Title: A commuting-vector-field approach to some dispersive estimates,
Abstract: We prove the pointwise decay of solutions to three linear equations: (i) the
transport equation in phase space generalizing the classical Vlasov equation,
(ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The
usual proofs use explicit representation formulae, and either obtain
$L^1$---$L^\infty$ decay through directly estimating the fundamental solution
in physical space, or by studying oscillatory integrals coming from the
representation in Fourier space. Our proof instead combines "vector field"
commutators that capture the inherent symmetries of the relevant equations with
conservation laws for mass and energy to get space-time weighted energy
estimates. Combined with a simple version of Sobolev's inequality this gives
pointwise decay as desired. In the case of the Vlasov and Schrodinger equations
we can recover sharp pointwise decay; in the Schrodinger case we also show how
to obtain local energy decay as well as Strichartz-type estimates. For the Airy
equation we obtain a local energy decay that is almost sharp from the scaling
point of view, but nonetheless misses the classical estimates by a gap. This
work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave
equations, as well as the recent work of Fajman, Joudioux, and Smulevici on
decay of mass distributions for the relativistic Vlasov equation.