Logarithmic Asymptotics of a Class of Mappings

Abstract

The asymptotic behavior of lower Q-homeomorphisms relative to a p-modulus in ℝⁿ, n ≥ 2, at a point is studied. A number of logarithmic estimates for the lower limits under various conditions imposed on the function Q are obtained. Some applications of these results to the Orlicz–Sobolev classes \( {W}_{\mathrm{loc}}^{1,\varphi } \) in ℝⁿ, n ≥ 3 under the Calderon-type condition imposed on the function φ and, in particular, to the Sobolev classes \( {W}_{\mathrm{loc}}^{1,p} \) for p > n – 1 are given. The example of a homeomorphism with finite distortion which shows the exactness of the found order of growth is constructed.