Regional inverse analyses of atmospheric trace gas observations quantify gridded two-dimensional surface fluxes by fitting the observations to simulated concentrations from a chemical transport model (CTM), usually by Bayesian optimization regularized by a gridded prior flux estimates. Regional inversions rely on the specification of background concentrations given by the boundary conditions (BCs) at the edges of the inversion domain, but biases in the BCs propagate to biases in the optimized fluxes. We develop a theoretical framework to explain how errors in the BCs influence the optimized fluxes as a function of the prior and observing system error statistics and of CTM transport. We derive a preview metric to estimate the BC-induced errors before conducting an inversion to support domain specification and a diagnostic metric to accurately quantify these errors after solving the inversion. We compare two methods to correct BC biases as part of an inversion, either directly by optimizing BC concentrations (boundary method) or indirectly by correcting grid cell fluxes outside the domain of interest (buffer method). We demonstrate that the boundary method is generally more accurate, physically grounded, and computationally tractable.