A Bounded Linear Extension Operator for $L^{2,p}(\R^2)$

TL;DR

This paper introduces a new linear extension operator for Sobolev spaces $L^{2,p}( ^2)$, solving the Whitney extension problem by defining a novel consistency notion among local extensions, with potential generalizations to higher dimensions.

Contribution

It presents a bounded linear extension operator for $L^{2,p}( ^2)$ that minimizes the semi-norm, introducing a new consistency concept for local Sobolev extensions.

Findings

Constructed a linear extension operator with near-minimal semi-norm

Developed a new notion of consistency for local Sobolev extensions

Method generalizes to higher dimensions and smoothness levels

Abstract

For a finite $E \subset \R^2$, $f:E \rightarrow \R$, and $p>2$, we produce a continuous $F:\R^2 \rightarrow \R$ depending linearly on $f$, taking the same values as $f$ on $E$, and with $L^{2,p}(\R^2)$ semi-norm minimal up to a factor $C=C(p)$. This solves the Whitney extension problem for the Sobolev space $L^{2,p}(\R^2)$. A standard method for solving extension problems is to find a collection of local extensions, each defined on a small square, which if chosen to be mutually consistent can be patched together to form a global extension defined on the entire plane. For Sobolev spaces the standard form of consistency is not applicable due to the (generically) non-local structure of the trace norm. In this paper, we define a new notion of consistency among local Sobolev extensions and apply it toward constructing a bounded linear extension operator. Our methods generalize to produce similar results for the $n$-dimensional case, and may be applicable toward understanding higher smoothness Sobolev extension problems.