Composite quasianalytic functions

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.

Pathological Phenomena in Denjoy–Carleman Classes

Let M denote a Denjoy–Carleman class of ∞ functions (for a given logarithmically-convex sequence M = (Mn)). We construct: (1) a function in M((−1, 1)) that is nowhere in any smaller class; (2) a function on ℝ that is formally M at every point, but not in M (ℝ); (3) (under the assumption of quasianalyticity) a smooth function on ℝp (p ≥ 2) that is M on every M curve, but not inM (ℝp).

About a conjecture on difference equations in quasianalytic Carleman classes

We consider the difference equation ?q j=1 aj (x)?(x+?j) = ?(x) where ?1 < ??? < ?q (q ? 3) are given real constants, aj(j=1,...,q) are given holomorphic functions on a strip R? (? > 0) such that a1 and aq vanish nowhere on it, and ? is a function belonging to a quasianalytic Carleman class CM{R}. We prove, under a growth condition on the functions aj, that the difference equation above is solvable in CM{R}.

A quasianalytic singular spectrum with respect to the Denjoy-Carleman class

Making use of the FBI (Fourier-Bros-Iagolnitzer) transforms we simplify the quasianalytic singular spectrum for the Fourier hyperfunctions, which was defined for distributions by Hörmander as follows; for any Fourier hyperfunction u, (x0, ξ0) does not belong to the quasianalytic singular spectrum W FM(u) if and only if there exist positive constants C, γ and N, and a neighborhood of x0 and a conic neighborhood Г of ξ0 such thatfor all x ∈ U, |ξ| ∈ Γ and |ξ| ≥ N, where M(t) is the associated function of the defining sequence Mp. This result simplifies Hörmander’s definition and unify the singular spectra for the C∞ class, the analytic class and the Denjoy-Carleman class, both quasianalytic and nonquasianalytic.