# On the LS-category of homomorphisms of groups with torsion

Nursultan Kuanyshov  Nursultan Kuanyshov, Department of Mathematics,
University of Florida, 358 Little Hall, Gainesville, FL 32611-8105, USA
<EMAIL_ADDRESS>
###### Abstract.

We prove the equality $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)$ for
homomorphisms $\phi:\Gamma\to\Lambda$ of any finitely generated abelian group
$\Gamma$.

In addition, we prove that the Lusternik-Schnirelmann category and the
cohomological dimension of any nonzero homomorphism of a torsion group cannot
be finite.

###### Key words and phrases:

Cohomological dimension, group homomorphism

###### 2010 Mathematics Subject Classification:

Primary 20J06; Secondary 20K45, 20K30, 20K10

##  1\. Introduction

The (reduced) Lusternik-Schnirelmann category (for short, LS-category),
$\operatorname{cat}(X)$, of a topological space $X$ is the minimal number $k$
such that there is an open cover $\\{U_{0},U_{1},\dots,U_{k}\\}$ of $X$ by
$k+1$ contractible in $X$. The LS-category gives a lower bound on the number
of critical points for a smooth real-valued function on a closed manifold
[LS], [CLOT].

Since it is a homotopy invariant, it can be defined for discrete groups
$\Gamma$ as $\operatorname{cat}(\Gamma)=\operatorname{cat}(B\Gamma)$ where
$B\Gamma=K(\Gamma,1)$ is a classifying space. Computation of the LS-category
of spaces presents a great challenge even spaces are nice such as manifolds
[DS]. In the 50s Eilenberg and Ganea [EG] proved that the LS-category of a
discrete group equals its cohomological dimension,
$\operatorname{cat}(\pi)=\operatorname{cd}(\pi)$. We recall that the
cohomological dimension of a group $\Gamma$ is defined as follows,

$\operatorname{cd}(\Gamma)=\sup\\{k\mid H^{k}(\Gamma,M)\neq 0\\}$

where $M$ is some ${\mathbb{Z}}\Gamma$-module [Br]. From the work of
Dranishnikov and Rudyak [DR] it follows that

$\operatorname{cd}(\Gamma)=\max\\{k\mid(\beta_{\Gamma})^{k}\neq 0\\}$

where $\beta_{\Gamma}\in H^{1}(\Gamma,I(\Gamma))$ is the Berstein-Schwarz
class of $\Gamma$ [Be].

The LS-category of the map $f:X\to Y$, $\operatorname{cat}(f)$, is the minimal
number $k$ such that X admits an open cover by $k+1$ open sets
$U_{0},U_{2},...,U_{k}$ with nullhomotopic restrictions $f|_{U_{i}}:U_{i}\to
Y$ for all $i$. The LS-category $\operatorname{cat}(\phi)$ of a group
homomorphism $\phi:\Gamma\to\pi$ is defined as $\operatorname{cat}(f)$ where
the map $f:B\Gamma\to B\pi$ induces the homomorphism $\phi$ for the
fundamental groups.

The cohomological dimension $\operatorname{cd}(\phi)$ of a group homomorphism
$\phi:\Gamma\to\Lambda$ was introduced by Mark Grant [Gr] as the maximum of
$k$ such that there is a ${\mathbb{Z}}\Lambda$-module $M$ with the nonzero
induced homomorphism $\phi^{*}:H^{k}(\Lambda,M)\to H^{k}(\Gamma,M)$. In view
of universality of the Berstein-Schwarz class [DR], for any homomorphism
$\phi:\Gamma\to\Lambda$

$\operatorname{cd}(\phi)=\max\\{k\mid\phi^{*}(\beta_{\Lambda})^{k}\neq 0\\}.$

This together with the cup-length lower bound for the LS-category brings the
inequality $\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)$ for all group
homomorphisms.

In view of the Eilenberg-Ganea equality
$\operatorname{cd}(\Gamma)=\operatorname{cat}(\Gamma)$, the following
conjecture seems to be natural:

######  Conjecture 1.1.

For any group homomorphism $\phi:\Gamma\to\Lambda$ always
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)$.

In [Sc] Jamie Scott considered this conjecture for geometrically finite groups
and he proved it for monomorphisms of any groups and for homomorphisms of free
and free abelian groups. In [Gr] Tom Goodwillie gave an example of an
epimorphism of an infinitely generated group $\phi:G\to{\mathbb{Z}}^{2}$ with
$\operatorname{cd}(\phi)=1$ that disproves the conjecture.

In the joint paper with Dranishnikov [DK] we reduced the conjecture from
arbitrary homomorphisms to epimorphisms and we gave a counterexample to the
conjecture with epimorphism between geometrically finite groups. Also, we
proved the conjecture for epimorphisms between finitely generated, torsion-
free nilpotent groups. It is a natural question to ask if one can remove the
torsion-free restriction in our result. In this paper we do it in the cases
abelian groups.

######  Theorem 1.2 (Theorem 4.4).

Let $\phi:\Gamma\to\Lambda$ be any homomorphism between finitely generated
abelian groups. Then $\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$

Moreover, in the above case the number $\operatorname{cat}(\phi)$ can
explicitely computed. For that one needs to present the above homomorphism as
a direct sum $\phi=\phi_{1}\oplus\phi_{2}$ where $\phi_{1}$ is a homomorphism
between free abelian groups and $\phi_{2}$ is from free abelian to finite
abelian group $T(\Lambda)=Torsion(\Lambda)$. Then
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=rank(\phi_{1})+k(T(\Lambda)),$
where $k(T(\Lambda))$ is the Smith Normal number of $T(\Lambda)$.

Dealing with torsion we investigated a satellite question whether it is
possible to have a homomorphism of finite groups $\phi:\Gamma\to\Lambda$ with
$\operatorname{cd}(\phi)<\infty.$ We give a negative answer to this question
in the following theorem.

######  Theorem 1.3 (Theorem 3.2).

Let $\phi:G\to H$ be a nonzero homomorphism of a torsion group $G.$ Then
$\operatorname{cd}(\phi)=\infty.$ In particular,
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$

##  2\. Preliminaries

In this section we recall some classic theorems used in the paper.

Given positive integers $m,n\geq 1,$ we denote by $M_{m\times
n}({\mathbb{Z}})$the set of $m\times n$ matrices with integer entries.

######  Theorem 2.1 (The Smith Normal Form).

Given a nonzero matrix $A\in M_{m\times n}({\mathbb{Z}})$, there exist
invertible matrices $P\in M_{m\times m}({\mathbb{Z}})$ and $Q\in M_{n\times
n}({\mathbb{Z}})$ such that

$PAQ=\left[{\begin{array}[]{ccccccc}n_{1}&0&\cdots&0&0&\cdots&0\\\
0&n_{2}&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&n_{k}&0&\cdots&0\\\ 0&0&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&0&0&\cdots&0\\\
\end{array}}\right]$

where the integer $n_{i}\geq 1$ are unique up to sign and satisfy
$n_{1}|n_{2}|\cdots|n_{k}.$ Further, one can compute the integers $n_{i}$ by
the recursive formula $n_{i}=\dfrac{d_{i}}{d_{i-1}},$ where $d_{i}$ is the
greatest common divisor of all $i\times i$-minors of the matrix A and $d_{0}$
is defined to be 1.

The proof of Theorem 2.1 can be found in [Hu][Proposition 2.11, p.339].

######  Corollary 2.2.

Given a matrix $A\in M_{n\times n}({\mathbb{Z}})$ with $det(A)\neq 0,$ there
exist invertible matrices $P\in M_{n\times n}({\mathbb{Z}})$ and $Q\in
M_{n\times n}({\mathbb{Z}})$ such that

$PAQ=\left[{\begin{array}[]{ccccccc}1&0&\cdots&0&0&\cdots&0\\\
0&1&\cdots&0&0&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&1&0&\cdots&0\\\ 0&0&\cdots&0&n_{1}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&0&0&\cdots&n_{k}\\\ \end{array}}\right]$

where the integer $n_{i}\geq 2$ are unique up to sign and satisfy
$n_{1}|n_{2}|\cdots|n_{k}.$

######  Theorem 2.3 (Invariant Factor Decomposition (IDF) for Finite Abelian
Groups).

Every finite abelian group G can be written uniquely as
$G=Z_{n_{1}}\times...\times Z_{n_{k}}$ where the integers $n_{i}\geq 2$ are
the invariant factors of G that satisfy $n_{1}|n_{2}|...|n_{k}.$

The proof of Theorem 2.3 can be found in [DF][Theorem 3, p.158].
Alternatively, one can apply Corollary 2.2 and get the result.

######  Definition 2.4.

Given a finite abelian group $G$, the Smith Normal number $k(G)$ of $G$ is the
number $k$ from Theorem 2.3.

In the proof of our main result about homomorphism between finite groups we
use Shapiro’s Lemma [Br][Proposition 6.2, page 73].

######  Theorem 2.5 (“Shapiro’s Lemma”).

If $i:H\to G$ is an monomorphism and $M$ is an $H$-module, then the through
homomorphism

$H^{*}(G,Coind_{H}^{G}M)\stackrel{{\scriptstyle
i^{*}}}{{\to}}H^{*}(H,Coind_{H}^{G}M)\stackrel{{\scriptstyle\alpha_{*}}}{{\to}}H^{*}(H,M)$

is an isomorphism, where $Coind_{H}^{G}M=Hom_{{\mathbb{Z}}H}({\mathbb{Z}}G,M)$
and the homomorphism of coefficients
$\alpha:Hom_{{\mathbb{Z}}H}({\mathbb{Z}}G,M)\to M$ is defined as
$\alpha(f)=f(e)$.

In the paper we use the notation $H^{*}(\Gamma,A)$ for the cohomology of a
group $\Gamma$ with coefficient in a $\Gamma$-module $A$. The cohomology
groups of a space $X$ with the fundamental group $\Gamma$ we denote as
$H^{*}(X;A)$. Thus, $H^{*}(\Gamma,A)=H^{*}(B\Gamma;A)$ where
$B\Gamma=K(\Gamma,1)$.

We say that a CW-complex $X$ is of finite type if each $n$-skeleton $X^{(n)}$
of $X$ is finite.

The Kunneth Formula Theorem for cohomology of product of two spaces with field
coefficient $F$ can be founded in Spanier [Sp][Theorem 5.5.11] or Dold
[Do][Proposition VI.12.16]. We state a special which will be used in the
paper.

######  Theorem 2.6 (The Kunneth Formula Theorem).

Let $F$ be a field. If the CW-complexes X, Y are of finite type, then the
cross product

$\underset{k}{\bigoplus}H^{k}(X;F)\otimes
H^{n-k}(Y;F)\overset{\times}{\to}H^{n}(X\times Y;F)$

is an isomorphism.

We recall the Universal Coefficient Formula (UCF) for cohomology (see
[Sp][Theorem 5.5.10]).

UCF: Let the CW-complex X be of finite type and a coefficient group $G,$ then
for each n there is the short exact sequence

$0\to H^{n}(X)\otimes G\to H^{n}(X;G)\to Tor(H^{n+1}(X),G)\to 0,$

which splits.

##  3\. Finite Groups

A group $G$ is called a torsion group if every element $g\in G$ has finite
order.

For the proof of the main result of this section (Theorem 3.2) we need the
following easy lemma.

######  Lemma 3.1.

Let $\phi:G\to{\mathbb{Z}}_{p}$ be an epimorphism where $p$ is prime and $G$
is a torsion group. Then $G$ contains ${\mathbb{Z}}_{p^{k}}$ for some k such
that the restriction
$\psi=\phi|_{{\mathbb{Z}}_{p^{k}}}:{\mathbb{Z}}_{p^{k}}\to{\mathbb{Z}}_{p}$ is
surjective.

###### Proof.

Let $a$ be a generator of ${\mathbb{Z}}_{p}.$ We pick $g\in G$ with
$\phi(g)=a$. Then $g$ has a finite order $n.$ Since $g^{n}=0,$ it follows that
$\phi(g)^{n}=na=0.$ Hence, $n$ is divisible by $p$. Let $n=p^{k}m$ where $m$
is not divisible by $p.$ Then the order of element $g^{m}$ is $p^{k}$. Thus,
$g^{m}$ generates a subgroup ${\mathbb{Z}}_{p^{k}}\subset G$. Since
$\phi(g^{m})=ma\neq 0$ in ${\mathbb{Z}}_{p},$ the element $ma$ generates
${\mathbb{Z}}_{p}.$ Therefore, $\psi$ is surjective,
$\phi({\mathbb{Z}}_{p^{k}})={\mathbb{Z}}_{p}.$

∎

######  Theorem 3.2.

Let $\phi:G\to H$ be a nonzero homomorphism of a torsion group $G.$ Then
$cat(\phi)=\operatorname{cd}(\phi)=\infty.$

###### Proof.

In view of Theorem 3.2 and Theorem 3.3 [DK] we may assume that $\phi$ is an
epimorphism.

