Toeplitz operators with symbols in \(L^1(\textbf{D})\) on Bergman spaces with variable exponent

We establish a boundary condition on the variable exponent p, for which the operators \(U_z:f\mapsto (f\circ \varphi _z)\varphi '_z\) are bounded in \(A^{p(\cdot )}(\textbf{D})\). This boundary condition enables us to investigate the boundedness and compactness of Toeplitz operators \(T_\varphi\) with symbols \(\varphi\) in \(L^1(\textbf{D})\), via the functions \(z\mapsto \Vert U_zT_\varphi U_z(\mathbbm {1})\Vert _{L^{p(\cdot )}(\textbf{D})}\) and \(z\mapsto \Vert U_zT_{\overline{\varphi }}U_z(\mathbbm {1})\Vert _{L^{p(\cdot )}(\textbf{D})}\).