弱型空间的鞅不等式及其应用——纪念李国平院士吴新谋教授诞辰100周年

Abstract

Abstract:

In recent years, the weak-type spaces have received more and more attention in Harmonic analysis and Martingale theory. In the paper, we intend to clarify some our study results in weak-type martingale spaces. It consists of four aspects: (1) The atomic decomposition of Lourentz martingale spaces; (2) The strong-type and weak-type weighted inequality for Orlicz martingales; (3) The weak-Orlicz martingale spaces and their quasi-normed inequalities; (4) Some applications in Banach space theory and the dyadic harmonic analysis.

Key words: Martingale theory, Atomic decomposition, Weighted inequality, Quasi-normed space

CLC Number:

60G42

LIU Pei-De. The Martingale Inequalities in Weak-type Spaces and Their Applications[J].Acta mathematica scientia,Series A, 2010, 30(5): 1292-1305.

Trendmd

References

[1] Carro M J, Raposo J A, Soria J. Recent developments in the theory of Lorentz spaces and weighted inequalities. Barcelna: Dept Appl Math and Analysis, Univ of Barcelna, 2000

[2] Chen W, Liu P D. Several weak-type weighted inequalities in Orlicz martingale classes. Acta Math Scientia, to appear

[3] Chen W, Liu P D. Weighted integral inequalities for the maximal geometric mean operator of martingales. J of Math Anal and Appl, 2010, 371: 821--831

[4] Chen W, Liu P D. Weighted mixed norm inequalities in martingale spaces. Math Inequalities and Appl, to appear

[5] Cwikel M. The dual of weak Lp. Ann Inst Fourier(Grenoble), 1975, 25: 85--126

[6] Cwikel M, Fefferman C. The canonical seminorm on weak L₁. Studia Math, 1984, 78: 275--278

[7] Fefferman R, Soria F. The space wH₁. Studia Math, 1987, 85: 1--16

[8] Grafakos L. Classical and modern Fourier analysis. Pearson Edu, Inc, 2004

[9] Herz C. Bounded mean oscillation and regulated martingales. Trans Amer Math Soc, 1974, 193: 199--215

[10] Hou Y L, Ren Y B. Vector-valued weak martingale Hardy spaces and atomic decompositions. Acta Math Hungar, 2007, 115(3): 235--246

[11] Jiao Y, Peng L H, Liu P D. Atom decompositions of Lorentz martingale spaces and applications. J of Func Spaces and Appl, 2009, 7(2): 153--166

[12] Jiao Y, Fan L P, Liu P D. Interpolation theorems on weighted Lorentz martingale spaces. Science in China, 2007, 50A(9): 1217--1226

[13] Jiao Y, Chen W, Liu P D. Interpolation for weak martingale Hardy spaces. Acta Math Sinica, 2009, 25B(8): 1297--1304

[14] Jiao Y, Liu P D, Peng L H. Interpolation for martingale Hardy spaces over weighted measure spaces. Acta Math Hungar, 2008, 120: 127--139

[15] Kalton N J. Linear operators on L_p for 0<p<1. Trans Amer Math Soc, 1980, 259: 319--355

[16] Kalton N J. Quasi-Banach Spaces, Handbook of the Geometry of Banach Spaces. Amsterdam: Elseviser, 2003: 1100--1130

[17] Kikuchi M. On weighted weak type maximal inequalities for martingales. Math Ineq & Appl, 2003, 6(1): 163--175

[18] Kikuchi M. New martingale inequalities in rearrangement-invariant function spaces. Proc of the Edinburgh Math Society, 2004, 47: 633--657

[19] Kokilashvili V, Krbec M. Weighted inequalities in Lorentz and Orlicz spaces. Singapore: World Sci Pub Co Pte Ltd, 1991

[20] Li Y F, Liu P D. Weak Atomic decomposition for B-valued martingales with two-parameters. Acta Math Hungar, 2010, 127(3): 225--238

[21] Li Y F, Liu P D. Atomic decomposition for weak martingales with two indexes. Chinese Ann of Math, 2008, 29A: 333--342

[22] Liu N, Liu P D. A Marcinkiewicz-type interpolation theorem for wL_p spaces and its applications. J Syst Sci and Math Sci, to appear

[23] Liu N, Ye Y. Several basic convergence theorems in weak Orlicz space. Acta Math Scientia, 2010, 30B(5): 1439--1452

[24] Liu P D. Martingales and Geometry of Banach Spaces. Beijing: Science Press, 2007

