# On Tempered Ultradistributions in Classical Sobolev Spaces

A. U. Amaonyeiro1 and M.E. Egwe2∗
Department of Mathematics, University, Ibadan, Nigeria
<EMAIL_ADDRESS><EMAIL_ADDRESS>
* Corresponding author

###### Abstract

We study the inclusion of tempered ultradistributions (or functions of slow
growth) in the notion of classical Sobolev spaces. We investigate basically
the properties of tempered ultradistribution spaces in Sobolev spaces. Our new
Sobolev space preserving the original properties and condition whose
derivatives are linear continuous operators embedding in $L^{p}$ for $1\leq
p\leq\infty$ is characterized. Moreover, we also consider some Sobolev
embedding theorems involving rapidly decreasing functions, and finally, we
prove the extension of Rellich’s compactness theorem.

keywords:Tempered ultradistributions, Classical Sobolev spaces, Embedding,
Compactness

Mathematics Subject Classification(2020): 46F10, 46E10, 46F05, 46F20, 46F25

##  1 Introduction

We conduct a survey on the extension of a special type of generalized
functions called the class of tempered ultradistributions in the classical
setting of Sobolev spaces.
The notion of classical Sobolev spaces $\mathcal{W}^{l,p}$ was introduced by
S. L. Sobolev in the 1930s basically developed for the framework of modern
theory of partial differential equations. The development of these spaces has
a foundation in the theory of generalized functions. For details about the
spaces, refer to [1], [2] and [3]. This theory was facilitated by the
existence of embedding between many of the function spaces like the
$L^{p}$-spaces.
In [3] one sees the space $\mathcal{W}^{l,p}$ with constant exponent $p$ as
the space of all functions in $L^{p}$ whose distributional partial derivatives
belong to $L^{p}$, that is, for all $j\in\mathbb{N}^{n}$ there are
$\psi_{j}\in L^{p}$ such that

$\int_{\Omega}\mu\frac{\partial^{\alpha}\eta}{\partial
y_{j}^{\alpha}}dy=(-1)^{|\alpha|}\int_{\Omega}\psi_{j}\eta
dy\quad\forall\quad\eta\in C_{c}^{\infty}\quad\text{for}\quad p\in[1,\infty]$

In this regard, $\psi_{j}$ is viewed as the weak derivative or the
distributional derivative of $\mu\in L^{p}$. The work on Sobolev space was
extended using the defining functions as tempered distributions in [4]. In the
work, it is defined that a tempered distribution $f\in\mathcal{S}$ belongs to
the Sobolev space $\mathcal{W}^{l,p}\equiv\mathcal{W}^{l,p}(\mathbb{R}^{n})$,
$l\in\mathbb{R}$ if $\hat{\mu}(x)$ is a function such that the following
condition is satisfied:

$\int|\hat{\mu}(x)|^{2}(1+|x|^{2})^{l}dx<\infty$ (1)

The notion of generalized functions bridge the relationship between many
function spaces and harmonic analysis. For instance, this theory provides the
preservation of properties of special class of ultradistributions named the
tempered ultradistributions and some related function space. For details on
tempered ultradistributions, see references Friedman [1], Silva [5] and
Roumieu [6].
This paper is concerned with the inclusion of tempered ultradistributions in
the classical setting of Sobolev spaces and see the behavioral properties of
the underlying functions. The aim can be viewed as taking fundamental Sobolev
spaces of infinitely ultradifferentiable tempered ultradistributions with slow
growth conditions whose derivatives belong to the $L^{p}$-spaces.
This paper is organized as follows: section 2 gives some detailed information
about the classical Sobolev spaces and tempered ultradistributions, and
present some auxiliary statements which play significant roles in the proof of
the main results. In section 3, we present the inclusion of tempered
ultradistributions in Sobolev spaces with their underlying structure and
properties giving emphasis on the exponent. Finally, in section 4, some main
results are presented in form of embeddings between the function spaces under
considerations, and we present the extension of the Rellich’s compactness
theorem involving the newly defined classical Sobolev spaces.

##  2 Preliminaries

###  2.1 On Classical Sobolev Space

Throughout this paper, let $\Omega$ denotes a non-empty open domain
$n$-dimensional space basically for the differentiability and integrability of
functions. We assume that $\Omega$ is an open subset of $\mathbb{C}^{n}$, the
complex plane. We introduce some of the notations to be used in this work.
Basically we outline below some concepts of the theory of topological vector
spaces with its important roles in the study of temperd ultradistributions and
Sobolev spaces. For detailed discussion of these topics, refer to [3] and [5].

######  Notation 2.1.

Denote by $\text{supp}(\mu)$ the support of functions $\mu$;
$\mathcal{D}(\Omega)$ or $C_{0}^{\infty}$ the space of all
$C^{\infty}$-functions with compact support in $\Omega$; the function spaces
$C^{l}(\Omega)$, $C^{l}(\bar{\Omega})$ and $C_{0}^{l}$ of all functions with
derivatives of order $l$; and the classes $C^{l,\beta}(\Omega)$,
$C^{l,\beta}(\bar{\Omega})$ and $C_{0}^{l,\beta}(\Omega)$ of all functions
with derivatives of order $l$ satisfying H$\ddot{\text{o}}$lder inequality of
exponent $\beta\in(0,1]$.
Let $\beta\in\mathbb{N}^{n}$ be a multi-index such that
$\displaystyle|\beta|=\beta_{1}+\cdots+\beta_{n}$,
$\beta!=(\beta_{1}!,\cdots,\beta_{n}!)$,
$D^{\beta}=D_{\xi_{1}}^{\beta_{1}},\cdots,D_{\xi_{n}}^{\beta_{n}}$, where
$\displaystyle D_{\xi_{i}}=\frac{\partial}{\partial\xi_{i}}$, and
$\displaystyle\xi^{\beta}=\Big{(}\xi_{i},\cdots,\xi_{n}\Big{)}^{(\beta_{1},\cdots,\beta_{n})}=\xi^{\beta_{1}},\cdots,\xi^{\beta_{n}}$.

######  Definition 2.2.

We define the space $\mathcal{D}^{\prime}(\Omega)$ as a dual space of
$\mathcal{D}$ consisting of distributions according to L. Schwartz [7]. The
space, denoted by $L^{p}_{\text{loc}}(\Omega)$, is defined as a space of all
$p$th locally integrable functions $\mu$ such that the following condition is
satisfied:

$\int_{\Omega}|\mu|^{p}d\xi<\infty.$

There are functions $\eta\in C^{1}(\mathbb{R}^{n})\setminus
C^{2}(\mathbb{R}^{n})$. We present the $\beta$th partial derivative of $\mu\in
L_{\text{loc}}^{1}(\Omega)$ as defined below:

######  Definition 2.3.

Let $\mu\in L_{\text{loc}}^{1}(\Omega)$ and let $\beta\in\mathbb{N}^{n}$ be a
multi-index. We define $\eta\in L_{\text{loc}}^{1}(\Omega)$ as the $\beta$the
partial derivative of $\mu$, denoted $D^{\beta}\mu=\eta$, if the following
condition is satisfied:

$\int_{\Omega}\mu D^{\beta}\psi d\xi=(-1)^{|\beta|}\int_{\Omega}\eta\psi
d\xi\quad\forall\quad\psi\in C_{0}^{\infty}(\Omega)$

From definition 2.3, we observed that for $\beta=0=(0,\cdots,0)$, we have
$D^{0}\mu=D^{(0,\cdots,0)}\mu=\mu$. Similarly then we have that

$D_{i}\mu=\frac{\partial\mu}{\partial\xi_{i}}=D^{(0,\cdots,1,\cdots,0)}\mu,\quad\text{for}\quad
i\in\mathbb{N}^{n}$

In particular $D\mu$ is termed the weak gradient of $\mu$.

######  Remark 2.4.

Weak derivatives are basically functions satisfying the condition of
integration by parts, and change of of any function on a measure of zero has
no effect on its weak derivatives. Recall that weak derivatives generalize the
classical derivative, and that $D^{\beta}\mu=0$ almost everywhere also implies
that $\mu\equiv 0$ almost everywhere.

The following lemma will back up our claim in Remark 2.4.

######  Lemma 2.5.

[8] A weak $\alpha$th partial derivative of $\mu$, if it exists, is uniquely
defined up to a set of measure zero.

We now present the definition of classical Sobolev spaces below.

######  Definition 2.6.

Let $\Omega$ be an open subset of $\mathbb{R}^{n}$, and let $\beta$ be a
multi-index such that $|\beta|\leq l$ for $l\in\mathbb{R}$. We define the
classical Sobolev space, denoted $\mathcal{W}^{l,p}(\Omega)$, as a function
space consisting of functions $\mu\in L^{p}(\Omega)$ such that
$D^{\beta}\mu\in L^{p}(\Omega)$ provided the weak derivative $D^{\beta}\mu$
exists. That is, we have

$\mathcal{W}^{l,p}(\Omega)\equiv\Big{\\{}\mu\in L^{p}(\Omega):D^{\beta}\mu\in
L^{p}(\Omega),\quad|\beta|\leq l\Big{\\}}$ (2)

The defining norm for $\mathcal{W}^{l,p}$ is given by

$\|\mu\|_{\mathcal{W}^{l,p}(\Omega)}=\Big{(}\sum_{|\beta|\leq
l}\int_{\Omega}|D^{\beta}\mu(\xi)|^{p}d\xi\Big{)}^{\frac{1}{p}},\quad\mu\in\mathcal{W}^{l,p}(\Omega),\quad
1\leq p<\infty$ (3)

and for $p=\infty$, its norm is given as

$\|\mu\|_{\mathcal{W}^{l,\infty}(\Omega)}=\sum_{|\beta|\leq
l}\|D^{\beta}\mu\|_{L^{\infty}(\Omega)}=\sum_{|\beta|\leq
k}\underset{\Omega}{\text{esssup}}|D^{\beta}\mu|\quad\mu\in\mathcal{W}^{l,\infty}(\Omega)$
(4)

where we can define the norm as
$\displaystyle\|D^{\beta}\mu\|_{L^{\infty}(\Omega)}=\underset{\Omega}{\text{esssup}}|D^{\beta}\mu|$.
In summary the Sobolev space $\mathcal{W}^{l,p}(\Omega)$ is a function space
consisting of functions that possesses some properties of distributional
derivatives up to order $l$ and belongs to $L^{p}$-space. The functions in
$\mathcal{W}^{l,p}$ are equal almost everywhere in $L^{p}$. We identify
different ways of describing norms for the classical Sobolev spaces. For more
details in different norms on $\mathcal{W}^{l,p}$ for $1\leq p<\infty$ and
$l=1,2,\cdots$ etc we refer, for instance to [3].

