Commutators Of Operators Research Articles

Characterizations for the commutator of parabolic nonsingular integral operator on parabolic generalized Orlicz-Morrey spaces

We show continuity in parabolic generalized Orlicz-Morrey spaces $M^{\Phi,\varphi}$ of commutator of parabolic nonsingular integral operators. We shall give necessary and sufficient conditions for the boundedness of the commutator of parabolic nonsingular integral operator on $M^{\Phi,\varphi}$ spaces with $BMO$ functions.

Commutators of certain fractional type operators with Hörmander conditions, one-weighted and two-weighted inequalities

Fil: Ibanez Firnkorn, Gonzalo Hugo. Universidad Nacional de Cordoba; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Cordoba. Centro de Investigacion y Estudios de Matematica. Universidad Nacional de Cordoba. Centro de Investigacion y Estudios de Matematica; Argentina

The Boundedness for Commutators of Anisotropic Calderón-Zygmund Operators

Let T be an anisotropic Calderon-Zygmund operator and φ : ℝn × [0, ∞) → [0, ∞) be an anisotropic Musielak-Orlicz function with φ(x, ·) being an Orlicz function and φ(·,t) being a Muckenhoupt A∞ (A) weight. In this paper, our goal is to study two boundedness theorems for commutators of anisotropic Calderon-Zygmund operators. Precisely, when b ∈ BMOw(ℝn, A) (a proper subspace of anisotropic bounded mean oscillation space BMO(ℝn, A)), the commutator [b, T] is bounded from anisotropic weighted Hardy space Hl(ℝn, A) to weighted Lebesgue space Lw1 (ℝn) and when b ∈ BMO(ℝn) (bounded mean oscillation space), the commutator [b, T] is bounded on Musielak-Orlicz space Lφ(ℝn), which are extensions of the isotropic setting.

Boundedness of the potential operators and their commutators in the local “complementary” generalized variable exponent Morrey spaces on unbounded sets

In this paper we prove a Sobolev–Spanne type $${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega } (\varOmega )\rightarrow {\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{q(\cdot ),\omega } (\varOmega )$$-theorem for the potential operators $$I^{\alpha }$$, where $${\,^{^{\complement }}\!\mathcal M}_{\{x_0\}}^{p(\cdot ),\omega }(\varOmega )$$ is local “complementary” generalized Morrey spaces with variable exponent p(x), $$\omega (r)$$ is a general function defining the Morrey-type norm and $$\varOmega $$ is an open unbounded subset of $${{\mathbb {R}}^n}$$. In addition, we prove the boundedness of the commutator of potential operators $$[b,I^{\alpha }]$$ in these spaces. In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on $$\omega (x,r)$$, which do not assume any assumption on monotonicity of $$\omega (x,r)$$ in r.

Weak type endpoint estimates for the commutators of rough singular integral operators

Let $\Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{\Omega}$ be the convolution singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$. For $b\in{\rm BMO}(\mathbb{R}^n)$, let $T_{\Omega,\,b}$ be the commutator of $T_{\Omega}$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $L\log L$ type for $T_{\Omega,\,b}$ when $\Omega\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$.

Commutators of the B-maximal operator and B-maximal commutators

In this paper we consider the commutator of the B-maximal operator and the B-maximal commutator associated with the Laplace-Bessel differential operator. The boundedness of the commutator of the B-maximal operator with BMO symbols on weighted Lebesque space and weak-type inequality for the commutator of the B-maximal operator are proved.

О неограниченных комплексных операторах

The concept of an unbounded complex operator as an operator acting in the pull-back of a Banach space is introduced. It is proved that each such operator is linear. Linear operations of addition and multiplication by a number and also the operation of multiplication are determined on the set of unbounded complex operators. The conditions for commutability of operators from this set are indicated. The product of complex conjugate operators and the properties of the conjugation operation are considered. Invertibility questions are studied: two contractions of an unbounded complex operator that have an inverse operator are proposed, and an explicit form of the inverse operator is found for one of these restrictions. It is noted that unbounded complex operators can find application in the study of a linear homogeneous differential equation with constant unbounded operator coefficients in a Banach space.

Separating Poor Playmates: Untangling Commuter Rail from Freight

North American commuter railroads run their trains largely on tracks shared with freights (and in the northeastern US, often shared with intercity passenger trains). This sharing makes commuter rail relatively affordable to implement. However, as freight traffic grows, so does the potential for delays. If freight and commuter operations affect each other’s performance, it may be necessary to consider separation measures, not adding to the existing shared-track capacity. Temporal separation can work on low-volume operations. Busier lines may require physical separation, which can be implemented in several forms for trunk lines (grade separation at junctions, freight bypass, partial trunk line separation, and full trunk line separation), as well as physical separation in downtown areas. This article identifies these separation options, describes their attributes, presents representative examples, and outlines criteria for evaluation.

Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

On Compact Perturbations of Hankel Operators and Commutators of Toeplitz and Hankel Operators

Motivated by results in uniform algebras, a distance localization formula in C*-algebras is established in the framework of the Allan–Douglas localization principle, and is used to derive a locality result for products of Hankel operators as compact perturbations of Hankel operators. Using localization and certain QC functions, it is proved that the essential spectrum of the commutator of a Toeplitz and a Hankel operator is antipodal symmetric under a mild condition on the Hankel symbol function. Under the same condition the essential spectrum of a Hankel operator also exhibits this symmetry. Conjugates of interpolating Blaschke products and characteristic functions are constructed that satisfy the condition, while examples show the condition is only sufficient.

Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrödinger Operator and Their Commutators

Let L = −Δ + V(x) be a Schrodinger operator, where Δ is the Laplacian on ℝn, while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators. Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schroodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schroodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood (H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Given C⁎-algebras A and B, consider the Banach algebra A⊗γB, where ⊗γ denotes the projective Banach space tensor product. If A and B are commutative, this is the Varopoulos algebra VA,B; we write VA for VA,A. It has been an open problem for almost 40 years to determine precisely when A⊗γB is Arens regular; see, e.g., [33], [48], [49]. We solve this classical question for arbitrary C⁎-algebras. Indeed, we show that A⊗γB is Arens regular if and only if A or B has the Phillips property; note that A has the latter property if and only if it is scattered and has the Dunford–Pettis Property. A further equivalent condition is that A⁎ has the Schur property, or, again equivalently, the enveloping von Neumann algebra A⁎⁎ is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A⊗γB is entirely encoded in the geometry of the C⁎-algebras. In case A and B are von Neumann algebra, we conclude that A⊗γB is Arens regular (if and) only if A or B is finite-dimensional. We also show that this characterization does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C⁎-algebras A and B, we obtain that VA,B is Arens regular if and only if A or B is scattered. We further describe the centre Z(VA⁎⁎), showing that it is Banach algebra isomorphic to A⁎⁎⊗ehA⁎⁎, where ⊗eh denotes the extended Haagerup tensor product. We deduce that VA is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, VA is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered; this is the case, e.g., for A=ℓ∞.

Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators.

Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents

The aim of this paper is to establish the boundedness of the commutator $$[b_{1}, b_{2}, T_{\sigma }]$$ generated by the bilinear pseudo-differential operator $$T_{\sigma }$$ with smooth symbols and $$b_{1},\ b_{2}\in \mathrm {BMO}({\mathbb {R}}^{n})$$ on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator $$T_{\sigma }$$ is bounded from the Lebesgue space $$L^{p}({\mathbb {R}}^{n})$$ into $$h^{p_{1}(\cdot )}({\mathbb {R}}^{n})\times h^{p_{2}(\cdot )}({\mathbb {R}}^{n})$$ . Moreover, the boundedness of the commutator $$[b_{1}, b_{2}, T_{\sigma }]$$ on product of local Hardy spaces with variable exponents is also obtained.

Electromagnetic field quantization near a dielectric slab and spontaneous emission rate determination

The Green function approach of the electromagnetic field quantization is used to quantize the electromagnetic field near dispersive and dissipative dielectric slabs. The dielectric function of the slab is an arbitrary complex function of frequency-satisfying Kramers–Kronig relations. For the atoms with the electric dipole moment perpendicular and parallel to the slab, the vector potential operator commutation relations are used to drive the spontaneous emission rate of an atom via Fermi’s golden rule. By choosing a Lorentz model for the dielectric function of the slab, the typical spatial variations of the spontaneous emission rates near the slab are shown.

Single power-conversion DAB microinverter with safe commutation and high efficiency for PV power applications

This paper presents a single power-conversion dual-active-bridge (DAB) microinverter with safe commutation and high efficiency for PV power applications. In DAB microinverters, the commutation problem is a crucial issue which must be solved in order to achieve a higher power capability. Owing to its novel circuit structure, the proposed microinverter can provide the commutation path in any switching case, so that it avoids the high voltage spikes from the commutation problem and provides high reliability. Moreover, the proposed microinverter has a circuit structure that is simpler and requires fewer auxiliary circuits compared to conventional DAB microinverters. In addition, the proposed microinverter achieves simple closed-loop current control and zero-voltage-switching turn-on of all switches owing to its control algorithm. Because these advantages are obtained with only a single power-conversion, the proposed microinverter has high efficiency and simple structure with high reliability. The operation principle and commutation issue of the proposed microinverter are theoretically analyzed in detail, and experimentally validated on a 400-W prototype.

IIB flux non-commutativity and the global structure of field theories

We discuss the origin of the choice of global structure for six dimensional (2, 0) theories and their compactifications in terms of their realization from IIB string theory on ALE spaces. We find that the ambiguity in the choice of global structure on the field theory side can be traced back to a subtle effect that needs to be taken into account when specifying boundary conditions at infinity in the IIB orbifold, namely the known non-commutativity of RR fluxes in spaces with torsion. As an example, we show how the classification of mathcal{N} = 4 theories by Aharony, Seiberg and Tachikawa can be understood in terms of choices of boundary conditions for RR fields in IIB. Along the way we encounter a formula for the fractional instanton number of mathcal{N} = 4 ADE theories in terms of the torsional linking pairing for rational homology spheres. We also consider six-dimensional (1, 0) theories, clarifying the rules for determining commutators of flux operators for discrete 2-form symmetries. Finally, we analyze the issue of global structure for four dimensional theories in the presence of duality defects.

Counterexamples Related to Commutators of Unbounded Operators

The present paper is exclusively devoted to examples and counterexamples about commutators and self commutators of unbounded operators on a Hilbert space. As a bonus, we provide a simpler counterexample than McIntosh’s famous example obtained some while ago.

Commutative algebraic monoid structures on affine spaces

We study commutative associative polynomial operations [Formula: see text] with unit on the affine space [Formula: see text] over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

Einstein gravity from ANEC correlators

We study correlation functions with multiple averaged null energy (ANEC) operators in conformal field theories. For large N CFTs with a large gap to higher spin operators, we show that the OPE between a local operator and the ANEC can be recast as a particularly simple differential operator acting on the local operator. This operator is simple enough that we can resum it and obtain the finite distance OPE. Under the large N - large gap assumptions, the vanishing of the commutator of ANEC operators tightly constrains the OPE coefficients of the theory. An important example of this phenomenon is the conclusion that a = c in d = 4. This implies that the bulk dual of such a CFT is semi-classical Einstein-gravity with minimally coupled matter.