We do the proof in two steps:

Step 1. Suppose that $H$ is a cyclic group of order $p,$ say
$H\cong{\mathbb{Z}}_{p},$ where $p$ is prime. Then by Lemma 3.1 the group $G$
contains ${\mathbb{Z}}_{p^{k}}$ for some $k$ such that
$\psi=\phi|_{{\mathbb{Z}}_{p^{k}}}$ maps ${\mathbb{Z}}_{p^{k}}$ onto
${\mathbb{Z}}_{p}$.

We claim that $\operatorname{cd}(\psi)=\infty$ with ${\mathbb{Z}}$
coefficient. It suffices to show
$\psi^{*}:H^{n}(B{\mathbb{Z}}_{p};{\mathbb{Z}})\to
H^{n}(B{\mathbb{Z}}_{p^{k}};{\mathbb{Z}})$ is not trivial for all even numbers
$n$. The reduced integral cohomology groups of $B{\mathbb{Z}}_{p}$ and
$B{\mathbb{Z}}_{p^{k}}$ are ${\mathbb{Z}}_{p}$ and ${\mathbb{Z}}_{p^{k}}$
respectively in even cases and zero in odd cases [Ha]. So, we consider the
case of even $n$.

Since for any $i$ the CW-complex $B{\mathbb{Z}}_{p^{i}}$ can be taken to be
the infinite lens space $S^{\infty}/{\mathbb{Z}}_{p^{i}},$ we may assume that
the $(n+1)$-dimensional lens space
$L^{n+1}_{p^{i}}=S^{n+1}/{\mathbb{Z}}_{p^{i}}$ is the $(n+1)$-skeleton of
$B{\mathbb{Z}}_{p^{i}}$. Hence it suffices to show that the map
$f:L_{p^{k}}^{n+1}\to L_{p}^{n+1}$ induces nonzero homomorphism

$f^{*}:H^{n}(L_{p}^{n+1};{\mathbb{Z}})\to
H^{n}(L_{p^{k}}^{n+1};{\mathbb{Z}}).$

First we note that the map $f$ has degree one. We may assume that each of our
lens spaces $L^{n+1}_{p^{i}}$, $i=1,k$ is the orbit space of a free
${\mathbb{Z}}_{p^{i}}$-action on the odd dimensional sphere
$S^{n+1}=S^{1}\ast\dots\ast S^{1}$ which is presented as the join product of
circles. We may assume the ${\mathbb{Z}}_{p^{i}}$-action takes place only on
the first factor. Let $\pi_{i}:S^{n+1}\to L_{p^{i}}^{n+1}$ be corresponding
maps. Let

$\bar{f}:S^{1}\ast\dots\ast S^{1}\to S^{1}\ast\dots\ast S^{1}$

be defined as $\xi\ast 1\ast\dots\ast 1$ where $\xi:S^{1}\to S^{1}$ is taking
unit complex numbers to the $p^{k-1}$ power, $\xi:z\mapsto z^{p^{k-1}}$. The
map $\bar{f}$ defines the map of the orbit spaces $f:L^{n+1}_{p^{k}}\to
L^{n+1}_{p}$ with $f_{*}=\psi$ such that the following diagram

${S^{n+1}}$${S^{n+1}}$${L_{p^{k}}^{n+1}}$${L_{p}^{n+1}}$$\scriptstyle{\pi_{k}}$$\scriptstyle{\bar{f}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{f}$

is commutative. The degrees of maps $\pi_{i}$ and $\bar{f}$ equal the degree
of their restrictions to the first circle in $S^{1}\ast\dots S^{1}$ and they
are $\deg(\pi_{i})=p^{i}$, $\deg(\bar{f})=p^{k-1}$. It follows from the
equality $\deg(\pi_{1})\deg(\bar{f})=\deg(f)\deg(\pi_{k})$ that $\deg(f)=1$.

For nonzero $\alpha\in H^{n}(L_{p}^{n+1};{\mathbb{Z}})$ by the Poincare
Duality and the naturality of the cap product it follows that

$0\neq[L^{n+1}_{p}]\cap\alpha=f_{*}([L_{p^{k}}^{n+1}])\cap\alpha=f_{*}([L_{p}^{n+1}]\cap
f^{*}(\alpha))$

where $[L_{p}^{n+1}]$ and $[L_{p^{k}}^{n+1}]$ are the fundamental classes.
Thus, we obtain that $f^{*}(\alpha)\neq 0.$

Claim: If $\operatorname{cd}(\psi)=\infty,$ then
$\operatorname{cd}(\phi)=\infty$ where $\psi:=\phi|_{{\mathbb{Z}}_{p^{k}}}.$

Suppose $\operatorname{cd}(\phi)=n<\infty.$ Since $\phi^{*}(a)=0$ for all
$a\in H^{2n}(B{\mathbb{Z}}_{p};{\mathbb{Z}}),$ the restriction homomorphism
$i^{*}:H^{2n}(BG;{\mathbb{Z}})\to H^{2n}(B{\mathbb{Z}}_{p^{k}};{\mathbb{Z}})$
is not trivial, but $i^{*}(\phi^{*}(a))=i^{*}(0)=0.$ Since
$\psi^{*}=i^{*}\circ\phi^{*},$ we get $\psi^{*}(a)=0$ for all $a\in
H^{2n}(B{\mathbb{Z}}_{p};{\mathbb{Z}}).$ This is contradiction, hence we prove
the claim.

Step 2. Let $H$ be an arbitrary torsion group. Pick a nonzero element $h\in H$
generating a cyclic group ${\mathbb{Z}}_{p}$ for some prime $p.$ The preimage
of the subgroup ${\mathbb{Z}}_{p}$ of $H$ is a torsion subgroup of $G$. We
apply Lemma 3.1 to find a cyclic subgroup ${\mathbb{Z}}_{p^{k}}\subset G$ that
maps onto ${\mathbb{Z}}_{p}.$ We have the following commutative diagrams:

${G}$${H}$${{\mathbb{Z}}_{p^{k}}}$${{\mathbb{Z}}_{p}}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$${BG}$${BH}$${B{\mathbb{Z}}_{p^{k}}}$${B{\mathbb{Z}}_{p}}$$\scriptstyle{B\phi}$$\scriptstyle{B\psi}$

where $\phi$ and $\psi$ are the homomorphisms of the fundamental groups
induced by the maps $B\phi$ and $B\psi$.

Let
$\alpha:Coind_{{\mathbb{Z}}_{p}}^{H}{\mathbb{Z}}=Hom_{{\mathbb{Z}}{\mathbb{Z}}_{p}}({\mathbb{Z}}H,{\mathbb{Z}})\to{\mathbb{Z}}$
be the canonical ${\mathbb{Z}}{\mathbb{Z}}_{p}$-homomorphism from Theorem 2.5.
Consider the following commutative diagram

${H^{*}(BH;Coind_{{\mathbb{Z}}_{p}}^{H}{\mathbb{Z}})}$${H^{*}(B{\mathbb{Z}}_{p};Coind_{{\mathbb{Z}}_{p}}^{H}{\mathbb{Z}})}$${H^{*}(B{\mathbb{Z}}_{p};{\mathbb{Z}})}$${H^{*}(BG;Coind_{{\mathbb{Z}}_{p}}^{H}{\mathbb{Z}})}$${H^{*}(B{\mathbb{Z}}_{p^{k}};Coind_{{\mathbb{Z}}_{p}}^{H}{\mathbb{Z}})}$${H^{*}(B{\mathbb{Z}}_{p^{k}};{\mathbb{Z}})}$$\scriptstyle{\phi^{8}}$$\scriptstyle{i^{*}}$$\scriptstyle{\psi^{*}}$$\scriptstyle{\alpha_{*}}$$\scriptstyle{\psi^{*}}$$\scriptstyle{i^{*}}$$\scriptstyle{\alpha_{*}}$

where $\alpha_{*}$ is the coefficient homomorphism generated by $\alpha$.

We claim that $\operatorname{cd}(\phi)=\infty.$ Assume the contrary,
$\operatorname{cd}(\phi)<n$ for some even number $n$. By Step 1, $\psi$ is a
nonzero homomorphism in dimension $n$. We pick a nonzero element $a\in
H^{n}(B{\mathbb{Z}}_{p};{\mathbb{Z}})$ with $\psi^{*}(a)\neq 0$. By Theorem
2.5, the top row through homomorphism

$\alpha_{*}i^{*}:H^{*}(BH;Coind_{\mathbb{Z}_{p}}^{H}\mathbb{Z})\to
H^{*}(B{\mathbb{Z}}_{p};{\mathbb{Z}})$

is an isomorphism. Let $a=\alpha_{*}i^{*}(b)$. Since by the assumption
$\phi^{*}(b)=0$, we obtain a contradiction:

$0\neq\psi^{*}(a)=\psi^{*}\alpha_{*}i^{*}(b)=\alpha_{*}i^{*}\phi^{*}(b)=\alpha_{*}i^{*}(0)=0.$

∎

##  4\. Finitely Generated Abelian Groups

######  Lemma 4.1.

Given an epimorphism $\phi:{\mathbb{Z}}^{n}\to G$ with a finite group $G$
there is an epimorphism $\pi:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{k}$ such that
$\phi=\psi\circ\pi$ where $k=k(G)$ is the Smith normal number for $G$,

$\psi=\prod_{i=1}^{k}(\psi_{i}:{\mathbb{Z}}\to{\mathbb{Z}}_{n_{i}}),$

and the numbers $n_{1}|\dots|n_{k}$ are taken from IFD for $G$ from Theorem
2.3.

###### Proof.

Being a subgroup of ${\mathbb{Z}}^{n}$, the kernel $\ker\phi$ is a free
abelian group. Since $G$ is finite, $\ker\phi$ is isomorphic to
${\mathbb{Z}}^{n}$. We fix a basis in $\ker\phi$. Let
$A:{\mathbb{Z}}^{n}\to\ker\phi$ be an isomorphism. We regard
$A:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{n}$ as the embedding. Then $A$ is given by
$n\times n$ matrix the columns of which form our basis. We apply Corollary 2.2
(Smith Normal Form) to get matrices $Q$ and $P$ that change in a special way
the bases in the domain of $A$ and the range of $\phi$ respectively. Thus,
$AQ({\mathbb{Z}}^{n})=A({\mathbb{Z}}^{n})=\ker\phi$. Then
$PAQ({\mathbb{Z}}^{n})=\ker(\phi P^{-1})$. Then

$G\cong(\phi P^{-1})({\mathbb{Z}}^{n})={\mathbb{Z}}^{n}\Big{/}\ker(\phi
P^{-1})={\mathbb{Z}}^{n}\Big{/}PAQ({\mathbb{Z}}^{n})=$
$=\left({\mathbb{Z}}^{n-k}\Big{/}\langle 1\rangle\times\cdots\times\langle
1\rangle\right)\times\left({\mathbb{Z}}^{k}\Big{/}\langle
n_{1}\rangle\times\langle n_{2}\rangle\times\cdots\times\langle
n_{k}\rangle\right)=$
$=\left({\mathbb{Z}}\Big{/}{\mathbb{Z}}\times\cdots{\mathbb{Z}}\Big{/}{\mathbb{Z}}\right)\times\left({\mathbb{Z}}\Big{/}n_{1}{\mathbb{Z}}\times\cdots\times{\mathbb{Z}}\Big{/}n_{k}{\mathbb{Z}}\right)=$
$=pr_{k}({\mathbb{Z}}^{n})\Big{/}\langle n_{1}\rangle\times\cdots\times\langle
n_{k}\rangle=\psi pr_{k}({\mathbb{Z}}^{n})$

where $pr_{k}:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{k}$ is the projection onto the
last k coordinates. Thus, $\phi P^{-1}=\psi pr_{k}$. Then $\phi=\psi\pi$ with
$\pi=pr_{k}P$.

∎

######  Lemma 4.2.

Let $\phi:{\mathbb{Z}}^{n}\to G$ be an epimorphism, where $G$ is a finite
abelian group. Then $\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$ In
particular, $\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=k(G)$ where k is
the Smith Normal number for given a finite abelian group G.

###### Proof.

Since $\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)$ for any group
homomorphism [DK], we just need to show two inequalities, i.e
$\operatorname{cat}(\phi)\leq k(G)$ and $k(G)\leq\operatorname{cd}(\phi).$
Then, observing the chain inequalities
$k(G)\leq\operatorname{cd}(\phi)\leq\operatorname{cat}(\phi)\leq k(G),$ we
obtain the conclusion of Lemma 4.2.

Let us show the first inequality $\operatorname{cat}(\phi)\leq k(G).$ Let
$k=k(G).$ By Lemma 4.1, there exists an epimorphisms
$\pi:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{k}$ and $\psi:{\mathbb{Z}}^{k}\to G$
such that we have the following commutative diagram:

${{\mathbb{Z}}^{n}}$${{\mathbb{Z}}^{k}}$${G}$${0.}$$\scriptstyle{\pi}$$\scriptstyle{\phi}$$\scriptstyle{\psi}$

Using well-known facts on the LS-category $\operatorname{cat}$ [CLOT], we
obtain:

$\operatorname{cat}(\phi)\leq\min\\{\operatorname{cat}(\psi),\operatorname{cat}(\pi))\\}\leq\operatorname{cat}(\psi)\leq\min\\{\operatorname{cat}(T^{k},\operatorname{cat}(BG))\\}\leq\operatorname{cat}(T^{k})=k$

where $T^{k}$ is the $k$ dimensional torus.

