# On-diagonal asymptotics for heat kernels of a class of inhomogeneous partial
differential operators

Evan Randles Corresponding author<EMAIL_ADDRESS>Address:
Department of Mathematics, Colby College, Waterville, Maine 04901, USA
Department of Mathematics, Cornell University, Ithaca, New York 14853-4201,
USA  Laurent Saloff-Coste  Department of Mathematics, Cornell University,
Ithaca, New York 14853-4201, USA

###### Abstract

We consider certain constant-coefficient differential operators on
$\mathbb{R}^{d}$ with positive-definite symbols. Each such operator $\Lambda$
with symbol $P$ defines a semigroup $e^{-t\Lambda}$ , $t>0$ , admitting a
convolution kernel $H^{t}_{P}$ for which the large-time behavior of
$H_{P}^{t}(0)$ cannot be deduced by basic scaling arguments. The simplest
example has symbol $P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$ ,
$\xi=(\eta,\zeta)\in\mathbb{R}^{2}$ . We devise a method to establish large-
time asymptotics of $H^{t}_{P}(0)$ for several classes of examples of this
type and we show that these asymptotics are preserved by perturbations by
certain higher-order differential operators. For the $P$ just given, it turns
out that $H^{t}_{P}(0)\sim c_{P}t^{-5/8}$ as $t\to\infty$ . We show how such
results are relevant to understand the convolution powers of certain complex
functions on $\mathbb{Z}^{d}$ . Our work represents a first basic step towards
a good understanding of the semigroups associated with these operators.
Obtaining meaningful off-diagonal upper bounds for $H_{P}^{t}$ remains an
interesting challenge.

2020 Mathematics Subject Classification: Primary 35K08, 35K25; Secondary
47D06, 42B99.

Keywords: On-diagonal heat kernel asymptotics, inhomogeneous partial
differential operators, convolution powers

##  1 Introduction

On $\mathbb{R}^{2}$, consider the constant-coefficient partial differential
operator

$\Lambda=\partial_{x_{1}}^{4}+\partial_{x_{2}}^{4}+2i\partial_{x_{1}}\partial_{x_{2}}^{2}-\partial_{x_{1}}^{2}$

and its symbol

$P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$

defined for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. It is evident that $\Lambda$
is a non-negative, symmetric, and fourth-order elliptic operator. Thus, when
defined initially on the set of compactly supported smooth functions,
$C_{c}^{\infty}(\mathbb{R}^{2})$, $\Lambda$ extends uniquely to a non-negative
self-adjoint operator on $L^{2}(\mathbb{R}^{2})$ which, by an abuse of
notation, we denote by $\Lambda$. Via the spectral calculus or the Hille-
Yosida construction, $-\Lambda$ generates a continuous one-parameter semigroup
of contractions on $L^{2}(\mathbb{R}^{2})$ which is denoted by
$\\{e^{-t\Lambda}\\}$ and called the heat semigroup associated to $\Lambda$.
Thanks to the Fourier transform111On $\mathbb{R}^{d}$, we shall take the
Fourier transform $\mathcal{F}$ and inverse Fourier transform
$\mathcal{F}^{-1}$ to be given by
$\mathcal{F}(f)(\xi)=\widehat{f}(\xi)=\int_{\mathbb{R}^{d}}f(x)e^{ix\cdot\xi}\,dx$
for $f\in L^{2}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})$ and
$\mathcal{F}^{-1}(g)(x)=\check{g}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}g(\xi)e^{-ix\cdot\xi}\,d\xi$
for $g\in L^{2}(\mathbb{R}^{d})\cap L^{1}(\mathbb{R}^{d})$, respectively.,
this semigroup has the integral representation

$(e^{-t\Lambda}f)(x)=\int_{\mathbb{R}^{2}}H_{P}^{t}(x-y)f(y)\,dy$ (1)

for each $f\in L^{2}(\mathbb{R}^{2})$ where $H_{P}=H_{P}^{(\cdot)}(\cdot)$ is
called the heat kernel associated to $\Lambda$ and is given by

$H_{P}^{t}(x)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{-tP(\xi)}e^{-ix\cdot\xi}\,d\xi$
(2)

for $t>0$ and $x\in\mathbb{R}^{2}$. For its central role in the analysis
surrounding $\Lambda$, including its spectral theory, associated Sobolev
inequalities, and properties of the semigroup $\\{e^{-t\Lambda}\\}$, we are
interested in the behavior and properties of the heat kernel $H_{P}$. As we
demonstrate below, this curiosity is further spurred by the appearance of
$H_{P}$ as a scaled limit of convolution powers of complex-valued functions on
$\mathbb{Z}^{2}$, just as the Gaussian density appears as the scaled limit in
the local (central) limit theorem [RSC17].

Consider the function $\phi:\mathbb{Z}^{2}\to\mathbb{C}$ defined by

$\phi=\frac{1}{3840000}\left(\frac{1}{12}\phi_{1}+\frac{1}{8}\phi_{2}+\frac{1}{3}\phi_{3}\right)$
(3)

where

$\phi_{1}(x)=\begin{cases}41375061&(x_{1},x_{2})=(0,0)\\\ 1080000\pm
969232i&(x_{1},x_{2})=\pm(1,0)\\\ -165072&(x_{1},x_{2})=\pm(2,0)\\\ 72000\pm
9024&(x_{1},x_{2})=\mp(3,0)\\\ -38256&(x_{1},x_{2})=\pm(4,0)\\\
0&\mbox{else}\end{cases},\hskip
28.45274pt\phi_{2}(x)=\begin{cases}1228800&(x_{1},x_{2})=(0,\pm 1)\\\
-286328&(x_{1},x_{2})=(0,\pm 2)\\\ -9524&(x_{1},x_{2})=(0,\pm 4)\\\
2232&(x_{1},x_{2})=(0,\pm 6)\\\ -179&(x_{1},x_{2})=(0,\pm 8)\\\
0&\mbox{else}\end{cases}$

and

$\phi_{3}(x)=\begin{cases}\pm 115200i&(x_{1},x_{2})=(\mp 1,1),(\mp 1,-1)\\\
\pm 6939i&(x_{1},x_{2})=(\mp 1,2),(\mp 1,-2)\\\ 216&(x_{1},x_{2})=(\pm
2,2),(\pm 2,-2)\\\ 1128i&(x_{1},x_{2})=(\pm 3,2),(\pm 3,-2)\\\ \pm
1062i&(x_{1},x_{2})=(\pm 1,4),(\pm 1,-4)\\\ -54&(x_{1},x_{2})=(\pm 2,4),(\pm
2,-4)\\\ \pm 77i&(x_{1},x_{2})=(\mp 1,6),(\mp 1,-6)\\\
0&\mbox{else}\end{cases}$

for $x=(x_{1},x_{2})\in\mathbb{Z}^{2}$. With this function, we define its
iterated convolution powers $\phi^{(n)}:\mathbb{Z}^{2}\to\mathbb{C}$ by
putting $\phi^{(1)}=\phi$ and, for $n\geq 2$,

$\phi^{(n)}(x)=\sum_{y\in\mathbb{Z}^{2}}\phi^{(n-1)}(x-y)\phi(y)$

for $x\in\mathbb{Z}^{2}$. Motivated by applications to data smoothing and
partial differential equations, we are interested in the asymptotic behavior
of $\phi^{(n)}$ as $n\to\infty$. Given the nature of convolution, it is
reasonable to expect that the mass of $\phi^{(n)}$ “spreads out” on
$\mathbb{Z}^{2}$ as $n$ increases, however, exactly how it does this is not a
priori clear. In Section 5, we show that, for large $n$, $\phi^{(n)}$ is well
approximated by the heat kernel $H_{P}$ evaluated at $t=n/100$. This so-called
local limit theorem is illustrated in Figure 1 and it motivates us to
understand $H_{P}^{t}(\cdot)$ for large $t$.

(a) $\operatorname{Re}(\phi^{(n)})$ for $n=1000$

(b) $\operatorname{Re}(H_{P}^{t})$ for $t=n/100$

Figure 1: The graphs of $\operatorname{Re}(\phi^{(n)})$ and
$\operatorname{Re}(H_{P})$ for $n=1000$.

In light of the fast growth of $P(\xi)$ as $\xi\to\infty$, it is not difficult
to show that, for each $t>0$, $x\mapsto H_{P}^{t}(x)$ is a Schwartz function.
Given that $\Lambda$ is a fourth-order (uniformly) elliptic operator, $H_{P}$
is known to satisfy the following estimate: There are positive constants
$A,B$, and $C$ for which

$\absolutevalue{H_{P}^{t}(x)}\leq\frac{A}{t^{1/2}}\exp\left(-Bt\absolutevalue{\frac{x}{t}}^{4/3}+Ct\right)$
(4)

for $t>0$ and $x\in\mathbb{R}^{2}$ [Ei69, Da95]. Using this so-called off-
diagonal estimate, it is easily verified that the semigroup
$\\{e^{-t\Lambda}\\}$, initially defined on $L^{2}(\mathbb{R}^{2})$, extends
uniquely to a continuous semigroup $\\{e^{-t\Lambda_{p}}\\}$ on
$L^{p}(\mathbb{R}^{2})$ for all $1\leq p\leq\infty$. In fact, this estimate
also guarantees that the spectrum of $\Lambda=\Lambda_{p}$ independent of $p$,
c.f., [Da95, Theorem 20].

For the present article, we shall focus our attention on the so-called on-
diagonal behavior of $H_{P}^{t}(x)$. That is, we are interested in the
behavior of

$\varphi(t):=H_{P}^{t}(0)=\sup_{x\in\mathbb{R}^{2}}\absolutevalue{H_{P}^{t}(x)}$

defined for $t>0$. Given that $P$ is non-negative, $\varphi$ is evidently a
non-increasing function on $(0,\infty)$. In view of the representation (1), we
observe immediately that

$\|e^{-t\Lambda}\|_{1\to\infty}=\sup_{x\in\mathbb{R}^{d}}\absolutevalue{H_{P}^{t}(x)}=\varphi(t)$

for $t>0$, i.e., $e^{-t\Lambda}$ is a bounded operator from
$L^{1}(\mathbb{R}^{2})$ to $L^{\infty}(\mathbb{R}^{2})$ with operator norm
precisely $\varphi(t)$. By virtue of Plancherel’s theorem,

$\|H_{P}^{t}\|_{2}^{2}=\frac{1}{(2\pi)^{2}}\|e^{-tP}\|_{2}^{2}=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{-2tP(\xi)}\,d\xi=H_{P}^{2t}(0)=\varphi(2t)$

for $t>0$. From this it follows that $e^{-t\Lambda}$ is a bounded operator
from $L^{2}(\mathbb{R}^{2})$ to $L^{\infty}(\mathbb{R}^{2})$ with

$\|e^{-t\Lambda}\|_{2\to\infty}=\|H_{P}^{t}\|_{2}=\sqrt{\varphi(2t)}$

for $t>0$. In other words, the semigroup $\\{e^{-t\Lambda}\\}$ is
ultracontractive with bound characterized by $\varphi$. It is well known that
the ultracontractivity of the semigroup $\\{e^{-t\Lambda}\\}$ can be used to
establish various Sobolev-type inequalities associated to $\Lambda$, e.g.,
Sobolev, Nash, Gagliardo-Nirenberg [CKS87, Co96, Da89, RSC20, SC09, Va85]. To
do this, however, it is necessary to have a good understanding of the function
$\varphi(t)$. Further, given our motivation to understand the convolution
powers of the example $\phi$ given above, we are especially interested in
understanding the function $\varphi(t)$ for large values of $t$ and, in fact,
we will find that $\|\phi^{(n)}\|_{\infty}\asymp\varphi(n)$ for222Here and in
what follows, for real-valued functions $f$ and $g$ defined, at least, on a
non-empty set $X$, we shall write $f(x)\asymp g(x)$ for $x\in X$ to mean that
there are positive constants $C$ and $C^{\prime}$ for which $Cg(x)\leq
f(x)\leq C^{\prime}g(x)$ for all $x\in X$.
$n\in\mathbb{N}_{+}:=\\{1,2,\dots\\}$.

Analyzing $\varphi$ in small time is straightforward. Using (4), we see that

$\varphi(t)\leq\frac{C}{t^{1/2}}$ (5)

for all $0<t\leq 1$ where $C$ is a positive constant. We can, of course, do
better by employing the following elementary scaling argument: Upon noting
that

$\lim_{t\to 0}tP(t^{-1/4}\xi)=\lim_{t\to
0}tP(t^{-1/4}\eta,t^{-1/4}\zeta)=\eta^{4}+\zeta^{4}$ (6)

for each $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$, the change of variables
$\xi\mapsto t^{-1/4}\xi=(t^{-1/4}\eta,t^{-1/4}\zeta)$ in (2) yields

$\lim_{t\to
0}t^{1/2}\varphi(t)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{-(\eta^{4}+\zeta^{4})}\,d\eta\,d\zeta=\frac{1}{\pi^{2}}\Gamma(5/4)^{2}\approx
0.0832420$ (7)

where $\Gamma$ is Euler’s Gamma function. Consequently, $\varphi(t)\asymp
t^{-1/2}$ for $0<t\leq 1$. It is noteworthy that scaled limit (6) at $t=0$
“picks out” the homogeneous fourth-order polynomial
$P_{0}(\xi)=\eta^{4}+\zeta^{4}$ which is precisely the principal symbol of
$\Lambda$. Though not directly related, we refer the reader to the works of
Evgrafov and Postnikov [EP70], and Tintarev [Ti82] who established short-time
off-diagonal asymptotics similar to the right hand side in (4) for heat
kernels of higher-order elliptic operators. See also the related works of
Barbatis and Davies [BD96, Ba98, Ba01, Da95a].

Following Davies [Da95a] and driven by the motivations previously discussed,
we seek to understand the behavior of $\varphi(t)$ in large time. For this
goal, it is clear that the estimate (4) is useless for, in contrast to
$\varphi(t)$, the function $t\mapsto t^{-1/2}\exp(Ct)$ is increasing for
$t>1/2C$. In looking back through the scaling argument above, we wonder:
Perhaps there is a rescaling of the symbol $P$ that will yield an asymptotic
(or simply a useful estimate) for $\varphi$ in large time. In fact, this type
of approach was taken in [Da95a] to characterize the large-time on-diagonal
behavior of the heat kernel associated to the fourth-order elliptic operator
$f\mapsto\frac{d^{4}f}{dx^{4}}-\frac{d^{2}f}{dx^{2}}$ in one spatial
dimension. We note that Davies’ results addressed a question posed by M. van
den Berg concerning the necessity of the term $Ct$ in off-diagonal estimates
of the form (4). Given real numbers $\alpha$ and $\beta$, we make the change
of variables $\xi=(\eta,\zeta)\mapsto(t^{-\alpha}\eta,t^{-\beta}\zeta)$ in the
integral (2) to see that

$\varphi(t)=\frac{t^{-(\alpha+\beta})}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{-tP(t^{-\alpha}\eta,t^{-\beta}\zeta)}\,d\xi$

for $t>0$. With the aim of mimicking our small-time scaling argument, we seek
values of $\alpha$ and $\beta$ for which $tP(t^{-\alpha}\eta,t^{-\beta}\zeta)$
is sufficiently well behaved as $t\to\infty$. Upon noting that

$tP(t^{-\alpha}\eta,t^{-\beta}\zeta)=t^{1-2\alpha}\eta^{2}+t^{1-\alpha-2\beta}2\eta\zeta^{2}+t^{1-4\alpha}\eta^{4}+t^{1-4\beta}\zeta^{4}$

for $t>0$ and $(\eta,\zeta)\in\mathbb{R}^{2}$ and considering all
possibilities for $\alpha$ and $\beta$, we find that

$\lim_{t\to\infty}tP(t^{-\alpha}\eta,t^{-\beta}\zeta)=\begin{cases}(\eta+\zeta^{2})^{2}&\alpha=1/2,\,\beta=1/4\\\
\eta^{2}&\alpha=1/2,\,\beta>1/4\\\ \zeta^{4}&\alpha>1/2,\beta=1/4\\\
0\,\,\mbox{or}\,\,\infty&\mbox{otherwise}\end{cases}$

for almost every $(\eta,\zeta)\in\mathbb{R}^{2}$. Given that none of the
functions $(\eta,\zeta)\mapsto e^{-(\eta+\zeta^{2})^{2}}$,
$(\eta,\zeta)\mapsto e^{-\eta^{2}}$, or $(\eta,\zeta)\mapsto e^{-\zeta^{4}}$
are integrable, it follows that

$\lim_{t\to\infty}\int_{\mathbb{R}^{2}}e^{-tP(t^{-\alpha}\eta,t^{-\beta}\zeta)}\,d\xi$

is either $0$ or $\infty$ for all possible cases of $\alpha$ and $\beta$.
Consequently, the argument we used to establish the small-time asymptotics for
$\varphi$ is not helpful to us. In fact, it can be shown that no “reasonable”
scaling, which is linear in $\xi$, can be used to deduce the asymptotic
behavior of $\varphi(t)$ as $t\to\infty$. These observations are tied, in some
sense, to the absence of a well-behaved lower-order component of $P$
characterizing the behavior of $\varphi(t)$ in large time just as the
principal symbol $\eta^{4}+\zeta^{4}$ does for small time. Without a tractable
scaling method for large time, the nature of the decay of $\varphi(t)$ as
$t\to\infty$ – be it exponential, polynomial, or otherwise – is not a priori
clear. Our main theorem, Theorem 3.1, yields the (to us) surprising conclusion
that $\varphi(t)\asymp t^{-5/8}$ for $t\geq 1$. By an application of Theorem
3.4, we are able to obtain the “true asymptotic”,

$\lim_{t\to\infty}t^{5/8}\varphi(t)=\frac{1}{2\pi^{3/2}}\Gamma(9/8)\approx
0.0845624.$

Taking this example as motivation, we introduce and study a class of symbols
on $\mathbb{R}^{d}$ for which it is possible to establish on-diagonal heat
kernel asymptotics in large (and small) time. For these examples, the small-
time behavior is often characterized by a principal homogeneous term and we
focus on the interesting question of large-time behavior. In particular, for
positive integers $a$ and $b$ with $d=a+b$, we consider a polynomial of the
form

$P(\xi)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{d}=\mathbb{R}^{a}\times\mathbb{R}^{b}$
where $P_{1}$ and $P_{2}$ are real-valued positive homogeneous polynomials in
sense of [RSC17] and $Q:\mathbb{R}^{b}\to\mathbb{R}^{a}$ is a so-called
multivariate nondegenerate homogeneous polynomial. The notions of positive
homogeneous and multivariate nondegenerate homogeneous are presented in
Section 2; we remark that the prototypical example of a positive homogeneous
polynomial is a positive-definite and homogeneous semi-elliptic polynomial
[Hö83, RSC17, RSC17a, RSC20]. The homogeneous structure (and order) of $P_{1}$
and $P_{2}$ need not coincide and so $P$ is generally inhomogeneous and
further, as seen in our motivating example, no rescaling of $P$ in large time
yields a tractable homogeneous term. To the constant-coefficient operator
$\Lambda$ on $\mathbb{R}^{d}$ with symbol $P$, provided that $P$ grows
sufficiently fast as $\xi\to\infty$ (which will always be the case for us),
there corresponds a heat semigroup $\\{e^{-t\Lambda}\\}$ with heat kernel
given by

$H_{P}^{t}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-tP(\xi)}e^{-ix\cdot\xi}\,d\xi$

for $t>0$ and $x\in\mathbb{R}^{d}$ and, with this, we define
$\varphi(t)=H_{P}^{t}(0)$ for $t>0$. Under certain hypotheses concerning
$P_{1}$, $P_{2}$, and $Q$, our main theorem (Theorem 3.1) gives positive
numbers $\mu_{0}$ and $\mu_{\infty}$ for which

$\varphi(t)\asymp\begin{cases}t^{-\mu_{0}}&0<t\leq 1\\\
t^{-\mu_{\infty}}&t\geq 1\end{cases}$

for $t>0$; in particular, our result describes the elusive behavior of
$\varphi(t)$ in large time. Under one addition hypothesis, we also show that
the limits

$\lim_{t\to 0}t^{\mu_{0}}\varphi(t)\hskip 28.45274pt\mbox{and}\hskip
28.45274pt\lim_{t\to\infty}t^{\mu_{\infty}}\varphi(t)$

exist and are positive computable numbers depending only on $P_{1}$, $P_{2}$
and $Q$; this is Theorem 3.4. We note that Theorems 3.1 and 3.4 are stated in
terms of positive homogeneous functions $P_{1}$ and $P_{2}$ (in the sense of
[BR22]) and a multivariate nondegenerate homogeneous function $Q$ (introduced
in Section 2) and, correspondingly, $P$ need not be a polynomial nor $H_{P}$
correspond to a constant-coefficient partial differential operator. Following
Section 3, we treat a perturbation theory in which $P$ is replaced by $P+R$
where $R(\xi)=o(P(\xi))$ as333This “little-o” notation means that, for each
$\epsilon>0$, there is an open neighborhood
$\mathcal{O}\subseteq\mathbb{R}^{d}$ of $0$ for which
$\absolutevalue{R(\xi)}\leq\epsilon P(\xi)$ whenever $\xi\in\mathcal{O}$.
$\xi\to 0$ and, under certain conditions, we show that
$H_{P+R}^{t}(0)\asymp\varphi(t)\asymp t^{-\mu_{\infty}}$ for $t\geq 1$; this
is Theorem 4.1. Further, viewing it essentially as a perturbation problem, we
then apply our methods to the related problem of determining the asymptotic
behavior of the convolution powers of certain complex-valued functions $\phi$
on $\mathbb{Z}^{d}$. Our results in this direction, Theorem 5.1 and Corollary
5.2, describe the asymptotic behavior of $\phi^{(n)}$ in form of local limit
theorems and sup-norm asymptotics. Our theory provides an inhomogeneous
counterpart to the homogeneous theory developed in [RSC17], [BR22], and
[Ra22]. Specifically, Theorem 5.1 can be compared to Theorem 1.6 in [RSC17]
and Theorems 1.9 and 3.8 of [Ra22] and Corollary 5.2 can be compared to
Theorem 4.1 of [RSC17], Theorem 3.2 of [BR22], and Theorem 3.1 of [Ra22].
Applying our results to the $\phi$ discussed in this introduction, we find
that $\|\phi^{(n)}\|_{\infty}\asymp n^{-5/8}$ for $n\in\mathbb{N}_{+}$ and
obtain the value of the (existent) limit,
$\lim_{n\to\infty}n^{5/8}\|\phi^{(n)}\|_{\infty}$.

