A note on the sublinear Sobolev inequality

TL;DR

This paper demonstrates that the classical Sobolev inequality does not hold in its unrestricted form for exponents p in the interval (0,1), clarifying the limitations of the inequality.

Contribution

It establishes a fundamental limitation of the Sobolev inequality for certain exponents, clarifying its scope and applicability.

Findings

Sobolev inequality fails for p in (0,1)

Clarifies the boundary of classical inequality validity

Provides insight into Sobolev space limitations

Abstract

In this note we show that the classical Sobolev inequality cannot in unrestricted form hold for exponents $p \in (0,1)$.