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Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

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Języki publikacji
EN
Abstrakty
EN
Let (hk)k≥0 be the Haar system on [0,1]. We show that for any vectors ak from a separable Hilbert space H and any εk∈[−1,1], k=0,1,2,…, we have the sharp inequality ...[formula], where W([0,1]) is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ∥Y∥W(Ω)≤2∥X∥L∞(Ω), where X and Y stand for H-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.
Rocznik
Strony
187--196
Opis fizyczny
Bibliogr. 12 poz.
Twórcy
  • Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland, ados@mimuw.edu.pl
Bibliografia
  • [1] C. Bennett, R. A. DeVore and R. Sharpley, Weak-L1 and BMO, Ann. of Math. 113 (1981), 601–611.
  • [2] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, Boston, MA, 1988.
  • [3] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
  • [4] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), 997–1011.
  • [5] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647–702.
  • [6] D. L. Burkholder, Differential subordination of harmonic functions and martingales, in: Harmonic Analysis and Partial Differential Equations (El Escorial, 1987), Lecture Notes in Math. 1384, Springer, 1989, 1–23.
  • [7] C. Dellacherie and P. A. Meyer, Probabilities and Potential B, North-Holland, Amsterdam, 1982.
  • [8] J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84–96.
  • [9] B. Maurey, Système de Haar, Séminaire Maurey–Schwartz 1974–1975, École Polytechnique, Paris, 1975.
  • [10] R. E. A. C. Paley, A remarkable series of orthogonal functions. I, Proc. London Math. Soc. 34 (1932), 241–264.
  • [11] J. Suh, A sharp weak type (p; p) inequality (p > 2) for martingale transforms and other subordinate martingales, Trans. Amer. Math. Soc. 357 (2005), 1545–1564.
  • [12] G. Wang, Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities, Ann. Probab. 23 (1995), 522–551.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-f5ceb8a6-fb57-4eba-87fc-0f49759b29dc
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