by Meyer [15] for G = ]−1, 1[ or R, and by Rösner [16]) and since T µ is surjective, it follows from Vogt [18], …”

Analytic Solutions of Convolution Equations on Convex Sets with an Obstacle in the Boundary

“…We choose PD as above. Since PD admits a continuous linear right inverse, we see as above^but now applying Lemma 3.8 some step AQ m H , by the Laplace transformation 1.9, we obtain the following condition: If f j , j P N, is a sequence in A H and if there is n such that for each l sup zPC j "zf j zj expÀH n z À Lz À jzjalY j P NY is bounded, then there is some m such that also sup zPC jf j zj expÀH m z À jzjalY j P NY is bounded for each l P N. (See Meyer [18], Lemma 4.13, or [19], Lemma 3.12, for this kind of reasoning.) We will now argue by contradiction applying a well known procedure of Ehrenpreis.…”

Geometric characterization of interpolation in the space of Fourier–Laplace transforms of ultradistributions of Roumieu type

“…If the set of zeros is an interpolating variety then A {ω} / Im Mμ can be identified with the space of sequences with suitable growth, which is in turn isomorphic to the dual of the kernel of T μ . This idea lies behind results in [9,15,17,20,24], where sequential descriptions of kernels of convolution operators are given, surjectivity characterizations of convolution operators are obtained or existence of right inverses for convolution operators is established. This methodology was developed for instance in [19] (comp.…”