# Boundary values of zero solutions of hypoelliptic differential operators in
ultradistribution spaces

A. Debrouwere1 1Department of Mathematics: Analysis, Logic and Discrete
Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
<EMAIL_ADDRESS>and  T. Kalmes2 2Technische Universität Chemnitz,
Faculty of Mathematics, 09107 Chemnitz, Germany<EMAIL_ADDRESS>chemnitz.de

###### Abstract.

We study ultradistributional boundary values of zero solutions of a
hypoelliptic constant coefficient partial differential operator
$P(D)=P(D_{x},D_{t})$ on $\mathbb{R}^{d+1}$. Our work unifies and considerably
extends various classical results of Komatsu and Matsuzawa about boundary
values of holomorphic functions, harmonic functions and zero solutions of the
heat equation in ultradistribution spaces. We also give new proofs of various
results of Langenbruch [21] about distributional boundary values of zero
solutions of $P(D)$.

Keywords: Hypoelliptic and semi-elliptic differential operators; Boundary
values; Ultradistributions.

MSC 2020: 35G15; 35H10; 46F05; 46F20

##  1\. Introduction

The study of distributional and ultradistributional boundary values of
holomorphic functions, which goes back to the seminal works of Köthe [14] and
Tillmann [32] for distributions and of Komatsu [17] for ultradistributions, is
an important subject in the theory of generalized functions. We refer to the
survey article [26] and the references therein for an account of results on
this topic. Similarly, the boundary value behavior of harmonic functions [1,
6, 20, 33] and of zero solutions to the heat equation [4, 16, 24, 25] in
generalized function spaces has been thoroughly investigated.

In the distribution case, Langenbruch [21] generalized the above results by
developing a theory of distributional boundary values for the zero solutions
of a hypoelliptic (constant coefficient) partial differential operator
$P(D)=P(D_{x},D_{t})$ on $\mathbb{R}^{d+1}$. For ultradistributions much less
is known in this general setting: In [22, 23] Langenbruch studied spaces of
formal boundary values of zero solutions of $P(D)$, in the style of Bengel’s
approach to hyperfunctions [2]. In [23, Satz 2.4] he characterized the zero
solutions $f$ of $P(D)$ that admit a boundary value in a given
ultradistribution space in terms of the ultradistributional extendability
properties of $f$ (see also [18, Section 3]). Furthermore, he proved that, for
certain semi-elliptic operators $P(D)$, the space of formal boundary values of
solutions of $P(D)$ may be identified with the Cartesian product of a Gevrey
ultradistribution space of Roumieu type [23, Satz 4.9 and Satz 4.10]. The aim
of the present paper is to complement these results by extending Langenbruch’s
results from [21] for distributions to the framework of ultradistributions,
both of Beurling and Roumieu type, defined via weight sequences [17].

We now describe the content of the paper and state a sample of our main
result. For the sake of clarity, we consider here only Gevrey
ultradistribution spaces. Let $P(D)$ be a hypoelliptic partial differential
operator on $\mathbb{R}^{d+1}$. Let $X\subseteq\mathbb{R}^{d}$ be open. For
$\sigma>1$ we denote by $\mathscr{D}^{\prime(\sigma)}(X)$ and
$\mathscr{D}^{\prime\\{\sigma\\}}(X)$ the space of Gevrey ultradistributions
of Beurling type and of Roumieu type, respectively, and endow these spaces
with their strong topology. We use $\mathscr{D}^{\prime[\sigma]}(X)$ as a
common notation for $\mathscr{D}^{\prime(\sigma)}(X)$ and
$\mathscr{D}^{\prime\\{\sigma\\}}(X)$. Let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. Set $C^{\infty}_{P}(V\backslash
X)=\\{f\in C^{\infty}(V\backslash X)\,|\,P(D)f=0\\}$. The boundary value
$\operatorname{bv}(f)\in\mathscr{D}^{\prime[\sigma]}(X)$ of an element $f\in
C^{\infty}_{P}(V\backslash X)$ is defined as

$\langle\operatorname{bv}(f),\varphi\rangle:=\lim_{t,s\to
0^{+}}\int_{X}(f(x,t)-f(x,-s))\varphi(x)dx,\qquad\varphi\in\mathscr{D}^{[\sigma]}(X),$

provided that $\operatorname{bv}(f)\in\mathscr{D}^{\prime[\sigma]}(X)$ exists.
We have two main goals: Firstly, we characterize the elements of
$C^{\infty}_{P}(V\backslash X)$ that admit a boundary value in
$\mathscr{D}^{\prime[\sigma]}(X)$ in terms of their local growth properties
near $X$. Secondly, we study the uniqueness and the solvability of the Cauchy
type problem

$\begin{cases}P(D)f=0&\mbox{ on $V\backslash X$},\\\
\operatorname{bv}(D^{j}_{t}f)=T_{j}&\mbox{on $X$ for $j=0,\ldots,m-1$},\\\
\end{cases}$

where $f\in C^{\infty}(V\backslash X)$, $m=\deg_{t}P$ and the initial data
$T_{0},\ldots,T_{m-1}$ belong to $\mathscr{D}^{\prime[\sigma]}(X)$. This leads
to the representation of the $m$-fold Cartesian product of
$\mathscr{D}^{\prime[\sigma]}(X)$ by means of boundary values of elements of
$C^{\infty}_{P}(V\backslash X)$.

Let us explicitly state the above two results for certain semi-elliptic
partial differential operators. To this end, we introduce the following
weighted spaces of zero solutions of $P(D)$

$\displaystyle C^{\infty}_{P,(\sigma)}(V\backslash X):=\\{$ $\displaystyle
f\in C^{\infty}_{P}(V\backslash X)\,|\,\forall K\Subset V\,\exists h>0\,:\,$
$\displaystyle\|f\|_{K,h}:=\sup_{(x,t)\in K\backslash
K\cap\mathbb{R}^{d}}|f(x,t)|e^{-h|t|^{-\frac{1}{\sigma-1}}}<\infty\\},$
$C^{\infty}_{P,\\{\sigma\\}}(V\backslash X):=\\{f\in
C^{\infty}_{P}(V\backslash X)\,|\,\forall K\Subset V\,\forall
h>0\,:\,\|f\|_{K,h}<\infty\\},$

where $K\Subset V$ denotes that $K$ is a compact subset of $V$. We endow these
spaces with their natural locally convex topology. We then have:

######  Theorem 1.1.

Let $P(D)$ be a semi-elliptic partial differential operator on
$\mathbb{R}^{d+1}$ with
$\operatorname{deg}_{x_{1}}P=\cdots=\operatorname{deg}_{x_{d}}P=n$. Set
$m=\deg_{t}P$ and $a_{0}=n/m$. Let $X\subseteq\mathbb{R}^{d}$ be open and let
$V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$.

  * $(a)$

Let $\sigma>\max\\{1,1/a_{0}\\}$. For $f\in C^{\infty}_{P}(V\backslash X)$ the
following statements are equivalent:

    * $(i)$

$f\in C^{\infty}_{P,[a_{0}\sigma]}(V\backslash X)$.

    * $(ii)$

$\operatorname{bv}(f)\in\mathscr{D}^{\prime[\sigma]}(X)$ exists.

    * $(iii)$

For every relatively compact open subset $Y$ of $X$ there is $r>0$ such that
$\\{f(\,\cdot\,,t)\,|\,0<|t|<r\\}$ is bounded in
$\mathscr{D}^{\prime[\sigma]}(Y)$.

In such a case,
$\operatorname{bv}(D^{l}_{t}f)\in\mathscr{D}^{\prime[\sigma]}(X)$ exists for
all $l\in\mathbb{N}_{0}$. Next, define

(1) $\operatorname{bv}^{m}:C^{\infty}_{P,[a_{0}\sigma]}(V\backslash
X)\rightarrow\prod_{j=0}^{m-1}\mathscr{D}^{\prime[\sigma]}(X),\,f\mapsto(\operatorname{bv}(D^{j}_{t}f))_{0\leq
j\leq m-1}.$

Then, the sequence

$0\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}\phantom{,}}}C^{\infty}_{P}(V)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}\phantom{,}}}C^{\infty}_{P,[a_{0}\sigma]}(V\backslash
X)\xrightarrow{\phantom{,}\operatorname{bv}^{m}\phantom{,}}\prod_{j=0}^{m-1}\mathscr{D}^{\prime[\sigma]}(X)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}\phantom{,}}}0$

is exact and $\operatorname{bv}^{m}$ is a topological homomorphism, provided
that $V$ is $P$-convex for supports. In particular, this holds for all open
sets of the form $V=X\times I$, with $I\subseteq\mathbb{R}$ an open interval
containing $0$.

  * $(ii)$

Let $\sigma=1/a_{0}>1$. We have that
$\operatorname{bv}(f)\in\mathscr{D}^{\prime\\{\sigma\\}}(X)$ exists for all
$f\in C^{\infty}_{P}(V\backslash X)$. Define the mapping
$\operatorname{bv}^{m}$ similarly as in (1). Then, the sequence

$0\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}\phantom{,}}}C^{\infty}_{P}(V)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}\phantom{,}}}C^{\infty}_{P}(V\backslash
X)\xrightarrow{\phantom{,}\operatorname{bv}^{m}\phantom{,}}\prod_{j=0}^{m-1}\mathscr{D}^{\prime\\{\sigma\\}}(X)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}\phantom{,}}}0$

is exact and $\operatorname{bv}^{m}$ is a topological homomorphism, provided
that $V$ is $P$-convex for supports. In particular, this holds for all open
sets of the form $V=X\times I$, with $I\subseteq\mathbb{R}$ an open interval
containing $0$.

Since a partial differential operator $P(D)$ on $\mathbb{R}^{d+1}$ is elliptic
precisely when it semi-elliptic with
$\operatorname{deg}_{x_{1}}P=\cdots=\operatorname{deg}_{x_{d}}P=\deg_{t}P$,
Theorem 1.1$(a)$ is applicable to every elliptic partial differential
operator. Moreover, it is well-known that every open set
$V\subset\mathbb{R}^{d}$ is $P$-convex for support in such a case. Therefore,
Theorem 1.1$(a)$ comprises two classical results of Komatsu about the boundary
values of holomorphic functions [17, Section 11] and harmonic functions [20,
Chapter 2] in ultradistribution spaces. Also the heat operator and, more
generally, the $k$-parabolic operators in the sense of Petrowsky [31,
Definition 7.11] satisfy the assumptions of Theorem 1.1$(a)$. In particular,
this result extends Matsuzawa’s characterization [25, Theorem 2.1] of
compactly supported ultradistributions via boundary values of zero solutions
of the heat equation to general ultradistributions. Theorem 1.1$(b)$ states
that the space $\prod_{j=0}^{m-1}\mathscr{D}^{\prime\\{1/a_{0}\\}}(X)$ may be
identified with the space $C^{\infty}_{P}(V\backslash X)/C^{\infty}_{P}(V)$ of
formal boundary values. This partially covers the above mentioned result of
Langenbruch [23, Satz 4.9 and Satz 4.10] (the assumption
$\operatorname{deg}_{x_{1}}P=\cdots=\operatorname{deg}_{x_{d}}P$ is not needed
as he considers anisotropic Gevrey ultradistribution spaces).

Next, we comment on the main new technique used in this article. In [10, p.
64] Hörmander showed the existence of distributional boundary values of
holomorphic functions by combining Stokes’ theorem with almost analytic
extensions. Petzsche and Vogt [28] (see also [27]) extended this method to
ultradistributions by using descriptions of ultradifferentiable classes via
almost analytic extensions [5, 8, 27, 28]. We develop here a similar technique
to establish the existence of ultradistributional boundary values of zero
solutions of a hypoelliptic partial differential operator $P(D)$. Namely, we
combine a Stokes type theorem for $P(D)$ (Lemma 5.4) with the description of
tuples $(\varphi_{0},\ldots,\varphi_{m-1})$ of compactly supported
ultradifferentiable functions via functions
$\Phi\in\mathscr{D}(\mathbb{R}^{d+1})$ that are almost zero solutions of
$P(D)$ and satisfy $D_{t}^{j}\Phi(\,\cdot\,,0)=\varphi_{j}$ for
$j=0,\ldots,m-1$ (Proposition 4.1). Petzsche [27] constructed almost analytic
extensions of ultradifferentiable functions by means of modified Taylor
series. We use here the same basic idea, starting from a power series Ansatz
$\Phi$ that formally solves the Cauchy problem $P(D)\Phi=0$ on
$\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}$ and
$D_{t}^{j}\Phi(\,\cdot\,,0)=\varphi_{j}$ on $\mathbb{R}^{d}$ for
$j=0,\ldots,m-1$ [12, Section 4].

This paper is organized as follows. In the preliminary Section 2 we collect
several results about partial differential operators and ultradistributions
that will be used throughout this work. Next, in Section 3, we construct a
fundamental solution of a hypoelliptic partial differential operator $P(D)$
satisfying precise regularity and growth properties. This fundamental solution
will play an essential role in our work. We obtain it by a careful analysis of
the construction of a fundamental solution of $P(D)$ by Langenbruch [21, p.
12–14]. Section 4 is devoted to the description of ultradifferentiable classes
via almost zero solutions of $P(D)$. By combining this description with a
Stokes type theorem for $P(D)$, we show in Section 5 that zero solutions of
$P(D)$ satisfying certain growth estimates near $\mathbb{R}^{d}$ have
ultradistributional boundary values. We also establish the continuity of the
boundary value mapping here. Our main results are proven in Section 6. We
adapt several classical techniques used in the study of ultradistributional
boundary values of holomorphic functions and harmonic functions [17, 20]. In
Sections 4 – 6 we also indicate how our methods may be adapted to the
distribution case, thereby providing new proofs of several results of
Langenbruch [21]. Most notably, we obtain an elementary proof of the
continuity of the boundary value mapping (compare with [21, Section 3]).
Finally, in Section 7, we discuss our results for semi-elliptic operators and
prove Theorem 1.1.

##  2\. Preliminaries

###  2.1. Partial differential operators

Elements of $\mathbb{R}^{d+1}=\mathbb{R}^{d}\times\mathbb{R}$ are denoted by
$(x,t)=(x_{1}\ldots,x_{d},t)$. We often identify $\mathbb{R}^{d}$ with the
subspace $\mathbb{R}^{d}\times\\{0\\}$ of $\mathbb{R}^{d+1}$. We write
$D=(D_{x},D_{t})$, where $D_{x}=-i(\partial_{x_{1}},\ldots,\partial_{x_{d}})$
and $D_{t}=-i\partial_{t}$. Throughout this article
$P\in\mathbb{C}[x_{1},\ldots,x_{d},t]$ always stands for a non-constant
hypoelliptic polynomial. Moreover, we use the notation

$P(x,t)=\sum_{k=0}^{m}Q_{k}(x)t^{k}$

for suitable $m\in\mathbb{N}$ and polynomials
$Q_{k}\in\mathbb{C}[X_{1},\ldots,X_{d}]$, $0\leq k\leq m$, with $Q_{m}\neq 0$.
Then, $Q_{m}=c\in\mathbb{C}\backslash\\{0\\}$ [11, Example 11.2.8] and we
assume without loss of generality that $c=1$. As customary, we associate to
$P$ the constant coefficient partial differential operator $P(D)$ given by

$P(D)=P(D_{x},D_{t})=\sum_{k=0}^{m}Q_{k}(D_{x})D_{t}^{k}.$

Given $V\subseteq\mathbb{R}^{d+1}$ open, we define $C^{\infty}_{P}(V)=\\{f\in
C^{\infty}(V)\,|\,P(D)f=0\\}$ and endow this space with the relative topology
induced by $C^{\infty}(V)$.

We now introduce various indices of $P$ that play an important role in this
article.

######  Definition 2.1.

We define $(\gamma_{0},\mu_{0})=(\gamma_{0}(P),\mu_{0}(P))$ as the largest
pair of positive numbers satisfying the following property: There are $C,R>0$
such that for all $\alpha\in\mathbb{N}^{d}_{0}$ and $l\in\mathbb{N}_{0}$

$(|x|^{\gamma_{0}}+|t|^{\mu_{0}})^{|\alpha|+l}|D^{\alpha}_{x}D^{l}_{t}P(x,t)|\leq
C|P(x,t)|,\qquad|(x,t)|\geq R.$

The existence of the pair $(\gamma_{0},\mu_{0})$ follows from the fact that
$P$ is hypoelliptic; see [11, Section 11.4] and [31, Section 7.4].
Furthermore, it holds that $\gamma_{0},\mu_{0}\leq 1$. Let
$V\subseteq\mathbb{R}^{d+1}$ be open. For $\sigma,\tau\geq 1$ we denote by
$\Gamma^{\sigma,\tau}(V)$ the space consisting of all $\varphi\in
C^{\infty}(V)$ such that for all $K\Subset V$ there is $h>0$ such that

$\sup_{(\alpha,l)\in\mathbb{N}^{d+1}_{0}}\sup_{(x,t)\in
K}\frac{|D^{\alpha}_{x}D^{l}_{t}\varphi(x,t)|}{h^{|\alpha|+l}|\alpha|!^{\sigma}l!^{\tau}}<\infty.$

By [31, Theorem 7.4], we have that
$C^{\infty}_{P}(V)\subset\Gamma^{1/\gamma_{0},1/\mu_{0}}(V)$.

######  Lemma 2.2.

Set

$d(x)=\min\\{|\operatorname{Im}\zeta|\,|\,\zeta\in\mathbb{C},P(x,\zeta)=0\\},\qquad
x\in\mathbb{R}^{d}.$

For $a>0$ the following two statements are equivalent:

  * $(i)$

There are $C,R>0$ such that for all $l\in\mathbb{N}_{0}$

$|x|^{al}|D^{l}_{t}P(x,t)|\leq C|P(x,t)|,\qquad|x|\geq R,t\in\mathbb{R}.$

  * $(ii)$

There are $C,R>0$ such that

$|x|^{a}\leq Cd(x),\qquad|x|\geq R.$

###### Proof.

The proof is similar to the one of [11, Lemma 11.1.4] and therefore left to
the reader. ∎

######  Definition 2.3.

We define $a_{0}=a_{0}(P)$ as the largest positive number $a$ such that
conditions $(i)$ and $(ii)$ from Lemma 2.2 are valid.

The existence of $a_{0}$ follows from the fact that $P$ is hypoelliptic and
[11, Corollary A.2.6] (cf. the proof of [11, Theorem 11.1.3]). Note that
$\gamma_{0}\leq a_{0}$.

