Berezin transform and Toeplitz operators on polygonal domains

We consider reflexive Bergman spaces Ap(Ω) on polygonal domains Ω of the complex plane. With some restrictions to angles boundary Ω, we show that boundedness Toeplitz operator Tg:Ap(Ω)→Ap(Ω) with a positive symbol g is equivalent Berezin transform or times area measure being Carleson measure. The result also formulated for more general simply connected domains. main technical tool new weighted Forelli–Rudin-type estimate.