The sharp Riesz potential estimates in metric spaces

We introduce the Riesz potential operator for functions defined on a quasi-metric space endowed with a nonnegative doubling measure, satisfying a certain lower bound for the measure of a ball. For this operator we prove sharp Lorentz-norm estimates in the spirit of the results due to O'Neil and Peetre (in the non-limiting case) and to Hansson and Brezis-Wainger (in the limiting case).