Singular Support and Wavefront Sets

Singular support records where a distribution is not smooth. The wavefront set refines each singular point by the nonzero cotangent directions in which no localized Fourier transform decays rapidly. That directional information distinguishes singular distributions with the same singular support and supplies precise sufficient conditions for products and pullbacks.

Required background. Test-Function Spaces, Distributions, Support, and Convergence supplies localization and distributional smoothness; Tempered Distributions and Fourier Calculus supplies localized Fourier decay and its transformation rules.

This page gives the reusable definition, examples, and basic product and pullback tests. Propagation of singularities, Hadamard states, and the microlocal spectrum condition remain with their specialist QFT treatments.

Singular support forgets direction

For $u\in\mathcal D'(\Omega)$, the singular support is

$$\operatorname{sing\,supp}u = \Omega \setminus \bigcup \left\{ U\subseteq\Omega\text{ open}: u|_U\in C^\infty(U) \right\}.$$

It is closed. Outside this set, $u$ agrees locally with a smooth function; inside it, no smooth representative exists. For example,

$$\operatorname{sing\,supp}\delta_0 = \operatorname{sing\,supp}H = \{0\}$$

on $\mathbb R$, even though a point mass and a jump are different singularities.

To detect smoothness Fourier-theoretically, first localize. If $\chi\in C_c^\infty(\Omega)$, then $\chi u$ has compact support and its Fourier transform is the smooth, polynomially bounded function

$$\widetilde{\chi u}(p) = \left\langle \chi u,e^{+ip\cdot x} \right\rangle.$$

A local coordinate expression uses the canonical covector pairing $p\cdot x=p_jx^j$, not a metric contraction. A compactly supported distribution is smooth precisely when this function decays faster than every inverse power in all frequency directions. A wavefront set asks for that decay only in a cone.

Fredenhagen and Rejzner 2012, Appendix A, Definitions A.4 and A.6, PDF give the singular-support and localized Fourier definitions used here.

Directional regularity is rapid decay in a cone

An open set $\Gamma\subset\mathbb R^d\setminus\{0\}$ is positively conic if

$$p\in\Gamma,\ \lambda>0 \quad\Longrightarrow\quad \lambda p\in\Gamma.$$

It need not contain $-p$ when it contains $p$. Let $x_0\in\Omega$ and $p_0\neq0$. The pair $(x_0,p_0)$ is a regular directed point of $u$ if there are

• a cutoff $\chi\in C_c^\infty(\Omega)$ equal to $1$ on a neighborhood of $x_0$, and
• an open conic neighborhood $\Gamma$ of $p_0$

such that, for every $N\geq0$,

$$\boxed{ \sup_{p\in\Gamma} (1+|p|)^N \left| \widetilde{\chi u}(p) \right| <\infty. }$$

The wavefront set is the complement of all regular directed points:

$$\operatorname{WF}(u) \subset \Omega\times(\mathbb R^d\setminus\{0\}).$$

The cutoff must localize a neighborhood, and the estimate must hold in a whole open cone and for every $N$. Decay along one ray, or a global Fourier transform without localization, is not enough.

Throughout this page, $\operatorname{WF}$ uses the site’s transform $\mathcal F_+$ with phase $e^{+ip\cdot x}$. Hörmander uses $\mathcal F_-$, so

$$\mathcal F_-(\chi u)(\xi) = \mathcal F_+(\chi u)(-\xi).$$

Thus his fiber label is $\xi=-p$ relative to the one used here. Reflecting all fiber covectors translates between the conventions; the zero-sum product condition and the pullback condition are unchanged by that simultaneous reflection.

Brouder, Dang, and Hélein 2014, Definitions 8–12, PDF use the same positive Fourier phase and state the rapid-decay characterization.

The definition is geometric

On a manifold $M$, apply the definition to a chart representative after multiplying by a compactly supported cutoff. Under a coordinate change, frequency covectors transform by the transpose inverse Jacobian. Rapid decay in an open cone is preserved, so

$$\operatorname{WF}(u) \subset T^*M\setminus0_M$$

is well defined independently of the chart. It is a closed, positively conic set. The second entry is a covector, not a tangent velocity, and its definition needs no metric.

Multiplying a chart representative by a smooth nowhere-vanishing density does not change its wavefront set. Thus the scalar-distribution and distribution-density conventions from the preceding manifold page give the same microlocal directions.

