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

In this note we give a new proof of the sharp constant C=e1/2+01ex2/2dxC = 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 L\mathbb{L} and M\mathbb{M} related to the problem, and relies on certain relationships between L\mathbb{L} and M\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 M\mathbb{M} yields the optimal obstacle condition for L\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