Since $B\pi:T^{n}\to T^{k}$ is a retraction, $\pi$ is injective on cohomology,
so we have $\operatorname{cd}(\phi)=\operatorname{cd}(\psi).$ Then to prove
the second inequality $k(G)\leq\operatorname{cd}(\phi),$ it suffices to show
$k(G)\leq\operatorname{cd}(\psi).$

We do it by induction on $k=k(G).$ When k=1 we have $G={\mathbb{Z}}_{n_{1}}$.
Then the homomorphism
$\psi^{*}=\psi_{1}^{*}:H^{1}(B{\mathbb{Z}}_{n_{1}};Z_{n_{1}})\to
H^{1}(B{\mathbb{Z}};Z_{n_{1}})$ is nonzero, since
$\psi:{\mathbb{Z}}\to{\mathbb{Z}}_{n_{1}}$ is surjective.

Suppose the result holds true for all $l\leq k$. First we note that by Theorem
2.3 the group $G$ for $k(G)=k+1$ is written uniquely as
$G={\mathbb{Z}}_{n_{1}}\times\cdots{\mathbb{Z}}_{n_{k+1}}$ with
$n_{1}|\cdots|n_{k+1}.$ Note also that $BG$ can be presented as the product
$B{\mathbb{Z}}_{n_{1}}\times\dots\times B{\mathbb{Z}}_{n_{k+1}}$. Let $p$ be a
prime that divides $n_{1}$ and, hence, all $n_{i}$. We show that the induced
homomorphism

$\psi^{*}:H^{k+1}(B{\mathbb{Z}}_{n_{1}}\times...\times
B{\mathbb{Z}}_{n_{k+1}};{\mathbb{Z}}_{p})\to
H^{k+1}(T^{k+1};{\mathbb{Z}}_{p})$

is a nonzero homomorphism.

It is known that the integral cohomology groups
$H^{j}(B{\mathbb{Z}}_{m};{\mathbb{Z}})$ are ${\mathbb{Z}}_{m}$ if $j$ is even
and zero otherwise [Ha]. Note that for prime $p$ dividing $m$ by the Universal
Coefficient Formula
$H^{j}(B{\mathbb{Z}}_{m};{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}$ for all $j$,
since ${\mathbb{Z}}_{m}\otimes{\mathbb{Z}}_{p}={\mathbb{Z}}_{p}$ and
$\text{Tor}({\mathbb{Z}}_{m},{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}.$ Thus, for
prime $p$ dividing $n_{1}$ we obtain
$H^{j}(B{\mathbb{Z}}_{n_{i}};{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}$ for all $i$
and $j$. Since for each $n_{i}$ the CW-complex $B{\mathbb{Z}}_{n_{i}}$ are of
finite type, we can apply the Kunneth Formula 2.6. By the Kunneth Formula with
a field coefficient ${\mathbb{Z}}_{p},$ and induction, we get that
$H^{j}(B({\mathbb{Z}}_{n_{1}}\times...\times{\mathbb{Z}}_{n_{k+1}});{\mathbb{Z}}_{p})$
is not zero for all $j$. Clearly for the $(k+1)$-torus $T^{k+1}$ we have
$H^{k+1}(T^{k+1};{\mathbb{Z}}_{p})={\mathbb{Z}}_{p}.$ Using commutative
diagram below, we get that $\psi$ is a nonzero homomorphism for the mod $p$
cohomology in dimension $k+1$.

${H^{k}(B{\mathbb{Z}}_{n_{1}}\times...\times
B{\mathbb{Z}}_{n_{k}});{\mathbb{Z}}_{p})\otimes
H^{1}(B{\mathbb{Z}}_{n_{k+1}};{\mathbb{Z}}_{p})}$${H^{k+1}(B{\mathbb{Z}}_{n_{1}}\times...\times
B{\mathbb{Z}}_{n_{k+1}});{\mathbb{Z}}_{p})}$${H^{k}(T^{k};{\mathbb{Z}}_{p})\otimes
H^{1}(S^{1};{\mathbb{Z}}_{p})}$${H^{k+1}(T^{k+1};{\mathbb{Z}}_{p})}$$\scriptstyle{\times}$$\scriptstyle{\psi^{*}\otimes\psi^{*}}$$\scriptstyle{\psi^{*}}$$\scriptstyle{\times}$

Indeed, the horizontal maps are isomorphism, by the Kunneth Theorem. Thus the
Kunneth Formula isomorphism takes the tensor product to the cross product,
$a\otimes b\overset{\times}{\rightarrow}a\times b.$ Here the cross product is
defined as $a\times b=p^{*}_{1}(a)\cup p^{*}_{2}(b)$ where $p_{1}$ and $p_{2}$
are the projections of the product $X\times Y$ onto $X$and $Y$ respectively.
Using the naturality of the cup product and the induction assumption, we
obtain:

$\psi^{*}(a\times b)=\psi^{*}(p^{*}_{1}(a)\cup p^{*}_{2}(b))=(\psi^{*}\degree
p^{*}_{1})(a)\cup(\psi^{*}\degree
p^{*}_{2})(b)=\psi^{*}(a)\otimes\psi^{*}(b)\neq 0$

Hence, $\operatorname{cd}(\psi)=k+1.$ ∎

We use the notation $T(A)$ for the torsion subgroup of an abelian group $A$.
For finitely generated abelian groups we define the rank
$rank(A)=rank(A/T(A))$.

######  Lemma 4.3.

Every epimorphism $\phi:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{m}\oplus G$ splits as
the direct sum

$\phi=\psi_{1}\oplus\psi_{2}:{\mathbb{Z}}^{m}\oplus{\mathbb{Z}}^{n-m}\to{\mathbb{Z}}^{m}\oplus
G.$

###### Proof.

The epimorphism $\phi$ as a map to the product is defined by coordinate
functions, $\phi=(\phi_{1},\phi_{2})$ which are also epimorphisms. There
exists a section $s:{\mathbb{Z}}^{m}\to{\mathbb{Z}}^{n}$ of the epimorphism
$\phi_{1}$, since ${\mathbb{Z}}^{m}$ is free abelian. We show that
${\mathbb{Z}}^{n}$ splits as the direct sum
$s({\mathbb{Z}}^{m})\oplus\phi^{-1}(G).$ For each element
$x\in{\mathbb{Z}}^{n}$ we have
$\phi_{1}(x-s\phi_{1}(x))=\phi_{1}(x)-\phi_{1}s(\phi_{1}(x))=0.$ Therefore,
$x-s\phi_{1}(x)\in\phi^{-1}(G)$. Thus, every element $x\in{\mathbb{Z}}^{n}$
can be written as $s\phi_{1}(x)+(x-s\phi_{1}(x))$. Suppose that $y\in
s({\mathbb{Z}}^{m})\cap\phi^{-1}(G).$ Since $\phi(y)\in G$, we obtain
$\phi_{1}(y)=0$. Since $y\in s({\mathbb{Z}}^{m})$, we have $s\phi_{1}(y)=y$.
Hence $y=0$ and the sum $s({\mathbb{Z}}^{m})+\phi^{-1}(G)={\mathbb{Z}}^{n}$ is
a direct sum.

Note that $s({\mathbb{Z}}^{m})\cong{\mathbb{Z}}^{m}$. In view of the splitting
${\mathbb{Z}}^{n}={\mathbb{Z}}^{m}\oplus\phi^{-1}(G)$ it follows that
$rank(\phi^{-1}(G))=n-m$. We define $\psi_{1}$ and $\psi_{2}$ to be the
restrictions of $\phi_{1}$ and $\phi_{2}$ to corresponding direct summands.
i.e., $\psi_{1}:=\phi_{1}|_{s({\mathbb{Z}}^{m})}$ and
$\psi_{2}:=\phi_{2}|_{\phi^{-1}(G)}.$ ∎

######  Theorem 4.4.

Let $\phi:\Gamma\to\Lambda$ be an epimorphism between finitely generated
abelian groups. Then

$\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$

If $\Gamma$ is torsion free, then

$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=rank(\Lambda)+k(T(\Lambda)),$

where $k(T(\Lambda))$ is the Smith Normal number of $T(\Lambda)$.

###### Proof.

Since the groups $\Gamma$ and $\Lambda$ are finitely generated abelian groups,
we may assume $\Gamma={\mathbb{Z}}^{n}\oplus T(\Gamma)$ and
$\Lambda={\mathbb{Z}}^{m}\oplus T(\Lambda)$ for some $m$ and $n$.

By Theorem 3.2, if both groups $\Gamma$ have torsion and
$\phi(T(\Gamma))\neq\emptyset$, then $\operatorname{cd}(\phi)=\infty,$ so
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi).$ Thus, we consider $\phi$
with $\phi(T(\Lambda))=0$. Such $\phi$ factors through the epimorphism
$\bar{\phi}:{\mathbb{Z}}^{n}\to\Lambda$. In view of the retraction $B\Gamma\to
B{\mathbb{Z}}^{n}$ we obtain that
$\operatorname{cd}(\bar{\phi})=\operatorname{cd}(\phi)$ and
$\operatorname{cat}(\bar{\phi})=\operatorname{cat}(\phi)$. Thus, we may assume
that $\Gamma={\mathbb{Z}}^{n}$.

If $\Lambda$ has no torsion, then
$\operatorname{cat}(\phi)=\operatorname{cd}(\phi)=rank(\phi)=rank(\Lambda)=m$
by Jamie Scott’s result [Sc]. We consider the case $T(\Lambda)\neq 0$. Let
$\phi:{\mathbb{Z}}^{n}\to{\mathbb{Z}}^{m}\oplus T(\Lambda)$ be an epimorphism.
By Lemma 4.3 $\phi$ breaks into the direct sum $\psi_{1}\oplus\psi_{2}$ where
$\psi_{1}:{\mathbb{Z}}^{m}\to{\mathbb{Z}}^{m}$ is an isomorphism and an
epimorphism $\psi_{2}:{\mathbb{Z}}^{n-m}\to T(\Lambda).$

By the well-known inequality for LS category of product of maps in [CLOT], we
obtain

$\operatorname{cat}(\psi_{1}\oplus\psi_{2})\leq\operatorname{cat}(\psi_{1})+\operatorname{cat}(\psi_{2})=m+k,$

where $k=k(T(\Lambda))$, since by Lemma 4.2
$\operatorname{cat}(\psi_{2})=\operatorname{cd}(\psi_{2})=k(T(\Lambda))$.

To complete the proof, we show that $m+k\leq\operatorname{cd}(\phi)$. By
Theorem 2.3, the torsion group $T({\Lambda})$ admits a decomposition
$T(\Lambda)={\mathbb{Z}}_{n_{1}}\times..\times{\mathbb{Z}}_{n_{k}}$ where
$n_{1},\cdots,n_{k}$ are natural numbers with $n_{1}|\cdots|n_{k}.$ The proof
of Lemma 4.2 gives a nonzero homomorphism

$\psi_{2}^{*}:H^{k}(BT(\Lambda);{\mathbb{Z}}_{p})\to
H^{k}(T^{n-m};{\mathbb{Z}}_{p}).$

We apply the Kunneth Formula (Theorem 2.6) with ${\mathbb{Z}}_{p}$
coefficients for $p|n_{1}$ to obtain a nonzero homomorphism

$\phi^{*}:H^{m+k}(T^{m}\times BT(\Lambda);{\mathbb{Z}}_{p})\to
H^{m+k}(T^{m}\times T^{n-m};{\mathbb{Z}}_{p}).$

Therefore, $\operatorname{cd}(\phi)\geq m+k$. ∎

## Acknowledgments

I would like to thank my advisor, Alexander Dranishnikov, for all of his help
and encouragement throughout this project.

## References

  * [Be] I. Berstein, On the Lusternik-Schnirelmann category of Grassmannians. Math. Proc. Camb. Philos. Soc. 79 (1976) 129-134.
  * [Br] K. Brown, Cohomology of Groups. _Graduate Texts in Mathematics_ , 87 Springer, New York Heidelberg Berlin, 1994.
  * [CLOT] O. Cornea, G. Lupton, J. Oprea, D. Tanre,  Lusternik-Schnirelmann Category, AMS, 2003.
  * [DF] D. Dummit, R.Foote, _Abstract algebra_ , Wiley Hoboken, 2004.
  * [Do] A. Dold, Lectures on algebraic topology. Springer Science & Business Media; 2012.
  * [DK] A. Dranishnikov, N. Kuanyshov, On the LS category of homomorphism, preprint arXiv:2203.03734 [math.AT]
  * [DR] A. Dranishnikov, Yu. Rudyak, On the Berstein-Svarc theorem in dimension 2. Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 407-413.
  * [DS] A. Dranishnikov, R. Sadykov, The Lusternik–Schnirelmann category of a connected sum, Fundamenta Mathematicae 251 (2020), no. 3, 313-328.
  * [EG] S. Eilenberg, T. Ganea, On the Lusternik-Schnirelmann Category of Abstract Groups. Annals of Mathematics, 65, (1957), 517-518.
  * [Gr] M. Grant,  https://mathoverflow.net/questions/89178/cohomological-dimension-of-a-homomorphism
  * [Ha] A. Hatcher, Algebraic topology, 2005.
  * [Hu] T. Hungerford, _Algebra (Vol. 73)_ , Springer, 2012.
  * [LS] L. Lusternik, L. Schnirelmann, “Sur le probleme de trois geodesiques fermees sur les surfaces de genre 0”, Comptes Rendus de l’Academie des Sciences de Paris, 189: (1929) 269-271.
  * [Sch] A. Schwarz, The genus of a fibered space. Trudy Moscov. Mat. Obsc. 10, 11 (1961 and 1962), 217-272, 99-126.
  * [Sc] J. Scott. On the topological complexity of maps, Topology and its Applications 314 (2022), Paper No. 108094, 25 pp.
  * [Sp] E. Spanier, Algebraic topology, Springer Science & Business Media 1989.


