Abstract
In this paper, we study Hölder continuity of (p, q)-harmonic functions defined on the unit disc \({{\mathbb {D}}}\) as the Poisson type integral \(u=K_{p,q}[f]\) of a \(\beta \)-Hölder function \(f\in \Lambda _\beta ({{\mathbb {T}}})\) on the unit circle \({{\mathbb {T}}}\). Mainly, we consider three cases, when \(p+q>\beta -1\), we show that \(u\in \Lambda _\beta ({{\mathbb {D}}})\), whereas in the case \(p+q<\beta -1\), we prove that \(u\in \Lambda _{p+q+1}({{\mathbb {D}}})\), and when \(p+q=\beta -1\), we show that \(u\in \bigcap _{0<\alpha <\beta }\Lambda _\alpha ({{\mathbb {D}}})\). Finally, we show the stability of the exponents of f and u in their corresponding Lipschitz spaces under the condition u is K-quasiconformal.
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We would like to thank the referee for insightful comments which led to significant improvements in the paper.
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Communicated by H. Turgay Kaptanoglu.
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This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Andreas Seeger, Franz Luef and Serap Oztop.
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Khalfallah, A., Mhamdi, M. Hölder Continuity of Generalized Harmonic Functions in the Unit Disc. Complex Anal. Oper. Theory 16, 101 (2022). https://doi.org/10.1007/s11785-022-01279-8
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DOI: https://doi.org/10.1007/s11785-022-01279-8