Polynomially oscillatory multipliers on Gelfand–Shilov spaces

We study continuity of the multiplier operator acting on Gelfand-Shilov spaces, where is a polynomial on of degree at least two with real coefficients. In the parameter quadrant for the spaces, we identify a wedge that depends on the polynomial degree for which the operator is continuous. We also show that in a large part of the complement region the operator is not continuous in dimension one. The results give information on well-posedness for linear evolution equations that generalize the Schr & ouml;dinger equation for the free particle.