We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey sp...We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.展开更多
In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved.Some well-known results are generalized and unified.Applications are given to classical operato...In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved.Some well-known results are generalized and unified.Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales,then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces.These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces.Finally we give an equivalent characterization of UMD Banach lattices,and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.展开更多
Let B be a Banach space, φ1, φ2 be two generalized convex φ-functions and φ1, φ2 the Young complementary functions of ψ1, ψ2 respectively with∫t t0ψ2（s）/sds≤ds≤c0ψ1（c0t）（t〉t0）for some constants co ...Let B be a Banach space, φ1, φ2 be two generalized convex φ-functions and φ1, φ2 the Young complementary functions of ψ1, ψ2 respectively with∫t t0ψ2（s）/sds≤ds≤c0ψ1（c0t）（t〉t0）for some constants co 〉 0 and to 〉 0, where ψ1 and ψ2 are the left-continuous derivative functions of ψ1 and ψ2, respectively. We claim that： （i） If B is isomorphic to a p-uniformly smooth space （or q-uniformly convex space, respectively）, then there exists a constant c 〉 0 such that for any B-valued martingale f = （fn）n≥0,｜｜f^＊｜｜φ1≤｜｜S^（p）（f）｜｜φ2（of｜｜S^（q）（f）｜｜φ1≤c｜｜f^＊｜｜φ2,respectively）,where f^＊ and S^（p） （f） are the maximal function and the p-variation function of f respectively; （ii） If B is a UMD space, Tvf is the martingale transform of f with respect to v = （Vn）z≥0 （V^＊ 〈 1）, then ｜｜（Tvf）^＊｜｜Ф1≤f^＊｜｜Ф2.展开更多
We establish a new characterization of AUMD （analytic unconditional martingale differences） spaces via biplurisubharmonic functions. That is, B∈AUMD iff there exists a bpsbh （biplurisubharmonic） function L ： B &...We establish a new characterization of AUMD （analytic unconditional martingale differences） spaces via biplurisubharmonic functions. That is, B∈AUMD iff there exists a bpsbh （biplurisubharmonic） function L ： B × B→[-∞,∞） satisfying L（x,0）, L（0,y）≥L（0,0）〉0,L（x,y）≤L（0,0）＋｜x-y｜ and L（x,y）≤｜x-y｜ for ｜x＋y｜＋｜x-y｜≥1. This provides an analogue of Piasecki＇s characterization of AUMS spaces. Our arguments are based on some special properties of zigzag analytic martingales and martingale transforms.展开更多
文摘We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.
基金supported by National Natural Science Foundation of China (GrantNo. 11001273)Research Fund for International Young Scientists (Grant No. 11150110456)+1 种基金Research Fundfor the Doctoral Program of Higher Education of China (Grant No. 20100162120035)Postdoctoral Science Foundation of China and Central South University
文摘In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved.Some well-known results are generalized and unified.Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales,then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces.These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces.Finally we give an equivalent characterization of UMD Banach lattices,and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.
基金supported by the National Natural Science Foundation of China (11071190)
文摘Let B be a Banach space, φ1, φ2 be two generalized convex φ-functions and φ1, φ2 the Young complementary functions of ψ1, ψ2 respectively with∫t t0ψ2（s）/sds≤ds≤c0ψ1（c0t）（t〉t0）for some constants co 〉 0 and to 〉 0, where ψ1 and ψ2 are the left-continuous derivative functions of ψ1 and ψ2, respectively. We claim that： （i） If B is isomorphic to a p-uniformly smooth space （or q-uniformly convex space, respectively）, then there exists a constant c 〉 0 such that for any B-valued martingale f = （fn）n≥0,｜｜f^＊｜｜φ1≤｜｜S^（p）（f）｜｜φ2（of｜｜S^（q）（f）｜｜φ1≤c｜｜f^＊｜｜φ2,respectively）,where f^＊ and S^（p） （f） are the maximal function and the p-variation function of f respectively; （ii） If B is a UMD space, Tvf is the martingale transform of f with respect to v = （Vn）z≥0 （V^＊ 〈 1）, then ｜｜（Tvf）^＊｜｜Ф1≤f^＊｜｜Ф2.
基金Supported by the National Natural Science Foun-dation of China (10371093)
文摘We establish a new characterization of AUMD （analytic unconditional martingale differences） spaces via biplurisubharmonic functions. That is, B∈AUMD iff there exists a bpsbh （biplurisubharmonic） function L ： B × B→[-∞,∞） satisfying L（x,0）, L（0,y）≥L（0,0）〉0,L（x,y）≤L（0,0）＋｜x-y｜ and L（x,y）≤｜x-y｜ for ｜x＋y｜＋｜x-y｜≥1. This provides an analogue of Piasecki＇s characterization of AUMS spaces. Our arguments are based on some special properties of zigzag analytic martingales and martingale transforms.
