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The sharp constant in the weak (1,1) inequality for the square function: a new proof
Abstract
In this note we give a new proof of the sharp constant C = e−1/2 + l 01 e−x2/2 dx 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.
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