Sobolev spaces on multiple cones

TL;DR

This paper investigates the properties of Sobolev spaces on multiple cones, focusing on density, interpolation, and extension, using Poincaré and Hardy inequalities to analyze their behavior.

Contribution

It provides new insights into Sobolev spaces on cones, especially regarding smooth function density and extension properties, combining inequalities in novel ways.

Findings

Density of smooth functions varies on multiple cones

Interpolation properties are characterized for these Sobolev spaces

Extension and restriction behaviors are analyzed using Hardy inequalities

Abstract

The purpose of this note is to discuss how various Sobolev spaces defined on multiple cones behave with respect to density of smooth functions, interpolation and extension/restriction to/from $\RR^n$. The analysis interestingly combines use of Poincar\'e inequalities and of some Hardy type inequalities.