Title: A critical topology for $L^p$-Carleman classes with $0<p<1$,
Abstract: 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 $0<p<1$.