Domination inequality for martingale transforms of a Rademacher sequence

Abstract

Letf n = Σ =1 v k r k ,n=1,…, be a martingale transform of a Rademacher sequence (r n)and let (r ′ ) 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.