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.
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:
(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
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
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
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
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 $f$:
(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 $f$
such that
on non-singular distributions $u_g$ corresponding to smooth functions $g \colon Y \to \mathbb{C}$ it acts by precomposition with $f$:
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
(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
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
where $i_\ast b \in C^\infty_{cp}(X)$ is the result of extending $b$ by zero to all of $X$.
composition of distributions?
Notions of pullback:
pullback, fiber product (limit over a cospan)
lax pullback, comma object (lax limit over a cospan)
(∞,1)-pullback, homotopy pullback, ((∞,1)-limit over a cospan)
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 November 22, 2023 at 13:50:26. See the history of this page for a list of all contributions to it.