Inequalities associated to Riesz potentials and non-doubling measures with applications

The main focus of this work is to study the classical Calderon-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling.We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis---the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in ( Rn , dmu), where mu is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.