Orlicz Sobolev Inequalities and the Doubling Condition

TL;DR

This paper demonstrates that very weak Orlicz-Sobolev inequalities still imply the doubling condition on measures, and introduces a 'superradius' concept to characterize when spaces support these inequalities without being doubling.

Contribution

It extends the understanding of Sobolev inequalities by showing weaker Orlicz versions imply doubling and introduces the superradius as a key parameter.

Findings

Weaker Orlicz-Sobolev inequalities imply doubling.

The superradius condition determines support for inequalities without doubling.

The paper characterizes the relationship between inequalities and measure doubling.

Abstract

In [12] it has been shown that $(p,q)$ Sobolev inequality with $p>q$ implies the doubling condition on the underlying measure. We show that even weaker Orlicz-Sobolev inequalities, where the gain on the left-hand side is smaller than any power bump, imply doubling. Moreover, we derive a condition on the quantity that should replace the radius on the righ-hand side (which we call `superradius'), that is necessary to ensure that the space can support the Orlicz-Sobolev inequality and simultaneously be non-doubling.