The entire theory developed in this article has a parallel version where the
large-time behavior of $\varphi(t)$ is easy to compute while the small-time
behavior is unclear. In fact, a key to our proof of Theorem 3.1 is that the
heat kernel $H_{P}$ associated to $\Lambda$ agrees on the diagonal with the
heat kernel $H_{\widetilde{P}}$ associated to a “dual” constant-coefficient
partial differential operator $\widetilde{\Lambda}$, i.e.,

$\varphi(t)=H_{P}^{t}(0)=H_{\widetilde{P}}^{t}(0)=\widetilde{\varphi}(t)$

for all $t>0$. The utility of this correspondence is that the large-time
behavior of $\widetilde{\varphi}$ is easily and directly computed. For the
fourth-order elliptic operator $\Lambda$ considered in this introduction,

$\widetilde{\Lambda}=-\partial_{x_{1}}^{2}+\partial_{x_{1}}^{4}+4i\partial_{x_{1}}^{3}\partial_{x_{2}}^{2}-6\partial_{x_{1}}^{2}\partial_{x_{2}}^{4}-4i\partial_{x_{1}}\partial_{x_{2}}^{6}+\partial_{x_{2}}^{8}$

which is not an elliptic operator, nor is it semi-elliptic or even
hypoelliptic. Consequently, the heat kernel corresponding to
$\widetilde{\Lambda}$, especially for small time, is not well understood. Akin
to the fact that $\Lambda$ has elliptic principal part
$\Lambda_{0}=\partial_{x_{1}}^{4}+\partial_{x_{2}}^{4}$ which determines the
small-time decay of $\varphi(t)$, $\widetilde{\Lambda}$ has “lowest-order”
component which is well behaved and determines the behavior of
$\widetilde{\varphi}(t)$ in large time. This component, which could be called
the principal symbol at infinity, is the operator

$\widetilde{\Lambda}_{\infty}=-\partial_{x_{1}}^{2}+\partial_{x_{2}}^{8}$

which is semi-elliptic and determines the $5/8=1/2+1/8$ exponent of polynomial
decay of $\widetilde{\varphi}=\varphi$ for large time. In view of this, the
authors see evidence for a useful notion of large-time semi-ellipticity for
operators and a theory surrounding it which is akin (and perhaps dual to) the
standard theory of elliptic/semi-elliptic operators. In Section 6, we discuss
this and future directions of the theory presented in this article.

##  2 Homogeneous Functions

In this section, we give a brief account of the theory of positive homogeneous
functions (presented more fully in [BR22]) and introduce a multivariate
generalization of such functions, which we will call nondegenerate
multivariate homogeneous functions. For a positive integer $d$, we shall
denote by $\operatorname{End}(\mathbb{R}^{d})$ the set of linear endomorphisms
of $\mathbb{R}^{d}$ and by $\operatorname{Gl}(\mathbb{R}^{d})$ the
corresponding subset of automorphisms. We shall take
$\operatorname{End}(\mathbb{R}^{d})$ to be equipped with the operator norm
$\|\cdot\|$ inherited from the standard Euclidean norm $\absolutevalue{\cdot}$
on $\mathbb{R}^{d}$. For a given $E\in\operatorname{End}(\mathbb{R}^{d})$, we
define $T:(0,\infty)\to\operatorname{Gl}(\mathbb{R}^{d})$ by

$T_{t}=t^{E}=\exp((\ln t)E)=\sum_{k=0}^{\infty}\frac{(\ln t)^{k}}{k!}E^{k}$

for $t>0$. It is straightforward to verify that $T$ is a Lie group
homomorphism from the set of positive real numbers under multiplication into
$\operatorname{Gl}(\mathbb{R}^{d})$. The collection
$\\{T_{t}:t>0\\}=\\{t^{E}:t>0\\}$, which we view both as a set and as a
subgroup of $\operatorname{Gl}(\mathbb{R}^{d})$, is called the continuous one-
parameter group generated by $E$; it will usually be written
$\\{t^{E}\\}_{t>0}$ or simply $\\{t^{E}\\}$. It is a standard fact that every
continuous one-parameter (sub)group of
$\\{T_{t}\\}\subseteq\operatorname{Gl}(\mathbb{R}^{d})$ is of this form, i.e.,
is generated by some $E\in\operatorname{End}(\mathbb{R}^{d})$. An account of
the theory of continuous one-parameter groups can be found in [EN00]. Two
notions of particular interest for us are captured by the following
definition; the first of which is equivalent to the so-called Lyapunov
stability of the one-parameter additive group $\mathbb{R}\ni t\to
T_{e^{-t}}=\exp(-tE)$ [EN00].

######  Definition 2.1.

Let $\\{T_{t}\\}\subseteq\operatorname{Gl}(\mathbb{R}^{d})$ be a continuous
one-parameter group.

  1. 1.

We say that $\\{T_{t}\\}$ is contracting if

$\lim_{t\to 0}\|T_{t}\|=0.$

  2. 2.

We say that $\\{T_{t}\\}$ is non-expanding if

$\sup_{0<t\leq 1}\|T_{t}\|<\infty.$

Thanks to the continuity of
$T:(0,\infty)\to\operatorname{Gl}(\mathbb{R}^{d})$, every contracting group is
non-expanding.

######  Proposition 2.2.

Let $\\{T_{t}\\}$ be a continuous one-parameter group generated by
$E\in\operatorname{End}(\mathbb{R}^{d})$. If $\\{T_{t}\\}$ is contracting,
then $\tr E>0$. If $\\{T_{t}\\}$ is non-expanding, then $\tr E\geq 0$.

###### Proof.

By virtue of the continuity of the determinant map and the fact that
$\operatorname{det}(t^{E})=t^{\tr E}$, we have

$\lim_{t\to 0}t^{\tr E}=\lim_{t\to
0}\operatorname{det}(t^{E})=\operatorname{det}(0)=0$

provided $\\{t^{E}\\}$ is contracting. In this case, it follows that $\tr
E>0$. If $\\{t^{E}\\}$ is expanding with $M=\sup_{0<t\leq 1}\|t^{E}\|$, we
have

$\sup_{0<t\leq 1}t^{\tr E}=\sup_{0<t\leq
1}\operatorname{det}(t^{E})\leq\sup_{\|A\|\leq
M}\absolutevalue{\operatorname{det}(A)}<\infty$

because $\\{A\in\operatorname{End}(\mathbb{R}^{d}):\|A\|\leq M\\}$ is a
compact set. Thus $\tr E\geq 0$. ∎

Given a function $P:\mathbb{R}^{d}\to\mathbb{R}$ and
$E\in\operatorname{End}(\mathbb{R}^{d})$, we shall say that $P$ is homogeneous
with respect to $E$ if

$tP(\xi)=P(t^{E}\xi)$

for all $t>0$ and $\xi\in\mathbb{R}^{d}$; in this case we say that $E$ is a
member of the exponent set of $P$ and write $E\in\operatorname{Exp}(P)$.
Central to the definition of positive homogeneous function given below is the
following characterization taken from [BR22].

######  Proposition 2.3.

Let $P:\mathbb{R}^{d}\to\mathbb{R}$ be continuous, positive-definite (in the
sense that $P\geq 0$ and $P(\xi)=0$ only when $\xi=0$), and have
$\operatorname{Exp}(P)\neq\varnothing$. Then the following are equivalent:

  1. 1.

The so-called unital level set of $P$,

$S_{P}:=\\{\xi\in\mathbb{R}^{d}:\absolutevalue{P(\xi)}=P(\xi)=1\\},$

is compact.

  2. 2.

There is a positive number $M$ for which $P(\xi)>1$ whenever
$\absolutevalue{\xi}>M$.

  3. 3.

For each $E\in\operatorname{Exp}(P)$, $\\{t^{E}\\}$ is contracting.

  4. 4.

There exists $E\in\operatorname{Exp}(P)$ for which $\\{t^{E}\\}$ is
contracting.

  5. 5.

We have

$\lim_{\absolutevalue{\xi}\to\infty}P(\xi)=\infty.$

######  Definition 2.4.

Let $P:\mathbb{R}^{d}\to\mathbb{R}$ be continuous, positive definite and have
$\operatorname{Exp}(P)\neq\varnothing$. If any one (and hence all) of the
equivalent conditions in Proposition 2.3 are fulfilled, we say that $P$ is
positive homogeneous. We will also say that $P$ is a positive homogeneous
function on $\mathbb{R}^{d}$.

The following proposition amasses some basic facts about positive homogeneous
functions and its proof can be found in Section 2 of [BR22].

######  Proposition 2.5.

Let $P$ be a positive homogeneous function and denote by
$\operatorname{Sym}(P)$ the set of $O\in\operatorname{End}(\mathbb{R}^{d})$
for which $P(O\xi)=P(\xi)$ for all $\xi\in\mathbb{R}^{d}$. We have

  1. 1.

$\operatorname{Sym}(P)$ is a compact subgroup of
$\operatorname{Gl}(\mathbb{R}^{d})$.

  2. 2.

For each $E,\widetilde{E}\in\operatorname{Exp}(P)$, we have

$\tr E=\tr\widetilde{E}>0.$

In view of the preceding proposition, we define the homogeneous order of $P$
to be the unique positive number $\mu_{P}$ for which

$\mu_{P}=\tr E$

for all $E\in\operatorname{Exp}(P)$.

######  Example 1:

For any $\alpha>0$, the map $\xi\mapsto\absolutevalue{\xi}^{\alpha}$ is a
positive homogeneous function on $\mathbb{R}^{d}$. Indeed, it is evident that
it is continuous and positive-definite and its unital level set is precisely
the unit sphere $S_{\absolutevalue{\cdot}^{\alpha}}=\mathbb{S}_{d}$ in
$\mathbb{R}^{d}$. Further,

$\operatorname{Exp}(\absolutevalue{\cdot}^{\alpha})=\frac{1}{\alpha}I+\mathfrak{o}_{d}$

where $I$ is the identity map on $\mathbb{R}^{d}$ and $\mathfrak{o}_{d}$ is
the Lie algebra of the orthogonal group $\mbox{O}(\mathbb{R}^{d})$ and is
characterized by the set of skew symmetric matrices. ∎

######  Example 2:

Given a $d$-tuple of positive even integers
$\mathbf{m}=(m_{1},m_{2},\dots,m_{d})$, we consider a polynomial of the form

$P(\xi)=\sum_{|\alpha:\mathbf{m}|=1}a_{\alpha}\xi^{\alpha}$ (8)

for $\xi=(\xi_{1},\xi_{2},\dots,\xi_{d})\in\mathbb{R}^{d}$ where, for each
multi-index
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbb{N}^{d}$,
$\xi^{\alpha}:=\xi_{1}^{\alpha_{1}}\xi_{2}^{\alpha_{2}}\cdots\xi_{d}^{\alpha_{d}}$
and

$|\alpha:\mathbf{m}|:=\sum_{k=1}^{d}\frac{\alpha_{k}}{m_{k}}.$

A polynomial of the form (8) is said to be semi-elliptic provided $P(\xi)$
vanishes only at $\xi=0$. Appearing in L. Hörmander’s treatise on linear
partial differential operators [Hö83], semi-elliptic polynomials are the
symbols of a class of hypoelliptic partial differential operators, called
semi-elliptic operators. For a semi-elliptic polynomial $P(\xi)$ of the form
(8), its corresponding semi-elliptic operator is the constant-coefficient
linear partial differential operator given by

$P(D)=\sum_{|\alpha:\mathbf{m}|=1}a_{\alpha}D^{\alpha}$

where we have written
$D=(-i\partial_{x_{1}},-i\partial_{x_{2}},\dots,-i\partial_{x_{d}})$ and, for
each multi-index
$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{d})\in\mathbb{N}^{d}$,
$D^{\alpha}=(-i\partial_{x_{1}})^{\alpha_{1}}(-i\partial_{x_{2}})^{\alpha_{2}}\cdots(-i\partial_{x_{d}})^{\alpha_{d}}$.
For a polynomial $P(\xi)$ of the form (8), observe that, for
$E\in\operatorname{End}(\mathbb{R}^{d})$ with standard matrix representation
$\operatorname{diag}(1/m_{1},1/m_{2},\dots,1/m_{d})$,

$P(t^{E}\xi)=\sum_{|\alpha:\mathbf{m}|=1}a_{\alpha}(t^{1/m_{1}}\xi_{1})^{\alpha_{1}}(t^{1/m_{2}}\xi_{2})^{\alpha_{2}}\cdots(t^{1/m_{d}}\xi_{d})^{\alpha_{d}}=\sum_{|\alpha:\mathbf{m}|=1}a_{\alpha}t^{|\alpha:\mathbf{m}|}\xi^{\alpha}=tP(\xi)$

for all $t>0$ and $\xi=(\xi_{1},\xi_{2},\dots,\xi_{d})\in\mathbb{R}^{d}$. Thus
$E\in\operatorname{Exp}(P)$ and, because $\\{t^{E}\\}$ is clearly contracting,
Proposition 2.3 guarantees that $P$ is positive homogeneous whenever it is
positive-definite. Thus, whenever a semi-elliptic polynomial $P$ of the form
(8) is positive-definite, it is positive homogeneous with homogeneous order

$\mu_{P}=\tr
E=|\mathbf{1}:\mathbf{m}|=\frac{1}{m_{1}}+\frac{1}{m_{2}}+\cdots+\frac{1}{m_{d}}.$

For two concrete examples, consider

$P_{1}(\xi)=\xi_{1}^{4}+\xi_{2}^{6}\hskip 28.45274pt\mbox{and}\hskip
28.45274ptP_{2}(\xi)=\xi_{1}^{2}+\xi_{1}\xi^{2}+\xi_{2}^{4}$

defined for $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}$. These are positive
homogeneous semi-elliptic polynomials on $\mathbb{R}^{2}$ with homogeneous
order $\mu_{P_{1}}=1/4+1/6=5/12$ and $\mu_{P_{2}}=1/2+1/4=3/4$, respectively.
∎

Before we conclude our treatment of positive homogeneous functions and turn
our attention to multivariate homogeneous functions, we present the following
lemma which is used several times throughout the course of this paper. Its
proof makes use of the generalized polar-coordinate integration formula
developed and presented in Theorem 1.4 of [BR22].

######  Lemma 2.6.

Let $P$ be a positive homogeneous function on $\mathbb{R}^{d}$ with relatively
compact unit ball $B_{P}=\\{\xi\in\mathbb{R}^{d}:P(\xi)<1\\}$ and homogeneous
order $\mu_{P}>0$. Then, for each $\epsilon>0$,

$\int_{\mathbb{R}^{d}}e^{-\epsilon
P(\xi)}\,d\xi=m(B_{P})\frac{\Gamma(\mu_{P}+1)}{\epsilon^{\mu_{P}}}$

where $m(B_{P})$ denoted the Lebesgue measure of $B_{P}$. In particular, for
each $\epsilon>0$, $\exp(-\epsilon P)\in L^{1}(\mathbb{R}^{d})$.

###### Proof.

By an appeal to Theorem 1.5 of [BR22], we obtain a Borel measure
$\sigma=\sigma_{P}$ on $S=S_{P}$ for which $\sigma(S)=\mu_{P}\cdot m(B_{P})$
and

$\int_{\mathbb{R}^{d}}e^{-\epsilon
P(\xi)}\,d\xi=\int_{S}\int_{0}^{\infty}e^{-\epsilon
P(t^{E}\eta)}t^{\mu_{P}-1}\,dt\,\sigma(d\eta)=\int_{S}\int_{0}^{\infty}e^{-\epsilon
t}t^{\mu_{P}-1}\,dt\,\sigma(d\eta)$

for any $E\in\operatorname{Exp}(P)$. Thus

$\int_{\mathbb{R}^{d}}e^{-\epsilon
P(\xi)}\,d\xi=\sigma(S)\int_{0}^{\infty}e^{-\epsilon
t}t^{\mu_{P}-1}\,dt=m(B_{P})\frac{\mu_{P}}{\epsilon^{\mu_{P}}}\int_{0}^{\infty}e^{-s}s^{\mu_{P}-1}\,ds=m(B_{P})\frac{\Gamma(\mu_{P}+1)}{\epsilon^{\mu_{P}}}$

where we have used the Laplace-transform representation of Euler’s Gamma
function and the property that
$\mu_{P}\cdot\Gamma(\mu_{P})=\Gamma(\mu_{P}+1)$. ∎

We now introduce a generalization of the notion of positive homogeneous
function. Given non-negative integers $a$ and $b$ and a function
$Q:\mathbb{R}^{b}\to\mathbb{R}^{a}$, we say that $Q$ is nondegenerate if
$Q(\zeta)\neq 0$ whenever $\zeta\neq 0$. Given a pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$,
we say that $Q$ is homogeneous with respect to the pair $(E,E^{\prime})$
provided that

$t^{E}Q(\zeta)=Q(t^{E^{\prime}}\zeta)$

for all $t>0$ and $\zeta\in\mathbb{R}^{{b}}$. Akin to Proposition 2.3, we have
the following:

######  Proposition 2.7.

Let $Q:\mathbb{R}^{{b}}\to\mathbb{R}^{a}$ be continuous, nondegenerate, and
homogeneous with respect to some pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $\\{t^{E}\\}_{t>0}$ is contracting. Then, the following are
equivalent:

  1. 1.

For any compact set $K\subseteq\mathbb{R}^{a}$,
$Q^{-1}(K)\subseteq\mathbb{R}^{{b}}$ is compact.

  2. 2.

The set

$S_{Q}=\\{\zeta\in\mathbb{R}^{{b}}:\absolutevalue{Q(\zeta)}=1\\}$

is compact.

  3. 3.

There is a number $M>0$ for which $\absolutevalue{Q(\zeta)}>1$ for all
$|\zeta|>M$.

  4. 4.

For each pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $Q$ is homogeneous and $\\{t^{E}\\}$ is contracting,
$\\{t^{E^{\prime}}\\}$ must also be contracting.

  5. 5.

There exists a pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $Q$ is homogeneous and $\\{t^{E}\\}$ and $\\{t^{E^{\prime}}\\}$ are
both contracting.

  6. 6.

We have

$\lim_{\absolutevalue{\zeta}\to\infty}\absolutevalue{Q(\zeta)}=\infty.$

###### Proof.

###### (1$\,\Rightarrow$2).

Since $S_{Q}=Q^{-1}(\mathbb{S}_{a})$ where $\mathbb{S}_{a}$ is the unit sphere
in $\mathbb{R}^{a}$, this is immediate. ∎

###### (2$\,\Rightarrow$3).

Assuming that $S_{Q}$ is compact, let $M>0$ be such that
$\absolutevalue{Q(\zeta)}\neq 1$ for all $\absolutevalue{\zeta}>M$. Denote by
$\overline{\mathbb{B}_{M}}$ the closed ball in $\mathbb{R}^{{b}}$ with center
$0$ and radius $M$ and by
$\mathcal{O}_{M}=\mathbb{R}^{{b}}\setminus\overline{\mathbb{B}_{M}}$ its
complement. Let us first treat the situation in which ${b}>1$. In this case,
the fact that $\mathcal{O}_{M}$ is path connected and $Q(\zeta)$ is continuous
ensures that there cannot be two elements $\zeta_{1}$ and $\zeta_{2}$ in
$\mathcal{O}_{M}$ with
$\absolutevalue{Q(\zeta_{1})}<1<\absolutevalue{Q(\zeta_{2})}$ for otherwise
the intermediate value theorem would imply that
$\absolutevalue{Q(\zeta_{3})}=1$ for some $\zeta_{3}\in\mathcal{O}_{M}$, an
impossibility. Thus, to prove the statement, we simply need to rule out the
possibility that $\absolutevalue{Q(\zeta)}<1$ for all
$\zeta\in\mathcal{O}_{M}$. Let us assume, to reach a contradiction, that
$\absolutevalue{Q(\zeta)}<1$ for all $\zeta\in\mathcal{O}_{M}$. Let
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
be a pair for which $Q$ is homogeneous and $\\{t^{E}\\}$ is contracting and
let $\zeta_{0}$ be a non-zero element in $\mathbb{R}^{{b}}$. The
nondegenerateness of $Q$ and the fact that $\\{t^{E}\\}$ is contracting
guarantees that

$\infty=\lim_{t\to\infty}\absolutevalue{t^{E}Q(\zeta_{0})}=\lim_{t\to\infty}\absolutevalue{Q(t^{E^{\prime}}\zeta_{0})}.$

In view of our hypothesis it follows that, for all sufficiently large $t$,
$t^{E^{\prime}}\zeta_{0}\in\overline{\mathbb{B}_{M}}$. Of course, this implies
that $\zeta\mapsto\absolutevalue{Q(\zeta)}$ is unbounded on the compact set
$\overline{\mathbb{B}_{M}}$ and this is impossible for we know that $Q$ is
continuous.

