Flat factors absorb polynomial derivative growth

Statement

Suppose $h\in C^\infty((-\varepsilon,\varepsilon))$ is flat at zero and $F\in C^\infty(U\times(0,\varepsilon))$ . Assume that, for each compact $K\subset U$ and each mixed derivative, some constants $C,N\ge0$ satisfy

\sup_{x\in K}|\partial_x^\alpha\partial_t^kF(x,t)|\le C t^{-N}

for sufficiently small positive $t$ . Then $h(t)F(x,t)$ , extended by zero for $t\le0$ , is smooth and flat on $t=0$ .

Proof

Taylor's theorem and flatness give $|h^{(j)}(t)|\le C_{j,M}t^M$ for every fixed $j,M$ . In the Leibniz formula for any derivative of $hF$ , choose $M$ larger than the finitely many growth exponents that occur. Each term then tends to zero locally uniformly in $x$ . Apply the zero-extension criterion.

Scope of the hypothesis

The exponent may depend on the derivative order and on $K$ . No conclusion holds for unrestricted growth: a factor $e^{1/t^2}$ cancels $e^{-1/t^2}$ exactly and leaves a nonvanishing product.