A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexi...A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.展开更多
If P(z) is a polynomial of degree at most n having all its zeros in , then it was recently claimed by Shah and Liman ([1], estimates for the family of $B$-operators, Operators and Matrices, (2011), 79-87) that for eve...If P(z) is a polynomial of degree at most n having all its zeros in , then it was recently claimed by Shah and Liman ([1], estimates for the family of $B$-operators, Operators and Matrices, (2011), 79-87) that for every?R≧1, p ≧ 1, where B is a Bn-operator with parameters in the sense of Rahman [2], and . Unfortunately the proof of this result is not correct. In this paper, we present certain more general sharp Lp-inequalities for Bn-operators which not only provide a correct proof of the above inequality as a special case but also extend them for 0≦p﹤1 as well.展开更多
In [3], they gave necessary and sufficient condition for T 1 C and then as applications T 1 C for weakly dependent sequences was established. In this note, based on Gozlan-L′eonard characterization for W 1 H -inequal...In [3], they gave necessary and sufficient condition for T 1 C and then as applications T 1 C for weakly dependent sequences was established. In this note, based on Gozlan-L′eonard characterization for W 1 H -inequalities, we extends this result to W 1 H inequalities.展开更多
In this paper, we consider an operator Da which maps a polynomial P（z） in to DaP（z）：= np（z）＋ （a-z）P＇（z）, where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Ou...In this paper, we consider an operator Da which maps a polynomial P（z） in to DaP（z）：= np（z）＋ （a-z）P＇（z）, where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Our results yields several generaliza- tions and refinements of many known results and also provide an alternative proof of a result due to Dewan et al. [7], which is independent of Laguerre＇s theorem.展开更多
In this paper, the conditions on pairs of weights (u,v) are given such that for the generalized Hardy operator Tf(x) = K(x,y)f(y) dy the following Φ-inequality holds: Φ-12( Φ2(Tf(x))V(x)dx≤CΦ-11( Φ1(f(x))U(x)...In this paper, the conditions on pairs of weights (u,v) are given such that for the generalized Hardy operator Tf(x) = K(x,y)f(y) dy the following Φ-inequality holds: Φ-12( Φ2(Tf(x))V(x)dx≤CΦ-11( Φ1(f(x))U(x)dx),where Φ1,Φ2 are Young function ;the corresponding weak type Φ-inequality for T is characterized.展开更多
Required by the application in the investigation of the Cauchy integral operators on Lipschitzsurfaces, the classical martingales are generalized to ones defined with respect to Clifford algebravalued measures. Meanwh...Required by the application in the investigation of the Cauchy integral operators on Lipschitzsurfaces, the classical martingales are generalized to ones defined with respect to Clifford algebravalued measures. Meanwhile, very general Φ-equivalences between S(f) and f* , the same as inthe classical case, are established too.展开更多
基金Supported by the Scientific Research Foundation of Hubei Province (D200613001)the National Natural Science Foundation of China (10371093)
文摘A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.
文摘If P(z) is a polynomial of degree at most n having all its zeros in , then it was recently claimed by Shah and Liman ([1], estimates for the family of $B$-operators, Operators and Matrices, (2011), 79-87) that for every?R≧1, p ≧ 1, where B is a Bn-operator with parameters in the sense of Rahman [2], and . Unfortunately the proof of this result is not correct. In this paper, we present certain more general sharp Lp-inequalities for Bn-operators which not only provide a correct proof of the above inequality as a special case but also extend them for 0≦p﹤1 as well.
文摘In [3], they gave necessary and sufficient condition for T 1 C and then as applications T 1 C for weakly dependent sequences was established. In this note, based on Gozlan-L′eonard characterization for W 1 H -inequalities, we extends this result to W 1 H inequalities.
文摘In this paper, we consider an operator Da which maps a polynomial P（z） in to DaP（z）：= np（z）＋ （a-z）P＇（z）, where and obtain some Lγ inequalities for lucanary polynomials having zeros in /z/ 〈 k 〈 1. Our results yields several generaliza- tions and refinements of many known results and also provide an alternative proof of a result due to Dewan et al. [7], which is independent of Laguerre＇s theorem.
文摘In this paper, the conditions on pairs of weights (u,v) are given such that for the generalized Hardy operator Tf(x) = K(x,y)f(y) dy the following Φ-inequality holds: Φ-12( Φ2(Tf(x))V(x)dx≤CΦ-11( Φ1(f(x))U(x)dx),where Φ1,Φ2 are Young function ;the corresponding weak type Φ-inequality for T is characterized.
文摘Required by the application in the investigation of the Cauchy integral operators on Lipschitzsurfaces, the classical martingales are generalized to ones defined with respect to Clifford algebravalued measures. Meanwhile, very general Φ-equivalences between S(f) and f* , the same as inthe classical case, are established too.
