MnLargeSymbols’164 MnLargeSymbols’171

#  The Fourier transform on valuations is the Fourier transform

Dmitry Faifman  School of Mathematical Sciences, Tel Aviv University, Tel Aviv
6997801, Israel<EMAIL_ADDRESS>and  Thomas Wannerer  Friedrich-
Schiller-Universitat Jena, Fakultät für Mathematik und Informatik, Institut
für Mathematik, Ernst-Abbe-Platz 2, 07743 Jena, Germany  thomas.wannerer@uni-
jena.de

###### Abstract.

Alesker has proved the existence of a remarkable isomorphism of the space of
translation-invariant smooth valuations that has the same functorial
properties as the classical Fourier transform. In this paper, we show how to
directly describe this isomorphism in terms of the Fourier transform on
functions. As a consequence, we obtain simple proofs of the main properties of
the Alesker–Fourier transform. One of these properties was previously only
conjectured by Alesker and is proved here for the first time.

###### 2020 Mathematics Subject Classification:

52B45, 53C65, 42B10, 43A32

DF was supported by the Israel Science Foundation grant No. 1750/20.

TW was supported by DFG grant WA 3510/3-1.

##  1\. Introduction

###  1.1. The Alesker–Fourier transform

Let $V$ be an $n$-dimensional real vector space and let $\mathcal{K}(V)$
denote the space of convex bodies, i.e., non-empty convex compact subsets of
$V$. In convex geometry, a valuation is a function
$\phi\colon\mathcal{K}(V)\to\mathbb{C}$ satisfying the property

$\phi(K\cup L)=\phi(K)+\phi(L)-\phi(K\cap L)$

whenever the union of $K$ and $L$ is convex. Valuations, in particular in
connection with dissection problems for polytopes and integral geometry, are a
classical part of convex geometry, see the books by Schneider [Schneider:BM]
and Gruber [Gruber:CDG] for more information.

About two decades ago, the revolutionary work of Alesker uncovered a
surprisingly rich algebraic structure of the space of translation-invariant
and continuous valuations. The latter space, denoted by
$\operatorname{Val}(V)$, has been intensively studied through the work of
Hadwiger [Hadwiger:Vorlesungen], McMullen [McMullen:Euler,
McMullen:Continuous], Schneider [Schneider:Simple], Goodey–Weil
[GoodeyWeil:Distributions], Klain [Klain:Short, Klain:Even], and others
[Schneider:BM, Chapter 6]. One of Alesker’s key insights was to identify a
natural dense subspace of smooth valuations
$\operatorname{Val}^{\infty}(V)\subset\operatorname{Val}(V)$ with remarkable
properties. The most important one is the existence of a product, now called
the Alesker product, that gives this space the structure of commutative graded
algebra with identity [Alesker:Product]. Working with smooth valuations is
crucial, as the Alesker product and most of the other operations do not
continuously extend to the larger space $\operatorname{Val}(V)$. Building on
Alesker’s work, Bernig and Fu [BernigFu:Convolution] discovered a second
multiplicative structure called the convolution of valuations.

Injective linear maps $f\colon V\hookrightarrow W$ induce a pullback of
valuations

$f^{*}\colon\operatorname{Val}^{\infty}(W)\to\operatorname{Val}^{\infty}(V),$

which respects the Alesker product, while surjective linear maps induce a
pushforward

$f_{*}\colon\operatorname{Val}^{\infty}(V)\otimes\mathrm{Dens}(V^{*})\to\operatorname{Val}^{\infty}(W)\otimes\mathrm{Dens}(W^{*}),$

which respects the Bernig–Fu convolution, see [Alesker:Fourier]. Here
$\mathrm{Dens}(V)$ denotes the one-dimensional space of densities on $V$, see
below for a precise definition. Product and convolution admit a common
description in terms of the exterior product of valuations

$\operatorname{Val}^{\infty}(V)\times\operatorname{Val}^{\infty}(W)\to\operatorname{Val}(V\times
W)$

denoted by $\phi\boxtimes\psi$. Namely, using the operations of pullback and
pushforward,

$\phi\cdot\psi=\Delta^{*}(\phi\boxtimes\psi)\quad\text{and}\quad\phi*\psi=a_{*}(\phi\boxtimes\psi),$

where $\Delta\colon V\to V\times V$ is the diagonal embedding, and $a\colon
V\times V\to V$ is the addition in the vector space $V$.

The Alesker–Fourier transform, or simply the Fourier transform on valuations
enriches this picture even further. First defined by Alesker for even
valuations in [Alesker:HL] and only later in [Alesker:Fourier] constructed in
full generality, it is an isomorphism

$\mathbb{F}:\operatorname{Val}^{\infty}(V)\to\operatorname{Val}^{\infty}(V^{*})\otimes\operatorname{Dens}(V)$

which interchanges the product with convolution, pullback with pushforward,
and satisfies the inversion formula: applying the Alesker–Fourier transform
twice equals the pullback by the antipodal map $x\mapsto-x$. Its construction
in the general case rested on Alesker’s highly non-trivial irreducibility
theorem, as well as sophisticated methods from infinite-dimensional
representation theory of $\operatorname{GL}_{n}(\mathbb{R})$.

The name “Fourier” was attached to $\mathbb{F}$ due to its functorial
properties, which are strongly reminiscent of the classical Fourier transform
on functions. More recently however, a special case of the Alesker–Fourier
transform was observed to be indeed linked to the Fourier transform of
functions [DorrekSchuster:AF, Corollary 3.9].

###  1.2. Our results

In this work, we give a description of the Alesker–Fourier transform that
directly derives it from the Fourier transform on functions, or rather
differential forms, in all cases, and directly deduces its key properties from
the corresponding properties of the Fourier transform. This solves a problem
posed by Alesker [AleskerFu:Barcelona, p. 17]. Moreover, our approach allows
us to establish an additional property of the Fourier transform, which were
conjectured by Alesker [AleskerFu:Barcelona, p. 17].

In particular, we make no use of the irreducibility theorem of Alesker
[Alesker:Irreducibility], assuming only the definition of smooth valuations
through integration over the normal cycle: A valuation
$\phi\in\operatorname{Val}(V)$ is smooth if there exists a smooth differential
form $\omega$ on $\mathbb{P}_{+}(V^{*})$, the oriented projectivization of
$V^{*}$, with values in $\mathbb{\wedge}^{\bullet}V^{*}\otimes\mathrm{or}(V)$,
and a density $\theta$ on $V$ such that

(1) $\phi(K)=\theta(K)+\int_{\operatorname{nc}(K)}\omega$

for all convex bodies $K$ in $V$. Here $\operatorname{nc}(K)$ denotes the
normal cycle of $K$, and $\operatorname{or}(V)$ denotes the one-dimensional
space of orientation functions on $V$.

In the first step of our construction of the Fourier transform on valuations,
which seems to be also of independent interest, we associate to every smooth
valuation $\phi\in\operatorname{Val}^{\infty}(V)$ a unique $0$-homogeneous
current $\tau=\tau(\phi)$ on $V^{*}$, which is smooth outside of the origin.
More precisely, if $\phi\in\operatorname{Val}_{k}^{\infty}(V)$ is given by
(1), then we define a generalized form
$\Omega_{-\infty}^{n-k}(V^{*},\mathbb{\wedge}^{k}V^{*}\otimes\mathrm{or}(V))$
by

$\tau(\phi)=\phi(\\{0\\})\cdot\delta_{0}+r^{0}((-1)^{n-k}a^{*}D\omega+\theta).$

Here $D$ is the Rumin differential of $\omega$, a natural second order
differential operator arising in contact geometry, $a$ is the antipodal map,
and $r^{0}$ denotes the $0$-homogeneous extension from $\mathbb{P}_{+}(V^{*})$
to $V^{*}$. For more details, see section 4.1. A closely related description
of valuations in terms of a pair of currents was introduced in
[AleskerBernig:Product] building on earlier work of Bernig and Bröcker
[BernigBroecker:Rumin].

The main point of this construction is that we are able to characterize all
generalized forms arising in this fashion by a short list of inevitable
properties (Proposition 4.2). Moreover, the operations of pullback,
pushforward, and exterior product of valuations admit simple descriptions in
terms of the $0$-homogeneous current $\tau(\phi)$, see Propositions 4.5, 4.6,
and 4.7. Furthermore, the $0$-homogeneous current of a valuation allows an
explicit description of the $0$-homogeneous component of the pushforward of a
valuation by an epimorphism, which is not readily available when using the
standard presentation by a pair of currents.

Let $F$ be a finite-dimensional vector spaces of over the reals, and let
$\Omega_{\mathcal{S}}^{k}(V,F)\subset\Omega^{k}(V,F)$ denote the Schwartz
space of differential $k$-forms on $V$ with values in $F$ and rapidly
decreasing coefficients. The Fourier transform of functions on $V$ extends
naturally to differential forms on $V$.

######  Definition 1.1.

The Fourier transform

$\mathcal{F}:\Omega_{\mathcal{S}}^{k}(V,F)\to\Omega_{\mathcal{S}}^{n-k}(V^{*},\operatorname{or}(V)\otimes
F)$

is defined as follows. For $\omega\in\Omega_{\mathcal{S}}^{k}(V,F)$, the Hodge
star isomorphism

$\ast:\mathbb{\wedge}^{k}V^{*}\xrightarrow{\sim}\mathbb{\wedge}^{n-k}V\otimes\mathbb{\wedge}^{n}V^{*}\simeq\mathbb{\wedge}^{n-k}V\otimes\operatorname{Dens}(V)\otimes\operatorname{or}(V)$

allows to consider $\omega$ as a map
$\widetilde{\omega}:V\to\mathbb{\wedge}^{n-k}V\otimes\operatorname{Dens}(V)\otimes\operatorname{or}(V)\otimes
F$. Hence we may define for $\xi\in V^{*}$

$\mathcal{F}(\omega)(\xi)=\int_{V}e^{2\pi\mathbf{i}\langle
x,\xi\rangle}\widetilde{\omega}(x)\in\mathbb{\wedge}^{n-k}V\otimes\operatorname{or}(V)\otimes
F.$

We remark that while different normalizations exist for the Fourier transform,
an involutive Fourier transform on $0$-homogeneous forms is unique up to sign.

We extend this definition also to differential forms with tempered
distributions as coefficients. In applications to valuations, the vector space
$F$ is $\mathbb{\wedge}^{\bullet}V^{*}\otimes\mathrm{or}(V)$. If we denote the
composition of $\mathcal{F}$ with the canonical isomorphism

$F\otimes\mathrm{or}(V)\simeq\mathbb{\wedge}^{\bullet}V\otimes\mathrm{Dens}(V)\otimes\operatorname{or}(V),$

by $\mathcal{F}^{0}$, then

$\mathcal{F}^{0}\tau(\phi)\in\Omega_{\mathcal{S}^{\prime}}(V,\mathbb{\wedge}^{\bullet}V\otimes\operatorname{or}(V))\otimes\mathrm{Dens}(V).$

Our first theorem states that this form defines a smooth valuation.

######  Theorem 1.2.

Let $\phi\in\operatorname{Val}^{\infty}(V)$ be a smooth valuation, and let
$\tau(\phi)$ be its $0$-homogeneous current. Then the Fourier transform
$\mathcal{F}^{0}\tau(\phi)$ is the $0$-homogeneous current of a unique smooth
valuation.

This theorem allows us to make the following definition.

######  Definition 1.3.

If $\phi\in\operatorname{Val}^{\infty}(V)$ is a smooth valuation, then its
$\mathcal{F}$-transform is the unique smooth valuation
$\mathcal{F}\phi\in\operatorname{Val}^{\infty}(V^{*})\otimes\mathrm{Dens}(V)$
satisfying

$\tau(\mathcal{F}\phi)=\mathcal{F}^{0}\tau(\phi).$

The same equation also defines the $\mathcal{F}$-transform of generalized
valuations $\phi\in\operatorname{Val}^{-\infty}(V)$. It will be seen in
Theorem 1.5 that the $\mathcal{F}$-transform coincides with the
Alesker–Fourier transform.

With this definition and the description of the algebraic operations of
valuations in terms of the $0$-homogeneous current, the main properties of the
Alesker–Fourier transform are for the $\mathcal{F}$-transform a direct
consequence of the corresponding properties of the Fourier transform of
functions. We write $\mathcal{F}_{V}$ for the $\mathcal{F}$-transform of
valuations on $V$ in case we consider several vector spaces at the same time.

######  Theorem 1.4.

The $\mathcal{F}$-transform of valuations has the following properties:

  1. (1)

$\mathcal{F}$ commutes with the natural action of $\mathrm{GL}(V)$.

  2. (2)

Inversion formula:
$(\mathcal{F}_{V^{*}}\times\operatorname{id})\circ\mathcal{F}_{V}\phi=(-\operatorname{id})^{*}\phi$.

  3. (3)

Let $i:V\hookrightarrow W$ be an injective linear map, and
$p=i^{\vee}:W^{*}\to V^{*}$. Then for $\phi\in\operatorname{Val}^{\infty}(W)$,

$\mathcal{F}_{V}(i^{*}\phi)=p_{*}(\mathcal{F}_{W}\phi).$

  4. (4)

$\mathcal{F}(\phi\boxtimes\psi)=\mathcal{F}\phi\boxtimes\mathcal{F}\psi$ for
$\phi\in\operatorname{Val}^{\infty}(V)$,
$\psi\in\operatorname{Val}^{\infty}(W)$.

  5. (5)

$\mathcal{F}$ intertwines product and convolution: for
$\phi,\psi\in\operatorname{Val}^{\infty}(V)$,

$\mathcal{F}(\phi\cdot\psi)=\mathcal{F}\phi\ast\mathcal{F}\psi.$

  6. (6)

Self-adjointness: $\langle\mathcal{F}u,\theta\rangle=\langle
u,\mathcal{F}\theta\rangle$ for $u\in\operatorname{Val}^{-\infty}(V)$,
$\theta\in\operatorname{Val}^{\infty}(V^{*})$.

  7. (7)

Let $p:V\to W$ be a surjective linear map, and $i=p^{\vee}:W^{*}\to V^{*}$.
Then for $\psi\in\operatorname{Val}^{-\infty}(W)$,

$\mathcal{F}_{V}(p^{*}\psi)=i_{*}(\mathcal{F}_{W}\psi).$

Using the above properties of the $\mathcal{F}$-transform we finally prove

######  Theorem 1.5.

The $\mathcal{F}$-transform and the Alesker–Fourier transform coincide:
$\mathcal{F}=\mathbb{F}$.

Taken together, the properties (3) and (7) imply the identity

$\mathcal{F}\circ f^{*}=(f^{\vee})_{*}\circ\mathcal{F}$

on smooth valuations for every linear map $f$ with dual map $f^{\vee}$.

The self-adjointness (6) of the Alesker–Fourier transform was already
established in the PhD thesis of the first-named author [faifman_thesis],
using Alesker’s original definition, and employed to extend the
Alesker–Fourier transform to the space of generalized valuations. This
extension gives formal meaning to properties (4) and (7), which were
previously conjectured by Alesker [AleskerFu:Barcelona, p. 17]. While
properties (3) and (7) are easily seen to be equivalent via self-adjointness
and the inversion formula, property (4) is proved here for the first time.

###  1.3. Relation to other work

The relevance of the classical Fourier transform for questions about volumes
of sections and projections of convex bodies was realized in 1990s, when it
led to a complete solution of the Busemann–Petty problem. The application of
Fourier analytic techniques to problems in convex geometry has developed into
a rich area of research, see the papers [Ball:Cube,
Gardner_etal:AnalyticSolution, Koldobsky:Intersection,
Koldobsky_etal:Projections, fourier_santalo, nazarov_mahler], the lecture
notes by Ryabogin and Zvavitch [RyaboginZvavitch:Methods], and Koldobsky’s
monograph [Koldobsky:FourierAnalysis] for more information.

As already mentioned above, that the Alesker–Fourier transform can be linked
to the Fourier transform on functions was first observed by Dorrek and
Schuster [DorrekSchuster:AF]. This special case motivated us to look for such
a connection in general.

The Alesker–Fourier transform has important applications in integral geometry.
Kinematic formulas are a centerpiece of integral geometry, first studied for
the orthogonal Euclidean group by Blaschke, Chern, and Santaló, see the books
by Santaló [Santalo] and Klain and Rota [KlainRota]. For convex bodies $A,B$
in $\mathbb{R}^{n}$ they come in two flavors, namely intersectional

$\int_{G\ltimes\mathbb{R}^{n}}\chi(A\cap
gB)\,dg=\sum_{i,j}c_{ij}\mu_{i}(A)\mu_{j}(B)$

and additive

$\int_{G}\operatorname{vol}(A+gB)\,dg=\sum_{i,j}d_{ij}\mu_{i}(A)\mu_{j}(B),$

where $G$ is a compact Lie group acting transitively on the unit sphere,
integration is with respect to the Haar measure, the $\mu_{i}$ are a basis of
$\operatorname{Val}^{G}$, the space of $G$-invariant valuations, and the
constants $c_{ij},d_{ij}$ are independent of $A$ and $B$. More generally, the
Euler characteristic and volume may be replaced by any of the $\mu_{i}$.

