D. Hajela

Copyright © 1990 D. Hajela. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let (Xn)n≥1 be a sequence of mean zero independent random variables. Let Wk={∏j=1kXij|1≤i1<i2…<ik}, Yk=⋃j≤kWj and let [Yk] be the linear span of Yk. Assume δ≤|Xn|≤K for some δ>0 and K>0 and let C(p,m)=16(52p2p−1)m−1plogp(Kδ)m for 1<p<∞. We show that for f∈[Ym] the following inequalities hold:‖f‖2≤‖f‖p≤C(p,m)‖f‖2                       for   2<p<∞‖f‖2≤C(q,m)‖f‖p≤C(q,m)‖f‖2     for   1<p<2,   1p+1q=1and ‖f‖2≤C(4,m)2‖f‖1≤C(4,m)2‖f‖2. These generalize various well known inequalities on Walsh functions.