Weighted L² estimates for maximal operators associated to dispersive equations

Let Tf(x, t) = e^2πitφ(D)f(x) be the solution of the general dispersive equation with phase φ and initial data f in the Sobolev space H^s. We prove a weighted L² estimate for the global maximal operator T** defined by taking the supremum over the time variable t ∈ R so that ∥T** f∥_{L2(w dx)} ≤ C∥f∥_Hs. The exponent s depends on the phase function φ, whose gradient may vanish or have singularities.