#  Weyl Transform on Nonunimodular Groups

Santosh Kumar Nayak  Santosh Kumar Nayak School of Mathematical Sciences
National Institute of Science Education and ResearchBhubaneswar - 752 050,
India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For $p>2$, B. Simon[15] studied the unboundedness of the Weyl transform for
symbol belonging to $L^{p}({\mathbb{R}^{n}\times\mathbb{R}^{n}})$. In this
article, we study the analog of unboundedness of the Weyl transform on some
nonunimodular groups, namely, the affine group, similitude group, and affine
Poincaré group.

###### Key words and phrases:

Representation theory, Nonunimodular group, Fourier Transform, Affine group,
Similitude group, Affine Poincaré group, Weyl transform

###### 1991 Mathematics Subject Classification:

Primary 47G10, 47G30,43A15, 43A30, Secondary 42B35

S. K. Nayak is supported by the Institute Fellowship.

##  1\. Introduction

Weyl transform is an operator introduced by H. Weyl in 1950, [19]. It is a
type of self-adjoint pseudo-differential operator on $\mathbb{R}^{n}$, [16].
H. Weyl studied this operator while solving quantization problems in quantum
mechanics, [11]. Physicists always love to work with self-adjoint operators.
The theory of the Weyl transforms covers a broad area of great interest in
both mathematical analysis and physics. In a number of problems like elliptic
theory, regularity problems, spectral asymptotic, etc., Weyl transforms have
proved to be a useful tool, [20]. The Weyl transform(operator) has been deeply
investigated mainly in the case where the symbol is a smooth function on
$\mathbb{R}^{n}\times\mathbb{R}^{n}$ belonging to some symbol classes. For
more details, we refer to [2, 3, 17, 18]. The Weyl transform is an operator
acting on $L^{2}(\mathbb{R}^{n})$ that belongs to Hilbert-Schmidt class when
the symbol lies in $L^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})$, [19]. And when
the symbol $\sigma$ lies in $L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{n}),1\leq
p\leq 2,$ the Weyl transform is bounded on $L^{2}(\mathbb{R}^{n})$. For $p>2$,
B. Simon[15] studied the unboundedness of the Weyl transform for symbol
belonging to $L^{p}({\mathbb{R}^{n}\times\mathbb{R}^{n}})$. The proof of
boundedness and unboundedness of the Weyl transform can be found in [20].

In this paper, we assume group G to be any of the three groups, namely, affine
group, similitude group, and affine Poincaré group, and their dual
$\widehat{G}$ are discrete sets.

Let $L^{p}(G\times\widehat{G},S_{p})$ be the set of all $S_{p}$ valued
functions such that

$\|f\|_{p,\mu}^{p}=\sum_{\rho\in\widehat{G}}\int_{G}\|f(x,\rho)K_{\rho}^{\frac{1}{p}}\|_{S_{p}}^{p}d\mu(x)<\infty,\quad
1\leq p<\infty,$ $\|f\|_{\infty,\mu}=\text{ess~{}~{}sup}_{(x,\rho)\in
G\times\widehat{G}}\|f(x,\rho)\|_{S_{\infty}}<\infty.$

######  Definition 1.1.

Let $f,g\in C_{c}(G)$. Then the Wigner transform associated to $f$ and $g$ is
given by

$W(f,g)(x,\rho)=\int_{G}f(x^{\prime})\tau_{x^{\prime}}g(x)\rho(x^{\prime})d\mu(x^{\prime}),$

where $\tau_{x}f(y)=f(x^{-1}y)$ is the left translation operator on
$L^{p}(G)$.

Proof of the following results is followed step by step from [10]. So we omit
those proofs from the article but to make it self-contained, we just have
stated.

######  Theorem 1.2.

Let $f,g\in C_{c}(G)$. Then for $2\leq p\leq\infty$, we have $W(f,g)\in
L^{p}(G\times\widehat{G},S_{p})$ and

$\|W(f,g)\|_{p,\mu}\leq\|f\|_{L^{2}(G)}\|g\|_{L^{2}(G)}.$

The following proposition represents the Fourier transform in terms of the
Wigner transform.

######  Proposition 1.3.

Let $f\in L^{1}(G)\cap L^{2}(G)$ and $C=\int\limits_{G}g(y)d\mu(y)\neq 0$.
Then

$\widehat{f}(\rho)=C^{-1}\int_{G}W(f,g)(y,\rho)d\mu(y),$

for all $\rho\in\widehat{G}.$

######  Corollary 1.4.

Let $f\in L^{1}(G)\cap A(G)$ and $g\in L^{1}(G)\cap L^{2}(G)$ with
$C=\int\limits_{G}g(x)d\mu(x)\neq 0$. Then

$f(x)=C^{-1}\sum_{\rho\in\widehat{G}}\operatorname{Tr}\left(\rho^{\ast}(x)\left(\int_{G}W(f,g)(x^{\prime},\rho)d\mu(x^{\prime})\right)K_{\rho}\right).$

######  Definition 1.5.

Let $\sigma\in L^{p}(G\times\widehat{G},S_{p}),1\leq p\leq 2.$ Then the Weyl
transform $W_{\sigma}:L^{2}(G)\to L^{2}(G)$ is defined by

$\langle W_{\sigma}f,\overline{g}\rangle=\langle
W(f,g),\sigma\rangle_{\mu}=\sum_{\rho\in\widehat{G}}\int_{G}\operatorname{Tr}\left(\sigma(x,\rho)^{\ast}W(f,g)(x,\rho)K_{\rho}\right)d\mu(x),$
(1.1)

where $f,g$ are in $L^{2}(G)$ and $\langle,\rangle_{\mu}$ denotes innerproduct
in $L^{2}(G\times\widehat{G},S_{2})$.

######  Theorem 1.6.

Let $\sigma\in L^{p}(G\times\widehat{G},S_{p}),1\leq p\leq 2$. Then

$\|W_{\sigma}\|\leq\|\sigma\|_{p,\mu}.$

For $G=\mathbb{R}^{n}$, B. Simon [15] gave the example of symbols in
$L^{p}(\mathbb{R}^{n}\times\mathbb{R}^{n}),p>2$, such that the corresponding
Weyl transform becomes unbounded. This idea has been motivating many authors
to study the unboundedness of Weyl transform in other settings like the
Heisenberg group [12], quaternion Heisenberg group [4], upper half plane [13],
motion groups [10], reduced Heisenberg group with multidimensional center [8],
and other Weyl transforms can be found in [21, 14]. In this article, we study
the unboundedness of Weyl transform on the affine group $U$, similitude group
$\mathbb{SIM}(2)$, and affine Poincaré group ${\mathcal{P_{\textrm{aff}}}}$.

The organization of the paper is as follows: we investigate the unboundedness
of Weyl transform on the affine group, similitude group, and affine Poicaré
group, respectively in Section 2, Section 3, and Section 4.

##  2\. Affine Group

In this section, we consider the group $G$ to be an affine group $U$. Here we
recall the Fourier analysis of the affine group, $U$, in detail. In this
setup, we will define the Weyl transform formula and show that it cannot be
extended as a bounded operator for the symbol in the corresponding $L^{p}$
spaces with $2<p<\infty$.

Let $U=\\{(b,a):b\in\mathbb{R},a>0\\}$ be the upper half plane. Then with
respect to the binary operation ‘$\cdot$’ on $U$, defined as

$(b,a)\cdot(c,d)=(b+ac,ad),\quad(b,a),(c,d)\in U,$ (2.1)

the upper half plane $U$ forms a non-abelian group. It can be shown
$(\frac{-b}{a},\frac{1}{a})$ is the inverse element of $(b,a)$ and $(0,1)$ is
the identity element in $U$. The left and right Haar measures on $U$ are
$d\mu=\frac{dbda}{a^{2}}$ and $d\gamma=\frac{dbda}{a}$ respectively. With
respect to the above multiplication ‘$\cdot$’ defined by (2.1), U is also a
locally compact and Hausdroff group on which the left Haar measure is
different from the right Haar measure, and hence $U$ is a non-unimodular
group.

Let $\mathcal{U}(L^{2}(\mathbb{R}_{\pm}))$ be the set of all unitary operators
on $L^{2}(\mathbb{R}_{\pm})$. Then the unitary irreducible representations of
$U$ on $L^{2}(\mathbb{R}_{\pm})$ are given by the mapping
$\rho_{\pm}:U\rightarrow\mathcal{U}(L^{2}(\mathbb{R}_{\pm}))$ defined as

$(\rho_{+}(b,a)u)(s)=a^{1/2}e^{-ibs}u(as),\quad
s\in\mathbb{R}_{+}=[0,\infty),$

for every $u\in L^{2}(\mathbb{R}_{+})$, and

$(\rho_{-}(b,a)u)(s)=a^{1/2}e^{-ibs}v(as),\quad
s\in\mathbb{R}_{-}=(-\infty,0],$

for every $v\in L^{2}(\mathbb{R}_{-}).$ These two $\\{\rho_{+},\rho_{-}\\}$
are only the irreducible inequvalent unitary representations, i.e.,
$\widehat{U}=\\{\rho_{+},\rho_{-}\\}$. The unbounded operators, $K_{\pm}$, on
$L^{2}(\mathbb{R}_{\pm}),$ are given by

$(K_{\pm}\phi)(s)=|s|\phi(s),\quad s\in\mathbb{R}_{\pm},$ (2.2)

are known as Duflo-Moore operators, [9].

Let $f\in L^{1}(U)\cap L^{2}(U)$, the Fourier transform

$\widehat{f}(\rho_{\pm})=\int_{U}f(b,a)\rho_{\pm}(b,a)\dfrac{dbda}{a^{2}}.$

Let $\phi\in L^{2}(\mathbb{R}_{\pm})$, then
$\widehat{f}(\rho_{\pm})\phi(s)=\int\limits_{0}^{\infty}\int\limits_{-\infty}^{\infty}f(b,a)\left(\rho_{\pm}(b,a)\phi\right)(s)\dfrac{dbda}{a^{2}},\phi\in
L^{2}(\mathbb{R}_{\pm})$. Define $L^{p}$-Fourier transform
${\mathcal{F}}_{p}(f)\rho_{\pm}=\widehat{f}(\rho_{\pm})K_{\pm}^{\frac{1}{p^{\prime}}},1\leq
p\leq\infty,\frac{1}{p}+\frac{1}{p^{\prime}}=1.$

######  Theorem 2.1 (Fourier inversion formula).

[5] Let $f\in L^{1}(U)$. Then

$f(x)=\sum_{j\in\\{\pm\\}}\operatorname{Tr}\left(\rho_{j}(b,a)^{\ast}\widehat{f}(\rho_{j})K_{j}\right).$

######  Theorem 2.2 (Plancherel Formula).

[5] Let $f\in L^{1}(U)\cap L^{2}(U)$. Then

$\int_{U}|f(b,a)|^{2}\dfrac{dbda}{a^{2}}=\sum_{j\in\\{\pm\\}}\|\widehat{f}(\rho_{j})K_{j}^{\frac{1}{2}}\|_{S_{2}}^{2}.$

For $G=U$, recall the definition of Weyl transform defined in (1.1), and the
Theorem 1.6 says that it is bounded when $1\leq p\leq 2$. Now we prove the
unboundedness of Weyl transform when $p>2$.

