Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

In this talk, we discuss Sobolev embeddings into rearrangement-invariant function spaces on (regular) domains in $\mathbb{R}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius, called $d$-Frostman measures. We show that these embeddings can be deduced from one-dimensional inequalities for an operator depending on $n$, $d$ and the order $m$ of the Sobolev space. We also point out an interesting feature of this theory - namely that the results take a substantially different form depending on whether the measure is decaying fast ($d\geq n-m$) or slowly ($d<n-m$). This is a
joint work with Andrea Cianchi and Lubos Pick.