Michael Ruzhansky Books

Evolution equations of hyperbolic or more general p-evolution type form an active field of current research. This volume aims to collect some recent advances in the area in order to allow a quick overview of ongoing research. The contributors are first rate mathematicians. This collection of research papers is centred around parametrix constructions  and microlocal analysis; asymptotic constructions of solutions; energy and dispersive estimates; and associated spectral transforms. Applications concerning elasticity and general relativity complement the volume. The book gives an overview of a variety of ongoing current research in the field and, therefore, allows researchers as well as students to grasp new aspects and broaden their understanding of the area. ​

This monograph focuses on the development of the theory of pseudo-differential operators on spaces with symmetries, including Euclidean space, the n-torus, compact Lie groups, and compact homogeneous spaces. It is structured in several parts, aiming to present new results while drawing parallels between various approaches to pseudo-differential operators across different spaces. The material is designed to be self-contained, making it accessible for newcomers. However, the diverse spaces necessitate different backgrounds. For instance, operators on Euclidean space in Chapter 2 rely on established Euclidean Fourier analysis, while those on the torus and general Lie groups in Chapters 4 and 10 require knowledge of discrete analysis and representation theory, which are introduced in Chapter 3 and Part III, respectively. Readers interested in pseudo-differential operators on Lie groups will benefit from a solid understanding of representation theory, which is covered in Part III to minimize the need for external resources. Additionally, the foundational material for the theory of pseudo-differential operators on S and SU(2) in Chapter 12 is provided in a self-contained manner in Chapter 11, ensuring immediate applicability.

The book provides a quick overview of a wide range of active research areas in partial differential equations. The book can serve as a useful source of information to mathematicians, scientists and engineers. The volume contains contributions from authors from a large variety of countries on different aspects of partial differential equations, such as evolution equations and estimates for their solutions, control theory, inverse problems, nonlinear equations, elliptic theory on singular domains, numerical approaches.

This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups. The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.

Different facets of interplay between harmonic analysis and approximation theory are covered in this volume. The topics included are Fourier analysis, function spaces, optimization theory, partial differential equations, and their links to modern developments in the approximation theory. The articles of this collection were originated from two events. The first event took place during the 9 th ISAAC Congress in Krakow, Poland, 5th-9th August 2013, at the section “Approximation Theory and Fourier Analysis”. The second event was the conference on Fourier Analysis and Approximation Theory in the Centre de Recerca Matemàtica (CRM), Barcelona, during 4th-8th November 2013, organized by the editors of this volume. All articles selected to be part of this collection were carefully reviewed.