  The aim of this article is to demonstrate how the vector field method of
Klainerman can be adapted to the study of transport equations. After an
illustration of the method for the free transport operator, we apply the vector
field method to the Vlasov-Poisson system in dimension 3 or greater. The main
results are optimal decay estimates and the propagation of global bounds for
commuted fields associated with the conservation laws of the free transport
operators, under some smallness assumption. Similar decay estimates had been
obtained previously by Hwang, Rendall and Vel\'azquez using the method of
characteristics, but the results presented here are the first to contain the
global bounds for commuted fields and the optimal spatial decay estimates. In
dimension 4 or greater, it suffices to use the standard vector fields commuting
with the free transport operator while in dimension 3, the rate of decay is
such that these vector fields would generate a logarithmic loss. Instead, we
construct modified vector fields where the modification depends on the solution
itself. The methods of this paper, being based on commutation vector fields and
conservation laws, are applicable in principle to a wide range of systems,
including the Einstein-Vlasov and the Vlasov-Nordstr\"om system.
