We transpose work by K.Yajima and by T.Mizumachi to prove dispersive and smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schr\"odinger equation (NLS) in 2D. As an application we extend to dimension 2D a result on asymptotic stability of ground states of NLS proved in the literature for all dimensions different from 2.
A revision of "On asymptotic stability in energy space of ground states of NLS in 1D"
Asymptotic stability in the energy space for dark solitons of the Landau-Lifshitz equation
Kink dynamics in the $φ^4$ model: asymptotic stability for odd perturbations in the energy space
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