Sobolev extension in a simple case

Abstract

Abstract In this paper, we establish the existence of a bounded, linear extension operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $T :{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 &lt; p &lt; 2 and E is a finite subset of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{2}$ contained in a line.

Locations

• Advanced Nonlinear Studies - View - PDF
• arXiv (Cornell University) - View - PDF