Some Sharp Estimates for the Haar System and Other Bases In L^1(0,1) - Osȩkowski, Adam (MATHEMATICA SCANDINAVICA, 2014)
Let h=(h_k)_k≥ 0 denote the Haar system of functions on [0,1]. It is well known that h forms an unconditional basis of L^p(0,1) if and only if 1<p<∞, and the purpose of this paper is to study a substitute for this property in the case p=1. Precisely, for any λ>0 we identify the best constant β=β_h(λ)[0,1] such that the following holds. If n is an arbitrary nonnegative integer and a_0, a_1, a_2, …, a_n are real numbers such that \|_k=0^n a_kh_k\|_1≤ 1, then \[ |{x[0,1]:|_k=0^n ε_ka_kh_k(x)|≥ λ}|≤ β, \] for any sequence ε_0, ε_1, ε_2,…, ε_n of signs. A related bound for an arbitrary basis of L^1(0,1) is also established. The proof rests on the construction of the Bellman function corresponding to the problem.