######  Example 2.7.

Given $n\geq 2$ and $\mu:B(0,1)\longrightarrow\mathbb{R}^{+}=[0,\infty]$ and
$\mu(\xi)=|\xi|^{-\beta},\quad\beta>0$. Then we have that $\mu$ is not bounded
within any neighborhood of the origin. That is, $\mu\in
C^{\infty}(B(0,1)\setminus\\{0\\})$, and it also means that $\mu$ is not $p$th
locally integrable. This also implies that some functions in
$\mathcal{W}^{1,p},\quad 1\leq p<n,\quad n\geq 2$, may not be bounded in every
open subset.

######  Example 2.8.

However the same function defined in Example 2.7 does not belong to
$\mathcal{W}^{1,n}\Big{(}B(0,1)\Big{)}$, and this brings some unbounded
functions in $\mathcal{W}^{1,n}$ for $n\geq 2$. However, we define the
following function $\mu:B(0,1)\longrightarrow\mathbb{R}$ by

$\mu(\xi)=\begin{cases}\log\Big{(}\log(1+\frac{1}{|\xi|})\Big{)},&\xi\neq 0\\\
0,&\xi=0\end{cases}$

It shows that $\mu\in\mathcal{W}^{1,n}\Big{(}B(0,1)\Big{)}$ when $n\geq 2$,
but $\mu\notin L^{\infty}\Big{(}B(0,1)\Big{)}$.

The space $L^{\infty}$ can be used to construct a function in
$\mathcal{W}^{1,n}\Big{(}B(0,1)\Big{)}$ that is unbounded in every open subset
of $B(0,1)$, that is, functions unbounded within the neighborhood of zero.
Thus functions defined in $\mathcal{W}^{1,p},\quad 1\leq p\leq n,\quad n\geq
2$ are not continuous. Every function defined in $\mathcal{W}^{1,p}$ with
$p>n$ coincide with a continuous function almost everywhere. Also the function
$\eta(0,1)\longrightarrow\mathbb{R}$, defined by

$\eta(\xi)=\eta(\xi_{1},\cdots,\xi_{n})=\begin{cases}1,&\xi_{n}\geq 0\\\
0,&\xi_{n}<0\end{cases}$

is a member of $\mathcal{W}^{1,p}\Big{(}B(0,1)\Big{)}$ for any $1\leq
p\leq\infty$.

###  2.2 On Some Properties of Sobolev Spaces

We assume that the domain under consideration is smooth. In this section we
are more concerned with the integral representation of functions in the
classical Sobolev space $\mathcal{W}^{l,p}$ which vanish on the boundary of
$\Omega$, and also this representation entails the properties of the space. We
refer for instance to [3] for the properties of classical Sobolev spaces.
Thus we present general properties of the classical Sobolev spaces involving
weak derivatives below.

######  Lemma 2.9.

[8] Assume that $\mu,\nu\in\mathcal{W}^{l,p}(\Omega)$ and $1\leq|\beta|\leq
l$. Then we have the following:

  * (i)

$\displaystyle D^{\beta}\mu\in\mathcal{W}^{l-|\beta|,p}(\Omega)$

  * (ii)

$\displaystyle D^{\beta}(D^{\alpha}\mu)=D^{\alpha}(D^{\beta}\mu)$ for all
multi-indices $\alpha,\beta$ with $|\beta|\leq l$

  * (iii)

$\displaystyle\forall\quad\lambda,t\in\mathbb{R},\quad\lambda\mu+t\nu\in\mathcal{W}^{l,p}(\Omega)$
and

$D^{\beta}(\lambda\mu+t\nu)=\lambda D^{\beta}\mu+tD^{\beta}\nu,$

  * (iv)

If $\Omega^{\prime}\subset\Omega$ is open, then
$\mu\in\mathcal{W}^{l,p}(\Omega^{\prime})$,

  * (v)

If $\eta\in C^{\infty}_{0}(\Omega)$, then
$\displaystyle\eta,\mu\in\mathcal{W}^{l,p}(\Omega)$ and

$D^{\alpha}(\eta\mu)=\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}D^{\beta}\eta
D^{\alpha-\beta}\mu$

where

$\binom{\alpha}{\beta}=\frac{\alpha!}{\beta!(\alpha-\beta!)},\quad\alpha!=\alpha_{1}!,\cdots,\alpha_{n}!$

and $\beta\leq\alpha$ means that $\beta_{i}\leq\alpha_{i}$ $\forall\quad
i=1,2,\cdots,n$.

######  Remark 2.10.

The condition (v) of Lemma 2.9 is called Leibnitz formula. Hence theories of
weak derivatives and classical derivatives have the same properties.

For further properties of the classical Sobolev spaces we refer, for instance
to [3] and [9].
We now introduce a somewhat different class of generalized functions called
tempered ultradistributions.

###  2.3 On The Spaces $\mathcal{U}$ and $\mathcal{U}^{\prime}$

We introduce a special class of generalized functions called tempered
ultradistributions defined on the spaces of rapidly decreasing
ultradifferentiable functions. We denote by $\mathcal{U}^{\prime}$ the strong
dual of the space $\mathcal{U}$ and call its Fourier distributions of
exponential type or tempered ultradistributions as seen by Silva [5].
We introduce the following definitions.

######  Definition 2.11.

We define the space $\mathcal{U}(\mathbb{C}^{n})$ as the function space of all
rapidly decreasing ultradifferentiable functions
$\mu\in\mathcal{U}\subset\mathcal{S}$ such that the condition below is
satisfied:

$\|\mu\|_{p}=\sup_{|\text{Im}(\xi)|<p;\xi\in\mathbb{C}}\Big{\\{}(1+|\xi|^{p})^{l}|\mu(\xi)|\Big{\\}}<\infty\quad\forall\quad
p\in\mathbb{N}\quad\text{and}\quad|\beta|\leq l,\quad l\in\mathbb{R}.$ (5)

The dual of the space $\mathcal{U}$ is the space of all tempered
ultradistributions, denoted by $\mathcal{U}^{\prime}$. The continuous linear
functional $\mu:\mathcal{U}\longrightarrow\mathbb{C}^{n}$ can be described as
the Fourier transform of exponential distributions or as finite order
derivatives of bounded functions with exponential growth. The spaces of
tempered ultradistributions can be represented by analytical functions. Next
we present the vanishing condition and compact supported tempered
ultradistributions.

######  Definition 2.12.

A tempered ultradistribution $\mu\in\mathcal{U}^{\prime}$ is said to vanish in
an open subset $\Omega$ if
$\displaystyle(1+|\xi|^{p})^{l}\mu(x+iy)-(1+|\xi|^{p})^{l}\mu(x-iy)\longrightarrow
0$ as $|\xi|\to 0$ and $y\to 0$.

######  Definition 2.13.

A tempered ultradistribution $\mu\in\mathcal{U}^{\prime}$ has a compact
support if there is a representative function $\hat{\mu}$ vanishing at
$\infty$. Equivalently, $\mu$ vanishes outside a compact support.

For more details about the underlying topology, properties about the spaces
$\mathcal{U}$, $\mathcal{U}^{\prime}$, $\mathcal{D}$, $\mathcal{S}$ and
$\mathcal{E}$ we refer, for instance to [5] and [10].

##  3 On Inclusion of Tempered Ultradistribution in Classical Setting of
Sobolev Spaces

We include tempered ultradistributions in the classical setting of Sobolev
space framework. Some of the basic properties of both spaces have been
introduced in the previous sections. We discuss when the two spaces of
respective underlying defining functions agree. Intuitively tempered
ultradistributions spaces agree with the classical Sobolev spaces if their
respective linear functionals are continuous and if the differentiable or
ultradifferentiable functions are dense in the spaces. In particular this
property reflects the fact the continuity of the respective linear functionals
defined on some different spaces of test functions or rapidly decreasing
ultradifferentiable functions is built into the definition of the classical
Sobolev space combined with the density of the defining functions.
We define the class of tempered ultradistributions in Sobolev space.

######  Definition 3.1.

Let $\Omega\subset\mathbb{C}^{n}$ be an open subset. Then the Sobolev space
$W_{\mathcal{U}}^{l,p}(\Omega)$ is the space of all tempered
ultradistributions $\mu\in\mathcal{U}^{\prime}$ whose ultradistributional
(generalized) gradient $\nabla\mu$ belongs to $L^{p}(\Omega;\mathbb{C}^{n})$
for $|\beta|\leq l$.

We present the similar definition below

######  Definition 3.2.

Let $p\in[1,\infty)$, $l\geq 0$ integer and $\beta$ multi-index be given. Then
the tempered ultradistribution $\mu\in\mathcal{U}^{\prime}(\mathbb{C})$
belongs to $\mathcal{W}^{l,p}(\Omega)$, denoted
$\mathcal{W}^{l,p}_{\mathcal{U}}$, if $\hat{\mu}$ is a function in
$\mathcal{D}^{\prime}$ and

$\|\mu\|_{\mathcal{W}^{l,p}_{\mathcal{U}}}=\Bigg{[}\int_{\mathbb{C}^{n}}(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|^{p}d\xi\Bigg{]}^{\frac{1}{p}}<\infty\quad\text{for}\quad|\beta|\leq
l$ (6)

Similarly we have

$\|\mu\|_{\mathcal{W}^{l,p}_{\mathcal{U}}}=\Bigg{[}\int_{\mathbb{C}^{n}}(1+|\xi|^{p})^{l}|\partial^{\beta}\hat{\mu}(\xi)|^{p}d\xi\Bigg{]}^{\frac{1}{p}}<\infty\quad\text{for}\quad|\beta|\leq
l$ (7)

The classical tempered space
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}(\Omega)$ can be represented by:

$\mathcal{W}_{\mathcal{U}}^{l,p}(\Omega):=\Big{\\{}\mu\in\mathcal{U}^{\prime}:(1+|\xi|^{p})^{l}D^{\beta}\mu\in
L^{p}\quad\text{for}\quad|\beta|\leq l\Big{\\}}$

where $\mathcal{D}^{\prime}$ is the space of distributions.

We observed that the space $\mathcal{S}$ (the space of all tempered
distributions) is contained in
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}(\Omega)$. Then it follows that
the space $\mathcal{U}$ is contained in
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}(\Omega)$ for
$\mathcal{U}\subset\mathcal{S}$. For $l=0$, then
$\displaystyle\mathcal{W}_{\mathcal{U}}^{0,p}$ coincides with classical
$L^{p}$.
We see that $\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}(\Omega)$ has a
structure of Hilbert space as presented in Theorem 3.3.

