# nLab Hörmander's criterion

Contents

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

# Contents

## Idea

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

$\left( WF(u_1) + WF(u_2) \right) \underset{T^\ast X}{\times} X \;=\; \emptyset$

i.e.

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

See at product of distributions for details.

## References

• 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 15:38:52. See the history of this page for a list of all contributions to it.