A mean value theorem for harmonic functions on a domain in Hilbert space. (English. Russian original) Zbl 0531.31009
Mosc. Univ. Math. Bull. 37, No. 5, 38-42 (1982); translation from Vestn. Mosk. Univ., Ser. I 1982, No. 5, 32-35 (1982).
The author proves that the value at a point \(x\in H\) of a harmonic function whose domain is an open set in a real separable Hilbert space H is equal to its mean value (with respect to a suitable measure) on a ball centered at x; the principal tool is an integral representation of the author [Vestn. Mosk. Univ., Ser. I 1981, No.6, 44-47 (1981; Zbl 0472.31006)] for such functions. As a corollary he proves that a bounded harmonic function defined everywhere on H is necessarily constant.
Reviewer: D.A.Brannan
MSC:
| 31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |
| 31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |
