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.

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