Smooth Whitney extension on compact sets

The empty-set case has the zero extension. Otherwise use the fixed Whitney cubes and partition ${\psi }_Q$ constructed in [F1]: outside $A$, their side lengths satisfy $\mathrm{\ell }(Q)\asymp \mathrm{dist}(Q,A)$, enlarged cubes have bounded overlap and comparable sizes when they meet, and $\mathrm{\mid }{D}^{\alpha }{\psi }_Q\mathrm{\mid }\le C_{\alpha }\mathrm{\ell }(Q{)}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$. These facts follow from the fixed maximal dyadic retention test and rescaled bump quotient in that proof, independently of the jet order. Put $\sigma (z)={\sum }_Q\mathrm{\ell }(Q){\psi }_Q(z)$. This is smooth and positive off $A$, and for $t=\mathrm{dist}(z,A)$ it satisfies $ct\le \sigma (z)\le Ct$ and $\mathrm{\mid }{D}^{\alpha }\sigma (z)\mathrm{\mid }\le C_{\alpha }{t}^{1-\mathrm{\mid }\alpha \mathrm{\mid }}$. Each derivative estimate follows by bounded overlap and the side-length comparison. Fix $\chi \in C^{\mathrm{\infty }}(\mathbb{R})$ with $\chi =1$ on $(-\mathrm{\infty },1]$ and $\chi =0$ on $[2,\mathrm{\infty })$. Then ${\eta }_{\delta }(z)=\chi (\sigma (z)\mathrm{/}\delta )$ equals one for $t\le \delta \mathrm{/}C$, vanishes for $t\ge 2\delta \mathrm{/}c$, and obeys $\mathrm{\mid }{D}^{\alpha }{\eta }_{\delta }\mathrm{\mid }\le B_{\alpha }{t}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ on its transition, with constants independent of $\delta$.

For each integer $k\ge 1$, take the actual extension $H_k$ produced by [F1]'s fixed cube formula for $P_a^k$. It is $C^k$ everywhere and smooth off $A$, since there it is a locally finite sum of smooth partition functions times polynomials. All these extensions are thereby specified without selecting elements of unspecified nonempty sets. Put ${\mathrm{\Delta }}_k=H_{k+1}-H_k$. It is $C^k$ and all its derivatives through order $k$ vanish on $A$. Taylor's formula at a nearest point of $A$, on a fixed compact neighborhood of $A$, gives for every $\mathrm{\mid }\beta \mathrm{\mid }\le k$ $\mathrm{\mid }{D}^{\beta }{\mathrm{\Delta }}_k(z)\mathrm{\mid }=o(t^{k-\mathrm{\mid }\beta \mathrm{\mid }})(t=\mathrm{dist}(z,A)\downarrow 0),$ uniformly near $A$. Indeed the top derivative is uniformly continuous and zero on $A$, and the lower estimates follow by Taylor's integral remainder along the segment from the nearest point. In the parameter-family case these remainders are uniform: the finite-order cube proof bounds $D^{\alpha }{H}_k-D^{\alpha }{P}_{a_z}^k$ by a constant times the specified Whitney remainder modulus; away from $A$ on a compact set its locally finite formula bounds every derivative by the coefficient bounds. Hence the $C^k$ moduli at $A$ used here are common to the family.

The product rule and step 1.1 show that, for $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, $D^{\alpha }({\eta }_{\delta }{\mathrm{\Delta }}_k)$ is bounded by a constant $B_k$ times ${\max }_{\mathrm{\mid }\beta \mathrm{\mid }\le k}{t}^{\mathrm{\mid }\beta \mathrm{\mid }-\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{\mid }{D}^{\beta }{\mathrm{\Delta }}_k\mathrm{\mid }$, wherever ${\eta }_{\delta }$ is nonzero. Choose ${\delta }_k$ as the largest dyadic number $2^{-j}$ satisfying ${\delta }_k\le 2^{-k}$, ${\delta }_k\le {\delta }_{k-1}\mathrm{/}2$ when $k>1$, and the finitely many uniform inequalities $B_k\mathrm{\mid }{D}^{\beta }{\mathrm{\Delta }}_k(z)\mathrm{\mid }\le 2^{-k}{t}^{k-\mathrm{\mid }\beta \mathrm{\mid }}(0<t\le 2{\delta }_k\mathrm{/}c,\mathrm{\mid }\beta \mathrm{\mid }\le k)\mathrm{.}$ The set of admissible integers $j$ is nonempty by step 2.1 and has a least element, so these scales are deterministic. Enlarge $B_k$ to include the finitely many binomial constants. Then $\mathrm{\mid }{D}^{\alpha }({\eta }_{{\delta }_k}{\mathrm{\Delta }}_k)(z)\mathrm{\mid }\le 2^{-k}{t}^{k-\mathrm{\mid }\alpha \mathrm{\mid }}(\mathrm{\mid }\alpha \mathrm{\mid }\le k)\mathrm{.}$ Uniform moduli in step 2.1 give the same scales for the parameter family.

Define off $A$ $F=H_1+{\sum }_{k\ge 1}{\eta }_{{\delta }_k}(H_{k+1}-H_k),$ and set $F(a)=f_0(a)$ on $A$. At each point off $A$ the sum is locally finite because ${\delta }_k\to 0$, so $F$ is smooth there. Fix $K\ge 1$. Sufficiently near $A$, all factors ${\eta }_{{\delta }_k}$ with $k<K$ equal one, and telescoping gives $F-H_K={\sum }_{k\ge K}{\eta }_{{\delta }_k}{\mathrm{\Delta }}_k$. For $\mathrm{\mid }\alpha \mathrm{\mid }\le K$, the single $k=K$ term is $o(t^{K-\mathrm{\mid }\alpha \mathrm{\mid }})$ by step 2.1 and the product rule, while the sum over $k>K$ is $O(t^{K+1-\mathrm{\mid }\alpha \mathrm{\mid }})$ by step 3.1 and a geometric series, taking $t<1\mathrm{/}2$. Thus every derivative through order $K$ of this difference off $A$ is $o(t^{K-\mathrm{\mid }\alpha \mathrm{\mid }})$. Extend the difference by zero on $A$. These estimates make its order-$K$ derivatives continuous and zero there; for lower orders the first-order difference quotient at any $a\in A$ tends to zero since $t\le \mathrm{\mid }z-a\mathrm{\mid }$, so induction verifies that the continuous derivative extensions are actual derivatives. Therefore $F-H_K$ is $C^K$ with zero $K$-jet on $A$. Since $H_K$ has the prescribed $K$-jet and $K$ is arbitrary, $F$ is smooth and realizes the entire field. The same estimates are uniform for the parameter family near $A$, and local finiteness gives uniform bounds on every compact set away from $A$. This proves the uniform assertion and completes the choice-free construction.