In the case that ${b}=1$, we first argue that $Q(\zeta)>1$ for all
$\zeta>M>0$. Of course, since $(M,\infty)$ is connected, an argument analogous
to that above for ${b}>1$ guarantees that it suffices to rule out the case
that $Q(\zeta)<1$ for all $\zeta>M$. We therefore assume, to reach a
contradiction, that $Q(\zeta)<1$ for all $\zeta>M$ and select a pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $Q$ is homogeneous and $\\{t^{E}\\}$ contracting. Due to the
simplicity of
$\operatorname{End}(\mathbb{R}^{{b}})=\operatorname{End}(\mathbb{R})$,
$E^{\prime}=\alpha^{\prime}I$ for some $\alpha^{\prime}\in\mathbb{R}$ and so
$t^{E^{\prime}}\zeta=t^{\alpha^{\prime}}\zeta$ for all $t>0$ and
$\zeta\in\mathbb{R}=\mathbb{R}^{{b}}$. Using the fact that $\\{t^{E}\\}$ is
contracting, we have

$\infty=\lim_{t\to\infty}\absolutevalue{t^{E}Q(1)}=\lim_{t\to\infty}\absolutevalue{Q(t^{\alpha^{\prime}})}$

and so, in view of our supposition, it follows that $0<t^{\alpha^{\prime}}\leq
M$ for all sufficiently large $t$ (which, at the same time guarantees that
$\alpha^{\prime}<0$). This, however, contradicts that fact that $Q$ is
continuous at $0$. Hence, $Q(\zeta)>1$ for all $\zeta>M$. A similar argument
shows that $Q(\zeta)>1$ for all $\zeta<-M$. Thus $Q(\zeta)>1$ for all
$\absolutevalue{\zeta}>M$, as was asserted. ∎

###### (3$\,\Rightarrow$4).

We prove the contrapositive statement. Let us assume that there is a pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $Q$ is homogeneous and $\\{t^{E}\\}$ is contracting, but
$\\{t^{E^{\prime}}\\}$ is not contracting. It follows that, for some
$\zeta\in\mathbb{R}^{{b}}$ and $\epsilon>0$, there is a sequence $t_{k}\to 0$
for which

$\absolutevalue{t_{k}^{E^{\prime}}\zeta}\geq\epsilon$

for all $k$. In the case that $\\{t_{k}^{E^{\prime}}\zeta\\}$ is bounded, we
assume without loss of generality (by passing to a subsequence, if needed)
that $\lim_{k\to\infty}t_{k}^{E^{\prime}}\zeta=\zeta_{0}$ where
$\absolutevalue{\zeta_{0}}\geq\epsilon$. Consequently,

$Q(\zeta_{0})=\lim_{k\to\infty}Q(t_{k}^{E^{\prime}}\zeta)=\lim_{k\to\infty}t_{k}^{E}Q(\zeta)=0$

since $t_{k}\to 0$ and $\\{t^{E}\\}$ is contracting. Since $\zeta_{0}\neq 0$,
this is impossible for we know that $Q$ is nondegenerate. Thus, we must
conclude that $\\{t_{k}^{E^{\prime}}\zeta\\}$ is unbounded. Without loss of
generality, we may assume (by passing to a subsequence, if necessary) that
$\zeta_{k}=t_{k}^{E^{\prime}}\zeta\to\infty$ as $k\to\infty$. Consequently,
there is a sequence $\zeta_{k}\to\infty$ for which

$\lim_{k\to\infty}Q(\zeta_{k})=\lim_{k\to\infty}t_{k}^{E}Q(\zeta)=0.$

This shows that Item 3 cannot hold, as was asserted. ∎

###### (4$\,\Rightarrow$5).

In view of the hypotheses, this is immediate. ∎

###### (5$\,\Rightarrow$6).

We fix a pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $Q$ is homogeneous and both $\\{t^{E}\\}$ and $\\{t^{E^{\prime}}\\}$
are contracting. Let $\\{\zeta_{k}\\}\subseteq\mathbb{R}^{{b}}$ be a sequence
with $\zeta_{k}\to\infty$ as $k\to\infty$. Since $\\{t^{E^{\prime}}\\}$ is
contracting and in view of Proposition A.5 of [BR22], we can write
$\zeta_{k}=t_{k}^{E^{\prime}}\eta_{k}$ where $\absolutevalue{\eta_{k}}=1$ for
each $k$ and $t_{k}\to\infty$. We claim that

$\lim_{k\to\infty}\absolutevalue{t_{k}^{E}Q(\eta_{k})}=\infty.$

To see this, we assume, to reach a contradiction, that a subsequence has the
property that $\absolutevalue{t_{k_{j}}^{E}Q(\eta_{k_{j}})}\leq M$ for some
$M$. In this case, we see that

$\absolutevalue{Q(\eta_{k_{j}})}=\absolutevalue{(1/t_{k_{j}})^{E}t_{k_{j}}^{E}Q(\eta_{k_{j})}}\leq
M\|(1/t_{k_{j}})^{E}\|$

for all $j$ and, since $1/t_{k_{j}}\to 0$ as $j\to\infty$, the fact that
$\\{t^{E}\\}$ is contracting implies that

$0=\inf_{\absolutevalue{\zeta}=1}\absolutevalue{Q(\zeta)}.$

This is, however, impossible because $Q$ is continuous and nonvanishing on the
compact unit sphere in $\mathbb{R}^{{b}}$. We have therefore substantiated our
claim and so it follows that

$\lim_{k\to\infty}\absolutevalue{Q(\zeta_{k})}=\lim_{k\to\infty}\absolutevalue{Q(t_{k}^{E^{\prime}}\eta_{k})}=\lim_{k\to\infty}\absolutevalue{t_{k}^{E}Q(\eta_{k})}=\infty,$

as desired. ∎

###### (6$\,\Rightarrow$1).

Let $K\subseteq\mathbb{R}^{a}$ be compact. The fact that $Q$ is continuous
ensures that $Q^{-1}(K)$ is necessarily closed. If $Q^{-1}(K)$ were unbounded,
we could find a sequence $\zeta_{k}\in Q^{-1}(K)$ for which
$\absolutevalue{\zeta_{k}}\to\infty$ yet $Q(\zeta_{k})\in K$ for all $k$. Of
course, this is impossible in light of our assumption. Hence $Q^{-1}(K)$ must
be bounded and therefore compact in view of the Heine-Borel theorem. ∎

∎

######  Definition 2.8.

Let $Q:\mathbb{R}^{{b}}\to\mathbb{R}^{a}$ be continuous, nondegenerate and
homogeneous with respect to a pair
$(E,E^{\prime})\in\operatorname{End}(\mathbb{R}^{a})\times\operatorname{End}(\mathbb{R}^{{b}})$
for which $\\{t^{E}\\}$ is contracting. If any (hence, every) of the
equivalent conditions listed in Proposition 2.7 are satisfied, we say that $Q$
is nondegenerate multivariate homogeneous.

######  Example 3:

For a given positive integer $\alpha$, consider $Q:\mathbb{R}\to\mathbb{R}$
defined by $Q(\zeta)=\zeta^{\alpha}$. It is clear that $Q(\zeta)$ is
continuous and nondegenerate. We observe further that

$Q(t^{I}\zeta)=Q(t\zeta)=t^{\alpha}\zeta^{\alpha}=t^{\alpha I}\zeta^{\alpha}$

for all $t>0$ and $\zeta\in\mathbb{R}$ where $I$ is the identity
transformation on $\mathbb{R}$. Thus $Q$ is homogeneous with respect to
$(\alpha I,I)$ and, further, it is clear that $\\{t^{\alpha
I}\\}=\\{t^{\alpha}\\}$ and $\\{t^{I}\\}=\\{t\\}$ are both contracting. Thus,
by virtue of Proposition 2.7, we conclude that $Q$ is nondegenerate
multivariate homogeneous. ∎

######  Example 4:

Given positive integers $a$ and $b$, let
$\sigma_{1},\sigma_{2},\dots,\sigma_{a}$ be a collection of positive integers
for which, as sets,
$\\{1,2,\dots,{b}\\}=\\{\sigma_{1},\sigma_{2},\dots,\sigma_{a}\\}$ and, given
positive integers $\alpha_{1},\alpha_{2},\dots,\alpha_{a}$, define
$Q:\mathbb{R}^{{b}}\to\mathbb{R}^{a}$ by

$Q(\zeta)=Q(\zeta_{1},\zeta_{2},\dots,\zeta_{{b}})=(\zeta_{\sigma_{1}}^{\alpha_{1}},\zeta_{\sigma_{2}}^{\alpha_{2}},\dots,\zeta_{\sigma_{a}}^{\alpha_{a}})$
(9)

for $\zeta=(\zeta_{1},\zeta_{2},\dots,\zeta_{{b}})\in\mathbb{R}^{{b}}$. We
claim that $Q$ is nondegenerate multivariate homogeneous. To see this, we
observe first that $Q$ is clearly continuous (in fact, $Q\in C^{\infty}$) and
$Q(\zeta)=0$ if and only if $\zeta_{\sigma_{j}}=0$ for all $j=1,2,\dots,d$. In
view of the condition that
$\\{\sigma_{1},\sigma_{2},\dots,\sigma_{d}\\}=\\{1,2,\dots,{b}\\}$, we
conclude that $Q(\zeta)=0$ if and only if
$\zeta=(\zeta_{1},\zeta_{2},\dots,\zeta_{{b}})=0$ and so $Q$ is nondegenerate.
Observe that

$Q(t^{I}\zeta)=Q(t\zeta)=(t^{\alpha_{1}}\zeta_{\sigma_{1}}^{\alpha_{1}},t^{\alpha_{2}}\zeta_{\sigma_{2}}^{\alpha_{2}},\dots,t^{\alpha_{a}}\zeta_{\sigma_{a}}^{\alpha_{a}})=t^{E}Q(\zeta)$

where $I$ is the identity on $\mathbb{R}^{{b}}$ and
$E\in\operatorname{End}(\mathbb{R}^{a})$ has standard representation
$\operatorname{diag}(\alpha_{1},\alpha_{2},\dots,\alpha_{a})$. Consequently,
$Q$ is homogeneous with respect to $(E,I)$ and, since $\\{t^{E}\\}$ and
$\\{t^{I}\\}$ are contracting, we conclude that $Q$ is nondegenerate
multivariate homogeneous. For a concrete example, consider
$Q:\mathbb{R}^{2}\to\mathbb{R}^{3}$ defined by

$Q(\zeta_{1},\zeta_{2})=(\zeta_{2},\zeta_{1}^{4},\zeta_{2}^{3})$

for $\zeta=(\zeta_{1},\zeta_{2})\in\mathbb{R}^{2}$. This is a nondegenerate
multivariate homogeneous function of the above form with
$\sigma_{1}=\sigma_{3}=2$, $\sigma_{2}=1$, $\alpha_{1}=1$, $\alpha_{2}=4$, and
$\alpha_{3}=3$. ∎

We remark that, for any $Q$ of the form (9), we have $b\leq a$. The following
example generalizes that above and allows for $b>a$. In addition, the example
shows that all positive homogeneous functions are nondegenerate multivariate
homogeneous.

######  Example 5:

For positive integers $a$ and $b$, let
$q_{1}(\zeta),q_{2}(\zeta),\dots,q_{a}(\zeta)$ be a collection of continuous
real-valued functions on $\mathbb{R}^{{b}}$ which satisfy the following
conditions:

  1. 1.

If $\zeta\neq 0$, then $q_{k}(\zeta)\neq 0$ for at least one $k=1,2,\dots,a$.

  2. 2.

There exists $E^{\prime}\in\operatorname{End}(\mathbb{R}^{{b}})$ for which
$\\{t^{E^{\prime}}\\}$ is contracting and
$E^{\prime}\in\operatorname{Exp}(q_{k})$ for all $k=1,2,\dots,a$.

Given such a collection, let $\alpha_{1},\alpha_{2},\dots,\alpha_{a}$ be
positive integers and define $Q:\mathbb{R}^{{b}}\to\mathbb{R}^{a}$ by

$Q(\zeta)=(q_{1}(\zeta)^{\alpha_{1}},q_{2}(\zeta)^{\alpha_{2}},\dots,q_{a}(\zeta)^{\alpha_{a}})$
(10)

for $\zeta\in\mathbb{R}^{{b}}$. We claim that $Q$ is nondegenerate
multivariate homogeneous. Indeed, $Q$ is continuous and nondegenerate in view
of Condition 1. Upon taking
$E^{\prime}\in\operatorname{End}(\mathbb{R}^{{b}})$ satisfying Condition 2, we
observe that $Q$ is homogeneous with respect to the pair $(E,E^{\prime})$
where $E\in\operatorname{End}(\mathbb{R}^{d})$ has standard representation
$\operatorname{diag}(\alpha_{1},\alpha_{2},\dots,\alpha_{a})$. Since
$\\{t^{E}\\}$ and $\\{t^{E^{\prime}}\\}$ are contracting, we conclude that $Q$
is nondegenerate multivariate homogeneous in view of Proposition 2.7.

In the case that $a=1$, $\alpha_{1}=1$, and $q_{1}(\zeta)=P(\zeta)$ is a
positive homogeneous function, the above conditions are automatically
satisfied for any $E^{\prime}\in\operatorname{Exp}(P)$. From this we conclude
that positive homogeneous functions are nondegenerate multivariate
homogeneous. We remark that, for any $E^{\prime}\in\operatorname{Exp}(P)$,
$Q(\zeta)=q_{1}(\zeta)=P(\zeta)$ is homogeneous with respect to the pair
$(I,E^{\prime})$ where $I$ is the identity on $\mathbb{R}$.

For a concrete example of a multivariate homogeneous function of the form (10)
(and for which $1<a<{b}$), consider $Q:\mathbb{R}^{3}\to\mathbb{R}^{2}$
defined by

$Q(\zeta)=(\zeta_{1}^{2}+\zeta_{2}^{4},(\zeta_{1}+\zeta_{3}^{3})^{5})$

for $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})\in\mathbb{R}^{3}$. This can be
written equivalently as

$Q(\zeta)=\left(\left(\zeta_{1}^{2}+\zeta_{2}^{4})^{1/2}\right)^{2},\left(\zeta_{1}+\zeta_{3}^{3}\right)^{5}\right)$

for $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})\in\mathbb{R}^{3}$. This is clearly
of the form (10) with $q_{1}(\zeta)=(\zeta_{1}^{2}+\zeta_{2}^{4})^{1/2}$,
$q_{2}(\zeta)=\zeta_{1}+\zeta_{3}^{3}$, $\alpha_{1}=2$ and $\alpha_{2}=5$. In
this case, it is easy to see that $q_{1}(\zeta)$ and $q_{2}(\zeta)$ satisfy
Condition 1. Further, observe that, for
$E^{\prime}\in\operatorname{End}(\mathbb{R}^{3})$ with standard representation
$\operatorname{diag}(1,1/2,1/3)$, we have
$E^{\prime}\in\operatorname{Exp}(q_{1})\cap\operatorname{Exp}(q_{2})$. Since
$\\{t^{E^{\prime}}\\}$ is clearly contracting, we may conclude that $Q$ is
nondegenerate multivariate homogeneous and homogeneous with respect to
$(E,E^{\prime})$ where $E^{\prime}$ is that above and
$E\in\operatorname{End}(\mathbb{R}^{2})$ has standard representation
$\operatorname{diag}(2,5)$. Of course, we can confirm the homogeneity of $Q$
with respect to the pair $(E,E^{\prime})$ directly: For $t>0$ and
$\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})$,

$\displaystyle Q(t^{E^{\prime}}\zeta)$ $\displaystyle=$ $\displaystyle
Q(t^{1}\zeta_{1},t^{1/2}\zeta_{2},t^{1/3}\zeta_{3})$ $\displaystyle=$
$\displaystyle\left((t\zeta_{1})^{2}+(t^{1/2}\zeta_{2})^{4},\left((t\zeta_{1})+(t^{1/3}\zeta_{3})^{3}\right)^{5}\right)$
$\displaystyle=$
$\displaystyle\left(t^{2}\left(\zeta_{1}^{2}+\zeta_{2}^{4}\right),t^{5}\left(\zeta_{1}+\zeta_{3}^{3}\right)^{5}\right)$
$\displaystyle=$ $\displaystyle t^{E}Q(\zeta).$

∎

######  Proposition 2.9.

Suppose that $P$ is a positive homogeneous function on $\mathbb{R}^{a}$,
$Q:\mathbb{R}^{{b}}\to\mathbb{R}^{a}$ is a nondegenerate multivariate
homogeneous function and, for some $E\in\operatorname{Exp}(P)$ and
$E^{\prime}\in\operatorname{End}(\mathbb{R}^{{b}})$, $Q$ is homogeneous with
respect to the pair $(E,E^{\prime})$. Then $P\circ Q$ and $P\circ(-Q)$ are
positive homogeneous functions on $\mathbb{R}^{{b}}$ whose exponent sets
contain $E^{\prime}$. In particular, $\mu_{P\circ Q}=\mu_{P\circ(-Q)}=\tr
E^{\prime}$.

###### Proof.

It is clear that $Q$ is nondegenerate multivariate homogeneous if and only if
$-Q$ is nondegenerate multivariate homogeneous and $Q$ is homogeneous with
respect to a pair $(E,E^{\prime})$ if and only if $-Q$ is. Thus, to prove the
proposition, it suffices to prove that $P\circ Q$ is positive homogeneous with
$E^{\prime}\in\operatorname{Exp}(P\circ Q)$. Because $P$ is continuous and
positive-definite and $Q$ is continuous and nondegenerate, it evident that
$P\circ Q$ is continuous and positive-definite. Given the pair
$(E,E^{\prime})$ as in the statement of the proposition, we observe that

$(P\circ Q)(t^{E^{\prime}}\zeta)=P(t^{E}Q(\zeta))=tP(Q(\zeta))=t(P\circ
Q)(\zeta)$

for all $t>0$ and $\zeta\in\mathbb{R}^{{b}}$. Thus,
$E^{\prime}\in\operatorname{Exp}(P\circ Q)$. Finally, since $\\{t^{E}\\}$ is
contracting thanks to Proposition 2.3, $\\{t^{E^{\prime}}\\}$ must also be
contracting in view or Proposition 2.7 and so we conclude that $P\circ Q$ is
positive homogeneous in light of Proposition 2.3. ∎

######  Example 6:

Consider the nondegenerate multivariate homogeneous function
$Q:\mathbb{R}^{3}\to\mathbb{R}^{2}$ from the previous example given by

$Q(\zeta)=(\zeta_{1}^{2}+\zeta_{2}^{4},(\zeta_{1}+\zeta_{3}^{3})^{5})$

for $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})\in\mathbb{R}^{3}$. Also, consider
the positive homogeneous function $P:\mathbb{R}^{2}\to\mathbb{R}$ defined by

$P(\xi)=P(\xi_{1},\xi_{2})=\xi_{1}^{5}+\xi_{2}^{2}$

for $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{2}$. We see that $P$ is homogeneous
with respect to $E\in\operatorname{End}(\mathbb{R}^{2})$ with standard
representation $\operatorname{diag}(1/5,1/2)$. Consider also
$E^{\prime}\in\operatorname{End}(\mathbb{R}^{3})$ with standard representation
$\operatorname{diag}(1/10,1/20,1/30)$ and observe that

$\displaystyle Q(t^{E^{\prime}}\zeta)$ $\displaystyle=$ $\displaystyle
Q(t^{1/10}\zeta_{1},t^{1/20}\zeta_{2},t^{1/30}\zeta_{3})$ $\displaystyle=$
$\displaystyle\left(t^{1/5}(\zeta_{1}^{2}+\zeta_{2}^{4}),t^{1/2}(\zeta_{1}+\zeta_{3}^{3})^{5}\right)$
$\displaystyle=$ $\displaystyle t^{E}Q(\zeta)$

for $t>0$ and $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})\in\mathbb{R}^{3}$. Thus,
by the preceding proposition, we have that $P\circ Q$ is positive homogeneous
with $\mu_{P\circ Q}=\tr E^{\prime}=11/60$. Of course, this can be verified
directly by simplification of $P\circ Q$. We have

$(P\circ
Q)(\zeta)=(\zeta_{1}^{2}+\zeta_{2}^{4})^{5}+(\zeta_{1}+\zeta_{3}^{3})^{10}$

which is clearly positive homogeneous with
$E^{\prime}\in\operatorname{Exp}(P\circ Q)$. ∎

##  3 On-Diagonal Asymptotics

######  Theorem 3.1.

Given positive integers ${a}$ and ${b}$, let $P_{1}$ and $P_{2}$ be positive
homogeneous functions on $\mathbb{R}^{{a}}$ with homogeneous orders
$\mu_{P_{1}}$ and $\mu_{P_{2}}$, respectively, and let
$Q:\mathbb{R}^{{b}}\to\mathbb{R}^{{a}}$ be a $C^{1}$ function which is
nondegenerate multivariate homogeneous. Set $d={a}+{b}$ and define
$P:\mathbb{R}^{d}\to\mathbb{R}$ by

$P(\xi)=P(\eta,\zeta)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

for
$\xi=(\eta,\zeta)\in\mathbb{R}^{{a}}\times\mathbb{R}^{{b}}=\mathbb{R}^{d}$.
Suppose that there exist $E_{1}\in\operatorname{Exp}(P_{1})$,
$E_{2}\in\operatorname{Exp}(P_{2})$, and
$F_{1},F_{2}\in\operatorname{End}(\mathbb{R}^{{b}})$ for which the following
conditions hold:

  1. 1.

For $k=1,2$, $Q$ is homogeneous with respect to the pair $(E_{k},F_{k})$.

  2. 2.

We have $[E_{1},E_{2}]=E_{1}E_{2}-E_{2}E_{1}=0$ and $\\{t^{E_{1}-E_{2}}\\}$ is
non-expanding.

In particular, $P_{1}\circ Q$ and $P_{2}\circ(-Q)$ are positive homogeneous on
$\mathbb{R}^{{b}}$ (by virtue of Proposition 2.9) with homogeneous orders
$\mu_{P_{1}\circ Q}$ and $\mu_{P_{2}\circ(-Q)}$, respectively. Then the heat
kernel

$H_{P}^{t}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-tP(\xi)}e^{-ix\cdot\xi}\,d\xi$

exists for each $t>0$ and $x\in\mathbb{R}^{d}$. Upon setting
$\varphi(t):=H_{P}^{t}(0)$ for $t>0$, we have the following on-diagonal
asymptotics:

$\varphi(t)\asymp\begin{cases}t^{-\mu_{0}}&0<t\leq 1\\\
t^{-\mu_{\infty}}&1\leq t<\infty\end{cases}$

for $t>0$ where $\mu_{0}=\mu_{P_{2}}+\mu_{P_{1}\circ Q}$ and
$\mu_{\infty}=\mu_{P_{1}}+\mu_{P_{2}\circ(-Q)}$.

