The boundedness result above was used by Howe (1980) to establish the more general inequality of Alberto Calderón and Remi Vaillancourt for pseudodifferential operators. An alternative proof that applies more generally to Fourier integral operators was given by Howe (1988). He showed that such operators can be expressed as integrals over the oscillator semigroup and then estimated using the Cotlar-Stein lemma.