# Asymptotic properties of solutions to a certain ultrahyperbolic equation

M. N. Demchenko111St. Petersburg Department of V. A. Steklov Institute of
Mathematics of the Russian Academy of Sciences, 27 Fontanka, St. Petersburg,
Russia. E-mail<EMAIL_ADDRESS>
The research was supported by Russian Foundation for Basic Research, grant No.
20-01-00627-a.

###### Abstract

We consider a certain ultrahyperbolic equation in a Euclidean space being a
generalization of Klein-Gordon-Fock equation. The behavior of solutions at
points tending to infinity along timelike directions is studied. We examine
the issue of existence of solutions possessing given asymptotic properties at
infinity.

Keywords: ultrahyperbolic equation, Klein-Gordon-Fock equation, relativistic
wave equation, asymptotic behavior at infinity, scattering problem.

##  1 Introduction

We consider the following ultrahyperbolic equation

$(\Delta_{t}-\Delta_{x}+m^{2})u=f,$ (1)

where a solution $u(x,t)$ and a function $f(x,t)$ are defined in
${\mathbb{R}}^{d}\times{\mathbb{R}}^{n}$, $d,n\geqslant 1$,

$\Delta_{x}=\partial_{x_{1}}^{2}+\ldots+\partial_{x_{d}}^{2},\quad\Delta_{t}=\partial_{t_{1}}^{2}+\ldots+\partial_{t_{n}}^{2},$

$m$ is a positive constant. In the particular case $n=1$, $f=0$, this is
(hyperbolic) Klein-Gordon-Fock equation describing free motion of a
relativistic spinless particle with rest mass $m$.

In the present paper, we consider solutions $u(x,t)$ with the following
asymptotic property

$u(s\theta,s\omega)=s^{-(d+n-1)/2}\sum_{\pm}U_{\pm}(\theta,\omega)e^{\pm
is\,m\sqrt{1-\theta^{2}}}+O(s^{-(d+n+1)/2}),\quad s\to+\infty.$ (2)

Here $(\theta,\omega)\in B^{d}\times S^{n-1}$,
$B^{d}=\\{\theta\in{\mathbb{R}}^{d}\,|\,|\theta|<1\\}$,
$S^{n-1}=\\{\omega\in{\mathbb{R}}^{n}\,|\,|\omega|=1\\}$, and
$\theta^{2}:=|\theta|^{2}$. Relation (2) characterizes the behavior of a
solution at the infinity along timelike directions. We will find a family of
solutions, which exhibit such behavior. Besides, for given functions $U_{\pm}$
(more precisely, either $U_{+}$ or $U_{-}$ will be given) and for a given
function $f$, we will construct a solution to equation (1) satisfying (2).

Asymptotic properties of solutions to the wave equation (the latter being the
particular case of (1) for $n=1$, $m=0$) and its vector analogues were studied
in a number of papers (e.g., in [1, 2, 3, 4, 5, 6]). However, we point out the
following considerable difference between the results obtained there and the
asymptotic formula (2) being valid in the case $m>0$. Consider, for example,
the case $d=3$, $n=1$. The solution to the Cauchy problem for the wave
equation with compactly supported initial data and $f$ vanishes when $(x,t)$
tends to infinity along a timelike direction (Huygens principle). On the other
hand, in the case when $(x,t)$ tends to infinity along a characteristic
direction, the solution exhibits nontrivial asymptotic behavior, which can be
described in terms of the Radon transform of the initial data and a certain
integral transform of the function $f$. Besides, the solution is uniquely
determined by the corresponding asymptotic coefficients [2]. We will show that
in the case of Klein-Gordon-Fock equation (when $m>0$) the situation is quite
the reverse: the solution to the Cauchy problem decay faster than any power of
time and distance along characteristic directions and possesses the asymptotic
property (2) along timelike directions. This will be established for a certain
family of solutions in the general case $n\geqslant 1$ avoiding analysis of
the Cauchy problem, which becomes ill-posed.

The set of known well-posed problems for ultrahyperbolic equations is less
than that for elliptic, parabolic, and hyperbolic equations. Paper [7]
concerns the characteristic problem for equation of the form (1) with $m=0$,
$f=0$, in which the solution $u(x,t)$ is determined in the region $|x|<|t|$
(or $|x|>|t|$) from its values on the characteristic cone $|x|=|t|$. There
existence and uniqueness of the solution in a certain class of functions were
proved for arbitrary $d,n\geqslant 1$, and an integral formula for $u$ was
obtained. In the present paper, we consider the problem for equation (1) with
$m>0$, in which one of the coefficients $U_{\pm}$ from relation (2) is given
as the data. We establish the existence of the solution to this problem.

The author is thankful to A. P. Kiselev for pointing to related results
concerning the wave equation, and to M. I. Belishev and A. F. Vakulenko for
helpful discussions.

##  2 Construction of solutions possessing property (2)

We use the following definition of Fourier transform

$\displaystyle\hat{v}(\xi)=\int_{{\mathbb{R}}^{d}}e^{-i\langle{}x,\xi\rangle}v(x)dx$

(here $\langle\cdot,\cdot\rangle$ is the standard inner product in a real
Euclidean space; further the angle brackets will also denote the pairing of a
distribution with a test function). For a function $v(x,t)$ defined in
“spacetime” ${\mathbb{R}}^{d}\times{\mathbb{R}}^{n}$, it is convenient to
define Fourier transform as follows

$\displaystyle\hat{v}(\xi,\tau)=\int_{{\mathbb{R}}^{d+n}}e^{i(-\langle{}x,\xi\rangle+\langle{}t,\tau\rangle)}v(x,t)dxdt.$

Then inverse Fourier transform has the following form

$\displaystyle
v(x,t)=(2\pi)^{-d-n}\int_{{\mathbb{R}}^{d+n}}e^{i(\langle{}x,\xi\rangle-\langle{}t,\tau\rangle)}\hat{v}(\xi,\tau)d\xi
d\tau.$

We will always assume that the function $f$ on the right-hand side of equation
(1) belongs to Schwartz space $\mathcal{S}({\mathbb{R}}^{d+n})$. By applying
Fourier transform to equation (1) we get

$(\xi^{2}+m^{2}-\tau^{2})\hat{u}(\xi,\tau)=\hat{f}(\xi,\tau).$

Then we may formally represent $\hat{u}$ as follows

$\displaystyle\hat{u}=\hat{u}^{f}+\hat{u}^{a},$ (3)
$\displaystyle\hat{u}^{f}(\xi,\tau)=\frac{\hat{f}(\xi,\tau)}{\xi^{2}+m^{2}-\tau^{2}},\quad\hat{u}^{a}(\xi,\tau)=\delta(\xi^{2}+m^{2}-\tau^{2})a(\xi,\tau),$

which is essentially a sum of a particular solution to the inhomogeneous
equation ($u^{f}$) and a general solution to the homogeneous equation ($u^{a}$
with an arbitrary density $a$). Later we will give rigorous definitions of the
functions $u^{f}$ and $u^{a}$. In particular, the factor
$(\xi^{2}+m^{2}-\tau^{2})^{-1}$, which is singular at the hypersurface

$\Sigma_{m}=\\{(\xi,\tau)\in{\mathbb{R}}^{d+n}\,|\,\xi^{2}+m^{2}-\tau^{2}=0\\},$

will be regularized by means of the Cauchy principal value.

Now we formally apply inverse Fourier transform to (3):

$\displaystyle u=u^{f}+u^{a},$ (4) $\displaystyle
u^{f}(x,t)=(2\pi)^{-d-n}\,\int_{{\mathbb{R}}^{d+n}}e^{i(\langle{}x,\xi\rangle-\langle{}t,\tau\rangle)}\frac{\hat{f}(\xi,\tau)}{\xi^{2}+m^{2}-\tau^{2}}\,d\xi
d\tau,$ $\displaystyle
u^{a}(x,t)=(2\pi)^{-d-n}\int_{{\mathbb{R}}^{d+n}}\,e^{i(\langle{}x,\xi\rangle-\langle{}t,\tau\rangle)}\delta(\xi^{2}+m^{2}-\tau^{2})a(\xi,\tau)\,d\xi
d\tau.$

By additional formal transformations, one can bring the expression for $u^{f}$
to the form

$\displaystyle
u^{f}(x,t)=(2\pi)^{-d-n}\,\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}\frac{e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}F(\xi,\sigma,\rho)}{1-\rho}\,d\xi
dS_{\sigma}d\rho,$ (5) $\displaystyle
F(\xi,\sigma,\rho)=(\xi^{2}+m^{2})^{n/2-1}\hat{f}(\xi,\rho\sigma\sqrt{\xi^{2}+m^{2}})\frac{\rho^{n-1}}{1+\rho},$

where we have chosen a certain regularization of the singularity
$(1-\rho)^{-1}$ arising from the singularity $(\xi^{2}+m^{2}-\tau^{2})^{-1}$
in the original expression. Namely, the integral in (5) is understood in the
sense of Cauchy principal value. Here and further $dS$ is the measure on the
unit sphere $S^{n-1}$ induced by the Euclidean metrics in ${\mathbb{R}}^{n}$;
for the sphere of dimension zero, we set

$dS_{\sigma}=\delta(\sigma-1)+\delta(\sigma+1).$

Expression (5), which is taken for the definition of the function $u^{f}$,
will be discussed in sec. 3 in more details. In particular, we will show that
$u^{f}$ is a smooth solution to equation (1). It is possible to chose a
regularization of the integral in (5) that differs from Cauchy principal
value. A solution we would have thus obtained differs from the chosen one by a
solution of the homogeneous equation of the form (1).

Next the expression for the term $u^{a}$ in (4) can be formally transformed to

$u^{a}(x,t)=(2\pi)^{-d-n}\int_{{\mathbb{R}}^{d}\times
S^{n-1}}e^{i\left(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\sqrt{\xi^{2}+m^{2}}\right)}A(\xi,\sigma)\,d\xi
dS_{\sigma},$ (6)

where the function $A(\xi,\sigma)$ is related to $a(\xi,\tau)$ as follows

$A(\xi,\sigma)=\frac{1}{2}(\xi^{2}+m^{2})^{n/2-1}a\left(\xi,\sigma\sqrt{\xi^{2}+m^{2}}\right).$

The function $A$ is defined in ${\mathbb{R}}^{d}\times S^{n-1}$. This
function, and so $u^{a}$, depends on the values of $a$ only on $\Sigma_{m}$.
Define $\mathcal{S}(\Sigma_{m})$ as the set of functions $a$ on the
hypersurface $\Sigma_{m}$ that correspond to functions $A$ from Schwartz space
$\mathcal{S}({\mathbb{R}}^{d}\times S^{n-1})$. When this condition is
fulfilled, we assume relation (6) as a definition of the function
$u^{a}(x,t)$. As can be easily verified, it is a smooth solution of
homogeneous equation of the form (1).

The assumptions on the coefficients $U_{\pm}$ in (2) will involve the class of
functions in $B^{d}\times S^{n-1}$ that have “zero of infinite order” on the
boundary $\partial B^{d}$. Namely, we say that $U$ belongs to
$\mathcal{S}(B^{d}\times S^{n-1})$ if it coincides on $B^{d}\times S^{n-1}$
with a function from $C^{\infty}({\mathbb{R}}^{d}\times S^{n-1})$ supported in
$\overline{B^{d}}\times S^{n-1}$.

######  Theorem 1.

Let $f\in\mathcal{S}({\mathbb{R}}^{d+n})$, $a\in\mathcal{S}(\Sigma_{m})$. Then
the function $u(x,t)$ given by (4) is a smooth solution of equation (1), and
asymptotic formula (2) holds true. The coefficients $U_{\pm}$ are expressed in
terms of $f$, $a$ as follows

$U_{\pm}(\theta,\omega)=\frac{e^{\pm i\pi(d-n+1)/4}}{4\pi
m}\left(\frac{m}{2\pi\sqrt{1-\theta^{2}}}\right)^{(d+n-1)/2}\,(a\mp
i\pi\hat{f})\left(\frac{\mp m\theta}{\sqrt{1-\theta^{2}}},\frac{\mp
m\omega}{\sqrt{1-\theta^{2}}}\right)$ (7)

(note that for any $(\theta,\omega)\in B^{d}\times S^{n-1}$, the arguments of
the function $a\mp i\pi\hat{f}$ on the right-hand side correspond to a point
on $\Sigma_{m}$).

