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Moduli of continuity of harmonic quasiregular mappings in \(\mathbb{B}^n\). (English) Zbl 1219.30008

If \(f\) is analytic in the unit disk \(D\) and continuous in its closure, then the modulus of continuity of \(f\) in \(D\) is the same as the modulus of continuity of \(f|_{\partial D}\) [L. A. Rubel, A. L. Shields and B. A. Taylor, J. Approx. Theory 15, 23–40 (1975; Zbl 0313.30036)]. This result fails for harmonic functions in the case of Lipschitz continuity, see e.g. [H. Aikawa, Bull. Lond. Math. Soc. 34, No. 6, 691–702 (2002; Zbl 1036.31003)], although it holds for the Hölder modulus of continuity.
In the plane, the situation for harmonic quasiregular mappings, see [S. Rickman, Quasiregular mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 26. Berlin: Springer (1993; Zbl 0816.30017)], is the same as for analytic functions, see [V. Kojić and M. Pavlović, J. Math. Anal. Appl. 342, No. 1, 742–746 (2008; Zbl 1162.31002)].
In this paper, the authors extend this result to \(\mathbb{R}^n\), \(n \geq 3\). As in the aforementioned paper, the proof is based on the subharmonicity of \(|f|^q\), \(q = q(n, K)\), where \(f\) is the \(K\)-quasiregular harmonic (Poisson) extension of a continuous mapping \(f\) on the boundary of the unit ball.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
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