In this paper we prove that if is a -quasiconformal map, with , and is a compact set contained in a ball , then
where stands for the analytic capacity and is a capacity associated to a nonlinear Riesz potential. As a consequence, if is not -removable (i.e., removable for bounded -quasiregular maps), it has positive capacity . This improves previous results that assert that must have non--finite Hausdorff measure of dimension . We also show that the indices , are sharp, and that Hausdorff gauge functions do not appropriately discriminate which sets are -removable. So essentially we solve the problem of finding sharp “metric” conditions for -removability.
Xavier Tolsa. Ignacio Uriarte-Tuero. "Quasiconformal maps, analytic capacity, and non linear potentials." Duke Math. J. 162 (8) 1503 - 1566, 1 June 2013. https://doi.org/10.1215/00127094-2208869