######  Theorem 3.3.

The space classical tempered
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}(\Omega)$ exhibit the structure
of Hilbert space with inner product defined by

$\langle\mu,\varphi\rangle_{l}=\int_{\Omega}(1+|\xi|^{p})^{l}\hat{\mu}(\xi)\overline{\hat{\nu}(\xi)}d\xi$
(8)

###### Proof.

It is trivial that the defined inner product is an inner product on
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}$ and since $\displaystyle
L^{2}\Big{(}\mathbb{C}^{n},(1+|\xi|^{p})^{l}d\xi\Big{)}$ is complete and the
Fourier transform of distributions of exponential type is a holomorphism of
$\mathcal{U}^{\prime}$, it follows that $\mathcal{W}_{\mathcal{U}}^{l,p}$ is
Banach. ∎

We note that every ultradifferential operator $P$ with constant coefficient
maps $\mathcal{W}^{l,p}_{\mathcal{U}}$ to $W^{l-|\beta|,p}_{\mathcal{U}}$ if
the degree of the polynomial is less than or equal to $|\beta|$. We present
the continuity of the ultradifferential operator in the classical Sobolev
space as reflected in the result below:

######  Theorem 3.4.

Let $\displaystyle P(D)=\sum_{|\beta|\leq l}b_{\beta}D^{\beta}$ be an
ultradifferential operator of infinite order such that there are constants
$M>0$ and $H>0$ for which the following condition holds:

$M=\sup\Bigg{(}\frac{\beta!|b_{\beta}|}{\Big{(}\frac{H}{\sqrt{H}}\Big{)}^{|\beta|}}\Bigg{)}\quad\forall\quad\beta\quad\text{and}\quad|\beta|\leq
l$ (9)

If $\displaystyle\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ then $P$ belongs to
$\mathcal{W}^{l-t,p}$.
Also, the map $\displaystyle
P:\mathcal{W}^{l,p}_{\mathcal{U}}\longrightarrow\mathcal{W}^{l-t,p}_{\mathcal{U}}$
is continuous where $t$ represents the degree of the polynomial $P$.

###### Proof.

We prove that $\displaystyle|b_{\beta}|\leq
M\frac{\Big{(}\frac{H}{\sqrt{H}}\Big{)}^{|\beta|}}{\beta!}\quad\forall\quad\beta\quad\text{and}\quad|\beta|\leq
l$, and the necessary and sufficient condition of (9) for the proof of the
result to hold is only when

$|P(\xi)|\leq M(1+|\xi|^{p})^{l}$

for all $\xi\in\mathbb{C}^{n}$. Hence the result follows. ∎

We give an example of an ultradifferential operator of infinite order of rapid
decrease as contained below.

######  Lemma 3.5.

Let $l$ be a non-negative integer, $\xi\in\mathbb{C}^{n}$, and let
$(1+|\xi|^{p})^{l}$ be an ultrapolynomial of degree $l$, for $1\leq p<\infty$.
The map $\displaystyle\varphi\mapsto(1+|\xi|^{p})^{l}|\varphi|$ is an
isomorphism from the space $\mathcal{W}^{l,p}_{\mathcal{U}}$ onto
$\mathcal{W}_{\mathcal{U}}^{l-|\beta|,p}$ for $|\beta|\leq l$.

###### Proof.

Let $\varphi\in\mathcal{W}_{\mathcal{U}}^{l,p}$ such that $|\beta|\leq l$ and
for $\leq p<\infty$, we get

$\Big{|}(1+|\xi|^{p})^{l}((1+|\xi|^{p})^{-l}D)^{\beta_{1}}(1+|\xi|^{p})^{t-(u+m)-1}\Big{(}(1+|\xi|^{p})^{m}\varphi(\xi)\Big{)}\Big{|}=\Big{|}(1+|\xi|^{p})^{l}((1+|\xi|^{p})^{-1}D)^{t-u-m}\varphi(\xi)\Big{|}$
(10)

Since
$\displaystyle\sup_{\xi\in\mathbb{C}^{n}}\Big{\\{}(1+|\xi|^{p})^{l}|\varphi(\xi)|\Big{\\}}<\infty$
then multiplying (10) by the operator $[(B+\beta)b_{i}]^{-1}$ with supremum
$\forall$ $\xi\in(0,n)$ for $n>1$, we obtain

$\partial^{t,u+m}_{l,\alpha}\Big{[}(1+|\xi|^{p})^{l}\varphi(\xi)\Big{]}=\partial^{t,u}_{l_{1},\alpha}(\varphi)$
(11)

Hence the result. ∎

The relationship between $\mathcal{W}_{\mathcal{U}}^{l,p}$ and
$\mathcal{W}_{\mathcal{U}}^{-l,p}$ can be defined via the pairing

$\langle\mu,\varphi\rangle=\int(1+|\xi|^{p})^{l}\hat{\mu}(\xi)\bar{\hat{\varphi}}(\xi)d\xi=\int(1+|\xi|^{p})^{l}|\hat{\gamma}(\xi)|^{p}d\xi$
(12)

where
$\displaystyle\hat{\mu}(\xi)\bar{\hat{\varphi}}(\xi)=|\hat{\gamma}(\xi)|$ for
$\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ and
$\varphi\in\mathcal{W}_{\mathcal{U}}^{-l,p}$. We obtain the following

$\Big{|}\langle\mu,\varphi\rangle\Big{|}\leq\Big{(}\int(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|^{p}d\xi\Big{)}^{\frac{1}{p}}\Big{(}\int(1+|\xi|^{p})^{-l}|\hat{\varphi}(\xi)|^{p^{\prime}}d\xi\Big{)}^{\frac{1}{p^{\prime}}}=\|\mu\|_{l}\|\varphi\|_{-l}$

for $\frac{1}{p}+\frac{1}{p^{\prime}}=1$.

######  Theorem 3.6.

There exists a canonical isometric isomorphism between
$\mathcal{W}_{\mathcal{U}}^{-l,p}$ and
$\Big{(}\mathcal{W}_{\mathcal{U}}^{l,p}\Big{)}^{\prime}$ (dual of
$\mathcal{W}_{\mathcal{U}}^{k,p}$) with the pairing (12).

###### Proof.

Let $\displaystyle\mu\in\mathcal{W}^{-l,p}_{\mathcal{U}}$ be fixed such that
$\mu:\mathcal{W}^{-l,p}_{\mathcal{U}}\longrightarrow\mathbb{C}$ defined by
$\displaystyle\eta\mapsto\langle\mu,\eta\rangle=\mu(\eta)$ is a continuous
linear functional on $\mathcal{W}^{l,p}_{\mathcal{U}}$ whose norm does not
exceed the number $\|\cdot\|_{-l}$. Since $\mu$ is a tempered
ultradistribution, it is evident to take
$\eta_{0}=\mathcal{F}\Big{[}(1+|\xi|^{p})^{l}\hat{\mu}(\xi)\Big{]}\in\mathcal{W}_{\mathcal{U}}^{l,p}$.
Thus we obtain that

$\langle\mu,\eta_{0}\rangle=\int(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|^{p}d\xi=\|\mu\|_{-l}\quad\text{for}\quad
1\leq p<\infty$

. Hence $\eta\mapsto\langle\mu,\eta\rangle=\|\mu\|_{-l}$ and isometry
obtained.
Next we show that the isometry is surjective and hence isomorphism, let
$\mu^{\prime}\in\Big{(}\mathcal{W}_{\mathcal{U}}^{l,p}\Big{)}^{*}$.
Thus there is a $\phi\in\mathcal{W}^{l,p}_{\mathcal{U}}$ such that

$\displaystyle\mu^{\prime}(\phi)$
$\displaystyle=\langle\mu^{\prime},\phi\rangle_{l}=\langle\phi,\eta\rangle_{l}$
$\displaystyle=\int(1+|\xi|^{p})^{l}\hat{\phi}(\xi)\bar{\eta}(\nu)d\xi$
$\displaystyle=\int(1+|\xi|^{p})^{l}|\widehat{\Phi}(\xi)|^{p}d\xi\quad\forall\quad\eta\in\mathcal{W}_{\mathcal{U}}^{l,p}.$

This concludes the proof. ∎

##  4 Some Structure Theorems and Embeddings

In this section we consider some embedding results in the classical setting of
Sobolev spaces involving tempered ultradistributions, and we extend the
Rellich’s compactness to the newly constructed classical Sobolev spaces since
the underling functions satisfy both vanishing and growth conditions. In the
framework of $\mathcal{W}^{l,p}_{\mathcal{U}}$, one can prove the
$L^{p}$-property of large orders of derivatives with consideration given to
the controlled growth of the defining functions.
We begin with following results’

######  Theorem 4.1.

Let $\mathfrak{K}=\\{z\in\mathbb{C}^{n}:|\text{Im}(z)|\leq l\\}$ for $l>0$, be
an infinite strip, and let $\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$. Then $\mu$
is a holomorphic function in $\mathfrak{K}$.

###### Proof.

Let $\displaystyle\mu(z)=(2\pi i)^{-n}\int(1+|\xi|^{p})^{l}\hat{\mu}(\xi)d\xi$
where $z$ is a complex number. Then for $|\beta|\leq l$ we have

$\displaystyle(2\pi
i)^{-n}\int(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|^{p}d\xi\leq\|\mu\|_{p}\Big{(}\int(1+|\xi|^{p})^{l}d\xi\Big{)}^{\frac{1}{p}}$

Since the integral of the polynomial is integrable if $|\beta|\leq l$, the
result follows. ∎

######  Definition 4.2.

We define the topologies of
$\displaystyle\mathcal{W}_{\mathcal{U}}^{\infty,p}=\bigcap_{l\in\mathbb{R}}\mathcal{W}_{\mathcal{U}}^{l,p}$
and
$\displaystyle\mathcal{W}_{\mathcal{U}}^{-\infty,p}=\bigcup_{l\in\mathbb{R}}\mathcal{W}_{\mathcal{U}}^{l,p}$
as the weakest and strongest topology, respectively, such that the following
injections
$\mathcal{W}_{\mathcal{U}}^{\infty,p}\longrightarrow\mathcal{W}_{\mathcal{U}}^{l,p}$
and
$vmathcal{W}_{\mathcal{U}}^{-\infty,p}\longrightarrow\mathcal{W}_{\mathcal{U}}^{l,p}$)
are continuous.

######  Definition 4.3.

Let $l\in\mathbb{Z}^{+}$. We define the space of entire functions, denoted
$\mathcal{H}^{l}(\Omega)$ with the uniform convergence for which the following
condition holds:

$\int_{\Omega}(1+|\xi|^{p})^{l}|\widehat{\eta\mu(\xi)}|d\xi<\infty\quad\text{for
all}\quad\eta\in\mathcal{D}(\Omega)$

For the sake of the next result, let $\mathcal{K}^{\prime}$ be space of finite
order derivatives of some exponentially bounded continuous functions.

######  Proposition 4.4.

Let $l$ be non-negative integer. If
$\displaystyle\mathcal{U}^{\prime}=\mathcal{F}\\{\mathcal{K}^{\prime}\\}$.
Then $\displaystyle\mathcal{K}^{l}\subset C^{l}(\Omega)$.

