Positive powers of flat weighted profiles

Statement

Let $w>0$ be a flat interval weight in an open interval and let $\delta$ denote its positive boundary margin. Suppose a smooth positive $y$ , possibly with additional compact parameters, satisfies

y\ge c w,\qquad |\partial^\alpha y|\le C_\alpha w\delta^{-N_\alpha}

for every fixed multi-index, including zero. For every fixed $r>0$ , the function $y^r$ extends smoothly by zero across the interval endpoints.

Derivative estimate

Repeated chain and product rules express each derivative as a finite sum of terms

c\,y^{r-j}\prod_{\nu=1}^j\partial^{\alpha_\nu}y, \qquad |\alpha_\nu|\ge1.

The upper bound on $y$ controls positive powers; the lower bound controls negative powers. Each term is bounded by $Cw^r\delta^{-N}$ for some finite $N$ . A positive power of the flat weight beats every such inverse margin power, so all mixed derivatives tend to zero at the boundary. The zero-extension criterion applies. In particular $r=1/2$ justifies square roots under these quantitative hypotheses.