Solvability for the Ginzburg-Landau equation linearized at the degree-one vortex

We consider the Ginzburg-Landau equation in the plane linearized around the standard degree-one vortex solution $W\left(x\right)=w\left(r\right)e^{i\theta }$. Using explicit representation formulae for the Fourier modes in θ, we obtain sharp estimates for the inverse of the linearized operator which hold for a large class of right-hand sides. This theory can be applied, for example, to estimate the inverse after dropping the usual orthogonality conditions.