Arkiv för Matematik

  • Ark. Mat.
  • Volume 29, Number 1-2 (1991), 107-126.

Representations of bounded harmonic functions

T. S. Mountford and S. C. Port

Full-text: Open access

Abstract

An open subset D of Rd, d≧2, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a Greenian D to be Poissonian in terms of its Martin boundary.

Note

Supported by NSF DMS86-01800.

Article information

Source
Ark. Mat., Volume 29, Number 1-2 (1991), 107-126.

Dates
Received: 8 August 1990
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898033

Digital Object Identifier
doi:10.1007/BF02384334

Mathematical Reviews number (MathSciNet)
MR1115078

Zentralblatt MATH identifier
0728.31003

Rights
1991 © Institut Mittag-Leffler

Citation

Mountford, T. S.; Port, S. C. Representations of bounded harmonic functions. Ark. Mat. 29 (1991), no. 1-2, 107--126. doi:10.1007/BF02384334. https://projecteuclid.org/euclid.afm/1485898033


Export citation

References

  • Ancona, A., Sur un conjecture concernant la capacite et l'effilement, Seminar on harmonic analysis 1983–84, Publ. Math. Orsay 85-2. 56–91.
  • Bishop, C. J., Carleson, L., Garnett, J. B. and Jones, P. W., Harmonic measures supporte curves, Pacific J. Math. 138 (1989), 233–236.
  • Bishop, C. J., A characterization of Poissonian domains, Ark. Mat. 29 (1991), 1–24.
  • Port, S. C. and Stone, C. J., Brownian motion and classical potential theory, Academic Press, New York, 1978.