Bilinear Commutation Research Articles

A new vanishing Lipschitz-type subspace of BMO and compactness of bilinear commutators

A new vanishing Lipschitz-type subspace of BMO and compactness of bilinear commutators

Characterizations of Weighted BMO Space and Its Application

In this paper, we prove that the weighted BMO space $${\rm{BM}}{{\rm{O}}^p}(\omega ) = \left\{ {f \in L_{{\rm{loc}}}^1:\mathop {\sup }\limits_Q \left\| {{\chi _Q}} \right\|_{{L^p}(\omega )}^{ - 1}{{\left\| {(f - {f_Q}){\omega ^{ - 1}}{\chi _Q}} \right\|}_{{L^p}(\omega )}} < \infty } \right\}$$ is independent of the scale p ∈ (0, ∞) in sense of norm when ω ∈ A1. Moreover, we can replace Lp(ω) by Lp,∞(ω). As an application, we characterize this space by the boundedness of the bilinear commutators [b, T]j(j = 1, 2), generated by the bilinear convolution type Calderón-Zygmund operators and the symbol b, from \({L^{{p_1}}}(\omega ) \times {L^{{p_2}}}(\omega )\) to Lp(ω1−p) with 1 < p1,p2 < ∞ and 1/p =1/p1 + 1/p2. Thus we answer the open problem proposed by Chaffee affirmatively.

XMO and Weighted Compact Bilinear Commutators

To study the compactness of bilinear commutators of certain bilinear Calderón–Zygmund operators which include (inhomogeneous) Coifman–Meyer bilinear Fourier multipliers and bilinear pseudodifferential operators as special examples, Torres and Xue (Rev Mat Iberoam 36:939–956, 2020) introduced a new subspace of BMO\(\,(\mathbb {R}^n)\), denoted by XMO\(\,(\mathbb {R}^n)\), and conjectured that it is just the space VMO\(\,(\mathbb {R}^n)\) introduced by D. Sarason. In this article, the authors give a negative answer to this conjecture by establishing an equivalent characterization of XMO\(\,(\mathbb {R}^n)\), which further clarifies that XMO\(\,(\mathbb {R}^n)\) is a proper subspace of VMO\(\,(\mathbb {R}^n)\). This equivalent characterization of XMO\(\,(\mathbb {R}^n)\) is formally similar to the corresponding one of CMO\(\,(\mathbb {R}^n)\) obtained by A. Uchiyama, but its proof needs some essential new techniques on dyadic cubes as well as some exquisite geometrical observations. As an application, the authors also obtain a weighted compactness result on such bilinear commutators, which optimizes the corresponding result in the unweighted setting.

Commutators of the Bilinear Hardy Operator on Herz Type Spaces with Variable Exponents

In this paper, we obtain the boundedness of bilinear commutators generated by the bilinear Hardy operator and BMO functions on products of Herz spaces and Herz-Morrey spaces with variable exponents.

Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces

In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that with modest assumptions on the underlying spaces and on the operator $T$, if the commutator $[b,T]$ is bounded, then the function $b$ is in $BMO$.

Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces

Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces

Compact bilinear commutators: The weighted case

Commutators of bilinear Calder\'on-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact on appropriate products of weighted Lebesgue spaces.

Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows

We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the 2n-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for n◊n matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato- Wilson relations. A reduction process leads to the AKNS, two-component Camassa-Holm and Cecotti-Vafa models and the formalism provides simple formulas for their solutions.

Neutrino statistics and non-standard commutation relations

Recently it was suggested that the neutrino may violate the Pauli exclusion principle (PEP). This renews interest in the systematic search for bilinear commutation relations that could describe deviations from PEP. In the context of this search we prove a no-go theorem which forbids a finite occupancy limit for an arbitrary system with a bilinear commutation relation. In other words, either the upper limit on the occupancy number is 1 (the ordinary fermionic case) or there is no upper limit at all. Some examples of the latter class include the usual Bose statistics, as well as non-standard quon statistics and infinite statistics.

Interpretation and extension of Green's ansatz for paraparticles

The anomalous bilinear commutation relations satisfied by the components of the Green's ansatz for paraparticles are shown to derive from the comultiplication of the paraboson or parafermion algebra. The same provides a generalization of the ansatz, wherein paraparticles of order p= ∑ α=1 r p α are constructed from r paraparticles of order p α , α=1, 2, …, r.

Realizations of U q ( su (1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrodinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(xi − xj).

Coherent states for parabosons

Taking into consideration that the Fock space of a parabose oscillator may be described by bilinear commutation relations involving creation and annihilation operators, we construct the coherent states for odd and even order parabosons.

Algebraic structure of parabose Fock space. I. The Green’s ansatz revisited

A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well-known Green’s ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green’s ansatz of two (three) terms. Analogs of the form in which the parabose Green’s ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q-deformed parabose oscillators to be derived.

A remark on possible violations of the Pauli principle

Recently Ignatiev and Kuzmin (1988) developed a theoretical model to implement small violations of the Pauli principle. The authors remark that the Pauli principle is unique among discrete symmetries, and that in consequence any apparent violation actually signals new physical degrees of freedom, with no violation of the Pauli principle. They construct an algebraic model which incorporates the algebra of Ignatiev and Kuzmin and leads to the same apparent violations, yet preserves the Pauli principle. Their algebra is that of a Jordan pair, an algebraic concept which provides a natural framework for structures that do not assume bilinear commutation relations.

Do trilinear commutation relations in quantum mechanics admit coordinate space realization in three dimensions?

The trilinear commutation relations involving coordinates and momenta introduced by Wigner [E. P. Wigner, Phys. Rev. 77, 711 (1950)] are generalized to three dimensions. It is shown that the only realizable coordinate space representation of the momenta implies the usual bilinear commutation relations.

Inconsistency in an external field of Dirac’s positive-energy wave equation with generalized internal variables

Sudarshan, Mukunda &amp; Chiang (1982) (see also Mukunda, Chiang &amp; Sudarshan 1982) have generalized Dirac's (1971) positive-energy relativistic wave equation by the adoption of internal variables satisfying trilinear relations rather than bilinear commutation relations. They have obtained an equation that also describes a spinless particle and admits only positive-energy solutions, and have argued that, unlike the equation proposed by Dirac, their equation remains consistent with minimal coupling to an external electromagnetic field. However, a relation between this equation and the equation for an electron is established here, and exploited to provide a proof of inconsistency for a constant and uniform external magnetic field. Analogous results are obtained in 2 + 1 -dimensional and 1 + 1-dimensional space–time, again contrary to the conclusions of Sudarshan et al .

Fermi-Bose Similarity

The usual theory of the Lie algebras and the Lie groups rs shown to be formally extended to the cases in which the group parameters commute and/or anticommute in the most general manner. It is then proved that the parafermi and parabose algebras for f degrees of freedom correspond to the Lie algebras of SO (2/+ 1) and of a graded version of Sp (2/, R), respectively. Further the trilinear and bilinear commutation relations for a general system comprising parafermi and parabose fields are shown to coincide with the Lie commutation relations of a certain group in the above-mentioned sense. Throughout our argumentation parafermi and parabose fields can formally be treated in an analogous man­ ner. It is thus concluded that the fermi-bose similarity that is known to hold in the ordi­ nary field theory persists also in parafielcl theory.