The sharp constant in the weak (1,1) inequality for the square function: a new proof (Q783778)

Summary: In this note we give a new proof of the sharp constant \(C=e^{-1/2}+\int_0^1e^{-x^2/2}dx\) in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions \(\mathbb{L}\) and \(\mathbb{M}\) related to the problem, and relies on certain relationships between \(\mathbb{L}\) and \(\mathbb{M} \), as well as the boundary values of these functions, which we find explicitly. Moreover, these Bellman functions exhibit an interesting behavior: the boundary solution for \(\mathbb{M}\) yields the optimal obstacle condition for \(\mathbb{L}\), and vice versa.