######  Remark 1:

In view of Propositions 2.5 and 2.9, $\mu_{P_{1}}=\tr E_{1}$, $\mu_{P_{2}}=\tr
E_{2}$, $\mu_{P_{1}\circ Q}=\tr F_{1}$, and $\mu_{P_{2}\circ(-Q)}=\tr F_{2}$
where $E_{1},E_{2},F_{1},F_{2}$ are those given in the hypotheses of Theorem
3.1 (or any which satisfy Conditions 1 and 2). In these terms, the asymptotics
for $\varphi(t)$ can be equivalently written

$\varphi(t)\asymp\begin{cases}t^{-\left(\tr E_{2}+\tr F_{1}\right)}&0<t\leq
1\\\ t^{-\left(\tr E_{1}+\tr F_{2}\right)}&1\leq t<\infty\end{cases}$

for $t>0$.

######  Remark 2:

Given our hypothesis that $Q$ is nondegenerate multivariate homogeneous, when
we ask that $Q$ be homogeneous with respect to the pair $(E_{1},F_{1})$ for
some $E_{1}\in\operatorname{Exp}(P_{1})$ and
$F_{1}\in\operatorname{End}(\mathbb{R}^{{b}})$, we are ensuring that
$\\{t^{F_{1}}\\}$ is contracting by virtue of Proposition 2.7 since it is
known that $\\{t^{E_{1}}\\}$ is contracting thanks to Proposition 2.3. For the
same reason, $\\{t^{F_{2}}\\}$ must also be contracting. Of course, these
observations also follow from Proposition 2.9 since
$F_{1}\in\operatorname{Exp}(P_{1}\circ Q)$ and
$F_{2}\in\operatorname{Exp}(P_{2}\circ(-Q))$.

In this direction, if Condition 1 were adjusted to include the hypothesis
that, for $F_{1}$ (or $F_{2}$), $\\{t^{F_{1}}\\}$ (or $\\{t^{F_{2}}\\}$) is
contracting, then the initial hypothesis could be weakened to ask only that
$Q:\mathbb{R}^{{a}}\to\mathbb{R}^{d}$ be nondegenerate and $C^{1}$. In this
case, the modified Condition 1 would ensure that $Q$ were nondegenerate
multivariate homogeneous by virtue of Item 5 of Proposition 2.7.

######  Remark 3:

If Condition 2 were replaced by the stronger condition that
$[E_{1},E_{2}]=E_{1}E_{2}-E_{2}E_{1}=0$ and $\\{t^{E_{1}-E_{2}}\\}$ is
contracting, then one finds that the small and large-time asymptotics
$t^{-\mu_{0}}$ and $t^{-\mu_{\infty}}$ are “true asymptotics” in the sense
that the limits $\lim_{t\to 0}t^{\mu_{0}}\varphi(t)$ and
$\lim_{t\to\infty}t^{\mu_{\infty}}\varphi(t)$ both exist and are positive
numbers. This result is presented in Theorem 3.4. The theorem, in fact, gives
us the precise value of these limits in terms of $P_{1}$, $P_{2}$, and $Q$.

######  Example 7:

Let $q$ and $l$ be even positive integers with $q\leq l$ and let
$p\in\mathbb{N}_{+}$. We consider $P:\mathbb{R}^{2}\to\mathbb{R}$ defined by

$P(\eta,\zeta)=(\eta+\zeta^{p})^{q}+\eta^{l}$

for $(\eta,\zeta)\in\mathbb{R}^{2}$ and the corresponding heat kernel $H_{P}$
defined by

$H_{P}^{t}(x,y)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{-tP(\eta,\zeta)}e^{-i(\eta,\zeta)\cdot(x,y)}\,d\zeta\,d\eta$

for $t>0$ and $(x,y)\in\mathbb{R}^{2}$. In this case, we can write

$P(\eta,\zeta)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

where $P_{1}(\eta)=\eta^{q}$ and $P_{2}(\eta)=\eta^{l}$ are positive
homogeneous on $\mathbb{R}=\mathbb{R}^{1}$ and $Q(\zeta)=\zeta^{p}$ is
evidently a nondegenerate multivariate homogeneous $C^{1}$ function from
$\mathbb{R}$ to itself. For $P_{1}$, we have
$E_{1}=I/q\in\operatorname{Exp}(P_{1})$ and $\mu_{P_{1}}=1/q$ and, for
$P_{2}$, we have $E_{2}=I/l$ and $\mu_{P_{2}}=1/l$. Further, we observe that

$t^{E_{1}}Q(\zeta)=t^{1/q}\zeta^{p}=(t^{1/qp}\zeta)^{p}=Q(t^{1/qp}\zeta)$

for $t>0$ and $\zeta\in\mathbb{R}$ and from this we see that $Q$ is
homogeneous with respect to the pair $(E_{1},F_{1})$ where $F_{1}=I/qp$.
Similarly,$Q$ is also homogeneous with respect to the pair $(E_{2},F_{2})$
where $F_{2}=1/lp$. Finally, we observe that $E_{1}$ and $E_{2}$ commute and,
since $q\leq l$, $t^{E_{1}-E_{2}}=t^{(1/q-1/l)I}$ is non-expanding. Thus, an
application of the theorem is valid and we conclude that

$\varphi(t)=H_{P}^{t}(0)\asymp\begin{cases}t^{-(1/l+1/qp)}&t\leq 1\\\
t^{-(1/q+1/lp)}&t\geq 1\end{cases}$

because $\mu_{0}=\mu_{P_{2}}+\mu_{P_{1}\circ Q}=\tr E_{2}+\tr F_{1}=1/l+1/qp$
and $\mu_{\infty}=\mu_{P_{1}}+\mu_{P_{2}\circ(-Q)}=\tr E_{1}+\tr
F_{2}=1/q+1/lp$.

We recognize that the motivating example in the introduction is of the above
form where $p=q=2$ and $l=4$, i.e.,

$P(\eta,\zeta)=(\eta+\zeta^{2})^{2}+\eta^{4}$

for $(\eta,\zeta)\in\mathbb{R}^{2}$. From this, we conclude that

$\varphi(t)\asymp\begin{cases}t^{-1/2}&0<t\leq 1\\\ t^{-5/8}&0<t\leq
1\end{cases}$

for $t>0$, as was asserted. Following Theorem 3.4, we revisit this example to
obtain precise values of $\lim_{t\to 0}t^{1/2}\varphi(t)$ and
$\lim_{t\to\infty}t^{5/8}\varphi(t)$. Necessarily, the $t\to 0$ limit is
precisely that obtained by the scaling argument presented in the introduction.
∎

######  Example 8:

In view of Example 4, let $Q:\mathbb{R}^{a}\to\mathbb{R}^{a}$ be a
nondegenerate multivariate homogeneous function of the form

$Q(\zeta)=Q(\zeta_{1},\zeta_{2},\dots,\zeta_{a})=(\zeta_{\sigma(1)}^{\alpha_{1}},\zeta_{\sigma(2)}^{\alpha_{2}},\dots,\zeta_{\sigma(a)}^{\alpha_{a}})$

where $\alpha_{1},\alpha_{2},\dots,\alpha_{a}\in\mathbb{N}_{+}$ and $\sigma$
is a permutation of $\\{1,2,\dots,a\\}$; in particular,
$\\{\sigma(1),\sigma(2),\dots,\sigma(a)\\}=\\{1,2,\dots,a\\}.$ We remark that
$Q$ is clearly smooth. Now, let $P_{1}$ and $P_{2}$ be positive homogeneous
functions on $\mathbb{R}^{a}$ and suppose that, for $k=1,2$,
$\operatorname{Exp}(P_{k})$ contains
$E_{k}\in\operatorname{End}(\mathbb{R}^{a})$ having standard matrix
representation
$\operatorname{diag}(\lambda_{k,1},\lambda_{k,2},\dots,\lambda_{k,a})$ where
$\lambda_{k,j}>0$ for $j=1,2,\dots,a$. If, for $k=1,2$, we consider $F_{k}$
with standard matrix representation

$\operatorname{diag}\left(\frac{\lambda_{k,\sigma^{-1}(1)}}{\alpha_{\sigma^{-1}(1)}},\frac{\lambda_{k,\sigma^{-1}(2)}}{\alpha_{\sigma^{-1}(2)}},\dots,\frac{\lambda_{k,\sigma^{-1}(a)}}{\alpha_{\sigma^{-1}(a)}}\right),$

we see that

$t^{E_{k}}Q(\zeta)=\left(t^{\lambda_{k,1}}\zeta_{\sigma(1)}^{\alpha_{1}},t^{\lambda_{k,2}}\zeta_{\sigma(2)}^{\alpha_{2}},\dots,t^{\lambda_{k,2}}\zeta_{\sigma(a)}^{\alpha_{a}}\right)=Q(t^{F_{k}}\zeta)$

for all $t>0$ and $\zeta\in\mathbb{R}^{a}$. Correspondingly, $Q$ is
homogeneous with respect to the pair $(E_{k},F_{k})$ for $k=1,2$. We have the
following result.

######  Proposition 3.2.

If $\lambda_{1,j}\geq\lambda_{2,j}$ for $j=1,2,\dots,a$, then for
$P:\mathbb{R}^{2a}\to\mathbb{R}$ defined by

$P(\eta,\zeta)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

for $(\eta,\zeta)\in\mathbb{R}^{2a}$, we have

$\varphi(t)\asymp\begin{cases}t^{-\mu_{0}}&t\leq 1\\\ t^{-\mu_{\infty}}&t\geq
1\end{cases}$

for $t>0$ where

$\mu_{0}=\sum_{j=1}^{a}\left(\lambda_{2,j}+\frac{\lambda_{1,j}}{\alpha_{j}}\right)$

and

$\mu_{\infty}=\sum_{j=1}^{a}\left(\lambda_{1,j}+\frac{\lambda_{2,j}}{\alpha_{j}}\right).$

###### Proof.

The hypothesis guarantees that $t^{E_{1}-E_{2}}$ is non-expanding and so, in
view of Remark 1, we only need to verify that the exponents $\mu_{0}$ and
$\mu_{\infty}$ are as stated. Of course, since $\sigma$ is a permutation of
$\\{1,2,\dots,a\\}$,

$\mu_{0}=\tr E_{2}+\tr
F_{1}=\sum_{j=1}^{a}\lambda_{2,j}+\sum_{j^{\prime}=1}^{a}\frac{\lambda_{1,\sigma^{-1}(j^{\prime})}}{\alpha_{\sigma^{-1}(j^{\prime})}}=\sum_{j=1}^{a}\lambda_{2,j}+\sum_{j=1}^{a}\frac{\lambda_{1,j}}{\alpha_{j}}.$

The computation for $\mu_{\infty}$ is done analogously. ∎

In this example, we have assumed that $a=b$. We remark that the results found
above can be extended to arbitrary dimensions $a$ and $b$ by consider $Q$’s of
the form found in Examples 4 and 5. We leave these details to the reader. ∎

Our proof of Theorem 3.1 makes use of the following lemma, whose proof can be
found in Appendix A.

######  Lemma 3.3.

Let $P_{1}$ and $P_{2}$ be positive homogeneous functions on
$\mathbb{R}^{{a}}$. If there exist $E_{1}\in\operatorname{Exp}(P_{1})$ and
$E_{2}\in\operatorname{Exp}(P_{2})$ for which $[E_{1},E_{2}]=0$ and
$\\{t^{E_{1}-E_{2}}\\}$ is non-expanding, then the following statements hold:

  1. 1.

There are positive constants $C,C^{\prime},M,M^{\prime}$ for which

$C(P_{1}(\xi)+P_{2}(\eta))-M\leq
P_{1}(t^{E_{1}-E_{2}}\eta+\xi)+P_{2}(\eta)\leq
C^{\prime}(P_{1}(\xi)+P_{2}(\eta))+M^{\prime}$ (11)

for all $\xi,\eta\in\mathbb{R}^{{a}}$ and $0<t\leq 1$.

  2. 2.

There are positive constants $C,C^{\prime},M,M^{\prime}$ for which

$C(P_{1}(\eta)+P_{1}(\xi))-M\leq
P_{1}(\eta)+P_{2}(t^{E_{1}-E_{2}}\eta+\xi)\leq
C^{\prime}(P_{2}(\eta)+P_{2}(\xi))+M^{\prime}$ (12)

for all $\eta,\xi\in\mathbb{R}^{{a}}$ and $0<t\leq 1$.

As the reader will see, this lemma is fundamental to our work throughout the
article and, in addition to its appearance in the proof of Theorem 3.1, it is
used essentially in the proofs of Theorems 4.1 and 5.1. For the aid of the
reader, we find it useful to state this result in the context of our
introductory example where $P(\eta,\zeta)=(\eta+\zeta^{2})^{2}+\eta^{4}$. In
this case (using $\xi=Q(\zeta)=\zeta^{2}$), the lemma gives the following
estimates.

  1. 1.

There are positive constants $C,\,C^{\prime},\,M,\,M^{\prime}$ for which

$C(\zeta^{4}+\eta^{4})-M\leq(t^{1/4}\eta+\zeta^{2})^{2}+\eta^{4}\leq
C^{\prime}(\zeta^{4}+\eta^{4})+M^{\prime}$

for all $\eta,\zeta\in\mathbb{R}$ and $0<t\leq 1$.

  2. 2.

There are positive constants $C,\,C^{\prime},\,M,\,M^{\prime}$ for which

$C(\eta^{2}+\zeta^{4})-M\leq\eta^{2}+(t^{1/4}\eta-\zeta^{2})^{4}\leq
C^{\prime}(\eta^{4}+\zeta^{8})+M^{\prime}$

for all $\eta,\zeta\in\mathbb{R}$ and $0<t\leq 1$.

###### Proof of Theorem 3.1.

We first treat the $t\leq 1$ behavior. Using the homogeneity of $P_{1}$ and
$P_{2}$, we observe that

$tP(\eta,\zeta)=P_{1}\left(t^{E_{1}}\eta+t^{E_{1}}Q(\zeta)\right)+P_{2}(t^{E_{2}}\eta)=P_{1}\left(t^{E_{1}}\eta+Q(t^{F_{1}}\zeta)\right)+P_{2}(t^{E_{2}}\eta)$

for all $t>0$, $\eta\in\mathbb{R}^{{a}}$ and $\zeta\in\mathbb{R}^{{b}}$. By
making the change of variables
$(\eta,\zeta)\mapsto(t^{-E_{2}}\eta,t^{-F_{1}}\zeta)$, we see that

$\varphi(t)=H_{P}^{t}(0)=\frac{t^{-(\tr E_{2}+\tr
F_{1})}}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-\left(P_{1}\left(t^{E_{1}-E_{2}}\eta+Q(\zeta)\right)+P_{2}(\eta)\right)}\,d\eta\,d\zeta$
(13)

for $t>0$. By virtue of Lemma 3.3, we find that there are positive constants
$C,C^{\prime},M,M^{\prime}$ for which

$-M^{\prime}-C^{\prime}(P_{1}(Q(\zeta))+P_{2}(\eta))\leq-
P_{1}(t^{E_{1}-E_{2}}\eta+Q(\zeta))-P_{2}(\eta)\leq
M-C(P_{1}(Q(\zeta))+P_{2}(\eta))$ (14)

for all $t\leq 1$, $\eta\in\mathbb{R}^{{a}}$, and $\zeta\in\mathbb{R}^{{b}}$.
Upon noting that $(\eta,\zeta)\mapsto P_{1}(Q(\zeta))+P_{2}(\eta)$ is a
positive homogeneous function on $\mathbb{R}^{d}$ in view of Proposition 2.9,
it follows from Lemma 2.6 that

$\int_{\mathbb{R}^{d}}e^{-\epsilon(P_{1}(Q(\zeta))+P_{2}(\eta))}\,d\eta\,d\zeta$

is a positive finite number for each $\epsilon>0$. With this in mind, the
inequality (14) guarantees constants $C,C^{\prime}>0$ for which

$C\leq\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{1}\left(t^{E_{1}-E_{2}}\eta+Q(\zeta)\right)}e^{-P_{2}(\eta)}\,d\eta\,d\zeta\leq
C^{\prime}$

for all $0<t\leq 1$, and from (13) we conclude that

$\varphi(t)\asymp t^{-(\tr E_{2}+\tr F_{1})}=t^{-(\mu_{P_{2}}+\mu_{P_{1}\circ
Q})}$

for $t\leq 1$.

On the other hand, establishing the $t\geq 1$ asymptotics is more difficult. A
first hope would be to introduce the change of variables
$(\eta,\zeta)\mapsto(t^{-E_{1}}\eta,t^{-F_{1}}\zeta)$ and find that

$\varphi(t)=\frac{t^{-(\tr E_{1}+\tr
F_{1})}}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta+Q(\zeta))-P_{2}(t^{E_{2}-E_{1}}\eta)}\,d\eta\,d\zeta$

for $t>0$. This is problematic, however. To see this, let us assume
momentarily that $\\{t^{E_{1}-E_{2}}\\}$ is contracting and observe that

$\lim_{t\to\infty}\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta+Q(\zeta))-P_{2}(t^{E_{2}-E_{1}}\eta)}\,d\eta\,d\zeta=\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta+Q(\eta))}\,d\eta\,d\zeta=\infty$

by virtue of Fatou’s lemma and the fact that

$\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta+Q(\zeta))}\,d\eta\,d\zeta=\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta^{\prime})}\,d\eta^{\prime}\,d\zeta^{\prime}=\infty.$

Consequently, in the case that $\\{t^{E_{1}-E_{2}}\\}$ is contracting, the
only information found by this argument is that $\varphi(t)$ decays more
slowly than $t^{-(\tr E_{1}+\tr F_{1})}=t^{-(\mu_{P_{1}}+\mu_{P_{1}\circ Q})}$
as $t\to\infty$.

Instead, we return to the assumption that $\\{t^{E_{1}-E_{2}}\\}$ is non-
expanding and make the non-linear change of variables
$(\eta,\zeta)\mapsto(\eta-Q(\zeta),\zeta)$. Denoting this transformation by
$T$, we find that

$DT(\eta,\zeta)=\begin{pmatrix}I_{{a}}&-D_{\zeta}Q\\\ 0&I_{{b}}\end{pmatrix}$

where $DT$ and $D_{\zeta}Q$ are the Jacobian matrices for $T$ and $Q$,
respectively, and $I_{{a}}$ and $I_{{b}}$ are the identity matrices on
$\mathbb{R}^{{a}}$ and $\mathbb{R}^{{b}}$, respectively. With this, it is easy
to see that $T$ is measure preserving and consequently

$\displaystyle\varphi(t)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-t(P\circ
T)(\eta,\zeta)}\operatorname{det}(DT(\eta,\zeta))\,d\eta\,d\zeta$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-t\widetilde{P}(\eta,\zeta)}\,d\eta\,d\zeta$

for $t>0$, where

$\widetilde{P}(\eta,\zeta)=(P\circ
T)(\eta,\zeta)=P_{1}(\eta)+P_{2}(\eta-Q(\zeta))$

for $(\eta,\zeta)\in\mathbb{R}^{d}$. Observe that

$t\widetilde{P}(\eta,\zeta)=P_{1}(t^{E_{1}}\eta)+P_{2}(t^{E_{2}}\eta-Q(t^{F_{2}}\zeta))$

for $t>0$ and $(\eta,\zeta)\in\mathbb{R}^{d}$ and, upon making the change of
variables $(\eta,\zeta)\mapsto(t^{-E_{1}}\eta,t^{-F_{2}}\zeta)$, it follows
that

$\varphi(t)=\frac{t^{-(\tr E_{1}+\tr
F_{2})}}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta)-P_{2}(t^{E_{2}-E_{1}}\eta-Q(\zeta))}\,d\eta\,d\zeta$
(15)

for $t>0$. By virtue of Lemma 3.3, there are positive constants
$C,C^{\prime},M,M^{\prime}$ for which

$M^{\prime}-C^{\prime}(P_{2}(\eta)+P_{2}(-Q(\zeta))\leq-(P_{1}(\eta)+P_{2}(t^{E_{2}-E_{1}}\eta-Q(\zeta))\leq
M-C(P_{1}(\eta)+P_{1}(-Q(\zeta)))$

for all $\eta\in\mathbb{R}^{{a}}$, $\zeta\in\mathbb{R}^{{b}}$, and $1\leq
t<\infty$ since $\\{t^{E_{1}-E_{2}}\\}$ is non-expanding and
$t^{E_{2}-E_{1}}=(1/t)^{E_{1}-E_{2}}$. With the observation that
$(\eta,\zeta)\mapsto P_{1}(\zeta)+P_{1}(-Q(\eta))$ and $(\eta,\zeta)\mapsto
P_{2}(\zeta)+P_{2}(-Q(\eta))$ are positive homogeneous functions on
$\mathbb{R}^{d}$, it follows from (12) and Lemma 2.6 that there are positive
numbers $C,C^{\prime}$ for which

$C\leq\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta)-P_{2}(t^{E_{2}-E_{1}}\eta-Q(\zeta))}\,d\eta\,d\zeta\leq
C^{\prime}$

for all $1\leq 1<\infty$. Hence,

$\varphi(t)\asymp t^{-(\tr E_{1}+\tr
F_{2})}=t^{-\left(\mu_{P_{1}}+\mu_{P_{2}\circ(-Q)}\right)}$

for $t\geq 1$. ∎

Our final result in this section shows that, in the case that
$\\{t^{E_{1}-E_{2}}\\}$ is contracting, the asymptotics of Theorem 3.1 are
“true asymptotics”.

######  Theorem 3.4.

Let $P:\mathbb{R}^{2}\to\mathbb{R}$ satisfy the hypotheses of Theorem 3.1 and,
for $P_{1}$, $P_{2}$, and $Q$ as in the theorem’s statement, put

$P_{0}(\xi)=P_{1}(Q(\zeta))+P_{2}(\eta)\hskip 28.45274pt\mbox{and}\hskip
28.45274ptP_{\infty}(\xi)=P_{1}(\eta)+P_{2}(-Q(\zeta))$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{d}$. Also, let $\varphi(t)$, $\mu_{0}$,
$\mu_{\infty}$, $E_{1}$, and $E_{2}$ be as they appear in the statement of
Theorem 3.1. If $\\{t^{E_{1}-E_{2}}\\}$ is contracting, then

$\lim_{t\to
0}t^{\mu_{0}}\varphi(t)=H_{P_{0}}^{1}(0)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{0}(\xi)}\,d\xi$

and

$\lim_{t\to\infty}t^{\mu_{\infty}}\varphi(t)=H_{P_{\infty}}^{1}(0)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{\infty}(\xi)}\,d\xi.$

In particular, $\lim_{t\to\infty}t^{\mu_{\infty}}\varphi(t)$ exists, is a
positive number, and can be computed using Lemma 2.6.

