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

### Irina Holmes

Texas A&M University, College Station, USA### Paata Ivanisvili

Princeton University, USA and University of California, Irvine, USA### Alexander Volberg

Michigan State University, East Lansing, USA

## 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.

## Cite this article

Irina Holmes, Paata Ivanisvili, Alexander Volberg, The sharp constant in the weak (1,1) inequality for the square function: a new proof. Rev. Mat. Iberoam. 36 (2020), no. 3, pp. 741–770

DOI 10.4171/RMI/1147