Global Weighted $L^2$ $\dbar$-solvability on Noncompact Pseudoconvex Complex Lie Groups

We prove global weighted $L^2$ solvability for the $\bar\partial$-equation on any noncompact pseudoconvex complex Lie group. If $G$ is a connected noncompact complex Lie group admitting a continuous plurisubharmonic (psh) exhaustion $\rho$, then for every $t \ge 0$, $p \ge 0$ and $q \ge 1$ the weighted $L^2$ Dolbeault cohomology $H^{p,q}_{\bar\partial,(2),t}(G)$ with respect to the weight $e^{-t\rho}$ vanishes and one has a global \emph{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 $\rho$ 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.

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