Explicitly determining the constants in the kinematic formulas is a challenge.
A key insight due to Fu [Fu:Unitary] is that they are the structure constants
of the algebras of $G$-invariant valuations for product and convolution.
Dually to the Alesker–Fourier transform intertwining product and convolution,
it intertwines intersectional and additive kinematic formulas. Since the
convolution is typically simpler to compute than the Alesker product, this
allows at least in principle to deduce both types of kinematic formulas from
the convolution table of the invariant valuations.

For the unitary groups $\mathrm{U}(n)$ and $\mathrm{SU}(n)$, this program has
been successfully completed by Bernig and Fu [BernigFu:Hig, Bernig:SU]. While
determining the Alesker–Fourier transform is relatively easy in these cases,
in the works of Bernig and Solanes [BernigSolanes:ClassificationH2,
BernigSolanes:IGH2] on the integral geometry of the quaternionic plane and
Bernig and Hug [BernigHug:Tensor] on the integral geometry of tensor
valuations, this step relies on a sizable and difficult argument. We have not
attempted to do this, but it seems plausible that our explicit description of
the Alesker–Fourier transform of valuations could simplify these arguments.

The Alesker–Fourier transform can also be used to construct the
Holmes–Thompson intrinsic volumes in a normed space. Crofton formulas in
normed spaces were first constructed Schneider and Wieacker
[SchneiderWieacker:CroftonFinsler] and Alvarez-Paiva and Fernandes
[APF:CroftonFinsler], which were then used by Bernig [Bernig:CroftonFinsler]
to construct a natural family of valuations on all normed spaces, now called
intrinsic volumes, extending the Holmes–Thompson volume on all flats. As
observed by Fu [Fu:Barcelona, Theorem 2.3.22], the simplest definition of
those valuations is in fact as the (inverse) Alesker–Fourier transform of the
mixed volume with the polar of the unit ball, explicitly

$V^{\mathrm{HT}}_{k}=\frac{1}{\omega_{k}}\binom{n}{k}\mathbb{F}^{-1}V(B^{\circ}[k],\bullet[n-k]),$

where $\omega_{k}$ denotes the volume of the $k$-dimensional Euclidean unit
ball. Moreover, for non-symmetric normed spaces the latter is the only known
way to define the intrinsic volumes so that they are generated from the first
one under the Alesker product, see [WannererFaifman:Finsler].

Let us finally point out that the Alesker–Fourier transform was used to deduce
new inequalities for mixed volumes of convex bodies from the recently
established Hodge–Riemann relations on the space of smooth valuations
[Alesker:Kotrbaty, Kotrbaty:HR, KotrbatyWannerer:AF].

###  1.4. Acknowledgements

We are grateful to Semyon Alesker for his valuable comments on an earlier
draft.

##  2\. Preliminaries

If $V$ is an $n$-dimensional real vector space, we denote by
$\mathbb{\wedge}^{\bullet}V=\bigoplus_{k=0}^{n}\mathbb{\wedge}^{k}V$ the
exterior algebra of $V$. A density on $V$ is a function
$\mu\colon\mathbb{\wedge}^{n}V\to\mathbb{C}$ with the property that
$\mu(t\omega)=|t|\mu(\omega)$ holds for all $t\in\mathbb{R}$. The
$1$-dimensional space of densities is denoted by $\operatorname{Dens}(V)$, and
its nonzero elements are naturally identified with the complex-valued Lebesgue
measures on $V$. An orientation function on $V$ is a function
$\epsilon\colon\mathbb{\wedge}^{n}V\setminus\\{0\\}\to\mathbb{C}$ satisfying
$\epsilon(t\omega)=\operatorname{sgn}(t)\epsilon(\omega)$ for all
$t\in\mathbb{R}\setminus\\{0\\}$. The $1$-dimensional space of orientation
functions is denoted by $\operatorname{or}(V)$. We will make use of the
following canonical isomorphisms

$\displaystyle\operatorname{Dens}(V)\otimes\operatorname{or}(V)$
$\displaystyle\simeq\mathbb{\wedge}^{n}V^{*},$
$\displaystyle\operatorname{or}(V)$
$\displaystyle\simeq\operatorname{or}(V^{*}),$
$\displaystyle\mathbb{\wedge}^{k}V$
$\displaystyle\simeq\mathbb{\wedge}^{n-k}V^{*}\otimes\mathbb{\wedge}^{n}V.$

Moreover, if $0\to U\to V\to W\to 0$ is exact, then

$\operatorname{Dens}(U)\otimes\operatorname{Dens}(W)\simeq\operatorname{Dens}(V)$

and

$\operatorname{or}(U)\otimes\operatorname{or}(W)\simeq\operatorname{or}(V).$

The Euler vector field $E(x)=x$ is defined on any vector space $V$.

###  2.1. Generalized forms

The theory of distributions on a smooth manifold is a bit more subtle than
that of distributions on open subsets of $\mathbb{R}^{n}$. We recall in this
section the relevant concepts and refer the reader to [GuilleminSternberg:GA,
Chapter VI] for more information.

By $\operatorname{Dens}(M)$ we denote the line bundle over $M$ with fiber
$\operatorname{Dens}(T_{x}M)$ over $x\in M$. Its smooth sections are naturally
identified with the smooth measures on $M$. The orientation bundle
$\operatorname{or}(M)$ of $M$ is defined analogously. The space of generalized
sections of a vector bundle $\mathcal{E}\to M$ is defined by
$C^{-\infty}(M,\mathcal{E})=(C_{c}^{\infty}(M,\mathcal{E}^{*}\otimes\operatorname{Dens}(M)))^{*}$.
The space of generalized differential $k$-forms on a manifold $M$ is therefore

$\Omega^{k}_{-\infty}(M)=\Omega_{c}^{n-k}(M,\operatorname{or}(M))^{*}=C_{c}^{\infty}(M,\mathbb{\wedge}^{n-k}T^{*}M\otimes\operatorname{or}(M))^{*}.$

It contains the dense subspace of smooth forms, given by

$\langle\omega,\psi\rangle=\int_{M}\omega\wedge\psi,\quad\omega\in\Omega^{k}(M),\psi\in\Omega_{c}^{n-k}(M,\operatorname{or}(M)).$

Most operations on smooth forms extend to generalized forms, including the
pullback under a smooth proper submersion $f\colon M\to N$, the exterior
derivative $d$, and the contraction with a vector field $i_{X}$. For
$\omega\in\Omega^{k}_{-\infty}(M)$, test forms
$\phi\in\Omega^{n-k}_{c}(M,\operatorname{or}(M))$,
$\eta\in\Omega^{n-k+1}_{c}(M,\operatorname{or}(M))$,
$\psi\in\Omega^{n-k-1}_{c}(M,\operatorname{or}(M))$, and $X$ a vector field we
have

$\displaystyle\langle f^{*}\omega,\phi\rangle$
$\displaystyle=\langle\omega,f_{*}\phi\rangle,$ $\displaystyle\langle
i_{X}\omega,\eta\rangle$
$\displaystyle=(-1)^{k+1}\langle\omega,i_{X}\eta\rangle,$
$\displaystyle\langle d\omega,\psi\rangle$
$\displaystyle=(-1)^{k+1}\langle\omega,d_{\nabla}\psi\rangle,$

where $f_{*}$ denotes integration along the fibers and $\nabla$ is the
canonical connection on $\operatorname{or}(M)$.

Let $V,F$ be finite-dimensional real vector spaces. By a tempered generalized
differential $k$-form on $V$ with values in $F$ we understand a continuous
linear functional on
$\Omega_{\mathcal{S}}^{n-k}(V,F^{*}\otimes\operatorname{or}(V))$. We denote it
by
$\Omega_{\mathcal{S}^{\prime}}^{k}(V,F)=\mathcal{S}^{\prime}(V,\wedge^{k}V^{*}\otimes
F)$. Equivalently, a tempered generalized form is a generalized form with
tempered distributions as coefficients with respect to a basis. Thus

$\Omega_{\mathcal{S}^{\prime}}^{k}(V,F)\subset\Omega^{k}_{-\infty}(V,F).$

In the following we will sometimes have to replace the vector space $F$ by a
canonically isomorphic vector space $G$. Let us be clear that this means that
if $\omega\in\Omega^{k}_{\mathcal{S}^{\prime}}(V,F)$ is a generalized form,
$f\colon F\to G$ is a linear isomorphism, and
$\eta\in\Omega^{n-k}_{\mathcal{S}}(V,G^{*}\otimes\operatorname{or}(V))$ is a
test form, then we define

(2) $\langle f\circ\omega,\eta\rangle=\langle\omega,f^{\vee}\circ\eta\rangle,$

where $f^{\vee}\colon G^{*}\to F^{*}$ denotes the dual map.

A generalized form $\omega\in\Omega_{-\infty}(V,F)$ is $r$-homogeneous if
$m_{\lambda}^{*}\omega=\lambda^{r}\omega$, for all $\lambda>0$. For any $r$,
an $r$-homogeneous generalized form is tempered.

We denote by $\mathbb{P}_{+}(V)$ the oriented projectivization of $V$. The
manifold $\mathbb{P}_{V}=V\times\mathbb{P}_{+}(V^{*})$ carries a contact
structure defined by the hyperplanes $H_{p,[\xi]}=\operatorname{ker}\xi\circ
d\pi$ in $T_{p,[\xi]}\mathbb{P}_{V}$, where $\pi\colon\mathbb{P}_{V}\to V$
denotes the projection to the first factor. A smooth differential form
$\omega$ on $\mathbb{P}_{V}$ is said to be vertical, if
$\omega|_{H_{p,[\xi]}}=0$ for all $p,[\xi]$. A generalized form $\omega$ is
called vertical if $\langle\omega,\psi\rangle=0$ for all vertical test forms
$\psi$.

By $\delta_{p}$ we denote the delta measure at $p$. The corresponding
generalized top form, which depends on a choice of orientation $\sigma_{p}$ at
$p$, is $\delta_{p}\otimes\sigma_{p}$. Often we will simply write $\delta_{p}$
also for the top form when the choice of $\sigma_{p}$ is clear or not
important.

##  3\. $0$-homogeneous forms

####  3.0.1. Homogeneous extension and restriction

Let us recall how to extend (generalized) forms on $\mathbb{P}_{+}(V)$ to
$0$-homogeneous tempered generalized forms on $V$. We moreover discuss a
characterization of such extensions.

Let $\pi\colon V\setminus\\{0\\}\to\mathbb{P}_{+}(V)$ denote the canonical
projection. Observe that there is a well-defined push-forward (fiber
integration) of twisted differential forms

$\pi_{*}\colon\Omega_{\mathcal{S}}^{k}(V,\operatorname{or}(V)\otimes
F)\to\Omega^{k-1}(\mathbb{P}_{+}(V),\operatorname{or}(\mathbb{P}_{+}(V))\otimes
F).$

Explicitly, assuming $F=\mathbb{R}$, if $\omega=f(r,\theta)dr\wedge\pi^{*}\xi$
is a $k$-form, where $\xi\in\Omega^{k-1}(\mathbb{P}_{+}(V))$, then
$\pi_{*}\omega=(-1)^{n-k}(\int_{0}^{\infty}fdr)\xi$, so that
$\langle\pi^{*}\phi,\omega\rangle=\langle\phi,\pi_{*}\omega\rangle$ for all
$\phi\in\Omega^{n-k}(\mathbb{P}_{+}(V))$.

######  Definition 3.1.

Let $T\in\Omega^{k}_{-\infty}(\mathbb{P}_{+}(V),F)$. We define the
$0$-homogeneous extension of $T$ to $V$, denoted
$r^{0}T\in\Omega^{k}_{\mathcal{S}^{\prime}}(V,F)$ by

$\langle r^{0}T,\omega\rangle=\langle
T,\pi_{*}\omega\rangle,\quad\omega\in\Omega_{\mathcal{S}}^{n-k}(V,\operatorname{or}(V)\otimes
F^{*}).$

Clearly the restriction of $r^{0}T$ to $V\setminus\\{0\\}$ is $\pi^{*}T$.

######  Remark 3.2.

The notation $r^{0}T$ is inspired by Gelfand–Shilov [GelfandShilov:I, Section
3.5].

######  Lemma 3.3.

Let $S,T\in\Omega^{k}_{-\infty}(V)$ be $0$-homogeneous generalized forms of
degree $k<n$. If $S-T$ is supported at the origin, then $S=T$.

###### Proof.

It suffices to prove that $T=0$ if $T$ is supported at the origin. Fixing any
basis of $V$, the corresponding coefficients of $T$ are $(-k)$-homogeneous
generalized functions that are supported at the origin. Any such function is a
linear combination of the delta function and its derivatives, and consequently
cannot be $(-k)$ homogeneous for $k<n$, unless it is zero. Hence $T=0$ as
claimed. ∎

Recall that $E$ denotes the Euler vector field $E(x)=x$.

######  Proposition 3.4.

Let $T\in\Omega^{k}_{-\infty}(\mathbb{P}_{+}(V))$. Then
$T_{0}=r^{0}T\in\Omega^{k}_{-\infty}(V)$ has the following properties

  1. (1)

$i_{E}T_{0}=0$.

  2. (2)

$T_{0}$ is $0$-homogeneous.

Conversely, if $T_{0}\in\Omega^{k}_{-\infty}(V)$ satisfies the above
properties, then either $0\leq k\leq n-1$, whence there exists a unique $T$
such that $r^{0}T=T_{0}$, or $k=n$ and $T$ is a multiple of a delta $n$-form
supported at the origin.

###### Proof.

It is clear that $T_{0}=r^{0}T$ has the second stated property, and it holds
that $i_{E}(T_{0}|_{V\setminus\\{0\\}})=0$. Thus $i_{E}T_{0}$ is a
$0$-homogeneous $(k-1)$-form supported at the origin, but this is impossible
for $k-1<n$. Therefore, $i_{E}T_{0}=0$.

For the converse statement, fix a Euclidean structure on $V=\mathbb{R}^{n}$,
and assume first $k\leq n-1$. The uniqueness of $T$ is straightforward. To
prove existence let us first observe that (1) and (2) imply

$i_{E}dT_{0}=\mathcal{L}_{E}T_{0}=0.$

We define $\langle T,\omega\rangle:=\langle T_{0},\phi\rangle$ for
$\omega=\pi_{*}\phi$, where
$\phi\in\Omega^{n-k}_{c}(\mathbb{R}^{n}\setminus\\{0\\})$ is a compactly
supported test form. For this to be well-defined it suffices to show that
$\langle T_{0},\phi\rangle=0$ if $\pi_{*}\phi=0$. We may introduce polar
coordinates $r>0,u\in S^{n-1}$ on $V\setminus\\{0\\}$ and write

$\phi=f(r,u)dr\wedge\xi+\eta,$

where $\xi,\eta\in u^{*}(\Omega(S^{n-1}))$. As a consequence,
$\pi_{*}\phi=(-1)^{k}\left(\int_{0}^{\infty}fdr\right)\xi$. By assumption we
have $\int_{0}^{\infty}fdr=0$, so that $f=\frac{\partial h}{\partial r}$ for
the compactly supported function $h(r,u)=\int_{0}^{r}f(t,u)dt$. Put
$\zeta=h\xi$. Observe that whenever $i_{E}\psi=0$ and $\theta\in
u^{*}\Omega(S^{n-1})$, and $\psi\wedge\theta$ is a top degree form, one has
the pointwise equality $\psi\wedge\theta=0$. Using that $i_{E}T_{0}=0$ and
$i_{E}dT_{0}=0$, we find

$\langle T_{0},\phi\rangle=\langle T_{0},\frac{\partial h}{\partial
r}dr\wedge\xi\rangle=\langle T_{0},d\zeta\rangle=(-1)^{k+1}\langle
dT_{0},\zeta\rangle=0$

as required. By construction, $r^{0}T-T_{0}$ is supported at the origin.
Applying Lemma 3.3 yields $r^{0}T=T_{0}$ as desired.

Now if $k=n$, we may write $T_{0}=f(x)dx_{1}\wedge\dots\wedge dx_{n}$ for some
$f\in C^{-\infty}(\mathbb{R}^{n})$. The condition $i_{E}T_{0}=0$ readily
implies that $f$ is supported at the origin. As it is also $(-n)$-homogeneous,
it must be a multiple of the delta function, completing the proof.

∎

Note that if $T\in\Omega^{k}(\mathbb{P}_{+}(V))$ is a smooth form, then
$r^{0}T$ is smooth on $V\setminus\\{0\\}$.

######  Lemma 3.5.

Let $\tau\in\Omega^{k}(\mathbb{P}_{+}(V))$ and
$\omega=r^{0}\tau\in\Omega^{k}_{-\infty}(V)$. Then $\omega$ is locally
integrable.

###### Proof.

Fix a Euclidean structure on $\mathbb{R}^{n}$. On
$\mathbb{R}^{n}\setminus\\{0\\}$ we may write $\omega=\omega^{\prime}$, where
$\omega^{\prime}=\frac{1}{|x|^{k}}\sum_{|I|=k}g_{I}(\frac{x}{|x|})dx_{I}$ with
$g_{I}\in C^{\infty}(S^{n-1})$. Since $k<n$, the coefficients of
$\omega^{\prime}$ are locally integrable. Applying Lemma 3.3 yields
$\omega=\omega^{\prime}$ on $\mathbb{R}^{n}$. ∎

###  3.1. Operations on $0$-homogeneous forms

We need to know how to pullback and pushforward $0$-homogeneous generalized
forms under linear maps.