Now we indicate necessary conditions on $U_{\pm}$, which follow from (7). In
the case $f=0$, we have

$U_{\pm}(-\theta,-\omega)=(\pm i)^{d-n+1}U_{\mp}(\theta,\omega),$ (8)

whereas in the case $a=0$, we have

$U_{\pm}(-\theta,-\omega)=(\pm i)^{d-n-1}U_{\mp}(\theta,\omega).$ (9)

Thus the coefficients $U_{\pm}$ in asymptotic formula (2) can not be arbitrary
functions. At the same time, the following theorem holds true.

######  Theorem 2.

Let $f\in\mathcal{S}({\mathbb{R}}^{d+n})$, $U_{+}\in\mathcal{S}(B^{d}\times
S^{n-1})$. Then the function

$a(\xi,\tau)=4\pi
m\,e^{-i\pi(d-n+1)/4}\left(\frac{2\pi}{|\tau|}\right)^{(d+n-1)/2}U_{+}\left(\frac{-\xi}{|\tau|},\frac{-\tau}{|\tau|}\right)+i\pi\hat{f}(\xi,\tau)$

of variables $(\xi,\tau)\in\Sigma_{m}$ belongs to $\mathcal{S}(\Sigma_{m})$,
and the corresponding solution $u(x,t)$ of the form (4) exhibits the
asymptotic behavior (2) with the given function $U_{+}$ and a function
$U_{-}\in\mathcal{S}(B^{d}\times S^{n-1})$ determined by relation (7).
Moreover, this is the unique solution of the form (4) with density $a$ from
$\mathcal{S}(\Sigma_{m})$, for which the coefficient $U_{+}$ in (2) coincides
with the given function.

In Theorem 2, the functions $U_{+}$ and $U_{-}$ are interchangeable, i.e., the
solution $u$ can be constructed from a given function $U_{-}$ (and $f$). In
this case, the corresponding expression for $a$ takes the form

$a(\xi,\tau)=4\pi
m\,e^{i\pi(d-n+1)/4}\left(\frac{2\pi}{|\tau|}\right)^{(d+n-1)/2}U_{-}\left(\frac{\xi}{|\tau|},\frac{\tau}{|\tau|}\right)-i\pi\hat{f}(\xi,\tau).$

We point out that there is a number of results concerning fundamental
solutions to ultrahyperbolic equations. We mention only the pioneering paper
[8], where equation (1) with $m=0$ was considered, and paper [9] concerned
with the case of arbitrary $m$. One can use fundamental solutions to construct
particular solutions to the inhomogeneous equation (1). However, for the study
of asymptotic properties of solutions, the Fourier analysis and the method of
stationary phase used in the present paper seem to be the most appropriate
tools.

##  3 Auxiliary assertions

For a function $F\in\mathcal{S}({\mathbb{R}}^{n})$, we give the following
definition of the integral in the sense of Cauchy principal value

$\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x)dx=\lim_{\varepsilon\to
0+}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x)\,dx.$ (10)

We will need some estimates related to definition (10) involving the following
functionals on functions in ${\mathbb{R}}^{n}$ ($k,p$ are nonnegative
integers):

$|F|_{k,p}=\sup_{|\alpha|\leqslant
k,\,x\in{\mathbb{R}}^{n}}(1+|x|)^{p}|\partial^{\alpha}F(x)|.$ (11)

For $0<\varepsilon^{\prime}\leqslant\varepsilon$, we have

$\int_{|x_{n}|>\varepsilon}-\int_{|x_{n}|>\varepsilon^{\prime}}\frac{1}{x_{n}}F(x)\,dx=\int_{{\mathbb{R}}^{n-1}}dx^{\prime}\int_{\varepsilon^{\prime}<|x_{n}|<\varepsilon}\frac{1}{x_{n}}F(x)\,dx_{n},$

where we use the notation $x^{\prime}=(x_{1},\ldots,x_{n-1})$. Now show that

$\left|\int_{\varepsilon^{\prime}<|x_{n}|<\varepsilon}\frac{1}{x_{n}}F(x)\,dx_{n}\right|\leqslant
2(\varepsilon-\varepsilon^{\prime})(1+|x^{\prime}|)^{-n}|F|_{1,n}.$ (12)

Indeed, the left-hand side equals

$\left|\int_{\varepsilon^{\prime}<|x_{n}|<\varepsilon}\frac{1}{x_{n}}(F(x)-F(x^{\prime},0))\,dx_{n}\right|\leqslant
2(\varepsilon-\varepsilon^{\prime})\sup_{x_{n}}|\partial_{x_{n}}F(x^{\prime},x_{n})|,$

which implies (12). Now the absolute value of the integral with respect to
$x^{\prime}$ occurring in the preceding relation can be estimated by
$C_{n}(\varepsilon-\varepsilon^{\prime})|F|_{1,n}$. This implies the existence
of the limit on the right-hand side of (10) and the estimate

$\left|\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x)\,dx-\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x)dx\right|\leqslant
C_{n}\varepsilon|F|_{1,n}.$ (13)

Next for $0<\varepsilon\leqslant 1$ we have

$\left|\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x)\,dx_{n}\right|\leqslant\left|\int_{|x_{n}|>1}\frac{1}{x_{n}}F(x)\,dx_{n}\right|+\left|\int_{\varepsilon<|x_{n}|<1}\frac{1}{x_{n}}F(x)\,dx_{n}\right|.$

The first term is estimated by

$C\sup_{x_{n}}((1+|x_{n}|)\,|F(x^{\prime},x_{n})|)\leqslant
C_{n}(1+|x^{\prime}|)^{-n}|F|_{0,n+1},$

while the second one can be estimated using (12). After integrating with
respect to $x^{\prime}$ we obtain

$\left|\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x)\,dx\right|\leqslant
C_{n}|F|_{1,n+1}.$ (14)

Our calculations also imply the equality

$\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x)\,dx=\int_{{\mathbb{R}}^{n-1}}dx^{\prime}\,\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{x_{n}}F(x)\,dx_{n}.$
(15)

Suppose that the function $F$ in the integral (10) depends on the parameter
$\lambda$. For a fixed $\varepsilon>0$ we have

$\partial_{\lambda}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x,\lambda)\,dx=\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}\partial_{\lambda}F(x,\lambda)\,dx,$

as soon as functionals $|F(\cdot,\lambda)|_{0,n+1}$,
$|\partial_{\lambda}F(\cdot,\lambda)|_{0,n+1}$ are uniformly bounded. This
follows from the fact that the specified assumption implies that the integrals
in this relation exist and the improper integral on the right-hand side
converges uniformly. Assume also that
$|\partial_{\lambda}F(\cdot,\lambda)|_{1,n}$ is uniformly bounded. Then due to
(13) the integral on the right-hand side of the last equality, and so the
left-hand side, tends to its limit as $\varepsilon\to 0$ uniformly with
respect to $\lambda$. This implies that

$\lim_{\varepsilon\to
0+}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}\partial_{\lambda}F(x,\lambda)\,dx=\lim_{\varepsilon\to
0+}\partial_{\lambda}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x,\lambda)\,dx\\\
=\partial_{\lambda}\left(\lim_{\varepsilon\to
0+}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x,\lambda)\,dx\right),$

whence

$\partial_{\lambda}\left(\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x,\lambda)\,dx\right)=\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}\partial_{\lambda}F(x,\lambda)\,dx.$

The analogous relation for higher order derivatives with respect to $\lambda$
is also valid:

$\partial_{\lambda}^{\alpha}\left(\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x,\lambda)\,dx\right)=\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}\partial_{\lambda}^{\alpha}F(x,\lambda)\,dx,$
(16)

providing that for all $\alpha$ we have
$|\partial_{\lambda}^{\alpha}F(\cdot,\lambda)|_{1,n+1}\leqslant C_{\alpha}$.

Now show that the integral of the form (10) allows changing of variable. We
will consider only the case when $n=1$, and the function $F$ belongs to
$C_{0}^{\infty}(I)$, where $I\subset{\mathbb{R}}$ is an open interval
containing zero. Suppose that $z(x)$ is a smooth function on $\overline{I}$
such that

$z(0)=0,\quad z^{\prime}|_{\overline{I}}\neq 0.$ (17)

By $x(z)$ we denote the mapping inverse to $z(x)$. We will establish the
equality

$\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{x}F(x)\,dx=\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{z}E(z)\,dz,\quad
E(z)=\frac{z}{x(z)}|x^{\prime}(z)|\,F(x(z)).$ (18)

The function $E(z)$ is defined by the specified expression on the interval
$z(I)$ and is continued by zero on its complement. Note that due to conditions
(17) the function $E(z)$ is regular at zero, and so it belongs to
$C_{0}^{\infty}({\mathbb{R}})$.

Suppose that $z^{\prime}>0$ (the opposite case $z^{\prime}<0$ is treated
analogously). For small enough $\varepsilon>0$ we have

$\int_{|x|>\varepsilon}\frac{1}{x}F(x)\,dx=\int_{z(\varepsilon)}^{\infty}+\int_{-\infty}^{z(-\varepsilon)}\frac{1}{z}E(z)\,dz.$
(19)

Set $h:=z^{\prime}(0)$. Then

$\left|\int_{z(\varepsilon)}^{\infty}-\int_{h\varepsilon}^{\infty}\frac{1}{z}E(z)\,dz\right|\leqslant\frac{|z(\varepsilon)-h\varepsilon|\,\sup|E|}{\min(z(\varepsilon),h\varepsilon)}\leqslant
C\varepsilon\sup|E|.$

The integral over the set $z<z(-\varepsilon)$ is analogously approximated by
the integral over the set $z<-h\varepsilon$. Then in view of (19) we obtain
that

$\left|\int_{|x|>\varepsilon}\frac{1}{x}F(x)\,dx-\int_{|z|>h\varepsilon}\frac{1}{z}E(z)\,dz\right|\leqslant
C\varepsilon\sup|E|.$

Sending $\varepsilon$ to zero in this inequality gives (18).

Now turn to expression (5) for the function $u^{f}(x,t)$. The form of this
expression differs from (10), since the set of integration is not a Euclidean
space. However, the definition of the integral in the sense of Cauchy
principal value is adapted to this case in an obvious way. The function $F$ in
(10) should be substituted by the function

$e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}F(\xi,\sigma,\rho)$
(20)

defined on the set $(\xi,\sigma,\rho)\in{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)$ and depending on the parameters $x$, $t$. We say that
a function defined on the specified set belongs to Schwartz space if the
corresponding values of all of the functionals $|\cdot|_{k,p}$ are finite, the
latter being defined (analogously to (11)) as follows. Chose an arbitrary
finite atlas on the sphere $S^{n-1}$. Denote by $\gamma$ a chart from this
atlas, i.e., a diffeomorphism of an open subset of the sphere (which will be
called the domain of definition of a chart) onto an open subset of
${\mathbb{R}}^{n-1}$ ($\gamma$ will also denote points from this subset of
${\mathbb{R}}^{n-1}$, i.e., coordinates). To each chart $\gamma$, we associate
a smooth function $\chi_{\gamma}(\sigma)$ on the sphere supported in the
domain of definition of $\gamma$, such that

$\sum_{\gamma}\chi_{\gamma}\equiv 1.$

Set

$|F|_{k,p}=\sup(1+|\xi|+\rho)^{p}\,|\partial^{\alpha}_{\xi}\partial^{\alpha^{\prime}}_{\gamma}\partial^{\alpha^{\prime\prime}}_{\rho}(\chi_{\gamma}(\sigma(\gamma))F(\xi,\sigma(\gamma)\,\rho))|,$

where the supremum is taken over all charts $\gamma$ of the atlas, points
$\xi,\gamma,\rho$, and multi-indices
$\alpha,\alpha^{\prime},\alpha^{\prime\prime}$, such that
$|\alpha|+|\alpha^{\prime}|+\alpha^{\prime\prime}\leqslant k$.

Note that the function (20) belongs to Schwartz space, as soon as so does
$\hat{f}(\xi,\tau)$ (see (5)). Moreover, the functionals $|\cdot|_{k,p}$ of
this function and its derivatives with respect to $(x,t)$ are locally
uniformly bounded, which allows for applying formula (16). We have

$(2\pi)^{d+n}(\Delta_{t}-\Delta_{x}+m^{2})u^{f}(x,t)=\\\
=\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}\frac{(\Delta_{t}-\Delta_{x}+m^{2})\left(e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}\right)F(\xi,\sigma,\rho)}{1-\rho}\,d\xi
dS_{\sigma}d\rho\\\ =\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}\frac{(1-\rho^{2})(\xi^{2}+m^{2})e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}F(\xi,\sigma,\rho)}{1-\rho}\,d\xi
dS_{\sigma}d\rho\\\ =\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}(\xi^{2}+m^{2})^{n/2}e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}\hat{f}(\xi,\sigma\rho\sqrt{\xi^{2}+m^{2}})\rho^{n-1}\,d\xi
dS_{\sigma}d\rho\\\
=\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{n}}(\xi^{2}+m^{2})^{n/2}e^{i(\langle{}x,\xi\rangle-\langle{}t,\tau\rangle\sqrt{\xi^{2}+m^{2}})}\hat{f}(\xi,\tau\sqrt{\xi^{2}+m^{2}})\,d\xi
d\tau\\\
=\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{n}}e^{i(\langle{}x,\xi\rangle-\langle{}t,\tau\rangle)}\hat{f}(\xi,\tau)\,d\xi
d\tau=(2\pi)^{d+n}f(x,t).$

Thus the function $u^{f}$ is a solution to equation (1).

