Special functions of quasiconformal theory (Q1262404)

Two basic functions of the theory of quasiconformal mappings in the n- space (n\(\geq 2)\), the capacities of the Grötzsch and Teichmüller rings, are discussed in this paper. Several functional inequalities of these special functions are given for all dimensions \(n\geq 2\), even though in the higher-dimensional case \(n\geq 3\) neither explicit expressions nor functional identities are known. The authors develop a systematic theory of these special functions in the n-space, which contains several new results even for \(n=2\). A list of open questions is given at the end of the paper.