On removable sets for quasiconformal mappings

Abstract

we call E removable for QCH if every f in QCH(E) is quasiconformal in R n. In R 2, the following theorems ([C], [G], [K]) are known: T h e o r e m A. (Carleson, Gehring) If S C R 1 is compact, then S• [0, 1] is removable for QCH if and only if S is countable. T h e o r e m B. (Kaufman) If S C R 1 is uncountable, then S• [0, 1] contains a graph E which is not removable for QCH. In R '~, n>2, Cantor type sets are removable ([HK]): T h e o r e m C. (Heinonen and Koskela) Let E be a closed set in R% Suppose that there exist a > l , and {rj}, rj--+O as j--*c~, such that at each x E E , the annular regions { y : a l r j m > 2 and S is an {ak}-porous set in R m with ~ a k = C ~ , then S • n-m is removable for QCH in R n. I f S is closed in R 1, then S • n-1 is removable for QCH in R n if and only if S is countable. Given {ak}, 0 < a k < l , a set SC_R n is called {ak}-porous if there exists a sequence of coverings gk -={Bk,j ~B(xk , j , rk,j)} of S, by balls with mutually disjoint interiors, so that each Bk,j S contains a ring Rk,j =-{ x: (1 -(~k ) rk,j < IX-Xk,j I < rk,j },