Order Commutators Research Articles

Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators

We deal with mixed weak estimates of Fefferman-Stein type for higher order commutators of Calderón-Zygmund operators with BMO symbol. The results obtained are Fefferman-Stein inequalities that include the estimates proved in [7] for the case of singular integral operators, as well as the classical weak endpoint estimate for commutators given in [26]. We also consider commutators of operators involving less regular kernels satisfying an LΦ–Hörmander condition. Particularly, the obtained results contain some previous estimates proved in [7] and [17].

Higher order corrections to KPV: The nonabelian brane stack perspective

In this work, we study the decay of D3¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{D3} $$\\end{document}-branes in the setup of Kachru, Pearson, and Verlinde (KPV) at higher order in α′ from the perspective of a nonabelian D3¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{D3} $$\\end{document}-brane stack. We extend the leading order analysis of KPV by including higher order commutators as well as higher derivative corrections. Recently, the KPV setup has been studied at higher order in α′ from the NS5-brane perspective. It was found that in order to control α′ corrections the quantity gsM2 determining the amount of warping in the Klebanov-Strassler throat has to be much larger than expected. This leads to serious issues when using the D3¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{D3} $$\\end{document}-branes as an uplift to dS. The benefit of the analysis in this work is that the D3¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{D3} $$\\end{document}-brane perspective is controlled when the distance between the branes inside the brane stack is substringy which is a regime not controlled on the NS5-brane side. As a main result, we find that the strong bound gsM2 ~ O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(100) obtained on the NS5-brane also holds in the regime accessible from the D3¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{D3} $$\\end{document}-brane perspective. We also show that the novel way of uplifting proposed in the recent work on α′ corrections to the KPV setup can only work for small warped throats.

Bloom weighted bounds for sparse forms associated to commutators

In this paper we consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner–Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón–Zygmund operators, the previously known quantitative Bloom bounds are not sharp for the second and higher order commutators.

Intermittency and Lower Dimensional Dissipation in Incompressible Fluids

In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as “intermittency”, and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents ζp=p/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _p={p}/{3}$$\\end{document} might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in every p-th order structure function for p>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>3$$\\end{document}. By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor’s frozen turbulence hypothesis, our strongest upper bound on ζp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta _p$$\\end{document} coincides with the β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally, we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on the p-th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.

On Sobolev theorem for higher commutators of fractional integrals in grand variable Herz spaces

The higher order commutators of fractional integral operator Iζ(⋅),Vm of variable order is shown to be bounded from the grand variable Herz spaces K̇p(⋅)a(⋅),u,θ(Rn) into the weighted space K̇ρ,q(⋅)a(⋅),u,θ(Rn), where ρ=(1+|z1|)−λ and 1q(z)=1p(z)−β(z)n when p(z) is not necessarily constant at infinity.

IS LIFE IMPRISONMENT WITHOUT PAROLE OR COMMUTATION AN EFFECTIVE ANTI-CORRUPTION MEASURE FOR CHINA?

New legislation adopting a tough criminal stance on the crimes of corruption and bribery responds to the need for strengthening Chinese anti-corruption work. The ninth amendment to the Chinese Criminal Law Code, which adds life imprisonment without parole or commutation, has received broad support from all sectors of society. The aim of the amendment, as stated by the legislature, is to safeguard judicial fairness and prevent criminals convicted of the most serious corruption offences from having their prison sentences shortened through commutation. This stated legislative aim is not acceptable. Whether from the perspective of deterrence or alternative measures to the death penalty, the approach of adding life imprisonment without parole or commutation in order to punish the corrupt is not ideal. This article argues that, given the particular nature of crimes of corruption and given the disadvantages of life imprisonment without parole or commutation, restraint in the use of the death penalty does not mean that life imprisonment has to be used as an alternative measure. Moreover, the death penalty with a two-year reprieve is sufficient for realising retribution and prevention as aims of punishment.

Global well-posedness of the viscous Camassa–Holm equation with gradient noise

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa--Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-It\^{o} correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional It\^{o} formula and a DiPerna--Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

English

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G' = C(3^e) x C(3), 2 <= e <= 6, according to the algebraic invariant kappa(G), admits conclusions concerning the length of the Hilbert 3-class field tower F(3,infty,K) of imaginary quadratic number fields K.

