  The purpose of this paper is to study local and global well-posedness of
initial value problem for generalized Korteweg-de Vries (gKdV) equation in
^L^r. We show (large data) local well-posedness, small data global
well-posedness, and small data scattering for gKdV equation in the scale
critical ^L^r space. A key ingredient is a Stein-Tomas type inequality for the
Airy equation, which generalizes usual Strichartz estimates for ^L^r-framework.
