Weighted maximal estimates along curve associated with dispersive equations

In the present paper, we give the local and global weighted L-q maximal estimate for the operator e(it phi)(root- Delta ) f(gamma( x, t)), which is defined by e(it phi)( root-Delta ) f(gamma( x, t)) = (2 pi)(-1) integral(R) e(i gamma( x, t). xi) (+) (it phi(broken vertical bar xi broken vertical bar)) f(xi)d xi, where phi satisfies some growth conditions, phi(root-Delta) is a pseudo-differential operator with symbol phi(broken vertical bar xi broken vertical bar) and gamma : R x R -> R satisfies Holder's condition of order a and bilipschitz conditions. As a corollary of the above conclusions, we show that if f epsilon H (delta)(R) for s >= 1/ 4 and 1/2 <= alpha <= 1, then lim(t -> 0) e(it phi)(root-Delta) f(gamma( x, t)) = f(x), a. e. x epsilon R. In particular, if taking phi(r) = r(2), then this improves a result in [C. Cho, S. Lee and A. Vargas, Problems on pointwise convergence of solutions to the Schrodinger equation, J. Fourier Anal. Appl. 18 (2012) 972-994], where (*) holds for s > 1/4 only.