Maximal functions and transference for groups of operators

Let a be the Laplace operator on R-d and 1 < delta < 2. Using transference methods we show that, for max{q, q/(q - 1)} < 4d/(2d + 1 - delta), the maxi mal function sup(t>0)\e(it Delta)f\ for the Schrodinger group is in L-q, fo r f is an element of L-q with Delta(delta/2) f is an element of L-q. We obt ain a similar result for the Airy group exp it Delta(3/2). An abstract vers ion of these results is obtained for bounded C-0-groups e(itL) on subspaces of L-p spaces. Certain results extend to maximal functions defined for fun ctions with values in UMD Banach spaces.