LAUSR

KΩ(z, z) is the normalized Bergman kernel as ‖kz ‖L2 = 1 for all z ∈ Ω where ‖.‖Lp denotes the Lp norm on Ω. Next we will define the Berezin transform of a bounded operator T : A2(Ω) → A2(Ω). The Berezin transform BΩT(z) of T at z ∈ Ω is defined as BΩT(z) = 〈Tkz , kz 〉. where 〈., .〉 denotes inner product on L2(Ω). For a function φ on Ω we defined its Berezin transform as BΩφ(z) = 〈Tφkz , kz 〉 where Tφ : A 2(Ω) → A2(Ω) is the Toeplitz operator with symbol φ. Namely, Tφ f = PΩ(φ f ). We note that BΩφ(z) = ∫ φ(w)|kz (w)|2dV(w). Berezin transform has been an important tool in operator theory. For instance, Axler-Zheng theorem and its various extensions establish compactness of operators using boundary values of the Berezin transform (see, for instance, [AZ98, Suá07, Eng99, ČŞ13, ČŞZ18]). Berezin transform is also related to Schatten norm of Hankel (and Toeplitz) operators (see [Zhu91, Li93, GŞ18]). We refer the reader to [Zhu07] for more details about the Berezin transform. The fact that ‖kz ‖L2 = 1 implies that ‖BΩ‖L∞ ≤ 1 on any domain Ω. In this article we are interested in Lp-regularity of the Berezin transform for p < ∞. That is, we want to know on what domain Ω is the Berezin transform BΩ : L p(Ω) → Lp(Ω) a bounded operator? This problem has been studied in few cases. Dostanić in [Dos08] showed that the Berezin transform is Lp regular on the unit disc and computed the norm ‖BD‖Lp = π(p+ 1)/(p2 sin(π/p)) for 1 < p ≤ ∞. Later Liu and Zhou in [LZ12] and Marković in [Mar15] generalized Dostanić’s result to the unit ball in Cn. In case of the higher dimensions, on the bidisc D2, Lee in [Lee97, Proposition 3.2] showed that the Berezin transform is bounded on Lp(D2) for p > 1 and unbounded on L1(D2). In this