Reflection extension from a half-space

For $U=(0,\infty )\times {\mathbb{R}}^{n-1}$, the formula $Eu(x_1,x^{\prime })=u(|x_1|,x^{\prime })$ defines a bounded Sobolev extension operator. Its weak normal derivative changes sign across the boundary, while its tangential weak derivatives are reflected unchanged.

The odd reflection of a function on a half-space is $Eu(x^{\prime },x_n)=\mathrm{sgn}(x_n)u(x^{\prime },|x_n|)$. When $u$ has zero trace on the boundary, this extension preserves the relevant weak regularity and often converts a homogeneous boundary problem into an interior problem.