Recent work showed that a theorem of Joris (that a function $f$ is smooth if two coprime powers of $f$ are smooth) is valid in a wide variety of ultradifferentiable classes $mathcal C$. The core of the proof was essentially $1$-dimensional. In certai
n cases a multidimensional version resulted from subtle reduction arguments, but general validity, notably in the quasianalytic setting, remained open. In this paper we give a uniform proof which works in all cases and dimensions. It yields the result even on infinite dimensional Banach spaces and convenient vector spaces. We also consider more general nonlinear conditions, namely general analytic germs $Phi$ instead of the powers, and characterize when $Phi circ f in mathcal C$ implies $f in mathcal C$.
We prove lifting theorems for complex representations $V$ of finite groups $G$. Let $sigma=(sigma_1,dots,sigma_n)$ be a minimal system of homogeneous basic invariants and let $d$ be their maximal degree. We prove that any continuous map $overline{f}
colon {mathbb R}^m to V$ such that $f = sigma circ overline{f}$ is of class $C^{d-1,1}$ is locally of Sobolev class $W^{1,p}$ for all $1 le p<d/(d-1)$. In the case $m=1$ there always exists a continuous choice $overline{f}$ for given $fcolon {mathbb R} to sigma(V) subseteq {mathbb C}^n$. We give uniform bounds for the $W^{1,p}$-norm of $overline{f}$ in terms of the $C^{d-1,1}$-norm of $f$. The result is optimal: in general a lifting $overline{f}$ cannot have a higher Sobolev regularity and it even might not have bounded variation if $f$ is in a larger Holder class.
We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under th
e action of the special orthogonal group by conjugation, possess some ultradifferentiable regularity if and only if their restriction to diagonal matrices has the same regularity.
We characterize the validity of the Whitney extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity. Specifically, we show that in the main Theorem 1.3 of [15] condition (1.3) can be dropped. Moreover, we clarify some questions that remained open in [15].
