  The purpose of this paper is threefold. First the natural extension of Riesz
potentials to the context of quasi metric measure spaces for the class of upper
doubling measures are studied on Lebesgue spaces, obtaining necessary and
sufficient conditions on a upper doubling measure. Second, we exhibit a
geometric property of the measure of the ball which permit prove the
boundedness in a unified way, both in of the doubling as non doubling
situation. Third, we show that the result can be applied to a type Riesz
potential operator defined over a space formed by two components which are not
of necessarily equal dimensions.
