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 study multipliers associated to the Hermite operator H = −∆ + |x| 2 on modulation spaces M p,q (R d ). We prove that the operator m(H) is bounded on M p,q (R d ) under standard conditions on m, for suitable choice of p and q. As an application, we point out that the solutions to the free wave and Schrödinger equations associated to H with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to H are bounded on some modulation spaces.2010 Mathematics Subject Classification. Primary: 42B15, 42B35, Secondary: 35L05, 35Q55.
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