A Smoothing Property of Schrödinger Equations in the Critical Case

(Watanabe [17]). But we insist, in this paper, we can attain the critical case if we assume some structure condition on the operator σ(X,D). This case is crucial in dealing with further applications to the low regularity solutions to nonlinear problems, where the structure condition corresponds to the null structure of the nonlinearity. We explain our strategy by generalizing the equation, so that we can understand the meaning of the structure well. This will also emphasize geometric qualities responsible for smoothing in the critical case. We set