####  3.1.1. Pullback under linear monomorphisms

For a vector $v\in\mathbb{R}^{n}$, we denote by
$S_{v}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ the shift $x\mapsto x+v$.

######  Lemma 3.6.

Assume $\omega\in\Omega^{k}_{-\infty}(\mathbb{R}^{n})$ is smooth outside of
the origin, $0$-homogeneous. Let
$e:\mathbb{R}^{j}\hookrightarrow\mathbb{R}^{n}$ be a fixed monomorphism, and
assume $j>k$. Then the weak limit

$\omega^{\prime}:=\lim_{V\setminus e(\mathbb{R}^{j})\ni v\to
0}e^{*}S_{v}^{*}\omega$

exists, is $0$-homogeneous, and on $\mathbb{R}^{j}\setminus\\{0\\}$ coincides
with $e^{*}\omega$. If moreover $i_{E}\omega=0$ then also
$i_{E}\omega^{\prime}=0$. If $\omega=r^{0}\tau$ for
$\tau\in\Omega^{k}(\mathbb{P}_{+}(\mathbb{R}^{n}))$, then one has
$\omega^{\prime}=r^{0}(e^{*}\tau)$.

We will subsequently denote $\omega^{\prime}=e^{*}\omega$.

###### Proof.

By the smoothness outside of the origin, $\omega^{\prime}$ coincides with
$e^{*}\omega$ on $\mathbb{R}^{j}\setminus\\{0\\}$. Choose orthonormal
coordinates such that $\mathbb{R}^{j}=\operatorname{Span}(e_{1},\dots,e_{j})$.
Observe that since $\omega$ is $0$-homogeneous, Lemma 3.3 implies

$\omega=\frac{1}{|x|^{k}}\sum_{|I|=k}g_{I}(\frac{x}{|x|})dx_{I}$

for some $g_{I}\in C^{\infty}(S^{n-1})$. Then

$\displaystyle\omega^{\prime}$ $\displaystyle=\lim_{v\to
0}\frac{1}{|y+v|^{k}}\sum_{I\subset\\{1,\dots,j\\}}g_{I}(\frac{y+v}{|y+v|})\wedge_{j\in
I}dy_{j}$
$\displaystyle=\frac{1}{|y|^{k}}\sum_{I\subset\\{1,\dots,j\\}}g_{I}(\frac{y}{|y|})\wedge_{j\in
I}dy_{j}.$

Thus $\omega^{\prime}$ is locally integrable and $0$-homogeneous.

Assume now $i_{E}\omega=0$. By Proposition 3.4 we may write $\omega=r^{0}\tau$
for some $\tau\in\Omega^{k}(\mathbb{P}_{+}(\mathbb{R}^{n}))$. Define
$\omega_{1}=r^{0}(e^{*}\tau)$. Then $\omega_{1}$ coincides with
$\pi^{*}(e^{*}\tau)=e^{*}(\pi^{*}\tau)=e^{*}\omega=\omega^{\prime}$ outside of
the origin. Thus $\omega_{1}=\omega^{\prime}$ by Lemma 3.3 and therefore in
particular $i_{E}\omega^{\prime}=0$. This concludes the proof.

∎

######  Proposition 3.7.

Assume $\omega\in\Omega^{k}_{-\infty}(\mathbb{R}^{n})$ is smooth outside of
the origin, $0$-homogeneous, $k<n$, and $i_{E}\omega=0$. Let
$e:\mathbb{R}^{k}\hookrightarrow\mathbb{R}^{n}$ be a fixed monomorphism. Then
for $v\in\mathbb{R}^{n}\setminus e(\mathbb{R}^{k})$, the weak limit

$e^{*}_{v}\omega:=\lim_{\epsilon\to 0^{+}}e^{*}S_{\epsilon v}^{*}\omega$

exists, is uniform in $v\in S^{n-1}\setminus e(\mathbb{R}^{k})$, and equals
$c([v])\delta_{0}$ for some continuous function $c$ on
$\mathbb{P}_{+}(\mathbb{R}^{n}/e(\mathbb{R}^{k}))$. Furthermore, if $\omega$
is closed then $c([v])$ is constant.

In the latter situation, we denote the limit by $e^{*}\omega$.

###### Proof.

Write $V=\mathbb{R}^{n}$, $W=\mathbb{R}^{k}$. Without loss of generality we
may assume that $e$ is the inclusion of a subspace $W$ of $V$. We choose an
orthonormal basis of $V$ such that $W=\operatorname{Span}(e_{1},\dots,e_{k})$.

By Lemma 3.3 we can find functions $g_{J}\in C^{\infty}(S^{n-1})$ such that

$\omega=\frac{1}{|x|^{k}}\sum_{|J|=k}g_{J}(\frac{x}{|x|})dx_{J}.$

Consider first a unit vector $u\perp W$. Denoting $I=\\{1,\dots,k\\}$, it
holds that

$e^{*}(S_{\epsilon u}^{*}\omega)|_{y}=\frac{1}{|y+\epsilon
u|^{k}}g_{I}\left(\frac{y+\epsilon u}{|y+\epsilon
u|}\right)dy_{1}\wedge\dots\wedge dy_{k}.$

Since $i_{E}\omega=0$, it holds that $g_{I}(y)=0$ for $y\in W$. Hence

(3) $g_{I}(y+z)=\sum_{i=k+1}^{n}z_{i}h_{i}(y,z),\quad y\in W,\ z\in
W^{\perp},$

where

$h_{i}(y,z)=\int_{0}^{1}\frac{\partial g_{I}}{\partial x_{i}}(y+tz)dt.$

Denoting $W_{u}=W\oplus\operatorname{Span}(u)$, $g_{u}=g_{I}|_{S(W_{u})}$, we
may write $g_{u}(\theta)=\langle\theta,u\rangle h_{u}(\theta)$ for some
$h_{u}\in C^{\infty}(S(W_{u}))$, which also depends smoothly on $u$. Observe
that the family $\\{|h_{u}|:u\in S(W^{\perp})\\}$ can be uniformly bounded by
some constant $B$.

Now let $\psi\otimes\sigma\in C^{\infty}_{c}(W)\otimes\operatorname{or}(W)$ be
supported in a ball of radius $R$. Assume for simplicity that the coordinates
$y_{1},\dots,y_{k}$ are positively oriented with respect to $\sigma$, and we
omit $\sigma$ henceforth.

We may write

$\langle e^{*}(S_{\epsilon u}^{*}\omega),\psi\rangle=\int_{|y|\leq
R}\psi(y)\frac{\epsilon}{|y+\epsilon u|^{k+1}}h_{u}\left(\frac{y+\epsilon
u}{|y+\epsilon u|}\right)dy=I_{1}+I_{2},$

where

$\displaystyle I_{1}$ $\displaystyle=\epsilon\int_{|y|\leq
R}(\psi(y)-\psi(0))\frac{1}{|y+\epsilon u|^{k+1}}h_{u}\left(\frac{y+\epsilon
u}{|y+\epsilon u|}\right)dy,$ $\displaystyle I_{2}$
$\displaystyle=\psi(0)\epsilon\int_{|y|\leq R}\frac{1}{|y+\epsilon
u|^{k+1}}h_{u}\left(\frac{y+\epsilon u}{|y+\epsilon u|}\right)dy.$

Observe that $|\psi(y)-\psi(0)|\leq C_{0}|y|$ for some constant $C_{0}$. Put
$M=C_{0}B$. We do a change of variables, $y=\epsilon w$, to find that

$\displaystyle|I_{1}|$ $\displaystyle\leq\epsilon M\int_{|w|\leq
R/\epsilon}\frac{\epsilon|w|}{\epsilon^{k+1}|w+u|^{k+1}}\epsilon^{k}dw=\epsilon
M\int_{|w|\leq R/\epsilon}\frac{|w|}{(1+|w|^{2})^{(k+1)/2}}dw$
$\displaystyle\leq C_{1}\epsilon+\epsilon M\int_{1\leq|w|\leq
R/\epsilon}\frac{|w|}{(1+|w|^{2})^{(k+1)/2}}dw\leq
C_{1}\epsilon+C_{2}\epsilon\int_{1}^{R/\epsilon}\frac{r^{k}}{(1+r^{2})^{(k+1)/2}}dr$
$\displaystyle\leq
C_{1}\epsilon+C_{2}\epsilon\int_{1}^{R/\epsilon}\frac{1}{r}dr=C_{1}\epsilon+C_{2}\epsilon\log\frac{R}{\epsilon}.$

Thus $I_{1}\to 0$ as $\epsilon\to 0^{+}$, uniformly in $u\in S(W^{\perp})$.

Next we do the same change of variable $y=\epsilon w$ for $I_{2}$. We find
that

$\displaystyle I_{2}$ $\displaystyle=\psi(0)\epsilon\int_{|y|\leq
R}\frac{1}{|y+\epsilon u|^{k+1}}h_{u}\left(\frac{y+\epsilon u}{|y+\epsilon
u|}\right)dy$ $\displaystyle=\psi(0)\int_{|w|\leq
R/\epsilon}\frac{1}{(1+|w|^{2})^{(k+1)/2}}h_{u}\left(\frac{w+u}{|w+u|}\right)dw.$

The last integral converges as $\epsilon\to 0$, uniformly in $u\in
S(W^{\perp})$, to the absolutely convergent integral

(4)
$c(u):=\int_{\mathbb{R}^{k}}\frac{1}{(1+|w|^{2})^{(k+1)/2}}h_{u}\left(\frac{w+u}{|w+u|}\right)dw,$

and so $I_{2}\to c(u)\psi(0)$. We conclude that $e^{*}(S_{\epsilon
u}^{*}\omega)\to c(u)\delta_{0}$ as $\epsilon\to 0^{+}$, uniformly in $u\in
S(W^{\perp})$. As a trivial consequence we obtain that
$e^{*}(S_{\epsilon\lambda u}^{*}\omega)\to c(u)\delta_{0}$ uniformly in
$\\{(u,\lambda):u\in S(W^{\perp}),0<\lambda\leq 1\\}$.

Now for a general $v\in S(V)\setminus W$, there are unique $w\in W$,
$0<\lambda\leq 1$ and $u\in S(W^{\perp})$ such that $v=\lambda u+w$. We have
$S_{\epsilon v}=S_{\epsilon\lambda u}\circ S_{\epsilon w}$, so that

$\langle e^{*}(S_{\epsilon v}^{*}\omega),\psi\rangle=\langle e^{*}(S_{\epsilon
w}^{*}\circ S_{\epsilon\lambda u}^{*}\omega),\psi\rangle=\langle
e^{*}(S_{\epsilon\lambda u}^{*}\omega),S_{-\epsilon w}^{*}\psi\rangle.$

Tracing the proof above with $\psi$ replaced by $S_{-\epsilon w}^{*}\psi$, we
see that $|I_{1}|\to 0$ uniformly in $v$, while

$I_{2}=\psi(-\epsilon w)\int_{|w|\leq
R/(\lambda\epsilon)}\frac{1}{(1+|w|^{2})^{(k+1)/2}}h_{u}\left(\frac{w+u}{|w+u|}\right)dw$

converges to $c(u)\psi(0)$ uniformly in $v$. In particular, the limit only
depends on $[v]\in\mathbb{P}_{+}(V/W)$. The continuity of $c(u)$ on
$S(W^{\perp})$ is seen from formula (4).

Finally, assume that $\omega$ is closed. By Proposition 3.4 we have
$\omega=r^{0}\tau$ for some $\tau\in\Omega^{k}(S^{n-1})$. Since
$0=d\omega=d\pi^{*}\tau=\pi^{*}d\tau$ on $V\setminus\\{0\\}$, where $\pi\colon
V\setminus\\{0\\}\to S^{n-1}$ denotes the radial projection, we conclude that
$\tau$ is closed. Fix $\psi\in C^{\infty}_{c}(W)\otimes\operatorname{or}(W)$.

For every $v\in V\setminus W$ put $W_{v}=W\oplus\operatorname{span}(v)$. For
every $\epsilon>0$, the image of $\epsilon v+W$ under $\pi$ is the open
hemisphere $S(W_{v})$ containing $\pi(v)$, with boundary $S(W)$. Let
$S_{v}^{+}(W_{v})$ denote the corresponding closed hemisphere. Put
$\pi_{\epsilon,v}=\pi|_{\epsilon v+W}$. Let $\psi_{\epsilon,v}\in
C^{\infty}_{c}(S(W_{v}),\operatorname{or}(S(W_{v}))$ be such that
$(\pi_{\epsilon,v})^{*}\psi_{\epsilon,v}=S^{*}_{-\epsilon v}\psi$. It then
holds that

$\langle e^{*}S^{*}_{\epsilon v}\omega,\psi\rangle=\int_{\epsilon
v+W}\pi^{*}\tau\otimes\pi^{*}\psi_{\epsilon,v}=\int_{S_{v}^{+}(W_{v})}\tau\otimes\psi_{\epsilon,v}.$

Observe that

$\psi_{\epsilon,v}\to\psi(0)\cdot\mathbf{1}_{S_{v}^{+}(W_{v})}$

pointwise as $\epsilon$ goes to zero. Hence by the dominated convergence
theorem we obtain

(5) $c(v)=\lim_{\epsilon\to
0}\int_{S(W_{v})}\tau\otimes\psi_{\epsilon,v}\to\int_{S_{v}^{+}(W_{v})}\tau\otimes\sigma_{v}\otimes\psi(0),$

where $\sigma_{v}(v)=1$. Since $d\tau=0$, the latter integral remains constant
if $v$ follows a continuous path in $S^{n-1}\setminus W$. By the same token,
$\int_{S_{v}^{+}(W)}\tau+\int_{S_{-v}^{+}(W)}\tau=\int_{S(W_{v})}\tau$. Since
$\sigma_{-v}=-\sigma_{v}$, it follows that $c(-v)=c(v)$. This concludes the
proof.

∎

######  Lemma 3.8.

Assume $\omega\in\Omega^{k}_{-\infty}(\mathbb{R}^{n})$ is a closed generalized
form, smooth outside of the origin, $0$-homogeneous, and $i_{E}\omega=0$. Let
$e:\mathbb{R}^{j}\hookrightarrow\mathbb{R}^{n}$ be a fixed monomorphism. Let
$\mu_{\epsilon}(x)dx\in\mathcal{M}^{\infty}(\mathbb{R}^{n})$ be an approximate
identity, with $\mu_{\epsilon}$ a Schwartz function. Then the weak limit

$\lim_{\epsilon\to 0^{+}}e^{*}(\omega\ast\mu_{\epsilon})$

exists and equals $e^{*}\omega$.

###### Proof.

For $k<j$, this follows from Lemma 3.6, while for $k=j$ we use Proposition
3.7. Let us provide the details in the case $k=j$, which is more involved.

By Proposition 3.7, the limit
$e^{*}\omega=\lim_{\mathbb{R}^{n}\setminus\mathbb{R}^{j}\ni v\to
0}e^{*}S^{*}_{v}\omega$ exists. Let $\psi\in\Omega_{c}(\mathbb{R}^{k})$ be a
test form. It then holds that

$f(v):=\langle e^{*}S_{v}^{*}\omega,\psi\rangle-\langle
e^{*}\omega,\psi\rangle:\mathbb{R}^{n}\setminus\mathbb{R}^{j}\to\mathbb{R}$

satisfies $f(v)\to 0$ as $|v|\to 0$.

Furthermore, $f$ is bounded. Indeed, write $v=\lambda u+w$, where $w\in W$ and
$u\in S(W^{\perp})$. Write

$\langle e^{*}S_{v}^{*}\omega,\psi\rangle=\langle e^{*}S_{\lambda
u}^{*}\omega,S^{*}_{-w}\psi\rangle=I_{1}+I_{2}$

as in the proof of Proposition 3.7. As
$|S^{*}_{-w}\psi(y)-S^{*}_{-w}\psi(0)|\leq C_{0}|y|$ for some $C_{0}$ that is
independent of $w$, while $|\psi(-w)|$ is uniformly bounded, it follows that
$I_{1},I_{2}$ and therefore $\langle e^{*}S_{v}^{*}\omega,\psi\rangle$ is
bounded as $\lambda\to 0$, uniformly in $w$ and $u$. The same estimates for
$I_{1},I_{2}$ in the proof of Proposition 3.7 show that $I_{1},I_{2}$ remain
bounded uniformly in $u,w$ also for $|\lambda|\geq\lambda_{0}>0$.

It follows that $\langle e^{*}S_{v}^{*}\omega,\psi\rangle$ is bounded as a
function of $v\in\mathbb{R}^{n}\setminus\mathbb{R}^{j}$, and therefore so is
$f(v)$.

