Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$mathbb {C}^2$$

Let (Omega ) be a bounded, uniformly totally pseudoconvex domain in (mathbb {C}^2) with smooth boundary (bOmega ). Assume that (Omega ) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let (alpha ) be a d-closed (1, 1)-form in (Omega ). We study the Poincaré–Lelong equation

$$begin{aligned} ipartial bar{partial }u=alpha quad text {on}, Omega end{aligned}$$

in (L^1(bOmega )) norm by applying the (L^1(bOmega )) estimates for (bar{partial }_b)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in (Omega ).