###### Proof.

Given $\mu\in\mathcal{K}^{l}$ and $\xi\in\mathbb{C}$. Since
$\mathcal{D}\subset\mathcal{K}$ then choose $\eta\in\mathcal{D}(\Omega)$ with
$\eta=1$ within a neighborhood of $\xi$. Thus

$(\eta\mu)(\xi)=\int_{\Omega}(1+|\xi|^{p})^{l}\widehat{\eta\mu(\xi)}d\xi$ (13)

because $\eta\mu\in L^{1}(\mathbb{C})$. We have the inequality

$(1+|\xi|^{p})^{|\beta|}\leq(1+|\xi|^{p})^{l}$

for $|\beta|\leq l$.
Taking the derivative of the integral (13), we get

$\displaystyle D^{\beta}(\eta\mu)(\xi)$
$\displaystyle=\int_{\Omega}D^{\beta}(1+|\xi|^{p})^{l}\hat{\eta\mu(\xi)}d\xi$
$\displaystyle=(-1)^{|\beta|}\int_{\Omega}(1+|\xi|^{p})^{l}D^{\beta}\hat{\eta\mu(\xi)}d\xi$
$\displaystyle<\infty$

Hence $\mu\in C^{l}(\mathbb{C})$. ∎

From Theorem 3.4 we obtain the following result

######  Theorem 4.5.

Let $l>0$. Every function in $\mathcal{W}_{\mathcal{U}}^{-l,p}$ can be
represented

$\mu=\sum_{\beta\in\mathbb{N}_{0}^{n}}(-1)^{|\beta|}\frac{2\pi
i}{\beta!}D^{\beta}\varphi_{\beta}$

###### Proof.

Let $\mu\in\mathcal{W}_{\mathcal{U}}^{-l,p}$ then by definition we have

$(1+|\xi|^{p})^{l}\hat{\mu}(\xi)\in L^{2}(\mathbb{C})$

which gives

$\hat{\varphi}(\xi)=\frac{\hat{\mu}(\xi)}{\sum_{\beta}(-1)^{|\beta|}\frac{2\pi
i}{\beta!}|\xi^{\beta}|}\in L^{2}$

Thus

$\displaystyle\hat{\mu}(\xi)$ $\displaystyle=\sum_{|\beta|\leq
l}(-1)^{|\beta|}\frac{2\pi i}{\beta!}|\xi^{\beta}|\hat{\varphi}$
$\displaystyle=2\pi i\sum_{|\beta|\leq
l}\frac{(-1)^{|\beta|}}{\beta!}\xi^{\beta}\Big{(}\frac{|\xi^{\beta}|}{\xi^{\beta}}\hat{\varphi}\Big{)}$
$\displaystyle=2\pi i\sum_{|\beta|\leq
l}\frac{(-1)^{|\beta|}}{\beta!}\xi^{\beta}\hat{\varphi}_{|}$

Since
$\hat{\varphi}_{\beta}(\xi)=\Big{(}|\xi^{\beta}|/\xi^{\beta}\Big{)}\hat{\varphi}(\xi)\in
L^{2}(\mathbb{C})$, hence the proof. ∎

######  Remark 4.6.

We see that Theorem 4.19 entails that any
$\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ can be represented as an infinite sum
of partial derivatives of $L^{2}$-functions satisfying the growth conditions.

We observed that if $\mu\in\mathcal{W}^{l,p}_{\mathcal{U}}$ then by definition
3.2, $\hat{\mu}$ is also a tempered ultradistribution. From Plancherel’s
theorem, it implies that
$\mathcal{W}_{\mathcal{U}}^{0,2}=L^{2}(\mathbb{C}^{n})$. That is, if $\mu\in
L^{2}$ then $\hat{\mu}\in L^{2}$ and $\|\hat{\mu}\|_{2}=(2\pi
i)^{\frac{n}{2}}\|\mu\|_{2}$.

We present the continuity result as well as the continuous embedding.

######  Theorem 4.7.

  * (i)

Let $t<l$ then the embedding
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}\hookrightarrow\mathcal{W}_{\mathcal{U}}^{t,p}$
is continuous.

  * (ii)

Let the linear partial differential operator with constant coefficient
$\displaystyle
P(D):\mathcal{W}_{\mathcal{U}}^{l,p}\hookrightarrow\mathcal{W}_{\mathcal{U}}^{l-|\beta|}$
of order $|\beta|\leq l$ be given. Then $P(D)$ is continuous.

###### Proof.

  * (i)

Let $t<l$, then we have

$\displaystyle\|\mu\|_{\mathcal{W}_{\mathcal{U}}^{t,p}}$ $\displaystyle=(2\pi
i)^{-\frac{n}{2}}\int(1+|\xi|^{p})^{t}|\hat{\mu}(\xi)|^{p}d\xi$
$\displaystyle=(2\pi
i)^{-\frac{n}{2}}\int(1+|\xi|^{p})^{l}(1+|\xi|^{p})^{t-l}\Big{|}\hat{\mu}(\xi)\Big{|}d\xi$
$\displaystyle\leq\|\mu\|_{\mathcal{W}_{\mathcal{U}}^{l,p}}\quad\because(1+|\xi|^{p})^{t-l}\leq
1$

Hence the proof.

  * (ii)

Let $P(D)=\partial^{\beta}$ and $\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ be
given such that by the exchange formula, we observe that

$\displaystyle\Big{(}1+|\xi|^{\beta}\Big{)}^{l-|\beta|}\Big{|}\widehat{\partial^{\beta}\mu}\Big{|}$
$\displaystyle\leq\Big{(}1+|\xi|^{p}\Big{)}^{l-|\beta|}|\xi|^{|\beta|}|\hat{\mu}(\xi)|$
$\displaystyle\leq\Big{(}1+|\xi|^{p}\Big{)}^{l-|\beta|}\Big{(}1+|\xi|^{\beta}\Big{)}^{|\beta|}|\hat{\mu}(\xi)|$
$\displaystyle=\Big{(}1+|\xi|^{p}\Big{)}^{l}|\hat{\mu}(\xi)|$

Whence

$\Big{(}1+|\xi|^{\beta}\Big{)}^{l-|\beta|}\Big{|}\widehat{\partial^{\beta}\mu}\Big{|}\leq\Big{(}1+|\xi|^{p}\Big{)}^{l}|\hat{\mu}(\xi)|$

Therefore

$\|\partial^{\beta}\mu\|_{\mathcal{W}_{\mathcal{U}}^{l-|\beta|,p}}\leq\|\mu\|_{\mathcal{W}_{\mathcal{U}}^{l,p}}.$

∎

######  Remark 4.8.

The following inclusion arises from Theorem 4.7(i):

$\mathcal{U}\subset\mathcal{S}\subset\mathcal{W}_{\mathcal{U}}^{\infty,p}\subset\mathcal{W}_{\mathcal{U}}^{-\infty,p}\subseteq\mathcal{U}^{\prime}$

We present the result below called the Sobolev norm of derivatives satisfying
the growth conditions.

######  Lemma 4.9.

Let $l\in\mathbb{R}$ and $p\in[1,\infty)$ be given. Then
$\displaystyle\mu\in\mathcal{W}_{\mathcal{U}}^{l+1,p}$ if and only if the set
$\displaystyle\Big{\\{}(1+|\xi|^{p})^{\beta}\mu,(1+|\xi|^{p})^{\beta_{1}}\partial_{1}^{\beta_{1}}\mu,\cdots,(1+|\xi|^{p})^{\beta_{n}}\partial_{n}^{\beta_{n}}\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}\Big{\\}}$.
under the following norm

$\|\mu\|_{\mathcal{W}_{\mathcal{U}}^{l+1,p}}^{p}=\|\mu\|_{\mathcal{W}_{\mathcal{U}}^{l,p}}^{p}+\sum_{i\in\mathbb{N}}\|\partial_{i}^{\beta}\mu\|_{\mathcal{W}_{\mathcal{U}}^{l,p}}^{p}.$

###### Proof.

Assume that $\displaystyle\mu\in\mathcal{W}_{\mathcal{U}}^{l+1,p}$ then
$\displaystyle(1+|\xi|^{p})^{\beta}\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ and
$\displaystyle(1+|\xi|^{p})^{\beta}\partial_{i}^{\beta}\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$
for $i\in\mathbb{N}^{n}$. To see this, let $\displaystyle
1+|\xi|^{p}=1+\sum_{i\in\mathbb{N}}\xi^{p}$ which implies that
$\displaystyle(1+|\xi|^{p})^{l}=\Bigg{(}1+\sum_{i\in\mathbb{N}}\xi^{p}\Bigg{)}^{l}$
. Thus for $|\beta|+1\leq l+1$

$\displaystyle\Big{|}(1+|\xi|^{p})^{l+1}\hat{\mu}\Big{|}^{p}$
$\displaystyle=\Big{|}(1+|\xi|^{p})\Big{|}^{p}\Big{|}(1+|\xi|^{p})^{l}\hat{\mu}\Big{|}^{p}$
$\displaystyle=\Big{|}(1+|\xi|^{p})^{l}\hat{\mu}\Big{|}^{p}+\sum_{i=1}^{n}\Big{|}(1+|\xi|^{p})^{l}\xi_{i}\hat{\mu}\Big{|}^{p}$
$\displaystyle=\Big{|}(1+|\xi|^{p})^{l}\hat{\mu}\Big{|}^{p}+\sum_{i=1}^{n}\Big{|}(1+|\xi|^{p})^{l}\widehat{\partial_{i}^{\beta}\mu}\Big{|}^{p}$

as required for $|\beta|\leq l$, and since
$\sup_{\xi\in\mathbb{C}^{n}}\Big{\\{}(1+|\xi|^{p})^{l}|\partial_{i}^{\beta}\mu(\xi)|\Big{\\}}<\infty$,
it implies that $\mu,\partial_{i}^{\beta}\mu$ belong to
$\mathcal{W}_{\mathcal{U}}^{l+1,p}$. ∎

We have the characterization of the space
$\mathcal{W}_{\mathcal{U}}^{|\beta|,p}$ Theorem 4.10 below.

