In this paper, we consider the defocusing Hartree nonlinear Schrödinger equations on T3 with real-valued and even potential V and Fourier multiplier decaying such as |k|−β. By relying on the method of random averaging operators [Deng et al., arXiv:1910.08492 (2019)], we show that there exists β0, which is less than but close to 1, such that for β > β0, we have invariance of the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way, we extend Bourgain’s seminal result [J. Bourgain, J. Math. Pures Appl. 76, 649–702 (1997)], which requires β > 2 in this case.

Abstract. In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space FL s,r (T) with s ≥ 1 2 , 2 < r < 4, (s − 1)r < −1 and scaling like H 1 2 −ǫ (T), for small ǫ > 0. We also show the invariance of this measure.

We study the question of well-posedness of the Cauchy problem for Schrödinger maps from R 1 × R 2 to the sphere S 2 or to H 2 , the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equations and then study this modified Schrödinger map system (MSM). We then prove local well-posedness of the Cauchy problem for the MSM with minimal regularity assumptions on the data and outline a method to derive well-posedness of the Schrödinger map itself from it. In proving well-posedness of the MSM, the heart of the matter is resolved by considering truly quatrilinear forms of weighted L 2 -functions.

We construct a gauge theoretic change of variables for the wave map from R × R n into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equationn ≥ 4 -for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into compact Lie groups and symmetric spaces with small critical initial data and n ≥ 4.

We develop some connections between interpolation theory and the theory of bounded holomorphic functional calculi of operators in Hilbert spaces, via quadratic estimates. In particular we s h o w that an operator T of type ! has a bounded holomorphic functional calculus if and only if the Hilbert space is the complex interpolation space midway b etween the completion of its domain and of its range. We a l s o c haracterise the complex interpolation spaces of the domains of all the fractional powers of T , whether or not T has a bounded functional calculus. This treatment extends earlier ones for self{adjoint and maximal accretive operators. This work is motivated by the study of rst order elliptic systems which are related to the square root problem for non{degenerate second order operators under boundary conditions on an interval.