Quasianalytical classes of functions in convex domains

R. S. Yulmukhametov

Sbornik. Mathematics, Tome 58 (1987) no. 2, pp. 505-523

Résumé

Let $D$ be a bounded convex domain lying in the left-hand half-plane, with $0\in\overline D$. A class $H(D,M_n)$, consisting of functions analytic in $D$ and satisfying the inequalities $$ \max_{z\in D}|f^{(n)}(z)|\leqslant C_fM_n,\qquad n=0,1,\dots, $$ is said to be quasianalytic at $z=0$ if $H(D,M_n)$ contains no functions that vanish with all their derivatives at $z=0$. Let $h(\varphi)=\max_{\lambda\in D}\operatorname{Re}\lambda e^{i\varphi}$ and $h(\varphi)=0$, $\varphi\in[\sigma_-,\sigma_+]$, and let \begin{gather*} \Delta_+(\alpha)=\sqrt{\alpha-\sigma_+}\biggl(h'(\alpha)+\int^\alpha_{\sigma_+}h(\theta)\,d\theta\biggr),\qquad\sigma_+\alpha\frac\pi2, \\ \Delta_-(\alpha)=-\sqrt{\sigma_--\alpha}\biggl(h'(\alpha)+\int_{\sigma_-}^\alpha h(\theta)\,d\theta\biggr),\qquad-\frac\pi2\alpha\sigma_-, \\ v_1(x)=\exp\int_{x_1}^x\frac{2\pi-\Delta_+^{-1}(y)+\Delta_-^{-1}(y)}{-\pi+\Delta_+^{-1}(y)-\Delta_-^{-1}(y)}\cdot\frac{dy}y,\qquad x\to0,\quad x_1>0. \end{gather*} It is shown that the condition $$ \int_1^\infty\frac{\ln T(r)}{v(r)\cdot r^2}\,dr=+\infty, $$ where $T(r)=\sup r^nM_n^{-1}$ is the trace function of the sequence $(M_n)$, and $v(r)$ is the inverse of $v_1(x)$, is necessary and sufficient for the quasianalyticity of $H(D,M_n)$. This theorem generalizes the classical Denjoy–Carleman theorem. In the case when $D=\bigl\{z:|\arg z|\frac\pi{2\gamma}\bigr\}$ the theorem follows from Salinas's results of 1955. For $D=\{z:|z+1|=1\}$ the theorem was proved by Korenblyum in 1965. Bibliography: 9 titles.