Li-Yorke and distributionally chaotic operators

We study Li-Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li-Yorke chaos in terms of the existence of irregular vectors. Sufficient “computable” criteria for distributional and Li-Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.     

Parabolic Muckenhoupt weights in the Euclidean space

We introduce a parabolic analogue of Muckenhoupt?s Ap class. We show that these weights satisfy a reverse Hölder type inequality and also prove a John-Nirenberg inequality for a related BMO type class. Both results are multidimensional generalizations of known one-dimensional results. As an application of our methods, we extend a two-dimensional weighted norm inequality of A. Lerner and S. Ombrosi to dimensions n?3.     

Double integral characterizations of harmonic Bergman spaces

Recently Li et al. have characterized, except for a critical case, the weighted Bergman spaces over the complex ball by means of integrability conditions of double integrals associated with difference quotients of holomorphic functions. In this paper we extend those characterizations to the case of weighted harmonic Bergman spaces over the real ball and complement their results by providing a characterization for the missing critical case. We also investigate the possibility of extensions to the half-space setting. Our observations reveal an interesting half-space phenomenon caused by the unboundedness of the half-space.     

Existence of solution for the n-dimension second order semilinear hyperbolic equations

In the paper we establish the local and global existence of solution for the n-dimensional second order semilinear hyperbolic equation with a strongly singular coefficient which appears in the boundary-value problems of fluid dynamics. Based on the analysis about the loss of regularity on the line t=0 for the solution of the corresponding linear equation and the decay at infinity which caused by the singular coefficient, we obtain the existence of a small solution for the semilinear equation by use of fixed point theorem.     

Slant joint antieigenvalues and antieigenvectors of operators in normal subalgebras

We will study the slant joint antieigenvalues and antieigenvectors of pairs of operators that belong to the same closed normal subalgebra of the algebra of bounded operators on a separable Hilbert space. This extends the slant antieigenvalue theory from single normal operators to pairs of normal operators. Our results may be viewed as extensions of the Greub-Rheinboldt inequality from two positive operators to two normal operators.     

Weighted composition operators between Hardy and growth spaces on the upper half-plane

Bounded weighted composition operators acting from Hardy to growth spaces on the upper half-plane are characterized. We also give some necessary, as well as some sufficient conditions for the compactness of weighted composition operators between these spaces.     

Essential norm of composition operators between weighted Hardy spaces

An asymptotic formula for the essential norm of composition operators acting between two weighted Hardy spaces H_w1 and H_w2, where w₁ and w₂ are two admissible weight functions, is given. The boundedness of the operators is also characterized.     

Weighted statistical approximation by kantorovich type q-Szász-Mirakjan operators

In the present paper, we introduce a Kantorovich type modification of q-Szász-Mirakjan operators and obtain weighted statistical approximation properties of these operators. Also for introduced operators, we give a Voronovskaja type theorem related to q-derivatives.     

The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.