Let $\pi:T^*M\to M$ be the bundle projection. Then

$$\boxed{ \pi\bigl(\operatorname{WF}(u)\bigr) = \operatorname{sing\,supp}u. }$$

Indeed, rapid decay in every direction implies smoothness; compactness of the unit sphere reduces “every direction” to finitely many conic estimates. Consequently,

$$\operatorname{WF}(u)=\varnothing \quad\Longleftrightarrow\quad u\in C^\infty(M).$$

Several operations cannot create new singular directions:

$$\begin{aligned} \operatorname{WF}(u+v) &\subseteq \operatorname{WF}(u)\cup\operatorname{WF}(v), \\ \operatorname{WF}(au) &\subseteq \operatorname{WF}(u), \qquad a\in C^\infty(M), \\ \operatorname{WF}(Pu) &\subseteq \operatorname{WF}(u) \end{aligned}$$

for every smooth-coefficient differential operator $P$. The second inclusion is an equality over an open set where $a$ never vanishes. The first can be strict because singularities may cancel. These statements do not include a propagation theorem or its characteristic-flow hypotheses.

Hörmander 2003, § 8.1 gives the coordinate-invariant definition and these structural properties.

Examples that separate position from direction

Smooth functions and point masses

Every smooth function has empty wavefront set. For the delta distribution, choose $\chi$ with $\chi(0)=1$. Then

$$\widetilde{\chi\delta_0}(p)=1,$$

which has no rapid decay in any nonzero direction. Therefore

$$\operatorname{WF}(\delta_0) = \{(0,p):p\neq0\}.$$

Multiplication by a polynomial in $p$ does not change the missing conic decay, so every nonzero derivative of $\delta_0$ has the same wavefront set.

The Heaviside distribution satisfies $H'=\delta_0$. Since differentiation does not create directions,

$$\operatorname{WF}(\delta_0) \subseteq \operatorname{WF}(H).$$

Both objects are singular only at $0$, so

$$\operatorname{WF}(H) = \{(0,p):p\neq0\}.$$

The principal value $\operatorname{PV}(1/x)$ also has both nonzero directions at the origin.

Boundary values remember a one-sided direction

Define

$$u_+(x) = \frac1{x+i0} = \operatorname{PV}\frac1x-i\pi\delta_0.$$

With the site’s positive phase, the localized Fourier transform decays rapidly as $p\to+\infty$ but not as $p\to-\infty$. Hence

$$\operatorname{WF}(u_+) = \{(0,p):p<0\}.$$

Complex conjugation reverses the prescription:

$$u_-(x) = \frac1{x-i0}, \qquad \operatorname{WF}(u_-) = \{(0,p):p>0\}.$$

Both have singular support $\{0\}$, but their wavefront sets point in opposite fiber directions. Their sum restores both:

$$u_++u_- = 2\,\operatorname{PV}\frac1x.$$

Fredenhagen and Rejzner 2012, § 4, Eqs. (29)–(31), PDF compute the first one-sided cone explicitly in the same phase convention. This example is the simplest demonstration that singular support alone loses operationally important information.

When a product is licensed

Let $u,v\in\mathcal D'(M)$. Suppose there is no nonzero covector $p$ such that

$$(x,p)\in\operatorname{WF}(u) \quad\text{and}\quad (x,-p)\in\operatorname{WF}(v).$$

Equivalently, the fiberwise sum of the two wavefront sets does not meet the zero section. Then the Hörmander product $uv$ exists as the pullback of $u\boxtimes v$ along the diagonal.

To state the output bound, adjoin the relevant zero covectors:

$$\operatorname{WF}_0(u) = \operatorname{WF}(u) \cup \{(x,0):x\in\operatorname{supp}u\}.$$

Then

$$\boxed{ \operatorname{WF}(uv) \subseteq \left\{ (x,p+q): \begin{array}{l} (x,p)\in\operatorname{WF}_0(u),\\ (x,q)\in\operatorname{WF}_0(v),\\ p+q\neq0 \end{array} \right\}. }$$

The construction agrees with the ordinary pointwise product whenever that product already defines a distribution, in particular when one factor is smooth. Hörmander 2003, Theorem 8.2.10 and Brouder, Dang, and Hélein 2014, Theorem 13, PDF give the theorem and the wavefront bound.

