Michael Ruzhansky

Pseudo-Differential Operators and Symmetries

Preface.- Introduction.- Part I Foundations of Analysis.- A Sets, Topology and Metrics.- B Elementary Functional Analysis.- C Measure Theory and Integration.- D Algebras.- Part II Commutative Symmetries.- 1 Fourier Analysis on Rn.- 2 Pseudo-differential Operators on Rn.- 3 Periodic and Discrete Analysis.- 4 Pseudo-differential Operators on Tn.- 5 Commutator Characterisation of Pseudo-differential Operators.- Part III Representation Theory of Compact Groups.- 6 Groups.- 7 Topological Groups.- 8 Linear Lie Groups.- 9 Hopf Algebras.- Part IV Non-commutative Symmetries.- 10 Pseudo-differential Operators on Compact Lie Groups.- 11 Fourier Analysis on SU(2).- 12 Pseudo-differential Operators on SU(2).- 13 Pseudo-differential Operators on Homogeneous Spaces.- Bibliography.- Notation.- Index.

Global L 2-Boundedness Theorems for a Class of Fourier Integral Operators

ABSTRACT The local L 2-mapping property of Fourier integral operators has been established in Hörmander (1971) and in Eskin (1970). In this article, we treat the global L 2-boundedness for a class of operators that appears naturally in many problems. As a consequence, we improve known global results for several classes of pseudodifferential and Fourier integral operators, as well as extend previous results of Asada and Fujiwara (1978) or Kumano-go (1976). As an application, we show a global smoothing estimate for generalized Schrödinger equations which extends the results of Ben-Artzi and Devinatz (1991) and Walther (1999); (2002).

Global quantization of pseudo-differential operators on compact Lie groups, SU(2), 3-sphere, and homogeneous spaces

Global quantization of pseudo-differential operators on general compact Lie groups G is introduced relying on the representation theory of the group rather than on expressions in local coordinates. A new class of globally defined symbols is introduced and related to the usual Hormanders classes of operators Psi(m)(G). Properties of the new class and symbolic calculus are analyzed. Properties of symbols as well as L-2-boundedness and Sobolev L-2-boundedness of operators in this global quantization are established on general compact Lie groups. Operators on the three-dimensional sphere S-3 and on group SU(2) are analyzed in detail. An application is given to pseudo-differential operators on homogeneous spaces K backslash G. In particular, using the obtained global characterization of pseudo-differential operators on Lie groups, it is shown that every pseudo-differential operator in Psi(m)(K backslash G) can be lifted to a pseudo-differential operator in Psi(m)(G), extending the known results on invariant partial differential operators.

Quantization on Nilpotent Lie Groups

Preface.- Introduction.- Notation and conventions.- 1 Preliminaries on Lie groups.- 2 Quantization on compact Lie groups.- 3 Homogeneous Lie groups.- 4 Rockland operators and Sobolev spaces.- 5 Quantization on graded Lie groups.- 6 Pseudo-differential operators on the Heisenberg group.- A Miscellaneous.- B Group C* and von Neumann algebras.- Schrodinger representations and Weyl quantization.- Explicit symbolic calculus on the Heisenberg group.- List of quantizations.- Bibliography.- Index.

On the Fourier Analysis of Operators on the Torus

Basic properties of Fourier integral operators on the torus \( \mathbb{T}^n = (\mathbb{R}/2\pi \mathbb{Z})^n \) are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations.

Schatten classes on compact manifolds: Kernel conditions

Abstract In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

In this paper we give several global characterisations of the Hörmander class

Modulation Spaces and Nonlinear Evolution Equations

\Psi ^m(G)