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Boundedness of a variation of the positive harmonic function along the normals to the boundary


Authors: P. Mozolyako and V. P. Havin
Translated by: P. Mozolyako
Original publication: Algebra i Analiz, tom 28 (2016), nomer 3.
Journal: St. Petersburg Math. J. 28 (2017), 345-375
MSC (2010): Primary 31B05
DOI: https://doi.org/10.1090/spmj/1454
Published electronically: March 29, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ u$ be a positive harmonic function on the unit disk. Bourgain showed that the radial variation

$\displaystyle \mathrm {var}(u\vert _{[0,re^{i\theta }]}) = \int _0^1\vert u'(re^{i\theta })\vert\,dr $

of $ u$ is finite for many points $ \theta $, and moreover, that the set

$\displaystyle \mathcal {V}(u)=\big \{e^{i\theta }\,:\, \mathrm {var}\big (u\vert _{[0,re^{i\theta }]}\big )< +\infty \big \} $

is dense in the unit circle $ \mathbb{T}$ and its Hausdorff dimension equals one. In the paper, this result is generalized to a class of smooth domains in $ \mathbb{R}^d$, $ d\geq 3$.

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Additional Information

P. Mozolyako
Affiliation: Chebyshev Laboratory, St. Petersburg State University, V.O. 14th Line 29B, 199178 St. Petersburg, Russia
Email: pmzlcroak@gmail.com

V. P. Havin
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, 198504 St. Petersburg, Russia

DOI: https://doi.org/10.1090/spmj/1454
Keywords: Positive harmonic functions, normal variation, mean variation, Bourgain points
Received by editor(s): September 10, 2015
Published electronically: March 29, 2017
Additional Notes: Supported by the Russian Science Foundation grant no. 14-21-00035
The second author, V.P.Havin, is deceased.
Article copyright: © Copyright 2017 American Mathematical Society