######  Definition 2.4.

We define

$b_{0}=b_{0}(P):=\max\left\\{\frac{\operatorname{deg}Q_{k}}{m-k}\,|\,k=0,\ldots,m-1\right\\}.$

######  Remark 2.5.

By evaluating the inequalities from Lemma 2.2$(i)$ for $t=0$, we obtain that
there are $C,R>0$ such that for all $k=1,\ldots,m$

$|x|^{a_{0}k}|Q_{k}(x)|\leq C|Q_{0}(x)|,\qquad|x|\geq R.$

Hence,

(2) $|P(x,|x|^{a_{0}})|\leq C|Q_{0}(x)|,\qquad|x|\geq R,$

and

$a_{0}\leq\min\left\\{\frac{\operatorname{deg}Q_{0}-\operatorname{deg}Q_{k}}{k}\,|\,1\leq
k\leq m\right\\}\leq\frac{\operatorname{deg}Q_{0}}{m}\leq b_{0}.$

Since $a_{0}\geq 0$, we also have that
$\operatorname{deg}Q_{k}\leq\operatorname{deg}Q_{0}$ for all $k=1,\ldots,m$.

The main results of this article are valid under the assumption $a_{0}=b_{0}$.
In Section 7 we calculate $a_{0}(P),b_{0}(P),\gamma_{0}(P)$ and $\mu_{0}(P)$
for semi-elliptic polynomials $P$.

###  2.2. Weight sequences

Let $M=(M_{p})_{p\in\mathbb{N}_{0}}$ be a sequence of positive numbers. We
write $M_{\alpha}=M_{|\alpha|}$ for $\alpha\in\mathbb{N}^{d}_{0}$.
Furthermore, we set $m_{p}=M_{p}/M_{p-1}$ for $p\in\mathbb{N}$. Given $a>0$,
we define $M^{a}=(M^{a}_{p})_{p\in\mathbb{N}_{0}}$ and
$M^{a,*}=(M^{a}_{p}/p!)_{p\in\mathbb{N}_{0}}$.

We consider the following conditions on a positive sequence $M$:

  * $(M.1)$

$M^{2}_{p}\leq M_{p-1}M_{p+1}$, $p\in\mathbb{N}$.

  * $(M.2)$

$M_{p+q}\leq CH^{p+q}M_{p}M_{q}$, $p,q\in\mathbb{N}_{0}$, for some $C,H\geq
1$.

  * $(M.2)^{*}$

$2m_{p}\leq m_{Np}$, $p\geq p_{0}$, for some $p_{0},N\in\mathbb{N}$.

  * $(M.3)^{\prime}$

$\displaystyle\sum_{p=0}^{\infty}\frac{1}{M_{p}^{1/p}}<\infty$.

Note that $(M.1)$ means that the sequence $(m_{p})_{p\in\mathbb{N}}$ is non-
decreasing. Hence, any positive sequence $M$ with $M_{0}=M_{1}=1$ satisfying
$(M.1)$ is non-decreasing. We refer to [17] for the meaning of the conditions
$(M.1)$, $(M.2)$ and $(M.3)^{\prime}$. Condition $(M.2)^{*}$ was introduced in
[3] without a name.

######  Definition 2.6.

A sequence $M=(M_{p})_{p\in\mathbb{N}_{0}}$ of positive numbers is called a
_weight sequence_ if

  * $(i)$

$M_{0}=M_{1}=1$.

  * $(ii)$

$M$ satisfies $(M.1)$, $(M.2)$ and $(M.3)^{\prime}$.

The most important examples of weight sequences are the _Gevrey sequences_
$p!^{\sigma}$ with index $\sigma>1$.

Given two positive sequences $M$ and $N$, the relation $M\subset N$ means that
there are $C,L>0$ such that $M_{p}\leq CL^{p}N_{p}$ for all
$p\in\mathbb{N}_{0}$. The stronger relation $M\prec N$ means that the latter
inequality is valid for every $L>0$ and suitable $C>0$. We write $M\asymp N$
if both $M\subset N$ and $N\subset M$.

The next condition plays an important role in this article.

######  Definition 2.7.

Let $a>0$. A positive sequence $M$ is said to satisfy condition $(M.4)_{a}$ if
the sequence $(m^{a,*}_{p})_{p\in\mathbb{N}}$ is almost non-decreasing, i.e.,
$m^{a,*}_{p}\leq Cm_{q}^{a,*}$, $p,q\in\mathbb{N}$, $p\leq q$, for some $C>0$.

The Gevrey sequence $p!^{\sigma}$ satisfies $(M.4)_{a}$ if and only if
$\sigma\geq 1/a$. In the next lemma we state some consequences of the
condition $(M.4)_{a}$.

######  Lemma 2.8.

Let $a>0$. Let $M$ be a positive sequence satisfying $(M.4)_{a}$.

  * $(i)$

There exists a positive sequence $N$ with $N\asymp M$ such that $N^{a,*}$
satisfies $(M.1)$.

  * $(ii)$

Either $p!^{1/a}\prec M$ or $p!^{1/a}\asymp M$ holds.

  * $(iii)$

$M$ satisfies $(M.2)^{*}$.

###### Proof.

$(i)$ This follows from [30, Lemma 8].

$(ii)$ By $(i)$, we may assume without loss of generality that $M^{a,*}$
satisfies $(M.1)$, or equivalently, that $(m^{a,*}_{p})_{p\in\mathbb{N}}$ is
non-decreasing. If $\lim_{p\to\infty}m^{a,*}_{p}=\infty$, we have that also
$\lim_{p\to\infty}(M^{a,*}_{p})^{1/p}=\infty$ and thus $p!^{1/a}\prec M$. If
$\lim_{p\to\infty}m^{a,*}_{p}<\infty$, it is clear that $p!^{1/a}\asymp M$.

$(iii)$ This follows from [15, Theorem 3.11]. ∎

Let $M$ be a positive sequence with $\lim_{p\to\infty}M_{p}^{1/p}=\infty$. The
_associated function_ of $M$ is defined as

$\omega_{M}(\rho)=\sup_{p\in\mathbb{N}_{0}}\log\frac{\rho^{p}M_{0}}{M_{p}},\qquad\rho\geq
0.$

Note that $\log\rho=o(\omega_{M}(\rho))$ as $\rho\to\infty$. If $M$ satisfies
$(M.1)$ and $(M.2)$, then [17, Proposition 3.6]

$2\omega_{M}(\rho)\leq\omega_{M}(H\rho)+\log C,\qquad\rho\geq 0,$

where $C$ and $H$ are the constants occurring in $(M.2)$. Given two positive
sequences $M$ and $N$ satisfying $(M.1)$ with
$\lim_{p\to\infty}M_{p}^{1/p}=\lim_{p\to\infty}N_{p}^{1/p}=\infty$ , it holds
that $N\subset M$ if and only if

$\omega_{M}(t)\leq\omega_{N}(Lt)+\log C,\qquad t\geq 0.$

for some $C,L>0$ [17, Lemma 3.8]. Similarly, $N\prec M$ if and only if the
latter inequality remains valid for every $L>0$ and suitable $C>0$ [17, Lemma
3.10]. We have that $\omega_{p!^{\sigma}}(t)\sim t^{1/\sigma}$ (which means
that $\omega_{p!^{\sigma}}(t)=O(t^{1/\sigma})$ and
$t^{1/\sigma}=O(\omega_{p!^{\sigma}}(t))$).

######  Remark 2.9.

Let $N=(N_{p})_{p\in\mathbb{N}_{0}}$ be a sequence of positive numbers such
that $N\asymp 1$, we define the associated function of $N$ as

$\omega_{N}(\rho):=\begin{cases}0,&0\leq\rho\leq 1,\\\ \infty&\rho>1.\\\
\end{cases}$

Whenever we make use of this function we employ the convention
$0\cdot\infty=0$. This applies in particular to $N=M^{a,*}$, where $M$ is a
weight sequence satisfying $p!^{1/a}\asymp M$.

Finally, we present a lemma that will be used later on.

######  Lemma 2.10.

Let $a>0$ and let $M$ be a weight sequence. There are $C,L>0$ such that

$M_{\lfloor ap\rfloor}\leq CL^{p}M^{a}_{p},\qquad p\in\mathbb{N}_{0}.$

###### Proof.

Note that $\omega_{M^{a}}(\rho)=a\omega_{M}(\rho^{1/a})$ for $\rho\geq 0$.
Conditions $(M.1)$ and $(M.2)$ imply that there are $C,L>0$ such that

$a\omega_{M^{a}}(\rho)\leq\omega_{M^{a}}(L\rho)+\log C,\qquad\rho\geq 0.$

By [17, Proposition 3.2] and the fact that $M\geq 1$, we have that for all
$p\in\mathbb{N}_{0}$

$\displaystyle M_{\lfloor ap\rfloor}=\sup_{\rho\geq 1}\frac{\rho^{\lfloor
ap\rfloor}}{e^{\omega_{M}(\rho)}}\leq\sup_{\rho\geq
1}\frac{\rho^{ap}}{e^{\omega_{M}(\rho)}}=\sup_{\rho\geq
1}\frac{\rho^{p}}{e^{\omega_{M}(\rho^{1/a})}}=\sup_{\rho\geq
1}\frac{\rho^{p}}{e^{\frac{1}{a}\omega_{M^{a}}(\rho)}}$ $\displaystyle\leq
C\sup_{\rho\geq 1}\frac{\rho^{p}}{e^{\omega_{M^{a}}(\rho/L)}}\leq
CL^{p}\sup_{\rho>0}\frac{\rho^{p}}{e^{\omega_{M^{a}}(\rho)}}=CL^{p}M^{a}_{p}.$

∎

###  2.3. Ultradistributions

Fix a weight sequence $M$. Let $K\Subset\mathbb{R}^{d}$ be such that
$\overline{\operatorname{int}K}=K$ and let $h>0$. We write
$\mathscr{E}^{M,h}(K)$ for the Banach space consisting of all $\varphi\in
C^{\infty}(K)$ such that

$\|\varphi\|_{\mathscr{E}^{M,h}(K)}:=\sup_{\alpha\in\mathbb{N}_{0}^{d}}\sup_{x\in
K}\frac{|D^{\alpha}_{x}\varphi(x)|}{h^{|\alpha|}M_{|\alpha|}}<\infty.$

For $X\subseteq\mathbb{R}^{d}$ open we define the space of
_ultradifferentiable functions of class $(M)$ and $\\{M\\}$ (of Beurling and
Roumieu type) in $X$_ as

$\mathscr{E}^{(M)}(X):=\varprojlim_{K\Subset X}\varprojlim_{h\rightarrow
0^{+}}\mathscr{E}^{M,h}(K),\qquad\mathscr{E}^{\\{M\\}}(X):=\varprojlim_{K\Subset
X}\varinjlim_{h\rightarrow\infty}\mathscr{E}^{M,h}(K).$

We employ $[M]$ as a common notation for $(M)$ and $\\{M\\}$. In addition, we
often first state assertions for the Beurling case followed in parenthesis by
the corresponding ones for the Roumieu case.

Let $K\Subset\mathbb{R}^{d}$ and let $h>0$. We define $\mathscr{D}^{M,h}_{K}$
as the Banach space consisting of all $\varphi\in C^{\infty}(\mathbb{R}^{d})$
with $\operatorname{supp}\varphi\subseteq K$ such that
$\|\varphi\|_{\mathscr{E}^{M,h}(K)}<\infty$. We set

$\mathscr{D}^{(M)}_{K}:=\varprojlim_{h\rightarrow
0^{+}}\mathscr{D}^{M,h}_{K},\qquad\mathscr{D}^{\\{M\\}}_{K}:=\varinjlim_{h\rightarrow\infty}\mathscr{D}^{M,h}_{K}.$

For $X\subseteq\mathbb{R}^{d}$ open we define

$\mathscr{D}^{[M]}(X):=\varinjlim_{K\Subset X}\mathscr{D}^{[M]}_{K}.$

Following Komatsu [19, 20], we will sometimes make use of an alternative
description of the spaces of ultradifferentiable functions of Roumieu type. We
denote by $\mathfrak{R}$ the set of all non-decreasing sequences
$h=(h_{l})_{l\in\mathbb{N}_{0}}$ of positive numbers such that $h_{0}=h_{1}=1$
and $\lim_{l\to\infty}h_{l}=\infty$. This set is partially ordered and
directed by the pointwise order relation $\leq$ on sequences.

Let $K\Subset\mathbb{R}^{d}$ be such that $\overline{\operatorname{int}K}=K$
and let $h\in\mathfrak{R}$. We write $\mathscr{E}^{M,h}(K)$ for the Banach
space consisting of all $\varphi\in C^{\infty}(K)$ such that

$\|\varphi\|_{\mathscr{E}^{M,h}(K)}:=\sup_{\alpha\in\mathbb{N}_{0}^{d}}\sup_{x\in
K}\frac{|D^{\alpha}_{x}\varphi(x)|}{\prod^{|\alpha|}_{l=0}h_{l}M_{|\alpha|}}<\infty.$

Let $K\Subset\mathbb{R}^{d}$ and let $h\in\mathfrak{R}$. We define
$\mathscr{D}^{M,h}_{K}$ as the Banach space consisting of all $\varphi\in
C^{\infty}(\mathbb{R}^{d})$ with $\operatorname{supp}\varphi\subseteq K$ such
that $\|\varphi\|_{\mathscr{E}^{M,h}(K)}<\infty$.

######  Lemma 2.11.

Let $M$ be a weight sequence.

  * $(i)$

[19, Proposition 3.5] Let $X\subseteq\mathbb{R}^{d}$ be open. Then,

$\mathscr{E}^{\\{M\\}}(X)=\varprojlim_{K\Subset
X}\varprojlim_{h\in\mathfrak{R}}\mathscr{E}^{M,h}(K)$

as locally convex spaces.

  * $(ii)$

[20, Proposition 1.1] Let $K\Subset\mathbb{R}^{d}$. Then,

$\mathscr{D}^{\\{M\\}}_{K}=\varprojlim_{h\in\mathfrak{R}}\mathscr{D}^{M,h}_{K}$

as locally convex spaces.

Let $G(z)=\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}z^{\alpha}$,
$c_{\alpha}\in\mathbb{C}$, be an entire function on $\mathbb{C}^{d}$. The
associated infinite order constant coefficient partial differential operator
$G(D_{x})=\sum_{\alpha\in\mathbb{N}_{0}^{d}}c_{\alpha}D^{\alpha}_{x}$ is said
to be an _ultradifferential operator of class $[M]$_ if for some $h>0$ (for
all $h>0$)

$\sup_{\alpha\in\mathbb{N}_{0}^{d}}\frac{|c_{\alpha}|M_{\alpha}}{h^{|\alpha|}}<\infty.$

By [17, Proposition 4.5], $G(D_{x})$ is an ultradifferential operator of class
$[M]$ if and only if for some $h>0$ (for all $h>0$)

$\sup_{z\in\mathbb{C}^{d}}|G(z)|e^{-\omega_{M}(h|z|)}<\infty.$

Condition $(M.2)$ implies that, for all $X\subseteq\mathbb{R}^{d}$ open, the
linear mappings $G(D_{x}):\mathscr{E}^{[M]}(X)\rightarrow\mathscr{E}^{[M]}(X)$
and $G(D_{x}):\mathscr{D}^{[M]}(X)\rightarrow\mathscr{D}^{[M]}(X)$ are
continuous.

Let $X\subseteq\mathbb{R}^{d}$ be open. We denote by
$\mathscr{D}^{\prime[M]}(X)$ and $\mathscr{E}^{\prime[M]}(X)$ the strong duals
of $\mathscr{D}^{[M]}(X)$ and $\mathscr{E}^{[M]}(X)$, respectively. The
elements of $\mathscr{D}^{\prime[M]}(X)$ are called _ultradistributions of
class $[M]$ in $X$_. The space $\mathscr{E}^{\prime[M]}(X)$ may be identified
with the subspace of $\mathscr{D}^{\prime[M]}(X)$ consisting of compactly
supported elements. Let $G(D_{x})$ be an ultradifferential operator of class
$[M]$. We define the mapping
$G(D_{x}):\mathscr{D}^{\prime[M]}(X)\rightarrow\mathscr{D}^{\prime[M]}(X)$ via
transposition. Hence,
$G(D_{x}):\mathscr{D}^{\prime[M]}(X)\rightarrow\mathscr{D}^{\prime[M]}(X)$
becomes a continuous linear mapping.

Later on, we will make use of the following result about the existence of
parametrices, which is essentially shown in [9]. It improves a classical
result of Komatsu [20], namely, the strong non-quasianalyticity condition
$(M.3)$ [17] is relaxed to the weaker conditions $(M.2)^{*}$ and
$(M.3)^{\prime}$. Its proof requires the next lemma.

######  Lemma 2.12.

[29, Lemma 2.3] For every $h\in\mathfrak{R}$ there is
$h^{\prime}\in\mathfrak{R}$ such that $h^{\prime}_{l}\leq h_{l}$ for $l$ large
enough and $\prod_{l=0}^{p+q}h^{\prime}_{l}\leq
2^{p+q}\prod_{l=0}^{p}h^{\prime}_{l}\prod_{l=0}^{q}h^{\prime}_{l}$ for all
$p,q\in\mathbb{N}_{0}$.

######  Lemma 2.13.

Let $M$ be a weight sequence satisfying $(M.2)^{*}$. Let $h>0$
($h\in\mathfrak{R}$) and let $r>0$. There exist an ultradifferential operator
$G(D_{x})$ of class $[M]$, $u_{1}\in\mathscr{D}^{M,h}_{\overline{B}(0,r)}$ and
$u_{2}\in\mathscr{D}^{[M]}_{\overline{B}(0,r)}$ such that
$G(D_{x})u_{1}=\delta+u_{2}$ in $\mathscr{D}^{\prime[M]}(\mathbb{R}^{d})$.

