Optimal constants and extremisers for some smoothing estimates

We establish new results concerning the existence of extremisers for a broad class of smoothing estimates of the form $|psi(| abla|) exp(itphi(| abla|)f |_{L^2(w)} leq C|f|_{L^2}$, where the weight $w$ is radial and depends only on the spatial variable; such a smoothing estimate is of course equivalent to the $L^2$-boundedness of a certain oscillatory integral operator $S$ depending on $(w,psi,phi)$. Furthermore, when $w$ is homogeneous, and for certain $(psi,phi)$, we provide an explicit spectral decomposition of $S^*S$ and consequently recover an explicit formula for the optimal constant $C$ and a characterisation of extremisers. In certain well-studied cases when $w$ is inhomogeneous, we obtain new expressions for the optimal constant.