Open discrete mappings having local ACL n inverses

Abstract

We consider open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f , so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and . For this class of mappings (or even for larger classes of open, discrete mappings) we generalize the important modular inequality of Poleckii, also using the modular estimates of the spherical rings from Cristea (Local homeomorphism having local ACL n inverses, Complex Var. Elliptic Equ. 53(1) (2008), 77–99). We continue the work from the same paper by generalizing some basic facts from the theory of quasiregular mappings. We give equicontinuity results, Picard, Montel and Liouville type theorems, estimates of the modulus of continuity, analogues of Schwarz's lemma, eliminability results and boundary extensions theorems. Together with the multiple extensions of Zoric's theorem from the paper, we establish strong generalizations of one of the most important theorems from the theory of quasiregular mappings. We also extend similar results given in some recent classes of functions larger than the class of quasiregular functions, as the class of mappings of finite distortion and satisfying condition (𝒜), or the class of mappings of finite dilatation with dilatation in the bounded mean oscillation class.