Trace formula of the integro-differential operator on a quantum graph

In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.

AN ORDER PRESERVING INEQUALITY FOR THREE OPERATORS VIA FURUTA INEQUALITY

Furuta showed that if A≥B≥0,then for each r≥0,f（p）=（A^r/2 B^p A^r/2）^t＋r/p＋r is decreasing for p≥t≥0.Using this result,the following inequality（C^r/2（AB^2A）^δC^ r/2）^ p-1＋r/4δ＋r ≤C^p-1＋r is obtained for 0〈p ≤1,r≥1,1/4≤δ≤1 and three positive operators A, B, C satisfy（A^1/2BA^1/2）^p/2≤A^p,（B^1/2AB^1/2）^p/2≥B^p,（C^1/2AC^1/2）^p/2≤C^p,（A^1/2CA^1/2）^p/2≥A^p.

Bohr Inequality for Multiple Operators

An absolute value equation is established for linear combinations of two operators.When the parameters take special values, the parallelogram law of operator type is given. In addition, the operator equation in literature [3] and its equivalent deformation are obtained.Based on the equivalent deformation of the operator equation and using the properties of conjugate number as well as the operator, an absolute value identity of multiple operators is given by means of mathematical induction. As Corollaries, Bohr inequalities are extended to multiple operators and some related inequalities are reduced to, such as inequalities in [2]and [3].

Application Of Jessen＇s Type Inequality For Positive C0-Semigroup Of Operators

Recently in [4], the Jessen＇s type inequality for normalized positive C0-semigroups is obtained. In this note, we present few results of this inequality, yielding Holder＇s Type and Minkowski＇s type inequalities for corresponding semigroup. Moreover, a Dresher＇s type inequality for two-parameter family of means, is also proved.

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.

STRONG CONVERSE INEQUALITY FOR SZASZ-DURRMEYER OPERATORS

For Szasz-Durrmeyer operators L.(f,x),1<p ∞ we prove that, forsome m,where(x)2=x,M>0,is Ditzian-Totik modulus of smoothness.

Fekete-Szegö Inequality for a Subclass of Bi-univalent Functions Associated with Hohlov Operator and Quasi-subordination

In this paper, we introduce a new subclass of bi-univalent functions defined by quasi-subordination and Hohlov operator and obtain the coefficient estimates and Fekete-Szego inequality for function in this new subclass. The results presented in this paper improve or generalize the recent works of other authors.

An Operator Inequality for Three Operators

As a generalization of grand Furuta inequality,recently Furuta obtain:If A≥ B≥0 with A>0,then for t∈[0,1]and p1,p2,p3,p4≥1, A t 2[A- t 2{A t 2(A/ t 2 Bp 1A /t2 )p 2A t 2}p 3A /t2 ]p 4A t 2 1 [{(p1/t)p2+t}p3-t]p4+t]≤A. In this paper,we generalize this result for three operators as follow:If A≥B≥C≥0 with B>0,t∈[0,1]and p1,p2,···,p2n/1,p2n≥1 for a natural number n.Then the following inequalities hold for r≥t, A1/t+r≥ [A r 2[B /t 2{B t 2······[B /t 2{B t 2(B /t 2 ←B /t 2 n times Bt 2 n/1 times by turns Cp 1B /t 2)p 2B t 2}p 3B /t 2]p 4···B t 2}p 2n/1B /t 2 B /t 2 n times Bt 2 n/1 times by turns→ ]p 2nA r 2] 1/t+r q[2n]+r/t, where q[2n]≡{···[{[(p1/t)p2+t]p3/t}p4+t]p5/···/t}p2n+t /t and t alternately n times appear .

Strong Converse Inequality for the Meyer-Knig and Zeller-Durrmeyer Operators

In this paper we give a strong converse inequality of type B in terms of unified K-functional Kλα （f, t2） （0 ≤λ≤ 1, 0 〈 α 〈 2） for the Meyer-Knig and Zeller-Durrmeyer type operators.

