Boundary modulus of continuity and quasiconformal mappings

Abstract

Let D be a bounded domain in R-n, n >= 2, and let f be a continuous mapping of (D) over bar into R-n which is quasiconformal in D. Suppose that vertical bar f(x) - f(y)vertical bar LT = omega(vertical bar x-y vertical bar) for all x and y in partial derivative D, where omega is a non-negative non-decreasing function satisfying omega(2t) LT = 2w(t) for t >= 0. We prove, with an additional growth condition on omega, that vertical bar f(x) - f(y)vertical bar LT = C max{omega(vertical bar x - y vertical bar), vertical bar x - y vertical bar(alpha)} for all x, y is an element of D, where alpha = K-1(f)(1/(1-n)).