nLab
pullback of a distribution

Contents

Idea

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

Definition

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

If the map ff 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 nX_1, X_2 \subset \mathbb{R}^n are two open subsets of Euclidean space, and if

f:X 1X 2 f \;\colon\; X_1 \overset{}{\longrightarrow} X_2

is a submersion (i.e. its differential is a surjective function df x:T xX 1T f(x)X 2d f_x \;\colon\; T_x X_1 \to T_{f(x)} X_2 for all xX 1x \in X_1), then there is a unique continuous linear functional

f *:𝒟(X 2)𝒟(X 1) 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 bb it is given by pre-composition

f *b=bf. f^\ast b = b \circ f \,.

This is hence called the pullback of distributions.

If ff happens to be a diffeomorphism with inverse function f 1f^{-1} then f *uf^\ast u for u𝒟(X 2)u \in \mathcal{D}'(X_2) is explicitly given by

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

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

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

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

Proposition

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

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

f *:{u𝒟(Y)|WF(u)N f *=}𝒟(X) 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 gu_g corresponding to smooth functions g:Yg \colon Y \to \mathbb{C} it acts by precomposition with ff:

    f *(u g)=u gf f^\ast (u_g) = u_{g \circ f}
  2. for uu 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 *u)f *WF(u){(x,(df(x)) *k)|(f(x),η)WF(u)} 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

Examples

Example

(restriction of distributions)

Let i:YXi \;\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 *:𝒟(Y)𝒟(X) 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 bC c (Y)b \in C^\infty_c(Y) a bump function on YY, the restriction i *ui^\ast u of a distribution u𝒟(X)u \in \mathcal{D}'(X) acts by

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

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

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)

Last revised on February 1, 2018 at 07:25:14. See the history of this page for a list of all contributions to it.