Choose a function $\delta(\epsilon)>0$ such that $\delta(\epsilon)\to 0$ as
$\epsilon\to 0$, and
$\int_{|v|\geq\delta(\epsilon)}\mu_{\epsilon}(v)\leq\delta(\epsilon)$. It then
holds that

$\displaystyle\langle e^{*}(\omega\ast\mu_{\epsilon}),\psi\rangle-\langle
e^{*}\omega,\psi\rangle=\int\langle
e^{*}S_{v}^{*}\omega-e^{*}\omega,\psi\rangle
d\mu_{\epsilon}(v)=\int_{\mathbb{R}^{n}}f(v)d\mu_{\epsilon}(v).$

Seeing that
$\left|\int_{|v|\geq\delta(\epsilon)}f(v)d\mu_{\epsilon}(v)\right|\leq\sup_{\mathbb{R}^{n}}|f(v)|\delta(\epsilon)$,
while

$\int_{|v|\leq\delta(\epsilon)}f(v)d\mu_{\epsilon}(v)\leq\sup_{|v|\leq\delta(\epsilon)}|f(v)|\to
0,$

it follows that

$\langle e^{*}(\omega\ast\mu_{\epsilon}),\psi\rangle\to\langle
e^{*}\omega,\psi\rangle,$

as claimed. ∎

####  3.1.2. Pushforward under linear epimorphisms

In general, generalized forms do not pushforward under linear epimorphisms.
The following definition is motivated by the description of the pullback of
valuations in terms of differential forms given in [alesker_radon].

######  Definition 3.9.

Let $\omega\in\Omega^{k}_{-\infty}(V)$ be smooth outside of the origin,
$0$-homogeneous, and satisfy $i_{E}\omega=0$, and let $p:V\to W$ be a linear
epimorphism. We define the _pushforward_ of $\omega$ as

$p_{*}\omega:=r^{0}(\tilde{p}_{*}\tilde{\tau})\in\Omega^{k-s}_{-\infty}(W,\operatorname{or}(\operatorname{ker}p)),$

where $\tilde{p}:\widetilde{\mathbb{P}_{+}(V)}\to\mathbb{P}_{+}(W)$ is the
induced projection on the oriented blow-up of $\mathbb{P}_{+}(V)$ along
$\mathbb{P}_{+}(\operatorname{ker}p)$, $\tau$ satisfies $\omega=r^{0}\tau$,
and $\tilde{\tau}=\pi^{*}\tau$ where
$\pi:\widetilde{\mathbb{P}_{+}(V)}\to\mathbb{P}_{+}(V)$ is the natural
projection.

Furthermore, for $\omega\in\Omega^{n}_{-\infty}(V)$ equal to
$c\delta_{0}\otimes\sigma_{V}$, we define
$p_{*}\omega=c\delta_{0}\otimes\sigma_{W}\otimes\sigma_{p}$, where
$\sigma_{V}\in\operatorname{or}(V),\sigma_{W}\in\operatorname{or}(W),\sigma_{p}\in\operatorname{or}(\operatorname{ker}p)$
are such that $\sigma_{V}=\sigma_{W}\otimes\sigma_{p}$.

We will need the following description of the pushforward.

######  Lemma 3.10.

Let $p:V\to W$ be a linear epimorphism, $\dim V=n$, $\dim W=n-s$. Assume that
$\omega\in\Omega_{-\infty}^{k}(V)$ is $0$-homogeneous, smooth outside of the
origin, and $i_{E}\omega=0$. Let $\nu_{\epsilon}\in C^{\infty}(V)$ satisfy
$\nu_{\epsilon}\to 1$ smoothly on compact sets, with $\nu_{\epsilon}$ a
uniformly bounded family of Schwartz functions. Then
$p_{*}\omega\in\Omega_{-\infty}^{k-s}(W,\operatorname{or}(\operatorname{ker}p))$
is given by

$\langle\psi,p_{*}\omega\rangle=\lim_{\epsilon\to 0}\langle\nu_{\epsilon}\cdot
p^{*}\psi,\omega\rangle,$

for all compactly supported $\psi\in\Omega^{n-k}(W,\operatorname{or}(V))$.

###### Proof.

For $k=n$ we have $\omega=c\delta_{0}$, and both sides trivially equal
$c\psi(0)$.

Assume now $k<n$ and that $V=\mathbb{R}^{n}$, $W=\mathbb{R}^{n-s}$ and
coordinates are chosen so that
$\operatorname{ker}p=\\{x_{1}=\dots=x_{n-s}=0\\}$. Write

$\omega=\frac{1}{|x|^{k}}\sum_{|I|=k}g_{I}(\frac{x}{|x|})dx_{I}=:\sum\omega_{I}.$

We first show that

$\int_{\mathbb{R}^{n}}p^{*}\psi\wedge\omega$

is absolutely convergent, and so by the dominated convergence theorem, the
limit on the right hand side of the claimed equality exists and equals this
integral.

When $k>s$, this follows at once since $\frac{1}{|x|^{k}}$ is integrable over
$\mathbb{R}^{s}$.

Now assume $k=s$. Denoting $I=\\{n-k+1,\dots,n\\}$, we may assume
$\omega=\omega_{I}$. Examining the coefficient of $dx_{n-k+2}\wedge\dots\wedge
dx_{n}$ in the equality $i_{E}\omega=0$, and writing
$I_{j}=\\{j,n-k+2,\dots,n\\}$ for $1\leq j\leq n-k$, we find that

$g_{I}(\frac{x}{|x|})x_{n-k+1}=-\sum_{j=1}^{n-k}g_{I_{j}}(\frac{x}{|x|})x_{j}.$

Denoting $x=y+z$ where $y\in\mathbb{R}^{n-k}$ and $z\in\mathbb{R}^{k}$, we
write this equality as

$g_{I}(\frac{x}{|x|})=\frac{f_{1,y}(z)}{x_{n-k+1}}=\frac{f_{1,y}(z)}{z_{1}}$

for some $f_{1,y}(z)\in C^{\infty}(\mathbb{R}^{k}\setminus 0)$, which is
bounded for each fixed $y$, and smooth in $\mathbb{R}^{k}$ when $y\neq 0$. We
similarly can find $f_{i,y}(z)\in C^{\infty}(\mathbb{R}^{k}\setminus 0)$ for
$1\leq i\leq k$ such that

$g_{I}(\frac{x}{|x|})=\frac{f_{i,y}(z)}{z_{i}}.$

Note also that $|f_{i,y}(z)|\leq c_{0}|y|$ for some fixed $c_{0}>0$.

We have

$\displaystyle p^{*}\psi\wedge\omega$
$\displaystyle=\frac{1}{|x|^{k}}g_{I}(\frac{x}{|x|})p^{*}\psi\wedge
dx_{n-k+1}\wedge\dots\wedge dx_{n}$
$\displaystyle=h(y,z)dy_{1}\wedge\dots\wedge dy_{n-k}dz_{1}\wedge\dots\wedge
dz_{k}$

for some $h(y,z)$ smooth outside $(0,0)$ and supported in $|y|<R$ for some
$R>0$.

Thus for all $(y,z)\in\mathbb{R}^{n-k}\oplus\mathbb{R}^{k}$ with $y,z\neq 0$
it holds that

$|h(y,z)|\leq\frac{c_{0}|y|}{|y+z|^{k}\max(|z_{1}|,\dots,|z_{k}|)}\leq\frac{c_{1}|y|}{|y+z|^{k}|z|}.$

Consequently,

$\int_{\mathbb{R}^{k}}|h(y,z)|dz\leq
c_{1}|y|\int_{\mathbb{R}^{k}}\frac{dz}{|z||y+z|^{k}}=c_{2}|y|\int_{0}^{\infty}\frac{r^{k-2}dr}{(r^{2}+|y|^{2})^{k/2}}=c_{3}$

and so $\int_{\mathbb{R}^{n}}|h(y,z)|dydz\leq\int_{|y|\leq R}c_{3}dy$ is
finite.

Thus in all cases it remains to verify that

$\langle\psi,p_{*}\omega\rangle=\langle p^{*}\psi,\omega\rangle.$

Let $\omega=r^{0}\tau$. To avoid dealing with the blow-up, we may assume that
$\tau$ vanishes near $\operatorname{ker}p$, and conclude using approximation.
Denoting by $\pi:V\setminus\\{0\\}\to\mathbb{P}_{+}(V)$ the radial projection,
the left hand side is by definition

$\langle\psi,p_{*}\omega\rangle=\langle\psi,r^{0}p_{*}\tau\rangle=\langle\pi_{*}\psi,p_{*}\tau\rangle=\langle
p^{*}\pi_{*}\psi,\tau\rangle.$

On the other hand, since $\tau$ vanishes near $\operatorname{ker}p$,

$\langle p^{*}\psi,\omega\rangle=\langle\pi_{*}p^{*}\psi,\tau\rangle.$

It remains to observe that the top arrow in

${V\setminus\operatorname{ker}p}$${\mathbb{P}_{+}(V\setminus\operatorname{ker}p)}$${W}$${\mathbb{P}_{+}(W)}$$p$$\pi$$p$$\pi$

is a bundle map that restricts to a diffeomorphism on each fiber, and so
$\pi_{*}p^{*}\psi=p^{*}\pi_{*}\psi$. ∎

##  4\. The $0$-homogeneous current of a valuation

A key property of smooth valuations is their description through a pair of
currents $(C,T)$, obtained in [AleskerBernig:Product], equations (60)–(62).
Here we unify them into a single current in the translation-invariant case.
This simplifies the description of some operations on valuations such as
pullback and pushforward, and is very natural for describing the
Alesker–Fourier transform on valuations.

In the following we make frequent use of the Hodge star isomorphism

(6)
$\ast=\ast_{V}:\mathbb{\wedge}^{k}V^{*}\xrightarrow{\sim}\mathbb{\wedge}^{n-k}V\otimes\mathbb{\wedge}^{n}V^{*}\simeq\mathbb{\wedge}^{n-k}V\otimes\operatorname{Dens}(V)\otimes\operatorname{or}(V),$

given by
$\langle\eta,\zeta\rangle=\eta\wedge\ast\zeta,\quad\zeta\in\wedge^{k}V^{*},\eta\in\wedge^{k}V$.
In view of later computations, we remark that the Hodge star isomorphism
satisfies

$\zeta\wedge\theta=\langle\ast\zeta,\theta\rangle,\quad\zeta\in\mathbb{\wedge}^{k}V^{*},\theta\in\mathbb{\wedge}^{n-k}V^{*}$

and

$(\ast_{V^{*}}\otimes\operatorname{id})\circ\ast_{V}=(-1)^{k(n-k)}\operatorname{id}.$

###  4.1. The $0$-homogeneous current of a valuation

Recall that by definition a generalized valuation
$\phi\in\operatorname{Val}_{k}^{-\infty}(V)$ is a continuous linear functional
on $\operatorname{Val}_{n-k}^{\infty}(V)\otimes\mathrm{Dens}(V)^{*}$. Hence
$\phi$ defines for $k>0$ a linear functional on
$\Omega^{k-1}(\mathbb{P}_{+}(V^{*}),\mathbb{\wedge}^{n-k}V^{*}\otimes\operatorname{or}(V)\otimes\mathrm{Dens}(V)^{*})$,
that is a generalized form in
$S\in\Omega_{-\infty}^{n-k}(\mathbb{P}_{+}(V^{*}),\mathbb{\wedge}^{n-k}V\otimes\mathrm{Dens}(V))$.
The Hodge star isomorphism (6) allows to consider it as a generalized form

$T=T(\phi)=(-1)^{k(n-k)}*_{V^{*}}\circ
S\in\Omega_{-\infty}^{n-k}(\mathbb{P}_{+}(V^{*}),\mathbb{\wedge}^{k}V^{*}\otimes\operatorname{or}(V)).$

This is the current $T$ of a valuation mentioned above, specialized to the
translation-invariant case.

######  Remark 4.1.

Note that our convention differs slightly from that of the standard one, see
e.g. [AleskerBernig:Product]. Namely, let
$\phi\in\operatorname{Val}_{k}^{\infty}(V)$, $V=\mathbb{R}^{n}$, be given by
$\phi(K)=\int_{\operatorname{nc}(K)}\omega$ for
$\omega\in\Omega^{k,n-1-k}(V\times\mathbb{P}_{+}(V^{*}),\operatorname{or}(V))$.
Then

$T(\phi)\in\Omega^{n-k}(\mathbb{P}_{+}(V^{*}),\mathbb{\wedge}^{k}V^{*}\otimes\operatorname{or}(V))=\Omega^{k,n-k}(V\times\mathbb{P}_{+}(V^{*}),\operatorname{or}(V))$

is given by

(7) $T(\phi)=(-1)^{n-k}a^{*}D\omega,$

where $a$ is the antipodal map, and $D$ the Rumin differential. The extra sign
$(-1)^{n-k}=(-1)^{(n-k)^{2}}$ is due to the sign difference between the
natural pairing

$\displaystyle\Omega^{n-k}(\mathbb{P}_{+}(V^{*}),\mathbb{\wedge}^{k}V^{*}\otimes\operatorname{or}(V))$
$\displaystyle\otimes\Omega^{k-1}(\mathbb{P}_{+}(V^{*}),\mathbb{\wedge}^{n-k}V^{*}\otimes\operatorname{or}(V)\otimes\operatorname{Dens}(V)^{*})\to\mathbb{C},$
$\displaystyle(\alpha\otimes dx_{I})$ $\displaystyle\otimes(\beta\otimes
dx_{J})\mapsto\left(\int_{\mathbb{P}_{+}(V^{*})}\alpha\wedge\beta\right)\otimes(dx_{I}\wedge
dx_{J}),$

and the single wedge product pairing

$\Omega^{k,n-k}(V\times\mathbb{P}_{+}(V^{*}),\operatorname{or}(V))\otimes\Omega^{n-k,k-1}(V\times\mathbb{P}_{+}(V^{*}),\operatorname{or}(V)\otimes\operatorname{Dens}(V)^{*})\to\mathbb{C}.$

For a generalized translation-invariant valuation
$\phi\in\operatorname{Val}^{-\infty}_{k}(V)$, we define its _$0$ -homogeneous
current_
$\tau(\phi)\in\Omega_{-\infty}^{n-k}(V^{*},\wedge^{k}V^{*}\otimes\operatorname{or}(V))$
as follows. If $k>0$, then $\tau(\phi)=r^{0}T$. If $k=0$, then $\phi=c\chi$
and we set $\tau(\phi)=c\delta_{0}$, where $\delta_{0}$ is the delta measure
at the origin.

######  Proposition 4.2.

A generalized form
$\tau\in\Omega_{-\infty}^{n-k}(V^{*},\wedge^{k}V^{*}\otimes\operatorname{or}(V))$
is $\tau(\phi)$ for some $\phi\in\operatorname{Val}^{-\infty}_{k}(V^{*})$ if
and only if $\tau$ satisfies

  1. (1)

$0$-homogeneous.

  2. (2)

$i_{E}\tau=0$.

  3. (3)

$\tau\wedge E=0$.

  4. (4)

$d\tau=0$.

Moreover, $\phi$ is smooth if and only if $\tau(\phi)$ is smooth on
$V^{*}\setminus\\{0\\}$.

###### Proof.

Assume $\tau=\tau(\phi)$. For $k=0$ the verification is trivial, thus we
assume $k\geq 1$. Write $\tau=r^{0}T$ as in the definition of $\tau(\phi)$. By
Lemma 3.4, $\tau$ satisfies the first two properties.

It holds on $V^{*}\setminus\\{0\\}$ that $d\pi^{*}T=\pi^{*}dT=0$. Therefore,
the generalized form $d\tau$ is supported at the origin. For a fixed
functional $\psi\in(\wedge^{k}V^{*}\otimes\operatorname{or}(V))^{*}$,
$\phi(d\tau)$ a $0$-homogeneous generalized $(n-k+1)$-form, and so it must
hold that $n-k+1=n$, and $\phi(d\tau)$ is a multiple of the delta top form at
the origin, denoted $\delta_{0}$. But $\delta_{0}$ does not vanish on constant
functions, while $d\tau$ must vanish on constants. It follows that $d\tau=0$.

Finally, for a test form $\psi\in\Omega^{k}(V^{*},\wedge^{n-k-1}V^{*})$ it
holds that

$\langle\tau,\psi\wedge E\rangle=\langle T,\pi_{*}(\psi\wedge E)\rangle=0$

by the verticality of $T$. Therefore, $\tau\wedge E=0$.

Conversely, let $\tau$ satisfy all four properties. Assume first $k=0$. Since
$\tau\wedge E=0$, $\tau$ must be supported at the origin. Since it is
$0$-homogeneous, $\tau$ is a multiple of the delta measure, and so
$\tau=\tau(c\chi)$ for some constant $c$.

Assume now $k\geq 1$. By Lemma 3.4, $\tau=r^{0}\omega$ for a unique
generalized form $\omega$ on $\mathbb{P}_{+}(V^{*})$. It follows immediately
from $\tau\wedge E=0$ that $\omega$ is vertical. Also on
$V^{*}\setminus\\{0\\}$ we have $\pi^{*}d\omega=d\pi^{*}\omega=d\tau=0$, and
therefore $d\omega=0$. The existence of $\phi$ such that $\tau=\tau(\phi)$ now
follows from [AleskerBernig:Product].

