Some Sharp Estimates for the Haar System and Other Bases In $L^1(0,1)$

Let $h=(h_k)_{k\geq 0}$ 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 $1<p<\infty$, and the purpose of this paper is to study a substitute for this property in the case $p=1$. Precisely, for any $\lambda>0$ we identify the best constant $\beta=\beta_h(\lambda)\in [0,1]$ such that the following holds. If $n$ is an arbitrary nonnegative integer and $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers such that $\bigl\|\sum_{k=0}^n a_kh_k\bigr\|_1\leq 1$, then \[ \Bigl|\Bigl\{x\in [0,1]:\Bigl|\sum_{k=0}^n \varepsilon_ka_kh_k(x)\Bigr|\geq \lambda\Bigr\}\Bigr|\leq \beta, \] for any sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2,\ldots, \varepsilon_n$ 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.