Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

Abstract

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the ∂-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.

Nonlinear Conditions for Ultradifferentiability

“…In Sect. 5 we prove a general characterization result by holomorphic approximation for E [M] (Theorem 5.3) which extends step (i); it builds on the description by almost analytic extension presented in our recent paper [9]. Then we execute a version of step (ii) under a quite minimal set of assumptions, see Lemma 6.1.…”

“…There is a slight mismatch between our notation (also used in [9]) and that of [30] (and [22]). We write M j = m j j!…”

“…It is equivalent to a regular weight matrix S (that means E [ω] = E [S] ) if and only if ω is equivalent to a concave weight function. In fact, a E [ω] -version of the almost analytic extension theorem 4.1 holds if and only if ω is equivalent to a concave weight function; see [9,Theorem 4.8]. The weight matrix S = {S x } x>0 has the property that for each x > 0 the sequence s x is log-convex and satisfies mg(s x , s 2x ) =: H < ∞ and thus h s x (t) ≤ h s 2x (Ht) 2 for all x, t > 0; see [25,Proposition 3].…”

Transfer operators for ultradifferentiable expanding maps of the circle

“…Then, when n is large enough, we may split the sum in (14) between the sum over 0 ≤ m < m 0 and the sum over m 0 ≤ m < n. The first sum is independent of n and can consequently be ignored since the right-hand side of ( 14) tends to +∞ when n tends to +∞, according to Lemma 3. To bound the second sum, recall (8) to see that…”

Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity

“…[15][16][17][18]). Especially, we have to mention [19]. At the end of this introduction we will briefly comment on the approach in this paper and our approach.…”