BOUNDEDNESS AND COMPACTNESS FOR THE COMMUTATOR OF THEω-TYPE CALDER′ON-ZYGMUND OPERATOR ON LORENTZ SPACE

In this paper,the authors consider theω-type Calder´on-Zygmund operator T_(ω)and the commutator[b,T_(ω)]generated by a symbol function b on the Lorentz space L^(p,r)(X)over the homogeneous space(X,d,μ).The boundedness and the compactness of the commutator[b,T_(ω)]on Lorentz space L^(p,r)(X)are founded for any p∈(1,∞)and r∈[1,∞).

LITTLEWOOD-PALEY CHARACTERIZATIONS OF ANISOTROPIC HARDY-LORENTZ SPACES

Let p∈（0, 1], q∈（0, ∞] and A be a general expansive matrix on Rn. Let HAp,q （Rn） be the anisotropic Hardy-Lorentz spaces associated with A defined via the non-tangential grand maximal function. In this article, the authors characterize HAp,q（Rn） in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley gλ＊-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space Lp,q（Rn）. All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on Rn. Moreover, the range of λ in the gλ＊-function characterization of HAp,q （Rn） coincides with the best known one in the classical Hardy space Hp（Rn） or in the anisotropic Hardy space HAp （Rn）.

MULTIPLIERS AND RELATIVE COMPLETION IN WEIGHTED LORENTZ SPACES

Let G be a locally compact Abelian group with Haar measure μ. In the present paper, first the authors discussed some properties of weighted Lorentz space. Then they defined the relative completion A of a subspace A of the weighted Lorentz space, and showed that the space of the multipliers from L_w~1,(G) to A is algebrically isomorphic and homeomorphic to A.

some Results on Embedding Spaces of Sobolev Type into Lorentz Spaces

Lorentz spaces L^(s,l) provide a deeper insight into integrable functions than L^s spaces do. Various results of the embedding of the following anisotropic spaces B_(p,).~ _p^and Ⅳ_p~ in- to L^(s,l) together with a counterexample are obtained.

THE DUAL OF NONCOMMUTATIVE LORENTZ SPACES

We prove some noncommutative analogues of the classical results about dual spaces of the classical Lorentz spaces.

SZEG TYPE FACTORIZATION THEOREM FOR NONCOMMUTATIVE HARDY-LORENTZ SPACES

We introduce noncommutative Hardy-Lorentz spaces and give the Szegō and inner-outer type factorizations of these spaces.

WEAK ATOMIC DECOMPOSITIONS OF MARTINGALE HARDY-LORENTZ SPACES AND APPLICATIONS

In this article, we establish some atomic decomposition theorems for martin- gale Hardy-Lorentz spaces. As applications, with the help of weak atomic decompositions, some interpolation theorems for martingale Hardy-Lorentz spaces are proved.

MULTIPLIERS AND TENSOR PRODUCTS OF L(p,q) LORENTZ SPACES

Let G be a locally compact abelian group. The main purpose of this article is to find the space of multipliers from the Lorentz space. L（p1, q1）（G） to L（p＇2, q＇2）（G）. For this reason, the authors define the space A p1,q1^ p2,p2（G）, discuss its properties and prove that the space of multipliers from L（p1, q1）（G） to L（p＇2, q＇2）（G） is isometrically isomorphic to the dual of A p1,q1^p2,q2 （G）.

On M–Term Approximations of the Nikol'skii–Besov Class in the Lorentz Spaces

In this paper, we consider a Lorentz space with a mixed norm of periodic functions of many variables. We obtain the exact estimation of the best M-term approximations of Nikol＇skii＇s and Besov＇s classes in the Lorentz space with the mixed norm.

On Approximation of Function Classes in Lorentz Spaces with Anisotropic Norm

In this paper, Lorentz space of functions of several variables and Besov＇s class are considered. We establish an exact approximation order of Besov＇s class by partial sums of Fourier＇s series for multiple trigonometric system.

Complete Spacelike Hypersurfaces with Constant Scalar Curvature in Locally Symmetric Lorentz Spaces

The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L1 introduced by S Y Cheng and S T Yau [3].

Linear Weingarten spacelike hypersurface in locally symmetric Lorentz space

The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an estimate of the upper bound of the function L(nH)is given,where L is a second-order differential operator.Then,under the assumption that the square norm of the second fundamental form is bounded by a given positive constant,it is proved that Mn must be either totally umbilical or contain two distinct principle curvatures,one of which is simple.Moreover,a similar result is obtained for complete noncompact spacelike hypersurfaces in locally symmetric Einstein spacetime.Hence,some known rigidity results for hypersurface with constant scalar curvature are extended for the linear Weingarten case.

