Abstract. If an integrable function f on the Heisenberg group is supported on the set B × R where B ⊂ C n is compact and the group Fourier transform f (λ) is a finite rank operator for all λ ∈ R \ {0}, then f ≡ 0.

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The aim of this paper is two fold. We show that if a complex function F on C operates in the modulation spaces M p,1 (R n ) by composition, then F is real analytic on R 2 ≈ C. This answers negatively, the open question posed in [M. Ruzhansky, M. Sugimoto, B. Wang, Modulation Spaces and Nonlinear Evolution Equations, arXiv:1203.4651], regarding the general power type nonlinearity of the form |u| α u. We also characterise the functions that operate in the modulation space M 1,1 (R n ).The local well-posedness of the NLS, NLW and NLKG equations for the 'real entire' nonlinearities are also studied in some weighted modulation spaces M p,q s (R n ).

We establish the local well posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on C n in certain first order Sobolev space. Our approach is based on Strichartz type estimates, and is valid for a general class of nonlinearities including power type. The case n = 1 represents the magnetic Schrödinger equation in the plane with magnetic potential A(z) = iz, z ∈ C.2010 Mathematics Subject Classification. Primary 42B37, Secondary 35G20, 35G25.

We establish a Strichartz type estimate for the Schrödinger propagator e itL for the special Hermite operator L on C n . Our method relies on a regularization technique. We show that no admissibility condition is required on (q, p) when 1 ≤ q ≤ 2.