Roots, trace, and extendability of flat nonnegative smooth functions

Building on the univariate techniques developed by Ray and Schmidt-Hieber, we study the class ${\mathcal{F}}^s({\mathbb{R}}^n)$ of multivariate nonnegative smooth functions that are sufficiently flat near their zeroes, which guarantees that ${\phi }^r$ has Hölder differentiability $rs$ whenever $\phi \in {\mathcal{F}}^s$. We then construct a continuous Whitney extension map that recovers an ${\mathcal{F}}^s$ function from prescribed jets. Finally, we prove a Brudnyi-Shvartsman Finiteness Principle for the class ${\mathcal{F}}^s$, thereby providing a necessary and sufficient condition for a nonnegative function defined on an arbitrary subset of ${\mathbb{R}}^n$ to be ${\mathcal{F}}^s$-extendable to all of ${\mathbb{R}}^n$.