Convenient descriptions of weight functions in time-frequency analysis

Let $v$ be a submultiplicative weight. Then we prove that $v$ satisfies GRS-condition, if and only if $v\cdot e^{-\text{ep}\mathrm{\mid }\text{cdo}\mathrm{\mid }}$ is bounded for every positive $\text{ep}$. We use this equivalence to establish identification properties between weighted Lebesgue spaces, and between certain modulation spaces and Gelfand-Shilov spaces.