Non-commutative Harmonic Analysis on Solv-Manifolds - Ali Baklouti

This book presents a thorough and up-to-date treatment of recent advances in non-commutative harmonic analysis on homogeneous spaces, with a particular focus on the intricate interplay between representation theory and analytic methods on Lie groups. The work offers a comprehensive and structured exploration of harmonic analysis and representation theory on a broad class of non-abelian Lie groups, including nilpotent, solvable, exponential, and certain semi-direct and compact extensions. It builds from foundational topics, such as induced representations, Plancherel theory, and coadjoint orbits, toward more advanced developments, including localized Plancherel formulas, functional inequalities, and a variety of uncertainty principles in both classical and non-commutative settings. A major strength of the work lies in its original contributions to the non-commutative analogues of classical theorems, especially uncertainty principles of Hardy, Cowling–Price, Morgan, Beurling, and Miyachi, adapted to nilpotent and solvable Lie groups, Heisenberg groups, and more general settings. Furthermore, the volume features a comprehensive section that extends the Müntz–Szász theorems to non-commutative settings, using sophisticated machinery such as Fourier operators, Plancherel theory on Lie groups and their homogeneous spaces. The exposition is mathematically rigorous, yet constructive and accessible, blending deep theoretical results with explicit examples and analytic techniques.

As such, the book serves both as a reference work for researchers in harmonic analysis, Lie theory, and representation theory, and as a graduate-level textbook for advanced students seeking a modern entry point into the theory of non-commutative harmonic analysis.