The object of this monograph is to give an exposition of the
real-variable theory of Hardy spaces (HP spaces). This theory
has attracted considerable attention in recent years because it led to a
better understanding in Rn of such related topics as singular
integrals, multiplier operators, maximal functions, and real-variable
methods generally. Because of its fruitful development, a systematic
exposition of some of the main parts of the theory is now desirable. In
addition to this exposition, these notes contain a recasting of the
theory in the more general setting where the underlying Rn is
replaced by a homogeneous group.
The justification for this wider scope comes from two sources: 1) the
theory of semi-simple Lie groups and symmetric spaces, where such
homogeneous groups arise naturally as "boundaries," and 2) certain
classes of non-elliptic differential equations (in particular those
connected with several complex variables), where the model cases occur
on homogeneous groups. The example which has been most widely studied in
recent years is that of the Heisenberg group.