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

Orlicz Sobolev Inequalities and the Doubling Condition