ON POINTWISE CONVERGENCE OF SCHRÖDINGER MEANS

For functions in the Sobolev space H s and decreasing sequencest n → 0 we examine convergence almost everywhere of the generalized Schrödinger means on the real line, given byS a f ( x , t n ) = exp ( i t n( − ∂ x x ) a / 2 ) f ( x ) ; here a > 0 , a ≠ 1 . For decreasing convex sequences we obtain a simple characterization of convergence a.e. for all functions in H s when 0 < s < min { a / 4 , 1 / 4 } and a ≠ 1 . We prove sharp quantitative local and global estimates for the associated maximal functions. We also obtain sharp results for the case a = 1 .