$L^{2}$ estimates for $\bar{\partial}$ across a divisor with Poincaré-like singularities

Abstract

We present results on $L^{2}$ estimates for solutions of $\bar{\partial}$-equations on a Stein manifold with a divisor. The structure of the divisor allows us to introduce weights with certain types of singularities, and the geometry of the manifold near the divisor allows us, by exploiting twisted techniques, to weaken the usual curvature hypotheses that guarantee a solution. We investigate two situations; one in which the weights are not locally integrable, and another in which they can be.