Weighted L-2 estimates for maximal operators associated to dispersive equations

Abstract

Let Tf (x, t) = e(2 pi it phi(D)) f(x) be the solution of the general dispersive equation with phase phi and initial data f in the Sobolev space H-s. We prove a weighted L-2 estimate for the global maximal operator T** defined by taking the supremum over the time variable t is an element of R so that vertical bar vertical bar T**f vertical bar vertical bar L-2(w dx) <= C vertical bar vertical bar f vertical bar vertical bar H-3. The exponent s depends on the phase function 0, whose gradient may vanish or have singularities.