A problem of Baernstein on the equality of the -harmonic measure of a set and its closure

Authors:
Anders Björn, Jana Björn and Nageswari Shanmugalingam

Journal:
Proc. Amer. Math. Soc. **134** (2006), 509-519

MSC (2000):
Primary 31C45; Secondary 30C85, 31A25, 31B20, 31C15, 46E35

DOI:
https://doi.org/10.1090/S0002-9939-05-08187-6

Published electronically:
August 12, 2005

MathSciNet review:
2176020

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Abstract: A. Baernstein II (*Comparison of -harmonic measures of subsets of the unit circle*, St. Petersburg Math. J. **9** (1998), 543-551, p. 548), posed the following question: If is a union of open arcs on the boundary of the unit disc , then is , where denotes the -harmonic measure? (Strictly speaking he stated this question for the case .) For the positive answer to this question is well known. Recall that for the -harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense.

The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when . In the proof, using a deep trace result of Jonsson and Wallin, it is first shown that the characteristic function is the restriction to of a Sobolev function from .

For it is no longer true that belongs to the trace class. Nevertheless, we are able to show equality for the case of one arc for all , using a very elementary argument. A similar argument is used to obtain a result for starshaped domains.

Finally we show that in a certain sense the equality holds for almost all relatively open sets.

**1.**A. Baernstein II,*Comparison of 𝑝-harmonic measures of subsets of the unit circle*, Algebra i Analiz**9**(1997), no. 3, 141–149; English transl., St. Petersburg Math. J.**9**(1998), no. 3, 543–551. MR**1466798****2.**Anders Björn, Jana Björn, and Nageswari Shanmugalingam,*The Dirichlet problem for 𝑝-harmonic functions on metric spaces*, J. Reine Angew. Math.**556**(2003), 173–203. MR**1971145**, https://doi.org/10.1515/crll.2003.020**3.**Anders Björn, Jana Björn, and Nageswari Shanmugalingam,*The Perron method for 𝑝-harmonic functions in metric spaces*, J. Differential Equations**195**(2003), no. 2, 398–429. MR**2016818**, https://doi.org/10.1016/S0022-0396(03)00188-8**4.**-,*Sobolev extensions of Hölder continuous and characteristic functions on metric spaces*, in preparation.**5.**Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu,*Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds*, Harmonic analysis and boundary value problems (Fayetteville, AR, 2000) Contemp. Math., vol. 277, Amer. Math. Soc., Providence, RI, 2001, pp. 19–37. MR**1840425**, https://doi.org/10.1090/conm/277/04544**6.**-,*Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot-Carathéodory Spaces*, to appear in Mem. Amer. Math. Soc., Amer. Math. Soc., Providence, R.I.**7.**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****8.**Piotr Hajłasz and Olli Martio,*Traces of Sobolev functions on fractal type sets and characterization of extension domains*, J. Funct. Anal.**143**(1997), no. 1, 221–246. MR**1428124**, https://doi.org/10.1006/jfan.1996.2959**9.**Juha Heinonen, Tero Kilpeläinen, and Olli Martio,*Nonlinear potential theory of degenerate elliptic equations*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR**1207810****10.**A. Jonsson and H. Wallin,*Function Spaces on Subsets of*, Math. Rep.**2**:1, Harwood, London, 1984. MR**0820626 (87f:46056)****11.**Jukka Kurki,*Invariant sets for 𝐴-harmonic measure*, Ann. Acad. Sci. Fenn. Ser. A I Math.**20**(1995), no. 2, 433–436. MR**1346825****12.**J. J. Manfredi,*-harmonic functions in the plane*, Proc. Amer. Math. Soc.**103**(1988), 473-479. MR**0943069 (89f:35076)****13.**Pertti Mattila,*Geometry of sets and measures in Euclidean spaces*, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR**1333890****14.**William P. Ziemer,*Weakly differentiable functions*, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR**1014685**

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Additional Information

**Anders Björn**

Affiliation:
Department of Mathematics, Linköpings Universitet, SE-581 83 Linköping, Sweden

Email:
anbjo@mai.liu.se

**Jana Björn**

Affiliation:
Department of Mathematics, Linköpings Universitet, SE-581 83 Linköping, Sweden

Email:
jabjo@mai.liu.se

**Nageswari Shanmugalingam**

Affiliation:
Department of Mathematical Sciences, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221-0025

Email:
nages@math.uc.edu

DOI:
https://doi.org/10.1090/S0002-9939-05-08187-6

Keywords:
Ahlfors regular,
Dirichlet problem,
$d$-set,
Lipschitz domain,
Minkowski dimension,
$p\mspace{1mu}$-harmonic function,
$p\mspace{1mu}$-harmonic measure,
Sobolev function,
starshaped,
trace,
unit disc

Received by editor(s):
September 27, 2004

Published electronically:
August 12, 2005

Additional Notes:
We thank Juha Heinonen for drawing our attention to the question of Baernstein

The first two authors were supported by the Swedish Research Council and Gustaf Sigurd Magnuson’s fund of the Royal Swedish Academy of Sciences. The second author did this research while she was at Lund University

The third author was partly supported by NSF grant DMS 0243355.

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.