Toeplitz operators between distinct Bergman spaces

For $-1<\alpha<\infty$, let $\omega_\alpha(z)=(1+\alpha)(1-|z|^2)^\alpha$ be the standard weight on the unit disk. In this note, we provide descriptions of the boundedness and compactness for the Toeplitz operators $T_{\mu,\beta}$ between distinct weighted Bergman spaces $L_{a}^{p}(\omega_{\alpha})$ and $L_{a}^{q}(\omega_{\beta})$ when $0<p\leq1$, $q=1$, $-1<\alpha,\beta<\infty$ and $0<p\leq 1<q<\infty, -1<\beta\leq\alpha<\infty$, respectively. Our results can be viewed as extensions of Pau and Zhao's work in \cite{Pau}. Moreover, partial of main results are new even in the unweighted settings.