This book studies translation-invariant function spaces and algebras on homogeneous manifolds. The central topic is the relationship between the homogeneous structure of a manifold and the class of translation-invariant function spaces and algebras on the manifold. Agranovskiĭ obtains classifications of translation-invariant spaces and algebras of functions on semisimple and nilpotent Lie groups, Riemann symmetric spaces, and bounded symmetric domains. When such classifications are possible, they lead in many cases to new characterizations of the classical function spaces, from the point of view of their group of admissible changes of variable. The algebra of holomorphic functions plays an essential role in these classifications when a homogeneous complex or \(CR\)-structure exists on the manifold. This leads to new characterizations of holomorphic functions and their boundary values for one- and multidimensional complex domains.

Readership

Specialists in functional analysis, harmonic analysis, function theory of one and several complex variables, and Lie group representation.

Reviews

"Very well written and nicely translated."

-- Zentralblatt MATH

Table of Contents

• Function spaces and function algebras on differentiable manifolds and symmetric spaces of noncompact type
• Translation invariant function spaces and function algebras on noncompact Lie groups
• Möbius spaces and algebras on symmetric domains and their Shilov boundaries
• Holomorphy tests in symmetric domains involving the automorphism group. Related problems
• References