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Analytical foundations of the theory of quasiconformal mappings in \(R^ n\). (English) Zbl 0548.30016
The basic method in the paper is F. W. Gehring’s higher \(L^ p\)- integrability theory which is developed in a unified manner, especially a comprehensive study of various maximal functions is presented. The method is applied to the following problems for quasiregular (qr) mappings: Hölder continuity and differentiability, discreteness, composition, zeros of the Jacobian, and \(m(B_ f)=0=m(fB_ f)\) where \(B_ f\) is the branch set of f; for the original proofs see Yu. G. Reshetnyak [Sib. Mat. Zh. 8, 629-658 (1967; Zbl 0162.381)] and O. Martio, S. Rickman and J. Väisälä [Ann. Acad. Sci. Fenn., Ser. A I Math. 448, 40 p. (1969; Zbl 0189.092)]. Basic properties of solutions for a nonlinear Dirichlet problem \(\nabla\cdot A(x,\nabla u)=0\) are established (Harnack’s inequality, capacity estimates) and it is shown that if f is qr then l\(n| f|\) is a solution of an equation associated with f; for similar results see Yu. G. Reshetnyak [Sib. Mat. Zh. 9, 652-666 (1968; Zbl 0162.382)] and S. Granlund, P. Lindqvist and the reviewer [Trans. Am. Math. Soc. 277, 43-73 (1983; Zbl 0518.30024)].
Reviewer: O.Martio

30C62 Quasiconformal mappings in the complex plane
35J60 Nonlinear elliptic equations
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