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Domination inequality for martingale transforms of a Rademacher sequence

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Abstract

Letf n = Σ n k=1 v k r k ,n=1,…, be a martingale transform of a Rademacher sequence (r n)and let (r ′ n ) be an independent copy of (r n).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true:\(\left\| {\sum {v_k r_k } } \right\|_p \leqslant K\left\| {\sum {v_k r_k^\prime } } \right\|_p \)

In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars (a n)one has\(\left\| {\sum {v_k r_k } } \right\|_p \approx K_{1,2} ((a_n )),\sqrt p \) where\(K_{1,2} ((a_n ),\sqrt p ) = K_{1,2} ((a_n ),\sqrt p ;\ell _1 ,\ell _2 )\) is theK-interpolation norm between ℓ1 and ℓ2. We also derive a new exponential inequality for martingale transforms of a Rademacher sequence.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, 27695-8205, Raleigh, NC, USA

    Paweł Hitczenko

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  1. Paweł Hitczenko
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This research was supported in part by an NSF grant and an FRPD grant at NCSU.

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Hitczenko, P. Domination inequality for martingale transforms of a Rademacher sequence. Israel J. Math. 84, 161–178 (1993). https://doi.org/10.1007/BF02761698

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  • DOI: https://doi.org/10.1007/BF02761698

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