######  Theorem 4.10.

Let $\beta\in\mathbb{N}^{n}$ be a multi-index such that
$|\beta|=\beta_{1}+\cdots+\beta_{n}$. Then we have

$\mathcal{W}^{|\beta|}_{\mathcal{U}}(\mathbb{C}^{n})\equiv\Big{\\{}\mu\in\mathcal{U}^{\prime}(\mathbb{C}^{n}):\partial^{\beta}\mu\in
L^{p}\quad\forall\quad|\beta|\leq l\Big{\\}}$ (14)

###### Proof.

From definition of $\mathcal{W}_{\mathcal{U}}^{l,p}$, we show that
$\mathcal{W}_{\mathcal{U}}^{l,p}$ can be expressed as
$\mathcal{W}_{\mathcal{U}}^{|\beta|,p}$. To do this, let
$\mu\in\mathcal{U}^{\prime}$ then we have that
$\displaystyle(1+|\xi|^{p})^{l}\mu\in\mathcal{W}^{|\beta|}_{\mathcal{U}}(\mathbb{C}^{n})$
and this shows that
$\partial^{|\beta|}\mu\in\mathcal{W}^{|\beta|}_{\mathcal{U}}(\mathbb{C}^{n})$
and
$(1+|\xi|^{p})\partial^{|\beta|}\mu\in\mathcal{W}^{|\beta|}_{\mathcal{U}}(\mathbb{C}^{n})$.
Thus

$\displaystyle\Bigg{(}1+|\xi|^{p}\Bigg{)}^{(\beta_{1},\cdots,\beta_{n})}\partial_{i}^{\beta_{1}+\cdots+\beta_{n}}\mu=(1+|\xi|^{p})^{\beta_{1}}\partial^{\beta_{1}}_{i}\mu\cdots(1+|\xi|^{p})^{\beta_{n}}\partial^{\beta_{n}}_{i}\mu\in\mathcal{W}_{\mathcal{U}}^{|\beta|,p}$

provided $|\beta|\leq l$. ∎

consider the following embedding result.

######  Theorem 4.11.

  * (i)

If $n>2|\beta|$ for $l<\frac{n}{2}$ then
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}\subseteq
C^{0}_{0}(\mathbb{C}^{n})$

  * (ii)

Let $|\beta|<r+n/2$, $r\in\mathbb{R}$ for $l<r+\frac{n}{2}$ and $|\beta|\leq
l$. Then $\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}\subseteq
C_{0}^{l}(\mathbb{C}^{n})$.

###### Proof.

  * (i)

Assume that $\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ with $n<2|\beta|$. From
definition of $\mathcal{U}^{\prime}$, then we have that
$\displaystyle\xi\mapsto(1+|\xi|^{p})^{-l}\xi\in L^{p-1}$ for $p\geq 2$ and
therefore $\eta=(1+|\xi|^{p})^{l}\hat{\mu}$. An application of Cauchy-Schwartz
inequality yields

$\displaystyle\|\hat{\mu}\|_{L^{p-1}}$
$\displaystyle\leq\|\eta\|_{L^{p}}\Big{(}\int(1+|\xi|^{p})^{|\beta|}d\xi\Big{)}^{\frac{1}{p}}$
$\displaystyle\leq\|\eta\|_{L^{p}}\Big{(}\sup_{\xi\in\mathbb{C}}\int(1+|\xi|^{p})^{l}d\xi\Big{)}^{\frac{1}{p}}$
$\displaystyle\leq M\|\eta\|_{L^{p}}\leq
M\|\eta\|_{\mathcal{W}_{\mathcal{U}}^{l,p}}$

This shows that $\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}\hookrightarrow
L^{p}\hookrightarrow L^{p-1}$. Thus $\hat{\mu}\in L^{p-1}(\mathbb{C}^{n})$ and
$\mu=\mathcal{F}^{-1}(\hat{\mu})$ be classical inversion formula. By Riemann-
Lebesgue lemma, it implies that $\mu\in C^{0}_{0}$.

  * (ii)

From definition of $C_{0}^{l}$, we have $f\in C_{0}^{l}$ such that
$\displaystyle\lim_{|\xi|\to\infty}f(\xi)=0$ $\forall\quad|\beta|\leq r$.
Assume that $|\beta|<r+\frac{n}{2}$ and let
$\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$. Then
$\partial^{\beta}\mu\in\mathcal{W}^{l-r}(\mathbb{C}^{n})$ for all $|\beta|\leq
r$ then we see that $\partial^{\beta}\mu\in
C^{\infty}(\mathbb{C}^{n})\quad\forall\quad\beta$. Thus
$\partial^{\beta}\mu\in\mathcal{W}_{\mathcal{U}}^{l-r,p}\implies\partial^{\beta}\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$.

∎

######  Corollary 4.12.

If $\mu\in\mathcal{W}_{\mathcal{U}}^{\infty,p}(\mathbb{C}^{n})$ then $\mu\in
C_{0}^{\infty}$.

###### Proof.

We show that $\mu\in C_{0}^{\infty}$ which implies that $\mu\in C^{\infty}\cap
C_{0}$ such that $\text{supp}\mu$ has a compact support in $K=\bigcup K_{i}$
provided $\mu\in\mathcal{W}_{\mathcal{U}}^{\infty,p}$. To show this, let
$\mu\in\mathcal{W}_{\mathcal{U}}^{\infty,p}$ then
$\partial^{\beta}\mu\in\mathcal{W}_{\mathcal{U}}^{\infty,p}$ for
$|\beta|\leq\infty$ within $K$ as neighborhood of zero. Take $\eta\in
L^{\infty}$, then $\mu\eta\in L^{\infty}(\Omega)$. Thus

$(\eta\mu)(y)=\int_{K}(1+|\xi|^{p})^{l}\widehat{\eta\mu}(\xi)d\xi$

Using the inequality $|\beta|\leq l$ and $|\beta|\leq\infty$,

$|\xi|^{\beta}\leq(1+|\xi|^{p})^{|\beta|}\leq(1+|\xi|^{p})^{l}\leq(1+|\xi|^{p})^{l}\widehat{\eta\mu}(\xi)<\infty$

has the smallest value for which $|\beta|\leq\infty$ since $\eta\mu\in
L^{\infty}$ has the common tempered ultradistribution in
$\mathcal{W}_{\mathcal{U}}^{\infty,p}$ since it satisfies some growth
condition.
Alternatively, let $\mu\in\mathcal{W}_{\mathcal{U}}^{\infty,p}$ then

$|\hat{\mu}(\xi)|\leq(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|,\quad\xi\in\mathbb{C}^{n}$

Hence

$\displaystyle\int_{K}(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|^{p}d\xi$
$\displaystyle\leq M\int_{K}(1+|\xi|^{p})^{l}d\xi$ $\displaystyle\leq
M\int_{K}(1+|\xi|^{p})^{l+r}d\xi$

for $l+r<(-n/2)$. Hence the result. ∎

######  Remark 4.13.

The space $\mathcal{W}_{\mathcal{U}}^{l,p}$ is closed under multiplication of
functions with rapid decrease. One can see that if
$\mu\in\mathcal{W}^{l,p}_{\mathcal{U}}$ and $\phi\in\mathcal{U}$ then
$\phi\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}(\mathbb{C}^{n})$ with the map that
$\phi\mapsto\phi\mu$ is continuous on $\mathcal{W}_{\mathcal{U}}^{l,p}$ since
$\displaystyle\sup_{\xi\in\mathbb{C}^{n}}\Big{(}(1+|\xi|^{p})^{l}|\phi\mu(\xi)|^{p}\Big{)}<\infty$.

We present the following embedding theorem

######  Proposition 4.14.

The space $\mathcal{U}^{\prime}(\mathbb{C}^{n})$ is dense in
$\mathcal{W}^{-\infty,p}_{\mathcal{U}}(\mathbb{C}^{n})$. That is,
$\displaystyle\overline{\mathcal{U}^{\prime}}=\mathcal{W}^{-\infty,p}_{\mathcal{U}}$.

###### Proof.

We show that
$\displaystyle\overline{\mathcal{U}^{\prime}}=\mathcal{W}^{-\infty,p}_{\mathcal{U}}$.
Recall that
$\displaystyle\mathcal{W}^{-\infty,p}_{\mathcal{U}}=\bigcup_{l\in\mathbb{R}}\mathcal{W}_{\mathcal{U}}^{l,p}$.
To do this let $\Big{\\{}(1+|\xi|^{p})^{l}\mu(\xi)\Big{\\}}$ denotes a
countable convergent sequence of rapidly decreasing ultradifferentiable
functions $\mu$ satisfying the growth condition such that
$\displaystyle\sup_{\xi\in\mathbb{C}}\Big{\\{}(1+|\xi|^{p})^{l}\mu(\xi)\Big{\\}}<\infty$.
Then it follows that $\displaystyle|\hat{\mu}(\xi)|\leq
M(1+|\xi|^{p})^{l}|\eta\mu(\xi)|^{p}<\infty$ for all $\eta\in\mathcal{U}$.
Thus it follows that
$\displaystyle\lim_{|\xi|\to\infty}\Big{\\{}(1+|\xi|^{p})^{l}\mu(\xi)\Big{\\}}=0$
for $|\beta|\leq l$.
Therefore

$\int_{\Omega}(1+|\xi|^{p})^{l}|\eta\widehat{\mu}(\xi)|^{p}d\xi\leq
M\int_{\Omega}(1+|\xi|^{p})^{l-|\beta|}d\xi<\infty\quad\text{for}\quad|\beta|\leq
l\quad\text{if}\quad l-|\beta|<-(n/2)$

Then it shows that $\mu$ belongs to $\mathcal{W}_{\mathcal{U}}^{l,p}$ and
$\cup\mathcal{W}_{\mathcal{U}}^{l,p}$. ∎

######  Corollary 4.15.

If $\phi\in\mathcal{U}$ then the multiplication operator operator,
$T_{\phi}:\mathcal{U}\longrightarrow\mathcal{W}_{\mathcal{U}}^{l,p}$ defined
by $T_{\phi}\mu=\phi\mu$ is a bounded linear operator on
$\mathcal{W}_{\mathcal{U}}^{l,p}$ for all $l\in\mathbb{R}$.

######  Definition 4.16.

We denote denote by $\overline{\mathcal{W}}_{\mathcal{U}}^{l,p}$ the closure
of $\mathcal{D}$ in the $\mathcal{W}_{\mathcal{U}}^{l,p}$-norm for all
$l\in\mathbb{R}$.

Before we present the extension of the Rellich’s compactness theorem, the
trace theorem is needed for the proof. The importance of the Rellich’s theorem
establish the fact that under certain conditions the embedding
$\mathcal{W}^{l,p}_{\mathcal{U}}\hookrightarrow\mathcal{W}^{t,p}_{\mathcal{U}}$
for $l>t$ is compact. This means that from any bounded sequence in
$\mathcal{W}_{\mathcal{U}}^{l,p}$ one can obtain an
$\mathcal{W}^{t,p}_{\mathcal{U}}$-convergent subsequence. The elementary
result called Bolzano-Weierstrass theorem plays a significant role in this
compactness result.

######  Theorem 4.17.

If $\mu\in\overline{\mathcal{W}}_{\mathcal{U}}^{l,p}$ for some $l>n/2$, then
$\mu\equiv 0$ on $\partial\Omega$ for $\Omega\subset\mathbb{C}^{n}$.

