关于二元绝对收敛加权 Dirichlet 级数的反演
On inversion of absolutely convergent weighted Dirichlet series in two variables
AI总结：
本文研究了二元绝对收敛加权 Dirichlet 级数的反演问题，证明了在几乎单调权下，级数倒数及全纯复合仍具有绝对收敛的 Dirichlet 表示，并给出了倒数收敛的充要条件。
摘要：
设 $0<p\leq 1$，且 $\omega:\mathbb N^2 \to [1,\infty)$ 为一几乎单调权。设 $\mathbb H$ 为复平面中的闭右半平面。设 $\widetilde a$ 为 $\mathbb H^2$ 上的复值函数，满足对所有 $(s_1,s_2)\in \mathbb H^2$，$\widetilde a(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$，且 $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^p\omega(m,n)<\infty$。若 $\widetilde a$ 在 $\mathbb H^2$ 上有下界（远离零），则存在 $\mathbb N^2$ 上的几乎单调权 $\nu$，使得 $1\leq \nu\leq \omega$，$\nu$ 为常数当且仅当 $\omega$ 为常数，$\nu$ 是容许的当且仅当 $\omega$ 是容许的，其倒数 $\frac{1}{\widetilde a}$ 具有 Dirichlet 表示 $\frac{1}{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$（对所有 $(s_1,s_2)\in \mathbb H^2$），且 $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^p\nu(m,n)<\infty$。若 $\varphi$ 在 $\widetilde a$ 值域的闭包邻域内全纯，则存在 $\mathbb N^2$ 上的几乎单调权 $\xi$，使得 $1\leq \xi\leq \omega$，$\xi$ 为常数当且仅当 $\omega$ 为常数，$\xi$ 是容许的当且仅当 $\omega$ 是容许的，$\varphi\circ \widetilde a$ 具有 Dirichlet 级数表示 $(\varphi\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}$（对所有 $(s_1,s_2)\in \mathbb H^2$），且 $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^p\xi(m,n)<\infty$。设 $\omega$ 为 $\mathbb N^2$ 上的容许权，且 $\widetilde a$ 具有 $p$ 次幂 $\omega$-绝对收敛 Dirichlet 级数。则证明了 $\widetilde a$ 的倒数具有 $p$ 次幂 $\omega$-绝对收敛 Dirichlet 级数当且仅当 $\widetilde a$ 有下界（远离零）。
英文摘要：
Let $0<p\leq 1$, and let $ω:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that $\widetilde a(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ with $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^pω(m,n)<\infty$. If $\widetilde a$ is bounded away from zero on $\mathbb H^2$, then there is an almost monotone weight $ν$ on $\mathbb N^2$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $ν$ is admissible if and only if $ω$ is admissible, the reciprocal $\frac{1}{\widetilde a}$ has the Dirichlet representation $\frac{1}{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ and $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^pν(m,n)<\infty$. If $φ$ is holomorphic on a neighbourhood of the closure of range of $\widetilde a$, then there is an almost monotone weight $ξ$ on $\mathbb N^2$ such that $1\leq ξ\leq ω$, $ξ$ is constant if and only if $ω$ is constant, $ξ$ is admissible if and only if $ω$ is admissible, $φ\circ \widetilde a$ has the Dirichlet series representation $(φ\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\;((s_1,s_2)\in \mathbb H^2)$ and $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^pξ(m,n)<\infty$. Let $ω$ be an admissible weight on $\mathbb N^2$, and let $\widetilde a$ have $p$-th power $ω$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has $p$-th power $ω$- absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero.
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