  We study a weighted $\ell^1$-regularization technique for solving inverse
problems when the forward operator has a significant nullspace. In particular,
we prove that a sparse source can be exactly recovered as the regularization
parameter $\alpha$ tends to zero. Furthermore, for positive values of $\alpha$,
we show that the regularized inverse solution equals the true source multiplied
by a scalar $\gamma$, where $\gamma = 1 - c\alpha$. Our analysis is supported
by numerical experiments for cases with one and several local sources.
  This investigation is motivated by a PDE-constrained optimization problem,
but the theory is developed in terms of Euclidean spaces. Our results can
therefore be applied to many problems.
