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A critical topology...
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Hedenmalm, Håkan,1961-KTH,Matematik (Avd.)
(author)
A critical topology for L^p Carleman classes with 0
- Article/chapterEnglish2018
Publisher, publication year, extent ...
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2018-02-16
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Springer Science and Business Media LLC,2018
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electronicrdacarrier
Numbers
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LIBRIS-ID:oai:DiVA.org:kth-228086
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https://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-228086URI
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https://doi.org/10.1007/s00208-018-1654-3DOI
Supplementary language notes
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Language:English
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Summary in:English
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Classification
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Subject category:ref swepub-contenttype
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Subject category:art swepub-publicationtype
Notes
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QC 20180518
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In this paper, we explore a sharp phase transition phenomenon which occurs for (Formula presented.)-Carleman classes with exponents (Formula presented.). These classes are defined as for the standard Carleman classes, only the (Formula presented.)-bounds are replaced by corresponding (Formula presented.)-bounds. We study the quasinorms (Formula presented.)for some weight sequence (Formula presented.) of positive real numbers, and consider as the corresponding (Formula presented.)-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 (Formula presented.)-Carleman class. A particular degenerate instance is when (Formula presented.) for (Formula presented.) and (Formula presented.) for (Formula presented.). This would give the (Formula presented.)-Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these (Formula presented.)-Sobolev spaces are highly degenerate for (Formula presented.). Indeed, the canonical map (Formula presented.) fails to be injective, and there is even an isomorphism (Formula presented.)corresponding to the canonical map (Formula presented.) 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 (Formula presented.)-Carleman classes defined by a weight sequence (Formula presented.). If (Formula presented.) has some regularity properties, and if the given collection of test functions is what we call (Formula presented.)-tame, then we find that there is a sharp boundary, defined in terms of the weight (Formula presented.): on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of (Formula presented.)-smooth functions and the canonical map (Formula presented.) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the (Formula presented.) setting, with (Formula presented.).
Subject headings and genre
Added entries (persons, corporate bodies, meetings, titles ...)
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Wennman, AronKTH,Matematik (Avd.)(Swepub:kth)u1m7nuri
(author)
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KTHMatematik (Avd.)
(creator_code:org_t)
Related titles
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In:Mathematische Annalen: Springer Science and Business Media LLC371:3-4, s. 1803-18440025-58311432-1807
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