Existence of a smooth compactly supported function

Question

Let $U$ be a bounded domain in $\mathbb R^n$. Does there exist a smooth function $f$ with compact support in $U$ such that: $$ \| f\|_{W^{k,\infty}(U)} \leq (k!)^{2-\epsilon},$$ for some $\epsilon>0$?

Thanks,

Answer 1

The answer is yes if $\epsilon<1$, and no when $\epsilon\geq 1$. This follows from Carleman's quasianalyticity criterion, see for example, Hormander, Analysis of linear partial differential operators, Vol. I, Chap I, Section 1, Theorem 1.3.8.

(Carleman's original proof used Complex Analysis, and it was reproduced in the books on the subject. Hormander has an elementary proof, not using Complex Analysis. As far as I know, this proof was first published in the book I refer to).