###### Proof.

We first consider the Beurling case. By [3, Theorem 14], the function
$\omega_{M}$ is a weight function in the sense of [9, Definition 1.1].
Moreover, there is $n\in\mathbb{N}$ and $C>0$ such that [3, Equation (5)]

$e^{n\omega^{*}_{M}\left(\frac{p}{n}\right)}\leq C(h/2)^{p}M_{p},\qquad
p\in\mathbb{N}_{0},$

where $\omega^{*}_{M}$ denotes the Young conjugate of $\omega_{M}$. Hence, [9,
Proposition 2.5] and conditions $(M.1)$ and $(M.2)$ imply that there are an
ultradifferential operator $G(D_{x})$ of class $(M)$ and $u\in
C^{\infty}(\mathbb{R}^{d})$ such that

$\sup_{\alpha\in\mathbb{N}_{0}^{d}}\sup_{x\in\mathbb{R}^{d}}\frac{|D^{\alpha}_{x}u(x)|}{(h/2)^{|\alpha|}M_{\alpha}}<\infty,$

$u$ is real analytic in $\mathbb{R}^{d}\backslash\\{0\\}$ and
$G(D_{x})u=\delta$ in $\mathscr{D}^{\prime(M)}(\mathbb{R}^{d})$. Choose
$\psi\in\mathscr{D}^{(M)}_{\overline{B}(0,r)}$ such that $\psi=1$ in a
neighbourhood of $0$. Then, $u_{1}=\psi
u\in\mathscr{D}^{M,h}_{\overline{B}(0,r)}$ and $u_{2}=G(D_{x})(\psi
u-u)\in\mathscr{D}^{(M)}_{\overline{B}(0,r)}$ satisfy
$G(D_{x})u_{1}=\delta+u_{2}$ in $\mathscr{D}^{\prime(M)}(\mathbb{R}^{d})$.
Next, we treat the Roumieu case. Note that the positive sequence
$N=(\prod_{l=0}^{p}h_{l}M_{p})_{p\in\mathbb{N}_{0}}$ satisfies $(M.1)$,
$(M.3)^{\prime}$ and $(M.2)^{*}$ because $M$ does so. Furthermore, by Lemma
2.12, we may assume without loss of generality that $N$ also satisfies
$(M.2)$. Hence, the result can be shown by applying the same reasoning as in
the Beurling case to $M=N$ and $h=1$. ∎

Finally, we introduce some notation from classical distribution theory. For
$K\Subset\mathbb{R}^{d}$ we denote by $\mathscr{D}_{K}$ the Fréchet space
consisting of all smooth functions with support in $K$. For
$X\subseteq\mathbb{R}^{d}$ open we define

$\mathscr{D}(X):=\varinjlim_{K\Subset X}\mathscr{D}_{K}.$

For $l\in\mathbb{N}_{0}$ we define the norm

$\|\varphi\|_{l}=\sup_{x\in\mathbb{R}^{d}}\max_{|\alpha|\leq
l}|D^{\alpha}_{x}\varphi(x)|,\qquad\varphi\in\mathscr{D}(\mathbb{R}^{d}).$

In this context, we sometimes also write $\mathscr{E}(X)=C^{\infty}(X)$. We
denote by $\mathscr{D}^{\prime}(X)$ and $\mathscr{E}^{\prime}(X)$ the strong
duals of $\mathscr{D}(X)$ and $\mathscr{E}(X)$, respectively. The space
$\mathscr{E}^{\prime}(X)$ may be identified with the subspace of
$\mathscr{D}^{\prime}(X)$ consisting of compactly supported elements.

##  3\. A Fundamental solution with good regularity and growth properties

The goal of this auxiliary section is to construct a fundamental solution of
$P(D)$ with precise regularity and growth properties. The next result and its
proof are inspired by [21, p. 12–14].

######  Proposition 3.1.

There exists a fundamental solution
$E\in\mathscr{D}^{\prime}(\mathbb{R}^{d+1})$ of $P(D)$ satisfying the
following property: There are $A,L,S>0$ such that

(3)
$\sup_{(\alpha,l)\in\mathbb{N}_{0}^{d+1}}\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}\frac{|D^{\alpha}_{x}D^{l}_{t}E(x,t)||t|^{|\alpha|/a_{0}+l/\mu_{0}+S}e^{-A|t|}}{L^{|\alpha|+l}|\alpha|!^{1/a_{0}}l!^{1/\mu_{0}}}<\infty.$

In particular, there is $L>0$ such that for all $l\in\mathbb{N}_{0}$ there is
$S>0$ such that for all $r>0$

(4)
$\sup_{\alpha\in\mathbb{N}_{0}^{d}}\sup_{\begin{subarray}{c}x\in\mathbb{R}^{d}\\\
0<|t|\leq
r\end{subarray}}\frac{|D^{\alpha}_{x}D^{l}_{t}E(x,t)||t|^{|\alpha|/a_{0}+S}}{L^{|\alpha|}|\alpha|!^{1/a_{0}}}<\infty.$

We need some preparation for the proof of Proposition 3.1. Definition 2.1 and
Definition 2.3 imply that there are $C,R\geq 1$ such that for all
$l\in\mathbb{N}_{0}$

$(|x|^{a_{0}}+|t|^{\mu_{0}})^{l}|D^{l}_{t}P(x,t)|\leq C|P(x,t)|,\qquad|x|\geq
R,t\in\mathbb{R}.$

Throughout this section the constants $C$ and $R$ will always refer to those
occurring in the above inequality. We need the following lemma.

######  Lemma 3.2.

  *   * $(i)$

There are $C_{1},L_{1}>0$ such that for all $l,p\in\mathbb{N}_{0}$

$\left|D^{p}_{t}\left(\frac{t^{l}}{P(-x,-t)}\right)\right|\leq\frac{C_{1}L_{1}^{p+l}p!(|x|^{a_{0}}+\max\\{1,|t|\\})^{l}}{(|x|^{a_{0}}+|t|^{\mu_{0}})^{p}},\qquad|x|\geq
R,t\in\mathbb{R}.$

  * $(ii)$

There are $A>0$ and $C_{2},L_{2}>0$ such that for all $l,p\in\mathbb{N}_{0}$

$\left|D^{p}_{t}\left(\frac{(t+iA)^{l}}{P(-x,-t-iA)}\right)\right|\leq\frac{C_{2}L_{2}^{p+l}p!}{(1+|t|)^{p-l}}\qquad|x|\leq
R,t\in\mathbb{R}.$

###### Proof.

Note that $|P(x,t)|\geq 1/C$ for all $|x|\geq R$ and $t\in\mathbb{R}$. We
first show $(i)$. Set $r=\log(1+(2C)^{-1})$. Fix $x\in\mathbb{R}^{d}$ with
$|x|\geq R$ and $t\in\mathbb{R}$ arbitrary. For all $\zeta\in\mathbb{C}$ with
$|\zeta|\leq r(|x|^{a_{0}}+|t|^{\mu_{0}})$ it holds that

$\displaystyle|P(x,t+\zeta)-P(x,t)|\leq\sum_{l=1}^{m}|D^{l}_{t}P(x,t)|\frac{|\zeta|^{l}}{l!}\leq\frac{|P(x,t)|}{2}.$

Hence, $|P(x,t+\zeta)|\geq 1/(2C)$. Cauchy’s inequalities give that for all
$l,p\in\mathbb{N}_{0}$

$\displaystyle\left|D^{p}_{t}\left(\frac{t^{l}}{P(x,t)}\right)\right|$
$\displaystyle\leq\frac{p!}{r^{p}(|x|^{a_{0}}+|t|^{\mu_{0}})^{p}}\max_{|\zeta|\leq
r(|x|^{a_{0}}+|t|^{\mu_{0}})}\frac{|t+\zeta|^{l}}{|P(x,t+\zeta)|}$
$\displaystyle\leq\frac{2Cp!(|t|+r(|x|^{a_{0}}+|t|^{\mu_{0}}))^{l}}{r^{p}(|x|^{a_{0}}+|t|^{\mu_{0}})^{p}},$

from which $(i)$ follows.

Next, we prove $(ii)$. Set $S=\max_{|x|\leq R}\sum_{k=0}^{m-1}|Q_{k}(x)|$. For
all $x\in\mathbb{R}^{d}$ with $|x|\leq R$ and $\zeta\in\mathbb{C}$ with
$|\zeta|\geq\max\\{1,S+\frac{1}{2}\\}$ it holds that

$|P(x,\zeta)|\geq|\zeta|^{m}-\sum_{k=0}^{m-1}|Q_{k}(x)||\zeta|^{k}\geq\frac{1}{2}.$

Set $A=2\max\\{1,S+\frac{1}{2}\\}$. Fix $x\in\mathbb{R}^{d}$ with $|x|\leq R$
and $t\in\mathbb{R}$ arbitrary. Cauchy’s inequalities give that for all
$l,p\in\mathbb{N}_{0}$

$\displaystyle\left|D^{p}_{t}\left(\frac{(t+iA)^{l}}{P(x,t+iA)}\right)\right|$
$\displaystyle\leq\frac{2^{p}p!}{|t+iA|^{p}}\max_{|\zeta|\leq|t+iA|/2}\frac{|t+iA+\zeta|^{l}}{|P(x,t+iA+\zeta)|}$
$\displaystyle\leq\frac{2^{p+l+1}p!}{|t+iA|^{p-l}},$

from which $(ii)$ follows. ∎

###### Proof of Proposition 3.1.

Set $B=\\{x\in\mathbb{R}^{d}\,|\,|x|\leq R\\}$. With $A$ from Lemma 3.2$(ii)$,
we define $E\in\mathscr{D}^{\prime}(\mathbb{R}^{d+1})$ via

$\displaystyle\langle E,\varphi\rangle=$
$\displaystyle\frac{1}{(2\pi)^{d+1}}\int_{\mathbb{R}^{d}\backslash
B}\int_{-\infty}^{\infty}\frac{\widehat{\varphi}(\xi,\eta)}{P(-\xi,-\eta)}d\eta
d\xi$
$\displaystyle+\frac{1}{(2\pi)^{d+1}}\int_{B}\int_{-\infty}^{\infty}\frac{\widehat{\varphi}(\xi,\eta+iA)}{P(-\xi,-\eta-
iA)}d\eta d\xi,\qquad\varphi\in\mathscr{D}(\mathbb{R}^{d+1}),$

where $\widehat{\varphi}$ denotes the Fourier transform of $\varphi$. It is
clear that $E$ is a fundamental solution of $P(D)$. Since $P$ is hypoelliptic,
$E$ is smooth in $\mathbb{R}^{d+1}\backslash\\{0\\}$. We set

$p_{\alpha,l}=\left\lceil\frac{|\alpha|+d+1}{{a_{0}}}+\frac{l+2}{\mu_{0}}\right\rceil,\qquad\alpha\in\mathbb{N}_{0}^{d},l\in\mathbb{N}_{0}.$

There are $C_{3},L_{3}>0$ such that

(5) $p_{\alpha,l}!\leq
C_{3}L_{3}^{|\alpha|+l}|\alpha|!^{1/{a_{0}}}l!^{1/{\mu_{0}}},\qquad\alpha\in\mathbb{N}_{0}^{d},l\in\mathbb{N}_{0},$

and $C_{4},L_{4}>0$ such that

(6) $\int_{\mathbb{R}^{d}\backslash
B}|\xi|^{|\alpha|}\int_{-\infty}^{\infty}\frac{(|\xi|^{a_{0}}+\max\\{1,|\eta|\\})^{l}}{(|\xi|^{a_{0}}+|\eta|^{\mu_{0}})^{p_{\alpha,l}}}d\eta
d\xi\leq C_{4}L^{l}_{4},\quad\alpha\in\mathbb{N}_{0}^{d},l\in\mathbb{N}_{0}.$

Let $\psi\in\mathscr{D}(\mathbb{R}^{d})$ and $\chi\in\mathscr{D}(\mathbb{R})$
be arbitrary. Let $r_{\chi}>0$ be such that
$\operatorname{supp}\,\chi\subseteq[-r_{\chi},r_{\chi}]$. For all
$\alpha\in\mathbb{N}_{0}^{d}$ and $l\in\mathbb{N}_{0}$ it holds that

$\displaystyle(2\pi)^{d+1}|\langle
D_{x}^{\alpha}D^{l}_{t}E(x,t)t^{p_{\alpha,l}},\psi(x)\chi(t)\rangle|$
$\displaystyle=\int_{\mathbb{R}^{d}\backslash
B}|\xi|^{|\alpha|}|\widehat{\psi}(\xi)|\int_{-\infty}^{\infty}|\widehat{\chi}(\eta)|\left|D_{t}^{p_{\alpha,l}}\left(\frac{\eta^{l}}{P(-\xi,-\eta)}\right)\right|d\eta
d\xi$
$\displaystyle+\int_{B}|\xi|^{|\alpha|}|\widehat{\psi}(\xi)|\int_{-\infty}^{\infty}|\widehat{\chi}(\eta+iA)|\left|D^{p_{\alpha,l}}_{t}\left(\frac{(\eta+iA)^{l}}{P(-\xi,-\eta-
iA)}\right)\right|d\eta d\xi.$

Lemma 3.2$(i)$ and (6) imply that

$\displaystyle\int_{\mathbb{R}^{d}\backslash
B}|\xi|^{|\alpha|}|\widehat{\psi}(\xi)|\int_{-\infty}^{\infty}|\widehat{\chi}(\eta)|\left|D_{t}^{p_{\alpha,l}}\left(\frac{\eta^{l}}{P(-\xi,-\eta)}\right)\right|d\eta
d\xi$ $\displaystyle\leq
C_{1}L_{1}^{p_{\alpha,l}+l}p_{\alpha,l}!\|\psi\|_{L^{1}}\|\chi\|_{L^{1}}\int_{\mathbb{R}^{d}\backslash
B}|\xi|^{|\alpha|}\int_{-\infty}^{\infty}\frac{(|\xi|^{a_{0}}+\max\\{1,|\eta|\\})^{l}}{(|\xi|^{a_{0}}+|\eta|^{\mu_{0}})^{p_{\alpha,l}}}d\eta
d\xi$ $\displaystyle\leq
C_{1}C_{4}L_{1}^{p_{\alpha,l}+l}L_{4}^{l}p_{\alpha,l}!\|\psi\|_{L^{1}}\|\chi\|_{L^{1}}.$

Likewise, Lemma 3.2$(ii)$ yields that

$\displaystyle\int_{B}|\xi|^{|\alpha|}|\widehat{\psi}(\xi)|\int_{-\infty}^{\infty}|\widehat{\chi}(\eta+iA)|\left|D^{p_{\alpha,l}}_{t}\left(\frac{(\eta+iA)^{l}}{P(-\xi,-\eta-
iA)}\right)\right|d\eta d\xi$ $\displaystyle\leq
C_{2}L_{2}^{p_{\alpha,l}+l}p_{\alpha,l}!e^{r_{\chi}A}\|\psi\|_{L^{1}}\|\chi\|_{L^{1}}\int_{B}|\xi|^{|\alpha|}\int_{-\infty}^{\infty}\frac{1}{(1+|\eta|)^{p_{\alpha,l}-l}}d\eta
d\xi$ $\displaystyle\leq
2C_{2}|B|R^{|\alpha|}L_{2}^{p_{\alpha,l}+l}p_{\alpha,l}!e^{Ar_{\chi}}\|\psi\|_{L^{1}}\|\chi\|_{L^{1}}.$

By (5), there are $C,L>0$ with

$|\langle
D_{x}^{\alpha}D^{l}_{t}E(x,t)t^{p_{\alpha,l}},\psi(x)\chi(t)\rangle|\leq
CL^{|\alpha|+l}|\alpha|!^{1/{a_{0}}}l!^{1/{\mu_{0}}}e^{Ar_{\chi}}\|\psi\|_{L^{1}}\|\chi\|_{L^{1}}.$

Fix $x\in\mathbb{R}^{d}$ and $t\in\mathbb{R}\backslash\\{0\\}$ arbitrary.
Choose $\psi\in\mathscr{D}(\mathbb{R}^{d})$ non-negative with
$\int_{\mathbb{R}^{d}}\psi(y)dy=1$ and $\chi\in\mathscr{D}(\mathbb{R})$ non-
negative with $\int_{-\infty}^{\infty}\chi(s)ds=1$ and
$\operatorname{supp}\,\chi\subseteq[-1,1]$. For $\varepsilon>0$ we set

$\psi_{x,\varepsilon}(y)=\frac{1}{\varepsilon^{d}}\psi\left(\frac{x-y}{\varepsilon}\right),\qquad\chi_{t,\varepsilon}(s)=\frac{1}{\varepsilon}\psi\left(\frac{t-s}{\varepsilon}\right).$

Then, $\|\psi_{x,\varepsilon}\|_{L^{1}}=\|\psi\|_{L^{1}}=1$,
$\|\chi_{t,\varepsilon}\|_{L^{1}}=\|\chi\|_{L^{1}}=1$ and
$\operatorname{supp}\,\chi_{t,\varepsilon}\subseteq[-|t|-\varepsilon,|t|+\varepsilon]$.
We obtain that for all $\alpha\in\mathbb{N}_{0}^{d}$ and $l\in\mathbb{N}_{0}$

$\displaystyle|D_{x}^{\alpha}D^{l}_{t}E(x,t)t^{p_{\alpha,l}}|$
$\displaystyle=\lim_{\varepsilon\to 0^{+}}|\langle
D_{x}^{\alpha}D^{l}_{t}E(y,s)s^{p_{\alpha,l}},\psi_{x,\varepsilon}(y)\chi_{t,\varepsilon}(s)\rangle|$
$\displaystyle\leq
CL^{|\alpha|+l}|\alpha|!^{1/{a_{0}}}l!^{1/{\mu_{0}}}e^{A|t|},$

which implies (3). ∎

##  4\. Characterization of ultradifferentiable functions via (almost) zero
solutions of $P(D)$

In this section we characterize tuples $(\varphi_{0},\ldots,\varphi_{m-1})$ of
compactly supported ultradifferentiable functions via functions
$\Phi\in\mathscr{D}(\mathbb{R}^{d+1})$ that are (almost) zero solutions of
$P(D)$ and satisfy $D_{t}^{j}\Phi(\,\cdot\,,0)=\varphi_{j}$ for
$j=0,\ldots,m-1$. We start with the following result which is essential for
this article.

######  Proposition 4.1.

Let $M$ be a weight sequence satisfying $(M.4)_{b_{0}}$ and let
$K\Subset\mathbb{R}^{d}$. There is $L>0$ such that for all $h>0$ the following
property holds: For all
$\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}^{M,h}_{K}$ there exists
$\Phi=\Phi(\varphi_{0},\ldots,\varphi_{m-1})\in\mathscr{D}(\mathbb{R}^{d+1})$
with $\operatorname{supp}\,\Phi\subseteq K\times\mathbb{R}$ such that

  * $(i)$

$D^{j}_{t}\Phi(\,\cdot\,,0)=\varphi_{j}$ for $j=0,\ldots,m-1$.

  * $(ii)$

$\displaystyle\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}|P(D)\Phi(x,t)|e^{\omega_{M^{b_{0},*}}\left(\frac{1}{Lh^{b_{0}}|t|}\right)}<\infty$.

######  Remark 4.2.

By Lemma 2.8$(ii)$, $(M.4)_{b_{0}}$ implies that either $p!^{1/b_{0}}\prec M$
or $p!^{1/b_{0}}\asymp M$. If $p!^{1/b_{0}}\prec M$, the meaning of condition
$(ii)$ in Proposition 4.1 is clear. If $p!^{1/b_{0}}\asymp M$, this condition
means that $\Phi$ satisfies $P(D)\Phi=0$ on
$\mathbb{R}^{d}\times(-1/(Lh^{b_{0}}),1/(Lh^{b_{0}}))$ (cf. Remark 2.9). The
same convention will be tacitly used in the rest of this article.

