We establish new results concerning the existence of extremisers for a broad class of Kato-smoothing estimates of the form
$${left| {psi left( {left| nabla right|} right)exp left( {itphi left( {left| nabla right|} right)f} right)} right|_{{L^2}left( omega right)}} leqslant C{left| d right|_{{L^2}}}$$
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.