###### Proof.

Let $l>n/2$ and $\mu\in\overline{\mathcal{W}}_{\mathcal{U}}^{l,p}$ be given,
then we have a convergent sequence $\\{\mu_{m}\\}$ of functions in
$\mathcal{D}(\Omega)$ with compact support since
$\mathcal{D}\subset\mathcal{U}\subset\mathcal{S}$ such that
$\mu_{m}\longrightarrow m$ in $\mathcal{W}_{\mathcal{U}}^{l,p}$. By Theorem
4.11, we get

$\sup_{\xi\in\partial\Omega}\Big{|}(1+|\xi|^{p})^{l}\mu(\xi)\Big{|}=\sup_{\xi\in\partial\Omega}\Big{|}(1+|\xi|^{p})^{l}(\mu-\mu_{m})(\xi)\Big{|}\leq
M_{l}\|\mu_{m}-\mu\|_{l}\quad\text{for}\quad m=2,\cdots,$

Since $\displaystyle\|\mu_{m}-\mu\|$ is non-negative, it implies that the
equality occurs when $\mu\equiv 0$ on $\partial\Omega$. ∎

######  Remark 4.18.

Theorem 4.17 entails that function $\mu$ with slow growth condition vanishes
at infinity on the boundary of the open subset.
Also from Theorem 4.17, we characterize the trace of a Sobolev function
$\mu\in\mathcal{W}^{1,p}_{\mathcal{U}}$ namely, the “restriction“ of $\mu$ to
the boundary $\partial\Omega$.

######  Theorem 4.19.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded open subset, and let
$l<l^{\prime}$ be given. Then there exists a convergent subsequence in
$\mathcal{W}_{\mathcal{U}}^{l,p}$ for every bounded sequence of tempered
ultradistributions in
$\overline{\mathcal{W}}_{\mathcal{U}}^{l^{\prime},p}(\Omega)$. In particular,
the inclusion map
$\displaystyle\overline{\mathcal{W}}_{\mathcal{U}}^{l^{\prime},p}(\Omega)\longrightarrow\mathcal{W}_{\mathcal{U}}^{l,p}$
is compact.

###### Proof.

First we show that both sequences of rapidly decreasing ultradifferentiable
functions and its partial derivatives defining tempered ultradistribtions is
bounded. To do this, let $\\{\mu_{m}\\}$ be a bounded sequence in
$\overline{\mathcal{W}}_{\mathcal{U}}^{l^{\prime},p}(\Omega)$ and let $\chi$
be a local unit for $\bar{\Omega}$. Then we have $|\beta|\leq l^{\prime}$ and
$|\beta|\leq l$,

$\displaystyle(1+|\xi|^{p})^{l^{\prime}}|\hat{\mu}_{m}(\xi)|$
$\displaystyle\leq\int_{\Omega}(1+|\xi|^{p})^{l^{\prime}-l}|\hat{\chi}(\xi)|(1+|\xi|^{p})^{l}|\hat{\mu}_{m}(\xi)|d\xi$
$\displaystyle\leq\Bigg{[}\int_{\Omega}(1+|\xi|^{p})^{l^{\prime}-l}|\hat{\chi}(\xi)|^{p}d\xi\Bigg{]}^{1/p}\Bigg{[}\int_{\Omega}(1+|\xi|^{p})^{l}|\hat{\mu}_{m}(\xi)|^{p}d\xi\Bigg{]}^{1/p}$
$\displaystyle\leq\|\mu_{m}\|_{l}\Bigg{[}\int_{\Omega}(1+|\xi|^{p})^{l^{\prime}-l}|\hat{\chi}(\xi)|^{p}d\xi\Bigg{]}^{1/p}$
$\displaystyle\leq\|\mu_{m}\|_{l}\sup_{\xi\in\Omega}\Big{\\{}(1+|\xi|^{p})^{l^{\prime}-l}|\hat{\chi}(\xi)|^{p}\Big{\\}}$
$\displaystyle\leq M_{1}<\infty.$

for $M_{1}>0$ (not dependent on $\xi$ and $m$).
We also prove, in a similar way, the boundedness of the sequence of partial
derivatives of tempered ultradistributions. Whence,

$\displaystyle(1+|\xi|^{p})^{l^{\prime}}|\partial_{i}^{\beta}\hat{\mu}_{m}(\xi)|$
$\displaystyle\leq\int_{\Omega}(1+|\xi|^{p})^{l^{\prime}-l}|\partial^{\beta}\hat{\chi}(\xi)|(1+|\xi|^{p})^{l}|\partial^{\beta}\hat{\mu}_{m}(\xi)|d\xi$
$\displaystyle\leq\Bigg{[}\int_{\Omega}\partial^{\beta}(1+|\xi|^{p})^{l^{\prime}-l}|\hat{\chi}(\xi)|^{p}d\xi\Bigg{]}^{1/p}\Bigg{[}\int_{\Omega}\partial^{\beta}(1+|\xi|^{p})^{l}|\hat{\mu}_{m}(\xi)|^{p}d\xi\Bigg{]}^{1/p}$
$\displaystyle\leq\|\mu_{m}\|_{l}|\xi|^{p}\Bigg{[}\int_{\Omega}(1+|\xi|^{p})^{l^{\prime}-l}|\partial^{\beta}\hat{\chi}(\xi)|^{p}d\xi\Bigg{]}^{1/p}$
$\displaystyle\leq\|\mu_{m}\|_{l}\sup_{\xi\in\Omega}\Bigg{\\{}(1+|\xi|^{p})^{l^{\prime}-l}|\chi|^{\beta}|\hat{\chi}(\xi)|^{p}\Bigg{\\}}$
$\displaystyle\leq M_{2}<\infty.$

Next we show that $(1+|\xi|^{p})^{\beta}\hat{\mu}_{m}(\xi)$ is an
equicontinuous sequence of rapidly decreasing ultradifferentiable functions on
compact set. To do this, let $K$ be any compact subset of $\mathbb{C}^{n}$ and
$\displaystyle
M_{3}=\sup_{\xi\in\mathbb{C}}\Bigg{\\{}(1+|\xi|^{p})^{\beta}:\xi\in\overline{B(0,T)}\Bigg{\\}}$
where $\displaystyle T=\sup_{\xi\in K}\|\xi\|$. Then
$\displaystyle|\hat{\mu}_{m}|\leq M_{1}(1+|\xi|^{p})^{-l}\leq M_{2}M_{3}$ and
$\displaystyle|\partial^{\beta}\hat{\mu}_{m}|\leq M_{2}M_{3}$ for every $m>0$
and $\xi$ in $K$, $|\beta|\leq l$. Due to the uniform continuity nature of
polynomials $(1+|\xi|^{p})^{l}$ on $K$, then for every $\varepsilon>0$,
constant $\delta>0$, we have that

$\Big{|}(1+|\xi|^{p})^{l}-(1+|\eta|^{p})^{l}\Big{|}<\frac{\varepsilon}{2M_{1}M_{3}}\quad\text{whenever}\quad|\xi-\eta|<\delta_{1}$

Then from mean value theorem, we get

$\displaystyle\Big{|}\hat{\mu}_{m}(\xi)-\hat{\mu}_{m}(\eta)\Big{|}$
$\displaystyle\leq\sum_{i\in\mathbb{N}}\Big{|}\xi_{i}-\eta_{i}\Big{|}^{p}\Big{|}\partial_{i}^{\beta}\hat{\mu}_{m}(\xi-\eta)\Big{|}$
$\displaystyle\leq M_{4}\|\xi-\eta\|$

for some constant $M_{4}$, $m\in\mathbb{N}$ and $\xi,\eta\in K$.
Also, for every $\varepsilon>0$ there is a constant $\delta_{2}>0$ such that
$\displaystyle\Big{|}\hat{\mu}_{m}(\xi)-\hat{\mu}_{m}(\eta)\Big{|}<\frac{\varepsilon}{2M_{3}M_{4}}$
whenever $|\xi-\eta|\leq\delta_{2}$. Now choose
$\delta=\min\\{\delta_{1},\delta_{2}\\}$, we get

$\displaystyle\Big{|}(1+|\xi|^{p})^{l}\hat{\mu}_{m}(\xi)-(1+|\eta|^{p})^{l}\hat{\mu}_{m}(\eta)\Big{|}$
$\displaystyle\leq(1+|\xi|^{p})^{l}\Big{|}\hat{\mu}_{m}(\xi)-\hat{\mu}_{m}(\eta)\Big{|}$
$\displaystyle+\Big{|}(1+|\xi|^{p})^{l}-(1+|\eta|^{p})^{l}\Big{|}\Big{|}\hat{\mu}(\eta)\Big{|}$
$\displaystyle<\varepsilon$

provided $|\xi-\eta|<\delta$ and $\xi,\eta\in K$ which established the uniform
continuity of $(1+|\xi|^{p})^{l}$.
This shows that
$\displaystyle\Big{\\{}(1+|\xi|^{p})^{l}\hat{\mu}_{m}\Big{\\}}$ is a pointwise
bounded sequence in $\overline{W}_{\mathcal{U}}^{k,p}$ on $K$. The convergent
subsequence
$\displaystyle\Big{\\{}(1+|\xi|^{p})^{\beta}\hat{\mu}_{m_{i}}\Big{\\}}$ by
Arzela-Ascoli theorem and Bolzano-Weierstrass theorem exists uniformly on each
compact sets in $\mathbb{C}^{n}$.
For every $T>0$, we have

$\displaystyle\|\mu_{m_{i}}-\mu_{m_{j}}\|$
$\displaystyle=\int_{\Omega}(1+|\xi|^{p})^{2l}|(\mu_{m_{i}}-\mu_{m_{j}})(\xi)|^{p}d\xi$
$\displaystyle=\int_{\Omega}(1+|\xi|^{p})^{2l}(1+|\xi|^{p})^{2l-2k}|(\mu_{m_{i}}-\mu_{m_{j}})(\xi)|^{p}d\xi$
$\displaystyle\leq\int_{|\xi|\geq
T}(1+|\xi|^{p})^{2l-2k}|(\mu_{m_{i}}-\mu_{m_{j}})(\xi)|^{p}d\xi$
$\displaystyle+\int_{|\xi|\leq
T}(1+|\xi|^{p})^{2k}|(\mu_{m_{i}}-\mu_{m_{j}})(\xi)|^{p}d\xi$
$\displaystyle\leq\int_{|\xi|\leq
T}(1+|\xi|^{p})^{2k}|(\mu_{m_{i}}-\mu_{m_{j}})(\xi)|^{p}d\xi+M$ (15)

The inequality of (4) follows from the boundedness of $\\{\mu_{m}\\}$ in
$\overline{\mathcal{W}}_{\mathcal{U}}^{l,p}$. Now given $\varepsilon>0$, we
can take a large $T$ so that due to the uniform convergence on $B(0,T)$ of the
subsequence $\displaystyle(1+|\xi|^{p})^{l}\hat{\mu}_{m_{j}}(\xi)$ then given
a constant such that $m_{i},M_{j}\geq M$ for $i<j$ we have

$\int_{|\xi|\leq
T}(1+|\xi|^{p})^{l}|\mu_{m_{i}}-\mu_{m_{j}}|^{p}d\xi<\frac{\varepsilon^{p}}{2}+\frac{\varepsilon}{2}<\varepsilon$

Hence $\\{\mu_{j}\\}$ is a convergent subsequence of $\mu_{m}$ in
$W_{\mathcal{U}}^{t,p}$. ∎

######  Remark 4.20.

If $l=t$, then Theorem 4.19 will not hold since their orders of derivatives
are same.

