Interior estimate on the space

Abstract. We will prove an interior estimate for the mappings on the space, which is generalization of the Heinz-Bernstein theorem for the plane. In other worlds we prove the theorem. Let s : Bn → R (s : Bn → Bm) be a continuous function from the closed unit ball Bn ⊂ R into the real line (closed unit ball in R) satisfying the conditions: (i) s is C on U, (ii) s is C in ∂B = Sn−1 and (iii) |∆s| ≤ a|∇s|2 + b on B for some constants a < ∞ (a < 1/2 respectively) and b < ∞. Then the function |∇s| is bounded on B. An application of these results yields the theorem: A q.c. harmonic mapping of the unit ball onto a domain with smooth boundary is Lipschitz continuous. 2000 Mathematics Subject Classification. Primary 30C55. Secondary 30C62.