(untitled) (European Journal of Pure and Applied Mathematics, 2025)
We prove global weighted L^2 solvability for the -equation on any noncompact pseudoconvex complex Lie group. If G is a connected noncompact complex Lie group admitting a continuous plurisubharmonic (psh) exhaustion ρ, then for every t 0, p 0 and q 1 the weighted L^2 Dolbeault cohomology H^p,q_,(2),t(G) with respect to the weight e^-tρ vanishes and one has a global a priori estimate. The argument relies on two uniformities provided by the Lie group geometry: (i) a uniform exhaustion by smoothly bounded strictly pseudoconvex domains whose defining functions approximate ρ on fixed sublevels; (ii) strictly psh reference functions on these domains with a Levi lower bound independent of the exhaustion index. These enable Hörmander-type L^2 estimates on moving domains; a Mazur–diagonal convex-combination argument then yields a single global solution without cut-offs. Consequences include a Hartogs-type extension under weighted L^2 growth and richness of weighted Bergman spaces on strictly pseudoconvex sublevels.