Sharp inequalities for the square function of a nonnegative martingale

• Autorzy: Osękowski A.
• Języki publikacji: EN

Abstrakty

EN

We determine the optimal constants C_p and C*_p p such that the following holds: if f is a nonnegative martingale and S(f) and f* denote its square and maximal functions, respectively, then ǁS(f)ǁ_p ≤C_p ǁfǁ_p; p < 1; and ǁS(f)ǁ_p ≤C*_p ǁf*ǁ_p; p ≤1.

Słowa kluczowe

EN

martingale; square function; maximal function

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2010
• Tom: Vol. 30, Fasc. 1
• Strony: 61--72
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Osękowski A.

• Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Bibliografia

• [1] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), pp. 1494-1504.
• [2] D. L. Burkholder, Explorations in martingale theory and its applications, in: École d’Été de Probabilités de Saint-Flour XIX - 1989, Lecture Notes in Math. No 1464, Springer, Berlin 1991, pp. 1-66.
• [3] D. L. Burkholder, The best constant in the Davis inequality for the expectation of the martingale square function, Trans. Amer. Math. Soc. 354 (2002), pp. 91-105.
• [4] B. Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), pp. 187-190.
• [5] C. Dellacherie and P.-A. Meyer, Probabilities and Potential B: Theory of Martingales, North Holland, Amsterdam 1982.
• [6] A. Osękowski, Two inequalities for the first moment of a martingale, its square and maximal function, Bull. Polish Acad. Sci. Math. 53 (2005), pp. 441-449.
• [7] G. Pisier and Q. Xu, Non-commutative martingale inequalities, Commun. Math. Phys. 189 (1997), pp. 667-698.
• [8] N. Randrianantoanina, Square function inequalities for non-commutative martingales, Israel J. Math. 140 (2004), pp. 333-365.
• [9] E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7 (1982), pp. 359-376.
• [10] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.