## Sets of determination for harmonic functions

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- by Stephen J. Gardiner PDF
- Trans. Amer. Math. Soc.
**338**(1993), 233-243 Request permission

## Abstract:

Let $h$ denote a positive harmonic function on the open unit ball $B$ of Euclidean space ${{\mathbf {R}}^n}\;(n \geq 2)$. This paper characterizes those subsets $E$ of $B$ for which ${\sup _E}H/h = {\sup _B}H/h$ or ${\inf _E}H/h = {\inf _B}H/h$ for all harmonic functions $H$ belonging to a specified class. In this regard we consider the classes of positive harmonic functions, differences of positive harmonic functions, and harmonic functions with a one-sided quasi-boundedness condition. We also consider the closely related question of representing functions on the sphere $\partial B$ as sums of Poisson kernels corresponding to points in $E$.## References

- A. F. Beardon,
*Montel’s theorem for subharmonic functions and solutions of partial differential equations*, Proc. Cambridge Philos. Soc.**69**(1971), 123–150. MR**269863**, DOI 10.1017/s030500410004648x - Arne Beurling,
*A minimum principle for positive harmonic functions*, Ann. Acad. Sci. Fenn. Ser. A I No.**372**(1965), 7. MR**0188466** - F. F. Bonsall,
*Decompositions of functions as sums of elementary functions*, Quart. J. Math. Oxford Ser. (2)**37**(1986), no. 146, 129–136. MR**841422**, DOI 10.1093/qmath/37.2.129 - F. F. Bonsall,
*Domination of the supremum of a bounded harmonic function by its supremum over a countable subset*, Proc. Edinburgh Math. Soc. (2)**30**(1987), no. 3, 471–477. MR**908454**, DOI 10.1017/S0013091500026869 - F. F. Bonsall and D. Walsh,
*Vanishing $l^1$-sums of the Poisson kernel, and sums with positive coefficients*, Proc. Edinburgh Math. Soc. (2)**32**(1989), no. 3, 431–447. MR**1015486**, DOI 10.1017/S0013091500004685 - Björn Dahlberg,
*A minimum principle for positive harmonic functions*, Proc. London Math. Soc. (3)**33**(1976), no. 2, 238–250. MR**409847**, DOI 10.1112/plms/s3-33.2.238 - J. L. Doob,
*Classical potential theory and its probabilistic counterpart*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR**731258**, DOI 10.1007/978-1-4612-5208-5
N. Dudley-Ward, Doctoral thesis, University of York, 1991.
- Matts Essén,
*On minimal thinness, reduced functions and Green potentials*, Proc. Edinburgh Math. Soc. (2)**36**(1993), no. 1, 87–106. MR**1200189**, DOI 10.1017/S0013091500005915 - S. J. Gardiner and M. Klimek,
*Convexity and subsolutions of partial differential equations*, Bull. London Math. Soc.**18**(1986), no. 1, 41–43. MR**841366**, DOI 10.1112/blms/18.1.41 - W. K. Hayman,
*Atomic decompositions*, Recent advances in Fourier analysis and its applications (Il Ciocco, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 315, Kluwer Acad. Publ., Dordrecht, 1990, pp. 597–611. MR**1081364** - W. K. Hayman and T. J. Lyons,
*Bases for positive continuous functions*, J. London Math. Soc. (2)**42**(1990), no. 2, 292–308. MR**1083447**, DOI 10.1112/jlms/s2-42.2.292 - Peter Sjögren,
*Une propriété des fonctions harmoniques positives, d’après Dahlberg*, Séminaire de Théorie du Potentiel de Paris, No. 2 (Univ. Paris, Paris, 1975–1976) Lecture Notes in Math., Vol. 563, Springer, Berlin, 1976, pp. 275–282 (French). MR**0588344**

## Additional Information

- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**338**(1993), 233-243 - MSC: Primary 31B05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1100694-2
- MathSciNet review: 1100694