The proof of Proposition 4.1 requires some preparation. In [27] Petzsche
constructed almost analytic extensions of ultradifferentiable functions by
means of modified Taylor series. If $p!^{1/b_{0}}\prec M$, we use here a
similar idea to prove Proposition 4.1, starting from a power series Ansatz
$\Phi$ that formally solves the Cauchy problem $P(D)\Phi=0$ on
$\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}$ and
$D_{t}^{j}\Phi(\,\cdot\,,0)=\varphi_{j}$ on $\mathbb{R}^{d}$ for
$j=0,\ldots,m-1$ [12, Section 4] (see also the proof of [22, Satz 4.1]). If
$p!^{1/b_{0}}\asymp M$, we even show that $\Phi$ converges. We now recall the
definition and some basic properties of this formal power series solution; see
[12, Section 4] for details. For $l\in\mathbb{N}_{0}$ we recursively define
the mapping $\mathscr{C}_{l}:C^{\infty}(\mathbb{R}^{d})\rightarrow
C^{\infty}(\mathbb{R}^{d})$ as

$\mathscr{C}_{l}(\varphi)=\begin{cases}0,&l=0,\ldots,m-1,\\\
\varphi,&l=m-2,\\\
-\sum_{k=0}^{m-1}Q_{k}(D_{x})\mathscr{C}_{k+l-m}(\varphi),&l\geq
m.\end{cases}$

Since $Q_{m}=1$, we have that for all $l\in\mathbb{N}_{0}$

(7) $\sum_{k=0}^{m}Q_{k}(D_{x})\mathscr{C}_{k+l}(\varphi)=0,\qquad\varphi\in
C^{\infty}(\mathbb{R}^{d}).$

Moreover, as $\mathscr{C}_{l}$ is a constant coefficient partial differential
operator,
$\operatorname{supp}\,\mathscr{C}_{l}(\varphi)\subseteq\operatorname{supp}\,\varphi$
and
$D_{x}^{\alpha}\mathscr{C}_{l}(\varphi)=\mathscr{C}_{l}(D_{x}^{\alpha}\varphi)$
for all $\varphi\in C^{\infty}(\mathbb{R}^{d})$ and
$\alpha\in\mathbb{N}_{0}^{d}$. If $\varphi_{0},\ldots,\varphi_{m-1}\in
C^{\infty}(\mathbb{R}^{d})$ are such that

$S(\varphi_{j})(x,t)=\sum_{l=0}^{\infty}\mathscr{C}_{l}(\varphi_{j})(x)\frac{(it)^{l}}{l!},\qquad(x,t)\in\mathbb{R}^{d+1},$

converges in $C^{\infty}(\mathbb{R}^{d+1})$ for all $j=0,\ldots,m-1$, then

$\Phi=\sum_{j=0}^{m-1}\sum_{k=0}^{m-1-j}Q_{j+k+1}(D_{x})D_{t}^{k}S(\varphi_{j})$

solves the Cauchy problem $P(D)\Phi=0$ on
$\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}$ and
$D_{t}^{j}\Phi(\,\cdot\,,0)=\varphi_{j}$ on $\mathbb{R}^{d}$ for
$j=0,\ldots,m-1$ [12, Proposition 4.4].

We need the following lemma.

######  Lemma 4.3.

Let $M$ be a weight sequence and let $K\Subset\mathbb{R}^{d}$. There is
$L_{1}>0$ such that for all $h>0$ the following property holds: For all
$\alpha\in\mathbb{N}^{d}$ there is $C>0$ such that for all
$p\in\mathbb{N}_{0}$

$\|\mathscr{C}_{p}(D^{\alpha}_{x}\varphi)\|_{0}\leq
C(L_{1}h^{b_{0}})^{p}M^{b_{0}}_{p}\|\varphi\|_{\mathscr{E}^{M,h}(K)},\qquad\varphi\in\mathscr{D}^{M,h}_{K}.$

###### Proof.

By $(M.2)$, it suffices to consider the case $\alpha=0$. For
$l\in\mathbb{N}_{0}$ we define the auxiliary norm

$|\varphi|_{l}=\sup_{x\in\mathbb{R}^{d}}\max_{|\alpha|=l}|D^{\alpha}_{x}\varphi(x)|,\qquad\varphi\in\mathscr{D}(\mathbb{R}^{d}).$

There is $R\geq 1$ (depending on $K$) such that for all $l\in\mathbb{N}_{0}$

$\|\varphi\|_{l}\leq R^{l}|\varphi|_{l},\qquad\varphi\in\mathscr{D}_{K}.$

By Lemma 2.10, there are $C,L\geq 1$ such that for all $l\in\mathbb{N}_{0}$

$\displaystyle|\varphi|_{\lfloor b_{0}l\rfloor}\leq h^{\lfloor
b_{0}l\rfloor}M_{{\lfloor b_{0}l\rfloor}}\|\varphi\|_{\mathscr{E}^{M,h}(K)}$
$\displaystyle\leq\frac{C}{\min\\{1,h\\}}(Lh^{b_{0}})^{l}M^{b_{0}}_{l}\|\varphi\|_{\mathscr{E}^{M,h}(K)},\qquad\varphi\in\mathscr{D}^{M,h}_{K}.$

The recursively defined operators $\mathscr{C}_{m-1+l}$, $l\in\mathbb{N}_{0},$
have the following explicit representation [12, Proposition 4.5]

$\mathscr{C}_{m-1+l}(\varphi)=\sum_{\beta\in\mathbb{N}_{0}^{m},\sigma(\beta)=l}(-1)^{|\beta|}\binom{|\beta|}{\beta_{1},\ldots,\beta_{m}}\prod_{k=1}^{m}Q_{m-k}^{\beta_{k}}(D_{x})\varphi,\qquad\varphi\in
C^{\infty}(\mathbb{R}^{d}),$

where $\sigma(\beta)=\sum_{j=1}^{m}j\beta_{j}$. Choose $L_{2}\geq 1$ such that
for all $k=0,\ldots,m-1$

$\|Q_{k}(D_{x})\varphi\|_{0}\leq
L_{2}\|\varphi\|_{\operatorname{deg}Q_{k}},\qquad\varphi\in\mathscr{D}(\mathbb{R}^{d}).$

Let $h>0$ and $\varphi\in\mathscr{D}^{M,h}_{K}$ be arbitrary. For all
$\beta\in\mathbb{N}_{0}^{m}$ with $\sigma(\beta)=l$ it holds that

$\displaystyle\left\|\prod_{k=1}^{m}Q_{m-k}^{\beta_{k}}(D_{x})\varphi\right\|_{0}\leq
L^{|\beta|}_{2}\|\varphi\|_{\sum_{k=1}^{m}\beta_{k}\operatorname{deg}Q_{m-k}}\leq
L_{2}^{l}\|\varphi\|_{\lfloor b_{0}l\rfloor}$
$\displaystyle\leq(L_{2}R^{b_{0}})^{l}|\varphi|_{\lfloor
b_{0}l\rfloor}\leq\frac{C}{\min\\{1,h\\}}(LL_{2}R^{b_{0}}h^{b_{0}})^{l}M^{b_{0}}_{l}\|\varphi\|_{\mathscr{E}^{M,h}(K)}.$

Hence,

$\displaystyle\|\mathscr{C}_{m-1+l}(\varphi)\|_{0}$
$\displaystyle\leq\frac{C}{\min\\{1,h\\}}(LL_{2}R^{b_{0}}h^{b_{0}})^{l}M^{b_{0}}_{l}\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{\beta\in\mathbb{N}_{0}^{m},|\beta|\leq
l}\binom{|\beta|}{\beta_{1},\ldots,\beta_{m}}$
$\displaystyle\leq\frac{C}{\min\\{1,h\\}}((m+1)LL_{2}R^{b_{0}}h^{b_{0}})^{l}M^{b_{0}}_{l}\|\varphi\|_{\mathscr{E}^{M,h}(K)}.$
$\displaystyle\leq\frac{C}{\min\\{1,h^{1+b_{0}(m-1)}\\}}((m+1)L_{2}R^{b_{0}}h^{b_{0}})^{m-1+l}M^{b_{0}}_{m-1+l}\|\varphi\|_{\mathscr{E}^{M,h}(K)}.$

Since $\mathscr{C}_{l}=0$ for $l<m-1$, this shows the result. ∎

###### Proof of Proposition 4.1.

Lemma 2.8$(ii)$ yields that either $p!^{1/b_{0}}\prec M$ or
$p!^{1/b_{0}}\asymp M$. Suppose first that $p!^{1/b_{0}}\prec M$. By Lemma
2.8$(i)$, we may assume without loss of generality that
$m^{b_{0},*}_{p}\nearrow\infty$. Throughout this proof $C$ will denote a
positive constant that is independent of $\varphi_{0},\ldots,\varphi_{m-1}$
but may vary from place to place. Let $H$ be the constant occurring in $(M.2)$
for $M$. Then, $M^{b_{0},*}$ satisfies $(M.2)$ with $H^{b_{0}}$ instead of
$H$. Choose $\psi\in\mathscr{D}(\mathbb{R})$ such that
$\operatorname{supp}\psi\subseteq[-2,2]$ and $\psi=1$ on $[-1,1]$. Set
$A=8L_{1}H^{b_{0}}$, where $L_{1}$ denotes the constant from Lemma 4.3. Define
$\lambda_{p}=Ah^{b_{0}}m_{p+1}^{b_{0},*}$ for $p\in\mathbb{N}_{0}$. Note that
$\lambda_{p}\nearrow\infty$. For $\varphi\in\mathscr{D}^{M,h}_{K}$ we define

(8)
$S(x,t)=S(\varphi)(x,t)=\sum_{p=0}^{\infty}\mathscr{C}_{p}(\varphi)(x)\frac{(it)^{p}}{p!}\psi(\lambda_{p}t),\qquad(x,t)\in\mathbb{R}^{d+1}.$

Since $\lambda_{p}\nearrow\infty$, the above series is finite on
$\mathbb{R}^{d}\times\\{t\in\mathbb{R}\,|\,|t|\geq\varepsilon\\}$ for each
$\varepsilon>0$. Hence, $S\in
C^{\infty}(\mathbb{R}^{d}\times(\mathbb{R}\backslash\\{0\\}))$. Set
$L=H^{b_{0}}A$. We claim that for all $\alpha\in\mathbb{N}_{0}^{d}$ and
$l\in\mathbb{N}_{0}$

(9) $\limsup_{t\to
0}\sup_{x\in\mathbb{R}^{d}}|P(D)(D^{\alpha}_{x}D^{l}_{t}S)(x,t)|e^{\omega_{M^{b_{0},*}}\left(\frac{1}{Lh^{b_{0}}|t|}\right)}<\infty,$
(10) $\lim_{t\to
0}D^{\alpha}_{x}D^{l}_{t}S(x,t)=D^{\alpha}_{x}\mathscr{C}_{l}(\varphi)(x)\mbox{
uniformly for $x\in\mathbb{R}^{d}$}.$

Before we prove these claims, let us show how they entail the result. Property
(10) implies that $S\in\mathscr{D}(\mathbb{R}^{d+1})$ with
$\operatorname{supp}\,S\subseteq K\times\mathbb{R}$ such that

(11) $D^{l}_{t}S(\,\cdot\,,0)=\mathscr{C}_{l}(\varphi),\qquad
l\in\mathbb{N}_{0}.$

For $\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}^{M,h}_{K}$ we define

$\Phi=\Phi(\varphi_{0},\ldots,\varphi_{m-1})=\sum_{j=0}^{m-1}\sum_{k=0}^{m-1-j}Q_{j+k+1}(D_{x})D_{t}^{k}S(\varphi_{j}).$

Then, $\Phi\in\mathscr{D}(\mathbb{R}^{d+1})$ with
$\operatorname{supp}\,\Phi\subseteq K\times\mathbb{R}$. Moreover, (9) implies
that $\Phi$ satisfies $(ii)$. Next, we show $(i)$. In [12, Proposition 4.2] it
is shown that

$\sum_{j=0}^{n}\sum_{k=m-1-n}^{m-1-j}Q_{j+k+1}(D_{x})\mathscr{C}_{k+n}(\varphi_{j})=\varphi_{n},\qquad
n=0,\ldots,m-1.$

Combining this with (11) and the fact that $\mathscr{C}_{l}=0$ for $l<m-1$, we
obtain that for all $n=0,\ldots,m-1$

$\displaystyle D^{n}_{t}\Phi(\,\cdot\,,0)$
$\displaystyle=\sum_{j=0}^{m-1}\sum_{k=0}^{m-1-j}Q_{j+k+1}(D_{x})D_{t}^{k+n}S(\varphi_{j})(\,\cdot\,,0)$
$\displaystyle=\sum_{j=0}^{m-1}\sum_{k=0}^{m-1-j}Q_{j+k+1}(D_{x})\mathscr{C}_{k+n}(\varphi_{j})$
$\displaystyle=\sum_{j=0}^{n}\sum_{k=m-1-n}^{m-1-j}Q_{j+k+1}(D_{x})\mathscr{C}_{k+n}(\varphi_{j})=\varphi_{n}.$

We now show (9) and (10). Let $\alpha\in\mathbb{N}_{0}^{d}$ and
$l\in\mathbb{N}_{0}$ be arbitrary. We have that
$D^{\alpha}_{x}D^{l}_{t}S=S_{1}+S_{2}$, where

$S_{1}(x,t)=\sum_{n=0}^{l-1}\binom{l}{n}\sum_{p=0}^{\infty}\mathscr{C}_{p+n}(D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\lambda^{l-n}_{p+n}D^{l-n}_{t}\psi(\lambda_{p+n}t)$

and

$S_{2}(x,t)=\sum_{p=0}^{\infty}\mathscr{C}_{p+l}(D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\psi(\lambda_{p+l}t)$

We first prove (9). Note that

$\displaystyle P(D)S_{1}(x,t)$
$\displaystyle=\sum_{n=0}^{l-1}\binom{l}{n}\sum_{k=0}^{m}\sum_{j=0}^{k}\binom{k}{j}\sum_{p=0}^{\infty}\mathscr{C}_{p+j+n}\left(Q_{k}(D_{x})D^{\alpha}_{x}\varphi\right)(x)\frac{(it)^{p}}{p!}\times$
$\displaystyle\phantom{=}\lambda^{l-n+k-j}_{p+j+n}D^{l-n+k-j}_{t}\psi(\lambda_{p+j+n}t)$

and $P(D)S_{2}=T_{1}+T_{2}$, where

$T_{1}(x,t)=\sum_{k=0}^{m}\sum_{j=0}^{k-1}\binom{k}{j}\sum_{p=0}^{\infty}\mathscr{C}_{p+j+l}(Q_{k}(D_{x})D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\lambda^{k-j}_{p+j+l}D^{k-j}_{t}\psi(\lambda_{p+j+l}t)$

and

$T_{2}(x,t)=\sum_{p=0}^{\infty}\sum_{k=0}^{m}\mathscr{C}_{p+k+l}(Q_{k}(D_{x})D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\psi(\lambda_{p+k+l}t).$

In order to derive suitable estimates for the above expressions, we introduce
the auxiliary function

$\Gamma(\rho)=\min\left\\{p\in\mathbb{N}_{0}\,|\,m^{b_{0},*}_{p+1}\geq\frac{1}{\rho}\right\\},\qquad\rho>0.$

Fix $\rho>0$. For all $p\in\mathbb{N}$ we have that $p\leq\Gamma(\rho)$ if and
only if $\rho m_{p}^{b_{0},*}<1$. Hence, the sequence
$p\mapsto\rho^{p}M^{b_{0},*}_{p}$ is non-increasing for $p\leq\Gamma(\rho)$
and non-decreasing for $p\geq\Gamma(\rho)$. Consequently,
$\rho^{\Gamma(\rho)}M^{b_{0},*}_{\Gamma(\rho)}=e^{-\omega_{M^{b_{0},*}}\left(\frac{1}{\rho}\right)}$.
Furthermore, for all $t\in\mathbb{R}\backslash\\{0\\}$ and $p\in\mathbb{N}$ it
holds that $\psi(\lambda_{p}t)=1$ if $p<\Gamma(Ah^{b_{0}}|t|)$ and
$\psi(\lambda_{p}t)=0$ if $p\geq\Gamma(Ah^{b_{0}}|t|/2)$. For all
$x\in\mathbb{R}^{d}$ and $t\in\mathbb{R}\backslash\\{0\\}$ with $|t|$ small
enough it thus follows

$\displaystyle P(D)S_{1}(x,t)$
$\displaystyle=\sum_{n=0}^{l-1}\binom{l}{n}\sum_{k=0}^{m}\sum_{j=0}^{k}\binom{k}{j}\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+j+n}(Q_{k}(D_{x})D^{\alpha}_{x}\varphi)(x)\times$
$\displaystyle\phantom{=}\frac{(it)^{p}}{p!}\lambda^{l-n+k-j}_{p+j+n}D^{l-n+k-j}_{t}\psi(\lambda_{p+j+n}t),$
$\displaystyle T_{1}(x,t)$
$\displaystyle=\sum_{k=0}^{m}\sum_{j=0}^{k-1}\binom{k}{j}\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+j+l}(Q_{k}(D_{x})D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\times$
$\displaystyle\phantom{=}\lambda^{k-j}_{p+j+l}D^{k-j}_{t}\psi(\lambda_{p+j+l}t),$

and, by (7),

$T_{2}(x,t)=\sum_{k=0}^{m}\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+k+l}(Q_{k}(D_{x})D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\psi(\lambda_{p+k+l}t).$

In order to estimate the inner sums of $P(D)S_{1}$, $T_{1}$ and $T_{2}$, let
$n\leq l$, $k\leq m$ and $j\leq k$ be arbitrary. Lemma 4.3 and the inequality

(12) $(p+q)!\leq 2^{p+q}p!q!,\qquad p,q\in\mathbb{N}_{0},$

imply that for all $x\in\mathbb{R}^{d}$ and $t\in\mathbb{R}\backslash\\{0\\}$
with $|t|$ small enough

$\displaystyle\left|\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+j+n}(Q_{k}(D_{x})D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\lambda^{l-n+k-j}_{p+j+n}D^{l-n+k-j}_{t}\psi(\lambda_{p+j+n}t)\right|$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}(2L_{1}h^{b_{0}})^{p}M^{b_{0}}_{p+j+n}\lambda^{l-n+k-j}_{p+j+n}\frac{|t|^{p}}{(p+j+n)!}$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}(2L_{1}h^{b_{0}}|t|)^{p}M^{b_{0},*}_{p+l+k}$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{p=\Gamma(Ah^{b_{0}}|t|)-m-l}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\frac{1}{2^{p}}(Ah^{b_{0}}|t|/2)^{p}M^{b_{0},*}_{p}$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}(Ah^{b_{0}}|t|/2)^{\Gamma(Ah^{b_{0}}|t|)-m-l}M^{b_{0},*}_{\Gamma(Ah^{b_{0}}|t|)-m-l}$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}e^{-\omega_{M^{b_{0},*}}\left(\frac{1}{Ah^{b_{0}}|t|}\right)+(m+l)\log\left(\frac{1}{Ah^{b_{0}}|t|}\right)}$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}e^{-\frac{1}{2}\omega_{M^{b_{0},*}}\left(\frac{1}{Ah^{b_{0}}|t|}\right)}$
$\displaystyle\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}e^{-\omega_{M^{b_{0},*}}\left(\frac{1}{Lh^{b_{0}}|t|}\right)}.$

