In this project, we aim to make progress on two surjectivity problems for weighted spaces of smooth functions and distributions satisfying global decay estimates. On the one hand, we aspire to prove a Whitney-type extension theorem for weighted spaces of smooth functions on unbounded closed sets and to investigate the existence of continuous linear extension operators in this setting. On the other hand, we deal with the problem of division by a polynomial in the space of rapidly decreasing distributions. We intend to study both of these problems by making use of methods from modern abstract functional analysis, namely the theory of the derived projective limit functor and the splitting theory for Fréchet and (PLS)-spaces.