A Bounded Linear Extension Operator for L^2,p(\R²)

2010/11/02 by Arie Israel, Israel, Arie
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1011.0689

46 pages

arxiv created 2010/11/05 · arxiv updated 2010/11/08

Abstract

For a finite E ⊂ \R², f:E → \R, and p>2, we produce a continuous F:\R² → \R depending linearly on f, taking the same values as f on E, and with L^2,p(\R²) semi-norm minimal up to a factor C=C(p). This solves the Whitney extension problem for the Sobolev space L^2,p(\R²). 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.

Related

• Sobolev extension in a simple case
• An example related to Whitney's extension problem for L^2,p(ℝ²) when 1
• Extension criteria for homogeneous Sobolev space of functions of one\n variable
• Sobolev functions on closed subsets of the real line: long version
• The existence of a bounded linear extension operator for L^s,p(ℝⁿ) when (n)/(p)<\s\