  This paper is concerned with the global existence of small solutions to
pure-power nonlinear Schroedinger equations subject to radially symmetric data
with critical regularity. Under radial symmetry we focus our attention on the
case where the power of nonlinearity is somewhat smaller than the
pseudoconformal power and the initial data belong to the scale-invariant
homogeneous Sobolev space. In spite of the negative-order differentiability of
initial data the nonlinear Schroedinger equation has global in time solutions
provided that the initial data have the small norm. The key ingredient in the
proof of this result is an effective use of global weighted smoothing estimates
specific to radially symmetric solutions.
