Commutator (electric)

About: Commutator (electric) is a research topic. Over the lifetime, 4857 publications have been published within this topic receiving 31390 citations.

A boundedness criterion for generalized Caldéron-Zygmund operators

Abstract: In 1965, A. P. Calderon showed the L2-boundedness of the so-called first Calderon commutator. This is one of the first examples of a non-convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the Hilbert transform. These properties, together with the L2-boundedness, imply the LP-boundedness for all p's in ]1, + oo[. Many operators in analysis, such as certain classes of pseudo-differential operators and the Cauchy integral operator on a curve, are associated with kernels having these properties. For these operators, one of the major questions is if they are bounded on L2. We are going to give necessary and sufficient conditions for such an operator to be bounded on L2. They are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in BMO. In the case of the aforementioned first commutator this can be checked by an integration by parts. In the first section we present some basic notions and state the theorem, which is proved in Sections 2 and 3. In Section 4 we show how to recover some classical results. In Sections 5 and 6 we construct a functional calculus for small perturbations of A, and in Section 7 we show a connection between the theory of Calderon-Zygmund operators and Kato's conjecture. It is a pleasure to express our thanks to R. R. Coifman and Y. Meyer for suggesting many elegant simplifications in our proofs and most of the applications. We also wish to thank Stephen Semmes for several pertinent remarks.

Commutator theory for congruence modular varieties

Abstract: Introduction In the theory of groups, the important concepts of Abelian group, solvable group, nilpotent group, the center of a group and centraliz-ers, are all defined from the binary operation [x, y] = x −1 y −1 xy. Each of these notions, except centralizers of elements, may also be defined in terms of the commutator of normal subgroups. The commutator [M, N] (where M and N are normal subgroups of a group) is the (normal) subgroup generated by all the commutators [x, y] with x ∈ M, y ∈ N. Thus we have a binary operation in the lattice of normal subgroups. This binary operation, in combination with the lattice operations , carries much of the information about how a group is put together. The operation is also interesting in its own right. It is a com-mutative, monotone operation, completely distributive with respect to joins in the lattice. There is an operation naturally defined on the lattice of ideals of a ring, which has these properties. Namely, let [J, K] be the ideal generated by all the products jk and kj, with j ∈ J and k ∈ K. The congruity between these two contexts extends to the following facts: [M, M] is the smallest normal subgroup U of M for which M/U is a commutative group; [J, J] is the smallest ideal K of J for which the ring J/K is a commutative group; that is, a ring with trivial multiplication. Now it develops, amazingly, that a commutator can be defined rather naturally in the congruence lattices of every congruence modular variety. This operation has the same useful properties that the commutator for groups (which is a special case of it) possesses. The resulting theory has many general applications and, we feel, it is quite beautiful. In this book we present the basic theory of commutators in congruence modular varieties and some of its strongest applications. The book by H. P. Gumm [41] offers a quite different approach to the subject. Gumm developed a sustained analogy between commutator theory and affine geometry which allowed him to discover many of the basic facts about the commutator. We take a more algebraic approach, using some of the shortcuts that Taylor and others have discovered. 1 2 INTRODUCTION Historical remarks. The lattice of normal subgroups of a group, with the commutator operation, is a lattice ordered monoid. It is a residuated lattice …

Scattering theory of classical and quantum N-particle systems

Abstract: 0. Introduction.- 1. Classical Time-Decaying Forces.- 2. Classical 2-Body Hamiltonians.- 3. Quantum Time-Decaying Hamiltonians.- 4. Quantum 2-Body Hamiltonians.- 5. Classical N-Body Hamiltonians.- 6. Quantum N-Body Hamiltonians.- A. Miscellaneous Results in Real Analysis.- A.1 Some Inequalities.- A.2 The Fixed Point Theorem.- A.3 The Hamilton-Jacobi Equation.- A.4 Construction of Some Cutoff Functions.- A.5 Propagation Estimates.- A.6 Comparison of Two Dynamics.- A.7 Schwartz's Global Inversion Theorem.- B. Operators on Hilbert Spaces.- B.1 Self-adjoint Operators.- B.2 Convergence of Self-adjoint Operators.- B.3 Time-Dependent Hamiltonians.- B.4 Propagation Estimates.- B.5 Limits of Unitary Operators.- B.6 Schur's Lemma.- C. Estimates on Functions of Operators.- C.1 Basic Estimates of Commutators.- C.2 Almost-Analytic Extensions.- C.3 Commutator Expansions I.- C.4 Commutator Expansions II.- D. Pseudo-differential and Fourier Integral Operators.- D.0 Introduction.- D.1 Symbols of Operators.- D.2 Phase-Space Correlation Functions.- D.3 Symbols Associated with a Uniform Metric.- D.4 Pseudo-differential Operators Associated with a Uniform Metric.- D.5 Symbols and Operators Depending on a Parameter.- D.6 Weighted Spaces.- D.7 Symbols Associated with Some Non-uniform Metrics.- D.8 Pseudo-differential Operators Associated with the Metric 91.- D.9 Essential Support of Pseudo-differential Operators.- D.10 Ellipticity.- D.12 Non-stationary Phase Method.- D.13 FIO's Associated with a Uniform Metric.- D.14 FIO's Depending on a Parameter.- References.

Cauchy integrals on Lipschitz curves and related operators.

Abstract: In this note, we establish certain properties of the Cauchy integral on Lipschitz curves and prove the Lp-boundedness of some related operators. In particular, we obtain the recent results of R. R. Coifman and Y. Meyer [(1976) “Commutateurs d'integrales singulieres:” Analyse harmonique d'Orsay no 211, Universite Paris XI] on the continuity of the so-called commutator operators.