Remarks on Sobolev norms of fractional orders

TL;DR

This paper investigates how Sobolev norms of fractional order functions supported on subsets compare to their norms on larger domains, providing inequalities and counterexamples to clarify misconceptions.

Contribution

It establishes new inequalities and clarifies misconceptions about Sobolev norms of fractional order functions supported on subsets.

Findings

Derived inequalities relating norms on subsets and larger domains

Disproved common misconceptions with counterexamples

Identified dependence of constants on domain properties

Abstract

When a function belonging to a fractional-order Sobolev space is supported in a proper subset of the Lipschitz domain on which the Sobolev space is defined, how is its Sobolev norm as a function on the smaller set compared to its norm on the whole domain? On what do the comparison constants depend on? Do different norms behave differently? This article addresses these issues. We prove some inequalities and disprove some misconceptions by counter-examples.