Weak L1 norms of random sums

Let {g(j)} denote a sequence of measurable functions on R-n, and let ∥•∥(WL1) denote the weak L-1 norm. It is shown that ∥ E(|Σ(N)(j=1)ε(j)g(j)|)∥(WL1) ≲ Σ(N)(j=1)∥ g(j)∥(WL1) where {ε(j)} is a sequence of independent random variables taking on values +1 and -1 with equal probability. Moreover, it is shown that ∥ E(|Σ(N)(j=1)ε(j)g(j)|)∥(WL1) ≲ E(∥Σ(N)(j=1)ε(j)g(j)∥(WL1)) The paper concludes by providing an example indicating that, if ∥ g(1)∥(WL1) = ••• = ∥ g(N)∥(WL1) = 1, then the estimate E(∥Σ(N)(j=1)ε(j)g(j)∥(WL1)) ≲ N log N is the best possible.