Akman, Murat and Lewis, John and Vogel, Andrew
(2022)
*On a Theorem of Wolff Revisited.*
Journal d'Analyse Mathematique, 146 (2).
pp. 487-530.
DOI https://doi.org/10.1007/s11854-022-0200-0
(In Press)

Akman, Murat and Lewis, John and Vogel, Andrew
(2022)
*On a Theorem of Wolff Revisited.*
Journal d'Analyse Mathematique, 146 (2).
pp. 487-530.
DOI https://doi.org/10.1007/s11854-022-0200-0
(In Press)

Akman, Murat and Lewis, John and Vogel, Andrew
(2022)
*On a Theorem of Wolff Revisited.*
Journal d'Analyse Mathematique, 146 (2).
pp. 487-530.
DOI https://doi.org/10.1007/s11854-022-0200-0
(In Press)

## Abstract

We study $p$-harmonic functions, $ 1 < p\neq 2 < \infty$, in $ \mathbb{R}^{2}_+ = \{ z = x + i y : y > 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic function, $ V = V ( r e^{i\theta} )$, which is $ 2\pi/N $ periodic in the $ \theta $ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with Lipschitz norm $\leq c N$ on $ \partial B ( 0, 1 )$ satisfying $V(0)=0$ and $ c^{-1} \leq \int_{-\pi}^{\pi} V ( e^{i\theta} ) d \theta \leq c$. In case $2<p<\infty $ we give a more or less explicit example of $V$ and our work is an extension of a result of Wolff on $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$. Using our first result, we extend the work of Wolff on failure of Fatou type theorems for $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$ for $p$-harmonic functions, $1< p\neq 2<\infty$. Finally, we also outline the modifications needed for extending the work of Llorente, Manfredi, and Wu regarding failure of subadditivity of $p$-harmonic measure on $ \partial \mathbb{R}^{2}_+ $ to $\partial B (0, 1)$.

Item Type: | Article |
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Additional Information: | 40 pages, 1 figure |

Uncontrolled Keywords: | math.AP; math.CA; 35J60, 31B15, 39B62, 52A40, 35J20, 52A20, 35J92 |

Divisions: | Faculty of Science and Health > Mathematical Sciences, Department of |

SWORD Depositor: | Unnamed user with email elements@essex.ac.uk |

Depositing User: | Unnamed user with email elements@essex.ac.uk |

Date Deposited: | 27 Feb 2022 20:26 |

Last Modified: | 03 Jun 2023 01:00 |

URI: | http://repository.essex.ac.uk/id/eprint/32420 |

## Available files

**Filename:** ALV_Wolff_Final.pdf