Continuous functions on multipolar sets
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- by Ramasamy Jesuraj PDF
- Proc. Amer. Math. Soc. 99 (1987), 331-339 Request permission
Abstract:
Let $\Omega = {\Omega _1} \times \cdots \times {\Omega _n}(n > 1)$ be a product of $n$ Brelot harmonic spaces each of which has a bounded potential, and let $K$ be a compact subset of $\Omega$. Then, $K$ is an $n$-polar set with the property that every $i$-section $(1 \leqslant i < n)$ of $K$ through any point in $\Omega$ is $(n - i)$ polar if and only if every positive continuous function on $K$ can be extended to a continuous potential on $\Omega$. Further, it has been shown that if $f$ is a nonnegative continuus function on $\Omega$ with compact support, then $MRf$, the multireduced function of $f$ over $\Omega$, is also a continuous function on $\Omega$.References
- M. Brelot, Lectures on potential theory, Lectures on Mathematics, vol. 19, Tata Institute of Fundamental Research, Bombay, 1960. Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy. MR 0118980
- Corneliu Constantinescu and Aurel Cornea, Potential theory on harmonic spaces, Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. With a preface by H. Bauer. MR 0419799
- Kohur Gowrisankaran, Multiply harmonic functions, Nagoya Math. J. 28 (1966), 27–48. MR 209513
- Kohur Gowrisankaran, Multiply superharmonic functions, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 3-4, xv, 235–244 (English, with French summary). MR 402096
- R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571 (French). MR 139756 R. Jesuraj, Continuous functions and exceptional sets in potential theory, Ph.D. Thesis, McGill Univ., Montreal, Quebec, 1981.
- Ramasamy Jesuraj, Continuous functions on polar sets, Proc. Amer. Math. Soc. 93 (1985), no. 2, 262–266. MR 770533, DOI 10.1090/S0002-9939-1985-0770533-9
- David Singman, Exceptional sets in a product of harmonic spaces, Math. Ann. 262 (1983), no. 1, 29–43. MR 690005, DOI 10.1007/BF01474168
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 99 (1987), 331-339
- MSC: Primary 31D05
- DOI: https://doi.org/10.1090/S0002-9939-1987-0870796-7
- MathSciNet review: 870796