Estimates for the Gross-Pitaevskii equation linearized around a vortex

We consider the linearized two-dimensional Gross-Pitaevskii equation around a vortex of degree one, with data in the same equivariance class. Various estimates are proved for the solution; in particular, conditions for optimal decay in $L^{\mathrm{\infty }}$ and boundedness in $L^2$ are identified. The analysis relies on a full description of the spectral resolution of the linearized operator through the associated distorted Fourier transform.