The proof mechanism is local Fourier convolution. After cutting off both factors near $x$, the formal transform of their product contains

$$\frac{1}{(2\pi)^d} \int \widetilde{\chi u}(q)\, \widetilde{\chi v}(p-q)\, \mathrm d^d q.$$

A conic partition separates the large-$q$ regions. If the two slow-decay cones never contain opposite covectors, at least one factor is rapidly decreasing in every potentially divergent region, while the other grows only polynomially. This makes the localized construction converge. Geometrically, the normal covectors to the diagonal are $(q,-q)$, so the same obstruction is exactly the failure of the diagonal pullback.

The one-sided examples expose the criterion:

$$\operatorname{WF}(u_+) + \operatorname{WF}(u_+) \quad\text{does not meet zero},$$

so $u_+^2$ is defined and

$$u_+^2 = -\partial_xu_+.$$

By contrast, $u_+$ and $u_-$ have canceling directions. The criterion does not license $u_+u_-$. At finite $\varepsilon$,

$$\frac1{x+i\varepsilon} \frac1{x-i\varepsilon} = \frac1{x^2+\varepsilon^2},$$

whose pairing with a test function contains $\pi\varphi(0)/\varepsilon$ and has no unmodified distributional limit.

The wavefront condition is sufficient, not necessary. Its failure means this canonical theorem supplies no product; it does not prove that no separately prescribed extension can ever be chosen. Products, Scaling Degree, and Extensions of Singular Distributions explains the additional local data in such an extension.

Pullback must avoid normal singular covectors

Let $F:X\to Y$ be smooth. Its normal set is

$$N_F = \left\{ \bigl(F(x),\eta\bigr) \in T^*Y\setminus0_Y: (dF_x)^{\mathsf T}\eta=0 \right\}.$$

If

$$\boxed{ \operatorname{WF}(u)\cap N_F = \varnothing, }$$

then the ordinary pullback of smooth functions extends canonically to $F^*u$. Its wavefront set satisfies

$$\operatorname{WF}(F^*u) \subseteq \left\{ \bigl(x,(dF_x)^{\mathsf T}\eta\bigr): \bigl(F(x),\eta\bigr)\in\operatorname{WF}(u) \right\}.$$

No zero covector appears on the right because the hypothesis excludes it.

If $F$ is a submersion, $(dF_x)^{\mathsf T}$ is injective, so $N_F$ is empty and every distribution pulls back. An embedding generally has nontrivial normal covectors, which is why restriction to a submanifold can fail. This sharpens the sufficient submersion rule from Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards.

Hörmander 2003, Theorem 8.2.4 and Fredenhagen and Rejzner 2012, Theorem A.7, PDF are the structural and QFT-facing sources for this theorem. As with the product criterion, failure of the condition removes this general license; it is not a universal nonexistence proof for every specially structured distribution.

The local proof uses the oscillatory representation of a regularized distribution. Composing its phase with $F$ differentiates that phase in the $x$ variables to $(dF_x)^{\mathsf T}\eta$. When this covector stays nonzero on the singular cone, a nonstationary-phase integration-by-parts operator produces arbitrary inverse powers of $|\eta|$, which makes the pullback limit converge. The normal set records precisely where that argument would have a stationary direction.

First QFT application: two restrictions of an i0 kernel

Consider the relative-time pole distribution

$$u_+(r) = \frac1{r+i0}$$

and the difference map

$$F:\mathbb R^2\longrightarrow\mathbb R, \qquad F(t,t')=t-t'.$$

The map is a submersion, so the pullback

$$K(t,t') = F^*u_+ = \frac1{t-t'+i0}$$

is well defined. The transpose differential sends $p\mapsto(p,-p)$, giving

$$\operatorname{WF}(K) = \left\{ (t,t;p,-p): p<0 \right\}.$$

This is the difference-kernel rule in Brouder, Dang, and Hélein 2014, § 7(j), PDF.

Now compare two embeddings. Fixing the second time gives

$$j:\mathbb R\longrightarrow\mathbb R^2, \qquad j(t)=(t,0),$$

whose normal set is

$$N_j = \{(t,0;0,q):q\neq0\}.$$

It does not meet $\operatorname{WF}(K)$, because the first component $p$ of a wavefront covector of $K$ is nonzero. Therefore

$$j^*K(t) = \frac1{t+i0}$$

is a licensed restriction.

