Toeplitz and Hankel operators on Bergman spaces with doubling weights

In this paper, we focus on the boundedness and compactness of the Toeplitz operators and the Hankel operators on the $\mathcal{D}$-weighted Bergman spaces $A_{\omega }^p$ ($1\le p<\mathrm{\infty }$). In particular, the boundedness of Toeplitz operators with ${\mathrm{B}\mathrm{M}\mathrm{O}}_{\omega }^p$-symbols on $A_{\omega }^p$ is characterized in terms of the Berezin-type transform. Several sufficient conditions for the Toeplitz operators $T_f^{\omega }$ with locally integrable symbols to be bounded (resp. compact) for each $1\le p<\mathrm{\infty }$ are presented. Moreover, sufficient and necessary conditions for the boundedness of the Toeplitz operator $T_{\bar{f}}^{\omega }:A_{\omega }^1\to A_{\omega }^1$ and the Hankel operator $H_{\bar{f}}^{\omega }:A_{\omega }^1\to L_{\omega }^1$ with co-analytic symbol are established.