Revista Matemática Iberoamericana


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Published online first: 2019-10-15
DOI: 10.4171/rmi/1147

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

Irina Holmes[1], Paata Ivanisvili[2] and Alexander Volberg[3]

(1) Texas A&M University, College Station, USA
(2) Princeton University, USA and University of California, Irvine, USA
(3) Michigan State University, East Lansing, USA

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.

Keywords: Dyadic square function, Bellman function, weak inequality

Holmes Irina, Ivanisvili Paata, Volberg Alexander: The sharp constant in the weak (1,1) inequality for the square function: a new proof. Rev. Mat. Iberoam. Electronically published on October 15, 2019. doi: 10.4171/rmi/1147 (to appear in print)