nLab pullback of a distribution

Idea

The pullback of a distribution $u \in \mathcal{D}'(X)$ along a smooth function $f\colon Y \to X$ is supposed to like the composition $u \circ f$ if $u$ is thought of as a generalized function, such that it is exactly this in the case that $u$ actually is an ordinary function.

Definition

In general, the pullback of distributions along a smooth function $f$ is meant to be a unique extension to distributions of the operation which on bump functions is given by pre-composition with $f$. There are various different conditions that are sufficient for this being well defined.

If the map $f$ we are pulling back along is a submersion, the all distributions pull back:

Proposition

(pullback of any distribution along a submersion)

If $X_1, X_2 \subset \mathbb{R}^n$ are two open subsets of Euclidean space, and if

$$f \;\colon\; X_1 \overset{}{\longrightarrow} X_2$$

is a submersion (i.e. its differential is a surjective function $d f_x \;\colon\; T_x X_1 \to T_{f(x)} X_2$ for all $x \in X_1$), then there is a unique continuous linear functional

$$f^\ast \;\colon\; \mathcal{D}'(X_2) \longrightarrow \mathcal{D}'(X_1)$$

between spaces of distributions (this def.) which extends the pullback of functions in that on a non-singular distribution represented by a bump function $b$ it is given by pre-composition

$$f^\ast b = b \circ f \,.$$

This is hence called the pullback of distributions.

If $f$ happens to be a diffeomorphism with inverse function $f^{-1}$ then $f^\ast u$ for $u \in \mathcal{D}'(X_2)$ is explicitly given by

$$\langle f^\ast u , b \rangle \;=\; \langle u, \frac{1}{det(D f)} b \circ f^{-1} \rangle$$

where $det(D f)$ denotes the Jacobian determinant (the determinant of the derivative of $f$).

(Hörmander 90, theorem 6.1.2, Melrose 03, sections 4.17, 4.19 and 4.21)

If $f$ is not a submersion, then pullback is still defined on those distributions whose wave front set does not intersect the conormal bundle of $f$:

Proposition

(pullback if wave front set is disjoint from conormal bundle)

Given a smooth function $f \colon X \to Y$, then there is a unique continuous linear functional $f^\ast$ from the space of those distributions on $Y$ whose wave front set does not intersect the conormal bundle of $f$

$$f^\ast \;\colon\; \left\{ u \in \mathcal{D}'(Y) \;\vert\; WF(u) \cap N^\ast_f = \emptyset \right\} \longrightarrow \mathcal{D}'(X)$$

such that

1. on non-singular distributions $u_g$ corresponding to smooth functions $g \colon Y \to \mathbb{C}$ it acts by precomposition with $f$:

   $$f^\ast (u_g) = u_{g \circ f}$$

2. for $u$ a distribution in the domain on the left, the wave front set of its pullback is inside the pullback of its wave front set

(1)$$WF(f^\ast u) \subset f^\ast WF(u) \coloneqq \left\{ (x, (d f(x))^\ast k) \;\vert\; ( f(x), \eta ) \in WF(u) \right\}$$

(Hörmander 90, theorem 8.2.4)

Properties

• for continuity see at Hörmander topology

Examples

Example

(restriction of distributions)

Let $i \;\colon\; Y \hookrightarrow X$ be an open subset inclusion. This is clearly a submersion, in fact a local diffeomorphism, and hence prop. applies. The resulting pullback operation

$$i^\ast \;\colon\; \mathcal{D}'(Y) \longrightarrow \mathcal{D}'(X)$$

is also called restriction of distributions (Hörmander 90, first lines of section 2.2). For $b \in C^\infty_c(Y)$ a bump function on $Y$, the restriction $i^\ast u$ of a distribution $u \in \mathcal{D}'(X)$ acts by

$$\langle i^\ast u, b\rangle = \langle u, i_\ast b \rangle \,,$$

where $i_\ast b \in C^\infty_{cp}(X)$ is the result of extending $b$ by zero to all of $X$.

Related concepts

• change of integration variables

• composition of distributions?

• scaling degree of a distribution

• extension of distributions

• support of a distribution

Notions of pullback:

• pullback, fiber product (limit over a cospan)

• wide pullback

• lax pullback, comma object (lax limit over a cospan)

• (∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)

• base change, context extension

• pullback bundle

• contravariant functor

  • pullback in cohomology, d-invariant

  • pullback of differential forms

  • pullback of a distribution

References

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

• Richard Melrose, sections 4.17, 4.19 and 4.21 of Introduction to microlocal analysis, 2003 (pdf)

• Sergiu Klainerman, chapter 3, section 4 of Lecture notes in analysis, 2011 (pdf)