A support theorem for quasianalytic functionals

For a weight function ω and an open set G in ℝ N denote by ℰ( ω )( G ) (resp. E { ω } ( G )) the ω ‐ultradifferentiable functions of Beurling (resp. Roumieu) type on G . Using ideas of Hörmander it is shown that the functionals u in ℰ′( ω )( G ) and ℰ′ { ω } ( G ) can be embedded into the realanalytic functionals on ℝ N and that there is a smallest supporting set for u in the corresponding class which coincides with the realanalytic (hyperfunction) support of u . Moreover, if ω is quasianalytic and if a compact subset K of G is the union of the compact sets K 1 and K 2 then each u ∈ ℰ′ { ω } ( G ) which is supported by K can be decomposed as u = u 1 + u 2 , where u j is supported by K j for j = 1, 2. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)