Carleman-Sobolev classes for small exponents

This paper is devoted to the study of a generalization of Sobolev spaces for small $L^{p}$ exponents, i.e. $0<p<1$. We consider spaces defined as abstract completions of certain classes of smooth functions with respect to weighted quasi-norms, simultaneously inspired by Carleman classes and classical Sobolev spaces. If the class is restricted with a growth condition on the supremum norms of the derivatives, we prove that there exists a condition which guarantees that the resulting space can be embedded into $C^{\infty}(\mathbb{R})$. This is sharp up to some regularity conditions on the weight sequence. We also show that the growth condition is necessary, in the sense that if we drop it entirely we can naturally embed $L^p$ into the resulting completion.