∎

######  Definition 4.3.

A form satisfying the properties listed in Proposition 4.2 will be called
_valuation-type_. If it is smooth outside the origin, we shall call it _smooth
valuation-type_.

The following special case will be relevant for us later.

######  Lemma 4.4.

Let $U$ be a $k$-dimensional linear subspace of $V=\mathbb{R}^{n}$ and let
$P\colon V\to V/U$ denote the canonical projection. Let
$\operatorname{vol}_{V/U}$ be a density on $V/U$. Then
$\operatorname{vol}_{V/U}$ defines a continuous valuation

$\phi(K)=\operatorname{vol}_{V/U}(PK),\quad K\subset V.$

It then holds that

$\tau(\phi)=P^{*}\operatorname{vol}_{V/U}\otimes[[U^{\perp}]].$

We remark that
$[[U^{\perp}]]\in\Omega^{k}_{-\infty}(V^{*},\operatorname{or}(U^{\perp})\otimes\operatorname{or}(V))=\Omega^{k}_{-\infty}(V^{*},\operatorname{or}(U))$,
while
$P^{*}\operatorname{vol}_{V/U}\in\mathbb{\wedge}^{n-k}(V/U)^{*}\otimes\operatorname{or}(V/U)\subset\mathbb{\wedge}^{n-k}V^{*}\otimes\operatorname{or}(V)\otimes\operatorname{or}(U)^{*}$,
so that $P^{*}\operatorname{vol}_{V/U}\otimes[[U^{\perp}]]$ is well-defined.

###### Proof.

Let $V=\mathbb{R}^{n}$ with the standard Euclidean structure and orientation,
and suppose that $U=\\{x_{k+1}=\ldots=x_{n}=0\\}$ and
$\phi(K)=\operatorname{vol}_{U^{\perp}}(PK)$, where $P$ is the orthogonal
projection onto $U^{\perp}$. Let $L\subset U$ be a convex body of unit volume.
The valuation $\phi\in\operatorname{Val}_{n-k}(V)$ is continuous, but not
smooth. Its action on $\psi\in\operatorname{Val}_{k}^{\infty}(V)$ is
$\langle\phi,\psi\rangle=\psi(L)$. If
$\psi=\int_{\operatorname{nc}(K)}\omega$, then by eq. (7) we have

$\langle\phi,\psi\rangle=(-1)^{k}\int_{L\times S(U^{\perp})}\omega$

and hence

$T(\phi)=(-1)^{k+k(n-k)}dx_{k+1}\wedge\cdots\wedge
dx_{n}\otimes[[S(U^{\perp})]].$

We conclude that

(8) $\tau(\phi)=r^{0}(T(\phi))=(-1)^{k(n-k)}dx_{k+1}\wedge\cdots\wedge
dx_{n}\otimes[[U^{\perp}]].$

∎

###  4.2. Operations on valuations

Recall for the following that for any generalized form
$\omega\in\Omega_{-\infty}(V)$, $i_{*}\omega$ and $p^{*}\omega$ are well-
defined generalized forms, where $i:V\to W$ denotes any monomorphism, and
$p:W\to V$ any epimorphism.

Here we describe some operations on valuations in terms of the $0$-homogeneous
current representation. Given a linear map $f:V_{1}\to V_{2}$, one can define
the pullback $f^{*}:\operatorname{Val}(V_{2})\to\operatorname{Val}(V_{1})$,
and the pushforward
$f_{*}:\operatorname{Val}(V_{1})\otimes\operatorname{Dens}(V_{1}^{*})\to\operatorname{Val}(V_{2})\otimes\operatorname{Dens}(V_{2}^{*})$,
see [Alesker:Fourier].

Recall that if $f$ is a monomorphism, then $f^{*}$ maps smooth valuations to
smooth valuations, while if $f$ is an epimorphism, then $f^{*}$ extends by
continuity to a linear map

$f^{*}\colon\operatorname{Val}^{-\infty}_{k}(W)\to\operatorname{Val}^{-\infty}_{k}(V).$

Let $n_{V}=\dim V$ and $n_{W}=\dim W$ in the following.

######  Proposition 4.5.

Let $f\colon V\to W$ be a linear map. The pullback
$f^{*}\colon\operatorname{Val}^{\infty}_{k}(W)\to\operatorname{Val}^{-\infty}_{k}(V)$
is then given by

$\tau(f^{*}u)=f^{\vee}\circ(f^{\vee})_{*}\tau(u),$

where
$(f^{\vee})_{*}:\Omega_{-\infty}^{n_{W}-k}(W^{*},\wedge^{k}W^{*}\otimes\operatorname{or}(W))\to\Omega^{n_{V}-k}_{-\infty}(V^{*},\wedge^{k}W^{*}\otimes\operatorname{or}(V))$
is the pushforward by $f^{\vee}\colon W^{*}\to V^{*}$, and
$f^{\vee}:\wedge^{k}W^{*}\to\wedge^{k}V^{*}$ is applied to the value of the
form.

###### Proof.

Assume first $k=0$. We then have $\tau(\chi)=\delta_{0}=[[\\{0\\}]]$. If $f$
is either an epimorphism or a monomorphism, then

$(f^{\vee})_{*}(\delta_{0})=\delta_{0}.$

Indeed, in the first case the pushforward is the usual pushforward under
proper immersions. In the second case, our Definition 3.9 applies. Since
$f^{*}\chi=\chi$, the conclusion for $k=0$ follows. All other cases follow
immediately from [alesker_radon], adapted to the translation-invariant case
and bearing in mind the additional sign of eq. (7) imposed by our convention
for the current of a valuation. ∎

Recall that if $f$ is an epimorphism, then $f_{*}$ maps smooth valuations to
smooth valuations, while if $f$ is a monomorphism, then $f_{*}$ extends by
continuity to a linear map

$f_{*}\colon\operatorname{Val}^{-\infty}_{k}(V)\otimes\operatorname{Dens}(V^{*})\to\operatorname{Val}^{-\infty}_{k+n_{W}-n_{V}}(W)\otimes\operatorname{Dens}(W^{*}).$

######  Proposition 4.6.

Let $f\colon V\to W$ be a linear map. For $0\leq k\leq n_{V}$ and
$\phi\in\operatorname{Val}_{k}^{-\infty}(V)\otimes\operatorname{Dens}(V^{*})$,
we have

$\tau(\phi)\in\Omega^{n_{V}-k}_{-\infty}(V^{*},\wedge^{k}V^{*}\otimes\wedge^{n_{V}}V)$

and so

$\widetilde{\tau}(\phi):=\ast_{V}\circ\tau(\phi)\in\Omega_{-\infty}^{n_{V}-k}(V^{*},\wedge^{n_{V}-k}V).$

The pushforward

$f_{*}\colon\operatorname{Val}^{\infty}_{k}(V)\otimes\operatorname{Dens}(V^{*})\to\operatorname{Val}^{-\infty}_{k+n_{W}-n_{V}}(W)\otimes\operatorname{Dens}(W^{*})$

is then given by

(9)
$\widetilde{\tau}(f_{*}\phi)=f\circ(f^{\vee})^{*}\widetilde{\tau}(\phi)\in\Omega^{n_{V}-k}_{-\infty}(W^{*},\wedge^{n_{V}-k}W)$

if $k+n_{W}>n_{V}$. If $k+n_{W}=n_{V}$, then

(10)
$\widetilde{\tau}(f_{*}\phi)=f\circ(f^{\vee})_{v}^{*}\widetilde{\tau}(\phi)\in\Omega^{n_{W}}_{-\infty}(W^{*},\wedge^{n_{W}}W),$

where the choice of $v\in V^{*}\setminus f^{\vee}(W^{*})$ can be arbitrary.

###### Proof.

Suppose $k>0$ and $k+n_{W}-n_{V}>0$. If $f$ is an epimorphism, then the
statement follows directly from [KotrbatyWannerer:Harmonic, Lemma 7.4]. If $f$
is a monomorphism, then using the fact that $f_{*}$ is defined as the dual of
$f^{*}$, the claim follows from a short computation.

Let us verify the statement for $k=0$, that is for $f_{*}(\chi)$. Since every
linear map factors as $V\to f(V)\to W$, we have that $f_{*}\chi=0$ if $f$ is
not injective. In this case also right-hand side of (9) vanishes. Hence we may
assume that $f$ is a monomorphism. Fix orientations and Lebesgue measures on
$V$ and $W$ for simplicity. This also fixes orientations and Lebesgue measures
on $f(V)$ and $W/f(V)$ and elements in the top exterior powers of $V$ and
$f(V)$ that we denote by $\sigma_{V}$ and $\sigma_{f(V)}$. One has
$f\otimes(f^{\vee})^{*}(\sigma_{V}\otimes[[0]])=f(\sigma_{V})\otimes[[\operatorname{ker}f^{\vee}]]$
, while

$f_{*}\chi(K)=\int_{W/f(V)}\chi((f(V)+w)\cap
K)dw=\operatorname{vol}(P_{W/f(V)}K)$

and so by Lemma 4.4 one has
$\widetilde{\tau}(f_{*}\chi)=\sigma_{f(V)}\otimes[[f(V)^{\perp}]]=f(\sigma_{V})\otimes[[\operatorname{ker}f^{\vee}]]$,
as required.

It remains to verify the statement for $k=n_{V}-n_{W}$. Since every linear map
factors as $V\to f(V)\to W$, the pushforward $f_{*}\phi$ vanishes if $f$ is
not surjective. In this case also right-hand side of (10) vanishes. We assume
therefore that $f$ is an epimorphism. For convenience fix a Euclidean
structure on $V$, and orientations on $V$ and $W$, such that $v$ is orthogonal
to $f^{\vee}(W^{*})$. Assume
$\widetilde{\tau}=\widetilde{\tau}(\phi)=r^{0}\widetilde{\beta}$ with
$\widetilde{\beta}\in\Omega^{n_{W}}(S(V^{*}),\wedge^{n_{W}}V)$. Note that
$\widetilde{\beta}=d\widetilde{\omega}$ for some
$\widetilde{\omega}\in\Omega^{n_{W}-1}(S(V^{*}),\wedge^{n_{W}}V)$. Then
$f_{*}\phi=\chi\otimes\sigma$, where
$\sigma=f(\int_{S(W^{*})}\widetilde{\omega})\in\wedge^{n_{W}}W$. Hence

$\widetilde{\tau}(f_{*}\phi)=f(\int_{S(f^{\vee}(W^{*}))}\widetilde{\omega})\delta_{0},$

and comparing with (5) completes the proof.

∎

######  Proposition 4.7.

Let $\phi\in\operatorname{Val}_{k}^{\infty}(V)$ and
$\psi\in\operatorname{Val}_{l}^{\infty}(W)$. Then
$\tau(\phi\boxtimes\psi)=(-1)^{(n-k)l}\tau(\phi)\boxtimes\tau(\psi)$.

Here the factor $(-1)^{(n-k)l}$ accounts for the difference between viewing
forms on $\mathbb{P}_{+}(V^{*})$ with values in
$\mathbb{\wedge}^{\bullet}V^{*}$ as translation-invariant forms on the
cosphere bundle $V\times\mathbb{P}_{+}(V^{*})$, which is the setting of
[AleskerBernig:Product], when taking wedge products.

###### Proof.

We may assume that $\phi\in\operatorname{Val}^{\infty}_{k}(V)$,
$\psi\in\operatorname{Val}^{\infty}_{l}(W)$. Fix Euclidean inner products.
Assume first $k,l>0$, so that $\tau(\phi)$, $\tau(\psi)$ are locally
integrable. Let us recall the construction of the exterior product of
$\phi\in\operatorname{Val}^{\infty}(\mathbb{R}^{m})$ and
$\psi\in\operatorname{Val}^{\infty}(\mathbb{R}^{n})$ in these terms. On the
blow-up space

$\Sigma:=S^{m-1}\times[0,\pi/2]\times S^{n-1}$

one has the blow-down map

$F:\Sigma\to S^{m+n-1},\quad F(u,\theta,v)=(\cos(\theta)u,\sin(\theta)v)\in
S^{m+n-1}$

and the projection

$\Phi:\Sigma\to S^{m-1}\times S^{n-1},\quad\Phi(u,\theta,v)=(u,v).$

The defining generalized $n$-form of $\phi\boxtimes\psi$ is

$T(\phi\boxtimes\psi)=F_{*}\Phi^{*}(T(\phi)\boxtimes
T(\psi))\in\Omega_{-\infty}(\mathbb{P}_{+}(V^{*}\times
W^{*}),\wedge^{\bullet}(V^{*}\times W^{*})).$

We use $\pi$ to denote the radial projection in all spaces. Define

$F_{0}\colon(0,\infty)\times\Sigma\to\mathbb{R}^{m+n},\quad
F_{0}(t,u,\theta,v)=(t\cos(\theta)u,t\sin(\theta)v)$

so that $\pi\circ F_{0}=F$. Define also
$\beta:(0,\infty)\times\Sigma\to\Sigma$, $\beta(r,\sigma)=\sigma$.

$\Phi_{0}:(0,\infty)\times\Sigma\to S^{m-1}\times
S^{n-1},\quad\Phi_{0}=\Phi\circ\beta.$

Finally, define the map

$R\colon\mathbb{R}^{m}\setminus\\{0\\}\times\mathbb{R}^{n}\setminus\\{0\\}\to(0,\infty)\times\Sigma$

so that $F_{0}\circ R=\operatorname{id}$ and $\Phi_{0}\circ R=\pi\times\pi$.
Observe also that $F$, $R$, $F_{0}$ are all diffeomorphisms outside of
submanifolds of positive codimension.

As both sides of the claimed equality

$r^{0}F_{*}\Phi^{*}(T(\phi)\boxtimes
T(\psi))=(-1)^{(n-k)l}r^{0}T(\phi)\boxtimes r^{0}T(\psi)$

are locally integrable, it suffices to show that

$\pi^{*}F_{*}\Phi^{*}(T(\phi)\boxtimes
T(\psi))=(\pi\times\pi)^{*}(T(\phi)\boxtimes T(\psi))$

holds outside of $\mathbb{R}^{m}\times\\{0\\}\cup\\{0\\}\times\mathbb{R}^{n}$.
Noting that $(\pi\times\pi)^{*}=R^{*}\Phi_{0}^{*}$, it suffices to verify that

$\displaystyle
R^{*}\Phi_{0}^{*}=\pi^{*}F_{*}\Phi^{*}\iff\Phi_{0}^{*}=F_{0}^{*}\pi^{*}F_{*}\Phi^{*}.$

Since $\pi\circ F_{0}=F\circ\beta$, we find $F_{0}^{*}\pi^{*}=\beta^{*}F^{*}$
and we should verify

$\Phi_{0}^{*}=\beta^{*}F^{*}F_{*}\Phi^{*},$

and since $F^{*}F_{*}=\operatorname{id}$, we are left with the equality

$\Phi_{0}^{*}=\beta^{*}\Phi^{*},$

which holds since $\Phi_{0}=\Phi\circ\beta$.

Now assume $k>0,l=0$, so that $\psi=\chi$. This case then follows immediately
from [alesker_radon, equation 2.1.13]. The remaining case $k=0,l=0$ amounts to
the verification $\tau(\chi\boxtimes\chi)=\tau(\chi)\boxtimes\tau(\chi)$,
which follows from $\delta_{0}\boxtimes\delta_{0}=\delta_{0}$.

∎

###  4.3. Symmetry of closed vertical forms

Closed, vertical forms on the sphere bundle play a key role in valuation
theory. In the translation invariant case, such forms exhibit a certain
symmetry, as we now proceed to show.

######  Lemma 4.8.

If $\tau\in\Omega^{n-k}(S^{n-1},\mathbb{\wedge}^{k}(\mathbb{R}^{n})^{*})$ is
closed and vertical, then the form
$\overline{\tau}\in\Omega^{n-k}(S^{n-1},\mathbb{\wedge}^{n-k}(\mathbb{R}^{n})^{*})$,
corresponding to $\tau$ under the Hodge star
$\mathbb{\wedge}^{k}(\mathbb{R}^{n})^{*}\simeq\mathbb{\wedge}^{n-k}(\mathbb{R}^{n})^{*}$,
belongs to the subspace
$C^{\infty}(S^{n-1},\operatorname{Sym}^{2}(\mathbb{\wedge}^{n-k}T^{*}S^{n-1}))$.

###### Proof.

Let $(x_{1},\ldots,x_{n},\xi_{1},\ldots,\xi_{n})$ be the standard coordinates
on $\mathbb{R}^{n}\times\mathbb{R}^{n}\supset\mathbb{R}^{n}\times S^{n-1}$.
Let $R=\sum\xi_{i}\frac{\partial}{\partial x_{i}}$ denote the Reeb vector
field. Since $\tau$ is vertical, one has $i_{R}\overline{\tau}=0$. Hence,
using that $\tau$ is closed, one obtains

$0=d(i_{R}\overline{\tau})=\sum d\xi_{i}\wedge i_{\frac{\partial}{\partial
x_{i}}}\overline{\tau}.$

Hence by [Gray:SomeRelations, Proposition 2.2] (see also [BFSW:Tube, Remark
5.8]) we deduce that
$\tau_{\xi}\in\operatorname{Sym}^{2}(\mathbb{\wedge}^{n-k}T^{*}_{\xi}S^{n-1})$,
concluding the proof. ∎

Let
$\tau\in\Omega^{n-k}_{-\infty}(V^{*},\wedge^{k}V^{*}\otimes\operatorname{or}(V))$.
Using the identification
$\wedge^{k}V^{*}\simeq\wedge^{n-k}V\otimes\wedge^{n}V^{*}$, we define the
generalized form
$\overline{\tau}\in\Omega^{n-k}_{-\infty}(V^{*},\wedge^{n-k}V\otimes\operatorname{Dens}(V))$
corresponding to $\tau$.