######  Theorem 2.3.

For $2<p<\infty$, there exists an operator valued function $\sigma$ in
$L^{p}(U\times\widehat{U},S_{p})$ such that $W_{\sigma}$ is not a bounded
linear operator on $L^{2}(U)$.

We prove the Theorem 2.3 using the method of contrapositive in three
propositions.

######  Proposition 2.4.

Let $2<p<\infty$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Then for all
$\sigma\in L^{p}(U\times\widehat{U},S_{p})$, the Weyl transform $W_{\sigma}$
is bounded linear operator on $L^{2}(U)$ iff there exists a constant $C$ such
that

$\|W(f,g)\|_{p^{\prime},\mu}\leq C\|g\|_{L^{2}(U)}\|f\|_{L^{2}(U)},$

for all $f,g$ in $L^{2}(U)$.

###### Proof.

Assume that

$\|W(f,g)\|_{p^{\prime},\mu}\leq C\|g\|_{L^{2}(U)}\|f\|_{L^{2}(U)},\quad
f,g\in L^{2}(U),$

for some $C>0$. By the definition 1.5, we obtain the necessary part of the
theorem. Conversely, suppose that for each $\sigma\in
L^{p}(U\times\widehat{U},S_{p})$, $W_{\sigma}$ is bounded, i.e.,

$\|W_{\sigma}f\|_{L^{2}(U)}\leq C({\sigma})\|f\|_{L^{2}(U)},$

for all $f\in L^{2}(U)$. Define a family of linear operators
$M_{f,g}:L^{p}(U\times\widehat{U},S_{p})\to\mathbb{C}$ by

$M_{f,g}(\sigma)=\langle W_{\sigma}f,g\rangle,\quad f,g\in C_{c}(U).$

For each $f,g\in C_{c}(U)$, it is easy to check that this family is pointwise
bounded linear functional. By uniform bounded principle, there exists $C$ such
that $\|M_{f,g}\|\leq C$, for all $f,g\in C_{c}(U),\|f\|_{2}=\|g\|_{2}=1$.
Hence $|\langle W_{\sigma}f,g\rangle|\leq C\|\sigma\|$. Thus for $f,g\in
C_{c}(U)$,

$\displaystyle\|W(f,g)\|_{p^{\prime},\mu}$
$\displaystyle=\sup_{\|\sigma\|_{p,\mu}=1}|\langle
W(f,g),\sigma\rangle_{\mu}|$
$\displaystyle=\sup_{\|\sigma\|_{p,\mu}=1}|\langle
W_{\sigma}f,\overline{g}\rangle|\leq C\|f\|_{2}\|g\|_{2}.$

By density argument, we conclude the proposition.

∎

######  Proposition 2.5.

Let $2<p<\infty$ and $f$ be a square integrable, compactly supported function
on $U$ such that $\int\limits_{U}f(b,a)d\mu(b,a)\neq 0$. If $W_{\sigma}$ is a
bounded operator on $L^{2}(U)$ for all $\sigma$ in
$L^{p}(U\times\widehat{U},S_{p})$, then

$\|{\mathcal{F}}_{p}(f)\rho_{+}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\rho_{-}\|_{S_{p^{\prime}}}^{p^{\prime}}<\infty.$

###### Proof.

From Proposition 1.3, it is enough to show

$\sum_{j\in\\{\pm\\}}\|\int_{U}W(f,f)(b,a,\rho_{j})K_{j}^{\frac{1}{p^{\prime}}}d\mu(b,a)\|_{S_{p^{\prime}}}^{p^{\prime}}<\infty,$

for all square integrable, compactly supported function $f$ on $U$ such that
$\int\limits_{U}f(b,a)d\mu(b,a)\neq 0$. Suppose $f$ is supported in $A$ of
$U$, then $W(f,f)$ is supported in $AA\times\widehat{U}$. Now by Hölder’s
inequality, Minkowski’s integral inequality, and Proposition 2.4, we get

$\displaystyle\sum_{j\in\\{\pm\\}}\|\int_{U}W(f,f)(b,a,\rho_{j})K_{j}^{\frac{1}{p^{\prime}}}d\mu(b,a)\|_{S_{p^{\prime}}}^{p^{\prime}}$
$\displaystyle\leq\left(\int_{AA}\left(\sum_{j\in\\{\pm\\}}\|W(f,f)(b,a,\rho_{j})K_{j}^{\frac{1}{p^{\prime}}}\|_{S_{p^{\prime}}}^{p^{\prime}}\right)^{\frac{1}{p^{\prime}}}d\mu(b,a)\right)^{p^{\prime}}$
$\displaystyle\leq\left(\int_{AA}d\mu(b,a)\right)^{\frac{p^{\prime}}{p}}\int_{AA}\sum_{j\in\\{\pm\\}}\|W(f,f)(b,a,\rho_{j})K_{j}^{\frac{1}{p^{\prime}}}\|_{S_{p^{\prime}}}^{p^{\prime}}d\mu(b,a)<\infty.$

This completes the proof. ∎

######  Proposition 2.6.

For $p\in(2,\infty)$, does there exists a square-integrable, compactly
supported function $f$ on $U$ with
$\int\limits_{U}f(b,a)\dfrac{dbda}{a^{2}}\neq 0$ such that
$\|{\mathcal{F}}_{p}(f)\rho_{+}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\rho_{-}\|_{S_{p^{\prime}}}^{p^{\prime}}=\infty?$

The answer to this question is yes. We will certainly find a function
$f_{\alpha},0<\alpha<\frac{1}{2}$, at the end of the section, which is needed
for Proposition 2.6. Let $f\in L^{2}(U)$. Now for all $\phi\in
L^{2}(\mathbb{R}_{+})$,

$\displaystyle\left(\widehat{f}(\rho_{+})K_{+}^{\frac{1}{p^{\prime}}}\phi\right)(s)$
$\displaystyle=\int_{0}^{\infty}\int_{-\infty}^{\infty}f(b,a)\left(\rho_{+}(b,a)K_{+}^{\frac{1}{p^{\prime}}}\phi\right)(s)\frac{dbda}{a^{2}}$
$\displaystyle=\int_{0}^{\infty}\int_{-\infty}^{\infty}f(b,a)a^{\frac{1}{2}}e^{-ibs}\left(K_{+}^{\frac{1}{p^{\prime}}}\phi\right)(as)\frac{dbda}{a^{2}}$
$\displaystyle=\int_{0}^{\infty}({\mathcal{F}}_{1}f)(s,a)a^{\frac{1}{2}}|as|^{\frac{1}{p^{\prime}}}\phi(as)\frac{da}{a^{2}}$
$\displaystyle=\int_{0}^{\infty}({\mathcal{F}}_{1}f)(s,a)a^{\frac{1}{2}+{\frac{1}{p^{\prime}}}}|s|^{\frac{1}{p^{\prime}}}\phi(as)\frac{da}{a^{2}}.$

Now substituting $as=t$, we get $da=\frac{dt}{s}$.

$\displaystyle\left(\widehat{f}(\rho_{+})K_{+}^{\frac{1}{p^{\prime}}}\phi\right)(s)$
$\displaystyle=\int_{0}^{\infty}\left({\mathcal{F}}_{1}f\right)\left(s,\frac{t}{s}\right)\frac{t}{s}^{\frac{1}{2}+{\frac{1}{p^{\prime}}}}|s|^{\frac{1}{p^{\prime}}}\frac{s^{2}}{t^{2}}\phi(t)\frac{dt}{s}.$

Again for all $\phi\in L^{2}(\mathbb{R}_{-})$, we have

$\left(\widehat{f}(\rho_{-})K_{-}^{\frac{1}{p^{\prime}}}\phi\right)(s)=\int_{-\infty}^{0}\left({\mathcal{F}}_{1}f\right)\left(s,\frac{t}{s}\right)\frac{t}{s}^{\frac{1}{2}+{\frac{1}{p^{\prime}}}}|s|^{\frac{1}{p^{\prime}}}\frac{s^{2}}{t^{2}}\phi(t)\frac{dt}{s}.$

Now

$\displaystyle\|\widehat{f}(\rho_{+})K_{+}^{\frac{1}{p^{\prime}}}\|_{S_{2}}^{2}+\|\widehat{f}(\rho_{-})K_{-}^{\frac{1}{p^{\prime}}}\|_{S_{2}}^{2}$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\left|\left({\mathcal{F}}_{1}f\right)\left(s,\frac{t}{s}\right)\right|^{2}\frac{t}{s}^{1+{\frac{2}{p^{\prime}}}}|s|^{\frac{2}{p^{\prime}}-2}\frac{s^{4}}{t^{4}}dsdt$
$\displaystyle+\int_{-\infty}^{0}\int_{-\infty}^{0}\left|\left({\mathcal{F}}_{1}f\right)\left(s,\frac{t}{s}\right)\right|^{2}\frac{t}{s}^{1+{\frac{2}{p^{\prime}}}}|s|^{\frac{2}{p^{\prime}}-2}\frac{s^{4}}{t^{4}}dsdt.$

Substituting $\frac{t}{s}=a$, then we get $\frac{dt}{s}=da$.

