Boundedness and Compactness of Composition Operators on Variable Exponent Bergman Spaces

Abstract

Let $$\mathbb {D}$$ be the open unit disk in the complex plane $$\mathbb {C}$$, and let $$\varphi $$ be a holomorphic function from disk $$\mathbb {D}$$ into $$\mathbb {D}$$. We study the composition operator $$C_{\varphi }$$ on the variable exponent Bergman space in unit disk, and prove that this operator is bounded on $$A^{p(\cdot )}(\mathbb {D})$$. We give a sufficient condition for the compactness of this operator on $$A^{p(\cdot )}(\mathbb {D})$$ as well.