Some Remarks on weighted Estimates for ∂̄

Abstract

Let us start by recalling the following well-known theorem of Hörmander (see [H1]). Theorem 1. Let D be a pseudoconvex open set in Cⁿ, and let Øbe a strictly plurisubharmonic function in D. Denote by c a lower bound for the smallest eigenvalue of the complex hessian of Ø. Suppose f is a ∂̄-closed (0, l)-form in D. Then there is a solution to

$$\overline \partial u\, = \,f$$

((1))

such that

$$\int_D {\left| u \right|^2 e^{ - \varphi } } \, \leqslant \,\int_D {\left| f \right|^2 /c\,e^{ - \varphi } }$$

((2))

provided the right hand side is finite.