WEIGHTED L~2 ESTIMATES FOR MAXIMAL OPERATORS ASSOCIATED TO DISPERSIVE EQUATIONS

摘要

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 L~2 estimate for the global maximal operator T~(**) defined by taking the supremum over the time variable t ∈ R so that ‖T~(**)f‖_(L~2(w dx)) ≤ C ‖f‖_(H~s). The exponent s depends on the phase function φ, whose gradient may vanish or have singularities.