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We continue the study of Carleman-Sobolev classes from previous joint workwith G. Behm. We consider spaces denoted by $W_\mathcal{M}^p$, defined asabstract completions of sets of smooth functions with respect to a weightedSobolev-flavoured norm involving derivatives of all orders. Previously weshowed that these classes behaves very differently on two sides of a conditionon the weight sequence $\mathcal{M}$. Here we prove a conjecture made in thatpaper; under some regularity assumptions on the weight, we show that on oneside of the condition there will be a complete independence betweenderivatives, expressed as $$ W_\mathcal{M}^p\cong L^p\oplus W_{\mathcal{M}_1}^p$$ where $\mathcal{M}_1$ is the shifted sequence. On the other side, we alreadyknow that one can embed $W_\mathcal{M}^p$ into $C^{\infty}(\mathbb{R})$. Thusthis is an instance of a kind of phase transition.

Independence of derivatives in Carleman-Sobolev Classes for exponents $0