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

Irina Holmes ^[1] ; Paata Ivanisvili ^[2] ; Alexander Volberg ^[3]
1. [1] Texas A&M University
  
  Texas A&M University
  
  Estados Unidos
2. [2] Princeton University
  
  Princeton University
  
  Estados Unidos
3. [3] Michigan State University
  
  Michigan State University
  
  City of East Lansing, Estados Unidos
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Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 3, 2020, págs. 741-770
Idioma: inglés
DOI: 10.4171/rmi/1147
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Resumen
- In this note we give a new proof of the sharp constant C=e−1/2+∫10e−x2/2dx in the weak (1, 1) inequality for the dyadic square function. The proof makes use of two Bellman functions L and M related to the problem, and relies on certain relationships between L and 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 yields the optimal obstacle condition for L, and vice versa.