GLOBAL RANGE ESTIMATES FOR MAXIMAL OSCILLATORY INTEGRALS WITH RADIAL TEST FUNCTIONS

We consider the maximal function of oscillatory integrals S-a f where (S-a f)(t)(boolean AND)(xi) = exp(it|xi|(a))(f) over cap(xi) and a is an element of]0, 1[. For a fixed n >= 2 we prove the global estimate parallel to S-a f parallel to(L2(Rn, L infinity(-1,1))) <= C parallel to f parallel to(Hs(Rn)), s> a/4 with C independent of the radial function f. We also prove that this result is almost sharp with respect to the Sobolev regularity s. This extends work of Sjolin who proved these result for a > 1.