######  Proposition 4.9.

Let
$\tau\in\Omega^{n-k}_{-\infty}(V^{*},\mathbb{\wedge}^{k}V^{*}\otimes\operatorname{or}(V))$
be the $0$-homogeneous current of a valuation. Considered as an element of
$C^{-\infty}(V^{*},\mathbb{\wedge}^{n-k}V\otimes\mathbb{\wedge}^{n-k}V\otimes\operatorname{Dens}(V))$,
$\overline{\tau}$ belongs to the subspace
$C^{-\infty}(V^{*},\operatorname{Sym}^{2}(\mathbb{\wedge}^{n-k}V)\otimes\operatorname{Dens}(V))$.

###### Proof.

If $k=0$, there is nothing to prove. Suppose therefore that $k>0$. and use
Lemma 3.4 to write $\tau=r^{0}\tau^{\prime}$, where $\tau^{\prime}$ is
vertical and closed.

Assume first that $\tau^{\prime}$ is a smooth form. Fix a Euclidean inner
product on $V$. By Lemma 4.8 we know that $\overline{\tau}^{\prime}$ is
symmetric. It follows that $\overline{\tau}=\pi^{*}\overline{\tau}^{\prime}$
is symmetric on $V\setminus\\{0\\}$. Since $\overline{\tau}$ is locally
integrable, the symmetry of $\overline{\tau}$ follows.

In the general case, choose an approximate identity $\rho_{j}\in
C^{\infty}(\mathrm{SL}(V^{*}))$, and set
$\tau^{\prime}_{j}=\rho_{j}\ast\tau^{\prime}$. Then $\tau^{\prime}_{j}$ is
smooth, vertical and closed, and so
$\rho_{j}\ast\overline{\tau}=r^{0}(\rho_{j}\ast\overline{\tau}^{\prime})=r^{0}\overline{\tau_{j}^{\prime}}$
is symmetric. Taking $j\to\infty$ proves the proposition.

∎

##  5\. Fourier transform of differential forms

###  5.1. Fourier transform of smooth and generalized differential forms

Let $V$ and $F$ be finite-dimensional vector spaces of over the reals, and let
$\Omega_{\mathcal{S}}^{k}(V,F)=\mathcal{S}(V,\mathbb{\wedge}^{k}V^{*}\otimes
F)\subset\Omega^{k}(V,F)$ denote the Schwartz space of differential $k$-forms
on $V$ with values in $F$ and rapidly decreasing coefficients.

######  Definition 5.1.

The Fourier transform

$\mathcal{F}:\Omega_{\mathcal{S}}^{k}(V,F)\to\Omega_{\mathcal{S}}^{n-k}(V^{*},\operatorname{or}(V)\otimes
F)$

is defined as follows. For $\omega\in\Omega_{\mathcal{S}}^{k}(V,F)$, the Hodge
star isomorphism (6) allows to consider $\omega$ as a map
$\widetilde{\omega}:V\to\mathbb{\wedge}^{n-k}V\otimes\operatorname{Dens}(V)\otimes\operatorname{or}(V)\otimes
F$. Hence we may define for $\xi\in V^{*}$

$\mathcal{F}(\omega)(\xi)=\int_{V}e^{2\pi\mathbf{i}\langle
x,\xi\rangle}\widetilde{\omega}(x)\in\mathbb{\wedge}^{n-k}V\otimes\operatorname{or}(V)\otimes
F.$

For $\omega\in\Omega^{k}(V,F)$ and
$\eta\in\Omega_{c}^{n-k}(V,\operatorname{or}(V)\otimes F^{*})$, the
isomorphism

$\wedge^{k}V^{*}\otimes\wedge^{n-k}V^{*}\otimes\operatorname{or}(V)\simeq\operatorname{Dens}(V)$

allows to define the function $V\ni
x\mapsto\omega(x)\wedge\eta(x)\in\operatorname{Dens}(V)$ and hence the pairing

(11) $\langle\omega,\eta\rangle=\int_{V}\omega(x)\wedge\eta(x).$

Note that the inclusion of $\Omega^{k}(V,F)$ into $\Omega^{k}_{-\infty}(V,F)$
defined by this pairing coincides with the canonical inclusion of
$C^{\infty}(M,\mathcal{E})$ into $C^{-\infty}(M,\mathcal{E})$.

######  Lemma 5.2.

Let $\omega\in\Omega_{\mathcal{S}}^{k}(V,F)$ and
$\eta\in\Omega_{\mathcal{S}}^{k}(V^{*},F^{*})$. Then

$\langle\mathcal{F}\omega,\eta\rangle=(-1)^{k(n-k)}\langle\omega,\mathcal{F}\eta\rangle.$

###### Proof.

This follows immediately from the fact that pointwise we have

$\widetilde{\omega}_{x}\wedge\eta_{\xi}=(-1)^{k(n-k)}\omega_{x}\wedge\widetilde{\eta}_{\xi}\in\operatorname{Dens}(V)\otimes\operatorname{Dens}(V^{*}).$

∎

######  Definition 5.3.

The Fourier transform of tempered generalized forms on $V$ with values in $F$

$\mathcal{F}\colon\Omega_{\mathcal{S}^{\prime}}^{k}(V,F)\to\Omega_{\mathcal{S}^{\prime}}^{n-k}(V^{*},F\otimes\operatorname{or}(V))$

is defined by

$\langle\mathcal{F}T,\eta\rangle=(-1)^{k(n-k)}\langle
T,\mathcal{F}\eta\rangle,\quad\forall\eta\in\Omega_{\mathcal{S}}^{k}(V^{*},F^{*}).$

In view of Lemma 5.2, this definition continuously extends the Fourier
transform of smooth forms.

Let us discuss functorial properties of the Fourier transform.

######  Lemma 5.4.

The inversion formula

$(\mathcal{F}_{V^{*}}\times\operatorname{id})\circ\mathcal{F}_{V}(\omega)=(-1)^{kn}(-\operatorname{id})^{*}\omega$

holds for all $\omega\in\Omega^{k}_{\mathcal{S}^{\prime}}(V,F)$.

###### Proof.

This follows immediately from the classical inversion formula. ∎

######  Lemma 5.5.

Let $i\colon V\to W$ be a monomorphism, and $p:W\to V$ an epimorphism. Write
$n=\dim V$, $m=\dim W$. Then

$\mathcal{F}_{W}\circ
i_{*}=(-1)^{(m-n)(n-k)}(i^{\vee})^{*}\circ\mathcal{F}_{V}\colon\Omega_{\mathcal{S}^{\prime}}^{k}(V,F)\to\Omega_{\mathcal{S}^{\prime}}^{n-k}(W^{*},F\otimes\operatorname{or}(V)),$

and

$(p^{\vee})_{*}\circ\mathcal{F}_{V}=\mathcal{F}_{W}\circ
p^{*}\colon\Omega_{\mathcal{S}^{\prime}}^{k}(V,F)\to\Omega_{\mathcal{S}^{\prime}}^{m-k}(W^{*},F\otimes\operatorname{or}(V)).$

###### Proof.

We may assume $F=\mathbb{R}$. It suffices to show that

(12)
$i^{*}\mathcal{F}_{W^{*}}\varphi=(-1)^{(m-n)(n-k)}\mathcal{F}_{V^{*}}(i^{\vee})_{*}\varphi$

for every Schwartz test function
$\varphi\in\Omega_{\mathcal{S}}^{k+m-n}(W^{*},\operatorname{or}(V)\otimes\operatorname{or}(W))$.
Indeed, if this holds, then for every
$T\in\Omega_{\mathcal{S}^{\prime}}^{k}(V)$ one has

$\displaystyle\langle\mathcal{F}(i_{*}T),\varphi\rangle$
$\displaystyle=(-1)^{(k+m-n)(n-k)}\langle i_{*}T,\mathcal{F}\varphi\rangle$
$\displaystyle=(-1)^{(k+m-n)(n-k)+(m-n+k)(n-k)+k(n-k)}\langle
T,i^{*}\mathcal{F}\varphi\rangle$
$\displaystyle=(-1)^{k(n-k)+(m-n)(n-k)}\langle
T,\mathcal{F}((i^{\vee})_{*}\varphi)\rangle$
$\displaystyle=(-1)^{(m-n)(n-k)}\langle\mathcal{F}T,(i^{\vee})_{*}\varphi\rangle$
$\displaystyle=(-1)^{(m-n)(n-k)}\langle(i^{\vee})^{*}\mathcal{F}T,\varphi\rangle$

For the proof of (12) choose linear coordinates $(y_{1},\ldots,y_{m})$ on $W$
such that $i(V)=\\{y_{n+1}=\cdots=y_{m}=0\\}$ and let $x_{j}=i^{*}y_{j}$ for
$j=1,\ldots,n$. Let $(\eta_{1},\ldots,\eta_{m})$ and
$(\xi_{1},\ldots,\xi_{n})$ be dual coordinates on $W^{*}$ and $V^{*}$. For a
multi-index $I=(i_{1}<\dots<i_{r})$ we write
$dy_{I}=dy_{i_{1}}\wedge\dots\wedge dy_{i_{r}}$, and similarly for other
coordinates.

We may assume that $\varphi=f(\eta)d\eta_{I}\wedge
d\eta_{J}\otimes\sigma_{W}\otimes\sigma_{V}$ where $f$ is a Schwartz function,
$I=(1,\dots,k)$, $J=(n+1,\dots,m)$.

Then, writing $d\eta=d\eta_{1}\wedge\dots\wedge d\eta_{m}$, we have
$\ast(d\eta_{I}\wedge d\eta_{J})=(-1)^{(m-n)(n-k)}dy_{I^{c}}$, where
$I^{c}=(k+1,\dots,n)$. Thus

$i^{*}\circ\mathcal{F}_{W^{*}}\varphi=(-1)^{(m-n)(n-k)}\left(\int_{W^{*}}e^{2\pi\mathbf{i}\langle\eta,i(x)\rangle}f(\eta)\;d\eta\otimes\sigma_{W}\right)dx_{I^{c}}\otimes\sigma_{V}.$

Before we treat $\mathcal{F}_{V^{*}}(i^{\vee})_{*}\varphi$, recall that the
defining property of fiber integration along an epimorphism $p\colon W\to V$
is $\int_{W}p^{*}\omega\wedge\psi=\int_{V}\omega\wedge p_{*}\psi$, where
$\omega$ and $\psi$ are Schwartz test forms. Thus, if
$\psi\in\Omega_{\mathcal{S}}^{k}(W,\operatorname{or}(W))$, say
$\psi=f(y)dy_{I}\wedge dy_{J}\otimes\sigma_{V}$ where
$p(y)=(y_{1},\dots,y_{n})$ and $dy_{J}=dy_{n+1}\wedge\cdots\wedge dy_{m}$,
then

$p_{*}\psi=\left(\int_{y\in p^{-1}(x)}f(y)\;dy_{J}\otimes\sigma_{\ker
p}\right)dx_{I}\otimes\sigma_{V},$

where $\sigma_{V}\otimes\sigma_{\ker p}=\sigma_{W}$.

Therefore, writing $i^{\vee}=q$ we obtain

$(i^{\vee})_{*}\varphi=\left(\int_{\eta\in
q^{-1}(\xi)}f(\eta)d\eta_{J}\otimes\sigma_{\ker q}\right)d\xi_{I}.$

Hence, putting $d\xi=d\xi_{1}\wedge\dots\wedge d\xi_{n}$, we find

$\displaystyle\mathcal{F}\circ(i^{\vee})_{*}\varphi$
$\displaystyle=\left(\int_{V^{*}}\left(\int_{\eta\in
q^{-1}(\xi)}f(\eta)d\eta_{J}\otimes\sigma_{\ker
q}\right)e^{2\pi\mathbf{i}\langle\xi,x\rangle}d\xi\otimes\sigma_{V}\right)dx_{I^{c}}\otimes\sigma_{V}$
$\displaystyle=\left(\int_{W^{*}}e^{2\pi\mathbf{i}\langle\eta,i(x)\rangle}f(\eta)\;d\eta\otimes\sigma_{W}\right)dx_{I^{c}}\otimes\sigma_{V},$

as claimed.

Combined with the inversion formula, we deduce from the first identity for
$S\in\Omega^{k}_{\mathcal{S}^{\prime}}(V^{*})$ that

$\displaystyle(-1)^{(m-k)m}(-\operatorname{id})^{*}i_{*}\mathcal{F}_{V^{*}}S$
$\displaystyle=\mathcal{F}_{W^{*}}\mathcal{F}_{W}i_{*}\mathcal{F}_{V^{*}}S=(-1)^{(m-n)k}\mathcal{F}_{W^{*}}(i^{\vee})^{*}\mathcal{F}_{V}\mathcal{F}_{V^{*}}S$
$\displaystyle=(-1)^{kn+(m-n)k}\mathcal{F}_{W^{*}}(i^{\vee})^{*}(-\operatorname{id})^{*}S$
$\displaystyle=(-1)^{mk}\mathcal{F}_{W^{*}}(i^{\vee})^{*}(-\operatorname{id})^{*}S.$

Recall that
$i_{*}\mathcal{F}_{V^{*}}S\in\Omega_{\mathcal{S}^{\prime}}(W^{*},\operatorname{or}(W))$.
Accounting for the action of $(-\operatorname{id})^{*}$ on
$\operatorname{or}(W)$, which is given by $(-1)^{m}$, while the action of
$(-\operatorname{id})^{*}$ appearing in the inversion formula is only on the
form and not on its values, we have
$(-\operatorname{id})^{*}i_{*}\mathcal{F}_{V^{*}}S=(-1)^{m}i_{*}\mathcal{F}_{V^{*}}(-\operatorname{id})^{*}S$.
This concludes the proof of the second identity. ∎

######  Corollary 5.6.

If $\omega\in\Omega_{\mathcal{S}^{\prime}}^{k}(V,F)$ is $r$-homogeneous, then
$\mathcal{F}\omega\in\Omega_{\mathcal{S}^{\prime}}^{n-k}(V^{*},F\otimes\operatorname{or}(V))$
is $(-r)$-homogeneous.

######  Lemma 5.7.

For a tempered generalized form
$\omega\in\Omega^{k}_{\mathcal{S}^{\prime}}(V)$ one has

$\mathcal{F}(d\omega)=(-1)^{k+1}2\pi\mathbf{i}\cdot
i_{E}(\mathcal{F}\omega),\qquad
d(\mathcal{F}\omega)=(-1)^{k+1}2\pi\mathbf{i}\cdot\mathcal{F}(i_{E}\omega).$

###### Proof.

This follows immediately from basic properties of the Fourier transform:
$\mathcal{F}(\partial_{j}f)=-2\pi\mathbf{i}\xi_{j}\mathcal{F}f$,
$\partial_{j}(\mathcal{F}f)=2\pi\mathbf{i}\mathcal{F}(x_{j}f)$. ∎

###  5.2. Fourier transform of valuation-type differential forms

The goal of this section is to show that valuation-type differential forms are
closed under the Fourier transform.

First, we consider closed forms.

######  Proposition 5.8.

Let $i:W\hookrightarrow V$ be a monomorphism, and let $p\colon V\to W$ be an
epimorphism. Put $n=\dim V$, $m=\dim W$. Let
$\omega\in\Omega^{k}_{-\infty}(V)$ be closed, $0$-homogeneous, smooth outside
the origin and satisfying $i_{E}\omega=0$. Then $\mathcal{F}\omega$ has the
same properties, and the identities

$\mathcal{F}_{W}i^{*}\omega=(i^{\vee})_{*}\mathcal{F}_{V}\omega\quad\text{and}\quad(p^{\vee})^{*}\mathcal{F}_{V}\omega=(-1)^{(n-m)(n-k)}\mathcal{F}_{W}p_{*}\omega$

hold.

###### Proof.

Observe first that $\mathcal{F}_{V}\omega$ is closed, $0$-homogeneous, smooth
outside the origin, and satisfies $i_{E}\mathcal{F}_{V}\omega=0$. Indeed, it
is $0$-homogeneous by Corollary 5.6, and it is smooth outside of the origin by
[lemoine_homogeneous, Theorem 3.2.4]. We have $d\mathcal{F}_{V}\omega=0$ and
$i_{E}\mathcal{F}_{V}\omega=0$ by Lemma 5.7. In particular, both sides of the
asserted equalities are well-defined.