##  4 The method of stationary phase

In this section, we consider the behavior of the integral

$\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x)e^{is\Phi(x)}\,dx$
(21)

as $s\to+\infty$, assuming that

$F\in\mathcal{S}({\mathbb{R}}^{n}),\quad\Phi\in
C^{\infty}({\mathbb{R}}^{n};{\mathbb{R}}),\quad|\partial^{\alpha}\Phi(x)|\leqslant
C_{\alpha}(1+|x|)^{M_{|\alpha|}}\quad\forall\alpha,$ (22)

for some nonnegative integers $M_{k}$, where $k\geqslant 0$. First we give the
following elementary lemma.

######  Lemma 1.

Assume conditions (22) are satisfied. Suppose also that for all $x\in{\rm
supp}F$ we have

$|\partial\Phi(x)|\geqslant C(1+|x|)^{-M}.$ (23)

Then for any $N$ we have

$\int_{{\mathbb{R}}^{n}}F(x)e^{is\Phi(x)}dx=O(s^{-N}),\quad s\to\pm\infty.$

###### Proof.

By integrating by parts, we get that

$\int_{{\mathbb{R}}^{n}}F(x)e^{is\Phi(x)}dx=-is^{-1}\int_{{\mathbb{R}}^{n}}\frac{F(x)}{|\partial\Phi(x)|^{2}}\langle\partial\Phi(x),\partial\rangle
e^{is\Phi(x)}\,dx\\\
=is^{-1}\int_{{\mathbb{R}}^{n}}(\langle\partial\Phi(x),\partial\rangle+\Delta\Phi(x))\left(\frac{F(x)}{|\partial\Phi(x)|^{2}}\right)e^{is\Phi(x)}\,dx.$

In view of the assumptions on the functions $F$ and $\Phi$, the exponential
under the integral sign is multiplied by a function from Schwartz space in
${\mathbb{R}}^{n}$ supported on a set contained in ${\rm supp}F$. The latter
means that estimate (23) holds true on the support of this function, and so we
can apply our argument once more. After sufficient number of iterations, we
can obtain the factor $s^{-N}$ for any given $N$. ∎

Further, as in sec. 3, we will use the notation
$x^{\prime}=(x_{1},\ldots,x_{n-1})$.

######  Lemma 2.

Assume conditions (22) are satisfied. Suppose also that for all $x\in{\rm
supp}F$ we have

$|\partial_{x^{\prime}}\Phi(x)|\geqslant C(1+|x|)^{-M}.$ (24)

Then the integral (21) equals $O(s^{-N})$ as $s\to\pm\infty$ for any $N$.

###### Proof.

We will estimate the integral

$\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x)e^{is\Phi(x)}\,dx$ (25)

for an arbitrary $\varepsilon>0$. This integral equals

$-is^{-1}\int_{|x_{n}|>\varepsilon}\frac{F(x)}{x_{n}|\partial_{x^{\prime}}\Phi(x)|^{2}}\langle\partial_{x^{\prime}}\Phi(x),\partial_{x^{\prime}}\rangle
e^{is\Phi(x)}\,dx\\\
=is^{-1}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}(\langle\partial_{x^{\prime}}\Phi(x),\partial_{x^{\prime}}\rangle+\Delta_{x^{\prime}}\Phi(x))\left(\frac{F(x)}{|\partial_{x^{\prime}}\Phi(x)|^{2}}\right)e^{is\Phi(x)}\,dx.$

In view of the assumptions on the functions $F$ and $\Phi$, we obtain the
integral of the form (25), in which the function $F$ is substituted by a
certain function $F_{1}$ from Schwartz space in ${\mathbb{R}}^{n}$ supported
on a set contained in ${\rm supp}F$. The latter means that estimate (24) for
the functions $F_{1}$ and $\Phi$ holds true, and so we can iterate our
argument. After a sufficient number of iterations, we get that the integral
(25) equals

$s^{-N}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F_{N}(x)e^{is\Phi(x)}\,dx,$

where $F_{N}\in\mathcal{S}({\mathbb{R}}^{n})$. Then due to (14), we have

$\left|\lim_{\varepsilon\to
0}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F(x)e^{is\Phi(x)}\,dx\right|=|s|^{-N}\left|\lim_{\varepsilon\to
0}\int_{|x_{n}|>\varepsilon}\frac{1}{x_{n}}F_{N}(x)e^{is\Phi(x)}\,dx\right|\\\
\leqslant C|s|^{-N}|F_{N}e^{is\Phi}|_{1,n+1}.$

It remains to observe that

$|F_{N}e^{is\Phi}|_{1,n+1}\leqslant|s|\,|F_{N}|_{1,n+1+M_{0}+M_{1}},$

where $M_{0}$, $M_{1}$ are the exponents in the last inequality in hypothesis
(22). ∎

######  Lemma 3.

Let $F\in\mathcal{S}({\mathbb{R}})$. Then for any $N\geqslant 0$ and $s>0$ we
have

$\left|\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{z}F(z)e^{isz}\,dz-i\pi
F(0)\right|\leqslant C_{N}|F|_{N+1,2}\,s^{-N}.$ (26)

###### Proof.

The integral on the left-hand side of (26) equals the pairing of the
distribution $\mathcal{P}_{1/z}$ with the test function $F(z)e^{isz}$ of the
variable $z$:

$\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{z}F(z)e^{isz}\,dz=\left\langle\mathcal{P}_{1/z},F(z)e^{isz}\right\rangle_{z}.$

This expression equals the pairing of the distribution
$(2\pi)^{-1}\widehat{\mathcal{P}_{1/z}}$ on the inverse Fourier transform of
the specified test function. The latter (as a function of the variable $\xi$)
equals $\check{F}(\xi+s)$. Therefore, by applying formula

$\widehat{\mathcal{P}_{1/z}}(\xi)=-i\pi\,{\rm sgn}\xi,$

we get

$\left\langle\mathcal{P}_{1/z},F(z)e^{isz}\right\rangle_{z}=-i\pi(2\pi)^{-1}\int_{\mathbb{R}}\check{F}(-\xi-s)\,{\rm
sgn}\xi\,d\xi\\\
=i/2\int_{\mathbb{R}}\check{F}(\xi)d\xi-i\int_{0}^{\infty}\check{F}(-s-\xi)\,d\xi=i\pi
F(0)-i\int_{-\infty}^{-s}\check{F}(\xi)\,d\xi.$

The second term decays rapidly when $s$ grows. This term can easily be
estimated by the right-hand side of (26). ∎

######  Lemma 4.

Assume conditions (22) are satisfied, and, besides, the function $F$ is
compactly supported. Suppose also that $\Phi$ satisfies the following
conditions:

$\displaystyle\partial\Phi(x)\neq 0,\quad x\in V,$ (27)
$\displaystyle\partial_{x^{\prime}}\Phi(x^{\prime},0)\neq
0,\quad(x^{\prime},0)\in V\setminus\\{0\\},$ (28)
$\displaystyle\partial_{x^{\prime}}\Phi(0)=0,\quad\det\partial_{x^{\prime}}^{2}\Phi(0)\neq
0,$ (29)

where $V$ is a neighborhood of ${\rm supp}F\cup\\{0\\}$. Then we have (${\rm
sgn}(\partial_{x^{\prime}}^{2}\Phi(0))$ is the difference of number of
positive eigenvalues and negative eigenvalues of the matrix
$\partial_{x^{\prime}}^{2}\Phi(0)$)

$\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}F(x)e^{is\Phi(x)}\,dx=\\\
=i\pi\,{\rm
sgn}(\partial_{x_{n}}\Phi(0))\left(\frac{2\pi}{s}\right)^{(n-1)/2}|\det\partial_{x^{\prime}}^{2}\Phi(0)|^{-1/2}e^{i\pi\,{\rm
sgn}(\partial_{x^{\prime}}^{2}\Phi(0))/4}e^{is\Phi(0)}F(0)\\\
+O(s^{(n+1)/2}),\quad s\to+\infty.$ (30)

###### Proof.

Condition (27) and the first condition in (29) imply that
$\partial_{x_{n}}\Phi(0)\neq 0$. Let $\chi(x)$ be a smooth function, such that
$\partial_{x_{n}}\Phi\neq 0$ on its support. Assume also that $\chi(x)=1$ for
small $x$. The left-hand side in (30) can be represented as the following sum

$\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}(\chi
Fe^{is\Phi})(x)\,dx+\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}((1-\chi)Fe^{is\Phi})(x)\,dx.$
(31)

Now show that the second term equals $O(s^{-N})$ as $s\to+\infty$ for any $N$.
Decompose it as a sum of two terms:

$\mathrm{v.p.}\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}\zeta(x_{n})((1-\chi)Fe^{is\Phi})(x)\,dx+\int_{{\mathbb{R}}^{n}}\frac{1}{x_{n}}(1-\zeta(x_{n}))((1-\chi)Fe^{is\Phi})(x)\,dx,$

where $\zeta$ is a function from $C_{0}^{\infty}({\mathbb{R}})$ that equals
unity in the neighborhood of zero. In view of condition (27) and the
compactness of the support of $F$, the second term can be treated with the use
of Lemma 1. The function $F(x)$ in Lemma 1 should be substituted by the
expression

$(1-\zeta(x_{n}))(1-\chi(x))F(x)/x_{n}.$

Due to condition (28), the first term in the preceding expression can be
treated by Lemma 2, provided that $\zeta$ is supported in a sufficiently small
neighborhood of zero.

Now turn to the first term in (31). By (15) it can be written in the form

$\int_{{\mathbb{R}}^{n-1}}dx^{\prime}\,\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{x_{n}}(\chi
Fe^{is\Phi})(x)\,dx_{n}.$ (32)

The condition $\partial_{x_{n}}\Phi\neq 0$, which is satisfied on the support
of the integrand, allows for changing the variable $x_{n}\mapsto
z=\Phi(x^{\prime},x_{n})-\Phi(x^{\prime},0)$ (the inverse mapping will be
denoted by $x_{n}(x^{\prime},z))$ in the inner integral by formula (18), which
yields

$e^{is\Phi(x^{\prime},0)}\,\mathrm{v.p.}\int_{\mathbb{R}}\frac{1}{z}\frac{z}{x_{n}(x^{\prime},z)}|\partial_{z}x_{n}(x^{\prime},z)|\,\chi(x)F(x)e^{isz}\,dz.$

Now apply Lemma 3 to the integral in this expression. Since
$z/x_{n}|_{z=0}=\partial_{x_{n}}\Phi(x^{\prime},0)$, we have

$\frac{z}{x_{n}(x^{\prime},z)}|\partial_{z}x_{n}(x^{\prime},z)|\,\chi(x)F(x)\big{|}_{z=0}=(\chi
F\,{\rm sgn}\,\partial_{x_{n}}\Phi)(x^{\prime},0)=(\chi F)(x^{\prime},0)\,{\rm
sgn}\,\partial_{x_{n}}\Phi(0).$

We obtain that the integral (32) equals

$i\pi\,{\rm sgn}\,\partial_{x_{n}}\Phi(0)\int_{{\mathbb{R}}^{n-1}}(\chi
Fe^{is\Phi})(x^{\prime},0)\,dx^{\prime}+O(s^{-N}),$

as $s\to+\infty$ for arbitrary $N$. In view of (29), the integral in the last
expression can be treated by the standard method of stationary phase
(providing that the support of $\chi(x)$ is contained in a sufficiently small
neighborhood of the origin), which gives the following asymptotic expression:

$\left(\frac{2\pi}{s}\right)^{(n-1)/2}|\det\partial_{x^{\prime}}^{2}\Phi(0)|^{-1/2}e^{i\pi\,{\rm
sgn}(\partial_{x^{\prime}}^{2}\Phi(0))/4}e^{is\Phi(0)}F(0)+O(s^{(n+1)/2}).$

Thus we arrive at (30). ∎

##  5 Asymptotic behavior of the function $u^{f}$

Represent the integral in (5) as the following sum

$\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}\frac{e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}\zeta(\rho)F(\xi,\sigma,\rho)}{1-\rho}\,d\xi
dS_{\sigma}d\rho\\\ +\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}\frac{e^{i(\langle{}x,\xi\rangle-\langle{}t,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}})}(1-\zeta(\rho))F(\xi,\sigma,\rho)}{1-\rho}\,d\xi
dS_{\sigma}d\rho,$ (33)

where $\zeta(\rho)$ is a function from $C_{0}^{\infty}(0,+\infty)$ that is
equal to unity in a neighborhood of $\rho=1$. Substitution $\tau=\rho\sigma$
in the second term yields

$\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{n}}\frac{e^{i(\langle{}x,\xi\rangle-\langle{}t,\tau\rangle\sqrt{\xi^{2}+m^{2}})}(\xi^{2}+m^{2})^{n/2-1}(1-\zeta(|\tau|))\hat{f}(\xi,\tau\sqrt{\xi^{2}+m^{2}})}{1-\tau^{2}}\,d\xi
d\tau.$

For $x=s\theta$, $t=s\omega$, the exponent of the exponential in the integrand
takes the form

$is\Phi(\xi,\tau;\theta,\omega)=is\left(\langle\theta,\xi\rangle-\langle\omega,\tau\rangle\sqrt{\xi^{2}+m^{2}}\right).$

Under the integral sign, this exponential is multiplied by an expression,
which belongs to Schwartz space if considered as a function of variables
$(\xi,\tau)$. Besides,

$|\partial_{\xi,\tau}\Phi|\geqslant|\partial_{\tau}\Phi|=\left|\omega\sqrt{\xi^{2}+m^{2}}\right|\geqslant
m,$ (34)

so Lemma 1 applies to the integral in consideration, which means that it is
equal to $O(s^{-N})$ as $s\to\infty$ for arbitrary $N$.