$\displaystyle\|\widehat{f}(\rho_{+})K_{+}^{\frac{1}{p^{\prime}}}\|_{S_{2}}^{2}+\|\widehat{f}(\rho_{-})K_{-}^{\frac{1}{p^{\prime}}}\|_{S_{2}}^{2}$
$\displaystyle=\int_{0}^{\infty}\int_{0}^{\infty}\left|\left({\mathcal{F}}_{1}f\right)\left(s,a\right)\right|^{2}a^{1+{\frac{2}{p^{\prime}}}}|s|^{\frac{2}{p^{\prime}}-1}\frac{1}{a^{4}}dsda$
$\displaystyle+\int_{-\infty}^{0}\int_{0}^{\infty}\left|\left({\mathcal{F}}_{1}f\right)\left(s,a\right)\right|^{2}a^{1+{\frac{2}{p^{\prime}}}}|s|^{\frac{2}{p^{\prime}}-1}\frac{1}{a^{4}}dsda$
$\displaystyle=\int_{-\infty}^{\infty}\int_{0}^{\infty}\left|\left({\mathcal{F}}_{1}f\right)\left(s,a\right)\right|^{2}a^{1+{\frac{2}{p^{\prime}}}-4}|s|^{\frac{2}{p^{\prime}}-1}dsda.$
(2.3)

For $0<\alpha<\frac{1}{2}$, let us choose

$f_{\alpha}(b,a)=\begin{cases}|b|^{-\alpha}a^{2},\quad(b,a)\in Q\\\
0,\quad\quad\text{otherwise},\end{cases}$

where $Q=\\{(b,a):-p\leq b,a\leq p\\},p>0$. Hence

$({\mathcal{F}}_{1}f_{\alpha})(s,a)=\left[(2\pi)^{-1/2}\int_{-p}^{p}e^{-ibs}|b|^{-\alpha}db\right]a^{2}.$

Now for $s>0$,

$\int_{-p}^{p}e^{-ibs}|b|^{-\alpha}db=2\left(\int_{0}^{sp}t^{-\alpha}\cos
tdt\right)s^{-1+\alpha},\quad j=1,2.$

When $\alpha\in(0,\frac{1}{2})$, then $|\int\limits_{0}^{sp}t^{-\alpha}\cos
tdt|\geq B,$ for some $B>0$. Now

$\displaystyle\int_{-\infty}^{\infty}\int_{0}^{\infty}\left|\left({\mathcal{F}}_{1}f\right)\left(s,a\right)\right|^{2}a^{1+{\frac{2}{p^{\prime}}}-4}|s|^{\frac{2}{p^{\prime}}-1}dsda$
$\displaystyle\geq\dfrac{2B^{2}}{\pi}\int_{0}^{p}\int_{R}^{\infty}s^{2(-1+\alpha)+\frac{2}{p^{\prime}}-1}a^{1+{\frac{2}{p^{\prime}}}}dsda.$

But

$\int_{R}^{\infty}s^{2(-1+\alpha)+{\frac{2}{p^{\prime}}-1}}ds=\infty,$

when $\alpha>1-\frac{1}{p^{\prime}}$. Also $f_{\alpha}$ is square integrable
if $\alpha<\frac{1}{2}$. Let us choose
$\frac{1}{2}>\alpha>1-\frac{1}{p^{\prime}}$, then the Equation (2) becomes

$\|\widehat{f_{\alpha}}(\rho_{+})K_{+}^{\frac{1}{p^{\prime}}}\|_{S_{2}}^{2}+\|\widehat{f_{\alpha}}(\rho_{-})K_{-}^{\frac{1}{p^{\prime}}}\|_{S_{2}}^{2}=\infty.$

Hence for given $f_{\alpha},\frac{1}{2}>\alpha>1-\frac{1}{p^{\prime}}$, we get

$\|{\mathcal{F}}_{p}(f_{\alpha})\rho_{+}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f_{\alpha})\rho_{-}\|_{S_{p^{\prime}}}^{p^{\prime}}=\infty,1<p^{\prime}<2.$

##  3\. Similitude Group

This section details the Fourier analysis of the similitude group,
$\mathbb{SIM}(2)$. In this setup, we will define the Weyl transform formula
and show that it cannot be extended as a bounded operator for the symbol in
the corresponding $L^{p}$ spaces with $2<p<\infty$.

The similitude group, $\mathbb{SIM}(2)$, can be considered as the
generalization of the affine group. This group contains affine group as a
subgroup. Also one can consider $\mathbb{SIM}(2)$ as a complexification of the
affine group, $U$. This group arises in the study of $2$-dimensional wavelet
transforms. It is a four-parameter group that contains the translations in the
image plane, $\mathbb{R}^{2}$, global dilations (zooming in and out by $a>0$),
and rotations around the origin ($\theta\in[0,2\pi)$). The action on the plane
is given by,

$x=(b,a,\theta)y=aR_{\theta}y+b,\quad x,y\in\mathbb{R}^{2},$

where $b\in\mathbb{R}^{2}$, $a\in\mathbb{R}$ and $R_{\theta}$ is the $2\times
2$ rotation matrix,

$R_{\theta}=\left(\begin{matrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{matrix}\right).$ (3.1)

An useful representation of $(b,a,\theta)$ is a $3\times 3$ matrix given by

$(b,a,\theta)=\left(\begin{matrix}aR_{\theta}&b\\\
0^{\intercal}&1\end{matrix}\right),~{}~{}~{}0^{\intercal}=(0,0).$ (3.2)

Matrix multiplication then gives the composition of successive transformations
and thus the group law is derived from that,

$\displaystyle(b,a,\theta)\ast(b^{\prime},a^{\prime},\theta^{\prime})$
$\displaystyle=$
$\displaystyle(b+aR_{\theta}b^{\prime},aa^{\prime},\theta+\theta^{\prime})$
$\displaystyle e$ $\displaystyle=$ $\displaystyle(0,1,0)$
$\displaystyle(b,a,\theta)^{-1}$ $\displaystyle=$
$\displaystyle(-a^{-1}R_{-\theta}b,a^{-1},-\theta).$

With respect to the operation $\ast$, $\mathbb{SIM}(2)$ is a non-abelian group
in which $({0},1,0)$ is the identity element and
$(\frac{-1}{a}R_{-\theta}{b},\frac{1}{a},-\theta)$ is the inverse of
$({b},a,\theta)$ in $\mathbb{SIM}(2)$. Also, it can be shown that
$\mathbb{SIM}(2)$ is a non-unimodular group as its left and right Haar
measures given by

$d\mu_{L}({b},a,\theta)=\dfrac{d{b}dad\theta}{a^{3}},~{}~{}d\mu_{R}({b},a,\theta)=\dfrac{d{b}dad\theta}{a},$

respectively, are different.

Moreover, the similitude group $\mathbb{SIM}(2)$, has the structure of a
semidirect product:

$\mathbb{SIM}(2)=\mathbb{R}^{2}\rtimes(\mathbb{R}_{\ast}^{+}\times\mathrm{SO}(2)),$

where $\mathbb{R}^{2}$ is the subgroup of the translations,
$\mathbb{R}_{\ast}^{+}$ that of dilations and $\mathrm{SO}(2)$ of rotations.
Topologically, one can identify $\mathbb{R}^{2}$ with $\mathbb{C}$, the
complex plane and $\mathbb{R}_{\ast}^{+}\times\mathrm{SO}(2)$ with
$\mathbb{C}^{\ast},$ the complex plane with the origin is removed. Then

$\mathbb{SIM}(2)=\mathbb{C}\rtimes C^{\ast},$

consisting of $(z,w),$ where $z\in\mathbb{C}$ and $w\in\mathbb{C}^{\ast},$ the
group composition law is,

$(z_{1},w_{1})(z_{2},w_{2})=(z_{1}+w_{1}z_{2},w_{1}w_{2}).$

In particular, if one considers the elements $(z,w)$, with $z=b+ic,$ $c=0$ and
$w=ae^{i\theta},$ $\theta=0$ then these elements clearly constitute a
subgroup, which is just the affine group of the line. Thus the affine group
forms a subgroup of $\mathbb{SIM}(2)$.

Let ${\mathcal{U}}(L^{2}(\mathbb{R}^{2}))$ be the set of all unitary operators
on $L^{2}(\mathbb{R}^{2})$. We denote
$\pi:\mathbb{SIM}(2)\rightarrow{\mathcal{U}}(L^{2}(\mathbb{R}^{2}))$ be the
mapping of $\mathbb{SIM}(2)$ into the group
${\mathcal{U}}(L^{2}(\mathbb{R}^{2}))$, is given by

$\big{(}\pi({b},a,\theta)\phi\big{)}(x)=ae^{-i{b}\cdot
x}\phi(aR_{-\theta}x),\quad x\in\mathbb{R}^{2},$

for all $(b,a,\theta)$ in $\mathbb{SIM}(2)$ and all $\phi$ in
$L^{2}(\mathbb{R}^{2})$. Here $\pi$ is the only irreducible unitary
representation of $\mathbb{SIM}(2)$ on $L^{2}(\mathbb{R}^{2})$ upto
equivalence, [6]. i.e., dual of $\mathbb{SIM}(2)$ is just singleton.

Let $f\in L^{1}(\mathbb{SIM}(2))\cap L^{2}(\mathbb{SIM}(2))$ and define the
Fourier transform $\widehat{f}$ of $f$ on
$\widehat{\mathbb{SIM}(2)}=\\{\pi\\}$ by

$(\widehat{f}(\pi)\phi)(x)=\int\limits_{\mathbb{SIM}(2)}f({b},a,\theta)(\pi({b},a,\theta)\phi)(x)\frac{d{b}dad\theta}{a^{3}},\quad
x\in\mathbb{R}^{2},$

for all $\phi\in L^{2}(\mathbb{R}^{2}).$

Define the function $K_{\pi}\phi$ on $\mathbb{R}^{2}$ by,

$(K_{\pi}\phi)(x)=\|x\|_{2}^{2}\phi(x),\quad x\in\mathbb{R}^{2},$ (3.3)

where $\|x\|_{2}=\sqrt{x_{1}^{2}+x_{2}^{2}}$. Define the $L^{p}$-Fourier
transform
${\mathcal{F}}_{p}(f)\pi=\widehat{f}(\pi)K_{\pi}^{\frac{1}{p^{\prime}}},1\leq
p\leq\infty,\frac{1}{p}+\frac{1}{p^{\prime}}=1.$ Now recall the Fourier
inversion and Plancherel formula for the similitude group from [6].

######  Theorem 3.1.

(Inversion theorem)

Let $f$ be in $L^{1}(\mathbb{SIM}(2))\cap L^{2}(\mathbb{SIM}(2))$. Then

$f(b,a,\theta)=\operatorname{Tr}\left(\pi(b,a,\theta)^{\ast}\widehat{f}(\pi)K_{\pi}\right).$

Let $S_{r}$ be the space of all $r$-Schatten class operators, which is a
Banach space with the norm
$\|A\|_{S_{r}}^{r}=\operatorname{Tr}\left(A^{\ast}A\right)^{r/2}$ for $1\leq
p<\infty$, and for $r=\infty$, the norm is the usual operator norm.

######  Theorem 3.2.

(Plancherel formula) Let $f$ be in $L^{1}(\mathbb{SIM}(2))\cap
L^{2}(\mathbb{SIM}(2))$. Then

$\int_{\mathbb{SIM}(2)}|f(b,a,\theta)|^{2}\dfrac{dbdad\theta}{a^{3}}=\|\widehat{f}(\pi)K_{\pi}^{1/2}\|_{S_{2}}^{2}.$

For $G=\mathbb{SIM}(2)$, recall the definition of Weyl transform defined in
(1.1), and the Theorem 1.6 says that it is bounded when $1\leq p\leq 2$. Now
we prove the unboundedness of Weyl transforms when $p>2$ for the similitude
group.

######  Theorem 3.3.

For $2<p<\infty$, there exists an operator valued function $\sigma$ in
$L^{p}(\mathbb{SIM}(2)\times\widehat{\mathbb{SIM}(2)},S_{p})$ such that
$W_{\sigma}$ is not a bounded linear operator on $L^{2}(\mathbb{SIM}(2))$.