######  Corollary 4.21.

Let $\Omega\subset\mathbb{C}$ be a bounded subset and $l<t<t^{\prime}$. Then
we have the following continuous embedding

$\mathcal{U}\hookrightarrow\mathcal{W}^{t^{\prime},p}_{\mathcal{U}}\hookrightarrow\mathcal{W}_{\mathcal{U}}^{t,p}\hookrightarrow\mathcal{W}_{\mathcal{U}}^{l,p}\hookrightarrow\mathcal{U}^{\prime}$
(16)

###### Proof.

There is need to show that
$\displaystyle\mathcal{W}_{\mathcal{U}}^{l,p}\hookrightarrow\mathcal{U}^{\prime}$
since other inclusions are obvious. To see this, let
$\mu\in\mathcal{W}_{\mathcal{U}}^{l,p}$ such that
$\displaystyle\sup_{\xi\in\mathbb{C}}\Big{\\{}(1+|\xi|^{p})^{l}|\mu(\xi)|\Big{\\}}$
is finite for $|\beta|\leq l$ and
$\displaystyle\sup_{\xi\in\mathbb{C}}\Big{\\{}(1+|\xi|^{p})^{l}\partial^{\beta}\mu(\xi)\Big{\\}}<\infty$.
Whence

$\displaystyle(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|$
$\displaystyle\leq\Bigg{[}\int_{\Omega}(1+|\xi|^{p})^{|\beta|}|\hat{\mu}(\xi)|^{p}d\xi\Bigg{]}^{\frac{1}{p}}$
$\displaystyle\leq\sup_{\xi\in\mathbb{C}}\Bigg{[}\int_{\Omega}(1+|\xi|^{p})^{l}|\hat{\mu}(\xi)|^{p}d\xi\Bigg{]}^{\frac{1}{p}}$
$\displaystyle\leq\|\mu\|_{\mathcal{W}_{\mathcal{U}}^{l,p}}$

as required. ∎

Conflict of Interest: The authors demonstrated no conflict of interest towards
the publication of this manuscript.

## References

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50


# Calculations of the binding-energy differences for highly-charged Ho and Dy
ions

I. M. Savelyev<EMAIL_ADDRESS>Department of Physics, St. Petersburg
State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia  M.
Y. Kaygorodov  Department of Physics, St. Petersburg State University, 7/9
Universitetskaya nab., 199034 St. Petersburg, Russia  Y. S. Kozhedub
Department of Physics, St. Petersburg State University, 7/9 Universitetskaya
nab., 199034 St. Petersburg, Russia  I. I. Tupitsyn  Department of Physics,
St. Petersburg State University, 7/9 Universitetskaya nab., 199034 St.
Petersburg, Russia  V. M. Shabaev  Department of Physics, St. Petersburg State
University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia  B. P.
Konstantinov Petersburg Nuclear Physics Institute of National Research Centre
“Kurchatov Institute”, Gatchina, 188300 Leningrad District, Russia

###### Abstract

The binding-energy differences for ${}^{163}\mathrm{Ho}^{q+}$ and
${}^{163}\mathrm{Dy}^{q+}$ ions with ionization degrees $q=38$, $39$, and $40$
are calculated. The calculations are performed using the large-scale
relativistic configuration-interaction and relativistic coupled-clusters
methods. The contributions from quantum-electrodynamics, nuclear-recoil, and
frequency-dependent Breit-interaction effects are taken into account. The
final uncertainty does not exceed $1$ eV. Combining the obtained results with
the binding-energy difference for neutral atoms calculated in [Savelyev et
al., Phys. Rev. A 105, 012806 (2022)], we get the secondary differences of the
ion-atom binding energies. These values can be used to evaluate the amount of
energy released in the electron capture process in ${}^{163}\mathrm{Ho}$ atom
(the $Q$ value), provided mass differences of highly charged ions
${}^{163}\mathrm{Ho}^{q+}$ and ${}^{163}\mathrm{Dy}^{q+}$ is known from
experiment. The $Q$ value is required by experiments on the determination of
the absolute scale of the electron neutrino mass by studying the beta-decay
process.

The neutrinos are one of the most intriguing topics of the modern physics [1].
On the one hand, the neutrino is a massless particle within the Standard
Model, on the other hand, observations of neutrino oscillations established
that neutrino must have nonzero mass. Nevertheless, the oscillation
experiments are only sensitive to the absolute squared mass–differences
between the neutrino mass eigenstates and do not allow one to determine the
absolute mass–scale of neutrinos. Constraints on the sum of the neutrino
masses can be obtained from the analysis of the cosmological data [2, 3].
According to the CPT invariance of the Standard Model, the mass of the
neutrino must be exactly equal to the mass of the antineutrino. However, at
present, upper limits on the electron neutrino and antineutrino masses differ
by several orders of magnitude [4]. All these points rise undoubted interest
to direct model-independent experimental determination of the neutrino mass.

The best upper direct limit on the electron antineutrino mass is obtained by
KATRIN collaboration [5]. The value of $0.8$ eV is obtained by performing the
kinematic analysis of $\beta^{-}$ decay in tritium. The best upper laboratory
limit on the electron neutrino mass is about two orders of magnitude larger.
So, in [6], using the spectroscopy of X-rays emitted in the electron-capture
(EC) decay of isotope ${}^{163}\mathrm{Ho}$, a limit of $225$ eV was
established. Several collaborations [7, 8, 9] aim to improve the current
laboratory limit on the electron neutrino mass up to a few eV and make it
comparable with the value of the electron antineutrino limit. The experiments
are also based on study of the nuclear-electron-capture process in neutral
${}^{163}\mathrm{Ho}$ atom, but this time using a more accurate calorimetric
method. Recently, within the framework of ECHo experiment [9], the upper limit
for the electron neutrino mass has been improved to a value of about $150$ eV
[10].

In order to extract the limit on the electron neutrino mass with the eV
accuracy from the experiment, one has to know in advance the mass difference
between the decay isotopes 163Ho and ${}^{163}\mathrm{Dy}$, also called the
$Q$ value, at least, with the same accuracy. The measurement of the mass
difference at the required level of accuracy can be performed for highly
charged ions utilizing Penning-trap mass spectrometers [11, 12]. Recently, the
mass difference 163Ho and 163Dy ions were measured for ionization degrees
$38$, $39$, and $40$ [13].

To convert the mass difference of highly charged 163Ho and 163Dy ions to the
mass difference of neutral 163Ho and 163Dy atoms, which is the required $Q$
value, the binding-energy difference for the atoms and ions has to be
calculated at an appropriate level of uncertainty. In Ref. [14], the
calculations of the binding-energy differences for the
${}^{163}\mathrm{Ho}^{q+}$ and ${}^{163}\mathrm{Dy}^{q+}$ ions were performed
for ionization degrees $q=30$, $48$, and $56$. The aim of the present work is
to extend the results of Ref. [14] with the calculations for the ionization
degrees needed for the experiment [13]. The atomic units are used throughout
the paper.

We consider the mass difference $\Delta m^{q}$ of the ions 163Hoq+ and 163Dyq+
with the same ionization degree $q$,

$\Delta m^{q}=\Delta m_{n}+m_{e}+\Delta E^{q},$ (1)

where $\Delta m_{n}$ is the mass difference of the 163Ho and 163Dy nuclei,
$m_{e}$ is the electronic mass, $\Delta E^{q}$ is the difference of the total
electronic binding energies of the ions. The mass difference of the neutral
163Ho and 163Dy atoms corresponds to the case $q=0$:

$\Delta m^{0}=\Delta m_{n}+m_{e}+\Delta E^{0}.$ (2)

The mass difference of the neutral atoms $\Delta m^{0}$ is related to the mass
difference of the ions $\Delta m^{q}$ by

$\Delta m^{0}=\Delta m^{q}+\Delta E^{0,q},$ (3)

where the secondary difference of the binding energies,

$\Delta E^{0,q}=\Delta E^{0}-\Delta E^{q},$ (4)

is introduced.

The present calculations are based on the relativistic Dirac-Coulomb (DC) and
Dirac-Coulomb-Breit (DCB) Hamiltonians

$\displaystyle\hat{H}_{\textrm{DC}}$
$\displaystyle=\Lambda^{+}(\hat{H}_{\textrm{D}}+\hat{H}_{\textrm{C}})\Lambda^{+},$
(5) $\displaystyle\hat{H}_{\textrm{DCB}}$
$\displaystyle=\Lambda^{+}(\hat{H}_{\textrm{D}}+\hat{H}_{\textrm{C}}+\hat{H}_{\textrm{B}})\Lambda^{+},$
(6)

where $\hat{H}_{\textrm{D}}$ is the sum of the one-electron Dirac
Hamiltonians:

$\hat{H}_{\mathrm{D}}=\sum_{i=1}^{N}\left[(\bm{\alpha}_{i}\cdot\bm{p}_{i})c+(\beta-1)mc^{2}+V(r_{i})\right],$
(7)

and $\hat{H}_{\textrm{C}}$ and $\hat{H}_{\textrm{B}}$ are the two-electrons
operators of the Coulomb and Breit interactions, respectively,

$\hat{H}_{\mathrm{C}}=\frac{1}{2}\sum_{i\neq j}^{N}\frac{1}{r_{ij}},$ (8)
$\hat{H}_{\mathrm{B}}=-\frac{1}{2}\sum_{i\neq
j}^{N}\frac{1}{2r_{ij}}\Big{[}\bm{\alpha}_{i}\cdot\bm{\alpha}_{j}+\frac{(\bm{\alpha}_{i}\cdot\bm{r}_{ij})(\bm{\alpha}_{j}\cdot\bm{r}_{ij})}{r_{ij}^{2}}\Big{]}.$
(9)

Here $\bm{\alpha}$ is a vector of the Dirac matrices, $\bm{p}$ is the momentum
operator, $\bm{r}_{ij}$ is a position of the $i$-th electron relative to the
$j$-th one, $r_{ij}=|\bm{r}_{ij}|$, and $N$ is the total number of electrons.
The projector $\Lambda^{+}$ ensures that the Hamiltonian acts in space of
states which corresponds to the positive-energy spectrum of the Dirac-Fock
Hamiltonian.

