LOCAL WELL POSEDNESS FOR THE NONLOCAL NONLINEAR SCHRODINGER EQUATION BELOW THE ENERGY SPACE

We establish local well posedness for arbitrarily large initial data in the usual Sobolev spaces H-s (R), s > 1/2, for the Cauchy problem associated to the integro-differential equation partial derivative(t)u + i alpha partial derivative(2)(x)u = beta u (1 + iT(h)) partial derivative(x)(vertical bar u vertical bar(2)) + i gamma vertical bar u vertical bar(2)u, where u = u(x, t) is an element of C, x, t is an element of R, and T-h denotes the singular operator defined by T(h)f(x) = 1/2h p.v. integral(infinity)(-infinity) coth (pi(x-y)/2h) f(y)dy, when 0 < h <= infinity. Note that T-infinity = H is the Hilbert transform. Our method of proof relies on a gauge transformation localized in positive frequencies which allows us to weaken the high-low frequencies interaction in the nonlinearity.