nLab Hörmander's criterion


This entry is about Hörmander’s criterion on wave front sets. This is different from “Hörmander's condition” on tangent vector fields.



The Hörmander criterion (Hörmander 90, theorem 8.2.10) says that the product of two distributions u 1u 2u_1 \cdot u_2 (on some manifold XX) is well-defined if their wave front sets WF(u)WF(u) are such that for vT x *Xv \in T^\ast_x X a covector contained in one of the two wave front sets then the covector vT x *X-v \in T^\ast_x X with the opposite direction in not contained in the other wave front set, i.e. the intersection fiber product inside the cotangent bundle T *XT^\ast X of the pointwise sum of wave fronts with the zero section is empty:

(WF(u 1)+WF(u 2))×T *XX= \left( WF(u_1) + WF(u_2) \right) \underset{T^\ast X}{\times} X \;=\; \emptyset


WF(u 1)+WF(u 2) (pb) X 0 T *X \array{ && \emptyset \\ & \swarrow && \searrow \\ WF(u_1) + WF(u_2) && (pb) && X \\ & \searrow && \swarrow_{\mathrlap{0}} \\ && T^\ast X }

See at product of distributions for details.


  • Lars Hörmander, Fourier integral operators. I. Acta Mathematica 127, 79–183 (1971) (Euclid)

  • Lars Hörmander, section 8.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Last revised on December 22, 2017 at 20:38:52. See the history of this page for a list of all contributions to it.