Quasianalyticity Criterion of Salinas-Korenblyum Type for Convex Domains

Gaysin, R. A.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 3.

Abstract:
The quasianalyticity problem of the class \(C_{I}(M_n)\) for interval \(I\) is known to be solved by the Denjoy-Carleman theorem. It follows from well-known Men'shov example that not only this theorem but the very statement of the quasianalyticity problem of the class \(C_{K}(M_n)\) doesn't expand on the case of arbitrary continuum \(K\) of the complex plain. The quasianalyticity problem was studied for Jordan domains and rectifiable arcs including quasismooth arcs by a number of authors. We discuss in this article theorems of Denjoy-Carleman type in the convex domains of the complex plane, more precisely, connection between R. S. Yulmukhametov criterion of quasianalyticity of the Carleman class \(H(D,M_n)\) for arbitrary convex domain \(D\) and R. Salinas criterion for the class \(H(\Delta_{\alpha},M_n)\) with angle \(\Delta_{\alpha}=\{z: |\arg z|\leq\frac{\pi}{2}\alpha,\ \ 0<\alpha\leq1\}\). The problem of quasianalyticity of the class \(H(D,M_n)\) is to find necessary and sufficient conditions for sequence \(M_n\) and point \(z_0\in\partial D\) for quasianalyticity of the class \(H(D,M_n)\) at this point. The answer to question of simultaneous quasianalyticity or nonquasianalyticity these Carleman classes at a point \(z=0\) has been obtained in therms of special integral condition which characterizes the degree of proximity of the domain boundaries \(D\) and the angle \(\Delta_{\alpha}\) in the neighbourhood of origin. Geometric interpretation of this integral condition and explicit examples illustrating essentiality of this condition are given.

Keywords: Carleman class, convex domain, Salinas criterion, integral condition of local aboutness of the boundaries

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