  This article represents the second installment of a series of papers
concerned with low regularity solutions for the water wave equations in two
space dimensions. Our focus here is on global solutions for small and localized
data. Such solutions have been proved to exist earlier in [15, 7, 10, 12] in
much higher regularity. Our goal in this paper is to improve these results and
prove global well-posedness under minimal regularity and decay assumptions for
the initial data. One key ingredient here is represented by the balanced cubic
estimates in our first paper. Another is the nonlinear vector field Sobolev
inequalities, an idea first introduced by the last two authors in the context
of the Benjamin-Ono equations [14].