Now turn to the first term in (33). For $x=s\theta$, $t=s\omega$, it takes the
following form

$\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times
S^{n-1}\times(0,\infty)}\frac{1}{1-\rho}\,e^{is\Phi(\xi,\sigma,\rho;\theta,\omega)}\,F_{\zeta}(\xi,\sigma,\rho)\,d\xi
dS_{\sigma}d\rho,$ (35)

where

$\displaystyle\Phi(\xi,\sigma,\rho;\theta,\omega)=\langle\theta,\xi\rangle-\langle\omega,\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}},$
(36) $\displaystyle F_{\zeta}(\xi,\sigma,\rho)=\zeta(\rho)F(\xi,\sigma,\rho).$

Let $\gamma$ be a chart on the sphere $S^{n-1}$. By $\gamma$ we will also
denote the corresponding coordinates ranging an open subset of
${\mathbb{R}}^{n-1}$; by $\partial_{\gamma}\Phi$ we denote the derivative

$\partial_{\gamma}(\Phi(\xi,\sigma(\gamma),\rho;\theta,\omega)).$

(In the case $n=1$, we assume that $\gamma=\sigma=\pm 1$.) Let $\chi(\sigma)$
be a smooth function on the sphere $S^{n-1}$ supported in the domain of
definition of the chart $\gamma$. We pass from (35) to the integral of the
same form for the function

$F_{\chi\zeta}(\xi,\sigma,\rho)=\chi(\sigma)F_{\zeta}(\xi,\sigma,\rho).$

This integral can be written as follows

$\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{n-1}\times{\mathbb{R}}}\frac{1}{1-\rho}\,e^{is\Phi(\xi,\sigma(\gamma),\rho;\theta,\omega)}F_{\chi\zeta}(\xi,\sigma(\gamma),\rho)J(\gamma)\,d\xi
d\gamma d\rho,$ (37)

where $J=\partial S_{\sigma}/\partial\gamma$ is the Jacobian determinant
corresponding to the substitution $\sigma\mapsto\gamma(\sigma)$ (in the case
$n=1$, the integral with respect to $\gamma$ equals the sum with respect to
$\gamma=\pm 1$, and we also assume $J=1$, $\gamma(\sigma)=\sigma$). The set of
integration here is larger than the domain of definition of the integrand.
This expression, however, makes sense as the integrand can be smoothly
continued by zero to the set of integration.

In order to apply Lemma 4, we need to find points $(\xi,\sigma,\rho)$, in
which $\rho=1$, $\partial_{\xi}\Phi=0$, and (in the case $n>1$)
$\partial_{\gamma}\Phi=0$. For $n>1$ the last condition implies that
$\sigma(\gamma)=\pm\omega$. For $n=1$ the last equality holds automatically as
$\omega$ and $\sigma$ take values $\pm 1$. Next

$\partial_{\xi}\Phi=\theta-\frac{\langle\omega,\sigma\rangle\rho\,\xi}{\sqrt{\xi^{2}+m^{2}}}=\theta\mp\frac{\xi}{\sqrt{\xi^{2}+m^{2}}}.$
(38)

Hence the condition $\partial_{\xi}\Phi=0$ means that

$\xi^{2}=\frac{m^{2}\theta^{2}}{1-\theta^{2}},\quad\xi=\pm\frac{m\theta}{\sqrt{1-\theta^{2}}}.$

Thus we find two critical points $(\xi,\sigma,\rho)$:

$(\xi,\sigma)=\pm\left(\frac{m\theta}{\sqrt{1-\theta^{2}}},\omega\right),\quad\rho=1.$
(39)

We can cover the sphere $S^{n-1}$ by a finite set of charts and choose
corresponding functions $\chi(\sigma)$ in such a way that their sum is
identically unity on the sphere. Then the integral (35) equals the sum of
integrals of the form (37). Consider first a chart $\gamma$, whose domain of
definition does not contain the points $\pm\omega$ (this is necessary only in
the case $n>1$). We will check that the integral (37) equals $O(s^{-N})$ for
any $N$ in this case. In view of Lemma 2, this is valid if
$|\partial_{\xi,\gamma}\Phi|\geqslant c>0$ on the support of the integrand. We
have ($j\leqslant n-1$)

$\partial_{\gamma_{j}}\Phi=-\langle\omega,\partial_{\gamma_{j}}\sigma\rangle\rho\sqrt{\xi^{2}+m^{2}}.$

As $\sigma(\gamma)\neq\pm\omega$, we can pick $j$ such that
$\langle\omega,\partial_{\gamma_{j}}\sigma\rangle\neq 0$, since the hyperplane
tangent to the sphere $S^{n-1}$ at the point $\sigma$ is not orthogonal to
$\omega$. Besides, on the support of the integrand, the variable $\rho$ is
separated from zero, hence $|\partial_{\gamma_{j}}\Phi|\geqslant c>0$. This
means that Lemma 2 applies to the integral (37).

Thus, to describe the asymptotic behavior of the integral (35), it is
sufficient to consider the integral (37) for charts $\gamma$, whose domain of
definition contain $\pm\omega$. Let the domain of definition of a chart
$\gamma$ contain $\omega$, do not contain $-\omega$, and, besides, let domains
of definition of other charts of the atlas do not contain $\omega$. In
particular, this implies that the corresponding function $\chi$ satisfies
$\chi(\omega)=1$ (in the case $n=1$, when $\omega,\,\sigma=\pm 1$, we assume
that $\chi(\omega)=1$, $\chi(-\omega)=0$). Asymptotic properties of the
integral (37) are determined by the behavior of the integrand at the critical
point $\varkappa=(\xi,\sigma,\rho)$ given by equality (39) with the sign $+$.
This is due to Lemma 4. In order to apply the latter, however, we need to
localize the integral (37) with respect to $\xi$. To this end, we introduce
one more smooth compactly supported cut-off function $\eta(\xi)$ that is equal
to unity in a neighborhood of the point $\xi=m\theta/\sqrt{1-\theta^{2}}$ (see
(39)) and decompose the integral (37) as follows

$\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{n-1}\times{\mathbb{R}}}\frac{-1}{\rho-1}\,e^{is\Phi(\xi,\sigma(\gamma),\rho;\theta,\omega)}\,F_{\eta\chi\zeta}(\xi,\sigma(\gamma),\rho)J(\gamma)\,d\xi
d\gamma d\rho\\\
+\mathrm{v.p.}\int_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{n-1}\times{\mathbb{R}}}\frac{-1}{\rho-1}\,e^{is\Phi(\xi,\sigma(\gamma),\rho;\theta,\omega)}\,(1-\eta(\xi))F_{\chi\zeta}(\xi,\sigma(\gamma),\rho)J(\gamma)\,d\xi
d\gamma d\rho,$

where

$F_{\eta\chi\zeta}(\xi,\sigma,\rho)=\eta(\xi)F_{\chi\zeta}(\xi,\sigma,\rho).$

To estimate the second term, we again apply Lemma 2. Turn to the hypothesis
(24) of this lemma. As can be seen from the first equality in (38), if
$\langle\omega,\sigma\rangle$ and $\rho$ are sufficiently close to unity, and
$\xi$ is sufficiently large, then we have $|\partial_{\xi}\Phi|\geqslant c>0$,
since $|\theta|<1$. To fulfill these conditions, we demand that: the chart
$\gamma$ on $S^{n-1}$ be determined in a sufficiently small neighborhood of
the point $\omega$; the function $\zeta(\rho)$ be nonzero only in a
sufficiently small neighborhood of the point $\rho=1$; the difference
$1-\eta(\xi)$ be nonzero only for sufficiently large $\xi$.

We will treat the first term with the help of Lemma 4. In the conditions of
this lemma (more specifically, in (28), (29)), the critical point zero should
be substituted by $\varkappa$ in the present context. Condition (27) reads
$\partial_{\xi,\gamma,\rho}\Phi\neq 0$. This relation follows, however, from
(34). The first equality of condition (29) has already been checked. Condition
(28) says that the are no critical points other than $\varkappa$ in a
neighborhood of the support of the integrand. This is valid in our situation,
since the domain of definition of the chart $\gamma$ by assumption does not
contain $-\omega$, which corresponds to the second critical point in (39).

Finally, we turn to calculation of quantities occurring in (30). Choose the
coordinate axes in such a way that

$\theta=|\theta|e_{d}\in{\mathbb{R}}^{d},\quad\omega=e_{n}\in{\mathbb{R}}^{n}.$

Then

$\Phi(\xi,\sigma,\rho;\theta,\omega)=|\theta|\xi_{d}-\sigma_{n}\rho\sqrt{\xi^{2}+m^{2}},$

and the first equality in (38) now reads

$\partial_{\xi}\Phi=|\theta|e_{d}-\frac{\sigma_{n}\rho\xi}{\sqrt{\xi^{2}+m^{2}}}.$

The critical point in consideration, which is $\varkappa$, is given by
equalities

$\xi=\frac{m|\theta|e_{d}}{\sqrt{1-\theta^{2}}},\quad\sigma=e_{n},\quad\rho=1.$

We have

$\Phi(\varkappa;\theta,\omega)=-m\sqrt{1-\theta^{2}}.$

Next ($k,j\leqslant d$)

$\partial_{\xi_{k}}\partial_{\xi_{j}}\Phi=\frac{-\sigma_{n}\rho\delta_{k}^{j}}{\sqrt{\xi^{2}+m^{2}}}+\frac{\sigma_{n}\rho\xi_{k}\xi_{j}}{(\xi^{2}+m^{2})^{3/2}}.$

At the point $\varkappa$, we have

$\partial_{\xi_{k}}\partial_{\xi_{j}}\Phi(\varkappa;\theta,\omega)=\frac{-\delta_{k}^{j}}{\sqrt{\xi^{2}+m^{2}}}+\frac{\xi^{2}\delta_{k}^{d}\delta_{j}^{d}}{(\xi^{2}+m^{2})^{3/2}}=\frac{-\delta_{k}^{j}\sqrt{1-\theta^{2}}}{m}+\frac{\xi^{2}\delta_{k}^{d}\delta_{j}^{d}}{(\xi^{2}+m^{2})^{3/2}}\\\
=\frac{-\sqrt{1-\theta^{2}}}{m}(\delta_{k}^{j}-\theta^{2}\delta_{k}^{d}\delta_{j}^{d}).$

Next we calculate the derivatives of the function $\Phi$ with respect to
$\gamma$. This step is required only for $n>1$. It is convenient to choose a
chart $\gamma$ on the sphere in such a way that

$\sigma(\gamma)=(\gamma,\sqrt{1-\gamma^{2}}).$ (40)

We have ($k,j\leqslant n-1$)

$\displaystyle\partial_{\gamma_{j}}\Phi=-\rho\sqrt{\xi^{2}+m^{2}}\,\partial_{\gamma_{j}}\sigma_{n}=\rho\sqrt{\xi^{2}+m^{2}}\,\frac{\gamma_{j}}{\sqrt{1-\gamma^{2}}},$
$\displaystyle\partial_{\gamma_{k}}\partial_{\gamma_{j}}\Phi=\rho\sqrt{\xi^{2}+m^{2}}\left(\frac{\delta_{k}^{j}}{\sqrt{1-\gamma^{2}}}+\frac{\gamma_{j}\gamma_{k}}{(1-\gamma^{2})^{3/2}}\right).$

At the point $\varkappa$ we have $\gamma=0$, and so

$\partial_{\gamma_{k}}\partial_{\gamma_{j}}\Phi(\varkappa;\theta,\omega)=\frac{m\delta_{k}^{j}}{\sqrt{1-\theta^{2}}}.$

Next ($k\leqslant d$, $j\leqslant n-1$)