###### Proof.

We shall prove the statement involving the limit as $t\to\infty$; the $t\to 0$
statement is proved analogously. In view of (15), we have

$t^{\mu_{\infty}}\varphi(t)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{1}(\eta)-P_{2}(t^{E_{1}-E_{2}}\eta-Q(\zeta))}\,d\xi$

for $t>0$. Now, given that $\\{t^{E_{1}-E_{2}}\\}$ is contracting, we have

$\lim_{t\to\infty}e^{-P_{1}(\eta)-P_{2}(t^{E_{2}-E_{1}}\eta-Q(\zeta))}=e^{-P_{\infty}(\xi)}$

for each $\xi=(\eta,\zeta)\in\mathbb{R}^{d}$. As noted in the proof of Theorem
3.1 in the paragraph following (15), the integrands
$\xi\mapsto\exp(-P_{1}(\eta)-P_{2}(t^{E_{2}-E_{1}}\eta-Q(\zeta)))$ are
uniformly dominated, for $t\geq 1$, by the integrable function
$\xi\mapsto\exp(M-C(P_{1}(\eta)+P_{1}(-Q(\zeta))))$ and so our desired result
follows by an appeal to the dominated convergence theorem. ∎

######  Example 9:

In Example 7, we found that

$\varphi(t)=H_{P}^{t}(0)\asymp\begin{cases}t^{-(1/l+1/qp)}&0<t\leq 1\\\
t^{-(1/q+1/lp)}&t\geq 1\end{cases}$

for $t>0$ where
$P(\eta,\zeta)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)=(\eta+\zeta^{p})^{q}+\eta^{l}$
for $p\in\mathbb{N}_{+}$ and positive even integers $q$ and $l$ with $q\leq
l$. As we noted in the example, $E_{1}-E_{2}=(1/q-1/l)I$ where $I$ is the
identity transformation on $\mathbb{R}$ and therefore $\\{t^{E_{1}-E_{2}}\\}$
is contracting whenever $q<l$. Upon noting that
$P_{0}(\eta,\zeta)=\zeta^{qp}+\eta^{l}$ and
$P_{\infty}(\eta,\zeta)=\eta^{q}+\zeta^{lp}$ for
$(\eta,\zeta)\in\mathbb{R}^{2}$, Theorem 3.4 guarantees that

$\displaystyle\lim_{t\to 0}t^{(1/l+1/qp)}\varphi(t)=H_{P_{0}}^{1}(0)$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\left(\int_{\mathbb{R}}e^{-\zeta^{qp}}\,d\zeta\right)\left(\int_{\mathbb{R}}e^{-\eta^{l}}\,d\eta\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\Gamma\left(1+\frac{1}{qp}\right)\Gamma\left(1+\frac{1}{l}\right)$

and similarly

$\lim_{t\to\infty}t^{(1/q+1/lp)}\varphi(t)=H_{P_{\infty}}^{1}(0)=\frac{1}{\pi^{2}}\Gamma\left(1+\frac{1}{q}\right)\Gamma\left(1+\frac{1}{lp}\right)$

provided $q<l$. In particular, for our introductory example in which
$P(\eta,\zeta)=(\eta+\zeta^{2})^{2}+\eta^{4}$, i.e., where $2=p=q<l=4$, this
gives the (expected) limit (7) and, more interestingly,

$\lim_{t\to\infty}t^{5/8}\varphi(t)=H_{P_{\infty}}^{1}(0)=\frac{1}{2\pi^{3/2}}\Gamma(9/8)\approx
0.0845624.$

∎

##  4 A perturbation theory

Let us take the classical viewpoint that the theory of elliptic/semi-elliptic
operators is a “perturbation theory” in which a sufficiently well-behaved
partial differential operator is perturbed by adding operators whose order is
lower than that of the given operator. In that setting, one may investigate
properties of solutions to related partial differential equations which are
preserved under such perturbations. For example, in the theory of elliptic
operators, short-time heat kernel estimates for a uniformly elliptic operator
are determined by the operator’s principal symbol. In this way, perturbation
by lower-order operators – provided they are sufficiently well behaved – will
not essentially change the short-time behavior of heat kernels. In this short
section, we explore perturbation by higher-order operators/symbols. In
particular, we show that, under certain conditions, the large-time decay of
$\varphi(t)=H_{P}^{t}(0)$ is essentially unchanged when $P$ is replaced with
$P+R$ where $R(\xi)=o(P(\xi))$ as $\xi\to 0$. Given that our analysis is done
exclusively in the frequency domain, our results amount, essentially, to a
perturbation theory of constant-coefficient operators. We suspect that a
successful variable-coefficient theory is possible, however, we do not pursue
that here. Our main result is as follows.

######  Theorem 4.1.

Let $P$ satisfy the hypotheses of Theorem 3.1 and let
$R:\mathbb{R}^{d}\to\mathbb{C}$ be a continuous function for which
$R(\xi)=o(P(\xi))$ as $\xi\to 0$ and $\xi\mapsto\operatorname{Re}(R(\xi))$
non-negative. Then

$H_{P+R}^{t}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-t(P(\xi)+R(\xi))}e^{-ix\cdot\xi}\,d\xi$

exists for all $t>0$ and $x\in\mathbb{R}^{d}$. Further, for
$H_{P}=H_{P}^{(\cdot)}(\cdot)$ and $\mu_{\infty}$ as given in Theorem 3.1,

$H_{P+R}^{t}(x)=H_{P}^{t}(x)+o(t^{-\mu_{\infty}})$ (16)

uniformly for $x\in\mathbb{R}^{d}$ as $t\to\infty$. In particular, we have the
following large-time on-diagonal asymptotics:

  1. 1.

$\absolutevalue{H_{P+R}^{t}(0)}\asymp t^{-\mu_{\infty}}$ (17)

for $t\geq 1$.

  2. 2.

If $\\{t^{E_{1}-E_{2}}\\}$ is contracting (where $E_{1}$ and $E_{2}$ are as
given in the statement of Theorem 3.1), then

$\lim_{t\to\infty}t^{\mu_{\infty}}H_{P+R}^{t}(0)=H_{P_{\infty}}^{1}(0)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{\infty}(\xi)}\,d\xi$
(18)

where $P_{\infty}$ is as defined in Theorem 3.4.

Before proving the theorem, we shall first treat a technical lemma which will
also be found useful in our application to convolution powers of complex-
valued functions presented in Section 5. The lemma introduces the useful
notion of subhomogeneity on an ad hoc basis; for a more complete treatment, we
refer the reader to Section 2 of [BR22].

######  Lemma 4.2.

Let $P$ satisfy the hypotheses of Theorem 3.1 and take $E_{1}$ and $F_{2}$ as
in the statement of the theorem. For convenience of notation, we set
$G=E_{1}\oplus F_{2}$ and $\widetilde{P}=P\circ T$ where
$T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is the measure-preserving transformation
defined by $T(\eta,\zeta)=(\eta-Q(\zeta),\zeta)$ for
$(\eta,\zeta)\in\mathbb{R}^{a}\times\mathbb{R}^{b}=\mathbb{R}^{d}$. Finally,
given an open neighborhood $\mathcal{O}\subseteq\mathbb{R}^{d}$ of $0$, let
$R:\mathcal{O}\to\mathbb{C}$ be a continuous function and set
$\widetilde{R}=R\circ T$. Then the following statements hold.

  1. 1.

$R(\xi)=o(P(\xi))$ as $\xi\to 0$ if and only if
$\widetilde{R}(\xi)=o(\widetilde{P}(\xi))$ as $\xi\to 0$.

  2. 2.

If either of the preceding equivalent conditions are satisfied, then
$\widetilde{R}$ is subhomogeneous with respect to $G$ in the sense that, for
each $\epsilon>0$ and compact set $K\subseteq\mathbb{R}^{d}$, there exists
$t_{0}>0$ for which

$\absolutevalue{\widetilde{R}(t^{G}\xi)}\leq\epsilon t$

whenever $0<t\leq t_{0}$ and $\xi\in K$.

###### Proof.

Because $T$ is a homeomorphism with $T(0)=0$, the first assertion is
immediate. For the second assertion, let us fix a compact set $K$ and a
positive number $\epsilon$. Also, given that
$\widetilde{R}(\xi)=o(\widetilde{P}(\xi))$ as $\xi\to 0$, let $\delta>0$ be
such that
$\absolutevalue{\widetilde{R}(\xi)}\leq(\epsilon/M)\widetilde{P}(\xi)$ for
$\absolutevalue{\xi}\leq\delta$ where

$M=\sup_{\xi=(\eta,\zeta)\in
K}C^{\prime}(P_{2}(\eta)+P_{2}(-Q(\zeta))+M^{\prime}$

where $C^{\prime}$ and $M^{\prime}$ are those positive constants appearing in
(12) of Lemma 3.3. Using the fact that $\\{t^{G}\\}$ is a contracting group,
there exists $0<t_{0}\leq 1$ for which $\absolutevalue{t^{G}\xi}\leq\delta$
for all $\xi\in K$ and $0<t\leq t_{0}$. Consequently, for $\xi=(\eta,\zeta)\in
K$ and $0<t\leq t_{0}$, we have

$\displaystyle\absolutevalue{\widetilde{R}(t^{G}\xi)}$ $\displaystyle\leq$
$\displaystyle(\epsilon/M)\widetilde{P}(t^{G}\xi)$ $\displaystyle=$
$\displaystyle(\epsilon/M)\left(P_{1}(t^{E_{1}}\eta)+P_{2}(t^{E_{1}}\eta-t^{E_{2}}Q(\zeta))\right)$
$\displaystyle=$
$\displaystyle(\epsilon/M)\left(tP_{1}(\eta)+tP_{2}(r^{E_{1}-E_{2}}\eta-Q(\zeta))\right)$
$\displaystyle\leq$
$\displaystyle(\epsilon/M)t\left(C^{\prime}(P_{2}(\eta)+P_{2}(-Q(\zeta))+M^{\prime}\right)$
$\displaystyle\leq$ $\displaystyle\epsilon t$

thanks to Lemma 3.3. ∎

###### Proof of Theorem 4.1.

Given that

$\absolutevalue{e^{-t(P(\xi)+R(\xi)}}=e^{-tP(\xi)}e^{-t\operatorname{Re}(R(\xi))}\leq
e^{-tP(\xi)}$

for $\xi\in\mathbb{R}^{d}$, the first assertion follows immediately from
Theorem 3.1. For the second assertion, observe that

$\displaystyle H_{P+R}^{t}(x)-H_{P}^{t}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-tP(\xi)}\left(e^{-tR(\xi)}-1\right)e^{-ix\cdot\xi}\,d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-t\widetilde{P}(\xi)}\left(e^{-t\widetilde{R}(\xi)}-1\right)e^{-ix\cdot
T(\xi)}\,d\xi$ $\displaystyle=$
$\displaystyle\frac{t^{-\mu_{\infty}}}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-t\widetilde{P}(t^{-G}\xi)}\left(e^{-t\widetilde{R}(t^{-G}\xi)}-1\right)e^{-ix\cdot
T(t^{-G}\xi)}\,d\xi$

for $t>0$ and $x\in\mathbb{R}^{d}$ where $G=E_{1}\oplus F_{2}$ and
$\mu_{\infty}=\tr E_{1}+\tr F_{2}=\tr G$. By an appeal to Lemma 3.3, we find
that

$-t\widetilde{P}(t^{-G}\xi)=-(P_{1}(\eta)+P_{2}(t^{E_{2}-E_{1}}\eta-Q(\zeta))\leq
M-CP_{*}(\xi)$

for all $t\geq 1$ where $M$ and $C$ are positive constants and we have set
$P_{*}(\xi)=P_{1}(\eta)+P_{1}(-Q(\zeta))$ for
$\xi=(\eta,\zeta)\in\mathbb{R}^{d}$. Consequently,

$t^{\mu_{\infty}}\absolutevalue{H_{P+R}^{t}(x)-H_{P}^{t}(x)}\leq
e^{M}\int_{\mathbb{R}^{d}}e^{-CP_{*}(\xi)}\absolutevalue{e^{-t\widetilde{R}(t^{-G}\xi)}-1}\,d\xi$
(19)

for $t\geq 1$ and $x\in\mathbb{R}^{d}$. Given that $R(\xi)=o(P(\xi))$ as
$\xi\to 0$, Lemma 4.2 guarantees that

$\lim_{t\to\infty}e^{-\rho
P_{*}(\xi)}\absolutevalue{e^{-t\widetilde{R}(t^{-G}\xi)}-1}=\lim_{s\to
0}e^{-\rho P_{*}(\xi)}\absolutevalue{e^{-s\widetilde{R}(s^{-G}\xi)}-1}=0$

for each $\xi\in\mathbb{R}^{d}$. Upon noting that the integrand in (19) is
dominated by the integrable function $\xi\mapsto 2e^{-CP_{*}(\xi)}$ (see Lemma
2.6), an appeal to the dominated convergence theorem guarantees that, for each
$\epsilon>0$, there exists $t_{0}\geq 1$ for which

$t^{\mu_{\infty}}\absolutevalue{H_{P+R}^{t}(x)-H_{P}^{t}(x)}<\epsilon$

for all $x\in\mathbb{R}^{d}$ and $t\geq t_{0}$; this is precisely the uniform
limit (16). Applying this result at $x=0$, we immediately obtain (17) from
Theorem 3.1 and (18) from Theorem 3.4. ∎

######  Example 10:

For the operator

$\Lambda=\partial_{x_{1}}^{4}+\partial_{x_{2}}^{4}+2i\partial_{x_{1}}\partial_{x_{2}}^{2}-\partial_{x_{1}}^{2},$

we consider the perturbation $\Lambda+\Lambda^{2}$ with symbol $P(\xi)+R(\xi)$
where $P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$ and $R(\xi)=P(\xi)^{2}$ for
$\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. As shown in Example 7, $P$ satisfies the
hypotheses of Theorem 3.1 and, from the theorem, we obtain the large-time
asymptotic: $\varphi(t)=t^{-5/8}$ for $t\geq 1$. Since $P$ is continuous at
$0$ and $R(\xi)=P(\xi)^{2}\geq 0$, it is evident that $R$ satisfies the
hypotheses of Theorem 4.1. Consequently, the heat kernel

$H_{P+R}^{t}(x)=\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}e^{-t(P(\xi)+R(\xi))}e^{-ix\cdot\xi}\,d\xi$

associated to the operator $\Lambda+\Lambda^{2}$ has

$H_{P+R}^{t}(x)=H_{P}^{t}(x)+o(t^{-5/8})$

uniformly for $x\in\mathbb{R}^{2}$ as $t\to\infty$; here, $H_{P}$ is that
given in (2) and illustrated in Figure 1(b) for $t=10$. Also, in view of our
analysis in Example 9, Theorem 4.1 gives us the large-time asymptotics,
$\absolutevalue{H_{P+R}^{t}(0)}\asymp t^{-5/8}$ for $t\geq 1$ and

$\lim_{t\to\infty}t^{5/8}H_{P+R}^{t}(0)=\frac{1}{2\pi^{3/2}}\Gamma(9/8).$

We note that, by contrast, $H_{P+R}^{t}(0)$ does not obey the $t^{-1/2}$
small-time on-diagonal asymptotic of $H_{P}$. Indeed, $\Lambda+\Lambda^{2}$ is
an eighth-order elliptic operator and necessarily decays as $t^{-1/4}$ in
small time. ∎

The following example generalizes the previous one and places the result in
the context semigroups and ultracontractivity.

######  Example 11:

Let $\Lambda$ be a constant-coefficient partial differential operator on
$\mathbb{R}^{d}=\mathbb{R}^{a}\times\mathbb{R}^{b}$ with polynomial symbol

$P(\xi)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

satisfying the hypotheses of Theorem 3.1. Let $q(\lambda)$ be a real-valued
polynomial of a single real variable for which $q(0)=0$, $q^{\prime}(0)=1$ and
$q(\lambda)\geq\lambda$ for all $\lambda\geq 0$. Using the Fourier transform
or the spectral calculus, it is easy to see that $q(\Lambda)$ is a positive
self-adjoint operator on $L^{2}(\mathbb{R}^{d})$ and therefore $-q(\Lambda)$
generates a continuous semigroup $\\{e^{-tq(\Lambda)}\\}$ on
$L^{2}(\mathbb{R}^{d})$. Denoting by $E$ the spectral resolution of $\Lambda$,
observe that, for $f\in L^{2}(\mathbb{R}^{d})$,

$\|e^{-t(q(\Lambda)-\Lambda)}f\|_{2}^{2}=\int_{0}^{\infty}e^{-2t(q(\lambda)-\lambda)}\,dE_{f,f}(\lambda)\leq\|f\|_{2}^{2}$

and therefore, for each $t>0$, $\\{e^{-t(q(\Lambda)-\Lambda)}\\}$ is a
contraction on $L^{2}(\mathbb{R}^{d})$. Consequently,

$\|e^{-tq(\Lambda)}\|_{2\to\infty}=\|e^{-t\Lambda}e^{-t(q(\Lambda)-\Lambda)}\|_{2\to\infty}\leq\|e^{-t(q(\Lambda)-\Lambda)}\|_{2\to
2}\|e^{-t\Lambda}\|_{2\to\infty}\leq\sqrt{\varphi(2t)}\leq
C^{\prime}t^{-\mu_{\infty}}$

for $t\geq 1$ where $C^{\prime}$ is a positive constant and

$\mu_{\infty}=\mu_{P_{1}}+\mu_{P_{2}\circ(-Q)}=\tr E_{1}+\tr F_{2}$

as given in the statement of Theorem 3.1. By duality, we find that
$\|e^{-tq(\Lambda)}\|_{1\to\infty}\leq Ct^{-\mu_{\infty}}$ for $t\geq 1$ for
some positive constant $C$. It follows (see Lemma 2.1.2 of [Da89]) that
$\\{e^{-tq(\Lambda)}\\}$ has integral representation

$\left(e^{-tq(\Lambda)}f\right)(x)=\int_{\mathbb{R}^{d}}H^{t}(x,y)f(y)\,dy$

where

$\|e^{-tq(\Lambda)}\|_{1\to\infty}=\sup_{x,y}\absolutevalue{H^{t}(x,y)}\leq
Ct^{-\mu_{\infty}}$ (20)

for $t\geq 1$. Of course, given that $q$ is a polynomial, it is easy to verify
that, in fact, $H^{t}(x,y)=H_{q\circ P}^{t}(x-y)$ for $x,y\in\mathbb{R}^{d}$
and $t>0$ where

$H_{q\circ P}^{t}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-t(q\circ
P)(\xi)}e^{-ix\cdot\xi}\,d\xi$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{d}$. Thus,

$\|e^{-tq(\Lambda)}\|_{1\to\infty}=\sup_{x,y}\absolutevalue{H^{t}(x,y)}=H_{q\circ
P}^{t}(0).$

We claim that

$H_{q\circ P}^{t}(0)\asymp t^{-\mu_{\infty}}$

for $t\geq 1$ and so the upper bound in the ultracontractive estimate (20) is
optimal. Indeed, under the given hypotheses concerning $q$, we may write

$q(\Lambda)=\Lambda+r(\Lambda)$

where $r$ is a polynomial having $r(\lambda)\to 0$ as $\lambda\to 0$ and
$r(\lambda)\geq 0$ for $\lambda\geq 0$. From this it follows that

$(q\circ P)(\xi)=P(\xi)+R(\xi)$

where $R(\xi)=r(P(\xi))$ is continuous, non-negative, and has
$R(\xi)=o(P(\xi))$ as $\xi\to 0$. With this, our claim follows by an
application of Theorem 4.1. If we additionally assume that, for $E_{1}$ and
$E_{2}$ as they appear in the statement of Theorem 3.1,
$\\{t^{E_{1}-E_{2}}\\}$ is contracting, then Theorem 4.1 also guarantees that

$\lim_{t\to\infty}t^{\mu_{\infty}}H_{q\circ P}^{t}(0)=H_{P_{\infty}}^{1}(0).$

∎

In contrast to the preceding examples, we now consider a perturbation of an
operator $\Lambda$ by one which is not easily comparable to $\Lambda$.

