This is a self-contained textbook of the theory of Besov spaces and
Triebel-Lizorkin spaces oriented toward applications to partial
differential equations and problems of harmonic analysis. These include
a priori estimates of elliptic differential equations, the T1 theorem,
pseudo-differential operators, the generator of semi-group and spaces on
domains, and the Kato problem. Various function spaces are introduced to
overcome the shortcomings of Besov spaces and Triebel-Lizorkin spaces as
well. The only prior knowledge required of readers is familiarity with
integration theory and some elementary functional analysis.Illustrations
are included to show the complicated way in which spaces are defined.
Owing to that complexity, many definitions are required. The necessary
terminology is provided at the outset, and the theory of distributions,
L^p spaces, the Hardy-Littlewood maximal operator, and the singular
integral operators are called upon. One of the highlights is that the
proof of the Sobolev embedding theorem is extremely simple. There are
two types for each function space: a homogeneous one and an
inhomogeneous one. The theory of function spaces, which readers usually
learn in a standard course, can be readily applied to the inhomogeneous
one. However, that theory is not sufficient for a homogeneous space; it
needs to be reinforced with some knowledge of the theory of
distributions. This topic, however subtle, is also covered within this
volume. Additionally, related function spaces--Hardy spaces, bounded
mean oscillation spaces, and Hölder continuous spaces--are defined and
discussed, and it is shown that they are special cases of Besov spaces
and Triebel-Lizorkin spaces.
