Sharp estimates for maximal operators associated to the wave equation

The wave equation, partial derivative(tt)u=Delta u, in Rn+1, considered with initial data u(x, 0)= f is an element of H-s(R-n) and u'(x, 0)=0, has a solution which we denote by 1/2 (e(it root-Delta) f+e(-it root-Delta) f). We give almost sharp conditions under which sup(0<t<1)\e(+/-it root-Delta) f\ and sup(t is an element of R)\e(+/-it root-Delta) f\ are bounded from H-s (R-n) to L-q(R-n).