A Schwarz lemma for hyperbolic harmonic mappings in the unit ball


  • Jiaolong Chen
  • David Kalaj




Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).


Axler, S., Bourdon, P. and Ramey, W., Harmonic function theory, Springer-Verlag, New York, 1992. https://doi.org/10.1007/b97238

Burgeth, B., A Schwarz lemma for harmonic and hyperbolic-harmonic functions in higher dimensions, Manuscripta Math. 77 (1992), no. 2–3, 283–291. https://doi.org/10.1007/BF02567058

Chen, J., Huang, M., Rasila, A. and Wang, X., On Lipschitz continuity of solutions of hyperbolic Poisson's equation, Calc. Var. Partial Differential Equations 57 (2018), no. 1, Paper No. 13, 32 pp. https://doi.org/10.1007/s00526-017-1290-x

Kalaj, D., Schwarz lemma for harmonic mappings in the unit ball, Complex Anal. Oper. Theory 12 (2018), no. 2, 545–554. https://doi.org/10.1007/s11785-017-0723-z

Kalaj, D. and Pavlović, M., On quasiconformal self-mappings of the unit disk satisfying Poisson equation, Trans. Amer. Math. Soc. 16 (2011), no. 8, 4043–4061. https://doi.org/10.1090/S0002-9947-2011-05081-6

Kresin, G. and Maz'ya, V., Sharp pointwise estimates for directional derivatives of harmonic functions in a multidimensional ball, J. Math. Sci. (N.Y.) 169 (2010), no. 2, 167–187. https://doi.org/10.1007/s10958-010-0045-4

Macintyre, A. and Rogosinski, W., Extremum problems in the theory of analytic functions, Acta Math. 82 (1950), 275–325. https://doi.org/10.1007/BF02398280

Stoll, M., Harmonic and subharmonic function theory on the hyperbolic ball, London Mathematical Society Lecture Note Series, 431, Cambridge University Press, Cambridge 2016. https://doi.org/10.1017/CBO9781316341063

Rainville, E., Special functions, The Macmillan Co., New York 1960.

Talvila, E., Necessary and sufficient conditions for differentiating under the integral sign, Amer. Math. Monthly 108 (2001), no. 7, 544–548. https://doi.org/10.2307/2695709

Zhu, K., Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York 2005.



How to Cite

Chen, J., & Kalaj, D. (2021). A Schwarz lemma for hyperbolic harmonic mappings in the unit ball. MATHEMATICA SCANDINAVICA, 127(3). https://doi.org/10.7146/math.scand.a-128528