We prove the Theorem 3.3 using the method of contrapositive in three
propositions.

######  Proposition 3.4.

Let $2<p<\infty$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Then for all
$\sigma\in L^{p}(\mathbb{SIM}(2)\times\widehat{\mathbb{SIM}(2)},S_{p})$, the
Weyl transform $W_{\sigma}$ is bounded linear operator on
$L^{2}(\mathbb{SIM}(2))$ iff there exists a constant $C$ such that

$\|W(f,g)\|_{p^{\prime},\mu}\leq
C\|g\|_{L^{2}(\mathbb{SIM}(2))}\|f\|_{L^{2}(\mathbb{SIM}(2))},$

for all $f,g$ in $L^{2}(\mathbb{SIM}(2))$.

###### Proof.

The proof follows from Theorem 2.4. ∎

######  Proposition 3.5.

Let $2<p<\infty$ and $f$ be a square integrable, compactly supported function
on $\mathbb{SIM}(2)$ such that
$\int\limits_{\mathbb{SIM}(2)}f(b,a,\theta)d\mu_{L}(b,a,\theta)\neq 0$. If
$W_{\sigma}$ is a bounded operator on $L^{2}(\mathbb{SIM}(2))$ for all
$\sigma$ in $L^{p}(\mathbb{SIM}(2)\times\widehat{\mathbb{SIM}(2)},S_{p})$,
then

$\|{\mathcal{F}}_{p}(f)\pi\|_{S_{p^{\prime}}}^{p^{\prime}}<\infty.$

###### Proof.

The proof follows from Theorem 2.5. ∎

######  Proposition 3.6.

For $p\in(2,\infty)$, does there exists a square-integrable, compactly
supported function $f$ on $\mathbb{SIM}(2)$ with
$\int\limits_{\mathbb{SIM}(2)}f(b,a,\theta)d\mu(b,a,\theta)\neq 0$ such that
$\|{\mathcal{F}}_{p}(f)\pi\|_{S_{p^{\prime}}}^{p^{\prime}}=\infty?$

The answer to this question is yes. We will certainly find a function
$f_{\alpha},0<\alpha<\frac{1}{2}$, at the end of the section, which is needed
for Proposition 3.6. The $L^{p}$-Fourier transform, for $f\in
L^{1}(\mathbb{SIM}(2)\cap L^{p}(\mathbb{SIM}(2)$, is defined by

${\mathcal{F}}_{p}(f)\pi=\widehat{f}(\pi)K_{\pi}^{\frac{1}{p^{\prime}}},$

where $(K_{\pi}\phi)(x)=\|x\|_{2}^{2}\phi(x)$. The integral representation of
the operator ${\mathcal{F}}_{p}(f)\pi$ is given by

$\left(\widehat{f}(\pi)K_{\pi}^{\frac{1}{p^{\prime}}}\phi\right)(x)=\iint_{\mathbb{R}^{2}}({\mathcal{F}}_{1}f)\left(x,\dfrac{\|y\|}{\|x\|},\cos^{-1}\left(\dfrac{x\cdot
y}{\|x\|\|y\|}\right)\right)\|y\|^{\frac{2}{p^{\prime}}-1}\dfrac{\|x\|}{\|y\|^{2}}\phi(y)dy,$

and the kernel

$K^{f}(x,y)=\begin{cases}({\mathcal{F}}_{1}f)\left(x,\dfrac{\|y\|}{\|x\|},\cos^{-1}\left(\dfrac{x\cdot
y}{\|x\|\|y\|}\right)\right)\|y\|^{\frac{2}{p^{\prime}}-1}\dfrac{\|x\|}{\|y\|^{2}},&~{}~{}x\neq
0,y\neq 0,\\\ 0,&~{}~{}otherwise.\end{cases}$ (3.4)

Now

$\displaystyle\iint_{\mathbb{R}^{4}}|K^{f}(x,y)|^{2}dxdy$
$\displaystyle=\iint_{\mathbb{R}^{4}}\left|({\mathcal{F}}_{1}f)\left(x,\dfrac{\|y\|}{\|x\|},\cos^{-1}\left(\dfrac{x\cdot
y}{\|x\|\|y\|}\right)\right)\right|^{2}\|y\|^{\frac{4}{p^{\prime}}-2}\dfrac{\|x\|^{2}}{\|y\|^{4}}dxdy$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{0}^{\infty}\int_{0}^{2\pi}|({\mathcal{F}}_{1}f)(x,a,\theta)|^{2}(a\|x\|_{2})^{\frac{4}{p^{\prime}}-2}\dfrac{dxdad\theta}{a^{3}}.$

For $0<\alpha<\frac{1}{2}$, let us choose

$f_{\alpha}(b,a,\theta)=\begin{cases}|b_{1}|^{-\alpha}|b_{2}|^{-\alpha}a^{3}\theta,\quad(b,a,\theta)\in
Q\\\ 0,\quad\quad\text{otherwise},\end{cases}$

where $Q=\\{(b,a,\theta):-p\leq b_{1},b_{2},a\leq p,\theta\in[0,2\pi)\\}$.
Hence

$({\mathcal{F}}_{1}f_{\alpha})(\xi,a,\theta)=\left[(2\pi)^{-1}\int_{-p}^{p}e^{-ib_{1}\xi_{1}}|b_{1}|^{-\alpha}db_{1}\int_{-p}^{p}e^{-ib_{2}\xi_{2}}|b_{2}|^{-\alpha}db_{2}\right]a^{3}\theta.$
(3.5)

Now for $\xi_{1},\xi_{2}>0$,

$\int_{-p}^{p}e^{-ib_{j}\xi_{j}}|b_{j}|^{-\alpha}db_{j}=2\left(\int_{0}^{\xi_{j}p}t^{-\alpha}\cos
tdt\right)\xi_{j}^{-1+\alpha},\quad j=1,2.$ (3.6)

When $\alpha\in(0,\frac{1}{2})$, then
$|\int\limits_{0}^{\xi_{j}p}t^{-\alpha}\cos tdt|\geq A,$ for some $A>0$. Since
$\dfrac{1}{\sqrt{2}}(|\xi_{1}|+|\xi_{2}|)\leq\|\xi\|_{2}\leq(|\xi_{1}|+|\xi_{2}|)$,
then $\|\xi\|_{2}\geq\dfrac{1}{\sqrt{2}}|\xi_{1}|$. Now using the equations
(3.5) and (3.5), we get

$\displaystyle\iint_{\mathbb{R}^{4}}|K^{f_{\alpha}}(x,y)|^{2}dxdy$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{0}^{\infty}\int_{0}^{2\pi}|({\mathcal{F}}_{1}f_{\alpha})(\xi,a,\theta)|^{2}(a\|\xi\|_{2})^{\frac{4}{p^{\prime}}-2}\dfrac{d\xi
dad\theta}{a^{3}}$
$\displaystyle\geq(2\pi)^{-2}(2A)^{4}\int_{0}^{2\pi}\int_{0}^{\infty}\int_{R}^{\infty}\int_{R}^{\infty}\xi_{1}^{2(-1+\alpha)}\xi_{2}^{2(-1+\alpha)}a^{3}\theta\left(a|\xi_{1}|\right)^{\frac{4}{p^{\prime}}-2}\dfrac{d\xi
dad\theta}{a^{3}}$
$\displaystyle=\dfrac{4A^{4}}{\pi^{2}}C_{1}^{\frac{4}{p^{\prime}}-2}\int_{0}^{2\pi}\theta
d\theta\int_{0}^{\infty}a^{\frac{4}{p^{\prime}}-2}da\int_{R}^{\infty}\xi_{1}^{2(-1+\alpha)+{\frac{4}{p^{\prime}}-2}}d\xi_{1}\int_{R}^{\infty}\xi_{2}^{2(-1+\alpha)}d\xi_{2}.$

But

$\int_{R}^{\infty}\xi_{1}^{2(-1+\alpha)+{\frac{4}{p^{\prime}}-2}}d\xi_{1}=\infty,$

when $\alpha>\frac{3}{2}-\frac{2}{p^{\prime}}$. Also $f_{\alpha}$ is square
integrable if $\alpha<\frac{1}{2}$. Let us choose
$\frac{1}{2}>\alpha>\frac{3}{2}-\frac{2}{p^{\prime}},2>p^{\prime}>\frac{4}{3}$,
and
$\frac{1}{2}>\alpha>0>\frac{3}{2}-\frac{2}{p^{\prime}},1<p^{\prime}\leq\frac{4}{3}$,
then $\widehat{f_{\alpha}}(\pi)K_{\pi}^{\frac{1}{p^{\prime}}}\notin S_{2}$.
Thus for $2<p<\infty$,

$\|\widehat{f_{\alpha}}(\pi)K_{\pi}^{\frac{1}{p^{\prime}}}\|_{S_{p^{\prime}}}=\infty,\frac{1}{p}+\frac{1}{p^{\prime}}=1,$

where
$\frac{1}{2}>\alpha>\frac{3}{2}-\frac{2}{p^{\prime}},2>p^{\prime}>\frac{4}{3}$,
and
$\frac{1}{2}>\alpha>0>\frac{3}{2}-\frac{2}{p^{\prime}},1<p^{\prime}\leq\frac{4}{3}$.

##  4\. Affine Poincaré Group

This section covers the Fourier analysis of the affine Poincaré group,
${\mathcal{P_{\textrm{aff}}}}$, in detail. In this setup, we will define the
Weyl transform and show that it cannot be extended as a bounded operator for
the symbol in the corresponding $L^{p}$ spaces with $2<p<\infty$.

