Quasianalytic ultradifferentiability cannot be tested in lower dimensions

Rainer, Armin

doi:10.36045/bbms/1576206353

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Cited by 4 publications

(1 citation statement)

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“…In fact, quasianalytic ultradifferentiability cannot be tested on quasianalytic curves (or lower dimensional plots) even if the function in question is known to be smooth ( [10,20]). Hence, we think that it is interesting that, combining our proof with a description of certain quasianalytic classes E {M} as an intersection of suitable non-quasianalytic ones (due to [16]), we obtain that these quasianalytic classes have property (D) in all dimensions (see Theorem 2.7 and also Remarks 3.3 and 4.4).…”

Section: Resultsmentioning

confidence: 99%

Nonlinear Conditions for Ultradifferentiability

2021

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A remarkable theorem of Joris states that a function f is $$C^\infty $$ C ∞ if two relatively prime powers of f are $$C^\infty $$ C ∞ . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

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Section: Resultsmentioning

confidence: 99%