The Sharp Constant in the Weak (1,1) Inequality for the Square Function: A New Proof

TL;DR

This paper presents a new proof for the exact sharp constant in the weak (1,1) inequality for the dyadic square function, utilizing Bellman functions and their boundary behaviors.

Contribution

It introduces a novel proof approach using Bellman functions, explicitly finds their boundary values, and reveals their interrelated behavior in the context of the inequality.

Findings

Explicit sharp constant for the weak (1,1) inequality derived

Bellman functions $ ext{L}$ and $ ext{M}$ explicitly characterized

Interdependence of boundary solutions and obstacle conditions demonstrated

Abstract

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.