On Khintchine inequalities with a weight

Abstract

In this note we prove a weighted version of the Khintchine inequalities. Let (Ω,A,P) be a probability space and let (rn)n≥1 be a Rademacher sequence. For a random variable ξ : Ω → R and p > 0 we write ‖ξ‖p = (E|ξ|p)1/p. Our main result is the following weighted version of Khintchine’s inequality. We also allow the weight to be zero on a set of positive measure. Theorem 1. Let 0 p, and assume s := P(w 6= 0) > 2/3. Let ξ = ∑n≥1 rnxn with ∑ n≥1 x 2 n 0 such that (1) C−1 1 ( ∑ n≥1 xn ) 1 2 ≤ ‖wξ‖p ≤ C2 ( ∑ n≥1 xn ) 1 2 . Consequently, the p-th moments for 0 < p < q are all comparable. If w ≡ 1 the result reduces the Khintchine inequalities [4]. Although the weighted version of the result is easy to prove, to our knowledge it was not known, and potentially useful for others. We need a well-known L-version of Khintchine’s inequality. We provide the details to obtain explicit constants. Proposition 2. For all a ∈ (0, 1) and for all (xn)n≥1 in l, one has P (∣