Nonlinear Conditions for Ultradifferentiability

Abstract

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.

Nonlinear conditions for ultradifferentiability: a uniform approach

“…In the recent paper [15] Thilliez showed that this result carries over to Denjoy-Carleman classes of Roumieu type and, by refining Thilliez's method, we proved in [11] that it is valid in a wide variety of ultradifferentiable classes. See the introduction in [11] for more on the historical development.…”

“…A function f is smooth provided that two relatively prime powers or, equivalently, two consecutive powers of f are smooth, by a theorem of Joris [4]. In the recent paper [15] Thilliez showed that this result carries over to Denjoy-Carleman classes of Roumieu type and, by refining Thilliez's method, we proved in [11] that it is valid in a wide variety of ultradifferentiable classes. See the introduction in [11] for more on the historical development.…”

“…Consequently, general validity of the result in all dimensions remained open in cases, most notably the quasianalytic case, where the suitable reduction tools are not available. See the table in the introduction of [11] for a summary of the known results (in that paper the result in question was called property (D)).…”

“…Remark. In the terminology of [11] These are the minimal requirements needed for the tools used in our proof; see also Remark 4.3 in which the necessity of these assumptions is discussed. Note that [admissibility] of the weight matrix M neither implies nor obstructs quasianalyticity of the corresponding class E [M] .…”

“…As another special case we obtain the following result for Braun-Meise-Taylor classes E [ω] . For their definition and why it is a special case of Theorem 1.1 we refer the reader to [11,Section 4.5] and the references therein. By a weight function we mean a continuous increasing function ω…”