Optimal constants and extremisers for some smoothing estimates

We establish new results concerning the existence of extremisers for a broad class of Kato-smoothing estimates of the form

$${\left\| {\psi \left( {\left| \nabla \right|} \right)\exp \left( {it\phi \left( {\left| \nabla \right|} \right)f} \right)} \right\|_{{L^2}\left( \omega \right)}} \leqslant C{\left\| d \right\|_{{L^2}}}$$

for solutions of dispersive equations, where the weight ω 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 (ω, ψ, ?). Furthermore, when ω is homogeneous, and for certain (ψ, ?), 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 ω is inhomogeneous, we obtain new expressions for the optimal constant and the non-existence of extremisers.