Extension of a function from the edge.

Question

How can I extend the function $f\in W^{1-\frac{1}{p},p}(\mathbb R\times\lbrace0\rbrace)$ up to $g\in W^{1,p}(\mathbb R\times(0,\infty))$?

Answer 1

By convolution with a smooth kernel. I sketch the construction in Theorem 15.21 in A first course in Sobolev spaces by Leoni.

Choose a smooth function $\varphi$ on $\mathbb R$ so that the support of $\varphi$ is contained in $(-1,1)$ and the integral of $\varphi$ is $1$. For $y>0$, define $$ g(x,y) =e^{-y/p} \frac{1}{y} \int_{\mathbb R} \varphi\left(\frac{x-t}{y}\right)f(t)\,dt $$ The factor $e^{-y/p}$ is used to make things integrable over large $y$. The rest is typical mollification, like in the solution of the diffusion equation but with compactly supported kernel (which is more convenient).

I do not reproduce the integral estimates for $g$ and its derivatives: they take four pages in the book. (Leoni extends from $\mathbb R^{N-1}$ to $\mathbb R^N_+$, but the proof does not simplify much when $N=2$).