Carleman-Sobolev classes for small exponents, pre-print

Abstract. This paper is devoted to the study of a generalization of Sobolev spaces for small Lp 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 involving derivatives of all orders, simultaneously inspired by Carleman classes and classical Sobolev spaces. If the class is re-stricted with a growth condition on the supremum norms of the derivatives, we prove that there exists a condition on the weight sequence which guarantees that the resulting space can be embedded into C∞(R). This condition is nec-essary for such an embedding to be possible, 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 Lp into the resulting completion. The purpose of this paper is to introduce a class of functions, simultaneously inspired by Sobolev spaces on the real line and so-called Carleman classes from the study of quasi-analytic functions. This shall be done for small Lp-exponents, tha