It is a distinct pleasure to have the opportunity to introduce Professor
Malliavin's book to the English-speaking mathematical world. In recent
years there has been a noticeable retreat from the level of ab-
straction at which graduate-level courses in analysis were previously
taught in the United States and elsewhere. In contrast to the practices
used in the 1950s and 1960s, when great emphasis was placed on the most
general context for integration and operator theory, we have recently
witnessed an increased emphasis on detailed discussion of integration
over Euclidean space and related problems in probability theory,
harmonic analysis, and partial differential equations. Professor
Malliavin is uniquely qualified to introduce the student to anal- ysis
with the proper mix of abstract theories and concrete problems. His
mathematical career includes many notable contributions to harmonic
anal- ysis, complex analysis, and related problems in probability theory
and par- tial differential equations. Rather than developed as a
thing-in-itself, the abstract approach serves as a context into which
special models can be couched. For example, the general theory of
integration is developed at an abstract level, and only then specialized
to discuss the Lebesgue measure and integral on the real line. Another
important area is the entire theory of probability, where we prefer to
have the abstract model in mind, with no other specialization than total
unit mass. Generally, we learn to work at an abstract level so that we
can specialize when appropriate.