The main method used to calculate $\Delta E^{q}$ is the relativistic
configuration interaction method in the basis of the Dirac-Fock-Sturm orbitals
(CI-DFS) [15, 16, 17]. To take into account the quantum-electrodynamics (QED)
effects, the model-QED-operator approach [18, 19, 20] is employed. The model-
QED operator $V_{\mathrm{QED}}^{\mathrm{mod}}$ allows one to approximately
account for the lowest order vacuum-polarization and self-energy corrections
in many-electron systems. The operator $V_{\mathrm{QED}}^{\mathrm{mod}}$ is
used by including it in the many-electron DCB Hamiltonian

$\hat{H}_{\mathrm{DCBQ}}=\Lambda^{+}(\hat{H}_{\mathrm{D}}+\hat{H}_{\mathrm{C}}+\hat{H}_{\mathrm{B}}+\hat{V}_{\mathrm{QED}}^{\mathrm{mod}})\Lambda^{+}.$
(10)

The frequency-dependent Breit-interaction correction is evaluated as the
expectation value of the frequency dependent part of the one-photon-exchange
operator considered in the Coulomb gauge with the CI-DFS many-electron wave
function. The nuclear-recoil effect is evaluated as the expectation value of
the relativistic nuclear-recoil Hamiltonian [21, 22, 23, 24] with the many-
electron wave function constructed by CI-DFS.

To verify the accuracy of the CI-DFS correlation treatment within the DC
Hamiltonian framework, we performed the calculations using another approach,
namely the relativistic single-reference coupled-clusters method involving
fully iterative single, double and perturbatively triple cluster amplitudes
(CCSD(T)). The software package DIRAC23 [25, 26] is used for this purpose.

The Fermi model of the nuclear charge distribution is used in our CI-DFS
calculations, whereas in the CCSD(T) calculations, the Gaussian model is
employed, but for the properties under consideration the difference is
negligible. The values of the root-mean-square (RMS) radii are taken from Ref.
[27].

We approach the problem of calculating $\Delta E^{q}$ by progressive stages.
At the first stage, we evaluate the total binding-energy differences for Ho
and Dy ions at the DC Hamiltonian level by means of the CI-DFS method. The
ground-state configurations of the considered Ho and Dy ions with $q=38$,
$39$, and $40$ are given in Table 1.

Table 1: The ground-state configurations of Hoq+ and Dyq+ ions. $q$ | Hoq+ | Dyq+
---|---|---
Configuration | $J$ | Configuration | $J$
$38$ | $[\mathrm{Ar}]3d^{10}4s^{1}$ | $1/2$ | $[\mathrm{Ar}]3d^{10}$ | $0$
$39$ | $[\mathrm{Ar}]3d^{10}$ | $0$ | $[\mathrm{Ar}]3d^{9}$ | $5/2$
$40$ | $[\mathrm{Ar}]3d^{9}$ | $5/2$ | $[\mathrm{Ar}]3d^{8}$ | $4$

In the CI-DFS calculations, all the occupied orbitals are divided into the
valence and core ones. The $1s2s2p$ orbitals are assigned to the frozen core,
the remaining occupied orbitals are active. The single and double excitations
from the active orbitals to the virtual ones are considered. The convergence
of $\Delta E^{q}$ with respect to the number of virtual orbitals is analyzed.
We start with one virtual $s$ orbital and expand the set of the virtual
orbitals in two directions: by successively adding new virtual orbitals with
larger principal quantum number $n$ and with larger orbital quantum number
$l$. Our calculations include up to five virtual orbitals for each orbital
quantum number $l$ up to $i$. The final results are obtained by extrapolating
the values to the infinite basis-set limit.

At the next stage, we evaluate the contribution from the frozen core. To this
end, we perform the CI-DFS calculations using smaller number of virtual
orbitals adapted to correlate strongly bound $1s2s2p$ electrons.

Further, we verify the correctness of our CI-DFS results by performing the
CCSD(T) calculations. Within the CCSD(T) approach, all electrons are
correlated and the standard basis sets dyall.ae3z and dyall.ae4z are employed.
The convergence of the CCSD(T) results is studied in the same manner as in the
CI-DFS scheme, with the extrapolation to the infinite basis-set limit being
used. It should be noted that the contribution of the perturbative triple
cluster amplitude is negligible for the considered systems.

The differences of the ground-state energies for 163Hoq+ and 163Dyq+ ions with
$q=38$, $39$, and $40$ evaluated by means of the CI-DFS and CCSD(T) methods
using the DC Hamiltonian are presented in Table 2.

Table 2: The ground-state energy differences for 163Hoq+ and 163Dyq+ ions obtained using the CI-DFS and CCSD(T) methods with the DC Hamiltonian (a.u.). $q$ | CI-DFS | CCSD(T)
---|---|---
38 | $-461.298(7)$ | $-461.305(11)$
39 | $-502.088(16)$ | $-502.089(16)$
40 | $-500.931(17)$ | $-500.934(17)$

The results obtained using two conceptually different methods agree within the
estimated uncertainties.

To evaluate the Breit, frequency-dependent Breit, QED, and nuclear-recoil
corrections, a series of the CI-DFS calculations with active $3s3p3d$ (and
$4s$ for $\textrm{Ho}^{38+}$) orbitals is carried out. It turns out that the
sufficient accuracy for these corrections can be achieved using a smaller
basis set than for the DC results. The Breit and QED corrections are obtained
using the Hamiltonians $\hat{H}_{\mathrm{DCB}}$ and $\hat{H}_{\mathrm{DCBQ}}$,
respectively, whereas the frequency-dependent part of the Breit-interaction
correction and the nuclear-recoil correction are calculated as expectation
values of the corresponding operators. The errors associated with the choice
of the nuclear model and with the RMS radii are estimated by considering
different charge distributions and varying the RMS radii within their
experimental uncertainties.

The final results for the total binding-energy differences for 163Hoq+ and
163Dyq+ ions for various degrees of ionization $q=38$, $39$, and $40$, $\Delta
E^{q}$, as well as the additional contributions from the DCB Hamiltonian,
$\Delta E^{q}_{\mathrm{DCB}}$, the QED corrections, $\Delta
E^{q}_{\mathrm{QED}}$, and the frequency-dependent Breit-interaction
correction, $\Delta E^{q}_{\mathrm{BRFD}}$, are presented in Table 3. The
nuclear-recoil correction is found to be about $0.001$ a.u. for all considered
$q$. This correction is included in $\Delta E^{q}_{\mathrm{DCB}}$.

Table 3:  Contributions to the ground-state energy differences for 163Hoq+ and 163Dyq+ ions, $\Delta E^{q}$, from the electron-correlation effects within the Breit approximation, $\Delta E^{q}_{\mathrm{DCB}}$, from the frequency-dependent Breit interaction, $\Delta E^{q}_{\mathrm{BRFD}}$, and from QED, $\Delta E^{q}_{\mathrm{QED}}$ (a.u.). In the last column, the total ground-state energy differences are presented. The numbers in the first parentheses show numerical accuracy, whereas the numbers in the second parentheses represent the uncertainty associated with the finite nuclear size. $q$ | $\Delta E^{q}_{\mathrm{DCB}}$ | $\Delta E^{q}_{\mathrm{BRFD}}$ | $\Delta E^{q}_{\mathrm{QED}}$ | $\Delta E^{q}$
---|---|---|---|---
38 | $-460.678$ | $-0.0140$ | $0.439$ | $-460.253(9)(13)$
39 | $-501.446$ | $-0.0146$ | $0.405$ | $-501.056(16)(13)$
40 | $-500.291$ | $-0.0146$ | $0.405$ | $-499.901(17)(13)$

The values $\Delta E^{q}$ have two sources of uncertainty: in the first
parentheses, the uncertainty due to the correlation treatment within the CI-
DFS method is shown, meanwhile, in the the second parentheses, the uncertainty
due to the nuclear parameters is presented. The evaluated energy differences
for the ions, $\Delta E^{q}$, have smaller uncertainty associated with the
electronic correlation than the energy difference for the neutral atoms from
Ref. [14] due to simpler electronic structure of the ions.

Combining the calculated energy differences for the ions, $\Delta E^{q}$, with
the energy difference for the neutral atoms, $\Delta E^{0}$, evaluated in Ref.
[14], we obtain the secondary differences of the ion-atom binding energies,
$\Delta E^{0,q}$, which are presented in units of eV in Table 4.

Table 4: The contributions to the secondary differences of the ion-atom binding energies for ${}^{163}\mathrm{Ho}$ and ${}^{163}\mathrm{Dy}$ atoms, $\Delta E^{0,q}$, from the electronic correlations within the Breit approximation, $\Delta E^{0,q}_{\mathrm{DCB}}$, the frequency-dependent part of the Breit interaction, $\Delta E^{0,q}_{\mathrm{BRFD}}$, and the QED effects, $\Delta E^{0,q}_{\mathrm{QED}}$ (eV). The total secondary differences, $\Delta E^{0,q}$, are given in the last column. $q$ | $\Delta E^{0,q}_{\mathrm{DCB}}$ | $\Delta E^{0,q}_{\mathrm{BRFD}}$ | $\Delta E^{0,q}_{\mathrm{QED}}$ | $\Delta E^{0,q}$
---|---|---|---|---
38 | $37.7$ | $-0.03$ | $-0.77$ | $36.9(7)$
39 | $1147.0$ | $-0.01$ | $0.17$ | $1147.2(8)$
40 | $1115.6$ | $-0.01$ | $0.18$ | $1115.8(8)$

The uncertainties associated with the finite nuclear-size effects are canceled
in the secondary differences $\Delta E^{0,q}$ and the uncertainties of the
final results are mainly determined by the binding-energy difference error for
the neutral atoms.

In conclusion, the total binding-energy differences for
${}^{163}\mathrm{Ho}^{q+}$ and ${}^{163}\mathrm{Dy}^{q+}$ ions for degrees of
ionization $q=38$, $39$, and $40$ are calculated by means of the CI-DFS method
with sub-eV uncertainty. The electronic correlations are evaluated within the
framework of the DCB Hamiltonian. The QED, frequency-dependent Breit-
interaction, and nuclear-recoil corrections are taken into account. Combining
the calculated energies differences for ions with the energies difference for
neutral atoms reported in Ref. [14], we obtain the secondary differences of
the ion-atom binding energies. These results can be used to recalculate the
mass difference between ions ${}^{163}\mathrm{Ho}^{q+}$ and
${}^{163}\mathrm{Dy}^{q+}$ to the $Q$ value of the electron capture process in
${}^{163}\mathrm{Ho}$ atom, which is required for upcoming experiments on
improving the upper limit of the electron neutrino mass.

This work was supported by RSCF grant No. 22-62-00004.

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