A bounded linear extension operator for $L^{2,p}(\mathbb{R}^2)$

Abstract

Let $L^{2,p}(\mathbb{R}^2)$ be the Sobolev space of real-valued functions on the plane whose Hessian belongs to $L^p$. For any finite subset $E \subset \Bbb{R}^2$ and $p>2$, let $L^{2,p}(\Bbb{R}^2)|_E$ be the space of real-valued functions on $E$, equipped with the trace seminorm. In this paper we construct a bounded linear extension operator $T : L^{2,p}(\mathbb{R}^2)|_E \rightarrow L^{2,p}(\mathbb{R}^2)$. We also provide an explicit formula that approximates the $L^{2,p}(\mathbb{R}^2)|_E$ trace seminorm.