Solutions of quasianalytic equations

Silva, André Belotto da; Biborski, Iwo; Bierstone, Edward

doi:10.1007/s00029-017-0345-3

Cited by 8 publications

(15 citation statements)

References 21 publications

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“…Theorem 1.4 and Corollary 1.5 following will be proved in § 5. Our proof of Theorem 1.4 follows the reasoning used in [BdSBB17, Lemmas 3.1 and 3.4], which give more precise estimates in the special cases that is a power substitution or a blowing-up. See § 2 for the notation used in the theorem.…”

Section: Introductionmentioning

confidence: 91%

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Composite quasianalytic functions

Silva¹,

Bierstone²,

Chow³

2018

Compositio Math.

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We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function $f(x)$ in a quasianalytic Denjoy–Carleman class ${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping $y=\unicode[STIX]{x1D711}(x)$ of class ${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of $\unicode[STIX]{x1D711}$ can be written locally as $f=g\circ \unicode[STIX]{x1D711}$, where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\mathcal{Q}}_{M^{(p)}}$; (2) a statement on a similar loss of regularity for functions definable in the $o$-minimal structure given by expansion of the real field by restricted functions of quasianalytic class ${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a ${\mathcal{C}}^{\infty }$ solution $g$ of the equation $f=g\circ \unicode[STIX]{x1D711}$, with $f$ and $\unicode[STIX]{x1D711}$ as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

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Section: Introductionmentioning

confidence: 91%

“…The following theorem is a further development of quasianalytic continuation techniques introduced in [BdSBB17, § 4].…”

Section: Introductionmentioning

confidence: 99%

Composite quasianalytic functions

Silva¹,

Bierstone²,

Chow³

2018

Compositio Math.

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“…The converse of this statement is due to Mandelbrojt . In a Denjoy–Carleman class

Q_{M}

, closure under differentiation is equivalent to the axiom (1) of closure under division by a coordinate — the converse of Remark (1) is a simple consequence of the fundamental theorem of calculus; see [, § 3.1].…”

Section: Quasianalytic Classesmentioning

confidence: 99%

“…It seems plausible that, even under the latter assumption, a quasianalytic solution

g_{1}, \dots, g_{q}

may necessarily involve a certain loss of regularity (that is, belong to a larger quasianalytic class

Q^{'} \supseteq Q

) depending on

{normalΦ}_{1}, \dots, {normalΦ}_{q}

and

f

; cf. .…”

Section: Introductionmentioning

confidence: 99%

Malgrange division by quasianalytic functions

Bierstone

Milman

2017

J. London Math. Soc.

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Quasianalytic classes are classes of infinitely differentiable functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of infinitely differentiable functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of infinitely differentiable by real-analytic functions.Comment: 18 pages, 1 figur

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“…This follows immediately from the case n = 1 due to [22]; in this reference only the case ⋆ = {M } was treated, but the arguments apply to all cases. It seems to be unknown whether a similar result holds for E ⋆ and n > 1, but see [1].…”

Section: Introductionmentioning

confidence: 97%