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 Schrödinger equation for the free particle.