Toeplitz operators between distinct Bergman spaces

For $-1<\alpha <\mathrm{\infty }$, let ${\omega }_{\alpha }(z)=(1+\alpha )(1-\mathrm{\mid }z{\mathrm{\mid }}^2{)}^{\alpha }$ 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 $0<p\le 1$, $q=1$, $-1<\alpha ,\beta <\mathrm{\infty }$ and $0<p\le 1<q<\mathrm{\infty },-1<\beta \le \alpha <\mathrm{\infty }$, 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.