The Properties Of ω-Quasianalitic Functions And Weight Functions

In the sixties of last century, some notions of the ultradifferentiable functions and ultradistributions have been introduced by A.Beurling, G.Bjock, and H,komatsu etc using the weight functions. In 1980s, these ideas have been expanded to theω-ultradifferentiable functions and theω-ultradistributions by Bonet, Meise, Taylor etc. The construction and the properties of these spaces are decided by the weight func-tion. According to the theorem of Denjoy-Carleman, theseω-ultradifferentiable func-tions and w-ultradistributions are divided into two groups:the quasianalitic and the nonquasianalitic class. The w-ultradifferentiable functions and w-ultradistributions of nonquasianalitic class have been investigated systematically by Bonet, Braun, Meise, Taylor and Vogt etc since the eighties of last century, and based on them many signif-icance results in the theory of linear partial differential operators have been obtained. For w-ultradifferentiable functions and w-quasianalytic founctionals of quasianalitic class, they also have made some study in last years.In this paper, we will further discuss the properties ofω-ultradifferentiable func-tions and w-quasianalytic founctionals of quasianalitic class and the weight functions be related based on the works mentioned above and using some methods in the theory ofω-nonquasianalitic class. We get the following results:Theorem 1 Letωbc a weight function and assume that g:[0,∞)â†'[0,∞) satisfies g(t)= o(ω(t)) as t tends to oo.Then there exists a weight funtionσwith the following properties:(1) g(t)= o(σ(t)), as tâ†'∞(2)σ(t)= o(ω(t)), as tâ†'∞:(3) for every A> 1:limsupιâ†'∞σ(At)/σ(t)< limsupιâ†'∞ω(At)/ω(t).If in addition there is R≥1 such thatω｜[R,∞) is concave,then it is possible to makeσ｜[R,∞)concave, too.Theorem 2 Letωbe a weight function and letΩbe a convex open set in Rn. Then for each u∈ε'{ω}(Ω), there exist a weight funtionσsatisfyingσ(t)= o(ω(t)) as t tends to oo and U∈ε'(σ), such that u= U｜ε{ω}(Ω). Thereforewhere W0(ω)={σ｜σis a weight function,σ=0(ω)}.