Closed principal ideals in nonradial Hörmander algebras
Let \(p\) denote a nonradial weight function, that is \[ p(z)= u(\text{Re }z)+ v(\text{Im }z)\qquad (z\in\mathbb{C}), \] where \(u\), \(v\) are continuous nonnegative even functions on \(\mathbb{R}\) which are increasing on \(\mathbb{R}_ += \{x\in \mathbb{R}\mid x\geq 0\}\). The function \(v\) is assumed to be convex. Moreover, the following inequality should hold: \[ \log(1+ | z|)= O(p(z)),\;p(2z)= O(p(z))\quad (| z|\to\infty),\quad u(t)= o(v(t))\quad (t\to\infty). \] (For example: \(p(z)= |\text{Re }z|^ \alpha+ |\text{Im } z|^ \beta\), where \(\beta\geq 1\) and \(\beta> \alpha>0\).) The author considers principal ideals in the Hörmander algebra \(A_ p\) of all entire functions \(f\) with \(| f|_ k:= \sup_{z\in\mathbb{C}} | f(z)|\exp(- kp(z))\to\infty\) for some \(k\in \mathbb{N}\) endowed with the natural inductive limit topology. The main result characterizes the closed principal ideals. Theorem: The following assertions are equivalent for a given \(F\in A_ p\backslash\{0\}\): (i) The principal ideal \(FA_ p\) is closed in \(A_ p\). (ii) For each \(g\in A_ p\) such that \(f:= g/F\) is entire, we have \(f\in A_ p\). (iii) There are \(k\in \mathbb{N}\) and \(x_ 0> 0\) such that for each \(x\in \mathbb{R}\) with \(| x|\geq x_ 0\) there is \(t\in \mathbb{R}\) with \[ | t-x|\leq kv^{-1}(u(x))\quad\text{and}\quad | F(t)|\geq \exp(- ku(t)). \] The proof follows the work of \textit{R. Meise}, \textit{B. A. Taylor} and \textit{D. Vogt} [Ind. Univ. Math. J. 36, 729-756 (1987; Zbl 0637.46037)] and of \textit{L. Ehrenpreis} [Am. J. Math. 82, 522-588 (1960; Zbl 0098.084)].
principal ideals in the Hörmander algebra
scientific article; zbMATH DE number 62255
scientific article