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This article represents the first installment of a series of papers concernedwith low regularity solutions for the water wave equations in two spacedimensions. Our focus here is on sharp cubic energy estimates. Precisely, weintroduce and develop the techniques to prove a new class of energy estimates,which we call \emph{balanced cubic estimates}. This yields a key improvementover the earlier cubic estimates of Hunter-Ifrim-Tataru [15], while preservingtheir scale invariant character and their position-velocity potentialholomorphic coordinate formulation.  Even without using any Strichartz estimates, these results allow us tosignificantly 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].

Two dimensional gravity waves at low regularity I: Energy estimates