On the nontrapping condition and its role in the smoothing effect of dispersive operators

Abstract In this paper, we discuss the smoothing effect of some dispersive operators with variable coefficients in connection with the so called nontrapping condition (NTC). The operators under consideration are of order $$m=2,3$$ m = 2 , 3 , and we shall show that, in some cases, a global nontrapping property of an operator $$A(x,D_x)$$ A ( x , D x ) leads to the local smoothing effect for the dispersive operator $$D_t-A(x,D_x)$$ D t - A ( x , D x ) . On the other hand, smallness assumptions on $$A(x,D_x)$$ A ( x , D x ) imply the local smoothing effect without assuming the global nontrapping property. We shall start our exposition with some fundamental results on the microlocal smoothing effect of Schrödinger operators with variable coefficients, emphasizing the role of the nontrapping condition. Then, we will focus on the local smoothing, both homogeneous and inhomogeneous, of operators of order two and three with and without the nontrapping condition. This survey is based mainly on the results of the pioneering works [6, 11, 13, 14, 41] and of the recent paper [23].