This implies (9). Finally, we show (10). For all $x\in\mathbb{R}^{d}$ and
$t\in\mathbb{R}\backslash\\{0\\}$ with $|t|$ small enough it holds that

$S_{1}(x,t)=\sum_{n=0}^{l-1}\binom{l}{n}\sum_{p=1}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+n}(D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\lambda^{l-n}_{p+n}D^{l-n}_{t}\psi(\lambda_{p+n}t)$

and

$S_{2}(x,t)-D^{\alpha}_{x}\mathscr{C}_{l}(\varphi)(x)=\sum_{p=1}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+l}(D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\psi(\lambda_{p+l}t).$

Let $n\leq l$ be arbitrary. Lemma 4.3 and (12) imply that for all
$x\in\mathbb{R}^{d}$ and $t\in\mathbb{R}\backslash\\{0\\}$ with $|t|$ small
enough

$\displaystyle\left|\sum_{p=1}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\mathscr{C}_{p+n}(D^{\alpha}_{x}\varphi)(x)\frac{(it)^{p}}{p!}\lambda^{l-n}_{p+n}D^{l-n}_{t}\psi(\lambda_{p+n}t)\right|$
$\displaystyle\leq$ $\displaystyle
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{p=1}^{\Gamma(Ah^{b_{0}}|t|/2)-1}(2L_{1}h^{b_{0}})^{p}M^{b_{0}}_{p+n}\lambda^{l-n}_{p+n}\frac{|t|^{p}}{(p+n)!}$
$\displaystyle\leq$ $\displaystyle
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{p=1}^{\Gamma(Ah^{b_{0}}|t|/2)-1}(2L_{1}h^{b_{0}})^{p}M^{b_{0},*}_{p+l}|t|^{p}$
$\displaystyle\leq$ $\displaystyle
C\|\varphi\|_{\mathscr{E}^{M,h}(K)}\sum_{p=1}^{\Gamma(Ah^{b_{0}}|t|/2)-1}\frac{1}{2^{p}}(Ah^{b_{0}}|t|/2)^{p}M^{b_{0},*}_{p}$
$\displaystyle\leq$ $\displaystyle|t|C\|\varphi\|_{\mathscr{E}^{M,h}(K)},$

which implies (10) and finishes the proof of the claim in case of
$p!^{1/b_{0}}\prec M$.

Next, we suppose that $p!^{1/b_{0}}\asymp M$. For
$\varphi\in\mathscr{D}^{M,h}_{K}$ we define

$S(x,t)=S(\varphi)(x,t)=\sum_{p=0}^{\infty}\mathscr{C}_{p}(\varphi)(x)\frac{(it)^{p}}{p!},\qquad(x,t)\in\mathbb{R}^{d+1}.$

Lemma 4.3 implies that there is $L>0$ (independent of $h>0$) such that the
series $S(x,t)$ converges in
$C^{\infty}(\mathbb{R}^{d}\times(-2/(Lh^{b_{0}}),2/(Lh^{b_{0}})))$ and that
for all $\alpha\in\mathbb{N}_{0}^{d}$ and $l\in\mathbb{N}_{0}$ there is $C>0$
such that

$\sup_{\begin{subarray}{c}x\in\mathbb{R}^{d}\\\ |t|\leq
2/(Lh^{b_{0}})\end{subarray}}|D^{\alpha}_{x}D^{l}_{t}S(\varphi)(x,t)|\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)},\qquad\varphi\in\mathscr{D}^{M,h}_{K}.$

Furthermore, $S$ satisfies (11) and, by (7), $P(D)S=0$ on
$\mathbb{R}^{d}\times(-2/(Lh^{b_{0}}),2/(Lh^{b_{0}}))$. Choose
$\chi\in\mathscr{D}(\mathbb{R})$ with
$\operatorname{supp}\,\chi\subset(-2/(Lh^{b_{0}}),2/(Lh^{b_{0}}))$ and such
that $\chi=1$ on $[-1/(Lh^{b_{0}}),1/(Lh^{b_{0}})]$. For
$\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}^{M,h}_{K}$ we define

$\Phi(x,t)=\Phi(\varphi_{0},\ldots,\varphi_{m-1})(x,t)=\chi(t)\sum_{j=0}^{m-1}\sum_{k=0}^{m-1-j}Q_{j+k+1}(D_{x})D_{t}^{k}S(\varphi_{j})(x).$

Then, $\Phi\in\mathscr{D}(\mathbb{R}^{d+1})$ with
$\operatorname{supp}\,\Phi\subset K\times\mathbb{R}$. Moreover, $P(D)\Phi=0$
on $\mathbb{R}^{d}\times(-1/(Lh^{b_{0}}),1/(Lh^{b_{0}}))$ and the exact same
argument as in the first part of the proof shows that $\Phi$ satisfies $(i)$.
∎

######  Remark 4.4.

Choose $\chi\in\mathscr{D}(\mathbb{R})$ such that $\chi=1$ on a neighbourhood
of $0$. Let $\Phi$ be the function constructed in Proposition 4.1. Set

$|||\Phi|||=\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}|P(D)(\chi(t)\Phi(x,t))|e^{\omega_{M^{b_{0},*}}\left(\frac{1}{Lh^{b_{0}}|t|}\right)}<\infty.$

An inspection of the proof of Proposition 4.1 shows that there is $C>0$ with

$|||\Phi(\varphi_{0},\ldots,\varphi_{m-1})|||\leq
C\max_{j=0,\ldots,m-1}\|\varphi_{j}\|_{\mathscr{E}^{M,h}(K)},\qquad\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}^{M,h}_{K}.$

######  Proposition 4.5.

Let $M$ be a weight sequence satisfying $(M.4)_{a_{0}}$. There is $L>0$ such
that for all $h>0$ the following property holds: Let
$\Phi\in\mathscr{D}(\mathbb{R}^{d+1})$ be such that

(13)
$\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}|P(D)\Phi(x,t)|e^{\omega_{M^{a_{0},*}}\left(\frac{1}{h|t|}\right)}<\infty.$

Set $K=\operatorname{supp}\,\Phi(\,\cdot\,,0)$. Then,
$D_{t}^{l}\Phi(\,\cdot\,,0)\in\mathscr{D}^{M,Lh^{1/a_{0}}}_{K}$ for all
$l\in\mathbb{N}_{0}$.

######  Remark 4.6.

We would like to point out that Proposition 4.5 is not needed to prove the
main results of this article in Section 5 and Section 6 below. However, we
believe this result is interesting in its own right as it provides a complete
characterization of ultradifferentiable classes in terms of (almost) zero
solutions of $P(D)$ (provided that $a_{0}=b_{0}$); see Theorem 4.8 and Remark
4.9 below.

We need the following lemma to prove Proposition 4.5.

######  Lemma 4.7.

Let $K\Subset\mathbb{R}^{d}$, $r>0$ and $V\subseteq\mathbb{R}^{d+1}$ be open
such that $K\times[-r,r]\Subset V$. Let $T\in\mathscr{D}^{\prime}(V)$ be such
that $T_{|V\backslash V\cap\mathbb{R}^{d}}\in
L^{1}_{\operatorname{loc}}(V\backslash V\cap\mathbb{R}^{d})$ and suppose that
there is $N\in\mathbb{N}$ with

(14) $\sup_{(x,t)\in K\times[-r,r]\backslash K}|T(x,t)||t|^{N}<\infty.$

Furthermore, let $l\in\mathbb{N}_{0}$ and $C>0$ be such that

(15) $|\langle T,\varphi\rangle|\leq
C\|\varphi\|_{l},\qquad\varphi\in\mathscr{D}_{K\times[-r,r]}.$

Let $\varphi\in\mathscr{D}_{K\times[-r,r]}$ be such that

(16)
$\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}\frac{|\varphi(x,t)|}{|t|^{\max\\{N,l+1\\}}}<\infty$

Then,

$\langle
T,\varphi\rangle=\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}T(x,t)\varphi(x,t)dxdt.$

###### Proof.

In view of (16), Taylor’s theorem yields that
$D^{n}_{t}\varphi(\,\cdot\,,0)=0$ for all $n\leq l$. Hence, also
$D^{\alpha}_{x}D^{n}_{t}\varphi(\,\cdot\,,0)=0$ for all
$\alpha\in\mathbb{N}^{d}$ and $n\leq l$. By another application of Taylor’s
theorem, we obtain that

(17)
$\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}\frac{|D^{\alpha}_{x}D^{n-k}_{t}\varphi(x,t)|}{|t|^{k+1}}<\infty,\qquad\alpha\in\mathbb{N}^{d},k\leq
n\leq l.$

Choose $\psi\in\mathscr{D}(\mathbb{R})$ such that $\psi=1$ on a neighbourhood
of $0$. Set $\psi_{\varepsilon}(t)=\psi(t/\varepsilon)$ for $\varepsilon>0$.
Property (17) implies that

$\lim_{\varepsilon\to 0^{+}}\|\psi_{\varepsilon}(t)\varphi(x,t)\|_{l}=0.$

Hence, by (15),

$\displaystyle\langle T,\varphi\rangle$ $\displaystyle=\lim_{\varepsilon\to
0^{+}}\langle T(x,t),(1-\psi_{\varepsilon}(t))\varphi(x,t)\rangle$
$\displaystyle=\lim_{\varepsilon\to
0^{+}}\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}T(x,t)(1-\psi_{\varepsilon}(t))\varphi(x,t)dxdt$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}T(x,t)\varphi(x,t)dxdt,$

where the last equality is justified by Lebesgue’s dominated convergence
theorem and the inequalities (14) and (16). ∎

###### Proof of Proposition 4.5.

By Lemma 2.8$(ii)$, we have that either $p!^{1/a_{0}}\prec M$ or
$p!^{1/a_{0}}\asymp M$. We only consider the case $p!^{1/a_{0}}\prec M$ as the
case $p!^{1/a_{0}}\asymp M$ can be treated similarly. Let $E$ be the
fundamental solution of $P(D)$ constructed in Proposition 3.1. Let
$l\in\mathbb{N}_{0}$ be arbitrary. Choose $r>0$ such that
$\operatorname{supp}\,\Phi\subset\mathbb{R}^{d}\times[-r,r]$. There are
$C,S>0$ such that

(18) $\sup_{\begin{subarray}{c}x\in\mathbb{R}^{d}\\\ 0<|t|\leq
r\end{subarray}}|D^{\alpha}_{x}D^{l}_{t}E(x,t)||t|^{|\alpha|/a_{0}+S}\leq
C{L^{|\alpha|}|\alpha|!^{1/a_{0}}},\qquad\alpha\in\mathbb{N}^{d}_{0}.$

Set

$p_{\alpha}=\left\lceil\frac{|\alpha|}{a_{0}}+S\right\rceil,\qquad\alpha\in\mathbb{N}_{0}^{d}.$

There are $C_{1},L_{1}>0$ such that

$|\alpha|!^{1/a_{0}}\leq
C_{1}L_{1}^{|\alpha|}p_{\alpha}!,\qquad\alpha\in\mathbb{N}_{0}^{d}.$

Lemma 2.10 and $(M.2)$ imply that there are $C_{2},L_{2}>0$ such that

$M_{p_{\alpha}}\leq
C_{2}L_{2}^{|\alpha|}M^{1/a_{0}}_{\alpha},\qquad\alpha\in\mathbb{N}_{0}^{d}.$

The above two inequalities and (13) yield that there are $C_{3},L_{3}>0$ (with
$L_{3}$ independent of $l$) such that

(19)
$\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}\frac{|P(D)\Phi(x,t)|}{|t|^{|\alpha|/a_{0}+S}}\leq\frac{C_{3}(L_{3}h^{1/a_{0}})^{|\alpha|}M_{\alpha}}{|\alpha|!^{1/a_{0}}},\qquad\alpha\in\mathbb{N}^{d}.$

Lemma 4.7 implies that for all $x\in\mathbb{R}^{d}$ and
$\alpha\in\mathbb{N}_{0}^{d}$

$\displaystyle D^{\alpha}_{x}D^{l}_{t}\Phi(x,0)$
$\displaystyle=(D^{\alpha}_{x}D^{l}_{t}E\ast P(D)\Phi)(x,0)$
$\displaystyle=\langle D^{\alpha}_{x}D^{l}_{t}E(x-y,-t),P(D)\Phi(y,t)\rangle$
$\displaystyle=\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}D^{\alpha}_{x}D^{l}_{t}E(x-y,-t)P(D)\Phi(y,t)dydt.$

The inequalities (18) and (19) imply that for all
$\alpha\in\mathbb{N}_{0}^{d}$

$\displaystyle\sup_{x\in\mathbb{R}^{d}}|D^{\alpha}_{x}D^{l}_{t}\Phi(x,0)|$
$\displaystyle\leq\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}|D^{\alpha}_{x}D^{l}_{t}E(x-y,-t)||P(D)\Phi(y,t)|dydt$
$\displaystyle\leq|\operatorname{supp}\,\Phi|CC_{3}(LL_{3}h^{1/a_{0}})^{|\alpha|}M_{\alpha}.$

∎

Proposition 4.1 and Proposition 4.5 yield the following result.

######  Theorem 4.8.

Suppose that $a_{0}=b_{0}$. Let $M$ be a weight sequence satisfying
$(M.4)_{a_{0}}$ (and $p!^{1/a_{0}}\prec M$ in the Beurling case). Let
$X\subseteq\mathbb{R}^{d}$ be open and let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. Let
$\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}(X)$. Then,
$\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}^{[M]}(X)$ if and only if for
all $h>0$ (for some $h>0$) there exists $\Phi\in\mathscr{D}(V)$ such that

  * $(i)$

$D^{j}_{t}\Phi(\,\cdot\,,0)=\varphi_{j}$ for $j=0,\ldots,m-1$.

  * $(ii)$

$\displaystyle\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}|P(D)\Phi(x,t)|e^{\omega_{M^{a_{0},*}}\left(\frac{1}{h|t|}\right)}<\infty$.

######  Remark 4.9.

For $M\asymp p!^{1/a_{0}}$ the following analogue of Theorem 4.8 holds in the
Beurling case: _Suppose that $a_{0}=b_{0}$. Let $X\subseteq\mathbb{R}^{d}$ be
open. Let $\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}(X)$. Then,
$\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}^{(M)}(X)$ if and only if for
all $h>0$ there exists $\Phi\in\mathscr{D}(X\times\mathbb{R})$ such that_

  * $(i)$

$D^{j}_{t}\Phi(\,\cdot\,,0)=\varphi_{j}$_for_ $j=0,\ldots,m-1$.

  * $(ii)$

$P(D)\Phi(x,t)=0$_on_ $X\times(-1/h,1/h)$.

We end this section by giving an analogue of Proposition 4.1 and Remark 4.4
for compactly supported smooth functions.

######  Proposition 4.10.

Let $K\Subset\mathbb{R}^{d}$ and let $\varepsilon>0$. Let
$N\in\mathbb{N}_{0}$. For all
$\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}_{K}$ there exists
$\Phi=\Phi(\varphi_{0},\ldots,\varphi_{m-1})\in\mathscr{D}_{K\times[-\varepsilon,\varepsilon]}$
such that

  * $(i)$

$D^{j}_{t}\Phi(\,\cdot\,,0)=\varphi_{j}$ for $j=0,\ldots,m-1$.

  * $(ii)$

$|||\Phi|||=\displaystyle\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}\frac{|P(D)\Phi(x,t)|}{|t|^{N}}<\infty$.

Furthermore, there are $l\in\mathbb{N}_{0}$ and $C>0$ such that

$|||\Phi(\varphi_{0},\ldots,\varphi_{m-1})|||\leq
C\max_{j=0,\ldots,m-1}\|\varphi_{j}\|_{l},\qquad\varphi_{0},\ldots,\varphi_{m-1}\in\mathscr{D}_{K}.$

###### Proof.

Let $\psi\in\mathscr{D}_{[-\varepsilon,\varepsilon]}$ be such that $\psi=1$ on
a neighbourhood of $0$. For $n\in\mathbb{N}$ and $\varphi\in\mathscr{D}_{K}$
we define

$S_{n}(x,t)=S_{n}(\varphi)(x,t)=\psi(t)\sum_{p=0}^{n}\mathscr{C}_{p}(\varphi)(x)\frac{(it)^{p}}{p!},\qquad(x,t)\in\mathbb{R}^{d+1}.$

The result can now be shown in a similar way as Proposition 4.1 but starting
from $S=S_{n}$ with $n$ large enough instead of the function defined in (8). ∎

##  5\. Boundary values of zero solutions of $P(D)$

In this section we show that zero solutions of $P(D)$ satisfying suitable
growth estimates near $\mathbb{R}^{d}$ have boundary values in a given
ultradistribution space. We start with the following fundamental definition.

######  Definition 5.1.

Let $M$ be a weight sequence. Let $X\subseteq\mathbb{R}^{d}$ be open and let
$V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$. The
_boundary value_ $\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$ of an
element $f\in C^{\infty}_{P}(V\backslash X)$ is defined as

(20) $\langle\operatorname{bv}(f),\varphi\rangle:=\lim_{t,s\to
0^{+}}\int_{X}(f(x,t)-f(x,-s))\varphi(x)dx,\qquad\varphi\in\mathscr{D}^{[M]}(X),$

provided that $\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$ exists.

Since $\mathscr{D}^{[M]}(X)$ is barrelled,
$\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$ exists if and only if the
limit in the right-hand side of (20) exists and is finite for all
$\varphi\in\mathscr{D}^{[M]}(X)$.

######  Lemma 5.2.

Let $M$ be a weight sequence. Let $X\subseteq\mathbb{R}^{d}$ be open and let
$V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$. Let
$f\in C^{\infty}_{P}(V\backslash X)$ be such that
$\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$ exists. Then,

  * $(i)$

$\operatorname{bv}(D^{\alpha}_{x}f)=D^{\alpha}_{x}\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$
exists for all $\alpha\in\mathbb{N}_{0}^{d}$.

