The missing proof of Paley's theorem about lacunary coefficients
  We modify the standard proof of Paley's theorem about lacunary coefficients of functions in $H^1$ to work without analytic factorization. This leads to the first direct proof of the extension of Paley's theorem that we applied to the former Littlewood conjecture about $L^1$ norms of exponential sums. 
John J.F. Fournier
math.CA