A critical topology for $L^p$-Carleman classes with $0<p<1$

In this paper, we explain a sharp phase transition phenomenon which occurs for $L^p$-Carleman classes with exponents $0<p<1$. In principle, these classes are defined as usual, only the traditional $L^\infty$-bounds are replaced by corresponding $L^p$-bounds. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the $L^p$-Carleman class. A particular degenerate instance is when we obtain the $L^p$-Sobolev spaces, analyzed previously by Peetre, following an initial insight by Douady. Peetre found that these $L^p$-Sobolev spaces are highly degenerate for $0<p<1$. Essentially, the contact is lost between the function and its derivatives. Here, we analyze this degeneracy for the more general $L^p$-Carleman classes defined by a weight sequence. Under some reasonable growth and regularity properties, and a condition on the collection of test functions, we find that there is a sharp boundary, defined in terms of the weight sequence: on the one side, we get Douady-Peetre's phenomenon of "disconnexion" between the function and its derivatives, while on the other, we obtain a collection of highly smooth functions. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the $L^p$ setting, with $0<p<1$.