Local Sobolev space - Knowlpedia

For an open set $\Omega$ , integer $k\ge0$ , and $1\le p\le\infty$ , a function belongs to $W^{k,p}_{\mathrm{loc}}(\Omega)$ if its restriction belongs to the Sobolev space $W^{k,p}(U)$ for every open $U$ whose closure is a compact subset of $\Omega$ . Write $H^k_{\mathrm{loc}}=W^{k,2}_{\mathrm{loc}}$ .

Scope

Local membership imposes no uniform bound as the subdomains approach the boundary or fill an unbounded domain. For example, a smooth polynomial is in every local integer-order Sobolev space, but a nonzero polynomial is not in $L^2(\mathbb R^n)$ . Multiplying a locally Sobolev function by a test function gives a globally Sobolev function after zero extension.