Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces

Let p be a non-negative Borel measure on R-d. Fix a real number n, 0 < n less than or equal to d, and assume that mu is "n-dimensional" in the following sense: the measure of a cube is smaller than the length of its side raised to the n-th power. Calderon-Zygmund operators, Hardy and BMO spaces, and some other topics in Harmonic Analysis have been successfully handled in this setting recently, although the measure may be non-doubling. The aim of this paper is to study two-weight norm inequalities for radial fractional maximal functions associated to such mu. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak type inequalities. Sawyer and radial Muckenhoupt type conditions are respectively the solutions for these problems. Furthermore, if we strengthen Muckenhoupt conditions by adding a "power-bump" to the right-hand side weight or even by introducing a certain Orlicz norm, strong type inequalities can be achieved. As a consequence, two-weight norm inequalities for fractional integrals associated to mu are obtained. Finally, for the particular case of the Hardy-Littlewood radial maximal function, we show how, in contrast with the classical situation, radial Muckenhoupt weights may fail to satisfy a reverse Holder's inequality and also strong type inequalities do not necessarily hold for them.