Obstacle Problems Generated by the Estimates of Square Function

In this note we give the formula for Bellman function associated with problem considered by B. Davis (Duke Math J 43:697–704, 1976) in 1976. article estimates of type ?Sf?p ? Cp?f?p, p ? 2, were dyadic square operator S, and found sharp values constants Cp. However, along one can consider a more subtle characteristic above estimate. This quantity is called problem. It has never been proved that confluent hypergeometric from Davis’ paper (second page) gives us function. Here fill out gap finding exact unweighted Lp estimate S. We cast proofs language obstacle problems. For sake comparison, also find weak (1, 1) was suggested Bollobas (Math Proc Camb Phil Soc 87:377–382, 1980) Osekowski (Statist Probab Lett 79(13):1536–1538, 2009), so it not new, but like to emphasize common approach those two descriptions.