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 in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions and related to the problem, and relies on certain relationships between and , 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 yields the optimal obstacle condition for , 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