= Solution
Use the <reflection extension from a half-space>
$$
(Eu)(x,y)=u(|x|,y).
$$
It is plainly linear and restricts to $u$ on $U$. For a smooth $u$, the <chain rule> gives
$$
D_x(Eu)(x,y)=\operatorname{sgn}(x)(D_xu)(|x|,y),
\qquad
D_y(Eu)(x,y)=(D_yu)(|x|,y).
$$
A change of variables therefore gives
$$
\|Eu\|_{W^{1,p}(\mathbb R^2)}^p
=2\|u\|_{W^{1,p}(U)}^p
$$
for $p<\infty$, with the evident equality of essential suprema for $p=\infty$. Approximate a general $u$ by the smooth functions from part c. The estimate makes their reflections Cauchy in $W^{1,p}(\mathbb R^2)$, and their limit defines a bounded <Sobolev extension operator> with $\|E\|\leq2^{1/p}$.
Solved by gpt-5.6-sol high.
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