Sobolev mappings and moduli inequalities on Carnot groups

In the article we study mappings of Carnot groups satisfy moduli inequalities. We prove that homeomorphisms satisfy the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem we prove that mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{\nu,\text{loc}}(\Omega;\Omega')$ belong to the Sobolev class $W^1_{1,\text{loc}}(\Omega';\Omega)$.