  * $(ii)$

Assume additionally that $p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset
M$). Let $G(D_{x})$ be an ultradifferential operator of class $[M]$. Then,
$G(D_{x})f\in C^{\infty}_{P}(V\backslash X)$ and
$\operatorname{bv}(G(D_{x})f)=G(D_{x})\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$
exists.

###### Proof.

$(i)$ Obvious.

$(ii)$ The condition $p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset M$)
implies that $G(D_{x})$ is an ultradifferential operator of class
$\\{p!^{1/\gamma_{0}}\\}$. Since $f\in C^{\infty}_{P}(V\backslash
X)\subset\Gamma^{1/\gamma_{0},1/\mu_{0}}(V\backslash X)$ (see Subsection 2.1),
we have that $G(D_{x})f\in C^{\infty}(V\backslash X)$ and
$P(D)G(D_{x})f=G(D_{x})P(D)f=0$. Hence, $G(D_{x})f\in
C^{\infty}_{P}(V\backslash X)$. The second statement is clear. ∎

We now introduce various weighted spaces of zero solutions of $P(D)$. Let $M$
be a positive sequence. Suppose that $p!\prec M$. Let
$X\subseteq\mathbb{R}^{d}$ be open and let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. For $h>0$ we define
$\mathscr{B}_{P,M,h}(V\backslash X)$ as the Banach space consisting of all
$f\in C^{\infty}_{P}(V\backslash X)$ such that

$\|f\|_{\mathscr{B}_{P,M,h}(V\backslash X)}:=\sup_{(x,t)\in V\backslash
X}|f(x,t)|e^{-\omega_{M^{1,*}}\left(\frac{1}{h|t|}\right)}<\infty.$

Let $b>0$ and suppose that $p!^{1/b}\prec M$. Choose a sequence
$(V_{l})_{l\in\mathbb{N}_{0}}$ of relatively compact open subsets of $V$ such
that $\overline{V}_{l}\Subset V_{l+1}$ and
$V=\bigcup_{l\in\mathbb{N}_{0}}V_{l}$. Set $X_{l}=V_{l}\cap X$ for
$l\in\mathbb{N}_{0}$. We define

$\displaystyle C^{\infty}_{P,(M),b}(V\backslash
X):=\varprojlim_{l\in\mathbb{N}_{0}}\varinjlim_{h\to
0^{+}}\mathscr{B}_{P,M^{b},h}(V_{l}\backslash X_{l}),$ $\displaystyle
C^{\infty}_{P,\\{M\\},b}(V\backslash
X):=\varprojlim_{l\in\mathbb{N}_{0}}\varprojlim_{h\to\infty}\mathscr{B}_{P,M^{b},h}(V_{l}\backslash
X_{l}).$

This definition is independent of the chosen sequence
$(V_{l})_{l\in\mathbb{N}_{0}}$. Obviously, it holds that
$C^{\infty}_{P}(V)\subset C^{\infty}_{P,[M],b}(V\backslash X)$. If
$p!^{1/b}\asymp M$, we set $C^{\infty}_{P,\\{M\\},b}(V\backslash
X):=C^{\infty}_{P}(V\backslash X)$.

Next, we define for $j=1,\ldots,m$

$P_{(j)}(x,t)=\sum_{k=j}^{m}Q_{k}(x)t^{k-j}.$

Since $Q_{m}=1$, we have the recursion relations

(21) $t^{j}=\begin{cases}P_{(m)},&j=0,\\\
P_{(m-j)}-\sum_{k=0}^{j-1}Q_{k+m-j}t^{k},&j=1,\ldots,m-1.\\\ \end{cases}$

We are ready to discuss the existence of ultradistributional boundary values
of zero solutions of $P(D)$.

######  Proposition 5.3.

Let $M$ be a weight sequence satisfying $(M.4)_{b_{0}}$ (and
$p!^{1/b_{0}}\prec M$ in the Beurling case). Let $X\subseteq\mathbb{R}^{d}$ be
open and let $V\subseteq\mathbb{R}^{d+1}$ be open such that
$V\cap\mathbb{R}^{d}=X$. For all $j=0,\ldots,m-1$ the mapping

$C^{\infty}_{P,[M],b_{0}}(V\backslash
X)\rightarrow\mathscr{D}^{\prime[M]}(X),\,f\mapsto\operatorname{bv}(P_{(j+1)}(D)f)$

is well-defined and continuous.

We shall show Proposition 5.3 by combining the description of
ultradifferentiable functions via (almost) zero solutions of $P(D)$
(Proposition 4.1) with the identity obtained in the next lemma. As mentioned
in the introduction, this is a generalization of the method of establishing
the existence of (ultra)distributional boundary values of holomorphic
functions by combining almost analytic extensions with Stokes’ theorem (more
precisely, the formula in [10, Equation (3.1.9), p. 62]).

######  Lemma 5.4.

Let $X\subseteq\mathbb{R}^{d}$ be open and let $a,b\in\mathbb{R}$ with $a<b$.
For all $f\in C^{\infty}(X\times[a,b])$ and
$\Phi\in\mathscr{D}(X\times\mathbb{R})$ it holds that

$\displaystyle\int_{a}^{b}\int_{X}f(x,s)\check{P}(D)\Phi(x,s)dxds=\int_{a}^{b}\int_{X}P(D)f(x,s)\,\Phi(x,s)dxds$
$\displaystyle+i\sum_{j=0}^{m-1}(-1)^{j}\int_{X}P_{(j+1)}(D)f(x,b)D_{t}^{j}\Phi(x,b)-P_{(j+1)}(D)f(x,a)D_{t}^{j}\Phi(x,a)dx,$

where as usual $\check{P}(\xi)=P(-\xi)$.

###### Proof.

This follows by multiple partial integrations. ∎

###### Proof of Proposition 5.3.

We only show the Beurling case as the Roumieu case can be treated similarly.
Let $0\leq j\leq m-1$ be arbitrary. It suffices to show that for all
relatively compact open subsets $Y$ of $\mathbb{R}^{d}$, $r,h>0$ and $K\Subset
Y$ the mapping

$\mathscr{B}_{P,M^{b_{0}},h}(Y\times(-r,r)\backslash
Y)\rightarrow(\mathscr{D}^{(M)}_{K})^{\prime}_{b},\,f\mapsto\operatorname{bv}(P_{(j+1)}(D)f)$

is well-defined and continuous. Let
$f\in\mathscr{B}_{P,M^{b_{0}},h}(Y\times(-r,r)\backslash Y)$ be arbitrary. We
set $f_{t}(x,s)=f(x,s+t)$ for $0<t<r/2$. Then, $f_{t}\in
C^{\infty}(Y\times[0,r/2])$. For $\varphi\in\mathscr{D}^{(M)}_{K}$ we consider
the function $\Phi=\Phi(0,\ldots,\varphi_{j}=\varphi,\ldots,0)$ from
Proposition 4.1 (with $\check{P}$ and $(h/L)^{1/b_{0}}$ instead of $P$ and
$h$). Choose $\psi\in\mathscr{D}(\mathbb{R})$ with
$\operatorname{supp}\,\psi\subseteq[-r/2,r/2]$ and $\psi=1$ on a neighbourhood
of $0$. By applying Lemma 5.4 to $f_{t}$ and $\psi\Phi$, we obtain that for
$0<t<r/2$

$\int_{Y}P_{(j+1)}(D)f(x,t)\varphi(x)dx=(-1)^{j}i\int_{0}^{r/2}\int_{Y}f(x,s+t)\check{P}(D)(\psi(s)\Phi(x,s))dxds.$

Likewise, one may show that

$\int_{Y}P_{(j+1)}(D)f(x,-t)\varphi(x)dx=(-1)^{j+1}i\int_{-r/2}^{0}\int_{Y}f(x,s-t)\check{P}(D)(\psi(s)\Phi(x,s))dxds.$

Hence,

$\langle\operatorname{bv}(P_{(j+1)}(D)f),\varphi\rangle=(-1)^{j}i\int_{-r/2}^{r/2}\int_{Y}f(x,s)\check{P}(D)(\psi(s)\Phi(x,s))dxds,$

where the integral at the right hand-side is convergent by property $(ii)$ of
Proposition 4.1. The result now follows from Remark 4.4. ∎

######  Proposition 5.5.

Let $M$ be a weight sequence satisfying $(M.4)_{b_{0}}$ (and
$p!^{1/b_{0}}\prec M$ in the Beurling case). Let $X\subseteq\mathbb{R}^{d}$ be
open and let $V\subseteq\mathbb{R}^{d+1}$ be open such that
$V\cap\mathbb{R}^{d}=X$. For all $l\in\mathbb{N}_{0}$ the mapping

$C^{\infty}_{P,[M],b_{0}}(V\backslash
X)\rightarrow\mathscr{D}^{\prime[M]}(X),\,f\mapsto\operatorname{bv}(D^{l}_{t}f)$

is well-defined and continuous.

###### Proof.

For $l=0,\ldots,m-1$ this is a consequence of (21), Lemma 5.2 and Proposition
5.3. As $Q_{m}=1$, the result for $l\geq m$ then follows by using Lemma 5.2
recursively. ∎

Finally, we discuss the existence of distributional boundary values of zero
solutions of $P(D)$.

######  Definition 5.6.

Let $X\subseteq\mathbb{R}^{d}$ be open and let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. The boundary value
$\operatorname{bv}(f)\in\mathscr{D}^{\prime}(X)$ of an element $f\in
C^{\infty}_{P}(V\backslash X)$ is defined as

$\langle\operatorname{bv}(f),\varphi\rangle:=\lim_{t,s\to
0^{+}}\int_{X}(f(x,t)-f(x,-s))\varphi(x)dx,\qquad\varphi\in\mathscr{D}(X),$

provided that $\operatorname{bv}(f)\in\mathscr{D}^{\prime}(X)$ exists.

Let $V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$.
For $N\in\mathbb{N}_{0}$ we define $\mathscr{B}_{P,N}(V\backslash X)$ as the
Banach space consisting of all $f\in C^{\infty}_{P}(V\backslash X)$ such that

$\|f\|_{\mathscr{B}_{P,N}(V\backslash X)}:=\sup_{(x,t)\in V\backslash
X}|f(x,t)||t|^{N}<\infty.$

Choose a sequence $(V_{l})_{l\in\mathbb{N}_{0}}$ of relatively compact open
subsets of $V$ such that $\overline{V}_{l}\Subset V_{l+1}$ and
$V=\bigcup_{l\in\mathbb{N}_{0}}V_{l}$. Set $X_{l}=V_{l}\cap X$ for
$l\in\mathbb{N}_{0}$. We define

$C^{\infty}_{P,\operatorname{pol}}(V\backslash
X):=\varprojlim_{l\in\mathbb{N}_{0}}\varinjlim_{N\in\mathbb{N}_{0}}\mathscr{B}_{P,N}(V_{l}\backslash
X_{l})$

We then have:

######  Proposition 5.7.

_(cf.[21, Satz 2.4 and Satz 3.3])_ Let $X\subseteq\mathbb{R}^{d}$ be open and
let $V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$.
For all $l\in\mathbb{N}_{0}$ the mapping

$C^{\infty}_{P,\operatorname{pol}}(V\backslash
X)\rightarrow\mathscr{D}^{\prime}(X),\,f\mapsto\operatorname{bv}(D^{l}_{t}f)$

is well-defined and continuous.

###### Proof.

It suffices to show that for all $j=0,\ldots,m-1$ the mapping

$C^{\infty}_{P,\operatorname{pol}}(V\backslash
X)\rightarrow\mathscr{D}^{\prime}(X),\,f\mapsto\operatorname{bv}(P_{(j+1)}(D)f)$

is well-defined and continuous (cf. the proof of Proposition 5.5). This can be
shown in the same way as Proposition 5.3 but by using Proposition 4.10 instead
of Proposition 4.1 and Remark 4.4. ∎

##  6\. Main results

This section is devoted to the two main results of this article. Firstly, we
give various characterizations of zero solutions of $P(D)$ that admit a
boundary value in a given ultradistribution space. Secondly, we represent the
$m$-fold Cartesian product of an ultradistribution space as the quotient of
certain spaces of zero solutions of $P(D)$.

######  Definition 6.1.

Let $M$ be a weight sequence satisfying $(M.4)_{b_{0}}$ (and
$p!^{1/b_{0}}\prec M$ in the Beurling case). Let $X\subseteq\mathbb{R}^{d}$ be
open and let $V\subseteq\mathbb{R}^{d+1}$ be open such that
$V\cap\mathbb{R}^{d}=X$. We define

$\operatorname{bv}^{m}_{1}:C^{\infty}_{P,[M],b_{0}}(V\backslash
X)\rightarrow\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X),\,f\mapsto(\operatorname{bv}(P_{(j+1)}(D)f))_{0\leq
j\leq m-1}$

and

$\operatorname{bv}^{m}_{2}:C^{\infty}_{P,[M],b_{0}}(V\backslash
X)\rightarrow\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X),\,f\mapsto(\operatorname{bv}(D^{j}_{t}f))_{0\leq
j\leq m-1}.$

The mappings $\operatorname{bv}^{m}_{1}$ and $\operatorname{bv}^{m}_{2}$ are
well-defined and continuous by Proposition 5.3 and Proposition 5.5,
respectively.

We start by showing that $\ker\operatorname{bv}^{m}_{n}=C^{\infty}_{P}(V)$ for
$n=1,2$. The proof of the next result is inspired by Komatsu’s proof of the
Schwarz reflection principle for ultradistributions [20, Theorem 2.12].

######  Proposition 6.2.

Let $M$ be a weight sequence satisfying $(M.2)^{*}$ and
$p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset M$). Let
$X\subseteq\mathbb{R}^{d}$ be open and let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. Let $f\in C^{\infty}_{P}(V\backslash
X)$ be such that $\operatorname{bv}(D^{j}_{t}f)=0$ in
$\mathscr{D}^{\prime[M]}(X)$ for all $j=0,\ldots,m-1$. Then, $f\in
C^{\infty}_{P}(V)$. Consequently,
$\ker\operatorname{bv}^{m}_{n}=C^{\infty}_{P}(V)$ for $n=1,2$.

###### Proof.

It suffices to show that for every relatively compact open subset $Y$ of $X$
there is $r>0$ such that $f\in C^{\infty}_{P}(Y\times(-r,r))$. Set
$K=\overline{Y}$ and write $K_{\varepsilon}=K+\overline{B}(0,\varepsilon)$ for
$\varepsilon>0$. Choose $r,s>0$ such that $K_{2s}\times[-r,r]\Subset V$. We
claim that there is $h>0$ ($h\in\mathfrak{R}$) such that for all
$\varphi\in\mathscr{D}^{M,h}_{\overline{B}(0,s)}$

$f\ast_{x}\varphi(x,t)=\int_{\mathbb{R}^{d}}f(x-y,t)\varphi(y)dy,\qquad(x,t)\in
Y\times(-r,r)\backslash Y,$

belongs $C^{\infty}_{P}(Y\times(-r,r))$. Before we prove the claim, let us
show how it entails the result. By Lemma 2.13, there exist an
ultradifferential operator $G(D_{x})$ of class $[M]$,
$u_{1}\in\mathscr{D}^{M,h}_{\overline{B}(0,s)}$ and
$u_{2}\in\mathscr{D}^{[M]}_{\overline{B}(0,s)}$ such that
$G(D_{x})u_{1}=\delta+u_{2}$ in $\mathscr{D}^{\prime[M]}(\mathbb{R}^{d})$. The
claim yields that $f\ast_{x}u_{n}\in C^{\infty}_{P}(Y\times(-r,r))$ for
$n=1,2$. Since $p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset M$), we
have that $G(D_{x})(f\ast_{x}u_{1})\in C^{\infty}_{P}(Y\times(-r,r))$ (cf. the
proof of Lemma 5.2$(ii)$). Hence,

$f=G(D_{x})(f\ast_{x}u_{1})-f\ast_{x}u_{2}\in C^{\infty}_{P}(Y\times(-r,r)).$

We now show the claim. Let $H$ be the constant occurring in $(M.2)$. Our
assumption yields that there are $T_{j}\in\mathscr{D}^{\prime[M]}(X)$,
$j=0,\ldots,m-1$, such that

(22) $\lim_{t\to 0^{+}}D^{j}_{t}f(\,\cdot\,,t)=\lim_{t\to
0^{+}}D^{j}_{t}f(\,\cdot\,,-t)=T_{j}$

in the weak-$\ast$ topology on $(\mathscr{D}^{[M]}_{K_{2s}})^{\prime}$. Since
$\mathscr{D}^{[M]}_{K_{2s}}$ is an (FS)-space ((DFS)-space) and the set
$\\{D^{j}_{t}f(\,\cdot\,,t)\,|\,0<|t|<r,j=0,\ldots,m-1\\}$ is equicontinuous
in $(\mathscr{D}^{[M]}_{K_{2s}})^{\prime}$, we obtain that there is
$h^{\prime}>0$ ($h^{\prime}\in\mathfrak{R}$) such that (22) holds in the
topology of uniform convergence on the unit ball of
$\|\,\cdot\|_{\mathscr{E}^{M,h^{\prime}}(K_{2s})}$ (in the Roumieu case we
used Lemma 2.11$(ii)$). Furthermore, by Lemma 2.12, we may assume without loss
of generality that the weight sequence
$(\prod_{l=0}^{p}h^{\prime}_{l}M_{p})_{p\in\mathbb{N}_{0}}$ satisfies $(M.2)$
(with $2H$ instead of $H$). We obtain that there is a net $(c_{t})_{0<t<r}$ of
positive numbers tending to zero such that for all
$\psi\in\mathscr{D}^{[M]}_{K_{2s}}$ and $j=0,\ldots,m-1$

(23) $\left|\int_{\mathbb{R}^{d}}D^{j}_{t}f(x,t)\psi(x)dx-\langle
T_{j},\psi\rangle\right|\leq
c_{t}\|\psi\|_{\mathscr{E}^{M,h^{\prime}}(K_{2s})},\quad 0<|t|<r.$

By the Hahn-Banach theorem, we may extend $T_{j}$, $j=0,\ldots,m-1$, to a
continuous linear functional on $\mathscr{D}^{M,h^{\prime}}_{K_{2s}}$ (which
we still denote by $T_{j}$). We now show that (23) holds for all
$\psi\in\mathscr{D}^{M,h^{\prime}/H}_{K_{s}}$
$(\psi\in\mathscr{D}^{M,h^{\prime}/(2H)}_{K_{s}})$. Choose
$\chi\in\mathscr{D}^{[M]}_{\overline{B}(0,1)}$ with
$\int_{\mathbb{R}^{d}}\chi(x)dx=1$ and set
$\chi_{\varepsilon}(x)=\varepsilon^{-d}\chi(x/\varepsilon)$ for
$\varepsilon>0$. Since $\lim_{\varepsilon\to
0^{+}}\psi\ast\chi_{\varepsilon}=\psi$ in
$\mathscr{D}^{M,h^{\prime}}_{K_{2s}}$, (23) implies that

$\displaystyle\left|\int_{\mathbb{R}^{d}}D^{j}_{t}f(x,t)\psi(x)dx-\langle
T_{j},\psi\rangle\right|$ $\displaystyle=\lim_{\varepsilon\to
0^{+}}\left|\int_{\mathbb{R}^{d}}D^{j}_{t}f(x,t)\psi\ast\chi_{\varepsilon}(x)dx-\langle
T_{j},\psi\ast\chi_{\varepsilon}\rangle\right|$ $\displaystyle\leq
c_{t}\lim_{\varepsilon\to
0^{+}}\|\psi\ast\chi_{\varepsilon}\|_{\mathscr{E}^{M,h^{\prime}}(K_{2s})}=c_{t}\|\psi\|_{\mathscr{E}^{M,h^{\prime}}(K_{2s})}.$

Set $h=h^{\prime}/H^{2}$ ($h=h^{\prime}/(2H)^{2}$). We are ready to prove that
$f\ast_{x}\varphi\in C^{\infty}_{P}(Y\times(-r,r))$ for all
$\varphi\in\mathscr{D}^{M,h}_{\overline{B}(0,s)}$. It is clear that
$f\ast_{x}\varphi\in C^{\infty}_{P}(Y\times(-r,r)\backslash Y)$. Let
$\alpha\in\mathbb{N}_{0}^{d}$ be arbitrary. Since the set
$\\{D^{\alpha}_{x}\varphi(x-\,\cdot\,)\,|\,x\in Y\\}$ is bounded in
$\mathscr{D}^{M,h^{\prime}/H}_{K_{s}}$
$(\mathscr{D}^{M,h^{\prime}/(2H)}_{K_{s}})$, we infer from (23) that for
$j=0,\ldots,m-1$

$\lim_{t\to 0}D^{\alpha}_{x}D^{j}_{t}(f\ast_{x}\varphi)(x,t)=\lim_{t\to
0}\int_{\mathbb{R}^{d}}D^{j}_{t}f(y,t)D^{\alpha}_{x}\varphi(x-y)dy=T_{j}\ast
D^{\alpha}_{x}\varphi(x)$

uniformly for $x\in Y$. Hence, $f\ast_{x}\varphi$ is a function on
$Y\times(-r,r)$ that is $C^{\infty}$ in the $x$-direction and $C^{m-1}$ in the
$t$-direction. Since $f\ast_{x}\varphi\in
C^{\infty}_{P}(Y\times(-r,r)\backslash Y)$ and $Q_{m}=1$, we obtain that
$f\ast_{x}\varphi\in C^{\infty}_{P}(Y\times(-r,r))$. ∎

Proposition 6.2 implies the next result; it can also be shown directly and
easier by using the Schwartz parametrix method instead of Lemma 2.13.

