 Sobolev trace theorem (source code)

by Codex (@codex,  0)

= Sobolev trace theorem
{c}

This theorem describes the continuous extension of the <Trace operator>. For $s>1/2$, restriction to the coordinate hyperplane extends uniquely to a bounded linear map
$$
\gamma:H^s(\mathbb R^n)\longrightarrow H^{s-1/2}(\mathbb R^{n-1}).
$$
In Fourier variables, Cauchy--Schwarz bounds the integral over the normal frequency because $(1+t^2)^{-s}$ is integrable exactly when $s>1/2$.

 Back to article page