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

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.

author list (cited authors)

  • Holmes, I., Ivanisvili, P., & Volberg, A.

publication date

  • January 1, 2020 11:11 AM