New oscillation classes and two weight bump conditions for commutators

In this paper we consider two weight bump conditions for higher order commutators. Given b and a Calderón–Zygmund operator T, define the commutator $$T^1_bf=[T,b]f= bTf-T(bf)$$ , and for $$m\ge 2$$ define the iterated commutator $$T^m_b f = [b,T_b^{m-1}]f$$ . Traditionally, commutators are defined for functions $$b\in BMO$$ , but we show that if we replace BMO by an oscillation class first introduced by Pérez (J Funct Anal 128(1):163–185, 1995), we can give a range of sufficient conditions on a pair of weights (u, v) for $$T^m_b : L^p(v) \rightarrow L^p(u)$$ to be bounded. Our results generalize work of Cruz-Uribe and Moen (Publ Mat 56(1):147–190, 2012), and more recent work by Lerner et al. (J Funct Anal 281(8):46, 2021). We also prove necessary conditions for the iterated commutators to be bounded, generalizing results of Isralowitz et al. (Commutators in the two scalar and matrix weighted setting. Preprint, 2020. http://arxiv.org/abs/2001.11182 ).

On two weight estimates for iterated commutators

In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators Tbm. Our results are new even for the first order commutator Tb1. A new bump type necessary condition for the two-weighted boundedness of Tbm is obtained as well. We also provide some results related to a converse to Bloom's theorem.

Preference heterogeneity in trip makers’ perception and policy Issues: A study with reference to bus services in Kolkata

The bus services in most of the cities in emerging countries are facing challenges in terms of financial viability and ridership. On the other hand, the increasing usage of private vehicle is aggravating the congestion and pollution in the urban areas. Therefore, there is a need for improving the bus service in accordance to the requirement of commuters in order to make it an attractive mode of transport. The paper presents an investigation on trip makers’ preference heterogeneity towards various attributes of bus service using stated preference choices, identify the differences in trip makers’ requirements based on trip, socioeconomic, and demographic characteristics, and accordingly suggest suitable policy measures for improving bus service. The stated preference data were analyzed by developing random parameter logit (RPL) models accounting for heterogeneity, and trip makers’ Willingness-to-pay (WTP) for improving bus service attributes were calculated. The study reveals existence of significant heterogeneity in commuters’ perceptions towards improving different attributes of bus service with respect to trip purpose, age, gender, and frequency of using bus. Trip makers’ WTP for time savings was found higher for work trips indicating the need for adopting efficient traffic management measures during peak hours. Female commuters were found to value comfort during travel much higher as compared to male commuters which justifies the existing ‘female only’ seat reservation policy in Kolkata and emphasizes the need for further initiatives such as women special services, especially during peak hours.

P.509 Centenary Psychosis

The bus services in most of the cities in emerging countries are facing challenges in terms of financial viability and ridership. On the other hand, the increasing usage of private vehicle is aggravating the congestion and pollution in the urban areas. Therefore, there is a need for improving the bus service in accordance to the requirement of commuters in order to make it an attractive mode of transport. The paper presents an investigation on trip makers’ preference heterogeneity towards various attributes of bus service using stated preference choices, identify the differences in trip makers’ requirements based on trip, socioeconomic, and demographic characteristics, and accordingly suggest suitable policy measures for improving bus service. The stated preference data were analyzed by developing random parameter logit (RPL) models accounting for heterogeneity, and trip makers’ Willingness-to-pay (WTP) for improving bus service attributes were calculated. The study reveals existence of significant heterogeneity in commuters’ perceptions towards improving different attributes of bus service with respect to trip purpose, age, gender, and frequency of using bus. Trip makers’ WTP for time savings was found higher for work trips indicating the need for adopting efficient traffic management measures during peak hours. Female commuters were found to value comfort during travel much higher as compared to male commuters which justifies the existing ‘female only’ seat reservation policy in Kolkata and emphasizes the need for further initiatives such as women special services, especially during peak hours.

