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.

Ali Baklouti is a full professor of mathematics at the University of Sfax, Tunisia. He earned his Ph.D. in Mathematics from the University of Metz, France in 1995. He served as the Vice President of the University of Sfax from December 2020 to July 2024 and held the position of President of the Tunisian Mathematical Society for two consecutive terms (April 2016–March 2019 and April 2019–March 2022). Since January 2012, he has served as the Deputy Director of the Mediterranean Institute of Mathematical Sciences, an institution he co-founded in the same year. In December 2016, he was elected as a permanent member of the Tunisian Academy of Sciences, Letters, and Arts. Professor Baklouti has received numerous prestigious awards, including the AMU-PaCOM 2022 Award and Medal, Category A in Mathematics, and the Royal Society Africa Prize 2024. In the same year, he was also honored with the Order of Merit for Education and Teaching by the President of Tunisia. He is also the holder of the "Chaire Pays de Sud" in Mathematics for 2026, organized by the CIRM (France). He currently serves as the Co-Editor-in-Chief of the Tunisian Journal of Mathematics (published by MSP, USA) and as the Editor-in-Chief of Advances in Pure and Applied Mathematics (published by ISTE, UK). Additionally, he serves on the editorial boards of several journals, including the Graduate Journal of Mathematics (MIMS) and the Arabian Journal of Mathematics (Springer).