Global Cauchy problems for the nonlocal (derivative) NLS in Eσs

We consider the Cauchy problem for the nonlocal (derivative) NLS in super-critical function spaces Essfor which the norms are defined by ||f|| E-sigma(s) = || <xi >(sigma) 2(s|xi|) (f) over cap(xi)||(L2), s< 0, sigma is an element of R. Any Sobolev space H-r is a subspace of E-sigma(s), i.e., H-r subset of E-sigma(s) for any r, sigma is an element of R and s< 0. Let s < 0 and sigma > -1/2(sigma > 0) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to E-sigma(s) and their Fourier transforms are supported in (0, infinity), the smallness conditions on the initial data in E-sigma(s) are not required for the global solutions. (c) 2022 Elsevier Inc. All rights reserved.