Compact Composition Operators On Weighted Hardy and Bergman Spaces

Abstract

We study the action of composition operators on Sobolev spaces of analytic functions in some weighted Bergman and Hardy spaces on the unit disc. Composition operators mapping into the Hardy space are included by making particular choices for the weights. We show the construction of the translating of certain classical inequalities for Hardy spaces to inequalities for Bergman spaces and then how to translate them to special inequalities for Hardy spaces , moreover we prove a relation of the equality on the general Lebesgue space and calculate a recognizable estimate for the exterma of the best possible constant. We show that Hardy space be contained in the Bergman space when they are equivalent to Hilbert space by finding the multipliers of Hardy space into Bergman spaces in the unit disc and show a generalization of pointwise multipliers from weighted Bergman spaces and Hardy to weighted Bergman spaces.