On inversion of absolutely convergent weighted Dirichlet series in two variables

Abstract

Let $0<p\leq 1$, and let $\omega:\mathbb N^2 \to [1,\infty)$ 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 $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^p\omega(m,n)<\infty$. If $\widetilde a$ is bounded away from zero on $\mathbb H^2$, then there is an almost monotone weight $\nu$ on $\mathbb N^2$ such that $1\leq \nu\leq \omega$, $\nu$ is constant if and only if $\omega$ is constant, $\nu$ is admissible if and only if $\omega$ 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 $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^p\nu(m,n)<\infty$. If $\varphi$ is holomorphic on a neighbourhood of the closure of range of $\widetilde a$, then there is an almost monotone weight $\xi$ on $\mathbb N^2$ such that $1\leq \xi\leq \omega$, $\xi$ is constant if and only if $\omega$ is constant, $\xi$ is admissible if and only if $\omega$ is admissible, $\varphi \circ \widetilde a$ has the Dirichlet series representation $(\varphi\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\;((s_1,s_2)\in \mathbb H^2)$ and $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^p\xi(m,n)<\infty$. Let $\omega$ be an admissible weight on $\mathbb N^2$, and let $\widetilde a$ have $p$-th power $\omega$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has $p$-th power $\omega$- absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero.