Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem

Alexander R. Pruss Department of Philosophy, University of Pittsburgh

TBD mathscidoc:1701.332911

Arkiv for Matematik, 37, (1), 183-210, 1996.6
Let$I$be a union of finitely many closed intervals in [−1, 0). Let$I$^{↞}be a single interval of the form [−1, −a] chosen to have the same logarithmic length as$I$. Let$D$be the unit disc. Then, Beurling [8] has shown that the harmonic measure of the circle ∂$D$at the origin in the slit disc$D$/$I$is increased if$I$is replaced by$I$^{↞}. We prove a number of cognate results and extensions. For instance, we show that Beurling's result remains true if the intervals in$I$are not just one-dimensional, but if they in fact constitute polar rectangles centred on the negative real axis and having some fixed constant angular width. In doing this, we obtain a new proof of Beurling's result. We also discuss a conjecture of Matheson and Pruss [25] and some other open problems.
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@inproceedings{alexander1996radial,
  title={Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem},
  author={Alexander R. Pruss},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203555452690720},
  booktitle={Arkiv for Matematik},
  volume={37},
  number={1},
  pages={183-210},
  year={1996},
}
Alexander R. Pruss. Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem. 1996. Vol. 37. In Arkiv for Matematik. pp.183-210. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203555452690720.
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