On Turán type inequality with doubling weights and A weights

Let Hn be the set of real algebraic polynomials of degree n, whose zeros all lie in the interval [-1,1]. The well known Turán type inequalities tell us that forf（x）∈Hn, it holds ‖f＇‖≥C√n‖f‖. This note deals with the weighted Turán type inequalities with the weights having inner singularities under L^p norm for 0〈p≤∞. Our results essentially extend the result of Wang and Zhou （2002）, and the method used in this paper is simpler and more direct than that of Wang and Zhou （2002）. The results and methods have their own values in approximation theory and computation.

Vanishing Theorems on Smooth Metric Measure Spaces with a Weighted p-Poincare Inequality

This paper deals with vanishing results for Lf^2 harmonic p-forms on complete metric measure spaces with a weighted p-Poincare inequality.Some results are without curvature assumptions for 1■p■n-1 and the others are with assumptions on lower bound of the m-Bakry-Emery Ricci curvature for p=1.These are weighted version for the corresponding results of the present author(J.Math.Anal.Appl.,2020,490).

A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane

In this paper, by introducing some parameters and estimating the weight coefficient, we give a new Hilbert’s inequality with the integral in whole plane and with a non-homogeneous and the equivalent form is given as well. The best constant factor is calculated by the way of Complex Analysis.

A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES

In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form ■holds a.e. for uniformly integrable martingales f =(f_n)n≥0 with some constant C > 0,where Φ_1,Φ_2 are Young functions, w_i(i = 1,2,3, 4) are weights, f~* =sup n≥0 |f_n| and f_∞=lim n→∞ f_n a.e. As an application, two-weight weak type maximal inequalities of martingales are considered, and particularly a new equivalence condition is presented.

HARDY-SOBOLEV INEQUALITIES WITH GENERAL WEIGHTS AND REMAINDER TERMS

The Hardy-Sobolev inequality with general weights is established, and it is shown that the constant is optimal. The two weights in this inequality are determined by a Bernoulli equation. In addition, the authors obtain the Hardy-Sobolev inequality with general weights and remainder terms. By choosing special weights, it turns to be many versions of the Hardy-Sobolev inequality and the Caffarelli-Kohn-Nirenberg inequality with remainder terms in the literature.

Generalizations of mixed weighted power mean inequality

Tarnavas established mixed weighted power mean inequality in 1999. A separation of weighted power mean inequslity was derived in this paper. As its applications, some separations of other inequalities were given.

WEIGHTED LASSO ESTIMATES FOR SPARSE LOGISTIC REGRESSION:NON-ASYMPTOTIC PROPERTIES WITH MEASUREMENT ERRORS

For high-dimensional models with a focus on classification performance,the?1-penalized logistic regression is becoming important and popular.However,the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data.We propose two types of weighted Lasso estimates,depending upon covariates determined by the Mc Diarmid inequality.Given sample size n and a dimension of covariates p,the finite sample behavior of our proposed method with a diverging number of predictors is illustrated by non-asymptotic oracle inequalities such as the?1-estimation error and the squared prediction error of the unknown parameters.We compare the performance of our method with that of former weighted estimates on simulated data,then apply it to do real data analysis.

Hilbert-type integral inequality with the homogeneous kernel of 0-degree

Through using weight function, we give a new Hilbert-type integral inequality with two independent parameters and two pair of conjugate exponents, which is a best extension of a Hilbert-type integral inequality with the homogeneous kernel of 0-degree. The equivalent form, the reverses and some particular results are considered.

Extension of reverse Hilbert-type inequality with some parameters

By introducing some parameters and estimating the weight function,we obtain an extension of reverse Hilbert-type inequality with the best constant factor.As applications,we build its equivalent forms and some particular results.

On Weighted Approximation by Modified Bernstein Operators for Functions with Singularities

Della Vecchia et al. （see [2]） introduced a kind of modified Bernstein operators which can be used to approximate functions with singularities at endpoints on [0,1]. In the present paper, we obtain a kind of pointwise Stechkin-type inequalities for weighted approximation by the modified Bemsetin operators.

A Study of Weighted Polynomial Approximations with Several Variables (II)

In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomial. Then we will give some results relating to the Lagrange interpolation, the best approximation, the Markov-Bernstein inequality and the Nikolskii- type inequality.

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.

The Best Constant of Discrete Sobolev Inequality on a Weighted Truncated Tetrahedron

The best constant of discrete Sobolev inequality on the truncated tetrahedron with a weight which describes 2 kinds of spring constants or bond distances. Main results coincides with the ones of known results by Kametaka et al. under the assumption of uniformity of the spring constants. Since the buckyball fullerene C60 has 2 kinds of edges, destruction of uniformity makes us proceed the application to the chemistry of fullerenes.

Rearrangement and the weighted logarithmic Sobolev inequality

Here we consider some weighted logarithmic Sobolev inequalities which can be used in the theory of singular Riemanian manifolds.We give the necessary and sufficient conditions such that the 1-dimension weighted logarithmic Sobolev inequality is true and obtain a new estimate on the entropy.

Fundamental Solution for Weighted Baouendi-Grushin Type Operators and a Class of Weighted Hardy Inequality

In this paper we obtain the fundamental solution for a class of weighted BaouendiGrushin type operator L_(p,γ,α)u = ▽_γ·(|▽_γu|^(p-2)ρ~α▽_γu) on R^(m+n )with singularity at the origin,where ▽_γ is the gradient operator defined by ▽_γ =(▽_x,|x|~γ▽_y) and ρ is the distance function.As an application,we get some Hardy type inequalities associated with ▽_γ.

A New Weight Initialization Method Using Cauchy’s Inequality Based on Sensitivity Analysis

In this paper, an efficient weight initialization method is proposed using Cauchy’s inequality based on sensitivity analy- sis to improve the convergence speed in single hidden layer feedforward neural networks. The proposed method ensures that the outputs of hidden neurons are in the active region which increases the rate of convergence. Also the weights are learned by minimizing the sum of squared errors and obtained by solving linear system of equations. The proposed method is simulated on various problems. In all the problems the number of epochs and time required for the proposed method is found to be minimum compared with other weight initialization methods.

INTEGRAL INEQUALITIES WITH WEIGHTS FOR MAXIAMAL OPERATORS

Let μ be a measure on the upper half-space R+n+1,and v a weight on Rn,we give a characterization for the pair (v,μ) such that ||μ(fv)||L(μ)≤c||f||L(μ)where is an N-function satisfying Δ2 condition and uf(x,t) is the maximal function on R+n+1, which was introduced by Ruiz,F. and Torrea, J.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

TWO-WEIGHT WEAK-TYPE MAXIMAL INEQUALITIES FOR MARTINGALES

In this article, some necessary and sufficient conditions are shown in order that the inequality of the form Ф1（λ）Pu（f^＊〉λ）≤Ev （Ф2（C｜f∞｜）） holds with some constant C 〉 0 independent of martingale f = （fn）n≥0 and λ 〉 0, where Фl and Ф2 are a pair of Young functions, f^＊=sup n≥0｜fn｜ adn f∞=lim n→∞ fn a.e.

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