| Summary: | 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.
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