Fix a Euclidean structure on $V$. For a compactly supported form
$\eta\in\Omega^{k}(W)$ we must show

$\langle\mathcal{F}_{W}i^{*}\omega,\eta\rangle=\langle(i^{\vee})_{*}\mathcal{F}_{V}\omega,\eta\rangle.$

Choose an approximate identity
$\mu_{\epsilon}(x)dx\in\mathcal{M}^{\infty}(\mathbb{R}^{n})$ with
$\mu_{\epsilon}$ Schwartz. Using Lemma 5.5 we find

$\displaystyle\langle\mathcal{F}i^{*}\omega,\eta\rangle$
$\displaystyle=(-1)^{k(m-k)}\langle
i^{*}\omega,\mathcal{F}\eta\rangle=(-1)^{k(m-k)}\lim_{\epsilon\to 0}\langle
i^{*}(\omega\ast\mu_{\epsilon}),\mathcal{F}\eta\rangle$
$\displaystyle=(-1)^{k(m-k)}\lim_{\epsilon\to
0}\langle\omega\ast\mu_{\epsilon},i_{*}\mathcal{F}\eta\rangle=(-1)^{k(m-k)}\lim_{\epsilon\to
0}\langle\omega,\mathcal{F}(i^{\vee})^{*}\eta\ast\mu_{\epsilon}\rangle$
$\displaystyle=(-1)^{k(m-k)}\lim_{\epsilon\to
0}\langle\omega,\mathcal{F}((i^{\vee})^{*}\eta\cdot\mathcal{F}^{-1}\mu_{\epsilon})\rangle$
$\displaystyle=(-1)^{k(m-k)+k(n-k)}\lim_{\epsilon\to
0}\langle\mathcal{F}\omega,(i^{\vee})^{*}\eta\cdot\mathcal{F}^{-1}\mu_{\epsilon}\rangle$
$\displaystyle=(-1)^{k(m-k)+k(n-k)+k(m-k)+k(n-k)}\langle(i^{\vee})_{*}(\mathcal{F}\omega),\eta\rangle,$

concluding the proof of the first identity. The second follows from first via
the inversion formula. ∎

####  5.2.1. Differential forms related to valuations

We now focus our attention on the forms in

$\mathcal{S}^{\prime}_{n-k,k}(V^{*})=\Omega_{\mathcal{S}^{\prime}}^{n-k}(V^{*},\wedge^{k}V^{*}\otimes\operatorname{or}(V))$

which play an important role in valuation theory. Given
$\omega\in\mathcal{S}^{\prime}_{n-k,k}(V^{*})$, we can consider its Fourier
transform

$\mathcal{F}\omega\in\Omega^{k}_{\mathcal{S}^{\prime}}(V,\wedge^{k}V^{*}).$

######  Definition 5.9.

Given $\omega\in\mathcal{S}^{\prime}_{n-k,k}(V^{*})$, we define its Fourier
transform

$\mathcal{F}^{0}\omega\in\mathcal{S}^{\prime}_{k,n-k}(V)\otimes\operatorname{Dens}(V),$

as the Fourier transform $\mathcal{F}\omega$ of Definition 5.1, combined with
the Hodge star
$*_{V^{*}}\colon\mathbb{\wedge}^{k}V\to\mathbb{\wedge}^{n-k}V^{*}\otimes\mathrm{Dens}(V^{*})\otimes\operatorname{or}(V)$
according to our convention (2).

The inversion formula for $\mathcal{F}^{0}$ assumes the following form.

######  Lemma 5.10.

It holds for all $\omega\in\mathcal{S}^{\prime}_{n-k,k}(V^{*})$ that

$(\mathcal{F}^{0}_{V}\times\operatorname{id})\circ\mathcal{F}^{0}_{V^{*}}(\omega)=(-\operatorname{id})^{*}\omega.$

###### Proof.

It follows from Lemma 5.4 that
$(\mathcal{F}_{V}\times\operatorname{id})\circ\mathcal{F}_{V^{*}}(\omega)=(-1)^{n(n-k)}(-\operatorname{id})^{*}$,
while the composition of the Hodge star with itself contributes a factor of
$(-1)^{k(n-k)}$. It remains to note that $(-\operatorname{id})^{*}$ acts also
on the space $\wedge^{k}V^{*}\otimes\operatorname{or}(V)$ by the factor
$(-1)^{k+n}$, and $(-1)^{n(n-k)}(-1)^{k(n-k)}(-1)^{k+n}=1$. ∎

Furthermore, $\mathcal{F}^{0}$ is compatible with exterior products.

######  Lemma 5.11.

If $V,W$ are vector spaces of dimensions $n,m$, then

$\mathcal{F}^{0}(\omega\boxtimes\zeta)=(-1)^{km+ln}\mathcal{F}^{0}\omega\boxtimes\mathcal{F}^{0}\zeta$

holds for all $\omega\in\mathcal{S}^{\prime}_{n-k,k}(V^{*})$ and
$\zeta\in\mathcal{S}^{\prime}_{m-l,l}(W^{*})$

###### Proof.

First observe that if $F_{1},F_{2}$ are vector spaces and

$\displaystyle\mathcal{F}_{V}\colon\Omega_{\mathcal{S}^{\prime}}^{n-k}(V^{*},F_{1})\to\Omega_{\mathcal{S}^{\prime}}^{k}(V,or(V)\otimes
F_{1})$
$\displaystyle\mathcal{F}_{W}\colon\Omega_{\mathcal{S}^{\prime}}^{m-l}(W^{*},F_{2})\to\Omega_{\mathcal{S}^{\prime}}^{l}(W,or(W)\otimes
F_{2})$ $\displaystyle\mathcal{F}_{V\times
W}\colon\Omega_{\mathcal{S}^{\prime}}^{n-k,m-l}(V^{*}\times W^{*},F_{1}\otimes
F_{2})\to\Omega_{\mathcal{S}^{\prime}}^{k,l}(V\times W,or(V\times W)\otimes
F_{1}\otimes F_{2}),$

denote the corresponding Fourier transforms, then

$\mathcal{F}_{V\times
W}=(-1)^{k(m-l)}\mathcal{F}_{V}\boxtimes\mathcal{F}_{W}.$

Let $I_{V}$, $I_{W}$, $I_{V\times W}$ be the respective Hodge star
isomorphisms as in eq. (6). Then

$I_{V\times W}=(-1)^{l(n-k)}I_{V}\boxtimes I_{W}.$

This completes the proof. ∎

####  5.2.2. Existence of the Fourier transform on valuations

We are now ready to prove the main result of this section.

######  Proposition 5.12.

Assume that
$\omega\in\Omega^{n-k}_{-\infty}(V^{*},\wedge^{k}V^{*}\otimes\operatorname{or}(V))$
is valuation-type, i.e. it has the following properties

  1. (1)

$0$-homogeneous.

  2. (2)

$i_{E}\omega=0$.

  3. (3)

vertical: $\omega\wedge E=0$.

  4. (4)

closed: $d\omega=0$.

Then
$\mathcal{F}^{0}\omega\in\Omega^{k}_{-\infty}(V,\wedge^{n-k}V\otimes\operatorname{or}(V^{*})\otimes\operatorname{Dens}(V))$
is valuation-type.

Furthermore, if $\omega$ is smooth outside of the origin, then so is
$\mathcal{F}^{0}\omega$.

###### Proof.

The case of $k=0$ is straightforward. Since $\omega\wedge E=0$, it follows
that $\omega$ is supported at the origin. As it is $0$-homogeneous, it must be
a multiple of the delta measure at the origin. Then $\mathcal{F}^{0}\omega$ is
constant on $V$, and so trivially valuation-type.

Assume $k\geq 1$. Using the identification
$\wedge^{k}V^{*}=\wedge^{n-k}V\otimes\wedge^{n}V^{*}$, we define the
generalized form
$\overline{\omega}\in\Omega^{n-k}_{-\infty}(V^{*},\wedge^{n-k}V\otimes\operatorname{Dens}(V))$
corresponding to $\omega$. It holds that $d\overline{\omega}=0$.

Now

$\eta:=\mathcal{F}^{0}\omega=\mathcal{F}\overline{\omega}\in\Omega^{k}_{-\infty}(V,\wedge^{n-k}V\otimes\operatorname{Dens}(V)\otimes\operatorname{or}(V^{*}))$

is $0$-homogeneous by Corollary 5.6. The identification
$\wedge^{k}V^{*}=\wedge^{n-k}V\otimes\operatorname{Dens}(V)\otimes\operatorname{or}(V^{*})$
gives a corresponding $0$-homogeneous generalized form
$\overline{\eta}\in\Omega^{k}_{-\infty}(V,\wedge^{k}V^{*})$.

We have by Lemma 5.7

$d\eta=d\mathcal{F}\overline{\omega}=(-1)^{n-k+1}2\pi\mathbf{i}\mathcal{F}(i_{E}\overline{\omega})=0,$

that is $\eta$ is closed, and hence so is $\overline{\eta}$. Similarly,
$i_{E}\overline{\eta}=i_{E}\mathcal{F}\omega=(-1)^{n-k}\frac{1}{2\pi\mathbf{i}}\mathcal{F}d\omega=0$.

The symmetry of $\overline{\omega}$ established in Proposition 4.9 immediately
implies the symmetry of $\overline{\eta}$. Combined with the identity
$i_{E}\overline{\eta}=0$, it follows that $\eta\wedge E=0$.

Finally, if $\omega$ is smooth on $V^{*}\setminus\\{0\\}$ then so is
$\overline{\omega}$, and by [lemoine_homogeneous, Theorem 3.2.4], $\eta$ is
smooth on $V\setminus\\{0\\}$.

∎

##  6\. Alesker–Fourier transform of translation-invariant valuations

Proposition 5.12 allows to define the $\mathcal{F}$-transform of valuations as
in Definition 1.3. More precisely:

######  Definition 6.1.

Let $V$ be an $n$-dimensional linear space, and $0\leq k\leq n$. The
$\mathcal{F}$-transform of a valuation
$\phi\in\operatorname{Val}^{-\infty}_{k}(V)$ is the unique valuation
$\mathcal{F}\phi\in\operatorname{Val}^{-\infty}_{n-k}(V^{*})$ such that
$\tau(\mathcal{F}\phi)=\mathcal{F}^{0}\tau(\phi)$.

It follows from Proposition 5.12 that $\mathcal{F}\phi$ is a smooth valuation
whenever $\phi$ is a smooth valuation.

We will prove below that the $\mathcal{F}$-transform of valuations thus
defined coincides with the Fourier transform constructed by Alesker in
[Alesker:Fourier].

######  Lemma 6.2.

For $i=1,2$ consider the pairings

$P_{i}\colon\operatorname{Val}^{\infty}(V)\otimes\operatorname{Dens}(V^{*})\otimes\operatorname{Dens}(V)\times\operatorname{Val}^{\infty}(V)\otimes\operatorname{Dens}(V^{*})\to\mathbb{C}$

defined by

$P_{1}(\phi\otimes\alpha\otimes
v,\psi\otimes\beta)=\langle\phi\otimes\alpha*\psi\otimes\beta(\\{0\\}),v\rangle$

and

$P_{2}(\phi\otimes\alpha\otimes
v,\psi\otimes\beta)=\langle\alpha,v\rangle\langle((-\operatorname{id})^{*}\phi\cdot\psi)_{n},\beta\rangle.$

Then $P_{1}=P_{2}$.

###### Proof.

Choose a Euclidean inner product to identify $V$ with $\mathbb{R}^{n}$ and
$\operatorname{Dens}(V)\simeq\mathbb{C}\simeq\operatorname{Dens}(V^{*})$.
Choose $S\subset\mathbb{R}^{n}$ with volume $1$. Then

$\displaystyle\phi*\psi(\\{0\\})$
$\displaystyle=a_{*}(\phi\boxtimes\psi)(\\{0\\})$
$\displaystyle=\frac{1}{n!}\left.\frac{\partial^{n}}{\partial
t^{n}}\right|_{t=0}\phi\boxtimes\psi((-\operatorname{id},\operatorname{id})\circ\Delta(tS))$
$\displaystyle=(\Delta^{*}((-\operatorname{id})^{*}\phi\boxtimes\psi)_{n}$
$\displaystyle=((-\operatorname{id})^{*}\phi\cdot\psi)_{n}.$

∎

We are now ready to prove Theorem 1.4, which we restate below.

######  Theorem 6.3.

The $\mathcal{F}$-transform of valuations has the following properties:

  1. (1)

$\mathcal{F}$ commutes with the natural action of $\mathrm{GL}(V)$.

  2. (2)

Inversion formula:
$(\mathcal{F}_{V^{*}}\times\operatorname{id})\circ\mathcal{F}_{V}\phi=(-\operatorname{id})^{*}\phi$.

  3. (3)

Let $i:V\hookrightarrow W$ be an injective linear map, and
$p=i^{\vee}:W^{*}\to V^{*}$. Then for $\phi\in\operatorname{Val}^{\infty}(W)$,
$\mathcal{F}_{V}(i^{*}\phi)=p_{*}(\mathcal{F}_{W}\phi)$.

  4. (4)

$\mathcal{F}(\phi\boxtimes\psi)=\mathcal{F}\phi\boxtimes\mathcal{F}\psi$ for
$\phi\in\operatorname{Val}^{\infty}(V)$,
$\psi\in\operatorname{Val}^{\infty}(W)$.

  5. (5)

$\mathcal{F}$ intertwines product and convolution: for
$\phi,\psi\in\operatorname{Val}^{\infty}(V)$,
$\mathcal{F}(\phi\cdot\psi)=\mathcal{F}\phi\ast\mathcal{F}\psi$.

  6. (6)

Self-adjointness: $\langle\mathcal{F}u,\theta\rangle=\langle
u,\mathcal{F}\theta\rangle$ for $u\in\operatorname{Val}^{-\infty}(V)$,
$\theta\in\operatorname{Val}^{\infty}(V^{*})$.

  7. (7)

Let $p:V\to W$ be a surjective linear map, and $i=p^{\vee}:W^{*}\to V^{*}$.
Then for $\psi\in\operatorname{Val}^{-\infty}(W)$,
$\mathcal{F}_{V}(p^{*}\psi)=i_{*}(\mathcal{F}_{W}\psi)$.

###### Proof.

  1. (1)

This is clear by construction.

  2. (2)

Follows at once from Lemma 5.10.

  3. (3)

Follows from Propositions 4.5, 4.6 and 5.8. The first two propositions assert
that $i^{*}\phi$ corresponds to the pushforward under $p$ of the
$0$-homogeneous current of the valuation, while $p_{*}(\mathcal{F}_{W}\phi)$
is given by the restriction (using $i$) of the respective current. The last
proposition asserts the equality of the two currents.

  4. (4)

Follows from Proposition 4.7 and Lemma 5.11.

  5. (5)

Observe that $\Delta_{V}^{\vee}=a_{V^{*}}$, while
$\mathcal{F}(\phi\boxtimes\psi)=\mathcal{F}\phi\boxtimes\mathcal{F}\psi$ by
(2). Let $\mu_{j}\in\mathcal{M}^{\infty}_{c}(\operatorname{GL}(V\times V))$ be
an approximate identity such that $\mu_{j}\to\delta_{\operatorname{id}}$ as
$j\to\infty$. Denoting
$\zeta=\phi\boxtimes\psi\in\operatorname{Val}^{-\infty}(V\times V)$,
$\zeta_{j}=\zeta\ast\mu_{j}\in\operatorname{Val}^{\infty}(V\times V)$, we have

$\displaystyle\mathcal{F}(\phi\cdot\psi)$
$\displaystyle=\mathcal{F}(\Delta^{*}(\zeta))=\lim_{j\to\infty}\mathcal{F}(\Delta^{*}(\zeta_{j}))=\lim_{j\to\infty}a_{*}\mathcal{F}(\zeta_{j})$
$\displaystyle=\lim_{j\to\infty}a_{*}(\mathcal{F}(\zeta)\ast\mu_{j})=a_{*}(\mathcal{F}(\zeta))=\mathcal{F}(\phi)\ast\mathcal{F}(\psi),$

where the second equality follows from the sequential continuity of
$\Delta^{*}$ in the Hörmander topology [Duistermaat:FIO] of a subspace of
valuations with restricted wavefront to which $\zeta$ belongs, and in which
$\zeta_{j}$ converges, the third equality is property (3), the fourth equality
follows from the equivariance of the $\mathcal{F}$-transform on valuations
with respect to the general linear group, and the fifth equality follows from
the sequential continuity of $a_{*}$ in the Hörmander topology, similarly to
$\Delta^{*}$.

  6. (6)

By continuity, it is enough to prove the claim for
$u\in\operatorname{Val}^{\infty}(V)\subset\operatorname{Val}^{-\infty}(V)$.

