Positive harmonic function

Summary

A positive harmonic function is a harmonic function that takes only positive values on its domain. Positivity is a strong constraint: on a connected open set, the values of such a function at two points control each other, by Harnack's inequality. On the unit disc and on balls, positive harmonic functions are exactly the Poisson integrals of nonzero positive finite Borel measures on the boundary sphere. They are the starting point of the theory of the Martin boundary.