Analytic implementation of the duals of some spaces of infinitely differentiable functions

Abstract

Using the Fourier-Laplace transform for functionals, we describe the duals of some spaces of the infinitely differentiable functions given on convex compact sets or convex domains in R N and such that the growth of their derivatives is determined by weight sequences of a general form.A weight function (or a weight) is an arbitrary nondecreasing function ϕ : [0, ∞) → R that is convex from some t 0 ≥ 0 on and enjoys the following conditions:(Denote the class of all weights by W.Let K be a compact set in R N whose nonempty interior int K has the closure coinciding with K. Given a weight ϕ, define the Banach spaceHere and in the sequel we use the following notations: C ∞ (K) is the set of all infinitely differentiable functions on a compact set K; N 0 := N ∪ {0}; f (α) is the derivative of f corresponding to a multiindex α ∈ N N 0 ; and ϕ * (s) := sup{rs − ϕ(r) : r ≥ 0} is the monotone conjugate function of ϕ. Denote by W ↑ (W ↓ ) the collection of all sequences Φ = {ϕ n } ∞ n=1 of weight functions such that, for each n ∈ N, there is C n satisfyingfor every t ≥ 0. Given Φ in W ↑ (W ↓ ) and a compact set K in R N , we naturally define the space

Solvability of convolution equations in the Beurling spaces of ultradifferentiable functions of mean type on an interval

“…Prove that condition (i 3 ) of Proposition 2 fails for the so-constructed family {f j : j ∈ N}. Note that the first inequality in (16) implies that f j ∈ H 1 (ω),I , j ∈ N. Further, by (13), as j → ∞, we have…”

“…We now consider the functions μf j , j ∈ N. Since μ ∈ M 1 (ω) , there exists C 3 ≥ 1 such that |μ(z)| ≤ C 3 exp ε 0 δ 0 2 ω(z) + πε 0 | Im z| for all z ∈ C. Then, using this estimate with the first inequality of (16) for |z − a j | ≥ 2R j , and the second inequality in (16) and (8) for |z − a j | < 2R j , we infer…”

On solutions of convolution equations in spaces of ultradifferentiable functions

“…This space possesses a natural topology determined by the collection of seminorms {| • | ω,q,l : q ∈ (0, 1), l ∈ (0, a)}, and it is an (F S)-space with this topology. As is well known (see [9,Theorem 1]), the Fourier-Laplace transformation of the functionals F : ϕ ∈ E 1 (ω) (I) → ϕ(z) := ϕ x (e −ixz ), z ∈ C, establishes a topological isomorphism between the strong dual E 1 (ω) (I) β to E 1 (ω) (I) and the following space of entire functions:…”

Solvability of inhomogeneous Cauchy–Riemann equation in spaces of functions with a system of uniform weight estimates

“…In particular, it is used for solving of problems on generators in spaces of entire functions [2,3], problems on separation of singularities of holomorphic functions [3], interpolation problems [4], Borel-Whitney extension [5], in studies of convolution equations and ideals in various spaces [6,7], conjugated spaces [8,9] and so on.…”

“…In inductive spaces we have to satisfy only one estimate from a family, and the H¨ormander result is sufficient to this end. The H¨ormander theorem is applied in that way in the cited above papers [5][6][7][8][9].…”