The affine Poincaré group, ${\mathcal{P_{\textrm{aff}}}}$, is a generalization
of the affine group, or more precisely, a complexification of the affine
group. This group emerges from the investigation of the $2$-dimensional
wavelet transform. Similar to the four-parameter similitude group,
${\mathcal{P_{\textrm{aff}}}}$ contains the translations $b$ in the image
plane $\mathbb{R}^{2}$, global dilations (zooming in and out by $a>0$), but it
has hyperbolic rotations around the origin (${\vartheta}\in\mathbb{R}$). The
action on the plane is given by

$x=(b,a,{\vartheta})y=a\Lambda_{{\vartheta}}y+b,$

where $b\in\mathbb{R}^{2}$, $a>0$, and $\Lambda_{{\vartheta}}$ is the $2\times
2$ hyperbolic rotation matrix

$\Lambda_{{\vartheta}}=\left(\begin{matrix}\cosh{\vartheta}&\sinh{\vartheta}\\\
\sinh{\vartheta}&\cosh{\vartheta}\end{matrix}\right).$ (4.1)

A convenient representation of the joint transformation $(b,a,{\vartheta})$ is
in the form of $3\times 3$ matrices

$(b,a,{\vartheta})=\left(\begin{matrix}a\Lambda_{\vartheta}&b\\\
0^{T}&1\end{matrix}\right),~{}~{}~{}0^{T}=(0,0).$ (4.2)

Then the matrix multiplications gives the composition of successive
transformations and thus the group law is derived as

$\displaystyle(b,a,{\vartheta})\ast(b^{\prime},a^{\prime},{\vartheta}^{\prime})$
$\displaystyle=$
$\displaystyle(b+a\Lambda_{{\vartheta}}b^{\prime},aa^{\prime},{\vartheta}+{\vartheta}^{\prime}),$

With respect to the operation $\ast$, ${\mathcal{P_{\textrm{aff}}}}$ is a non-
abelian group in which $({0},1,0)$ is the identity element and
$(\frac{-1}{a}\Lambda_{-{\vartheta}}{b},\frac{1}{a},-{\vartheta})$ is the
inverse of $({b},a,{\vartheta})$ in ${\mathcal{P_{\textrm{aff}}}}$. Also, it
can be shown that ${\mathcal{P_{\textrm{aff}}}}$ is a non-unimodular group as
its left and right Haar measures

$d\mu_{L}({b},a,{\vartheta})=\dfrac{d{b}dad{\vartheta}}{a^{3}},~{}~{}d\mu_{R}({b},a,{\vartheta})=\dfrac{d{b}dad{\vartheta}}{a},$

respectively are different, and hence the modular function is given by
$\Delta(b,a,{\vartheta})=\dfrac{1}{a^{2}}$. Moreover, the affine Poincaré
group ${\mathcal{P_{\textrm{aff}}}}$ has the structure of a semi-direct
product:

${\mathcal{P_{\textrm{aff}}}}=\mathbb{R}^{2}\rtimes(\mathbb{R}_{\ast}^{+}\times\mathrm{SO}(1,1)),$

where $\mathbb{R}^{2}$ is the subgroup of the translations,
$\mathbb{R}^{\ast}_{+}$ that of dilations, and $\mathrm{SO}(1,1)$ of
hyperbolic rotations. Topologically, one can write
${\mathcal{P_{\textrm{aff}}}}=\mathbb{R}^{2}\times\mathcal{C}$, where
$\mathcal{C}$ is any one of the four cones:

$\displaystyle
C_{1}^{1}=\\{x\in\mathbb{R}^{2}:x_{1}^{2}>x_{2}^{2},+x_{1}>0\\}$ (4.3)
$\displaystyle
C_{2}^{1}=\\{x\in\mathbb{R}^{2}:x_{1}^{2}>x_{2}^{2},-x_{1}>0\\}$ (4.4)
$\displaystyle
C_{1}^{2}=\\{x\in\mathbb{R}^{2}:x_{1}^{2}<x_{2}^{2},+x_{1}>0\\}$ (4.5)
$\displaystyle
C_{2}^{2}=\\{x\in\mathbb{R}^{2}:x_{1}^{2}<x_{2}^{2},-x_{1}>0\\}.$ (4.6)

Let us define the Fourier transform ${\mathcal{F}}$ and inverse Fourier
transform ${\mathcal{F}}^{-1}$, by

$\displaystyle(\mathcal{F}\varphi)(\xi)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{i\langle\xi;x\rangle}\varphi(x)dx$
(4.7)
$\displaystyle(\mathcal{F}^{-1}\varphi)(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}e^{-i\langle\xi;x\rangle}({\mathcal{F}}{\varphi})(\xi)d\xi,$
(4.8)

for all $\varphi\in S(\mathbb{R}^{n})$, where $\langle
x;y\rangle=x_{1}y_{1}-x_{2}y_{2}$ is the Minkowski inner product. In this
section, we denote ${\mathcal{F}}$ and ${\mathcal{F}}^{-1}$ are the Minkowski-
Fourier transform and inverse Minkowski-Fourier transform on $\mathbb{R}^{2}$,
respectively.

######  Remark 1.

The relation between Euclidean Fourier transform and Minkowski-Fourier
transform on $\mathbb{R}^{2}$ is

$(\mathcal{F}\varphi)(\xi_{1},\xi_{2})=\widehat{\varphi}(-\xi_{1},\xi_{2}),({\mathcal{F}}^{-1}\varphi)(\xi_{1},\xi_{2})=\widecheck{\varphi}(-\xi_{1},\xi_{2}),$

where $\widehat{\varphi}$ and $\widecheck{\varphi}$ are the Euclidean Fourier
transform and inverse Euclidean Fourier transform $\varphi$ on
$\mathbb{R}^{2}$, respectively. The reader can easily verify that the
Minkowski-Fourier transform carries almost all properties similar to Euclidean
Fourier transform on $\mathbb{R}^{2}$, like the Plancherel formula, and
Parseval identity.

Define the mappings
$\pi_{i}^{j}:{\mathcal{P_{\textrm{aff}}}}\to{\mathcal{U}}(L^{2}(C_{i}^{j})),$
by

$\left(\pi_{i}^{j}(b,a,{\vartheta})\phi\right)(x)=ae^{\langle
x;b\rangle}\phi(a\Lambda_{{\vartheta}}x),$

for all $(b,a,{\vartheta})\in{\mathcal{P_{\textrm{aff}}}}$ and $\phi\in
L^{2}(C_{i}^{j}),i,j=1,2.$ Let $f\in L^{1}({\mathcal{P_{\textrm{aff}}}})\cap
L^{2}({\mathcal{P_{\textrm{aff}}}})$, the Fourier transform is defined by

$\left(\widehat{f}(\pi_{i}^{j})\phi\right)(x)=\int_{{\mathcal{P_{\textrm{aff}}}}}f(b,a,{\vartheta})\left(\pi_{i}^{j}(b,a,{\vartheta})\phi\right)(x)d\mu_{L}(b,a,{\vartheta}),$

for all $\phi\in L^{2}(C_{i}^{j}),i,j=1,2.$ Define the Duflo-Moore operators,
[7],

$(K_{i,j}\phi)(x)=\frac{1}{2\pi}|\langle x;x\rangle|\phi(x),i,j=1,2.$

Define the $L^{p}$-Fourier transform

${\mathcal{F}}_{p}f(\pi_{i}^{j})=\widehat{f}(\pi_{i}^{j})K_{i,j}^{\frac{1}{p^{\prime}}},i,j=1,2,$

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1,1\leq p\leq\infty$.

######  Theorem 4.1 (Plancherel Theorem).

[7] Let $f$ be in $L^{1}({\mathcal{P_{\textrm{aff}}}})\cap
L^{2}({\mathcal{P_{\textrm{aff}}}})$. Then

$\int_{{\mathcal{P_{\textrm{aff}}}}}|f(b,a,{\vartheta})|^{2}\dfrac{dbdad\theta}{a^{3}}=\sum_{i,j=1}^{2}\|\widehat{f}(\pi_{i}^{j})K_{i,j}^{\frac{1}{2}}\|_{S_{2}}^{2}.$

######  Theorem 4.2 (Inversion Theorem).

[7] Let $f$ be in $L^{1}({\mathcal{P_{\textrm{aff}}}})\cap
L^{2}({\mathcal{P_{\textrm{aff}}}})$. Then

$f(b,a,{\vartheta})=\sum_{i,j=1}^{2}\operatorname{Tr}\left(\pi_{i}^{j}(b,a,{\vartheta})^{\ast}\widehat{f}(\pi_{i}^{j})K_{i,j}\right).$

For $G={\mathcal{P_{\textrm{aff}}}}$, recall the definition of Weyl transform
defined in (1.1), and the Theorem 1.6 says that it is bounded when $1\leq
p\leq 2$. Now we prove the unboundedness of Weyl transform when $p>2$.

######  Theorem 4.3.

For $2<p<\infty$, there exists an operator valued function $\sigma$ in
$L^{p}(U\times\widehat{U},S_{p})$ such that $W_{\sigma}$ is not a bounded
linear operator on $L^{2}(U)$.

We prove the Theorem 4.3 using the method of contrapositive in three
propositions.

######  Proposition 4.4.

Let $2<p<\infty$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Then for all
$\sigma\in
L^{p}({\mathcal{P_{\textrm{aff}}}}\times\widehat{{\mathcal{P_{\textrm{aff}}}}},S_{p})$,
the Weyl transform $W_{\sigma}$ is bounded linear operator on
$L^{2}({\mathcal{P_{\textrm{aff}}}})$ iff there exists a constant $C$ such
that

$\|W(f,g)\|_{p^{\prime},\mu}\leq
C\|g\|_{L^{2}({\mathcal{P_{\textrm{aff}}}})}\|f\|_{L^{2}({\mathcal{P_{\textrm{aff}}}})},$

for all $f,g$ in $L^{2}({\mathcal{P_{\textrm{aff}}}})$.

###### Proof.

The proof is followed from Theorem 2.4. ∎

######  Proposition 4.5.

Let $2<p<\infty$ and $f$ be a square integrable, compactly supported function
on ${\mathcal{P_{\textrm{aff}}}}$ such that
$\int\limits_{{\mathcal{P_{\textrm{aff}}}}}f(b,a,{\vartheta})d\mu_{L}(b,a,{\vartheta})\neq
0$. If $W_{\sigma}$ is a bounded operator on
$L^{2}({\mathcal{P_{\textrm{aff}}}})$ for all $\sigma$ in
$L^{p}({\mathcal{P_{\textrm{aff}}}}\times\widehat{{\mathcal{P_{\textrm{aff}}}}},S_{p})$,
then

$\|{\mathcal{F}}_{p}(f)\pi_{1}^{1}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\pi_{2}^{2}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\pi_{2}^{1}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\pi_{1}^{2}\|_{S_{p^{\prime}}}^{p^{\prime}}<\infty.$

###### Proof.

The proof is followed from Theorem 2.5. ∎

######  Proposition 4.6.

For $p\in(2,\infty)$, does there exists a square-integrable, compactly
supported function $f$ on ${\mathcal{P_{\textrm{aff}}}}$ with
$\int\limits_{{\mathcal{P_{\textrm{aff}}}}}f(b,a,\vartheta)d\mu(b,a,{\vartheta})\neq
0$ such that

$\|{\mathcal{F}}_{p}(f)\pi_{1}^{1}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\pi_{2}^{2}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\pi_{2}^{1}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f)\pi_{1}^{2}\|_{S_{p^{\prime}}}^{p^{\prime}}=\infty?$

The answer to this question is yes. We will certainly find a function
$f_{\alpha},0<\alpha<\frac{1}{2}$, at the end of the section, which is needed
for Proposition 4.6. Let $f\in L^{2}({\mathcal{P_{\textrm{aff}}}})$. For
$i,j=1,2$, consider the operators

