PUB - Publikationen an der Universität Bielefeld

On a bounded domain U in R(d) containing the origin, probability measures mu which have a density w with respect to Lebesgue measure and satisfy h(0) = integral 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 is an element of C infinity (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 is an element of C infinity (U) tending to zero at partial derivative U.