Sharp Capacity Estimates and Volume Growth for Fractional Sobolev Spaces

We obtain sharp two-sided estimates for the relative capacity of fat and thin annuli in the fractional Sobolev spaces $W_{p,q}^s$, where $s\in (0,1)$, $p\in (0,\mathrm{\infty })$, $q\in (0,\mathrm{\infty }]$, in terms of an explicit volume growth quantity. The underlying space is a general metric measure space equipped with a doubling and reverse doubling measure. In particular, whenever the capacity is finite, the estimates are independent of the parameter $q$. For $p<q$, we also introduce a sharp condition on the underlying measure that is necessary for the existence of admissible functions in $W_{p,q}^s$ space.