In this note we give a new proof of the sharp constant $C = e^{-1/2} + \int_0^1 e^{-x^2/2}\,dx$ 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.

中文翻译：

---

平方函数的弱（1,1）不等式中的锐常数：一个新证明

在本说明中，我们给出了弱点（1，1）中的尖锐常数$ C = e ^ {-1/2} + \ int_0 ^ 1 e ^ {-x ^ 2/2} \，dx $的新证明。二进角平方函数的不等式。该证明利用与问题相关的两个Bellman函数$ \ mathbb {L} $和$ \ mathbb {M} $，并依赖于$ \ mathbb {L} $和$ \ mathbb {M} $之间的某些关系，以及这些函数的边界值，我们可以找到它们。而且，这些Bellman函数表现出有趣的行为：$ \ mathbb {M} $的边界解产生$ \ mathbb {L} $的最优障碍条件，反之亦然。