Boundedness theorems for flowers and sharps

We show that the Σ1/1- and Σ1/2-boundedness theorems extend to the category of continuous dilators. We then apply these results to conclude the corresponding theorems for the category of sharps of real numbers, thus establishing another connection between Proof Theory and Set Theory, and extending work of Girard-Normann [J. Symbolic Logic 57 (1992), pp. 659–676] and Kechris-Woodin [Ann. Pure Appl. Logic 52 (1991), pp. 93–97].