## Volume densities with the mean value property for harmonic functions

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- by W. Hansen and I. Netuka PDF
- Proc. Amer. Math. Soc.
**123**(1995), 135-140 Request permission

## Abstract:

On a bounded domain*U*in ${\mathbb {R}^d}$ containing the origin, probability measures $\mu$ which have a density

*w*with respect to Lebesgue measure and satisfy $h(0) = \smallint h\;d\mu$ for every bounded harmonic function on

*U*are studied. A domain

*U*is constructed such that $\inf w(U) = 0$ for any such measure. (This solves a problem proposed by A. Cornea.) If, however,

*U*has smooth boundary, then $\mu$ having a density $w \in {\mathcal {C}^\infty }(U)$ which is bounded away from zero on

*U*can be constructed. On the other hand, for arbitrary

*U*it is always possible to choose a strictly positive $w \in {\mathcal {C}^\infty }(U)$ tending to zero at $\partial U$.

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## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 135-140 - MSC: Primary 31A05; Secondary 31B05
- DOI: https://doi.org/10.1090/S0002-9939-1995-1213859-3
- MathSciNet review: 1213859