Let
$L_{0}\colon\operatorname{Val}^{\infty}(V^{*})\otimes\operatorname{Dens}(V)\otimes\operatorname{Dens}(V^{*})\to\mathbb{C}$
be defined by composition of evaluation at a point with the identification
$\operatorname{Dens}(V)\otimes\operatorname{Dens}(V^{*})\simeq\mathbb{C}$.
Similarly, let
$L_{n}\colon\operatorname{Val}^{\infty}(V)\otimes\operatorname{Dens}(V^{*})\to\mathbb{C}$
be the projection to the top-degree component, which is just
$\operatorname{Dens}(V)$, followed by the same identification. Using
$\mathcal{F}\chi=\operatorname{vol}\otimes\operatorname{vol}^{*}$, the
homomorphism property (5), Lemma 6.2, and the inversion formula (2), we obtain

$\displaystyle\langle\mathcal{F}_{V}u,\theta\rangle$
$\displaystyle=L_{n}(\mathcal{F}_{V}u\cdot\theta)$
$\displaystyle=L_{0}(\mathcal{F}_{V^{*}}\times\operatorname{id}(\mathcal{F}_{V}u\cdot\theta))$
$\displaystyle=L_{0}((\mathcal{F}_{V^{*}}\times\operatorname{id})\circ\mathcal{F}_{V}u)*\mathcal{F}_{V^{*}}\theta)$
$\displaystyle=P_{1}((\mathcal{F}_{V^{*}}\times\operatorname{id})\circ\mathcal{F}_{V}u,\mathcal{F}_{V^{*}}\theta)$
$\displaystyle=P_{2}((\mathcal{F}_{V^{*}}\times\operatorname{id})\circ\mathcal{F}_{V}u,\mathcal{F}_{V^{*}}\theta)$
$\displaystyle=L_{n}(u\cdot\mathcal{F}_{V^{*}}\theta)$ $\displaystyle=\langle
u,\mathcal{F}_{V^{*}}\theta\rangle,$

as claimed.

  7. (7)

For smooth valuations, this follows from Propositions 4.5 and 4.6 and Lemma
5.5. The general case then follows by continuity.

∎

It remains to show that our $\mathcal{F}$\- and Alesker’s Fourier transforms
coincide. Since this verification will be reduced to the two-dimensional case,
we will use the following statement about valuations in the plane.

######  Lemma 6.4.

Suppose that $\phi\in\operatorname{Val}^{-\infty}_{1}(\mathbb{R}^{2})$ is a
generalized valuation, and $g(e^{i\theta})d\theta$ a generalized measure such
that

$\phi(K)=\int_{0}^{2\pi}h_{K}(e^{\mathbf{i}\theta})g(e^{i\theta})d\theta$

for every smooth and strictly positively curved convex body $K$ in
$\mathbb{R}^{2}$. Then

$\tau(\phi)=\frac{g(-y/|y|)}{|y|^{3}}(y_{1}dx_{1}+y_{2}dx_{2})\otimes(y_{1}dy_{2}-y_{2}dy_{1}),$

where $(x,y)$ are the coordinates on
$\mathbb{R}^{2}\times\mathbb{R}^{2}\supset\mathbb{R}^{2}\times S^{1}$.

###### Proof.

If $\psi=\int_{\operatorname{nc}(K)}\omega$, where
$\omega=f_{1}(y)dx_{1}+f_{2}(y)dx_{2}\in\Omega^{1}(S\mathbb{R}^{2})$, then a
computation shows that

$\psi(K)=\int_{S^{1}}(-f_{1}(y)y_{2}+f_{2}(y)y_{1}\,)dS_{1}(K,y)=2V(K,-f_{1}y_{2}+f_{2}y_{1}).$

The definition of $\phi$ implies

$\phi\cdot\psi=\int_{S^{1}}h(e^{\mathbf{i}\theta})g(-e^{i\theta})d\theta$

for valuations of the form $\psi(K)=2V(K,h)$. Hence

$T(\phi)=(y_{1}dx_{1}+y_{2}dx_{2})\otimes g(-e^{i\theta})d\theta,$

from which the expression for $\tau(\phi)$ follows. ∎

###### Proof of Theorem 1.5.

The Alesker construction of the Fourier transform strongly relies on the
irreducibility theorem [Alesker:Irreducibility], and so we make use of it for
the verification. Denote the Alesker–Fourier transform by $\mathbb{F}$.
Observe that property (4) of Theorem 1.4 holds for both definitions, while
Alesker products of $k$-tuples of elements of
$\operatorname{Val}_{1}^{\infty}(V)$ span a dense subset of
$\operatorname{Val}_{k}^{\infty}(V)$ by the irreducibility theorem. It
therefore suffices to verify that $\mathcal{F}=\mathbb{F}$ on
$\operatorname{Val}_{1}^{\infty}(V)$. Furthermore, since
$\mathcal{F},\mathbb{F}:\operatorname{Val}_{1}^{\pm,\infty}(V)\to\operatorname{Val}_{n-1}^{\pm,\infty}(V^{*})\otimes\operatorname{Dens}(V)$
are both equivariant isomorphisms between irreducible
$\operatorname{GL}_{n}(\mathbb{R})$-modules, there exist constants
$c_{\pm}(V)$ such that $\mathcal{F}=c_{\pm}(V)\mathbb{F}$. Since by Theorem
1.4 and [Alesker:Fourier] both $\mathcal{F}$ and $\mathbb{F}$ intertwine
restrictions and pushforwards by projections, the number
$c_{\pm}(V)=c_{\pm}(n)$ depends only on the dimension of $V$. By the same
token, fixing an inclusion $i:\mathbb{R}^{k}\hookrightarrow\mathbb{R}^{n}$
shows that $c_{\pm}$ is independent of the dimension. Finally, the inversion
formulas of Theorem 1.4 and [Alesker:Fourier] imply that
$c_{+}^{2}=c_{-}^{2}=1$.

To determine the value of $c_{+}$, it suffices to note that on
$\mathbb{R}^{1}$,
$\mathcal{F}(\operatorname{vol}_{1})=\chi=\mathbb{F}(\operatorname{vol}_{1})$,
that is $c_{+}=1$.

Now consider $\mathbb{R}^{2}=\mathbb{C}$, and define
$\phi\in\operatorname{Val}_{1}(\mathbb{R}^{2})$ by

$\phi(K)=h_{K}(1)+h_{K}(e^{2\pi\mathbf{i}/3})+h_{K}(e^{4\pi\mathbf{i}/3}),$

which is a continuous, and therefore generalized, translation-invariant
valuation. Then $\psi(K):=\phi(K)-\phi(-K)$ lies in
$\operatorname{Val}_{1}^{-,\infty}(\mathbb{R}^{2})$.

According to Lemma 6.4 and denoting
$\omega=\operatorname{sign}(y_{1})\boxtimes\delta_{0}(y_{2})dy_{2}$, the
$0$-homogeneous current of $\phi$ is

$\tau(\psi)=dx_{1}\otimes\omega+(e^{2\pi\mathbf{i}/3})^{*}(dx_{1}\otimes\omega)+(e^{-2\pi\mathbf{i}/3})^{*}(dx_{1}\otimes\omega).$

We have

$\mathcal{F}^{0}(dx_{1}\otimes\omega)=d\xi_{2}\otimes\mathcal{F}(\omega),$

where

$\mathcal{F}(\omega)(\eta_{1},\eta_{2})=\frac{1}{\pi\mathbf{i}\eta_{1}}\boxtimes
1\,d\eta_{1}.$

As $\mathcal{F}$ commutes with rotations, a straightforward computation now
shows that

$\displaystyle\tau(\mathcal{F}\psi)=\frac{3}{\pi\mathbf{i}\eta_{1}(\eta_{1}^{2}-3\eta_{2}^{2})}(\eta_{1}d\xi_{1}+\eta_{2}d\xi_{2})\otimes(\eta_{1}d\eta_{2}-\eta_{2}d\eta_{1}),$

where $(\xi,\eta)$ are the coordinates on
$\mathbb{R}^{2}\times\mathbb{R}^{2}\supset\mathbb{R}^{2}\times S^{1}$.

Let us recall from [alesker2013fourier] the description of the Alesker–Fourier
transform in $\mathbb{R}^{2}=\mathbb{C}$. For

$\zeta(K)=\int_{0}^{2\pi}h_{K}(e^{\mathbf{i}\theta})f(\theta)d\theta,$

write $f=f_{+}+f_{-}$, $f_{-}=f_{-}^{hol}+f_{-}^{anti}$, where $f_{+},f_{-}$
are the even and odd parts of $f$ on the circle,
$f_{-}^{hol}(\theta)=\sum_{n\equiv
1(2),n>0}\widehat{f}(n)e^{\mathbf{i}n\theta}$ and
$f_{-}^{anti}(\theta)=\sum_{n\equiv
1(2),n<0}\widehat{f}(n)e^{\mathbf{i}n\theta}$ the holomorphic and anti-
holomorphic parts of $f_{-}$, respectively. Then

$\mathbb{F}\zeta(K)=\int_{0}^{2\pi}h_{K}(e^{\mathbf{i}\theta})\tilde{f}(\theta)d\theta,$

where

$\displaystyle\tilde{f}(\theta)$
$\displaystyle=f_{+}(\theta+\frac{\pi}{2})+f_{-}^{hol}(\theta+\frac{\pi}{2})-f_{-}^{anti}(\theta+\frac{\pi}{2})=\sum\mathbf{i}^{|n|}\widehat{f}(n)e^{\mathbf{i}n\theta}.$

Now $\psi$ is given by
$\psi(K)=\int_{0}^{2\pi}h_{K}(e^{\mathbf{i}\theta})f_{\psi}(\theta)d\theta$
with

$f_{\psi}(\theta)=\frac{1}{2\pi}(\delta_{0}(\theta)+\delta_{2\pi/3}(\theta)+\delta_{4\pi/3}(\theta)-\delta_{\pi}(\theta)-\delta_{\pi/3}(\theta)-\delta_{5\pi/3}(\theta)).$

The Fourier series of $\delta_{0}(\theta)$ is

$\delta_{0}(\theta)=\sum_{n=-\infty}^{\infty}e^{\mathbf{i}n\theta},$

implying that

$f_{\psi}(\theta)=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}(3e^{3\mathbf{i}n\theta}-3e^{3\mathbf{i}n(\theta+\pi)}).$

It follows that $\mathbb{F}\psi$ is given by

$\mathbb{F}\psi(K)=\int_{0}^{2\pi}h_{K}(e^{\mathbf{i}\theta})g_{\psi}(\theta)d\theta$

with

$\displaystyle g_{\psi}(\theta)$
$\displaystyle=\frac{1}{2\pi}\sum_{n=-\infty}^{\infty}(3\mathbf{i}^{3|n|}e^{3\mathbf{i}n\theta}-3\mathbf{i}^{3|n|}e^{3\mathbf{i}n(\theta+\pi)})=\frac{3}{\pi\mathbf{i}\cos
3\theta}.$

In view of Lemma 6.4 it remains to notice that on the unit circle,
$\eta_{1}(\eta_{1}^{2}-3\eta_{2}^{2})=4\cos^{3}\theta-3\cos\theta=\cos
3\theta$.

∎

###  6.1. Example: even valuations

The Fourier transform on even valuations, acting on the Klain section or the
Crofton measure of a valuation, amounts to the action of the orthogonal
complement on the Grassmannian. Let us recover this fact using our
construction.

We use the standard Euclidean structure and orientation on $\mathbb{R}^{n}$.
For $E\in\operatorname{Gr}_{k}(\mathbb{R}^{n})$, denote
$\phi_{E}(K)=\operatorname{vol}(P_{E^{\perp}}K)$, where $P_{E^{\perp}}$ is the
orthogonal projection onto $E^{\perp}$. Choose orthonormal coordinates on
$\mathbb{R}^{n}$ such that $E=\\{x_{k+1}=\dots=x_{n}=0\\}$. In view of Lemma
4.4, equation (8), we have

$\tau(\phi_{E})=dx_{1}\wedge\dots\wedge
dx_{k}\otimes\delta_{0}(y_{k+1})\cdots\delta_{0}(y_{n})dy_{k+1}\wedge\dots\wedge
dy_{n}.$

As

$\mathcal{F}(\delta_{0}(y_{k+1})\cdots\delta_{0}(y_{n})dy_{k+1}\wedge\dots\wedge
dy_{n})=(-1)^{k(n-k)}\delta_{0}(y_{1})\cdots\delta_{0}(y_{k})dy_{1}\wedge\dots\wedge
dy_{k},$

we find that

$\displaystyle\tau(\mathcal{F}(\phi_{E}))$
$\displaystyle=\mathcal{F}^{0}\tau(\phi_{E})$
$\displaystyle=(-1)^{k(n-k)}\ast(dx_{1}\wedge\dots\wedge
dx_{k})\otimes\delta_{0}(y_{1})\cdots\delta_{0}(y_{k})dy_{1}\wedge\dots\wedge
dy_{k}$ $\displaystyle=\tau(\phi_{E^{\perp}}).$

Now for any valuation given by the Crofton formula
$\phi(K)=\int_{\operatorname{Gr}_{k}(\mathbb{R}^{n})}\phi_{E}(K)dm(E)$, we
deduce by linearity and continuity of the Fourier transform that
$\mathcal{F}\phi(K)=\int_{\operatorname{Gr}_{k}(\mathbb{R}^{n})}\phi_{E^{\perp}}(K)dm(E),$
that is
$\mathcal{F}\phi(K)=\int_{\operatorname{Gr}_{n-k}(\mathbb{R}^{n})}\phi_{F}(K)dm^{\prime}(F)$,
where $m^{\prime}=\perp_{*}(m)$.

###  6.2. Example: intrinsic volumes

For the purpose of illustration, let us compute the Fourier transform of the
intrinsic volumes $V_{k}$ directly from our definition.

######  Lemma 6.5.

For $k=1,\ldots,n-1$,

$\tau(V_{k})=di_{E}\lambda_{k},$

where

$\lambda_{k}=\frac{1}{k!(n-k)!\operatorname{vol}(S^{n-k-1})}\frac{1}{|y|^{n-k}}\sum_{\pi}\operatorname{sign}(\pi)dx_{\pi_{1}}\cdots
dx_{\pi_{k}}dy_{\pi_{k+1}}\cdots dy_{\pi_{n}}.$

###### Proof.

Integration over the normal cycle of

$\kappa_{k}=c_{k,n}\sum_{\pi}\operatorname{sign}(\pi)y_{\pi_{1}}dx_{\pi_{2}}\cdots
dx_{\pi_{k+1}}dy_{\pi_{k+2}}\cdots dy_{\pi_{n}},$

where $c_{k,n}=\frac{1}{k!(n-k)!\omega_{n-k}}$ and the sum is over the
permutation group, yields the $k$th intrinsic volume. Note that

$d\kappa_{k}=c_{k,n}\sum_{\pi}\operatorname{sign}(\pi)dx_{\pi_{1}}\cdots
dx_{\pi_{k}}dy_{\pi_{k+1}}\cdots
dy_{\pi_{n}}=(-1)^{n-k}a^{*}D\kappa_{k}=T(V_{k}).$

We have to pull this form back to $\mathbb{R}^{n}\setminus\\{0\\}$. First
observe that since $j\circ\pi(y)=y/|y|$, where $j\colon
S^{n-1}\to\mathbb{R}^{n}$ denotes the inclusion and $\pi$ is the radial
projection, we have

$\pi^{*}j^{*}dy_{i}=d(y_{i}/|y|)=\frac{1}{|y|}dy_{i}-\frac{y_{i}}{|y|}\rho$

where $\rho=|y|^{-2}\sum_{j=1}^{n}y_{j}dy_{j}=\frac{1}{|y|}d(|y|)$. Thus

$\displaystyle\pi^{*}j^{*}(dy_{\pi_{k+1}}\cdots dy_{\pi_{n}})$
$\displaystyle=\frac{1}{|y|^{n-k}}dy_{\pi_{k+1}}\cdots
dy_{\pi_{n}}-\frac{1}{|y|^{n-k}}\rho\wedge\sum_{j=1}^{k}y_{\pi_{j}}i_{\partial_{y_{\pi_{j}}}}dy_{\pi_{k+1}}\cdots
dy_{\pi_{n}}$ $\displaystyle=\frac{1}{|y|^{n-k}}dy_{\pi_{k+1}}\cdots
dy_{\pi_{n}}-\frac{1}{|y|^{n-k}}\rho\wedge i_{E}dy_{\pi_{k+1}}\cdots
dy_{\pi_{n}}$ $\displaystyle=\frac{1}{|y|^{n-k}}i_{E}(\rho\wedge
dy_{\pi_{k+1}}\cdots dy_{\pi_{n}})$
$\displaystyle=-\frac{1}{n-k}i_{E}d(|y|^{-(n-k)}dy_{\pi_{k+1}}\cdots
dy_{\pi_{n}}).$

Since $i_{E}d\lambda_{k}=-di_{E}\lambda_{k}$ the claim follows. ∎

According to [Koldobsky:FourierAnalysis, Lemma 3.6],

$\mathcal{F}(|x|^{-n+k})(\xi)=\frac{\operatorname{vol}(S^{n-k-1})}{\operatorname{vol}(S^{k-1})}\frac{1}{|\xi|^{k}}$

and hence

$\mathcal{F}^{0}\lambda_{k}=\lambda_{n-k}.$

Using Lemmas 6.5 and 5.7 we conclude

$\mathcal{F}^{0}\tau(V_{k})=\mathcal{F}^{0}(di_{E}\lambda_{k})=-i_{E}d\mathcal{F}^{0}\lambda_{k}=-i_{E}d\lambda_{n-k}=di_{E}\lambda_{n-k}=\tau(V_{n-k}),$

that is, $\mathcal{F}V_{k}=V_{n-k}$.

## References