$\displaystyle\left(\widehat{f}(\pi_{i}^{j})K_{i,j}^{\frac{1}{p^{\prime}}}\phi\right)(x)$
$\displaystyle=\int_{{\mathcal{P_{\textrm{aff}}}}}f(b,a,{\vartheta})\left(\pi_{i}^{j}(b,a,{\vartheta})K_{i,j}^{\frac{1}{p^{\prime}}}\phi\right)(x)\dfrac{dbdad{\vartheta}}{a^{3}}$
$\displaystyle=\frac{1}{2\pi}\int_{{\mathcal{P_{\textrm{aff}}}}}f(b,a,{\vartheta})ae^{i\langle
x;b\rangle}|\langle
a\Lambda_{-{\vartheta}}x;a\Lambda_{-{\vartheta}}x|^{\frac{1}{p^{\prime}}}\phi(a\Lambda_{-{\vartheta}}x)\dfrac{dbdad{\vartheta}}{a^{3}}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}}a^{1+\frac{2}{p^{\prime}}}({\mathcal{F}}_{1}f)(x,a,{\vartheta})|\langle
x;x|^{\frac{1}{p^{\prime}}}\phi(a\Lambda_{-{\vartheta}}x)\dfrac{dad{\vartheta}}{a^{3}},$

where $\phi\in L^{2}(C_{i}^{j})$. Substitute $a\Lambda_{-{\vartheta}}x=y$,
then $a=\Bigg{(}\dfrac{\langle y;y\rangle}{\langle
x;x\rangle}\Bigg{)}^{\frac{1}{2}}$,
${\vartheta}=\cosh^{-1}\Bigg{(}\dfrac{\langle x;y\rangle}{\langle
x;x\rangle}\times\Bigg{(}\dfrac{\langle x;x\rangle}{\langle
y;y\rangle}\Bigg{)}^{\frac{1}{2}}\Bigg{)}$ and $dy=a|\langle
x;x\rangle|dad{\vartheta}.$

Hence the integral representations become

$\left(\widehat{f}(\pi_{i}^{j})K_{i,j}^{\frac{1}{p^{\prime}}}\phi\right)(x)=\int_{C_{i}^{j}}K_{i,j}^{f}(x,y)\phi(y)dy,$

where

$K_{i,j}^{f}(x,y)=(\mathcal{F}_{1}f)\Bigg{(}x,\Bigg{(}\dfrac{\langle
y;y\rangle}{\langle
x;x\rangle}\Bigg{)}^{\frac{1}{2}},\cosh^{-1}\Bigg{(}\dfrac{\langle
x;y\rangle}{\langle x;x\rangle}\times\Bigg{(}\dfrac{\langle
x;x\rangle}{\langle
y;y\rangle}\Bigg{)}^{\frac{1}{2}}\Bigg{)}\Bigg{(}\dfrac{\langle
y;y\rangle}{\langle
x;x\rangle}\Bigg{)}^{\frac{1}{2}+\frac{1}{p^{\prime}}}|\langle
x;x\rangle|^{\frac{1}{p^{\prime}}}\dfrac{|\langle x;x\rangle|}{|\langle
y,y\rangle|^{2}},$ (4.9)

for all $(x,y)\in C_{i}^{j}\times C_{i}^{j}$, $i,j=1,2$. Now using the
Plancherel formula for Minkowski-Fourier transform ${\mathcal{F}}$, we get

$\displaystyle\|{\mathcal{F}}_{p}(f)\pi_{1}^{1}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f)\pi_{1}^{2}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f)\pi_{2}^{1}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f)\pi_{2}^{2}\|_{S_{2}}^{2}$
$\displaystyle=\sum_{i,j=1}^{2}\int_{C_{i}^{j}}\int_{C_{i}^{j}}|K_{i,j}^{f}(x,y)|^{2}dxdy$
$\displaystyle=\sum_{i,j=1}^{2}\int_{C_{i}^{j}}\int_{C_{i}^{j}}\left|{(\mathcal{F}_{1}f)\Bigg{(}x,\Bigg{(}\dfrac{\langle
y;y\rangle}{\langle
x;x\rangle}\Bigg{)}^{\frac{1}{2}},\cosh^{-1}\Bigg{(}\dfrac{\langle
x;y\rangle}{\langle x;x\rangle}\times\Bigg{(}\dfrac{\langle
x;x\rangle}{\langle
y;y\rangle}\Bigg{)}^{\frac{1}{2}}\Bigg{)}}\right|^{2}\Bigg{(}\dfrac{\langle
y;y\rangle}{\langle x;x\rangle}\Bigg{)}^{1+\frac{2}{p^{\prime}}}$
$\displaystyle\times|\langle x;x\rangle|^{\frac{2}{p^{\prime}}}\dfrac{|\langle
x;x\rangle|^{2}}{|\langle y,y\rangle|^{4}}dxdy$
$\displaystyle=\sum_{i,j=1}^{2}\int_{C_{i}^{j}}\int_{0}^{\infty}\int_{\mathbb{R}}\left|(\mathcal{F}_{1}f)(x,a,{\vartheta})\right|^{2}a^{1+\frac{2}{p^{\prime}}}|\langle
x;x\rangle|^{\frac{2}{p^{\prime}}}\dfrac{|\langle x;x\rangle|^{2}}{|\langle
y,y\rangle|^{4}}a|\langle x;x\rangle|dad{\vartheta}dx$
$\displaystyle=\sum_{i,j=1}^{2}\int_{C_{i}^{j}}\int_{0}^{\infty}\int_{\mathbb{R}}\left|(\mathcal{F}_{1}f)(x,a,{\vartheta})\right|^{2}a^{\frac{2}{p^{\prime}}-6}|\langle
x;x\rangle|^{\frac{2}{p^{\prime}}-1}dad{\vartheta}dx$
$\displaystyle=\int_{\mathbb{R}^{2}}\int_{0}^{\infty}\int_{\mathbb{R}}\left|(\mathcal{F}_{1}f)(x,a,{\vartheta})\right|^{2}a^{\frac{2}{p^{\prime}}-6}|\langle
x;x\rangle|^{\frac{2}{p^{\prime}}-1}dxdad{\vartheta}.$ (4.10)

Let us consider a function on $\mathbb{R}^{2}$, with $0<\alpha<\frac{1}{2}$,

$\phi_{\alpha}(b)=\begin{cases}|b_{1}|^{-\alpha}|b_{2}|^{-\alpha},\quad b\in
Q,\\\ 0,\quad\quad\text{otherwise},\end{cases}$

where $Q=\\{b\in\mathbb{R}^{2}:-p\leq b_{j}\leq p,j=1,2\\}$. Then the
Minkowski-Fourier transform of $\phi_{\alpha}$ becomes

$({\mathcal{F}}\phi_{\alpha})(x)=\dfrac{1}{2\pi}\int_{-p}^{p}e^{b_{1}x_{1}}|b_{1}|^{-\alpha}db_{1}\int_{-p}^{p}e^{-b_{2}x_{2}}|b_{2}|^{-\alpha}db_{2}.$

Now for $x_{2}>0,0<\alpha<\dfrac{1}{2}$,

$\int_{-p}^{p}e^{-b_{2}x_{2}}|b_{2}|^{-\alpha}db_{2}=2\left(\int\limits_{0}^{px_{2}}t^{-\alpha}\cos
tdt\right)x_{2}^{-1+\alpha}\geq 2Ax_{2}^{-1+\alpha},$ (4.11)

and for $x_{1}>0,0<\alpha<\dfrac{1}{2}$,

$\int_{-p}^{p}e^{-b_{1}x_{1}}|b_{1}|^{-\alpha}db_{1}=2\left(\int\limits_{0}^{px_{1}}t^{-\alpha}\cos
tdt\right)x_{1}^{-1+\alpha}\geq 2Ax_{1}^{-1+\alpha}.$ (4.12)

Let us choose a function $f$ on ${\mathcal{P_{\textrm{aff}}}}$, with
$0<\alpha<\frac{1}{2}$,

$f_{\alpha}(b,a,{\vartheta})=\begin{cases}\phi_{\alpha}(b)a^{5}{\vartheta},\quad
b\in Q,0<a<p,-p\leq{\vartheta}\leq p,\\\
0,\quad\quad\text{otherwise}.\end{cases}$

Now using equations (4.11) and (4.12), the equation (4) becomes

$\displaystyle\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{1}^{1}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{1}^{2}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{2}^{1}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{2}^{2}\|_{S_{2}}^{2}$
$\displaystyle\geq
16A^{4}\int_{R}^{\infty}\int_{R}^{\infty}\int_{0}^{p}\int_{-p}^{p}x_{1}^{2(-1+\alpha)}x_{2}^{2(-1+\alpha)}a^{\frac{2}{p^{\prime}}-1}{\vartheta}|\langle
x;x\rangle|^{\frac{2}{p^{\prime}}-1}dxdad{\vartheta}.$ (4.13)

Let $A=\\{(x_{1},x_{2}):R<x_{1},x_{2}<\infty\\}$ and $B=\\{(x_{1},x_{2})\in
A:x_{2}<x_{1}-1\\}$. It is easy to check that

$\displaystyle x_{1}^{2}-x_{2}^{2}=x_{1}^{2}+x_{2}^{2}-2x_{2}^{2}\geq
2x_{1}x_{2}-2x_{2}^{2}=2x_{2}(x_{1}-x_{2})\geq 2x_{2},$

on $B$. Consider the left side of equation (4), we have

$\displaystyle\int_{R}^{\infty}\int_{R}^{\infty}x_{1}^{2(-1+\alpha)}x_{2}^{2(-1+\alpha)}|\langle
x;x\rangle|^{\frac{2}{p^{\prime}}-1}dx$
$\displaystyle\geq\iint_{B}x_{1}^{2(-1+\alpha)}x_{2}^{2(-1+\alpha)}(2x_{2})^{\frac{2}{p^{\prime}}-1}dx$
$\displaystyle=\int_{x_{1}=R}^{\infty}\int_{x_{2}=R}^{x_{2}=x_{1}-1}x_{1}^{2(-1+\alpha)}x_{2}^{2(-1+\alpha)}(2x_{2})^{\frac{2}{p^{\prime}}-1}dx_{1}dx_{2}$
$\displaystyle=2^{\frac{2}{p^{\prime}}-1}\int_{x_{1}=R}^{\infty}\int_{x_{2}=R}^{x_{2}=x_{1}-1}x_{1}^{2(-1+\alpha)}x_{2}^{2(-1+\alpha)+\frac{2}{p^{\prime}}-1}dx_{1}dx_{2}$
$\displaystyle=K\int_{x_{1}=R}^{\infty}x_{1}^{2(-1+\alpha)}(x_{1}-1)^{2(-1+\alpha)+\frac{2}{p^{\prime}}}dx_{1}$
$\displaystyle-
KR^{2(-1+\alpha)+\frac{2}{p^{\prime}}}\int_{R}^{\infty}x_{1}^{2(-1+\alpha)}dx_{1},$

where $K=\dfrac{1}{2(-1+\alpha)+\frac{2}{p^{\prime}}}$. The integral

$\int_{x_{1}=R}^{\infty}x_{1}^{2(-1+\alpha)}(x_{1}-1)^{2(-1+\alpha)+\frac{2}{p^{\prime}}}dx_{1}=\infty,$

when
$\frac{1}{2}>\alpha\geq\frac{3}{4}-\frac{1}{2p^{\prime}}>0,1<p^{\prime}<2$,
and the integral $\int\limits_{R}^{\infty}x_{1}^{2(-1+\alpha)}dx_{1}<\infty,$
when $\alpha<\frac{1}{2}$. Then the left-hand side of equation (4) becomes
infinity, and hence

$\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{1}^{1}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{1}^{2}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{2}^{1}\|_{S_{2}}^{2}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{2}^{2}\|_{S_{2}}^{2}=\infty,$

when
$\frac{1}{2}>\alpha\geq\frac{3}{4}-\frac{1}{2p^{\prime}}>0,1<p^{\prime}<2$.
Thus for these values of $\alpha$,

$\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{1}^{1}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{2}^{2}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{2}^{1}\|_{S_{p^{\prime}}}^{p^{\prime}}+\|{\mathcal{F}}_{p}(f_{\alpha})\pi_{1}^{2}\|_{S_{p^{\prime}}}^{p^{\prime}}=\infty.$

## acknowledgment

The author is thankful to Prof. Aparajita Dasgupta for many fruitful
discussions on the problem.

