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• W3014166444 abstract "In this paper, we explore a sharp phase transition phenomenon which occurs for $$L^p$$ -Carleman classes with exponents $$0<p<1$$ . These classes are defined as for the standard Carleman classes, only the $$L^infty $$ -bounds are replaced by corresponding $$L^p$$ -bounds. We study the quasinorms $$begin{aligned} left||uright||_{p,mathcal {M}}=sup _{nge 0}frac{||u^{(n)}||_p}{M_n}, end{aligned}$$ for some weight sequence $$mathcal {M}={M_n}_n$$ of positive real numbers, and consider as the corresponding $$L^p$$ -Carleman space the completion of a given collection of smooth test functions. 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 $$M_n=1$$ for $$0le nle k$$ and $$M_n=+infty $$ for $$n>k$$ . This would give the $$L^p$$ -Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these $$L^p$$ -Sobolev spaces are highly degenerate for $$0<p<1$$ . Indeed, the canonical map $$W^{k,p}rightarrow L^p$$ fails to be injective, and there is even an isomorphism $$begin{aligned} W^{k,p}cong L^poplus L^poplus cdots oplus L^p, end{aligned}$$ corresponding to the canonical map $$fmapsto (f,f',ldots ,f^{(k)})$$ acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they “disconnect”). Here, we analyze this degeneracy for the more general $$L^p$$ -Carleman classes defined by a weight sequence $$mathcal {M}$$ . If $$mathcal {M}$$ has some regularity properties, and if the given collection of test functions is what we call $$(p,theta )$$ -tame, then we find that there is a sharp boundary, defined in terms of the weight $$mathcal {M}$$ : on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of $$C^infty $$ -smooth functions and the canonical map $$fmapsto (f,f',f'',ldots )$$ is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the $$L^p$$ setting, with $$0<p<1$$ ." @default.
• W3014166444 created "2020-04-10" @default.
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• W3014166444 date "2018-02-16" @default.
• W3014166444 modified "2023-10-17" @default.
• W3014166444 title "A critical topology for $$L^p$$ L p -Carleman classes with $$0<p<1$$ 0 < p < 1" @default.
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• W3014166444 doi "https://doi.org/10.1007/s00208-018-1654-3" @default.
• W3014166444 hasPublicationYear "2018" @default.
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