Abstract
Let u(x, y) be a harmonic function in the halfspace \({\mathbb {R}}^n\times {\mathbb {R}}_+\) that grows near the boundary not faster than some fixed majorant w(y). Recently it was proven that an appropriate weighted average along the vertical lines of such a function satisfies the law of iterated logarithm (LIL). We extend this result to a class of Lipschitz domains in \({\mathbb {R}}^{n+1}\). In particular, we obtain the local version of this LIL for the upper halfspace. The proof is based on approximation of the weighted averages by a Bloch function, satisfying some additional condition determined by the weight w. The growth rate of such Bloch function depends on w and, for slowly increasing w, turns out to be slower than the one provided by LILs of Makarov and Llorente. We discuss the necessary condition for an arbitrary Bloch function to exhibit this type of behaviour.
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Acknowledgments
We would like to thank A. Nicolau and E. Malinnikova for fruitful discussions of the problem. This work was done while the author was working in the Department of Mathematical Sciences at the Norwegian University of Science and Technology. We are grateful to the Department for hospitality and great working conditions.
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Author was supported by Research Council of Norway, Grant 204726/V30.
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Mozolyako, P. Boundary oscillations of harmonic functions in Lipschitz domains. Collect. Math. 68, 359–376 (2017). https://doi.org/10.1007/s13348-016-0177-z
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DOI: https://doi.org/10.1007/s13348-016-0177-z