## References

  * [1]
  * [2] Boggiatto, B. Buzano, E. and Rodino, L. _Global ellipticity and spectral theory_. Mathematical Research, 92(1996).
  * [3] Boggiatto, P. and Rodino, L. “Quantization and pseudo-differential operators.” _Cubo Mathematical Eductional_ , _1_(5), (2003).
  * [4] Chen, L. and Zhao, J. “Weyl transform and generalized spectrogram associated with quaternion Heisenberg group.” _Bulletin des Sciences Mathématiques_ , _136_(2), 127–143(2012).
  * [5] Dasgupta, A. Nayak, S. K., and Wong M. W. “Hilbert-Schmidt and Trace Class Pseudo-Differential Operators and Weyl Transforms on the Affine Group.” _Journal of Pseudo-differential Operators_ , _12_(11), (2021).
  * [6] Dasgupta, A. and Nayak, S. K. “Pseudo-Differential Operators, Wigner Transform and Weyl Transforms on Similitude Group, $\mathbb{SIM}(2)$”, _Bulletin des sciences mathématiques_ , Volume 174, 103087, (2022).
  * [7] Dasgupta, A. and Nayak, S. K. “Pseudo-Differential Operators, Wigner Transform and Weyl Transform on the Affine Poincaré Group.” _Bulletin des sciences mathématiques_ , Volume 184, 103255, (2023).
  * [8] Dasgupta, A. and Nayak, S. K. “Localization Operator and Weyl Transform on Reduced Heisenberg Group with Multi-dimensional Center.” 2022. http://arxiv.org/abs/2211.06613
  * [9] Führ, H. _Abstract harmonic analysis of continuous wavelet transforms_. vol. 1963. Springer Science & Business Media, (2005).
  * [10] Ghosh, S. and Srivastav, R. K. “Unbounded Weyl transforms on the Euclidean motion group and Heisenberg motion group.” DOI: arXiv:2106.15704.
  * [11] Kohn, J. J. and Nirenberg, L. “An algebra of pseudo-differential operators.” _Communications on Pure and Applied mathematics_ , _18_(1), 269–305(1965).
  * [12] Peng, L. and Zhao, J. “ Weyl transforms associated with the Heisenberg group.” _Bulletin des Sciences Mathématiques_ , _132_ , 78–86(2008).
  * [13] Peng, L. and Zhao, J. “Weyl transforms on the upper half plane.” _Revista Matemática Complutense_ , _23_ , 77–95(2010).
  * [14] Rachdi, L. T. and Triméche, K. “Weyl transforms associated with the spherical mean operator.” _Analysis and Applications_ , _1_(2), 141–164(2003).
  * [15] Simon, B. “ The Weyl transforms and $L^{p}$ functions on phase space.” _Proceedings of the American Mathematical Society_ , _116_(4), 1045–1047(1992).
  * [16] Stien, E. M. and Murphy, T. S. _Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals_. Princeton University Press, New Jersey(1993).
  * [17] Toft, J. “Continuity properties for modulation spaces with applications to pseudo-differential calculus I.” _J. Funct. Anal._ ,_207_(2), 399–429(2004).
  * [18] Toft, J. “Continuity properties for modulation spaces with applications to pseudo-differential calculus II.” _Ann. Global Anal. Geom._ ,_26_ , 73–106(2004).
  * [19] Weyl, H. _The theory of groups and quantum mechanics_. Dover(1950).
  * [20] Wong, M. W. _The Weyl Transform_. Springer(1998).
  * [21] Zhao, J. M. and Peng, L. Z. “Wavelet and Weyl transform associated with the spherical mean operator.” _Integral Equation and Operator Theory_ , _50_ , 279–290(2004).


# The Tensor Pomeron and Low-x Deeply Virtual Compton Scattering ††thanks:
Presented by P. Lebiedowicz at XXIX Cracow Epiphany Conference on Physics at
the Electron-Ion Collider and Future Facilities, Kraków, Poland, January
16-19, 2023.

Piotr Lebiedowicz  Antoni Szczurek
Otto Nachtmann
Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152,
PL-31342 Kraków, Poland  Institut für Theoretische Physik, Universität
Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany

###### Abstract

The two-tensor-pomeron model is applied to deeply virtual Compton scattering
(DVCS) on a proton. A good description of the DVCS HERA data at small
Bjorken-$x$ is achieved due to a sizeable interference of soft and hard
pomeron contributions. We present two fits which differ somewhat in the
strength of the hard pomeron contribution. We describe, in the same framework,
both the low $Q^{2}$ and high $Q^{2}$ regimes and the transition between them.
We find that the soft-pomeron contribution is considerable up to $Q^{2}\sim
20$ GeV2. The reggeon exchange term is particularly relevant for describing
the scattering of a real photon on a proton measured at lower $\gamma p$
energies at FNAL. We find that the ratio of $\gamma^{*}p\to\gamma p$ cross-
sections for longitudinally and transversely polarized virtual photons
strongly increases with $t$. Our findings may be checked in future lepton-
nucleon scattering experiments in the low-$x$ regime, for instance, at a
future Electron-Ion Collider (EIC) at the BNL and LHeC at the LHC.

##  1 Introduction

This presentation summarizes some of the key results of [1] to which we refer
the reader for further details. We apply the two-tensor-pomeron model [2] to
deeply virtual Compton scattering (DVCS) on the proton, $\gamma^{*}p\to\gamma
p$. Our model can be used for large $\gamma^{*}p$ c.m. energy $W\gg
m_{p},\sqrt{|t|}$, $|t|\lesssim 1$ GeV2, and small Bjorken-$x$, say
$x=Q^{2}/(W^{2}+Q^{2}-m_{p}^{2})<0.02$. Here $m_{p}$ is the proton mass,
$Q^{2}$ is the photon virtuality, and $t$ is the squared momentum transfer at
the proton vertex.

The DVCS has been a subject of extensive experimental and theoretical
research; see Sec. 1 in [1]. Experimental program at the Electron-Ion Collider
(EIC) [4], and, if realized, at the Large Hadron Electron Collider (LHeC) at
the LHC [5], are expected to improve our knowledge of DVCS in a wide kinematic
range.

DVCS is a prime playground for the application of the generalized parton-
distribution (GPD) concept based on perturbative QCD (pQCD), cf. [3] for a
review. Here we discuss DVCS in the Regge approach where the scattering is
described using exchange objects. In the tensor-pomeron model, introduced for
soft high-energy reactions in [6] and extended to hard reactions in [2], the
charge-conjugation $C=+1$ soft ($\mathbb{P}_{1}$) and hard ($\mathbb{P}_{0}$)
pomeron and the reggeons ($\mathbb{R}_{+}=f_{2\mathbb{R}},a_{2\mathbb{R}}$)
are described as effective rank-2 symmetric tensor exchanges.

A two-pomeron description of low-$x$ DIS was first proposed in [8]. However,
there a vector nature of the pomerons was considered. It was shown in [2] that
considering these two pomerons as vector objects leads to the conclusion that
they decouple in the total photoabsorption cross-section on the proton and in
the structure functions of low-$x$ DIS. But experiment clearly shows pomeron-
exchange behaviour for these quantities at large $W$; see Fig. 5 of [2].

Applications of the tensor-pomeron approach were given for a number of
exclusive central-production reactions, see e.g. [9, 10, 11, 12, 13, 14, 15],
and for soft-photon radiation [16, 17, 18] in hadron-hadron collisions. For
some remarks on the history of tensor-pomeron concepts and corresponding
references see [7]. It is worth mentioning that the tensor-pomeron current
(Eq. (2.3) of [7]) cannot be universally proportional to the energy-momentum
tensor.

##  2 Formalism

We investigate the real and deeply virtual Compton scattering on a proton

$\gamma^{(*)}(q,\epsilon)+p(p,\lambda)\to\gamma(q^{\prime},\epsilon^{\prime})+p(p^{\prime},\lambda^{\prime})\,.$
(1)

The momenta are indicated in brackets,
$\lambda,\lambda^{\prime}\in\\{1/2,-1/2\\}$ are the proton helicities, and
$\epsilon,\epsilon^{\prime}$ are the photon polarization vectors. For an
initial virtual photon $\gamma^{*}$ the reaction (1) is extracted from $ep\to
ep\gamma$ scattering (see Fig. 1)

$e(k)+p(p,\lambda)\to
e(k^{\prime})+\gamma(q^{\prime},\epsilon^{\prime})+p(p^{\prime},\lambda^{\prime})\,.$
(2)

Here the Bethe-Heitler process and DVCS contribute with the latter
corresponding to electroproduction of the $\gamma p$ state.

Figure 1: DVCS contribution to $ep\to ep\gamma$ (2).