Existence of Weak Solutions of Two-dimensional Euler Equations with Initial Vorticity in Lorentz Space L_(2,1)(R^2)

Some estimates on 2-D Euler equations are given when initia l vorticity ω0 belongs to a Lorentz space L(2,1). Then based on these estimates, it is proved that there exist global weak solutions of two dim ensional Euler equations when ω0∈L(2,1).

Regularity for 3-D MHD Equations in Lorentz Space L^(3,∞)

.The regularity for 3-D MHD equations is considered in this paper,it is proved that the solutions(v,B,p)are Holder continuous if the velocity field v∈L^(∞)(0,T;L^(3,∞)_(x)(R^(3))with local small condition r^(-3)|{x∈B_(r)(x_(0):|v(x,t_(0))|>εr^(-1)}|≤ε and the magnetic field B∈L^(∞)(0,T;VMO^(-1)(R^(3)).

Cartan-Hadamard流形上关于Lorentz范数的Trudinger-Moser不等式

本文研究了Cartan-Hadamard流形上带Lorentz范数的Trudinger-Moser不等式.利用了相关格林函数的逐点估计以及O’Neil不等式,我们得到了该不等式的最佳常数,推广了相应欧氏空间上的结果.

The Vertical Projection Model of Hyperbolic Space and Geometric Illustration of Special Relativity

In this article, new visual and intuitive interpretations of Lorentz transformation and Einstein velocity addition are given. We first obtain geometric interpretations of isometries of vertical projection model of hyperbolic space, which are the analogues of the geometric construction of inversions with respect to a circle on the complex plane. These results are then applied to Lorentz transformation and Einstein velocity addition to obtain geometric illustrations. We gain new insights into the relationship between special relativity and hyperbolic geometry.

LORENTZ MARTINGALE SPACES AND INTERPOLATION

In this article, the authors introduce some new Lorentz spaces for martingales, which are extensions of Hardy spaces of martingales. Then they discuss their basic properties, embedding relationships, and interpolation spaces between them, during which the use of rearrangement good-λ-inequality plays an important role.

An Off-Diagonal Marcinkiewicz Interpolation Theorem on Lorentz Spaces

Let （X, μ） be a measure space. In this paper, using some ideas from Grafakos and Kalton, the authors establish an off-diagonal Marcinkiewicz interpolation theorem for a quasilinear operator T in Lorentz spaces L^P,q（X） with p, q ￡ （0, ∞], which is a corrected version of Theorem 1.4.19 in [Grafakos, L.： Classical Fourier Analysis, Second Edition, Graduate Texts in Math., No. 249, Springer, New York, 2008] and which, in the case that T is linear or nonnegative sublinear, p ￡ [1, ∞） and q ￡ [l,∞）, was obtained by Stein and Weiss [Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971].

Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H^(p,q)_A(R^n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H^(p1,q1)_A(Rn) and H^(p2,q2)_A(R^n) with 0 < p1 < p < p2 < ∞ and q1, q, q2 ∈(0, ∞], and also between H^(p,q1)_A(Rn) and H^(p,q2)_A(R^n) with p ∈(0, ∞)and 0 < q1 < q < q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H^(p,q)_A(R^n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H^(p,∞)_A(R^n) to the weak Lebesgue space L^(p,∞)(R^n)(or to H^p_A(R^n)) in the ln λcritical case, from H^(p,q)_A(R^n) to L^(p,q)(R^n)(or to H^(p,q)_A(R^n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H^(p,q)_A(R^n) to L^(p,∞)(R^n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.

On Mixed Pressure-Velocity Regularity Criteria to the Navier-Stokes Equations in Lorentz Spaces

In this paper the authors derive regular criteria in Lorentz spaces for LerayHopf weak solutions v of the three-dimensional Navier-Stokes equations based on the formal equivalence relationπ≌|v|^(2),whereπdenotes the fluid pressure and v denotes the fluid velocity.It is called the mixed pressure-velocity problem(the P-V problem for short).It is shown that if(π/(e-^|(x)|^(2)+|v|^(θ)∈L^(p)(0,T;L^(q,∞)),where 0≤θ≤1 and 2/p+3/q=2-θ,then v is regular on(0,T].Note that,ifΩ,is periodic,e^(-|x|)^(2) may be replaced by a positive constant.This result improves a 2018 statement obtained by one of the authors.Furthermore,as an integral part of the contribution,the authors give an overview on the known results on the P-V problem,and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin(L-P-S for short)type.