Weighted weak-type inequality for martingales

• Autorzy: Osękowski, A.
• Języki publikacji: EN

Abstrakty

EN

Let X = (X_t) t ≥ 0 be a bounded martingale and let Y = (Y_t) t ≥ 0 be differentially subordinate to X. We prove that if 1 ≤ p < ∞ and W = (W_t) t ≥ 0 is an A_p weight of characteristic [W] _Ap, then ∥Y∥L^{p, ∞} (W) ≤ C_p [W]A_p∥X∥L^∞(W). The linear dependence on [W]A_p is shown to be the best possible. The proof exploits a weighted exponential bound which is of independent interest. As an application, a related estimate for the Haar system is established.

Słowa kluczowe

EN

martingale; weight; weak type

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2017
• Tom: Vol. 65, no. 2
• Strony: 165--175
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Osękowski, A.

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

Bibliografia

• [1] R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transformations, Duke Math. J. 80 (1995), 575-600.
• [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. 1464, Springer, Berlin, 1991, 1-66.
• [3] C. Dellacherie and P.-A. Meyer, Probabilities and Potential B: Theory of Martingales, North-Holland, Amsterdam, 1982.
• [4] M. Izumisawa and N. Kazamaki, Weighted norm inequalities for martingales, Tôhoku Math. J. 29 (1977), 115-124.
• [5] A. Osękowski, Sharp Martingale and Semimartingale Inequalities, Monografie Mat. 72, Birkhäuser, 2012.
• [6] G. Wang, Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522-551.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).

Identyfikatory

DOI

10.4064/ba8096-11-2017