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.

HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING

In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.

THE BERNSTEIN TYPE INEQUALITY AND SIMULTANEOUS APPROXIMATION BY INTERPOLATION POLYNOMIALS IN COMPLEX DOMAIN

In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.

WEAK TYPE INEQUALITY FOR THE MAXIMAL OPERATOR OF WALSH-KACZMARZ-MARCINKIEWICZ MEANS

The main aim of this article is to prove that the maximal operator σ^k＊ of the Marcinkiewicz-Fejer means of the two-dimensional Fourier series with respect to Walsh- Kaczmarz system is bounded from the Hardy space H2/3 to the space weak-L2/3.

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.

Hàjek-Rèniy Type Inequality for Banach Space Valued Martingales and Its Applications

In this paper,we establish the Hàjek-Rèniy type inequality for Banach space valued martingales generalizing the recent results of Tómcs and L'ibor [1].Then p-uniformly smoothable Banach space is characterized in terms of the Hàjek-Rèniy type inequality for Banach space valued martingales.Those results generalize the recent results of Gan Shixin [2].

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.

PLANCHEREL-PoLYA TYPE INEQUALITY AND ITS APPLICATIONS

In this note, using the discrete Calderon type reproducing formula, we prove the Plancherct-Poly a type inequality with the minimum regularity. As a consequence, we give a new characterization of the Besov and Triebel -Lizorkin spaces by use of the minimum regularity.

MULTIVARIATE WEIGHTED BERNSTEIN-TYPE INEQUALITY AND ITS APPLICATIONS

Bernstein inequality played an important role in approximation theory and Fourier analysis. This article first introduces a general system of functions and the socalled multivariate weighted Bernstein, Nikol＇skii, and Ul＇yanov-type inequalities. Then, the relations among these three inequalities are discussed. Namely, it is proved that a family of functions equipped with Bernstein-type inequality satisfies Nikol＇skii-type and Ul＇yanov-type inequality. Finally, as applications, some classical inequalities are deduced from the obtained results.

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 Turan type inequalities tell us that forf(x)∈Hn, it holds ‖f′‖≥ C√n‖f‖. This note deals with the weighted Turan type inequalities with the weights having inner singularities under Lp 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.

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.

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.

Integral Inequality of Hardy-Type on Time Scales

Inequalities are essential in the study of Mathematics and are useful tools in the theory of analysis. They have been playing a critical role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations with their bounds. In this paper, we obtain new integral inequalities mainly by using some known inequalities. Various generalizations of Hardy's inequality are special cases of the results therein.

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.

Hájek-Rényi-type Inequality for a Class of Random Variable Sequences and Its Applications

In this paper, we obtain the Hejek-Renyi-type inequality for a class of random variable sequences and give some applications for associated random variable sequences, strongly positive dependent stochastic sequences and martingale difference sequences which generalize and improve the results of Prakasa Rao and Soo published in Statist. Probab. Lett., 57（2002） and 78（2008）. Using this result, we get the integrability of supremum and the strong law of large numbers for a class of random variable sequences.

Two Optimal Inequalities Related to the Sándor-Yang Type Mean and One-parameter Mean

In this paper, we establish two optimal the double inequalities for Sandor- Yang type mean and one-parameter mean.

Two-weight weak-type inequalities for potential type operators

For the potential type operatorTФf（x）=∫RnФ（x-y）f（y）dy,where Ф is a non-negative locally integrable function on R^n and satisfies weak growth condition, a two-weight weak-type （p,q） inequality for TФ is obtained.

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.

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.