######  Example 12:

Consider

$\Lambda+(-\Delta)^{5}$ (21)

where $\Delta=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}$ is the Laplacian on
$\mathbb{R}^{2}$ and
$\Lambda=\partial_{x_{1}}^{4}+\partial_{x_{2}}^{4}+2i\partial_{x_{1}}\partial_{x_{2}}^{2}-\partial_{x_{1}}^{2}.$
Associated to the operator (21) is the heat kernel

$H_{P+R}^{t}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{2}}e^{-t(P(\xi)+R(\xi))}e^{-ix\cdot\xi}\,d\xi$

where

$P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}\hskip 28.45274pt\mbox{and}\hskip
28.45274ptR(\xi)=(\eta^{2}+\zeta^{2})^{5}$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. We claim that $R(\xi)=o(P(\xi))$ as
$\xi\to 0$. To see this, we consider the open neighborhood
$\mathcal{O}=\\{\xi=(\eta,\zeta)\in\mathbb{R}^{2}:P(\xi)<1\\}$ of $0$ and
write $\mathcal{O}=\mathcal{R}_{1}\cup\mathcal{R}_{2}$ where

$\mathcal{R}_{1}=\left\\{\xi=(\eta,\zeta)\in\mathcal{O}:\absolutevalue{\eta+\zeta^{2}}\leq(1-1/\sqrt{2})\zeta^{2}\right\\}$
(22)

and

$\mathcal{R}_{2}=\left\\{\xi=(\eta,\zeta)\in\mathcal{O}:\absolutevalue{\eta+\zeta^{2}}\geq(1-1/\sqrt{2})\zeta^{2}\right\\}.$
(23)

For $\xi=(\eta,\zeta)\in\mathcal{R}_{1}$, observe that
$0\leq\zeta^{2}\leq-\sqrt{2}\eta$ and therefore

$4P(\xi)=4\left((\eta+\zeta^{2})^{2}+\eta^{4}\right)\geq
4\eta^{4}\geq\zeta^{8}.$

Thus,

$\eta^{2}+\zeta^{2}\leq
P(\xi)^{1/2}+\sqrt{2}P(\xi)^{1/4}=\left(P(\xi)^{1/4}+\sqrt{2}\right)P(\xi)^{1/4}\leq
5P(\xi)^{1/4}$ (24)

for $\xi=(\eta,\zeta)\in\mathcal{R}_{1}$. On $\mathcal{R}_{2}$, we find that

$P(\xi)\geq\max\left\\{\eta^{4},\left(1-\frac{1}{\sqrt{2}}\right)^{2}\zeta^{4}\right\\}$

so that

$\eta^{2}+\zeta^{2}\leq
P(\xi)^{1/2}+(2+\sqrt{2})P(\xi)^{1/2}=(3+\sqrt{2})P(\xi)^{1/2}\leq
5P(\xi)^{1/4}$ (25)

for $\xi=(\eta,\zeta)\in\mathcal{R}_{2}$. Since
$\mathcal{O}=\mathcal{R}_{1}\cup\mathcal{R}_{2}$, the estimates (24) and (24)
guarantee that, for each $\xi=(\eta,\zeta)\in\mathcal{O}$,

$\absolutevalue{R(\xi)}=(\eta^{2}+\zeta^{2})(\eta^{2}+\zeta^{2})^{4}\leq
625(\eta^{2}+\zeta^{2})P(\xi)$

and, from this, our claim follows immediately. Since $R(\xi)$ is non-negative,
an appeal to Theorem 4.1 is valid and we conclude that
$\absolutevalue{H_{P+R}^{t}(0)}\asymp t^{-5/8}$ for $t\geq 1$ and

$\lim_{t\to\infty}t^{5/8}H_{P+R}^{t}(0)=\frac{1}{2\pi^{3/2}}\Gamma(9/8)$

in view of Example 9. ∎

As evidenced by the preceding example, it isn’t straightforward to show that
$R(\xi)=o(P(\xi))$ as $\xi\to 0$. This is connected to the fact that $P$ is
generally inhomogeneous and so examining a polynomial $R$ along the coordinate
axes or by comparing the order of its terms against those of $P$ is not often
helpful. For $P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$, a careful study of the
example shows that $R(\xi)=o(P(\xi))$ as $\xi\to 0$ provided that $R$ is a
polynomial comprised of terms whose (multivariate) order is at least nine.
Still, polynomials $R$ with terms of lower order can decay as “little-o” of
$P$, e.g., $(\eta,\zeta)\mapsto\eta^{4}$, however, in general, it is difficult
to tell. For example, consider the polynomials

$R_{1}(\xi)=\eta^{2}\zeta^{2}+2\eta\zeta^{4}+\zeta^{6}\hskip
28.45274pt\mbox{and}\hskip 28.45274ptR_{2}(\xi)=\eta^{2}\zeta^{4}$

defined for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. Though the polynomial $R_{1}$
contains terms of lower order than $R_{2}$ and thus decays more slowly than
$R_{2}$ as $\xi\to 0$ (at least, along rays), $R_{1}(\xi)=o(P(\xi))$ as
$\xi\to 0$ while, by contrast, $R_{2}(\xi)\neq o(P(\xi))$ as $\xi\to 0$. For
sorting out these somewhat unintuitive statements, the following refinement of
Lemma 4.2 is helpful; its proof can be found in Appendix A.

######  Proposition 4.3.

Let $P$ satisfy the hypotheses of Theorem 3.1 and let $P_{1}$, $P_{2}$, $Q$,
$E_{1}$, $E_{2}$, $F_{1}$ and $F_{2}$ be as in the statement of the theorem.
Set $G=E_{1}\oplus F_{2}$ and $\widetilde{P}=P\circ T$ where
$T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is the measure-preserving transformation
defined by $T(\eta,\zeta)=(\eta-Q(\zeta),\zeta)$ for
$(\eta,\zeta)\in\mathbb{R}^{d}$. Finally, given an open neighborhood
$\mathcal{O}\subseteq\mathbb{R}^{d}$ of $0$, let $R:\mathcal{O}\to\mathbb{C}$
be a continuous function and set $\widetilde{R}=R\circ T$. If
$\\{t^{E_{1}-E_{2}}\\}$ is contracting, then the following statements are
equivalent:

  1. 1.

$R(\xi)=o(P(\xi))$ as $\xi\to 0$.

  2. 2.

$\widetilde{R}(\xi)=o(\widetilde{P}(\xi))$ as $\xi\to 0$.

  3. 3.

$\widetilde{R}$ is subhomogeneous with respect to $G$ in the sense that, for
each $\epsilon>0$ and compact set $K\subseteq\mathbb{R}^{d}$, there exists
$t_{0}>0$ for which

$\absolutevalue{\widetilde{R}(t^{G}\xi)}\leq\epsilon t$

whenever $0<t\leq t_{0}$ and $\xi\in K$.

We now use the proposition to prove the assertions made right before it.

######  Example 13:

Let $R_{1}$ and $R_{2}$ be as in the paragraph preceding the proposition and
let $P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$. As shown in Example 7, we have
$P_{1}(\eta)=\eta^{2}$, $P_{2}(\eta)=\eta^{4}$, $Q(\zeta)=\zeta^{2}$,
$E_{1}=I/2$, $E_{2}=I/4$, and $F_{2}=1/8$. Observe that, since
$E_{1}-E_{2}=I/4$, $\\{t^{E_{1}-E_{2}}\\}$ is contracting and so an
application of proposition is justified for $R_{1}$ and $R_{2}$. Focusing
first on $R_{1}$, we compute

$\widetilde{R}_{1}(\xi)=R_{1}(\eta-\zeta^{2},\zeta)=\left((\eta-\zeta^{2})^{2}+2(\eta-\zeta^{2})\zeta^{2}+\zeta^{4}\right)\zeta^{2}=\eta^{2}\zeta^{2}$

and, because $G=E_{1}\oplus F_{2}$ has standard matrix representation
$\operatorname{diag}(1/2,1/8)$,

$\widetilde{R}_{1}(t^{G}\xi)=(t^{1/2}\eta)^{2}(t^{1/8}\zeta)^{2}=t^{5/4}\eta^{2}\zeta^{2}$

for $t>0$ and $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. Consequently, given
$\epsilon>0$ and a compact set $K\subseteq\mathbb{R}^{2}$, we observe that

$\absolutevalue{\widetilde{R}_{1}(t^{G}\xi)}\leq t^{5/4}\eta^{2}\zeta^{2}\leq
tt_{0}^{1/4}\eta^{2}\zeta^{2}\leq\epsilon t$

whenever $\xi=(\eta,\zeta)\in K$ and $0<t\leq
t_{0}:=\epsilon^{4}/(1+\sup_{(\eta,\zeta)\in K}\eta^{2}\zeta^{2})^{4}$. Thus
$\widetilde{R}_{1}$ is subhomogeneous with respect to $G$ and from Proposition
4.3 we conclude that $R_{1}(\xi)=o(P(\xi))$ as $\xi\to 0$.

For $R_{2}$, we have

$\widetilde{R}_{2}(\xi)=(\eta-\zeta^{2})^{2}\zeta^{4}$

and therefore

$\widetilde{R}_{2}(t^{G}\xi)=\left(t^{1/2}\eta-(t^{1/8}\zeta)^{2}\right)^{2}\left(t^{1/8}\zeta\right)^{4}=t(t^{1/4}\eta-\zeta^{2})^{2}\zeta^{4}$

for $t>0$ and $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. Thus, for any
$\xi=(\eta,\zeta)\in\mathbb{R}^{2}$ for which $\zeta\neq 0$,

$\lim_{t\to 0}t^{-1}\widetilde{R}_{2}(t^{G}\xi)=\zeta^{8}\neq 0.$

Consequently, $\widetilde{R}_{2}$ is not subhomogeneous with respect to $G$
and, by virtue of the proposition, we conclude that $R_{2}(\xi)\neq o(P(\xi))$
as $\xi\to 0$. ∎

######  Example 14:

We return to the set-up of Example 8 and let
$Q:\mathbb{R}^{a}\to\mathbb{R}^{a}$ be a nondegenerate multivariate
homogeneous function given by

$Q(\zeta)=\left(\zeta_{\sigma(1)}^{\alpha_{1}},\zeta_{\sigma(2)}^{\alpha_{2}},\dots,\zeta_{\sigma(a)}^{\alpha_{a}}\right)$
(26)

for $\zeta=(\zeta_{1},\zeta_{2},\dots,\zeta_{a})\in\mathbb{R}^{a}$ where
$\alpha_{1},\alpha_{2},\dots,\alpha_{a}\in\mathbb{N}_{+}$ and $\sigma$ is a
permutation of the set $\\{1,2,\dots,a\\}$. As in that example we shall take
positive homogeneous functions $P_{1}$ and $P_{2}$ on $\mathbb{R}^{a}$ for
which, for $k=1,2$, $\operatorname{Exp}(P_{k})$ contains
$E_{k}\in\operatorname{End}(\mathbb{R}^{a})$ with standard matrix
representation
$\operatorname{diag}(\lambda_{k,1},\lambda_{k,2},\dots,\lambda_{k,a})$ where
$\lambda_{k,j}>0$ for $j=1,2,\dots,a$. As we showed in Example 8, the function
$P:\mathbb{R}^{2a}\to\mathbb{R}$ defined by

$P(\xi)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2a}$ satisfies the hypotheses of Theorem
3.1 provided that $\lambda_{1,j}\geq\lambda_{2,j}$ for all $j=1,2,\dots,a$. In
that case, we proved that

$\varphi(t)=H_{P}^{t}(0)\asymp t^{-\mu_{\infty}}$

for $t\geq 1$ where

$\mu_{\infty}=\sum_{j=1}^{a}\left(\lambda_{1,j}+\frac{\lambda_{2,j}}{\alpha_{j}}\right).$

Motivated by the example preceding Proposition 4.3, we shall perturb
$P:\mathbb{R}^{2a}\to\mathbb{R}$ by

$L_{k}(\xi)=\absolutevalue{\xi}^{2k}=(\absolutevalue{\eta}^{2}+\absolutevalue{\zeta}^{2})^{k}$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2a}$ where $k>0$. We have the following
result.

######  Proposition 4.4.

Suppose that $\lambda_{1,j}>\lambda_{2,j}$ for $j=1,2,\dots,a$. If

$k>\frac{1}{2\min_{j=1,2,\dots,a}(\lambda_{2,j}/\alpha_{j})},$

then $L_{k}(\xi)=o(P(\xi))$ as $\xi\to 0$.

###### Proof.

We have

$\widetilde{L}_{k}(\xi)=\left(\absolutevalue{\eta-Q(\zeta)}^{2}+\absolutevalue{\zeta}^{2}\right)^{k}$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2a}$. Using our analysis in Example 4, we
see that $F_{2}$ has standard matrix representation

$\operatorname{diag}\left(\frac{\lambda_{2,\sigma^{-1}}(1)}{\alpha_{\sigma^{-1}(1)}},\frac{\lambda_{2,\sigma^{-1}}(2)}{\alpha_{\sigma^{-1}(2)}},\dots,\frac{\lambda_{2,\sigma^{-1}}(a)}{\alpha_{\sigma^{-1}(a)}}\right).$

Thus, for $G=E_{1}\oplus F_{2}$, we have

$\displaystyle\widetilde{L}_{k}(t^{G}\xi)$ $\displaystyle=$
$\displaystyle\left(\absolutevalue{t^{E_{1}}\eta-Q(t^{F_{2}}\zeta)}^{2}+\absolutevalue{t^{F_{2}}\zeta}^{2}\right)^{k}$
$\displaystyle=$
$\displaystyle\left(\absolutevalue{t^{E_{2}}(t^{E_{1}-E_{2}}\eta-Q(\zeta))}^{2}+\absolutevalue{t^{F_{2}}\zeta}^{2}\right)^{k}$
$\displaystyle\leq$
$\displaystyle\left(\|t^{E_{2}}\|^{2}\absolutevalue{t^{E_{1}-E_{2}}\eta-Q(\zeta)}^{2}+\|t^{F_{2}}\|^{2}\absolutevalue{\zeta}^{2}\right)^{k}$
$\displaystyle\leq$
$\displaystyle\left(\|t^{E_{2}}\|^{2}\left(\|t^{E_{1}-E_{2}}\|\absolutevalue{\eta}+\absolutevalue{Q(\zeta)}\right)^{2}+\|t^{F_{2}}\|^{2}\absolutevalue{\zeta}^{2}\right)$

for $t>0$ and $\xi=(\eta,\zeta)\in\mathbb{R}^{2a}$. Given that $E_{1}$,
$E_{2}$, and $F_{2}$ are diagonal, our hypotheses guarantee that
$\|t^{E_{1}-E_{2}}\|\leq 1$, $\|t^{E_{2}}\|\leq t^{\omega}$, and
$\|t^{F_{2}}\|\leq t^{\omega}$ for $0<t\leq 1$ where

$\omega=\min\left(\operatorname{Spec}(F_{2})\right)=\min_{j=1,2,\dots,a}(\lambda_{2,j}/\alpha_{j})\leq\min_{j=1,2\dots,a}\lambda_{2,j}=\min\left(\operatorname{Spec}(E_{2})\right).$

Therefore,

$\widetilde{L}_{k}(t^{G}\xi)\leq\left(t^{2\omega}\left(\absolutevalue{\eta}+\absolutevalue{Q(\zeta)}\right)^{2}+t^{2\omega}\absolutevalue{\zeta}^{2}\right)^{k}\leq
t^{2k\omega}\left(\left(\absolutevalue{\eta}+\absolutevalue{Q(\zeta)}\right)^{2}+\absolutevalue{\zeta}^{2}\right)^{k}$

for $0<t\leq 1$ and $\xi=(\eta,\zeta)\in\mathbb{R}^{2a}$. Since
$2k\omega=2k\left(\min_{j=1,2,\dots,a}\lambda_{2,j}/\alpha_{j}\right)>1$, it
follows that, for each $\epsilon>0$ and compact set
$K\subseteq\mathbb{R}^{2a}$,

$\absolutevalue{\widetilde{L}_{k}(t^{G}\xi)}\leq\epsilon t$

for $\xi\in K$ and $0<t\leq t_{0}$ where $t_{0}>0$ is chosen so that

$t_{0}^{2k\omega-1}\sup_{\xi=(\eta,\zeta)\in
K}\left(\left(\absolutevalue{\eta}+\absolutevalue{Q(\zeta)}\right)^{2}+\absolutevalue{\zeta}^{2}\right)^{k}<\epsilon.$

Upon noting that $\\{t^{E_{1}-E_{2}}\\}$ is contracting because
$\lambda_{1,j}>\lambda_{2,j}$ for $j=1,2,\dots,a$, the desired result follows
immediately from Proposition 4.3. ∎

Upon noting that the hypotheses of Proposition 4.4 guarantee that
$\\{t^{E_{1}-E_{2}}\\}$ is contracting, by an application of Theorem 4.1 and
Proposition 4.4, we immediately obtain the following corollary.

######  Corollary 4.5.

Let $\Lambda$ be a constant-coefficient operator on $\mathbb{R}^{2a}$ with
symbol $P(\xi)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$ where $Q$ is given by (26)
and $P_{1}$ and $P_{2}$ are positive-definite semi-elliptic polynomials given
by

$P_{1}(\eta)=\sum_{|\beta:\mathbf{m}^{(1)}|=1}b_{1,\beta}\eta^{\beta}\hskip
28.45274pt\mbox{and}\hskip
28.45274ptP_{2}(\eta)=\sum_{|\beta:\mathbf{m}^{(2)}|=1}b_{2,\beta}\eta^{\beta}$

for $\eta\in\mathbb{R}^{a}$, respectively, where
$\mathbf{m}^{(1)}=\left(m_{1}^{(1)},m_{2}^{(1)},\dots,m_{a}^{(1)}\right)$ and
$\mathbf{m}^{(2)}=\left(m_{1}^{(2)},m_{2}^{(2)},\dots,m_{a}^{(2)}\right)$ are
$a$-tuples of even positive integers. Also, let
$\Delta=\partial_{x_{1}}^{2}+\partial_{x_{2}}^{2}+\cdots+\partial_{x_{2a}}^{2}$
denote the Laplacian on $\mathbb{R}^{2a}$. If $m_{j}^{(2)}>m_{j}^{(1)}$ for
all $j=1,2,\dots,a$, then, for any integer

$k>\left(\max_{j=1,2,\dots,a}\alpha_{j}m_{j}^{(2)}\right)/2,$

the heat kernel $H_{P+L_{k}}^{t}(x)$ associated to the operator
$\Lambda+(-\Delta)^{k}$ satisfies the on-diagonal large-time asymptotics,
$H_{P+L_{k}}^{t}(0)\asymp t^{-\mu_{\infty}}$ for $t\geq 1$ and

$\lim_{t\to\infty}t^{\mu_{\infty}}H_{P+L_{k}}^{t}(0)=H_{P_{\infty}}^{1}(0)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{\infty}(\xi)}\,d\xi$

where

$\mu_{\infty}=\sum_{j=1}^{a}\left(\frac{1}{m_{j}^{(1)}}+\frac{1}{\alpha_{j}m_{j}^{(2)}}\right)$

and

$P_{\infty}(\xi)=\sum_{|\beta:\mathbf{m}^{(1)}|=1}b_{1,\beta}\eta^{\beta}+\sum_{|\beta:\mathbf{m}^{(2)}|=1}b_{2,\beta}(-Q(\zeta))^{\beta}=\sum_{|\beta:\mathbf{m}^{(1)}|=1}b_{1,\beta}\eta^{\beta}+\sum_{|\beta:\mathbf{m}^{(2)}|=1}b_{2,\beta}(-1)^{|\beta|}\zeta_{\sigma}^{\alpha\odot\beta}$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2a}$ where $\zeta_{\sigma}$ denotes the
action of the permutation $\sigma$ on
$\zeta=(\zeta_{1},\zeta_{2},\dots,\zeta_{a})$ defined by $\zeta_{\sigma}\\\
=(\zeta_{\sigma(1)},\zeta_{\sigma(2)},\dots,\zeta_{\sigma(a)})$ and for each
multi-index $\beta$, we have set
$|\beta|=\beta_{1}+\beta_{2}+\cdots+\beta_{a}$,
$\alpha\odot\beta=(\alpha_{1}\beta_{2},\alpha_{2}\beta_{2},\dots,\alpha_{a}\beta_{a})$.

For an easy application of this corollary, consider our motivating example in
which $P(\xi)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)=(\eta+\zeta^{2})^{2}+\eta^{4}$
for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. In this case, $P_{1}$ and $P_{2}$ are
positive-definite and semi-elliptic with $\mathbf{m}^{(1)}=(m_{1}^{(1)})=(2)$
and $\mathbf{m}^{(2)}=(m_{1}^{(2)})=(4)$, respectively, and
$Q(\zeta)=\zeta^{2}$ is a nondegenerate multivariate homogeneous of the form
(26) with $\alpha_{1}=2$. Thus, for any integer

$k>\left(\max_{j}\alpha_{j}m_{j}^{(2)}\right)/2=8/2=4,$

the heat kernel $H_{P+L_{k}}$ associated to

$\Lambda+(-\Delta)^{k}=\partial_{x_{1}}^{4}+\partial_{x_{2}}^{4}+2i\partial_{x_{1}}\partial_{x_{2}}^{2}-\partial_{x_{1}}^{2}+(-\Delta)^{k}$

has $H_{P+L_{k}}^{t}(0)\asymp t^{-5/8}$ for $t\geq 1$ and
$\lim_{t\to\infty}t^{5/8}H_{P+L_{k}}^{t}(0)=\Gamma(9/8)/2\pi^{3/2}$. In
particular, this holds for $k=5$ and so we have recaptured the result of
Example 12. ∎

##  5 An application to the study of convolution powers of complex-valued
functions on $\mathbb{Z}^{d}$

Given a finitely-supported444We work with this condition for simplicity. One
can assume, more generally, that $\phi$ and all of its multivariate moments
are absolutely summable, c.f., [RSC17]. function
$\phi:\mathbb{Z}^{d}\to\mathbb{C}$, we are interested in the behavior of its
convolution powers $\phi^{(n)}:\mathbb{Z}^{d}\to\mathbb{C}$ defined
iteratively by putting $\phi^{(1)}=\phi$ and, for $n\geq 2$,

$\phi^{(n)}(x)=\sum_{y\in\mathbb{Z}^{d}}\phi^{(n-1)}(x-y)\phi(y)$

for $x\in\mathbb{Z}^{d}$. In the case that $\phi$ is non-negative and
$\sum_{x}\phi(x)=1$, the behavior of $\phi^{(n)}$ is well-known and is the
subject of the local central limit theorem [Sp64]. Beyond the probabilistic
setting, the study of convolution powers of complex-valued functions dates
back to the late nineteenth century and was initially investigated by E. L. de
Forest through its applications to data smoothing. During the explosion of
scientific computing in the mid-twentieth century, the study was reinvigorated
by applications to numerical solution algorithms to partial differential
equations. Early on, these studies focused almost entirely in one spatial
dimension, i.e., $d=1$, and, for an account of these results and a thorough
discussion of the early history, we refer the reader to the article [DSC14].
Recent developments in the context of one dimension can be found in [RSC15],
[CF22], and [Coe22]. Moving beyond one spatial dimension, the articles
[RSC17], [BR22], and [Ra22] develop a theory for convolution powers of
complex-valued functions on $\mathbb{Z}^{d}$ and, in particular, the article
[RSC17] establishes local limit theorems, sup-norm estimates, off-diagonal
estimates, and stability results in that context. As we only briefly discuss
the local limit theorems of [RSC17] and [Ra22] below, we refer the reader to
these articles, both of which provide history and a more thorough presentation
of the theory than is given here.

