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

Abstract

Let μ be a non-negative Borel measure on R d . Fix a real number n, 0 < n ≤ d, and assume that μ 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 p. 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 μ 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.