Abstract
The paper aims at the identification of the best constants in the weak-type (p; p) in- equalities for the martingale square function, 1 p < ¥ . To accomplish this, a related optimal stopping problem for the space-time Brownian motion is investigated. Interestingly, the analysis of the cases 1 p 2 and 2 < p < ¥ requires completely different methods. Namely, in the first case the corresponding value function can be written down explicitly; in the second case the approach rests on the careful analysis of an interesting, integral functional equation.
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