When two Radon measures on the half line are given, the harmonic mean of their Stieltjes transforms is again the Stieltjes transform of a Radon measure. We study the relationship between the asymptotic behavior of the resulting measure and those of the original ones. The problem comes from the spectral theory of second--order differential operators and the results are applied to linear diffusions neither boundaries of which is regular.
"Tauberian theorem for harmonic mean of Stieltjes transforms and its applications to linear diffusions." Osaka J. Math. 53 (1) 221 - 251, January 2016.