In this book we present a comprehensive introduction to the use of
Clifford algebras and Dirac operators in harmonic analysis and analy-
analysis more generally. In the past 30 years, Clifford algebras and Dirac
operators have played a key role in three of the most important ar-
areas of mathematical research during that time: the boundedness of the
Cauchy integral on Lipschitz surfaces, the realization of discrete series
representations of semi-simple Lie groups, and the celebrated Atiyah-
Singer index theorem. Much as an analyst would like to understand and
appreciate these developments, however, there are formidable technical
barriers to doing so, particularly for more classically trained analysts, as
we have found to our cost over the years. Thus our aim from the outset
has been to meld into a coherent and reasonably self-contained whole
a body of ideas from classical singular integral theory, representation
theory and analysis on manifolds, with a view to making this material
accessible to more classically trained analysts.