Sharp inequalities for the Haar system and martingale transforms

• Autorzy: Osękowski A.
• Języki publikacji: EN

Abstrakty

EN

A classical result of Paley and Marcinkiewicz asserts that the Haar system on [0, 1] forms an unconditional basis in L^p provided 1 < p < ∞. The purpose of the paper is to study related weak-type inequalities, which can be regarded as a version of this property for p = 1. Probabilistic counterparts, leading to some sharp estimates for martingale transforms, are presented.

Słowa kluczowe

EN

Haar system; martingale; weak-type inequality; Bellman function; best constants

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2016
• Tom: Vol. 36, Fasc. 1
• Strony: 147--164
• Opis fizyczny: Bibliogr. 16 poz., wykr.

Twórcy

autor

Osękowski A.

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

Bibliografia

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• [9] J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), pp. 84-96.
• [10] B. Maurey, Système de Haar, in: Séminaire Maurey-Schwartz (1974-1975), École Polytechnique, Paris 1975.
• [11] F. Nazarov and S. Treil, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analyz 8 (1996), pp. 32-162.
• [12] A. Osękowski, Survey article: Bellman function method and sharp inequalities for martingales, Rocky Mountain J. Math. 43 (2013), pp. 1759-1823.
• [13] A. Osękowski, Some sharp estimates for the Haar system and other bases in L1 (0, 1), Math. Scand. 115 (1) (2014), pp. 123-142.
• [14] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), pp. 241-264.
• [15] J. Schauder, Eine Eigenschaft des Haarschen Orthogonalsystems, Math. Z. 28 (1928), pp. 317-320.
• [16] J. Ville, Étude critique de la notion de collectif, Gauthier-Villars, Paris 1939.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).