Localization of ideals and asymptotic uniqueness theorems for functions with restrictions on growth

S. A. Apresyan

Sbornik. Mathematics, Tome 34 (1978) no. 5, pp. 561-592

Résumé

Let $\mathbf D=\{z\in\mathbf C:|z|<1\}$; let $U_\varphi(\mathbf D)$ be the set of all functions $u$, subharmonic in $\mathbf D$, for which $u(z)<C_u\varphi(1/(1-|z|))$; and let $A_\varphi(\mathbf D)$ be the algebra of all functions $f$, analytic in $\mathbf D$, for which $\log|f(z)|<C_f\varphi(1/(1-|z|))$. We prove the following theorems subject to known restrictions on the regularity of growth of the function $\varphi$. Theorem 1. If $\gamma$ is a continuous curve in $\mathbf D$ reaching out to the circle $\partial\mathbf D$ (i.e., $\gamma\cap\partial\mathbf D\ne\varnothing$), and if $$ \varlimsup_{z\in\gamma,|z|\to1}\frac{u(z)}{\varphi^*(1/(1-|z|))}=-\infty, $$ then $u\equiv-\infty$. Here, $\varphi^*(t)=t\bigl(\int_1^t(\varphi(x)/x^3)^{1/2}\,dx\bigr)^2$ for $a_\varphi\leqslant1$, $\varphi^*=\varphi$ for $1<a_\varphi\leqslant+\infty$; and $a_\varphi=\lim_{x\to\infty}\varphi'(x)x/\varphi(x)$. Theorem 2. {\it In order that every closed ideal of the algebra $A_\varphi(\mathbf D)$ be a divisor ideal, it is necessary and sufficient that the condition $\int_1^\infty(\varphi(x)/x^3)^{1/2}\,dx=+\infty$ be satisfied.} Here, we say that an ideal $I$ is a divisor ideal when $I=\{f\in A_\varphi(\mathbf D):k_f\geqslant k_I\}$, where $k_f(\xi)$ is the multiplicity of a zero of the function $f$ at the point $\xi$ and $k_I(\xi)=\min_{f\in I}k_f(\xi)$. Figures: 5. Bibliography: 33 titles.

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