WEIGHTED NORM INEQUALITIES FOR THE COMMUTATORS OF MULTILINEAR SINGULAR INTEGRAL OPERATORS

In this paper, the authors consider the weighted estimates for the commutators of multilinear Calderón-Zygmund operators.By introducing an operator which shifts the commutation, and establishing the weighted estimates for this new operator, the authors prove that, if p_1 ∈ （1,∞）, p_2,…,p_m ∈（1,∞], p ∈ （0,∞） with 1/p =Σ1≤k≤ m 1/pk, then for any weight w, the commutators of m-linear Galderón-Zygmund operator are bounded from L P1（R n,M_l（logL） σw）× p2（Rn,M~w）×...×Lpm（Rn,Mw） to Lp（Rn,w）with σ to be a constant depending only on p_1 and the order of commutator

Weighted Norm Inequalities of the Maximal Commutator of Quasiradial Bochner-Riesz Operators

In this paper, we obtain certain Lpw（Rn）-mapping properties for the maximal operator associated with the commutators of quasiradial Bochner-Riesz means with index δ under certain surface condition on Σed , provided that δ （n-1）/2, b ∈ BMO（Rn）, 1 p ∞ and w ∈ A1 . Moreover, if δ （n-1）/2, then we prove that the above maximal operator admits weak type （H1w（Rn）, L1w（Rn））-mapping properties for b ∈ BMO（Rn） and w ∈ A1 under the surface condition on Σed .

Weighted Norm Inequalities for Some Singular Integral Operators

In this paper,we consider the related weighted boundedness of the operators which were studied by J.Duoandikotxea and J.L.Ruhio de Francia.

Weighted Norm Inequalities with General Weights for the Commutator of Calderón

In this paper, by a sharp function estimate and an idea of Lerner, the authors establish some weighted estimates for the m-multilinear integral operator which is bounded from L1 （Rn） ×..…… L1 （Rn） to L1/m,∞（Rn）, and the associated kernel K（x; yl,.. , Ym） enjoys a regularity on the variable x. As an application, weighted estimates with general weights are given for the commutator of CalderSn.

Some Weighted Norm Inequalities on Manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property and the Gaussian upper bounds. Denote by A the Laplace-Beltrami operator and by V the Riemannian gradient. In this paper, the author proves the weighted reverse inequality ｜｜Δ^1/2f｜｜L^p（w）×C｜｜｜Δ↓｜｜｜L^p（w）, for some range ofp determined by M and w. Moreover, a weak type estimate is proved when p = 1. Some weighted vector-valued inequalities are also established.

A WEIGHTED NORM INEQUALITY FOR THETA(t)-TYPE OSCILLATORY SINGULAR INTEGRALS

The theta (t)-type oscillatory singular integral operators has been discussed. With the non-negative locally integrable weighted function, the weighted norm inequality of theta ( t) -type oscillatory singular integral operators is proved, and the weighted function has replaced by action of Hardy-Littlewood maximal operators several times.

VARIABLE STEP-SIZE BDF3 METHOD FOR ALLEN-CAHN EQUATION

In this work,we analyze the three-step backward differentiation formula(BDF3)method for solving the Allen-Cahn equation on variable grids.For BDF2 method,the discrete orthogonal convolution(DOC)kernels are positive,the stability and convergence analysis are well established in[Liao and Zhang,Math.Comp.,90(2021),1207–1226]and[Chen,Yu,and Zhang,arXiv:2108.02910,2021].However,the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels.By developing a novel spectral norm inequality,the unconditional stability and convergence are rigorously proved under the updated step ratio restriction rk:=τk/τk−1≤1.405 for BDF3 method.Finally,numerical experiments are performed to illustrate the theoretical results.To the best of our knowledge,this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.

TWO-WEIGHT NORM INEQUALITY FOR IMAGINARY POWERS OF A LAPLACE OPERATOR

We study two-weight norm inequality for imaginary powers of a Laplace operator in R^n, n ≥ 1, especially from weighted Lebesgue space Lv^p（R^n） to weighted Lebesgue space Lμ^p（R^n）, where 1 〈 p 〈 ∞. We prove that the two-weighted norm inequality holds whenever for some t 〉 1, （μ^t, v^t） ∈ Ap, or if （μ, v） ∈Ap, where μ and v^-1/（p-1） satisfy the growth condition and reverse doubling property.

L^2 NORM INEQUALITY WITH POWER WEIGHTS FOR THE MAXIMAL RIESZ SPHERICAL MEANS

We have pointed out in [1] that so far the L^2 norm inequalities with power weights for the Riesz means σ_R~δ(g)(x) of multiple Fourier integrals have been obtained only by Hirschman and J. L. Rubio de Francia respectively:

Notes on the Norm Estimates for the Sum of Two Matrices

This is a lecture note of my joint work with Chi-Kwong Li concerning various results on the norm structure of n 2 n matrices (as Hilbert-space operators). The main result says that the triangle inequality serves as the ultimate norm estimate for the upper bounds of summation of two matrices. In the case of summation of two normal matrices, the result turns out to be a norm estimate in terms of the spectral variation for normal matrices.

Weighted Estimates for Iterated Commutators of Multilinear Fractional Operators

The author establishes weighted strong type estimates for iterated commutators of multi- linear fractional operators.

Multiple weighted estimates for commutators of multilinear fractional integral operators

Multilinear commutators and iterated commutators of multilinear fractional integral operators with BMO functions are studied. Both strong type and weak type endpoint weighted estimates involving the multiple weights for such operators are established and the weak type endpoint results are sharp in some senses. In particular, we extend the results given by Cruz-Uribe and Fiorenza in 2003 and 2007 to the multilinear setting. Moreover, we modify the weak type of endpoint weighted estimates and improve the strong type of weighted norm inequalities on the multilinear commutators given by Chen and Xue in 2010 and 2011.

Littlewood-Paley Operators on Weighted Lipschitz Spaces

Littlewood-Paley operators, the g-function, the area integral and the function g^＊λ, are considered as operators on weighted Lipschitz spaces. It is proved that the image of a weighted Lipschitz function under one of these operators is either equal to infinity almost everywhere or is in weighted Lipschitz spaces.

Multi-weight, Weighted Weak Type Estimates for the Multilinear Calderón-Zygmund Operators

Let m be an integer and T be an m-linear Calderón-Zygmund operator, u, v1,..., vm be weights. In this paper, the authors give some sufficient conditions on the weights （u, vk） with 1 ≤ k ≤ m, such that T is bounded from Lp1（Rn, v1） × ··· × Lpm（Rn, vm） to Lp,∞（Rn, u）.

Weighted estimates for the multilinear singular integral operators with non-smooth kernels

In this paper, by sharp function estimates and certain weak type endpoint estimates, the authors establish some weighted norm inequalities with Ap weights for the multilinear singular integral operators with non-smooth kernels.