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 $\varphi^r$ has H\"older differentiability $rs$ whenever $\varphi \in \mathcal{F}^s$. 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 $\mathbb{R}^n$.