For our finitely-supported function $\phi:\mathbb{Z}^{d}\to\mathbb{C}$, we
define its Fourier transform $\widehat{\phi}$ by

$\widehat{\phi}(\xi)=\sum_{x\in\mathbb{Z}^{d}}\phi(x)e^{ix\cdot\xi}$

for $\xi\in\mathbb{R}^{d}$. With this, we obtain the representation

$\phi^{(n)}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{T}^{d}}\widehat{\phi}(\xi)^{n}e^{-ix\cdot\xi}\,d\xi$
(27)

for $x\in\mathbb{Z}^{d}$ and $n\in\mathbb{N}_{+}$ where
$\mathbb{T}^{d}=(-\pi,\pi]^{d}$ or, equivalently, any representation of the
$d$ dimensional torus of the form $(-\pi,\pi]^{d}+\xi^{\prime}$ for
$\xi^{\prime}\in\mathbb{R}^{d}$. Beyond the assumption of finite support, we
shall assume that $\phi$ has been normalized so that

$\max_{\xi}\absolutevalue{\widehat{\phi}(\xi)}=\sup_{\xi}\absolutevalue{\widehat{\phi}(\xi)}=1.$

Also, and though this assumption can be significantly weakened (c.f., [RSC17,
Ra22]), we shall assume that this maximum is attained at exactly one point
$\xi_{0}\in\mathbb{T}^{d}$ and that

$\Gamma_{\xi_{0}}(\xi):=\log\left(\frac{\widehat{\phi}(\xi+\xi_{0})}{\widehat{\phi}(\xi_{0})}\right)=i\alpha\cdot\xi-P(\xi)+R(\xi)$
(28)

where $\alpha\in\mathbb{R}^{d}$, $P$ is a positive-definite polynomial and $R$
is a smooth complex-valued function for which $R(\xi)=o(P(\xi))$ as $\xi\to
0$. Extending the one-dimensional results of [DSC14] and [RSC15], the article
[RSC17] establishes local limit theorems in the case that the polynomial $P$
is a positive homogeneous function on $\mathbb{R}^{d}$. Under the assumptions
above, if $P$ is a positive homogeneous polynomial on $\mathbb{R}^{d}$,
Theorem 1.6 of [RSC17] says that

$\phi^{(n)}(x)=\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}H_{P}^{n}(x-n\alpha)+o(n^{-\mu_{P}})$
(29)

uniformly for $x\in\mathbb{Z}^{d}$ as $n\to\infty$ where $\mu_{P}$ is the
homogeneous order of $P$ and, for each $x\in\mathbb{R}^{d}$ and
$n\in\mathbb{N}_{+}$,

$H_{P}^{n}(x)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-ix\cdot\xi}\,d\xi.$

Thanks to the homogeneity of $P$, it is easy to see that

$H_{P}^{n}(x)=n^{-\mu_{P}}H_{P}^{1}(n^{-E^{*}}x)$ (30)

for all $x\in\mathbb{R}^{d}$, $n\in\mathbb{N}_{+}$, and
$E\in\operatorname{Exp}(P)$ where $E^{*}$ denotes the adjoint of $E$. In this
way, the local limit theorem (29) can be written in terms of the single
rescaled attractor, $H_{P}^{1}$. We note that homogeneity and the resultant
space-time rescaling in (30) are central to the proof of Theorem 1.6 of
[RSC17]. Using homogeneity as a main ingredient and making use of the
generalized polar-coordinate integration formula developed in [BR22], the
recent article [Ra22] extends Theorem 1.6 of [RSC17] further to include the
case in which the positive homogeneous polynomial $P$ is replaced by $iQ(\xi)$
where $Q$ is a real-valued function on $\mathbb{R}^{d}$ for which
$\xi\mapsto\absolutevalue{Q(\xi)}$ is positive homogeneous; see Theorems 1.9
and Theorem 3.8 of [Ra22]. At present and to our knowledge, all known local
limit theorem on $\mathbb{Z}^{d}$ assume, in one way or another, that the
expansion (28) is dominated, at low order, by a homogeneous polynomial. The
following treats an example in which this assumption is not satisfied.

######  Theorem 5.1.

For positive integers $a$ and $b$, set $d=a+b$ and consider
$\phi:\mathbb{Z}^{d}\to\mathbb{C}$ as above, i.e., $\phi$ is finitely
supported on $\mathbb{Z}^{d}$, $\widehat{\phi}$ is maximized in absolute value
at a single point $\xi_{0}\in\mathbb{T}^{d}$ and the local approximation
$\Gamma_{\xi_{0}}(\xi)=i\alpha\cdot\xi-P(\xi)+R(\xi)$ is valid on the domain
of $\Gamma_{\xi_{0}}$ where $\alpha\in\mathbb{R}^{d}$, $P$ is a positive-
definite polynomial, and $R(\xi)=o(P(\xi))$ as $\xi\to 0$. Suppose,
additionally, that $P$ satisfies the hypotheses of Theorem 3.1. That is, we
assume that

$P(\xi)=P_{1}(\eta+Q(\zeta))+P_{2}(\eta)$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{{a}}\times\mathbb{R}^{{b}}=\mathbb{R}^{d}$
where:

  1. 1.

$P_{1}$ and $P_{2}$ are positive homogeneous functions on $\mathbb{R}^{{a}}$
with homogeneous orders $\mu_{P_{1}}$ and $\mu_{P_{2}}$, respectively.

  2. 2.

$Q:\mathbb{R}^{{b}}\to\mathbb{R}^{{a}}$ is nondegenerate multivariate
homogeneous.

  3. 3.

There exist $E_{1}\in\operatorname{Exp}(P_{1})$ and
$E_{2}\in\operatorname{Exp}(P_{2})$, and
$F_{1},F_{2}\in\operatorname{End}(\mathbb{R}^{{b}})$ for which

    1. (a)

For $k=1,2,$ $Q$ is homogeneous with respect to the pair $(E_{k},F_{k})$.

    2. (b)

We have $[E_{1},E_{2}]=E_{1}E_{2}-E_{2}E_{1}$ and $\\{t^{E_{1}-E_{2}}\\}$ is
non-expanding.

Upon setting $\mu_{\phi}=\mu_{P_{1}}+\mu_{P_{2}\circ(-Q)}=\tr E_{1}+\tr F_{2}$
and taking $H_{P}$ as in Theorem 3.1, we have the following local limit
theorem: For each $\epsilon>0$, there exists $N\in\mathbb{N}_{+}$ for which

$\absolutevalue{\phi^{(n)}(x)-\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}H_{P}^{n}(x-n\alpha)}<\epsilon
n^{-\mu_{\phi}}$

for all $n\geq N$ and $x\in\mathbb{Z}^{d}$. In other words,

$\phi^{(n)}(x)=\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}H_{P}^{n}(x-n\alpha)+o(n^{-\mu_{\phi}})$

uniformly for $x\in\mathbb{Z}^{d}$ as $n\to\infty$.

Combining this result with Theorem 3.1 and Theorem 3.4, we obtain the
following corollary.

######  Corollary 5.2.

Let $\phi$ satisfy the hypotheses of the above theorem. If $\alpha=0$, then

$\|\phi^{(n)}\|_{\infty}\asymp n^{-\mu_{\phi}}$ (31)

for $n\geq 1$. If, additionally, $\\{t^{E_{1}-E_{2}}\\}$ is contracting, then

$\lim_{n\to\infty}n^{\mu_{\phi}}\|\phi^{(n)}\|_{\infty}=H_{P_{\infty}}^{1}(0)=\frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{-P_{\infty}(\xi)}\,d\xi$
(32)

where $P_{\infty}(\xi)=P_{1}(\eta)+P_{2}(-Q(\zeta))$ for
$\xi=(\eta,\zeta)\in\mathbb{R}^{d}$.

###### Proof.

For each $\epsilon>0$, an application of Theorem 5.1 guarantees that

$\absolutevalue{\phi^{(n)}(x)}\leq\epsilon
n^{-\mu_{\phi}}+\absolutevalue{H_{P}^{n}(x)}\leq\epsilon
n^{-\mu_{\phi}}+H_{P}^{n}(0)$ (33)

for all $x\in\mathbb{Z}^{d}$ and sufficiently large values of $n$; here, we
have noted that
$\absolutevalue{\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}}=1$ and
$\absolutevalue{H_{P}^{n}(x)}\leq H_{P}^{n}(0)$ whenever $n\in\mathbb{N}_{+}$
and $x\in\mathbb{Z}^{d}$. By virtue of Theorem 3.1 and in view of the fact
that $\mu_{\phi}=\mu_{\infty}$, we have

$\|\phi^{(n)}\|_{\infty}=\sup_{x\in\mathbb{Z}^{d}}\absolutevalue{\phi^{(n)}(x)}\leq
Cn^{-\mu_{\phi}}$

for sufficiently large $n$ where $C$ is some positive constant. We now obtain
a matching lower bound. Using Theorem 3.1, $H_{P}^{n}(0)\geq
2C^{\prime}n^{-\mu_{\phi}}$ for all $n\in\mathbb{N}_{+}$ for some
$C^{\prime}>0$. An appeal to Theorem 5.1 (with $\epsilon=C^{\prime}$) now
guarantees that

$2C^{\prime}n^{-\mu_{\phi}}\leq\absolutevalue{\widehat{\phi}(\xi_{0})^{n}H_{P}^{n}(0)}\leq\absolutevalue{\widehat{\phi}(\xi_{0})^{n}H_{P}^{n}(0)-\phi^{(n)}(0)}+\absolutevalue{\phi^{(n)}(0)}\leq
C^{\prime}n^{-\mu_{\phi}}+\absolutevalue{\phi^{(n)}(0)}$

and consequently

$C^{\prime}n^{-\mu_{\phi}}\leq\absolutevalue{\phi^{(n)}(0)}$

for all sufficiently large $n$. Thus, for some $N\in\mathbb{N}_{+}$,

$C^{\prime}n^{-\mu_{\phi}}\leq\absolutevalue{\phi^{(n)}(0)}\leq\|\phi^{(n)}\|_{\infty}\leq
Cn^{-\mu_{\phi}}$

for $n\geq N$. With this, the asymptotic (31) follows by, if necessary,
adjusting $C$ and $C^{\prime}$ to account for those $n$ from $1$ to $N-1$ (and
while noting that $\|\phi^{(n)}\|_{\infty}$ cannot vanish for any such $n$ for
otherwise all subsequent convolution powers would vanish identically).

We now prove prove (32). In view of (33), for each $\epsilon>0$,

$n^{\mu_{\phi}}\|\phi^{(n)}\|_{\infty}=\sup_{x\in\mathbb{Z^{d}}}n^{\mu_{\phi}}\absolutevalue{\phi^{(n)}(x)}\leq\epsilon+n^{\mu_{\phi}}H_{P}^{n}(0)$

for all sufficiently large $n$. Upon noting that $\mu_{\phi}=\mu_{\infty}$, an
appeal to Theorem 3.4 guarantees that

$\limsup_{n\to\infty}n^{\mu_{\phi}}\|\phi^{(n)}\|_{\infty}\leq\epsilon+\limsup_{n\to\infty}n^{\mu_{\phi}}H_{P}^{n}(0)=\epsilon+H_{P_{\infty}}^{1}(0)$

for each $\epsilon>0$ and therefore

$\limsup_{n\to\infty}n^{\mu_{\phi}}\|\phi^{(n)}\|_{\infty}\leq
H_{P_{\infty}}^{1}(0).$

By virtue of Theorem 3.4 and Theorem 5.1, observe that

$H_{P_{\infty}}^{1}(0)=\lim_{n\to\infty}n^{\mu_{\phi}}H_{P}^{n}(0)=\lim_{n\to\infty}n^{\mu_{\phi}}\absolutevalue{\widehat{\phi}(\xi_{0})^{n}H_{P}^{n}(0)}=\lim_{n\to\infty}n^{\mu_{\phi}}\absolutevalue{\phi^{(n)}(0)}.$

Since $\absolutevalue{\phi^{(n)}(0)}\leq\|\phi^{(n)}\|_{\infty}$ for all $n$,
it follows that

$H_{P_{\infty}}^{1}(0)=\liminf_{n\to\infty}n^{\mu_{\phi}}\absolutevalue{\phi^{(n)}(0)}\leq\liminf_{n\to\infty}n^{\mu_{\phi}}\|\phi^{(n)}\|_{\infty}\leq\limsup_{n\to\infty}n^{\mu_{\phi}}\|\phi^{(n)}\|_{\infty}\leq
H_{P_{\infty}}^{1}(0)$

and, from this, (32) follows at once. ∎

As we did in the proofs of Theorem 3.1 and Theorem 4.1, our proof of Theorem
5.1 makes use of the measure-preserving transformation
$T(\eta,\zeta)=(\eta-Q(\zeta),\zeta)$ from
$\mathbb{R}^{d}=\mathbb{R}^{{a}}\times\mathbb{R}^{{b}}$ to itself. It is easy
to verify that $T:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a homeomorphism and
$T(0)=0$. With this transformation, we define $\widetilde{P}=P\circ T$ and
$\widetilde{R}=R\circ T$ and note that the domain of $\widetilde{R}$ coincides
with the preimage of $\Gamma_{\xi_{0}}$’s domain under $T$ and necessarily
contains an open neighborhood of $0$. Finally, for convenience of notation, we
shall set $G=E_{1}\oplus F_{2}$ and recall that $\\{t^{G}=t^{E_{1}}\oplus
t^{F_{2}}\\}$ is a contracting group. Before the proof of the theorem, we
present two lemmas; the first follows immediately from Lemma 4.2.

######  Lemma 5.3.

For any compact set $K\subseteq\mathbb{R}^{d}$ and $\epsilon>0$, there is a
natural number $N$ for which

$\absolutevalue{e^{n\widetilde{R}\left(n^{-G}\xi\right)}-1}<\epsilon$

for all $n\geq N$ and $\xi\in K$.

The follows lemma asserts that the collection
$\left\\{\xi\mapsto\exp(-n\widetilde{P}(n^{-G}\xi)/2)\right\\}_{n}$ is
uniformly integrable on $\mathbb{R}^{d}$.

######  Lemma 5.4.

For any $\epsilon>0$, there exists a compact set $K$ for which

$\int_{\mathbb{R}^{d}\setminus
K}e^{-n\widetilde{P}(n^{-G}\xi)/2}\,d\xi<\epsilon$

for all $n\in\mathbb{N}_{+}$.

###### Proof.

Fix $\epsilon>0$ and observe that, for $n\in\mathbb{N}_{+}$ and
$\xi=(\eta,\zeta)\in\mathbb{R}^{d}$,

$\displaystyle n\widetilde{P}(n^{-G}\xi)$ $\displaystyle=$ $\displaystyle
nP_{1}(n^{-E_{1}}\eta)+nP_{2}(n^{-E_{1}}\eta-Q(n^{-F_{2}}\zeta))$
$\displaystyle=$ $\displaystyle
P_{1}(\eta)+P_{2}(n^{E_{2}-E_{1}}\eta-Q(\zeta))$

where we have used the fact that $Q$ is homogeneous with respect to the pair
$(E_{2},F_{2})$. By virtue of Lemma 3.3, we can find positive constants $C$
and $M$ for which

$2C(P_{1}(\eta)+P_{1}(-Q(\zeta))-2M\leq
P_{1}(\eta)+P_{2}(n^{E_{2}-E_{1}}\eta-Q(\zeta))$

for all $(\eta,\zeta)\in\mathbb{R}^{d}$ and $n\in\mathbb{N}_{+}$.
Consequently,

$-n\widetilde{P}(n^{-G}\xi)/2=-(1/2)(P_{1}(\eta)+P_{2}(n^{E_{2}-E_{2}}\eta-Q(\zeta))\leq
M-C(P_{1}(\eta)+P_{1}(-Q(\zeta))$

for all $\xi=(\eta,\zeta)\in\mathbb{R}^{d}$ and $n\in\mathbb{N}_{+}$. Upon
noting that $\xi=(\eta,\zeta)\mapsto C(P_{1}(\eta)+P_{1}(-Q(\zeta))$ is a
positive homogeneous function, an appeal to Lemma 2.6 guarantees a compact set
$K\subseteq\mathbb{R}^{d}$ for which

$\int_{\mathbb{R}^{d}\setminus
K}e^{-n\widetilde{P}(n^{-G}\xi)/2}\,d\xi\leq\int_{\mathbb{R}^{d}\setminus
K}e^{M-C(P_{1}(\eta)+P_{1}(-Q(\eta)))}\,d\xi<\epsilon$

for all $n\in\mathbb{N}_{+}$. ∎

###### Proof of Theorem 5.1.

Fix $\epsilon>0$. Without loss of generality, we shall assume that $\xi_{0}$
lives on the interior of $\mathbb{T}^{d}$ for, otherwise, another
representation of the $d$-dimensional torus can be used as the domain of
integration in (27) and the proof proceeds without change. Let us select a
sufficiently small open neighborhood
$\mathcal{O}_{\xi_{0}}\subseteq\mathbb{T}^{d}$ of $\xi_{0}$ for which the
following properties hold:

  1. P1

The functions $\Gamma_{\xi_{0}}$ and $R$ are defined and smooth on the open
neighborhood $\mathcal{O}:=\mathcal{O}_{\xi_{0}}-\xi_{0}$ of $0$.

  2. P2

The open neighborhood
$\mathcal{U}:=T^{-1}(\mathcal{O})=T^{-1}(\mathcal{O}_{\xi_{0}}-\xi_{0})$ of
$0$ has

$\absolutevalue{\widetilde{R}(\xi)}\leq\widetilde{P}(\xi)/2$

for all $\xi\in\mathcal{U}$.

The fact that $\mathcal{O}_{\xi_{0}}$ can be chosen so that P1 holds is clear
from the definitions of $\Gamma_{\xi_{0}}$ and $R$. The ability to choose
$\mathcal{O}_{\xi_{0}}$ so that Property P2 also holds is a consequence of
Item 1 of Lemma 4.2. Using Lemma 5.4, let $K\subseteq\mathbb{R}^{d}$ be a
compact set for which

$\int_{\mathbb{R}^{d}\setminus
K}e^{-n\widetilde{P}(n^{-G}\xi)/2}\,d\xi<\epsilon/4$

for all $n\in\mathbb{N}_{+}$.

With the sets $\mathcal{O}_{\xi_{0}}$, $\mathcal{O}$, $\mathcal{U}$, and $K$
in hand, we shall now go about selecting $N$. First, because $\\{t^{G}\\}$ is
a contracting group and $\mathcal{U}$ is an open neighborhood of $0$, there
exists $N_{1}\in\mathbb{N}_{+}$ for which $n^{-G}(K)\subseteq\mathcal{U}$
(equivalently, $K\subseteq n^{G}(\mathcal{U})$) for all $n\geq N_{1}$. By an
appeal to Lemma 5.3, let $N_{2}\in\mathbb{N}_{+}$ be such that

$\absolutevalue{e^{n\widetilde{R}(n^{-G}\xi)}-1}<\frac{\epsilon}{4(m(K)+1)}$

for all $\xi\in K$ and $n\geq N_{1}$ where $m(K)$ is the $d$-dimensional
Lebesgue measure of $K$. Finally, given that
$\absolutevalue{\widehat{\phi}(\xi)}$ is maximized only at $\xi_{0}$,

$\rho:=\sup_{\xi\in\mathbb{T}^{d}\setminus\mathcal{O}_{\xi_{0}}}\absolutevalue{\widehat{\phi}(\xi)}<1$

and so we may choose a natural number $N\geq\max\\{N_{1},N_{2}\\}$ for which
$n^{\mu_{\phi}}\rho^{n}<\epsilon/4$ for all $n\geq N$.

