The Cauchy problem for the periodic Kadomtsev--Petviashvili--II equation below $L^2$

We extend Bourgain’s $L^2$-wellposedness result for the KP-II equation on ${\mathbb{T}}^2$ to initial data with negative Sobolev regularity. The key ingredient is a new linear $L^4$-Strichartz estimate which is effective on frequency-dependent time scales. The $L^4$-Strichartz estimates follow from combining an ${\mathrm{\ell }}^2$-decoupling inequality recently proved by Guth–Maldague–Oh with semiclassical Strichartz estimates. Moreover, we rely on a variant of Bourgain’s bilinear Strichartz estimate on frequency-dependent times, which is proved via the Córdoba–Fefferman square function estimate.