This open access book provides an extensive treatment of Hardy
inequalities and closely related topics from the point of view of
Folland and Stein's homogeneous (Lie) groups. The place where Hardy
inequalities and homogeneous groups meet is a beautiful area of
mathematics with links to many other subjects. While describing the
general theory of Hardy, Rellich, Caffarelli-Kohn-Nirenberg, Sobolev,
and other inequalities in the setting of general homogeneous groups, the
authors pay particular attention to the special class of stratified
groups. In this environment, the theory of Hardy inequalities becomes
intricately intertwined with the properties of sub-Laplacians and
subelliptic partial differential equations. These topics constitute the
core of this book and they are complemented by additional, closely
related topics such as uncertainty principles, function spaces on
homogeneous groups, the potential theory for stratified groups, and the
potential theory for general Hörmander's sums of squares and their
fundamental solutions.
This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize,
a prestigious award for books of expository nature presenting the latest
developments in an active area of research in mathematics. As can be
attested as the winner of such an award, it is a vital contribution to
literature of analysis not only because it presents a detailed account
of the recent developments in the field, but also because the book is
accessible to anyone with a basic level of understanding of analysis.
Undergraduate and graduate students as well as researchers from any
field of mathematical and physical sciences related to analysis
involving functional inequalities or analysis of homogeneous groups will
find the text beneficial to deepen their understanding.
