Remarks on nonlinear operations on modulation spaces

Abstract

Let F(f) be the composition of functions F and f. We consider the question ‘If f belongs to some function space, does F(f) belong to the same space again?’. The answers to this question for the Sobolev space and the Besov space are well known by virtue of the theory of paradifferential operators by Bony [Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), pp. 209–246] and Meyer [Remarques sur un théorème de J.-M. Bony, Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), Rend. Circ. Mat. Palermo Vol. 2, Suppl. 1, 1981, pp. 1–20]. This note is a trial to answer this question for the modulation space which is defined by using the short-time Fourier transform, a real variable reformulation of the Bargmann transform. The idea of the modulation space is to consider the space variable and the variable of its Fourier transform simultaneously to measure the decaying and regularity property. We give some partial answers to this question for this relatively new function space. We also give a less restrictive answer for the Wiener amalgam, a variant of the modulation space.

Keywords:

• Bargmann transform
• short-time Fourier transform
• nonlinear operation
• modulation space
• Wiener amalgam space

AMS Subject Classification :

• 44A15
• 42B35
• 42B37
• 35S50