$\partial_{\xi_{k}}\partial_{\gamma_{j}}\Phi=\frac{\rho\xi_{k}\gamma_{j}}{\sqrt{(\xi^{2}+m^{2})(1-\gamma^{2})}},\quad\partial_{\xi_{k}}\partial_{\gamma_{j}}\Phi(\varkappa;\theta,\omega)=0,$

whence

$\det\partial_{\xi,\gamma}^{2}\Phi(\varkappa;\theta,\omega)=(\det\partial_{\xi}^{2}\Phi\,\det\partial_{\gamma}^{2}\Phi)(\varkappa;\theta,\omega)$

(for $n=1$, we set by the definition
$\partial_{\xi,\gamma}^{2}\Phi=\partial_{\xi}^{2}\Phi$,
$\det\partial_{\gamma}^{2}\Phi=1$). It follows from the equalities obtained
above that

$\displaystyle\det\partial_{\xi}^{2}\Phi(\varkappa;\theta,\omega)=\left(\frac{-\sqrt{1-\theta^{2}}}{m}\right)^{d}(1-\theta^{2}),\quad\det\partial_{\gamma}^{2}\Phi(\varkappa;\theta,\omega)=\left(\frac{m}{\sqrt{1-\theta^{2}}}\right)^{n-1}.$

We summarize our calculations by the following equalities

$\displaystyle\Phi(\varkappa;\theta,\omega)=-m\sqrt{1-\theta^{2}},\quad|\det\partial_{\xi,\gamma}^{2}\Phi(\varkappa;\theta,\omega)|=\frac{(1-\theta^{2})^{(d-n+3)/2}}{m^{d-n+1}},$
$\displaystyle{\rm
sgn}(\partial_{\xi,\gamma}^{2}\Phi(\varkappa;\theta,\omega))=n-1-d.$ (41)

We also need to find the sign of $\partial_{\rho}\Phi$ at $\varkappa$:

$\partial_{\rho}\Phi(\varkappa;\theta,\omega)=-\sqrt{\xi^{2}+m^{2}}<0.$

Now find $J$ occurring in the integral (37). We have ($j\leqslant n-1$)

$\partial_{j}\sigma=e_{j}+e_{n}\partial_{j}\sigma_{n}=e_{j}-\frac{\gamma_{j}e_{n}}{\sqrt{1-\gamma^{2}}},$

therefore, at the point $\varkappa$, in which $\gamma=0$, we have

$J(0)=|\det\\{\langle\partial_{j}\sigma(0),\partial_{k}\sigma(0)\rangle\\}_{j,k}|^{1/2}=1.$

It remains to find the value of the function $F_{\eta\chi\zeta}$ at
$\varkappa$:

$F_{\eta\chi\zeta}(\varkappa)=F(\varkappa)=\frac{m^{n-2}}{2(1-\theta^{2})^{n/2-1}}\,\hat{f}\left(\frac{m\theta}{\sqrt{1-\theta^{2}}},\frac{m\omega}{\sqrt{1-\theta^{2}}}\right)$

(we have taken into account that $\eta=\chi=\zeta=1$ at this point). Thus we
derive the following asymptotic formula for the integral (37) as
$s\to+\infty$:

$i\pi\,\left(\frac{2\pi}{s}\right)^{(d+n-1)/2}\frac{m^{(d-n+1)/2}}{(1-\theta^{2})^{(d-n+3)/4}}e^{i\pi\,(n-d-1)/4}e^{-is\,m\sqrt{1-\theta^{2}}}F(\varkappa)\\\
+O(s^{(d+n+1)/2}).$

The leading term here can be written as

$\frac{(2\pi)^{d+n}}{s^{(d+n-1)/2}}\,e^{-is\,m\sqrt{1-\theta^{2}}}U_{-}^{f}(\theta,\omega),$

where

$U_{-}^{f}(\theta,\omega)=i\pi\,(2\pi)^{-(d+n+1)/2}\frac{m^{(d-n+1)/2}}{(1-\theta^{2})^{(d-n+3)/4}}e^{i\pi\,(n-d-1)/4}F(\varkappa)\\\
=\frac{e^{i\pi(n-d+1)/4}\,m^{(d+n-3)/2}}{4(2\pi)^{(d+n-1)/2}(1-\theta^{2})^{(d+n-1)/4}}\,\hat{f}\left(\frac{m\theta}{\sqrt{1-\theta^{2}}},\frac{m\omega}{\sqrt{1-\theta^{2}}}\right).$

Consider the second critical point in (39), which will be denoted by
$\varkappa^{\prime}$. Similarly to (40), we choose coordinates $\gamma$ as
follows

$\sigma(\gamma)=(\gamma,-\sqrt{1-\gamma^{2}}).$

Then the calculations analogous to those performed previously yield the
equalities

$\displaystyle\Phi(\varkappa^{\prime};\theta,\omega)=m\sqrt{1-\theta^{2}},\quad|\det\partial_{\xi,\gamma}^{2}\Phi(\varkappa^{\prime};\theta,\omega)|=\frac{(1-\theta^{2})^{(d-n+3)/2}}{m^{d-n+1}},$
$\displaystyle{\rm
sgn}(\partial_{\xi,\gamma}^{2}\Phi(\varkappa^{\prime};\theta,\omega))=d-n+1$
(42)

and, besides,

$\partial_{\rho}\Phi(\varkappa^{\prime};\theta,\omega)=\sqrt{\xi^{2}+m^{2}}>0.$

So the leading term in the asymptotic expansion of the integral (37) can be
written as

$\frac{(2\pi)^{d+n}}{s^{(d+n-1)/2}}\,e^{is\,m\sqrt{1-\theta^{2}}}U_{+}^{f}(\theta,\omega),$

where

$U_{+}^{f}(\theta,\omega)=\frac{e^{i\pi(d-n-1)/4}\,m^{(d+n-3)/2}}{4(2\pi)^{(d+n-1)/2}(1-\theta^{2})^{(d+n-1)/4}}\,\hat{f}\left(\frac{-m\theta}{\sqrt{1-\theta^{2}}},\frac{-m\omega}{\sqrt{1-\theta^{2}}}\right).$

We can rewrite the expressions for $U^{f}_{\pm}$ in the following form

$U_{\pm}^{f}(\theta,\omega)=\frac{e^{\pm
i\pi(d-n-1)/4}}{4m}\left(\frac{m}{2\pi\sqrt{1-\theta^{2}}}\right)^{(d+n-1)/2}\,\hat{f}\left(\frac{\mp
m\theta}{\sqrt{1-\theta^{2}}},\frac{\mp m\omega}{\sqrt{1-\theta^{2}}}\right).$

Relation (9) is a consequence of this formula.

##  6 Asymptotic behavior of the function $u^{a}$

For $x(s)=s\theta$, $t(s)=s\omega$, we have

$\displaystyle
u^{a}(s\theta,s\omega)=(2\pi)^{-d-n}\int_{{\mathbb{R}}^{d}\times
S^{n-1}}e^{is\Phi(\xi,\sigma;\theta,\omega)}A(\xi,\sigma)d\xi dS_{\sigma},$
$\displaystyle\Phi(\xi,\sigma;\theta,\omega)=\langle\theta,\xi\rangle-\langle\omega,\sigma\rangle\sqrt{\xi^{2}+m^{2}}.$
(43)

To describe asymptotic properties of this integral as $s\to+\infty$, we apply
the method of stationary phase. The phase function $\Phi$ coincides with the
function defined in (36) restricted to the set $\\{\rho=1\\}$. This allows us
to use the results of calculations from the previous section. In particular,
the critical points $(\xi,\sigma)$, in which $\partial_{\xi,\sigma}\Phi=0$,
are given by the first equality in (39). The values of the function $\Phi$ and
its derivatives at critical points, which are required for asymptotic
analysis, are given by formulas (41), (42). The justification of the method of
stationary phase (localization with respect to $\xi$, passing to local
coordinates on the sphere $S^{n-1}$) is quite analogous to that made in the
previous section, so we will not expose it here.

Thus the function $u^{a}$ has the asymptotic property of the form (2) with the
following coefficients

$U^{a}_{\pm}(\theta,\omega)=\frac{e^{\pm
i\pi(d-n+1)/4}}{(2\pi)^{(d+n+1)/2}}\frac{m^{(d-n+1)/2}}{(1-\theta^{2})^{(d-n+3)/4}}\,A\left(\frac{\mp
m\theta}{\sqrt{1-\theta^{2}}},\mp\omega\right).$

This relation can be written in terms of the density $a(\xi,\tau)$ as follows

$U^{a}_{\pm}(\theta,\omega)=\frac{e^{\pm i\pi(d-n+1)/4}}{4\pi
m}\left(\frac{m}{2\pi\sqrt{1-\theta^{2}}}\right)^{(d+n-1)/2}\,a\left(\frac{\mp
m\theta}{\sqrt{1-\theta^{2}}},\frac{\mp m\omega}{\sqrt{1-\theta^{2}}}\right).$

Relation (8) follows from this formula.

The formulas for $U^{f}_{\pm}$, $U^{a}_{\pm}$ obtained above prove Theorem 1.
Formula (7) allows expressing the density $a$ via the function $\hat{f}$ and
one of the functions $U_{\pm}$. This leads to the relation given in the
formulation of Theorem 2. Inclusion $a\in\mathcal{S}(\Sigma_{m})$ follows from
this relation. The density $a$ is determined uniquely at every point of
$\Sigma_{m}$, which leads to the uniqueness of solution of the form (4) that
has the asymptotic property (2) with a given coefficient $U_{+}$ (or $U_{-}$).
Thus Theorem 2 is proved.

##  7 Asymptotic behavior of the function $u$ along characteristic directions

Now we discuss the behavior of solution (4) when $(x,t)$ tends to infinity
along a characteristic direction, which will be parameterized by unit vectors
$(\theta,\omega)\in S^{d-1}\times S^{n-1}$ and a parameter $q\in{\mathbb{R}}$:

$x(s)=(s+q)\theta,\quad t(s)=s\omega.$ (44)

For simplicity, we will assume $f=0$. Then $u=u^{a}$, and we need to examine
the behavior of the integral

$u(x(s),t(s))=(2\pi)^{-d-n}\int_{{\mathbb{R}}^{d}\times
S^{n-1}}e^{is\Phi(\xi,\sigma;\theta,\omega)}e^{iq\langle\theta,\xi\rangle}a(\xi,\sigma)\,d\xi
dS_{\sigma}$

as $s\to\infty$, where the function $\Phi$ is given by (43). The product
$e^{iq\langle\theta,\xi\rangle}a(\xi,\sigma)$ considered as a function of
variables $(\xi,\sigma)$ belongs to Schwartz space. Next

$|\partial_{\xi}\Phi|=\left|\theta-\frac{\langle\omega,\sigma\rangle\xi}{\sqrt{\xi^{2}+m^{2}}}\right|\geqslant
1-\frac{|\xi|}{\sqrt{\xi^{2}+m^{2}}}\geqslant\frac{C_{m}}{1+|\xi|}.$

This means that Lemma 1 applies to the integral in the preceding formula
(after substituting $\sigma$ by local coordinates), since condition (23) is
satisfied for $M=-1$. Thus for any $N$ we have

$u(x(s),t(s))=O(s^{-N}),\quad s\to\infty.$

A particular case of equation (1) is Klein-Gordon-Fock equation ($n=1$,
$f=0$), for which the Cauchy problem with the initial data given, for example,
on the hypersurface $\\{t=0\\}$:

$u|_{t=0}=u_{0},\quad u|_{t=0}=u_{1},$

is well-posed. If the initial data $u_{0}$, $u_{1}$ belong to
$\mathcal{S}({\mathbb{R}}^{d})$, then the solution to this problem can be
written in the form (6) with the following density $A(\xi,\sigma)$ (in the
present context, $\sigma=\pm 1$)

$A(\xi,\pm
1)=\frac{\hat{u}_{0}(\xi)}{2}\pm\frac{i\hat{u}_{1}(\xi)}{2\sqrt{\xi^{2}+m^{2}}}.$

(In the case under consideration, when $n=1$, formula (6) with such a function
$A$ can, of course, be obtained by the standard Fourier method.) Thus our
results apply to a solution to the Cauchy problem. In particular, the last
equality together with formula (7) imply that the asymptotic behavior of the
solution along a timelike direction is described in terms of the Fourier
transform of the initial data. However, the solution decays faster than any
power of distance and time along a characteristic direction. Thus we see a
distinct contrast between behavior of solutions to Klein-Gordon-Fock equation
and that of the wave equation. Solutions to the analogous Cauchy problem for
the wave equation decay rapidly along timelike directions (for odd $d$, this
follows from Huygens principle), and exhibit nontrivial asymptotic behavior
along characteristic directions. In the case $d=3$, the asymptotic behavior is
described by the following relation [2]

$u(x(s),t(s))=\frac{1}{4\pi
s}\left(-\partial_{q}(Ru_{0}(\theta,q))+Ru_{1}(\theta,q)\right)+o(s^{-1}),\quad
s\to\pm\infty,$

in which $x(s)$, $t(s)$ are given by (44), $R$ is Radon transform:

$Rv(\theta,q)=\int_{\langle{}x,\theta\rangle=q}v(x)\,dS_{x}.$

Thus in the case of the wave equation, the behavior of a solution at infinity
is described in terms of Radon transform of the initial data (rather than
Fourier transform).

## References

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