The diagonal embedding

$$\iota(s)=(s,s)$$

has normal set

$$N_\iota = \{(s,s;q,-q):q\neq0\}.$$

This set intersects $\operatorname{WF}(K)$. The pullback theorem therefore does not define the coincident expression $K(s,s)$. The two restrictions meet the same singular kernel, but only one is transverse to its singular covectors.

Detailed covector bookkeeping for products, pullbacks, pushforwards, kernel composition, and QFT distributions belongs to Wavefront-Set Products, Pullbacks, and Pushforwards. That page also develops the physical applications; the present example is only the underlying mathematical diagnostic.

Common pitfalls and stop conditions

1. Including zero. The wavefront set lives in $T^*M\setminus0_M$. Zero covectors may appear in the auxiliary set $\operatorname{WF}_0$, but never in $\operatorname{WF}$ itself.
2. Replacing a cone by a ray. Rapid decay must hold uniformly in an open conic neighborhood for every inverse power.
3. Skipping localization. Global Fourier behavior mixes distant singularities and behavior at infinity. Multiply by a cutoff equal to $1$ near the point first.
4. Forgetting the Fourier sign. This page uses $e^{+ip\cdot x}$. A minus-phase source reflects every fiber covector.
5. Treating covectors as velocities. A wavefront direction lies in the cotangent fiber. Future, past, timelike, or null labels require additional Lorentzian structure not used in the definition.
6. Reading a failed criterion as impossibility. The product and pullback theorems are sufficient. Failure says the canonical construction is not licensed, not that no extension with extra data can exist.
7. Asking wavefront data to choose contact terms. A wavefront set controls directions of singularity; it does not select scaling-degree extension coefficients or a renormalization condition.
8. Crossing the scope boundary. No propagation, Hadamard, microlocal spectrum, pushforward, or kernel-composition conclusion follows from the definitions alone.

Exercises

1. Determine the wavefront set of $\partial^\alpha\delta_a$ for a nonzero multi-index $\alpha$.

   Check

   The distribution is supported at $a$, and its localized Fourier transform is a nonzero polynomial times $e^{ip\cdot a}$. It is not rapidly decreasing in any nonzero cone. Hence

   $$\operatorname{WF}(\partial^\alpha\delta_a) = \{(a,p):p\neq0\}.$$

2. A source using $e^{-i\xi x}$ states that $\operatorname{WF}((x+i0)^{-1})=\{(0,\xi):\xi>0\}$. Translate it to the convention of this page.

   Check

   Since $\xi=-p$, the positive $\xi$ half-line becomes the negative $p$ half-line:

   $$\operatorname{WF}\!\left(\frac1{x+i0}\right) = \{(0,p):p<0\}.$$

   The distribution has not changed; only the Fourier fiber label has.

3. Which of $u_+^2$ and $u_+u_-$ is licensed by the no-opposite-covector criterion?

   Check

   Both covectors in $u_+^2$ are negative, so their sum cannot vanish and the product exists. In $u_+u_-$, a negative and a positive covector can cancel, so the criterion fails. This failure does not itself prove an absolute nonexistence statement.

4. Compute $(d\iota_s)^{\mathsf T}$ for $\iota(s)=(s,s)$ and recover $N_\iota$.

   Check

   Since $d\iota_s(v)=(v,v)$, the transpose acts by

   $$(d\iota_s)^{\mathsf T}(p,q) = p+q.$$

   Its nonzero kernel consists of $(q,-q)$, so

   $$N_\iota = \{(s,s;q,-q):q\neq0\}.$$

References

• Christian Brouder, Nguyen Viet Dang, and Frédéric Hélein, A Smooth Introduction to the Wavefront Set, PDF, Journal of Physics A 47 (2014), 443001. Journal record. Definitions 8–12, Theorem 13, Examples 14–17, and § 7 provide a positive-phase treatment of conic decay, products, boundary-value examples, pullbacks, and difference kernels.
• Klaus Fredenhagen and Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, PDF, § 4 and Appendix A, 2012. Book record. This is the QFT-facing source for the one-sided $i0$ wavefront set, product criterion, normal set, pullback theorem, and diagonal-pullback interpretation.
• Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., §§ 8.1–8.2, Springer, 2003. Definition 8.1.2 and Theorems 8.2.4 and 8.2.10 are the structural sources for the wavefront set, pullback, and product. Hörmander’s negative Fourier phase reflects the fiber label relative to this page.