By virtue of the identity (27), we compute

$\displaystyle
n^{\mu_{\phi}}\absolutevalue{\phi^{(n)}(x)-\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}H_{P}^{n}(x-n\alpha)}$
(34) $\displaystyle=$
$\displaystyle\frac{n^{\mu_{\phi}}}{(2\pi)^{d}}\absolutevalue{\int_{\mathbb{T}^{d}}\widehat{\phi}(\xi)^{n}e^{-ix\cdot\xi}\,d\xi-\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi}$
$\displaystyle\leq$
$\displaystyle\frac{n^{\mu_{\phi}}}{(2\pi)^{d}}\int_{\mathbb{T}^{d}\setminus\mathcal{O}_{\xi_{0}}}\absolutevalue{\widehat{\phi}(\xi)}^{n}\,d\xi+$
$\displaystyle\hskip
56.9055ptn^{\mu_{\phi}}\absolutevalue{\int_{\mathcal{O}_{\xi_{0}}}\widehat{\phi}(\xi)^{n}e^{-ix\cdot\xi}\,d\xi-\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi}$
$\displaystyle\leq$ $\displaystyle
n^{\mu_{\phi}}\rho^{n}+n^{\mu_{\phi}}\absolutevalue{\int_{\mathcal{O}}\widehat{\phi}(\xi+\xi_{0})^{n}e^{-ix\cdot(\xi+\xi_{0})}\,d\xi-\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi}$
$\displaystyle\leq$
$\displaystyle\frac{\epsilon}{4}+n^{\mu_{\phi}}\absolutevalue{\int_{\mathcal{O}}e^{n\Gamma_{\xi_{0}}(\xi)}e^{-ix\cdot\xi}\,d\xi-\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi}$

for all $n\geq N$ and $x\in\mathbb{Z}^{d}$ where we have made of variables
$\xi\mapsto\xi+\xi_{0}$ and made use of the fact that
$\absolutevalue{\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}}=1$. Let us now
make the measure-preserving change of variables $\xi\mapsto T(\xi)$ to see
that

$\displaystyle\int_{\mathcal{O}}e^{n\Gamma_{\xi_{0}}(\xi)}e^{-ix\cdot\xi}\,d\xi$
$\displaystyle=$ $\displaystyle\int_{\mathcal{U}}e^{n(\Gamma_{\xi_{0}}\circ
T)(\xi)}e^{-ix\cdot T(\xi)}\,d\xi$ $\displaystyle=$
$\displaystyle\int_{n^{-G}(K)}e^{n(\Gamma_{\xi_{0}}\circ T)(\xi)}e^{-ix\cdot
T(\xi)}\,d\xi+\int_{\mathcal{U}\setminus n^{-G}(K)}e^{n(\Gamma_{\xi_{0}}\circ
T)(\xi)}e^{-ix\cdot T(\xi)}\,d\xi.$

for each $n\geq N\geq N_{1}$ and $x\in\mathbb{Z}^{d}$. Similarly,

$\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi=\int_{n^{-G}(K)}e^{-n\widetilde{P}(\xi)}e^{-i(x-n\alpha)\cdot
T(\xi)}\,d\xi+\int_{\mathbb{R}^{d}\setminus
n^{-G}(K)}e^{-n\widetilde{P}(\xi)}e^{-i(x-n\alpha)\cdot T(\xi)}\,d\xi$

for all $n\in\mathbb{N}_{+}$ and $x\in\mathbb{Z}^{d}$. Consequently, for
$x\in\mathbb{Z}^{d}$ and $n\geq N$,

$\displaystyle\absolutevalue{\int_{\mathcal{O}}e^{n\Gamma_{\xi_{0}}(\xi)}e^{-ix\cdot\xi}\,d\xi-\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi}$
$\displaystyle\leq$
$\displaystyle\absolutevalue{\int_{n^{-G}(K)}e^{n(\Gamma_{\xi_{0}}\circ
T)(\xi)}e^{-ix\cdot
T(\xi)}\,d\xi-\int_{n^{-G}(K)}e^{-n\widetilde{P}(\xi)}e^{-i(x-n\alpha)\cdot
T(\xi)}\,d\xi}$ (35) $\displaystyle\hskip
28.45274pt+\absolutevalue{\int_{\mathcal{U}\setminus
n^{-G}(K)}e^{n(\Gamma_{\xi_{0}}\circ T)(\xi)}e^{-ix\cdot
T(\xi)}\,d\xi}+\absolutevalue{\int_{\mathbb{R}^{d}\setminus
n^{-G}(K)}e^{-n\widetilde{P}(\xi)}e^{-i(x-n\alpha)\cdot T(\xi)}\,d\xi}$
$\displaystyle\leq$
$\displaystyle\absolutevalue{\int_{n^{-G}(K)}\left(e^{n\widetilde{R}(\xi)}-1\right)e^{-n\widetilde{P}(\xi)}e^{-i(x-n\alpha)\cdot
T(\xi)}\,d\xi}$ $\displaystyle\hskip 28.45274pt+\int_{\mathcal{U}\setminus
n^{-G}(K)}\absolutevalue{e^{n(\Gamma_{\xi_{0}}\circ T)(\xi)}e^{-ix\cdot
T(\xi)}}\,d\xi+\int_{\mathbb{R}^{d}\setminus
n^{-G}(K)}\absolutevalue{e^{-n\widetilde{P}(\xi)}e^{-i(x-n\alpha)\cdot
T(\xi)}}\,d\xi$ $\displaystyle\leq$
$\displaystyle\int_{n^{-G}(K)}\absolutevalue{e^{-n\widetilde{R}(\xi)}-1}\,d\xi$
$\displaystyle\hskip 28.45274pt+\int_{\mathcal{U}\setminus
n^{-G}(K)}e^{n\operatorname{Re}[(\Gamma_{\xi_{0}}\circ
T)(\xi)]}\,d\xi+\int_{\mathbb{R}^{d}\setminus
n^{-G}(K)}e^{-n\widetilde{P}(\xi)}\,d\xi$

where we have used the fact that $(\Gamma_{\xi_{0}}\circ T)(\xi)=i\alpha\cdot
T(\xi)-\widetilde{P}(\xi)+\widetilde{R}(\xi)$ and $\widetilde{P}$ is non-
negative. Upon making the change of variables $\xi\mapsto n^{-G}\xi$ and
recalling how $N_{2}$ was chosen, observe that

$\int_{n^{-G}(K)}\absolutevalue{e^{-n\widetilde{R}(\xi)}-1}\,d\xi=n^{-\tr
G}\int_{K}\absolutevalue{e^{-n\widetilde{R}(n^{-G}\xi)}-1}\,d\xi\leq
n^{-\mu_{\phi}}\frac{\epsilon}{4(m(K)+1)}m(K)\leq\frac{\epsilon
n^{-\mu_{\phi}}}{4}$ (36)

whenever $n\geq N\geq N_{2}$; here, we have recalled that $\tr
G=\tr(E_{1}\oplus F_{2})=\tr E_{1}+\tr E_{2}=\mu_{\phi}$. Recalling our choice
of $K$ and upon noting that

$\operatorname{Re}[(\Gamma_{\xi_{0}}\circ
T)(\xi)]=-\widetilde{P}(\xi)+\operatorname{Re}[\widetilde{R}(\xi)]\leq-\widetilde{P}(\xi)+\widetilde{P}(\xi)/2=-\widetilde{P}(\xi)/2$

for all $\xi\in\mathcal{U}$ thanks to P2, we find that

$\displaystyle\int_{\mathcal{U}\setminus
n^{-G}(K)}e^{n\operatorname{Re}[(\Gamma_{\xi_{0}}\circ T)(\xi)]}\,d\xi$
$\displaystyle\leq$ $\displaystyle\int_{\mathcal{U}\setminus
n^{-G}(K)}e^{-n\widetilde{P}(\xi)/2}\,d\xi$ (37) $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}^{d}\setminus
n^{-G}(K)}e^{-n\widetilde{P}(\xi)/2}\,d\xi$ $\displaystyle=$ $\displaystyle
n^{-\mu_{\phi}}\int_{\mathbb{R}^{d}\setminus
K}e^{-n\widetilde{P}(n^{-G}\xi)/2}\,d\xi<\frac{\epsilon n^{-\mu_{\phi}}}{4}$

for all $n\geq N$ where we have made the change of variables $\xi\mapsto
n^{-G}\xi$ between the final and penultimate lines. Finally, using the fact
that $\widetilde{P}$ is non-negative, a similar computation shows that

$\int_{\mathbb{R}^{d}\setminus n^{-G}(K)}e^{-n\widetilde{P}(\xi)}\,d\xi\leq
n^{-\mu_{\phi}}\int_{\mathbb{R}^{d}\setminus
K}e^{-n\widetilde{P}(n^{-G}\xi)/2}\,d\xi<\frac{\epsilon n^{-\mu_{\phi}}}{4}$
(38)

for all $n\in\mathbb{N}_{+}$. Combining (5), (36), (37), and (38) yields

$n^{\mu_{\phi}}\absolutevalue{\int_{\mathcal{O}}e^{n\Gamma_{\xi_{0}}(\xi)}e^{-ix\cdot\xi}\,d\xi-\int_{\mathbb{R}^{d}}e^{-nP(\xi)}e^{-i(x-n\alpha)\cdot\xi}\,d\xi}<\frac{3\epsilon}{4}$

for all $n\geq N$ and $x\in\mathbb{Z}^{d}$. Finally, substituting the above
estimate into (34) gives

$n^{\mu_{\phi}}\absolutevalue{\phi^{(n)}(x)-\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi}H_{P}^{n}(x-n\alpha)}<\epsilon$

for all $n\geq N$ and $x\in\mathbb{Z}^{d}$, as was asserted. ∎

######  Example 15 (The introductory Example):

For $\phi:\mathbb{Z}^{2}\to\mathbb{C}$ given by (3), we have

$\displaystyle\widehat{\phi}(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{100}\left(100-\left(\sin(\eta)+4\sin^{2}(\zeta/2)\right)^{2}-\frac{797}{600}\sin^{4}(\eta)-10\sin^{6}(\eta/2)-\frac{1}{6}\sin^{6}(\zeta)-\frac{179}{1200}\sin^{8}(\zeta)\right.$
$\displaystyle\left.-\frac{1}{6}\sin(\eta)\sin^{4}(\zeta)-\frac{77}{900}\sin(\eta)\sin^{6}(\zeta)-\frac{47}{150}\sin(\zeta)^{2}\sin(\eta)^{3}+\frac{3}{100}\sin^{2}(\eta)\sin^{4}(\zeta)\right)$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. A straightforward computation shows
that $\max_{\xi}|\widehat{\phi}(\xi)|=1$ and, within $\mathbb{T}^{2}$, this
maximum is attained only at the origin, $\xi_{0}=(0,0)$. To give a reader a
sense of $\widehat{\phi}$, we have illustrated its absolute value in Figure 2.

Figure 2: The graph of $|\widehat{\phi}(\xi)|$ for
$\xi=(\eta,\zeta)\in\mathbb{T}^{2}$.

It is easy to see that $\widehat{\phi}(\xi_{0})=1$ and we compute

$\Gamma_{\xi_{0}}(\xi)=\log(\widehat{\phi}(\xi))=-\frac{1}{100}P(\xi)+R(\xi)$

where $P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$ is the symbol discussed
throughout the introduction and

$R(\xi)=-\frac{1}{1000}\left(\frac{1}{5}\eta^{2}\zeta^{6}-\frac{47}{45}\eta^{3}\zeta^{4}-\frac{79}{400}\zeta^{4}\zeta^{4}-\frac{1201}{1000}\eta^{5}\zeta^{2}-\frac{60739}{9000}\eta^{6}-\frac{37233979}{16800000}\eta^{8}\right)+O(\absolutevalue{\xi}^{9})$

as $\xi\to 0$. Observe that $\xi\mapsto\absolutevalue{\xi}^{9}=L_{4.5}(\xi)$
where $L_{k}$ is defined in Example 12 and corresponds to the $k$-th power of
the Laplacian. As shown in that example (following Corollary 4.5),
$L_{4.5}(\xi)=o(P(\xi))$ as $\xi\to 0$ and so it follows that
$O(\absolutevalue{\xi}^{9})=o(P(\xi)/100)$ as $\xi\to 0$. Thus, to show that
$R(\xi)=o(P(\xi)/100)$ as $\xi\to 0$ so that we may apply Theorem 5.1 and
Corollary 5.2, we must show that the monomials
$R_{1}(\xi):=\eta^{2}\zeta^{6}$, $R_{2}(\xi):=\eta^{3}\zeta^{4}$,
$R_{3}(\xi):=\eta^{4}\zeta^{4}$, $R_{4}(\xi):=\eta^{5}\zeta^{2}$,
$R_{5}(\xi):=\eta^{6}$, and $R_{6}(\xi)=\eta^{8}$ are “little-o” of $P(\xi)$
as $\xi\to 0$ and, to this end, our approach will employ Proposition 4.3 just
as we did in Example 13. Indeed, we have

$\widetilde{R_{1}}(\xi)=(\rho_{1}\circ
T)(\eta,\zeta)=(\eta-\zeta^{2})^{2}\zeta^{6}$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$ where
$T(\eta,\zeta)=(\eta-\zeta^{2},\zeta)$. Therefore,

$\widetilde{R_{1}}(t^{G}\xi)=(t^{1/2}\eta-(t^{1/8}\zeta)^{2})^{2}(t^{1/8}\zeta)^{6}=t^{5/4}(t^{1/4}\eta-\zeta^{2})^{2}\zeta^{6}$

for $t>0$ and $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$ where $G=E_{1}\oplus F_{2}$
has standard matrix representation $\operatorname{diag}(1/2,1/8)$. From this
it follows (by the same argument used in Example 13) that $\widetilde{R_{1}}$
is subhomogeneous with respect to $G$ and therefore
$R_{1}(\xi)=o(P(\xi))=o(P(\xi)/100)$ as $\xi\to 0$ on account of Proposition
4.3. We leave it to the reader the verify this conclusion for $R_{k}$ for
$k=2,3,\dots,6$. Consequently, $R(\xi)=o(P(\xi)/100)$ as $\xi\to 0$.

Since $\xi\mapsto P(\xi)/100$ meets the hypotheses of Theorem 5.1 (as well as
Theorem 3.1) with $\mu_{\infty}=5/8$ and $R(\xi)=o(P(\xi)/100)$ as $\xi\to 0$,
an appeal to Theorem 5.1 guarantees that

$\phi^{(n)}(x)=\widehat{\phi}(\xi_{0})^{n}e^{-ix\cdot\xi_{0}}H_{P/100}^{n}(x-n\alpha)+o(n^{-5/8})=H_{P}^{n/100}(x)+o(n^{-5/8})$

uniformly for $x\in\mathbb{Z}^{2}$ as $n\to\infty$; here, we have noted that
$\mu_{\phi}=\mu_{\infty}=5/8$, $\xi_{0}=(0,0)$, $\widehat{\phi}(\xi_{0})=1$,
and $\alpha=(0,0)$. We recall that this conclusion was discussed in the
introduction and illustrated in Figure 1. Finally, we appeal to Corollary 5.2
to conclude that $\|\phi^{(n)}\|_{\infty}\asymp n^{-5/8}$ as $n\geq 1$ and

$\lim_{n\to\infty}n^{5/8}\|\phi^{(n)}\|_{\infty}=H_{(P/100)_{\infty}}^{1}(0).$

Of course, $(P/100)_{\infty}=P_{\infty}/100$ where
$P_{\infty}(\xi)=\eta^{2}+\zeta^{8}$ and therefore

$H_{(P/100)_{\infty}}^{1}(0)=H_{P_{\infty}}^{1/100}(0)=(100)^{5/8}H_{P_{\infty}}(0)$

where we have used the homogeneity of $P_{\infty}$ with respect to $G$. In
view of the calculations done in Example 9, we conclude that

$\lim_{n\to\infty}n^{5/8}\|\phi^{(n)}\|_{\infty}=(100)^{5/8}H_{P_{\infty}}^{1}(0)=\frac{(100)^{5/8}}{2\pi^{3/2}}\Gamma(9/8)\approx
1.50376.$

This is illustrated in Figure 3 wherein $\|\phi^{(n)}\|_{\infty}$ is compared
with $1.5\times n^{-5/8}$ on a log-log scale for $10^{1}\leq n\leq 10^{6}$.

Figure 3: A log-log plot of $\|\phi^{(n)}\|_{\infty}$ and $1.5\times n^{-5/8}$
for $10^{1}\leq n\leq 10^{6}$.

∎

######  Example 16 (A nearby non-example):

In this example, we consider a complex-valued function $\psi$ on
$\mathbb{Z}^{2}$ which is a close approximation to the function $\phi$ of the
preceding example, however, its convolution powers $\psi^{(n)}$ behave
distinctly from those of $\phi$. In particular, this example shows that the
hypotheses concerning $R(\xi)$ in the expansion (28) of Theorem 5.1 are
necessary and, further, it demonstrates the delicate nature of “higher order”
terms present in these expansions insofar as they affect the behavior of
convolution powers. In what follows, we shall assume the notation of the
preceding example and consider $\psi:\mathbb{Z}^{d}\to\mathbb{C}$ defined by

$\psi=\frac{1}{960000}\left(\psi_{1}+\psi_{2}+\psi_{3}\right)$

where

$\psi_{1}(x)=\begin{cases}862318&(x_{1},x_{2})=(0,0)\\\ 22500\pm
19200i&(x_{1},x_{2})=\pm(1,0)\\\ -3412&(x_{1},x_{2})=\pm(2,0)\\\
1500&(x_{1},x_{2})=\pm(3,0)\\\ -797&(x_{1},x_{2})=\pm(4,0)\\\
0&\mbox{else}\end{cases},\hskip
28.45274pt\psi_{2}(x)=\begin{cases}38400&(x_{1},x_{2})=(0,\pm 1)\\\
-9225&(x_{1},x_{2})=(0,\pm 2)\\\ -150&(x_{1},x_{2})=(0,\pm 4)\\\
25&(x_{1},x_{2})=(0,\pm 6)\\\ 0&\mbox{else}\end{cases}$

and

$\psi_{3}(x)=\begin{cases}\pm 9600i&(x_{1},x_{2})=(\mp 1,1),(\mp 1,-1)\\\
0&\mbox{else}\end{cases}$

for $x=(x_{1},x_{2})\in\mathbb{Z}^{2}$. We remark that $\psi$ is close to
$\phi$ in several senses. For example, we have
$\|\phi-\psi\|_{\infty}=51/20935\approx 0.0024\approx
0.0024\times\|\phi\|_{\infty}$ and $\|\phi-\psi\|_{2}=1/\sqrt{247293}\approx
0.0020\approx 0.0022\times\|\phi\|_{2}$. Of course, given that the behavior of
convolution powers $\psi^{(n)}$ is determined by the nature of the Fourier
transform of $\psi$ near points at which is it maximized in absolute value
(c.f., [RSC15, RSC17, BR22, Ra22]), we analyze $\widehat{\psi}$. We have

$\widehat{\psi}(\xi)=\frac{1}{100}\left(100-\left(\sin(\eta)+4\sin^{2}(\zeta/2)\right)^{2}-\frac{1}{6}\sin^{6}(\zeta)\right)$

for $\xi=(\eta,\zeta)\in\mathbb{R}^{2}$. As illustrated in Figure 4, the graph
of $|\widehat{\psi}(\xi)|$ is extremely similar to that of
$|\widehat{\phi}(\xi)|$ on $\mathbb{T}^{2}$.

(a) $|\widehat{\psi}(\xi)|$ for $\xi=(\eta,\zeta)\in\mathbb{T}^{2}$.

(b) $|\widehat{\phi}(\xi)|$ for $\xi=(\eta,\zeta)\in\mathbb{T}^{2}$.

Figure 4: Illustration of the Fourier transforms of $\psi$ and $\phi$ on
$\mathbb{T}^{2}$.

Akin to the previous example, $\max_{\xi}|\widehat{\psi}(\xi)|=1$ and this
maximum is attained only at $\xi_{0}=(0,0)$ in $\mathbb{T}^{2}$ where we have
$\widehat{\psi}(\xi_{0})=1$. We compute

$\Gamma_{\xi_{0}}(\xi)=\log(\widehat{\psi}(\xi))=-\frac{1}{100}P(\xi)+R(\xi)$

where $P(\xi)=(\eta+\zeta^{2})^{2}+\eta^{4}$ and

$\displaystyle R(\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{100}\left(\frac{179}{1200}\zeta^{8}+\frac{1}{6}\eta\zeta^{4}-\frac{23}{900}\eta\zeta^{6}-\frac{3}{100}\eta^{2}\zeta^{4}+\frac{47}{150}\eta^{3}\zeta^{2}\right)$
(39)
$\displaystyle+\frac{1}{10000}\left(\frac{1}{3}\eta^{2}\zeta^{6}-\frac{47}{18}\eta^{3}\zeta^{4}-\frac{227}{600}\eta^{4}\zeta^{4}-\frac{1003}{300}\eta^{5}\zeta^{2}+\frac{60739}{900}\eta^{6}-\frac{37233979}{1680000}\eta^{8}+O(\absolutevalue{\xi}^{9})\right)$
$\displaystyle=:$ $\displaystyle R_{1}(\xi)+R_{2}(\xi)$

as $\xi=(\eta,\zeta)\to 0$. In other words, akin to the analogous expansion
for $\widehat{\phi}$, $\Gamma_{\xi_{0}}$ is made up of the polynomial
$-P(\xi)/100$ and a series $R(\xi)$ consistent of “higher order terms”
relative to those of $P$. With the initial aim of applying Theorem 5.1 or
Corollary 5.2 to this example, we ask if $R(\xi)=o(P(\xi))$ as $\xi\to 0$. As
we did in the preceding example, we investigate this by composing by the non-
linear transformation $T$ and then checking if $R_{1}(\xi)$ and $R_{2}(\xi)$
are subhomogeneous with respect to $G=E_{1}\oplus F_{2}$. We find that

$\widetilde{R}_{1}(\xi)=\frac{1}{100}\left(\frac{179}{1200}\zeta^{8}+\frac{1}{6}(\eta-\zeta^{2})\\\
\zeta^{4}-\frac{23}{900}(\eta-\zeta^{2})\zeta^{6}-\frac{3}{100}(\eta-\zeta^{2})^{2}\zeta^{4}+\frac{47}{150}(\eta-\zeta^{2})^{3}\zeta^{2}\right)$

for $\xi=(\eta,\zeta)$. From this it follows that

$\lim_{t\to 0}t^{-1}\widetilde{R}_{1}(t^{G}\xi)=\lim_{t\to
0}\frac{1}{6}(\eta-t^{-1/4}\zeta^{2})\zeta^{4}=-\infty$ (40)

whenever $\xi=(\eta,\zeta)$ is such that $\zeta\neq 0$. Consequently,
$\widetilde{R}_{1}$ is not subhomogeneous with respect to $G$. By contrast, we
find that $\widetilde{R}_{2}$ is subhomogeneous with respect to $G$ by an
argument analogous to that used in Example 13. In view of Proposition 4.3, we
conclude that $R(\xi)\neq o(P(\xi))$ as $\xi\to 0$. We note that, though our
calculation above for $\widetilde{R}_{1}$ shows that $R_{1}$ is not well
controlled by $P(\xi)$ for small $\xi$, this loss of control only happens from
below and from this we will still be able to deduce something useful (see
Lemma 5.6 and Proposition 5.5 below).

Though the expansion of $\Gamma_{\xi_{0}}$ for $\widehat{\psi}$ agrees with
that for $\widehat{\phi}$ at the lowest order (both are $-P(\xi)/100$), we
cannot apply Theorem 5.1 or Corollary 5.2 to this example because $R(\xi)\neq
o(P(\xi))$ as $\xi\to 0$. In particular, for $\|\psi^{(n)}\|_{\infty}$, we are
not able to deduce from Corollary 5.2 the decay of $n^{-5/8}$ which is
characteristic of $H_{P/100}^{n}$ and $\|\phi^{(n)}\|_{\infty}$. Figure 5
presents strong numerical evidence that $\|\psi^{(n)}\|_{\infty}$ decays
faster than $n^{-5/8}$ as $n\to\infty$.

Figure 5: A log-log plot of $\|\psi^{(n)}\|_{\infty}$ and $1.5\times n^{-5/8}$
for $10^{1}\leq n\leq 10^{6}$.

In fact, the following proposition confirms that this is the case.

######  Proposition 5.5.

We have

$\lim_{n\to\infty}n^{5/8}\|\psi^{(n)}\|_{\infty}=0.$