  We consider an iterative resolution scheme for a broad class of active scalar
equations with a fractional power \gamma of the Laplacian and focus our
attention on the second iterate. The main objective of our work is to analyze
boundedness properties of the resulting bilinear operator, especially in the
super-critical regime. Our results are two-fold: we prove continuity of the
bilinear operator in BMO^{1-2\gamma} - a fractional analogue of the Koch-Tataru
space; for equations with an even symbol we show that the
B^{-\gamma}_{\infty,q} -regularity, where q > 2, is in a sense a minimal
necessary requirement on the solution.
