In 2014, Banjade and Trent proved an analog of Wolff's theorem for the multiplier algebra of the Dirichlet space, which gives conditions for the cube of a function to be in a generated ideal. This dissertation investigates conditions that guarantee the function itself is in the ideal. This result is then extended to the multiplier algebra of the weighted Dirichlet space, which is also seen as a refinement of a similar result by Banjade and Trent, but for the weighted space. Following this, discussion takes place on how to generalize the results obtainedin this dissertation.