  This article represents the first 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 sharp cubic energy estimates. Precisely, we
introduce and develop the techniques to prove a new class of energy estimates,
which we call \emph{balanced cubic estimates}. This yields a key improvement
over the earlier cubic estimates of Hunter-Ifrim-Tataru [15], while preserving
their scale invariant character and their position-velocity potential
holomorphic coordinate formulation.
  Even without using any Strichartz estimates, these results allow us to
significantly lower the Sobolev regularity threshold for local well-posedness,
drastically improving earlier results obtained by Alazard-Burq-Zuily [5, 6],
Hunter-Ifrim-Tataru [15] and Ai [2].