######  Proposition 6.3.

_(cf.[21, Satz 1.5])_ Let $X\subseteq\mathbb{R}^{d}$ be open and let
$V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$. Let
$f\in C^{\infty}_{P}(V\backslash X)$ be such that
$\operatorname{bv}(D^{j}_{t}f)=0$ in $\mathscr{D}^{\prime}(X)$ for all
$j=0,\ldots,m-1$. Then, $f\in C^{\infty}_{P}(V)$.

Next, we show that $\prod_{j=0}^{m-1}\mathscr{E}^{\prime[M]}(X)$ is contained
in the range of $bv^{m}_{n}$ for $n=1,2$ (provided that $a_{0}=b_{0}$). It
suffices to consider the case $X=\mathbb{R}^{d}$.

######  Proposition 6.4.

Let $M$ be a weight sequence satisfying $(M.4)_{a_{0}}$ and
$p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset M$). Let $E$ be the
fundamental solution of $P(D)$ constructed in Proposition 3.1. For
$T_{0},\ldots,T_{m-1}\in\mathscr{E}^{\prime[M]}(\mathbb{R}^{d})$ we define

$\displaystyle f(x,t)=f(T_{0},\ldots,T_{m-1})(x,t)=-i\sum_{j=0}^{m-1}\langle
T_{j}(y),D^{j}_{t}E(x-y,t)\rangle,\quad(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}.$

Then, $f\in
C^{\infty}_{P,[M],a_{0}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$ and
$\operatorname{bv}^{m}_{1}(f)=(T_{j})_{0\leq j\leq m-1}$. Consequently, for
$n=1,2$ the following statement holds: For all
$T_{0},\ldots,T_{m-1}\in\mathscr{E}^{\prime[M]}(\mathbb{R}^{d})$ there is
$g_{n}\in C^{\infty}_{P,[M],a_{0}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$
such that $\operatorname{bv}^{m}_{n}(g_{n})=(T_{j})_{0\leq j\leq m-1}$.

We first prove the following distributional variant of Proposition 6.4.

######  Proposition 6.5.

Let $T_{0},\ldots,T_{m-1}\in\mathscr{E}^{\prime}(\mathbb{R}^{d})$. Then,
$f=f(T_{0},\ldots,T_{m-1})\in
C^{\infty}_{P,\operatorname{pol}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$
and $\operatorname{bv}(P_{(j+1)}(D)f)=T_{j}$ in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$ for $j=0,\ldots,m-1$.

###### Proof.

It is clear that $f\in
C^{\infty}_{P}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$. The bound (4)
implies that there is $N\in\mathbb{N}_{0}$ such that for all $r>0$

$\sup_{\mathbb{R}^{d}\times[-r,r]\backslash\mathbb{R}^{d}}|f(x,t)||t|^{N}<\infty.$

In particular, $f\in
C^{\infty}_{P,\operatorname{pol}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$.
We now show that $\operatorname{bv}(P_{(j+1)}(D)f)=T_{j}$ in
$\mathscr{D}^{\prime}(\mathbb{R}^{d})$ for $j=0,\ldots,m-1$. We define

$T=\left(-i\sum_{j=0}^{m-1}T_{j}\otimes D^{j}_{t}\delta\right)\ast
E\in\mathscr{D}^{\prime}(\mathbb{R}^{d+1}).$

Note that $T_{|\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}=f$. Let $0\leq j\leq
m-1$ and $\varphi\in\mathscr{D}(\mathbb{R}^{d})$ be arbitrary. Set
$K=\operatorname{supp}\,\varphi$. Choose $l\in\mathbb{N}$ and $C>0$ such that

$|\langle T,\psi\rangle|\leq
C\|\psi\|_{l},\qquad\psi\in\mathscr{D}_{K\times[-1,1]}.$

By Proposition 4.10 (with $\check{P}$ instead of $P$), there exists
$\Phi\in\mathscr{D}_{K\times[-1,1]}$ such that
$D^{l}_{t}\Phi(\,\cdot\,,0)=\delta_{l,j}\varphi$ for $l=0,\ldots,m-1$ (where
$\delta$ denotes the Kronecker delta) and

$\sup_{(x,t)\in\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}}\frac{|\check{P}(D)\Phi(x,t)|}{|t|^{\max\\{N,l+1\\}}}<\infty.$

By employing Lemma 5.4 in the same way as in the proof of Proposition 5.3, we
obtain that

$\langle\operatorname{bv}(P_{(j+1)}(D)f),\varphi\rangle=(-1)^{j}i\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}f(x,s)\check{P}(D)\Phi(x,s)dxds.$

Lemma 4.7 implies that

$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{R}^{d}}f(x,s)\check{P}(D)\Phi(x,s)dxds$
$\displaystyle=\left\langle\left(-i\sum_{k=0}^{m-1}T_{j}\otimes
D^{k}_{t}\delta\right)\ast E,\check{P}(D)\Phi\right\rangle$
$\displaystyle=-i\sum_{k=0}^{m-1}\langle T_{k}\otimes
D^{k}_{t}\delta,\Phi\rangle$ $\displaystyle=-i(-1)^{j}\langle
T_{j},\varphi\rangle,$

which yields the result. ∎

###### Proof of Proposition 6.4.

Since $\gamma_{0}\leq a_{0}$, we have that $p!^{1/a_{0}}\prec M$ in the
Beurling case. The bound (4) implies that for all $l\in\mathbb{N}_{0}$ it
holds that
$D^{l}_{t}E(\,\cdot\,,t)\in\mathscr{E}^{\\{p!^{1/a_{0}}\\}}(\mathbb{R}^{d})\subset\mathscr{E}^{[M]}(\mathbb{R}^{d})$
for $t\in\mathbb{R}\backslash\\{0\\}$ fixed. Hence, the dual pairing in the
definition of $f$ makes sense. It is clear that $f\in
C^{\infty}_{P}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$. We now show that
$f\in C^{\infty}_{P,[M],a_{0}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$. By
Lemma 2.8$(ii)$, we have that either $p!^{1/a_{0}}\prec M$ or
$p!^{1/a_{0}}\asymp M$. If $p!^{1/a_{0}}\asymp M$, by the definition of
$C^{\infty}_{P,[M],a_{0}}\left(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d}\right)$,
there is nothing left to show. Therefore, we assume that $p!^{1/a_{0}}\prec
M$. There is $K\Subset\mathbb{R}^{d}$ such that for some $h>0$ (for all $h>0$)
there is $C>0$ such that for all $j=0,\ldots,m-1$

$|\langle T_{j},\varphi\rangle|\leq
C\|\varphi\|_{\mathscr{E}^{M,h}(K)},\qquad\varphi\in\mathscr{E}^{[M]}(\mathbb{R}^{d}).$

The bound (4) yields that there are $C_{1},L_{1},S>0$ such that for all
$\alpha\in\mathbb{N}_{0}^{d}$ and $j=0,\ldots m-1$

$|D^{\alpha}_{x}D^{j}_{t}E(x,t)|\leq\frac{C_{1}L_{1}^{|\alpha|}|\alpha|!^{1/a_{0}}}{|t|^{|\alpha|/a_{0}+S}},\qquad
x\in\mathbb{R}^{d},0<|t|\leq 1.$

Furthermore, $(M.2)$ implies that there are $C_{2},L_{2}>0$ such that

$\frac{2}{a_{0}}\omega_{M^{a_{0},*}}(t)\leq\omega_{M^{a_{0},*}}(L_{2}t)+\log
C_{2},\qquad t\geq 0.$

For all $x\in\mathbb{R}^{d}$ and $t\in\mathbb{R}\backslash\\{0\\}$ with $|t|$
small enough it holds that

$\displaystyle|f(x,t)|\leq
C\sum_{j=0}^{m-1}\|D^{j}_{t}E(x-\,\cdot\,,t)\|_{\mathscr{E}^{M,h}(K)}\leq\frac{mCC_{1}}{|t|^{S}}\sup_{\alpha\in\mathbb{N}_{0}^{d}}\frac{L_{1}^{|\alpha|}|\alpha|!^{1/a_{0}}}{h^{|\alpha|}M_{\alpha}|t|^{|\alpha|/a_{0}}}$
$\displaystyle\leq\frac{mCC_{1}}{|t|^{S}}\sup_{\alpha\in\mathbb{N}_{0}^{d}}\left(\frac{(L_{1}/h)^{a_{0}|\alpha|}}{M^{a_{0},*}_{\alpha}|t|^{|\alpha|}}\right)^{1/a_{0}}=mCC_{1}e^{\frac{1}{a_{0}}\omega_{M^{a_{0},\ast}}\left(\frac{1}{(h/L_{1})^{a_{0}}|t|}\right)+S\log\left(\frac{1}{|t|}\right)}$
$\displaystyle\leq
mCC_{1}e^{\frac{2}{a_{0}}\omega_{M^{a_{0},\ast}}\left(\frac{1}{(h/L_{1})^{a_{0}}|t|}\right)}\leq
mCC_{1}C_{2}e^{\omega_{M^{a_{0},\ast}}\left(\frac{L_{2}}{(h/L_{1})^{a_{0}}|t|}\right)}$

for some $h>0$ (for all $h>0$). This shows that $f\in
C^{\infty}_{P,[M],a_{0}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$. Finally,
we prove that $\operatorname{bv}^{m}_{1}(f)=(T_{j})_{0\leq j\leq m-1}$. By
Lemma 2.13 (and Lemma 2.11$(i)$ in the Roumieu case), there are an
ultradifferential operator $G(D_{x})$ of class $[M]$ and $T_{q,j}\in
C_{c}(\mathbb{R}^{d})\subset\mathscr{E}^{\prime}(\mathbb{R}^{d})$ for $q=1,2$
and $j=0,\ldots,m-1$ such that $T_{j}=G(D_{x})T_{1,j}+T_{2,j}$ in
$\mathscr{D}^{\prime[M]}(\mathbb{R}^{d})$. Since

$f(T_{0},\ldots,T_{m})=G(D_{x})f(T_{1,0},\ldots,T_{1,m})+f(T_{2,0},\ldots,T_{2,m}),$

the result follows from Lemma 5.2 and Proposition 6.5. The result obviously
shows that for all
$T_{0},\ldots,T_{m-1}\in\mathscr{E}^{\prime[M]}(\mathbb{R}^{d})$ there is
$g\in C^{\infty}_{P,[M],a_{0}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$ such
that $\operatorname{bv}^{m}_{1}(g)=(T_{j})_{0\leq j\leq m-1}$. The
corresponding statement for $\operatorname{bv}^{m}_{2}$ follows from it by
(21). ∎

We are ready to prove the two main results of this article.

######  Theorem 6.6.

Suppose that $a_{0}=b_{0}$. Let $M$ be a weight sequence satisfying
$(M.4)_{a_{0}}$ and $p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset M$).
Let $X\subseteq\mathbb{R}^{d}$ be open and let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. Let $f\in C^{\infty}_{P}(V\backslash
X)$. The following statements are equivalent:

  * $(i)$

$f\in C^{\infty}_{P,[M],a_{0}}(V\backslash X)$.

  * $(ii)$

$\operatorname{bv}(D^{l}_{t}f)\in\mathscr{D}^{\prime[M]}(X)$ exists for all
$l\in\mathbb{N}_{0}$.

  * $(iii)$

$\operatorname{bv}(f)\in\mathscr{D}^{\prime[M]}(X)$ exists.

  * $(iv)$

For every relatively compact open subset $Y$ of $X$ there is $r>0$ such that
$\\{f(\,\cdot\,,t)\,|\,0<|t|<r\\}$ is bounded in $\mathscr{D}^{\prime[M]}(Y)$.

  * $(v)$

For every relatively compact open subset $Y$ of $X$ there is $r>0$ such that
$f=G(D_{x})f_{1}+f_{2}$ on $Y\times(-r,r)\backslash Y$, where $G(D_{x})$ is an
ultradifferential operator of class $[M]$ and $f_{q}\in
C^{\infty}_{P}(Y\times(-r,r)\backslash Y)\cap L^{\infty}(Y\times(-r,r))$ for
$q=1,2$.

###### Proof.

$(i)\Rightarrow(ii)$ This has been shown in Proposition 5.5.

$(ii)\Rightarrow(i)$ It suffices to show that for every relatively compact
open subset $Y$ of $X$ and $r>0$ with $\overline{Y}\times[-r,r]\Subset V$ it
holds that $f\in C^{\infty}_{P,[M],a_{0}}(Y\times(-r,r)\backslash Y)$. Choose
$\psi\in\mathscr{D}^{[M]}(X)$ such that $\psi=1$ on $Y$. Set
$T_{j}=\psi\operatorname{bv}(D^{j}_{t}f)\in\mathscr{E}^{\prime[M]}(\mathbb{R}^{d})$
for $j=0,\ldots,m-1$. By Proposition 6.4, there is $g\in
C^{\infty}_{P,[M],a_{0}}(\mathbb{R}^{d+1}\backslash\mathbb{R}^{d})$ such that
$\operatorname{bv}(D^{j}_{t}g)=T_{j}$ in
$\mathscr{D}^{\prime[M]}(\mathbb{R}^{d})$ and thus
$\operatorname{bv}(D^{j}_{t}g)=T_{j}=\operatorname{bv}(D^{j}_{t}f)$ in
$\mathscr{D}^{\prime[M]}(Y)$ for $j=0,\ldots,m-1$. Proposition 6.2 implies
that $f-g\in C^{\infty}_{P}(Y\times(-r,r))$, whence $f\in
C^{\infty}_{P,[M],a_{0}}(Y\times(-r,r)\backslash Y)$.

$(ii)\Rightarrow(iii)$ _and_ $(iii)\Rightarrow(iv)$ Obvious.

$(iv)\Rightarrow(v)$ This can be shown by using the parametrix method
(Proposition 2.13) in a similar way as in the proof of Proposition 6.2.

$(v)\Rightarrow(ii)$ This follows from Lemma 5.2$(ii)$ and Proposition 5.5 as
obviously $f_{q}\in C_{P,[M],a_{0}}^{\infty}(Y\times(-r,r)\backslash Y)$ for
$q=1,2$. ∎

######  Theorem 6.7.

Suppose that $a_{0}=b_{0}$. Let $M$ be a weight sequence satisfying
$(M.4)_{a_{0}}$ and $p!^{1/\gamma_{0}}\prec M$ ($p!^{1/\gamma_{0}}\subset M$).
Let $X\subseteq\mathbb{R}^{d}$ be open and let $V\subseteq\mathbb{R}^{d+1}$ be
open such that $V\cap\mathbb{R}^{d}=X$. Suppose that $V$ is $P$-convex for
supports. For $n=1,2$ the sequence

$0\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}_{n}\phantom{,}}}C^{\infty}_{P}(V)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}_{i}\phantom{,}}}C^{\infty}_{P,[M],a_{0}}(V\backslash
X)\xrightarrow{\phantom{,}\operatorname{bv}^{m}_{n}\phantom{,}}\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}_{n}\phantom{,}}}0$

is exact and $\operatorname{bv}^{m}_{n}$ is a topological homomorphism.

