Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main focus of this work is to study the classical Calder\'n-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 $(\rn, d\mu)$, where $\mu$ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

The presented work contains a survey of the so far known results about the operator inequalities of the type "good λ", "better good λ" and "rearranged good λ" on the function spaces over the Euclidean space with the Lebesgue measure and their corollaries in the form of more complex operator inequal- ities and norm estimates. However, the main aim is to build similar theory for the Riesz potential operator on the function spaces over the quasi-metric space with the so-called "doubling" measure. Combining the corollaries of this theory with the known norm estimates we obtain the boundedness for the Riesz potential operator on the Lebesgue and Lorentz spaces.