The Definition of Universal Momentum Operator of Quantum Mechanics and the Essence of Micro-Particle’s Spin——To Reveal the Real Reason That the Bell Inequality Is Not Supported by Experiments

The definition of momentum operator in quantum mechanics has some foundational problems and needs to be improved. For example, the results are different in general by using momentum operator and kinetic operator to calculate microparticle’s kinetic energy. In the curved coordinate systems, momentum operators can not be defined properly. When momentum operator is acted on non-eigen wave functions in coordinate space, the resulting non-eigen values are complex numbers in general. In this case, momentum operator is not the Hermitian operator again. The average values of momentum operator are complex numbers unless they are zero. The same problems exist for angle momentum operator. Universal momentum operator is proposed in this paper. Based on it, all problems above can be solved well. The logical foundation of quantum mechanics becomes more complete and the EPY momentum paradox can be eliminated thoroughly. By considering the fact that there exist a difference between the theoretical value and the real value of momentum, the concepts of auxiliary momentum and auxiliary angle momentum are introduced. The relation between auxiliary angle momentum and spin is deduced and the essence of micro-particle’s spin is revealed. In this way, the fact that spin gyro-magnetic ratio is two times of orbit gyro-magnetic ratio, as well as why the electrons of ground state without obit angle momentum do not fall into atomic nuclear can be explained well. The real reason that the Bell inequality is not supported by experiments is revealed, which has nothing to do with whether or not hidden variables exist, as well as whether or not locality is violated in microcosmic processes.

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.

HARDY-LITTLEWOOD-POLYA INEQUALITY FOR A LINEAR DIFFERENTIAL OPERATOR AND SOME RELATED OPTIMAL PROBLEMS

In this paper a generalized version of the classical Hardy-Littlewood-Polya inequality is given.Furthermore,the Stechkin's problem for a linear differential operator is solved in L_2(R), and the optimal recovery problem for such differential operator is considered.

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.

(L^p, L^q)-Boundedness of Hausdorff Operators with Power Weight on Euclidean Spaces

In this paper, we prove the （L^p, L^q）-boundedness of （fractional） Hausdorff operators with power weight on Euclidean spaces. As special cases, we can obtain some well known results about Hardy operators.

Parameter Conditions for Hilbert-Type Operator Bounded with Homogeneous Kernels between Sequence Space l and Function Space L

By using the technique of real analysis,the parameter conditions for Hilberttype series operator and integral operator T_(1)(˜a)(x)=∑∞n=1 K(n,x)a_(n),T_(2)(f)_(n)=∫+∞0 K(n,x)f(x)dx.bounded with homogeneous kernels are discussed.The necessary and sufficient conditions for T_(1):l^(α)_(p)→L^(β)_(p)^((1−p))p(0,+∞)and T_(2):L_(q)^(β)(0,+∞)→l^(α(1−q))_(q)bounded are obtained,and their norm expressions are established under certain conditions.

NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CESàRO TYPE

This paper deals with a general class of weighted multilinear Hardy-Cesaro op- erators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesaro oper- ators （with symbols in central BMO spaces） are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.

Rellich type inequalities related to Grushin type operator and Greiner type operator

We prove some sharp Hardy inequality associated with the gradient △γ=（△x,｜x｜γ [x△y） by a direct and simple approach. Moreover, similar method is applied to ob- tain some weighted sharp Rellich inequality related to the Grushin operator in the setting of L^p. We also get some weighted Hardy and Rellich type inequalities related to a class of Greiner type operators.

Improved Parameter-dependent H_∞ Filtering for Polytopic Delta Operator Systems

The problem of H∞ filtering for polytopic Delta operator linear systems is investigated. An improved H∞ performance criterion is presented based on the bounded real lemma. Upon the improved performance criterion, a sufficient condition for the existence of parameter-dependent H∞ filtering is derived in terms of linear matrix inequalities. The designed filter can be obtained from the solution of a convex optimization problem. The filter design makes full use of the parameter-dependent approach, which leads to a less conservative result than conventional design methods. A numerical example is given to illustrate the effectiveness of the proposed approach.

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.

Minimizers of curl prescribed full trace

This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain.The existence of the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a constructed auxiliary variational problem.And the H2 estimate of the type II divergence-free part of the minimizers is established by div-curl-gradient type estimates of vector fields.