###### Proof.

The mappings $\operatorname{bv}^{m}_{1}$ and $\operatorname{bv}^{m}_{2}$ are
continuous by Proposition 5.3 and Proposition 5.5, respectively. Proposition
6.2 yields that $\ker\operatorname{bv}^{m}_{n}=C^{\infty}_{P}(V)$. Next, we
show that $\operatorname{bv}^{m}_{n}$ is surjective. To this end, we shall use
some basic facts about the derived projective limit functor; we refer to the
book [34] for more information. Choose a sequence
$(V_{l})_{l\in\mathbb{N}_{0}}$ of relatively compact open subsets of $V$ such
that $\overline{V}_{l}\Subset V_{l+1}$ and
$V=\bigcup_{l\in\mathbb{N}_{0}}V_{l}$. Set $X_{l}=V_{l}\cap X$ and
$\operatorname{bv}^{m}_{n,l}=\operatorname{bv}^{m}_{n}:C^{\infty}_{P,[M],a_{0}}(V_{l}\backslash
X_{l})\rightarrow\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X_{l})$ for
$l\in\mathbb{N}_{0}$. We define the projective spectra

$\mathscr{X}=(C^{\infty}_{P}(V_{l}))_{l\in\mathbb{N}_{0}},\quad\mathscr{Y}=(C^{\infty}_{P,[M],a_{0}}(V_{l}\backslash
X_{l}))_{l\in\mathbb{N}_{0}},\quad\mathscr{Z}=\left(\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X_{l})\right)_{l\in\mathbb{N}_{0}},$

(with restriction as spectral mappings) and the morphism
$(\operatorname{bv}^{m}_{n,l})_{l\in\mathbb{N}_{0}}:\mathscr{Y}\rightarrow\mathscr{Z}$.
We need to show that the mapping

$\operatorname{bv}^{m}_{n}=\operatorname{Proj}\,(\operatorname{bv}^{m}_{n,l})_{l\in\mathbb{N}_{0}}:C^{\infty}_{P,[M],a_{0}}(V\backslash
X)=\operatorname{Proj}\,\mathscr{Y}\rightarrow\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X)=\operatorname{Proj}\,\mathscr{Z}$

is surjective. Proposition 6.2 yields that
$\mathscr{X}=\ker(\operatorname{bv}^{m}_{n,l})_{l\in\mathbb{N}_{0}}$. By [34,
Proposition 3.1.8], it suffices to show:

  * $(i)$

For all $l\in\mathbb{N}_{0}$ and
$T_{0},\ldots,T_{m-1}\in\mathscr{D}^{\prime[M]}(X_{l+1})$ there is $f\in
C^{\infty}_{P,[M],a_{0}}(V_{l}\backslash X_{l})$ such that
$\operatorname{bv}^{m}_{n,l}(f)=(T_{0|X_{l}},\ldots T_{m-1|X_{l}})$.

  * $(ii)$

$\operatorname{Proj}^{1}\,\mathscr{X}=0$.

Property $(i)$ is a consequence of Proposition 6.4 (cf. the proof of
$(ii)\Rightarrow(i)$ in Theorem 6.6), whereas $(ii)$ follows from the fact
that $V$ is $P$-convex for supports [34, Section 3.4.4]. Finally, since
$C^{\infty}_{P,[M],a_{0}}(V\backslash X)$ is webbed and
$\prod_{j=0}^{m-1}\mathscr{D}^{\prime[M]}(X)$ is ultrabornological,
$\operatorname{bv}^{m}_{n}$ is a topological homomorphism by the open mapping
theorem of De Wilde. ∎

We end this section by giving the analogues of Theorem 6.6 and Theorem 6.7 for
distributions.

######  Theorem 6.8.

_(cf.[21, Satz 2.4])_ Let $X\subseteq\mathbb{R}^{d}$ be open and let
$V\subseteq\mathbb{R}^{d+1}$ be open such that $V\cap\mathbb{R}^{d}=X$. Let
$f\in C^{\infty}_{P}(V\backslash X)$. The following statements are equivalent:

  * $(i)$

$f\in C^{\infty}_{P,\operatorname{pol}}(V\backslash X)$.

  * $(ii)$

$\operatorname{bv}(D^{l}_{t}f)\in\mathscr{D}^{\prime}(X)$ exists for all
$l\in\mathbb{N}_{0}$.

  * $(iii)$

$\operatorname{bv}(f)\in\mathscr{D}^{\prime}(X)$ exists.

  * $(iv)$

For every relatively compact open subset $Y$ of $X$ there is $r>0$ such that
$\\{f(\,\cdot\,,t)\,|\,0<|t|<r\\}$ is bounded in $\mathscr{D}^{\prime}(Y)$.

  * $(v)$

For every relatively compact open subset $Y$ of $X$ there is
$l\in\mathbb{N}_{0}$ such that $f=\Delta^{l}_{x}f_{1}+f_{2}$ on
$Y\times(-r,r)\backslash Y$, where $f_{q}\in
C^{\infty}_{P}(Y\times(-r,r)\backslash Y)\cap L^{\infty}(Y\times(-r,r))$ for
$q=1,2$.

###### Proof.

This can be shown in a similar way as Theorem 6.6. ∎

The mappings $\operatorname{bv}^{m}_{1}$ and $\operatorname{bv}^{m}_{2}$ for
distributions are defined similarly as in Definition 6.1. We then have:

######  Theorem 6.9.

_(cf.[21, Satz 3.3 and the subsequent remark])_ Let $X\subseteq\mathbb{R}^{d}$
be open and let $V\subseteq\mathbb{R}^{d+1}$ be open such that
$V\cap\mathbb{R}^{d}=X$. Suppose that $V$ is $P$-convex for supports. For
$n=1,2$ the sequence

$0\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}_{n}\phantom{,}}}C^{\infty}_{P}(V)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}_{i}\phantom{,}}}C^{\infty}_{P,\operatorname{pol}}(V\backslash
X)\xrightarrow{\phantom{,}\operatorname{bv}^{m}_{n}\phantom{,}}\prod_{j=0}^{m-1}\mathscr{D}^{\prime}(X)\xrightarrow{\phantom{\phantom{,}\operatorname{bv}^{m}_{n}\phantom{,}}}0$

is exact and $\operatorname{bv}^{m}_{n}$ is a topological homomorphism.

###### Proof.

This can be shown in a similar way as Theorem 6.7. ∎

##  7\. Semi-elliptic polynomials

In this section we discuss the assumptions of our main results (Theorem 6.6
and Theorem 6.7) for semi-elliptic polynomials. For
$\beta\in\mathbb{N}_{0}^{d+1}$ and ${\bf{n}}\in\mathbb{N}^{d+1}$ we write
$|\beta:{\bf{n}}|:=\sum_{j=1}^{d+1}\frac{\beta_{j}}{n_{j}}$.

######  Definition 7.1.

A polynomial $P\in\mathbb{C}[\xi_{1},\ldots,\xi_{d+1}]$ is called semi-
elliptic if there is ${\bf{n}}\in\mathbb{N}^{d+1}$ such that $P$ can be
written as

$P(\xi)=\sum_{|\beta:{\bf{n}}|\leq 1}c_{\beta}\xi^{\beta}$

and

$P^{0}(\xi):=\sum_{|\beta:{\bf{n}}|=1}c_{\beta}\xi^{\beta}\neq 0$

for all $\xi\in\mathbb{R}^{d+1}\backslash\\{0\\}$.

If $P$ is semi-elliptic, the multi-index ${\bf{n}}$ in the above definition is
unique. More precisely, $n_{j}=\operatorname{deg}_{\xi_{j}}P$,
$j=1,\ldots,d+1$[31, Lemma 7.14]. Furthermore, $\deg
P=\max\\{\operatorname{deg}_{x_{j}}P\,|\,1\leq j\leq d+1\\}$ [7, Proposition
2]. It is well-known that semi-elliptic polynomials are hypoelliptic; see [11,
Theorem 11.1.11] and [31, Theorem 7.7].

We now determine the values $a_{0}(P)$, $b_{0}(P)$ and $\gamma_{0}(P)$ for
semi-elliptic polynomials $P$.

######  Proposition 7.2.

Let $P(x,t)=\sum_{k=0}^{m}Q_{k}(x)t^{k}$ be semi-elliptic. Then,

$\displaystyle a_{0}(P)=\frac{\min\\{\operatorname{deg}_{x_{j}}P\,|\,1\leq
j\leq d\\}}{m},$ $\displaystyle
b_{0}(P)=\frac{\max\\{\operatorname{deg}_{x_{j}}P\,|\,1\leq j\leq
d\\}}{m}=\frac{\deg Q_{0}}{m}$
$\displaystyle\gamma_{0}(P)=\frac{\min\\{\operatorname{deg}_{x_{j}}P\,|\,1\leq
j\leq d\\}}{\operatorname{deg}P},$

###### Proof.

The result for $\gamma_{0}$ follows from [11, Theorem 11.4.15] (see also [31,
Theorem 7.7]). Set $n_{j}=\deg_{x_{j}}P$, $j=1,\ldots,d$, $n_{d+1}=m$ and
${\bf{n}}=(n_{1},\ldots,n_{d+1})$. Then, $P(\xi)=\sum_{|\beta:{\bf{n}}|\leq
1}c_{\beta}\xi^{\beta}$, $\xi\in\mathbb{R}^{d+1}$, for suitable
$c_{\beta}\in\mathbb{C}$ and
$P^{0}(\xi)=\sum_{|\beta:{\bf{n}}|=1}c_{\beta}\xi^{\beta}\neq 0$ for all
$\xi\in\mathbb{R}^{d+1}\backslash\\{0\\}$. We define
$|\xi|_{{\bf{n}}}=\sum_{j=1}^{d+1}|\xi_{j}|^{n_{j}},\xi\in\mathbb{R}^{d+1}$.
Note that for all $\beta\in\mathbb{N}_{0}^{d+1}$

(24)
$\,|\xi^{\beta}|\leq|\xi|_{{\bf{n}}}^{|\beta:{\bf{n}}|},\qquad\xi\in\mathbb{R}^{d+1}.$

Furthermore, there are $C,R\geq 1$ such that for all
$\beta\in\mathbb{N}_{0}^{d+1}$

(25) $|\xi|_{{\bf{n}}}^{|\beta:{\bf{n}}|}|D^{\beta}P(\xi)|\leq
C|P(\xi)|,\qquad|\xi|\geq R,$

as essentially follows from (24) and the fact that $P$ is semi-elliptic; see
the proof of [31, Theorem 7.7] for details. We now determine $a_{0}$. For all
$x\in\mathbb{R}^{d}$ with $|x|\geq R$, $t\in\mathbb{R}$ and
$l\in\mathbb{N}_{0}$ it holds that

$\displaystyle|(x,t)|_{{\bf{n}}}^{|(0,\ldots,0,l):{\bf{n}}|}$
$\displaystyle\geq\left(\sum_{j=1}^{d}|x_{j}|^{n_{j}}\right)^{\frac{l}{m}}\geq\left(\sum_{j=1}^{d}\left(\frac{|x_{j}|}{|x|}\right)^{n_{j}}|x|^{\min\\{n_{j}\,|\,1\leq
j\leq d\\}}\right)^{\frac{l}{m}}$
$\displaystyle\geq\inf_{y\in\mathbb{R}^{d},|y|=1}\left(\sum_{j=1}^{d}|y_{j}|^{n_{j}}\right)^{\frac{l}{m}}|x|^{l\frac{\min\\{n_{j}\,|\,1\leq
j\leq d\\}}{m}},$

which, by (25), shows that $a_{0}\geq\min\\{n_{j}\,|\,1\leq j\leq d\\}/m$. For
the converse inequality, note that

(26) $Q_{0}(x)=\sum_{|(\alpha,0):{\bf{n}}|\leq 1}c_{(\alpha,0)}x^{\alpha}.$

Hence, (24) implies that

$|Q_{0}(x)|\leq C_{0}\sum_{j=1}^{d}|x_{j}|^{n_{j}},\qquad
x\in\mathbb{R}^{d},|x|\geq 1,$

for some $C_{0}>0$. Inequality (25) particularly shows that

$|\xi|_{{\bf{n}}}\leq C_{1}|P(\xi)|,\qquad|\xi|\geq R,$

for some $C_{1},R>0$. The above two inequalities together with (2) yield that
for $x\in\mathbb{R}^{d}$ with $|x|$ sufficiently large

$\sum_{j=1}^{d}|x_{j}|^{n_{j}}+|x|^{a_{0}m}=|(x,|x|^{a_{0}})|_{{\bf{n}}}\leq
C_{1}|P(x,|x|^{a_{0}})|\leq C\sum_{j=1}^{d}|x_{j}|^{n_{j}}$

for some $C>0$, which implies that $a_{0}\leq\min\\{n_{j}\,|\,1\leq j\leq
d\\}/m$. This proves the result for $a_{0}$. Next, we determine $b_{0}$. Let
$0\leq k\leq m-1$ with $Q_{k}\neq 0$ be arbitrary. Then, there is
$\alpha\in\mathbb{N}^{d}$ with $|\alpha|=\deg Q_{k}$ such that
$c_{(\alpha,k)}\neq 0$ and, thus, $|(\alpha,k):{\bf{n}}|\leq 1$. Hence,

$1\geq\sum_{j=1}^{d}\frac{\alpha_{j}}{n_{j}}+\frac{k}{m}\geq\frac{\deg
Q_{k}}{\max\\{n_{j}\,|\,1\leq j\leq d\\}}+\frac{k}{m}.$

This implies that $\operatorname{deg}Q_{k}/(m-k)\leq\max\\{n_{j}\,|\,1\leq
j\leq d\\}/m$ and, thus, $b_{0}\leq\max\\{n_{j}\,|\,1\leq j\leq d\\}/m$. Let
$1\leq j\leq d$ be arbitrary. Consider
$\alpha^{j}=(0\ldots,\alpha^{j}_{j}=n_{j},\ldots 0)\ \in\mathbb{N}^{d}$. Then,
$c_{(\alpha^{j},0)}\neq 0$ for otherwise $(\delta_{l,j})_{1\leq l\leq d+1}\neq
0$ would be a zero of $P^{0}$. By (26), we obtain that
$n_{j}=\deg_{x_{j}}Q_{0}$. Since $Q_{0}$ is semi-elliptic (on
$\mathbb{R}^{d}$) as $P$ is so, we have that
$\operatorname{deg}Q_{0}=\max\\{\deg_{x_{j}}Q_{0}\,|\,1\leq j\leq d\\}$.
Hence, $\max\\{n_{j}\,|\,1\leq j\leq d\\}/m=\deg Q_{0}/m$. Clearly, $\deg
Q_{0}/m\leq b_{0}$. This proves the result for $b_{0}$. ∎

######  Remark 7.3.

Let $P(x,t)=\sum_{k=0}^{m}Q_{k}(x)t^{k}$ be semi-elliptic. By [11, Theorem
11.4.15], we have that $\mu_{0}(P)=m/\operatorname{deg}P$. Since the exact
value of $\mu_{0}(P)$ is not important to us, we do not consider it in what
follows.

######  Corollary 7.4.

Let $P(x,t)=\sum_{k=0}^{m}Q_{k}(x)t^{k}$ be a semi-elliptic polynomial with
$\operatorname{deg}_{x_{1}}P=\cdots=\operatorname{deg}_{x_{d}}P=n$. Then,

$a_{0}(P)=b_{0}(P)=\frac{n}{m}=\frac{\deg
Q_{0}}{m}\qquad\mbox{and}\qquad\gamma_{0}(P)=\min\left\\{1,\frac{n}{m}\right\\}.$

Next, given a semi-elliptic polynomial $P$, we present a sufficient geometric
condition on an open set $V\subseteq\mathbb{R}^{d+1}$ to be $P$-convex for
supports [7, 13]. A function $f:V\rightarrow\mathbb{R}$ is said to satisfy the
minimum principle in a closed subset $F$ of $\mathbb{R}^{d+1}$ if for all
$K\Subset F\cap V$ it holds that

$\min_{x\in K}f(x)=\min_{x\in\partial_{F}K}f(x),$

where $\partial_{F}K$ denotes the boundary of $K$ in $F$.

######  Proposition 7.5.

( [13, Theorem 1] _and_ [7, Proposition 2$(iii)$]) Let $P$ be a semi-elliptic
polynomial on $\mathbb{R}^{d+1}$. Then, an open set
$V\subseteq\mathbb{R}^{d+1}$ is $P$-convex for supports if its boundary
distance function

$d_{V}\,:\,V\rightarrow\mathbb{R},\,\xi\mapsto
d(\xi,\mathbb{R}^{d+1}\backslash V)$

satisfies the minimum principle in every affine subspace parallel to the
subspace

$\\{\xi\in\mathbb{R}^{d+1}|\xi_{j}=0\mbox{ for all }1\leq j\leq d+1\mbox{ with
}\operatorname{deg}_{\xi_{j}}P<\operatorname{deg}P\\},$

where $d(\xi,\mathbb{R}^{d+1}\backslash
V)=\inf\\{|\xi-\eta|;\,\eta\in\mathbb{R}^{d+1}\backslash V\\}$.

Corollary 7.4 implies that Theorem 6.6 and Theorem 6.7 are applicable to every
semi-elliptic polynomial with
$\operatorname{deg}_{x_{1}}P=\cdots=\operatorname{deg}_{x_{d}}P=n$. Moreover,
Proposition 7.5 gives a sufficient condition on the open sets
$V\subset\mathbb{R}^{d+1}$ for which Theorem 6.7 is valid. We now discuss this
class of polynomials in some more detail. We distinguish three cases:

$(i)$ $m=n=\deg P$: Then, $P$ is elliptic and, conversely, every elliptic
polynomial is of this form. Corollary 7.4 implies that

$a_{0}(P)=b_{0}(P)=\gamma_{0}(P)=1,$

while Proposition 7.5 yields the well-known fact that every open set
$V\subseteq\mathbb{R}^{d+1}$ is $P$-convex for supports.

$(ii)$ $m<n=\deg P$: The heat operator and, more generally, the $k$-parabolic
operators in the sense of Petrowsky [31, Definition 7.11] are of this form.
Corollary 7.4 implies that

$a_{0}(P)=b_{0}(P)=n/m\qquad\mbox{and}\qquad\gamma_{0}(P)=1.$

By Proposition 7.5, we have that an open set $V\subseteq\mathbb{R}^{d+1}$ is
$P$-convex for supports if $d_{V}$ satisfies the minimum principle in every
characteristic hyperplane, i.e., in every hyperplane of the form
$\\{(x,\tau)|\,x\in\mathbb{R}^{d}\\},\tau\in\mathbb{R}$. We mention that, by
[13, Corollary 5] (see also [11, Theorem 10.8.1]), the converse holds true as
well in this case. In particular, all open sets of the form $V=X\times I$,
$X\subseteq\mathbb{R}^{d}$ and $I\subseteq\mathbb{R}$ open, are $P$-convex for
supports.

$(iii)$ $m=\deg P>n$: Corollary 7.4 implies that

$a_{0}(P)=b_{0}(P)=\gamma_{0}(P)=n/m.$

By Proposition 7.5, we have that an open set $V\subseteq\mathbb{R}^{d+1}$ is
$P$-convex for supports if $d_{V}$ satisfies the minimum principle in every
line $\\{(x,\tau)|\,\tau\in\mathbb{R}\\},x\in\mathbb{R}^{d}$. Again, all open
sets of the form $V=X\times I$, $X\subseteq\mathbb{R}^{d}$ and
$I\subseteq\mathbb{R}$ open, are $P$-convex for supports.

Finally, note that Theorem 1.1 stated in the introduction follows from Theorem
6.6, Theorem 6.7 and the above remarks.

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