  In this note we study some basic properties of general fractional derivatives
induced by weighted Bergman kernels. As an application we demonstrate a method
for generating pre-images of analytic functions under weighted Bergman
projections. This approach is useful for proving the surjectivity of weighted
Bergman projections in cases when the target space is not a subspace of the
domain space (such situations arise often when dealing with Bloch and Besov
spaces). We also discuss a fractional Littlewood-Paley formula.
