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 $A_ω^p$ ($1\leq p<\infty$). In particular, the boundedness of Toeplitz operators with $\mathrm{BMO}_ω^p$-symbols on $A_ω^p$ is characterized in terms of the Berezin-type transform. Several sufficient conditions for the Toeplitz operators $T_f^ω$ with locally integrable symbols to be bounded (resp. compact) for each $1\leq p<\infty$ are presented. Moreover, sufficient and necessary conditions for the boundedness of the Toeplitz operator $T_{\bar{f}}^ω:A_ω^1\rightarrow A_ω^1$ and the Hankel operator $H_{\bar{f}}^ω:A_ω^1\rightarrow L_ω^1$ with co-analytic symbol are established.
math.FA↗