[25] Liu P D, Hou Y L. Atomic decompositions for Banach-space-valued martingales. Sciences in China, 1998, 41A: 884--892

[26] Liu P D, Yu L. Atomic decompositions and small index spaces of Banach-space-valued martingales. Science in China, 2001, 44A: 615--625

[27] Liu P D, Hou Y L, Wang M F. Weak Orlicz space and its applications to martingale theory. Science China Mathematics, 2010, 53(4): 905--916

[28] Long R L. Martingale Spaces and Inequalities. Beijing: Peking Univ Press, 1993

[29] Ma T, Liu P D. Atomic decomposition and duals of weak Hardy spaces of B-valued martingales. Acta Math Scientia, 2009, 29B(5): 1439--1452

[30] Mei T, Liu P D. On the maximal inequalities for martingales involving two functions. Proc Amer Math Soc, 2001, 130: 883--892

[31] Nakai E. Orlicz-Morrey spaces and the Hardy-Littlewood maxmal function. Studia Math, 2008, 188: 193--221

[32] Ren Y B, Hou Y L. Two-weighted weak-maximal inequalities for martingales. Acta Math Scientia, 2009, 29B(2): 402--408

[33] Weisz F. Martingale Hardy spaces and their applications in Fourier analysis. LNM, 1994: 1568

[34] Weisz F. Weak martingale Hardy spaces. Prob Math Stat, 1998, 18: 133--148

[35] Weisz F. Bounded operators on weak Hardy spaces and applications. Acta Math (Hungar), 1998, 80: 249--264

[36] Zhang C Z, Liu P D. Weighted averange of Dirichlet kernals for Vilenkin-like system. Acta Math Scientia, 2009, 29B(1): 45--55

[37] Zhang C Z, Zhang X Y. Weighted average of Dirichlet kernels for two dimensional Vilenkin-like system. Science China Mathematics, 2010, 40(6): 593--602

[38] Zhang C Z, Chen L H, Liu P D. B-valued dyadic dwrivative. Wuhan Univ J Nat Sci, 2007, 12(6): 961--964

The Martingale Inequalities in Weak-type Spaces and Their Applications

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 10

Metrics

Comments

Recommended 10

[1]	ZHANG Xue-Jun, XI Li-Hua, FAN Hai-Xia, LI Jun-Feng. ATOMIC DECOMPOSITION OF μ-BERGMAN SPACE IN Cⁿ [J]. Acta mathematica scientia,Series A, 2014, 34(3): 779-789.
[2]	WANG Hua. Weak Type Estimates of Marcinkiewicz Integrals on the Weighted Hardy and Herz-type Hardy Spaces [J]. Acta mathematica scientia,Series A, 2012, 32(5): 840-850.
[3]	YAN Xue-Fang, LI Wen-Ming. Coifman Type Weighted Inequalities for Commutators of Generalized Fractional Integral Operators [J]. Acta mathematica scientia,Series A, 2011, 31(3): 776-784.
[4]	XU Jing-Shi. Atomic Decomposition of Herz-type Besov and Triebel-Lizorkin Spaces [J]. Acta mathematica scientia,Series A, 2009, 29(6): 1500-1507.
[5]	LI De-Feng, YANG Li-Jun. Generalized Perturbation Results of Banach Frames and Atomic Decompositions on Banach Spaces [J]. Acta mathematica scientia,Series A, 2009, 29(3): 613-622.
[6]	MA Cong-Bian, HOU You-Liang. Atomic Decompositions of B -valued Quasi-martingales with Respect to Complex Measures [J]. Acta mathematica scientia,Series A, 2009, 29(3): 584-592.
[7]	Li Wenming. WEIGHTED INEQUALITIES FOR MULTILINEAR POTENTIAL TYPE OPERATORS [J]. Acta mathematica scientia,Series A, 2006, 26(6): 1144-.
[8]	YANG Ai-Xiang, CHENG Zheng-Xin, BANG Li-Zhong. Uniform Characterization of Function Spaces by Wavelets [J]. Acta mathematica scientia,Series A, 2005, 25(1): 130-144.
[9]	ZHU Yue-Ping. Weighted local hardy spaces on locally compact vilenkin groups [J]. Acta mathematica scientia,Series A, 2000, 20(zk): 616-624.
[10]	Wei Dan. On Boundedness of a Class of Singular Integral Operator in Banach-Valued Hardy Space [J]. Acta mathematica scientia,Series A, 1997, 17(2): 235-240.