Estimation of intermediate derivatives and a Bang-type theorem. I

Abstract

Certain estimates for intermediate derivatives on a quasismooth arc are proved and applied. For arcs of bounded slope, the corresponding results by Bang and Leont ev are generalized.

Regularization of sequences in sense of E. M. Dyn'kin

“…It was shown in [5] that if { } is a regular sequence, then class ( ) is quasi-analytic if and only if the associated weight ℎ introduced by (2) satisfies condition (1). Thus, in the present case, Denjoy-Carleman theorem remains true for ( ).…”

“…In studying Carleman classes ( ) on arbitrary continuums of the complex plane, a special role is played by regular in some sense sequences { }; many statements are proven exactly for such sequences. It is happened that if is an arc of a bounded slope, as in the case of the segment a = [0, 1], in the Bang type theorems, sequence of numbers > 0 can be arbitrary [1].…”

Noncomplete systems of exponentials on arcs and Carleman nonquasianalytic classes. II

“…γ , integrating 2n times by parts (this is legitimate because γ is rectifiable, see [18]), we obtain…”

“…We show that, in fact, ϕ ∈ C γ ( M n−2 ). For this, we use the estimates for intermediate derivatives of ϕ on γ established in [18]. We state the result of [18] in a convenient form.…”

“…where N = c(3d + |γ| + 1), d is the diameter of γ, |γ| is its length, and c = c(γ) ≥ 1 is the constant occurring in the definition of quasismoothness in [18]: for every pair of points z, ξ ∈ γ we have γ zξ ≤ c|z − ξ| (γ zξ is the part of γ between z and ξ, and |γ zξ | is the length of this γ zξ ).…”