# Existence of a smooth compactly supported function

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,

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.