Groups with boundedly many commutators of maximal order

By a result of Cocke and Venkataraman we know that if G is a group with at most m elements of maximal order, then |G| is m-bounded. In this paper we consider the following related setting. Suppose G is a group with at most m commutators whose order is maximal among all commutators. What can we say about the structure of the group G?

A revisit on the compactness of commutators

AbstractA new characterization of$\text {CMO}(\mathbb R^n)$is established replying upon local mean oscillations. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.

On the optimal numerical parameters related with two weighted estimates for commutators of classical operators and extrapolation results

We give two-weighted norm estimates for higher order commutator of classical operators such as singular integral and fractional type operators, between weighted $$L^p$$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required. Finally, we exhibit an extrapolation result that allows us to obtain boundedness results of the type described above in the variable setting and for a great variety of operators, by starting from analogous inequalities in the classical context. In order to get this result we prove a Calderon–Scott type inequality with weights that connects adequately the spaces involved.

Off-diagonal estimates for the first order commutators in higher dimensions

In this paper we study natural generalizations of the first order Calderón commutator in higher dimensions d≥2. We study the bilinear operator Tm which is given byTm(f,g)(x):=∬R2d[∫01m(ξ+tη)dt]fˆ(ξ)gˆ(η)e2πix⋅(ξ+η)dξdη. Our results are obtained under two different conditions of the multiplier m. The first result is that when K∈S′∩Lloc1(Rd∖{0}) is a regular Calderón-Zygmund convolution kernel of regularity 0<δ≤1, TKˆ maps Lp(Rd)×Lq(Rd) into Lr(Rd) for all 1<p,q≤∞, 1r=1p+1q as long as r>dd+1. The second result is that when the multiplier m∈Cd+1(Rd∖{0}) satisfies the Hörmander derivative conditions|∂ξαm(ξ)|≤Dα|ξ|−|α| for all ξ≠0, and for all multi-indices α with |α|≤d+1, Tm maps Lp(Rd)×Lq(Rd) into Lr(Rd) for all 1<p,q≤∞, 1r=1p+1q as long as r>dd+1. These two results are sharp except for the endpoint case r=dd+1. In case d=1 and K(x)=1/x, it is well-known that TKˆ maps Lp(R)×Lq(R) into Lr(R) for 1<p,q≤∞, 1r=1p+1q as long as r>1/2. In higher dimensional case d≥2, in 2016, when Kˆ(ξ)=ξj/|ξ|d+1 is the Riesz multiplier on Rd, P. W. Fong, in his Ph.D. Thesis [9], obtained‖TKˆ(f,g)‖r≤C‖f‖p‖g‖q for 1<p,q≤∞ as long as r>d/(d+1). As far as we know, except for this special case, there has been no general results for the off-diagonal case r<1 in higher dimensions d≥2. To establish our results we develop ideas of C. Muscalu and W. Schlag [18,19] with new methods.

Hilbert-Haar coordinates and Miranda's theorem in Lie groups

We study the interior regularity of solutions to a class of quasilinear equations of non-degenerate p-Laplacian type on Lie groups that admit a system of Hilbert-Haar coordinates. These are coordinates with respect to which every linear function has zero symmetrized second order horizontal derivatives. All Carnot groups of rank two are in this category, as well as the Engel group, the Goursat type groups, and those general Carnot groups of step three for which the non-zero commutators of order three are linearly independent.

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrödinger Type Operators

Fractional maximal operator and its higher order commutators in generalized weighted Morrey spaces on Heisenberg group

Fractional maximal operator and its higher order commutators in generalized weighted Morrey spaces on Heisenberg group

Riesz potential, Marcinkiewicz integral and their commutators on mixed Morrey spaces

This paper deals with the boundedness of integral operators and their commutators in the framework of mixed Morrey spaces. Precisely, we study the mixed boundedness of the commutator [b,I?], where I? denotes the fractional integral operator of order ? and b belongs to a suitable homogeneous Lipschitz class. Some results related to the higher order commutator [b,I?]k are also shown. Furthermore, we examine some boundedness properties of the Marcinkiewicz-type integral ?? and the commutator [b,??] when b belongs to the BMO class.