Systematically constructing an optimal theory this monograph develops and explores several
approaches to Hardy spaces in the setting of Alhlfors-regular quasi-metric spaces. The text is
divided into two main parts with the first part providing atomic molecular and grand maximal
function characterizations of Hardy spaces and formulates sharp versions of basic analytical
tools for quasi-metric spaces such as a Lebesgue differentiation theorem with minimal demands
on the underlying measure a maximally smooth approximation to the identity and a
Calderon-Zygmund decomposition for distributions. These results are of independent interest.
The second part establishes very general criteria guaranteeing that a linear operator acts
continuously from a Hardy space into a topological vector space emphasizing the role of the
action of the operator on atoms. Applications include the solvability of the Dirichlet problem
for elliptic systems in the upper-half space with boundary data from Hardy spaces. The tools
established in the first part are then used to develop a sharp theory of Besov and
Triebel-Lizorkin spaces in Ahlfors-regular quasi-metric spaces. The monograph is largely
self-contained and is intended for mathematicians graduate students and professionals with a
mathematical background who are interested in the interplay between analysis and geometry.
