Representation of Generalized Translations of the Product of Two Functions on Signed Hypergroups on the Real Line
Néjib Ben Salem & Nabila Hamda
10.3969/j.issn.1672-4070.2012.03.001
Anal. Theory Appl., 28 (2012), pp. 201-223.
In this paper, we consider the generalized translations associated with the Dunkl and the Jacobi-Dunkl differential-difference operators on the real line which provide the structure of signed hypergroups on $\mathbf{R}$. Especially, we study the representation of the generalized translations of the product of two functions for these signed hypergroups.
BMO Estimates for Multilinear Fractional Integrals
Xiangxing Tao & Yunpin Wu
In this paper, the authors prove that the multilinear fractional integral operator $T_{\Omega,\alpha}^{A_1,A_2}$ and the relevant maximal operator $M_{\Omega,\alpha}^{A_1,A_2}$ with rough kernel are both bounded from $L^p$ $(1 < p < \infty)$ to $L^q$ and from $L^p$ to $L^{n/(n−\alpha),\infty}$ with power weight, respectively, where$$T_{\Omega,\alpha}^{A_1,A_2}(f)(x) =\int_{\mathbf{R}^n}\frac{R_{m_1}(A_1;x,y)R_{m_2}(A_2;x,y)}{|x-y|^{n-\alpha+m_1+m_2-2}}\Omega(x-y)f(y)dy$$and$$M_{\Omega,\alpha}^{A_1,A_2}(f)(x)= \sup_{r > 0}\frac{1}{r^{n−\alpha+m_1+m_2−2}}\int_{|x−y|<r}\prod_{i=1}^{2}{R_{m_i}}(A_i;x,y)\Omega(x-y)f(y)dy$$ and $0<\alpha<n$, $\Omega \in L^s(S^{n−1})$ $(s \geq 1)$ is a homogeneous function of degree zero in $\mathbf{R}^n$, $A_i$ is a function defined on $\mathbf{R}^n$ and $R_{m_i} (A_i;x,y)$ denotes the $m_i$−th remainder of Taylor series of $A_i$ at $x$ about $y$. More precisely, $R_{m_i} (A_i;x,y) = A_i(x)− \sum\limits_{|\gamma|< m_i}\frac{1}{\gamma!}D^{\gamma}A_i(y)(x−y)^r,$ where $D^{\gamma} (A_i) \in BMO(\mathbf{R}^n)$ for $|\gamma | = m_i−1(m_i > 1)$, $i = 1,2$.
Stancu Polynomials Based on the $q$-Integers
Xueyan Xiang
A new generalization of Stancu polynomials based on the $q$-integers and a nonnegative integer $s$ is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are also investigated.
On the Location of Zeros of a Polynomial
M. H. Gulzar
In this paper we extend Enestrom-Kakeya theorem to a large class of polynomials with complex coefficients by putting less restrictions on the coefficients. Our results generalise and extend many known results in this direction.
Generalizations of the Suzuki and Kannan Fixed Point Theorems in G-Cone Metric Spaces
Mohammad Sadegh Asgari & Zohreh Abbasbigi
In this paper we develop the Banach contraction principle and Kannan fixed point theorem on generalized cone metric spaces. We prove a version of Suzuki and Kannan type generalizations of fixed point theorems in generalized cone metric spaces.
A Characterization for Fractional Integrals on Generalized Morrey Spaces
Eridani, M. I. Utoyo & H. Gunawan
This paper concerns with the fractional integrals, which are also known as the Riesz potentials. A characterization for the boundedness of the fractional integral operators on generalized Morrey spaces will be presented. Our results can be viewed as a refinement of Nakai's[7].
Some New Iterated Function Systems Consisting of Generalized Contractive Mappings
Shaoyuan Xu, Wangbin Xu & Dingxing Zhong
Iterated function systems (IFS) were introduced by Hutchinson in 1981 as a natural generalization of the well-known Banach contraction principle. In 2010, D. R. Sahu and A. Chakraborty introduced K-Iterated Function System using Kannan mapping which would cover a larger range of mappings. In this paper, following Hutchinson, D. R. Sahu and A. Chakraborty, we present some new iterated function systems by using the so-called generalized contractive mappings, which will also cover a large range of mappings. Our purpose is to prove the existence and uniqueness of attractors for such class of iterated function systems by virtue of a Banach-like fixed point theorem concerning generalized contractive mappings.
Weighted Estimates for Multivariate Hausdorff Operators
Yanling Sun & Yinsheng Jiang
In this paper, some weighted estimates for the multivariate Hausdorff operators are obtained. It is proved that the multivariate Hausdorff operators are bounded on $L^p$ spaces with power weights, which is based on the boundedness of multivariate Hausdorff operators on Herz spaces, and are bounded on weighted $L^p$ spaces with the weights satisfying the homogeneity of degree zero.
The Boundedness for a Class of Rough Fractional Integral Operators on Variable Exponent Lebesgue Spaces
Huiling Wu & Jiacheng Lan
In this paper, we will discuss the behavior of a class of rough fractional integral operators on variable exponent Lebesgue spaces, and establish their boundedness from $L^{p_1(\cdot)}(\mathbf{R}^n)$ to $L^{p_2(\cdot)}(\mathbf{R}^n)$.
Existence Problems of Additive Selection Maps for Another Type Subadditive Set-Valued Map
Yongjie Piao & Hailan Jin
In this paper, we consider the following subadditive set-valued map $F : X \to P_0(Y):$ $$F(\sum_{i=1}^rx_i+\sum_{j=1}^sx_{r+j})\subseteq rF(\frac{\sum\limits_{i=1}^rx_i}{r})+sF(\frac{\sum\limits_{j=1}^sx_{r+j}}{s}), \forall x_i\in X, i=1,2,\cdots,r+s,$$ where $r$ and $s$ are two natural numbers. And we discuss the existence and unique problem of additive selection maps for the above set-valued map.