Singular support of a distribution

Let $T$ be a distribution on an open set $\Omega\subseteq\mathbb R^n$ . A point $x\in\Omega$ is regular for $T$ if some neighborhood $U$ of $x$ and some $f\in C^\infty(U)$ satisfy

\langle T,\varphi\rangle=\int_U f(y)\varphi(y)\,dy

for every test function $\varphi$ supported in $U$ . The singular support $\operatorname{singsupp}T$ is the complement in $\Omega$ of the set of regular points. It is a relatively closed subset of the support of $T$ and records where $T$ fails to be locally a smooth function. This definition depends only on the restriction of $T$ to arbitrarily small neighborhoods.

Local character and basic operations

Regularity is local, so $T$ is induced by a smooth function precisely when $\operatorname{singsupp}T=\varnothing$ . Multiplication by a smooth function and distributional differentiation satisfy

\operatorname{singsupp}(aT)\subseteq\operatorname{singsupp}T, \qquad \operatorname{singsupp}(\partial^\alpha T) \subseteq\operatorname{singsupp}T.

Either inclusion may be strict: multiplication can cancel a singularity, and differentiation in a direction along which a singular distribution is invariant can annihilate it.

Examples and relation to wavefront set

A smooth function, regarded as a regular distribution, has empty singular support. The Dirac distribution $\delta_a$ and all its nonzero derivatives have singular support $\{a\}$ . A piecewise smooth function has singular support contained in the locus where its smooth pieces fail to fit smoothly.

The wavefront set refines singular support by retaining cotangent directions: the projection of $\operatorname{WF}(T)$ to $\Omega$ is $\operatorname{singsupp}T$ .

Conventions and scope

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: Chapter 2, support, local regularity, and singular support of distributions.
Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999. Publisher record. Relevant: Chapter 9, distributions and their local behavior.