We assume for (2) unpolarized initial particles and no observation of the
polarization of final state particles. The standard kinematic variables are
(see Table 1 of [19])

$\displaystyle
q=k-k^{\prime}\,,\;q^{2}=-Q^{2}\,,\;s=(p+k)^{2}\,,\;t=(p-p^{\prime})^{2}\,,\;W^{2}=(p+q)^{2}\,,$
$\displaystyle x=\frac{Q^{2}}{2p\cdot
q}=\frac{Q^{2}}{W^{2}+Q^{2}-m_{p}^{2}}\,,\;y=\frac{p\cdot q}{p\cdot
k}=\frac{W^{2}+Q^{2}-m_{p}^{2}}{s-m_{p}^{2}}\,.$ (3)

Adapting (3.20) and (3.21) of [19] to the reaction (2) and integrating over
the azimuthal angle $\varphi$ defined in (2.1) of [19] we get for the DVCS
part:

$\displaystyle\frac{d\sigma(ep\to
ep\gamma)}{dydQ^{2}dt}=\Gamma_{ep\gamma}\left(\frac{d\sigma_{\rm
T}}{dt}(Q^{2},W^{2},t)+\varepsilon\frac{d\sigma_{\rm
L}}{dt}(Q^{2},W^{2},t)\right)$ (4)

with $\Gamma_{ep\gamma}$ the $\gamma^{*}$ flux factor [1]. The differential
cross-sections for $\gamma^{*}p\to\gamma p$ for transverse and longitudinal
polarization of the $\gamma^{*}$ are

$\displaystyle\frac{d\sigma_{\rm
T}}{dt}(Q^{2},W^{2},t)=\frac{1}{2}\left(\frac{d\sigma_{++}}{dt}(Q^{2},W^{2},t)+\frac{d\sigma_{--}}{dt}(Q^{2},W^{2},t)\right)\,,$
$\displaystyle\frac{d\sigma_{\rm
L}}{dt}(Q^{2},W^{2},t)=\frac{d\sigma_{00}}{dt}(Q^{2},W^{2},t)\,,$ (5)

where

$\displaystyle\frac{d\sigma_{mm}}{dt}(Q^{2},W^{2},t)=\frac{1}{16\pi(W^{2}-m_{p}^{2})\sqrt{(W^{2}-m_{p}^{2}+Q^{2})^{2}+4m_{p}^{2}Q^{2}}}$
$\displaystyle\qquad\qquad\times\frac{1}{2}\sum_{\lambda,\lambda^{\prime},a}\big{|}\Braket{\gamma(q^{\prime},\epsilon^{\prime}_{a}),p(p^{\prime},\lambda^{\prime}),{\rm
out}}{eJ_{\mu}(0)\epsilon_{m}^{\mu}}{p(p,\lambda)}\big{|}^{2}\,.$ (6)

Here $eJ_{\mu}$ is the electromagnetic current, $\epsilon_{m}(m=\pm,0)$ are
the standard $\gamma^{*}$ polarization vectors for right and left circular and
longitudinal polarization (see (3.11)–(3.14) of [19]) and
$\epsilon^{\prime}_{a}(a=1,2)$ are the polarization vectors of the real photon
in the final state.

We describe the amplitude for (1) in terms of the $C=+1$ exchanges of soft
($\mathbb{P}_{1}$) and hard ($\mathbb{P}_{0}$) pomeron and the reggeons
$f_{2\mathbb{R}}$ and $a_{2\mathbb{R}}$. We get

$\displaystyle\Braket{\gamma(q^{\prime},\epsilon^{\prime}),p(p^{\prime},\lambda^{\prime}),{\rm
out}}{eJ_{\nu}(0)\epsilon^{\nu}}{p(p,\lambda)}\equiv(\epsilon^{\prime\mu})^{*}{\cal
M}_{\mu\nu,\lambda^{\prime}\lambda}\,\epsilon^{\nu}$
$\displaystyle=-(\epsilon^{\prime\mu})^{*}\sum_{j=0,1}\Gamma_{\mu\nu\kappa\rho}^{(\mathbb{P}_{j}\gamma^{*}\gamma^{*})}(q^{\prime},q)\,\epsilon^{\nu}\,\Delta^{(\mathbb{P}_{j})\,\kappa\rho,\alpha\beta}(W^{2},t)\,\bar{u}_{\lambda^{\prime}}(p^{\prime})\Gamma_{\alpha\beta}^{(\mathbb{P}_{j}pp)}(p^{\prime},p)u_{\lambda}(p)$
$\displaystyle\quad+(\mathbb{P}_{j}\to f_{2\mathbb{R}},a_{2\mathbb{R}})\,.$
(7)

Here $\Delta^{(\mathbb{P}_{j})}$, $\Gamma^{(\mathbb{P}_{j}pp)}$, and
$\Gamma^{(\mathbb{P}_{j}\gamma^{*}\gamma^{*})}_{\mu\nu\kappa\rho}(q^{\prime},q)$
denote the effective propagator, the proton vertex function, and the
$\mathbb{P}_{j}\gamma^{*}\gamma^{*}$ ($j=0,1$) vertex, respectively, for the
tensor pomerons $\mathbb{P}_{j}$. The Ansätze for the $f_{2\mathbb{R}}$ and
$a_{2\mathbb{R}}$ reggeons ($j=2$) are analogous. The properties of
$\mathbb{P}_{0},\mathbb{P}_{1},f_{2\mathbb{R}}$, and $a_{2\mathbb{R}}$, and
their couplings to protons and photons will be taken as in [2]. We get

$\displaystyle{\cal
M}_{\mu\nu,\lambda^{\prime}\lambda}=-i\sum_{j=0,1,2}\frac{3\beta_{jpp}}{2W^{2}}(-iW^{2}\alpha^{\prime}_{j})^{\alpha_{j}(t)-1}\,F_{1}^{(j)}(t)\bar{u}_{\lambda^{\prime}}(p^{\prime})\gamma^{\kappa}(p^{\prime}+p)^{\rho}u_{\lambda}(p)$
$\displaystyle\qquad\times\left[2a_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)\,\Gamma^{(0)}_{\mu\nu\kappa\rho}(q^{\prime},-q)\,-b_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)\,\Gamma^{(2)}_{\mu\nu\kappa\rho}(q^{\prime},-q)\right]\,.$
(8)

For the coupling constants $\beta_{jpp}$ of the pomerons $(j=0,1)$ and reegeon
$(j=2)$ to protons we take $\beta_{0pp}=\beta_{1pp}=1.87$ GeV-1,
$\beta_{2pp}=3.68$ GeV-1. The pomeron and reggeon trajectory functions are
assumed to be of linear form

$\displaystyle\alpha_{j}(t)=\alpha_{j}(0)+\alpha^{\prime}_{j}\,t\,,\quad\alpha_{j}(0)=1+\epsilon_{j}\,,\quad(j=0,1,2)\,.$
(9)

The slope parameters $\alpha^{\prime}_{j}$ are taken as the default values
from [2]: $\alpha^{\prime}_{1}=\alpha^{\prime}_{0}=0.25\;{\rm GeV}^{-2}$,
$\alpha^{\prime}_{2}=0.9\;{\rm GeV}^{-2}$. The intercept parameters of the
trajectories (9) were determined from detailed comparison of the model with
the DIS HERA data and photoproduction data in [2]:

$\displaystyle{\rm soft\;pomeron}\;\mathbb{P}_{1}:\quad$
$\displaystyle\epsilon_{1}=0.0935(^{+76}_{-64})\,,$ (10) $\displaystyle{\rm
hard\;pomeron}\;\mathbb{P}_{0}:\quad$
$\displaystyle\epsilon_{0}=0.3008(^{+73}_{-84})\,,$ (11) $\displaystyle{\rm
reggeon}\;\mathbb{R}_{+}:\quad$
$\displaystyle\alpha_{2}(0)=0.485(^{+88}_{-90})\,.$ (12)

The coupling functions $a_{j\gamma^{*}\gamma^{*}}$ and
$b_{j\gamma^{*}\gamma^{*}}$ in (8) are (see (2.21)–(2.23) of [1])

$\displaystyle
a_{j\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{a}_{j}(Q^{2})F^{(j)}(t)\,,\quad
j=0,1,2\,,$ $\displaystyle
b_{2\gamma^{*}\gamma^{*}}(q^{2},0,t)=e^{2}\hat{b}_{2}(Q^{2})F^{(2)}(t)\,.$
(13)

Here, $\hat{a}_{j}(Q^{2})$ and $\hat{b}_{j}(Q^{2})$ were determined in [2]
from the global fit to HERA inclusive DIS data for $Q^{2}<50$ GeV2 and
$x<0.01$ and the ($Q^{2}=0$) photoproduction data. All coupling functions
$\hat{a}_{j}$ and $\hat{b}_{j}$ are plotted in Fig. 2 of [1]. For small
$Q^{2}$, the soft pomeron function $b_{1\gamma^{*}\gamma^{*}}$ gives a larger
contribution to the cross-section than the corresponding hard one
$b_{0\gamma^{*}\gamma^{*}}$. In the large $Q^{2}$ region the reverse is found.
In Sec. 3, we shall show two alternative fits for $b_{1\gamma^{*}\gamma^{*}}$
and $b_{0\gamma^{*}\gamma^{*}}$ obtained from a comparison to HERA DVCS data.

We use the combined form-factor functions for a given $j$ $(j=0,1,2)$

$F_{\rm eff}^{(j)}(t)=F^{(j)}(t)\times F_{1}^{(j)}(t)=\exp(-b_{j}|t|/2)\,,$
(14)

assuming the same $t$ dependence for both $a$ and $b$ coupling functions. We
take $b_{1}=b_{2}=5.0\;{\rm GeV}^{-2}$ and $b_{0}=1.0\;{\rm GeV}^{-2}$ from
[1].

##  3 Results

We shall restrict our discussion to experimental results that satisfy the
conditions $x\approx Q^{2}/W^{2}<0.02$ and $|t|\lesssim 1$ GeV2 where our
model should be reliable.

In Fig. 2 we compare the tensor-pomeron model results, FIT 1 (left panel) and
FIT 2 (right panel), to the FNAL data [20] on real-photon-proton scattering
($\gamma p\to\gamma p$), and to HERA data [23, 21, 24, 22] on DVCS
($\gamma^{*}(Q^{2})p\to\gamma p$) for different averaged $W$ and $Q^{2}$. The
complete cross-section is a coherent sum of soft and hard components in the
amplitude. For real Compton scattering ($Q^{2}=0$) the cross-section is
dominated by soft-pomeron exchange with an additional contribution from
reggeon exchange at lower energies. The hard-pomeron contribution is
negligibly small there. The dominant contribution comes from the $b$-type
coupling functions $b_{j\gamma^{*}\gamma^{*}}$ ($j=0,1$). Their size differs
in the two fits. We see from the bottom panels of Fig. 2 that for higher
$Q^{2}$ the soft component slowly decreases relative to the hard one. A
significant constructive interference effect between the soft and hard
components is clearly visible. Here and in the following, the interference
term is calculated as the difference of coherent and incoherent cross-sections
of the $\mathbb{P}_{1}$, $\mathbb{P}_{0}$, and $\mathbb{R}$ contributions. The
Fits 1 and 2 hardly differ for the $W$ region where there are data. But for
higher $W$ values FIT 2, where the contribution from the hard pomeron is
enhanced, gives a steeper rise of the cross-section with $W$ and especially so
for larger $Q^{2}$. For FIT 1, we see that the soft pomeron survives to
relatively large $Q^{2}$ and at $Q^{2}\simeq 50$ GeV2 the interference term
plays an important role in the description of the data.

Figure 2: Top panels: Total cross-sections as a function of the c.m. energy
$W$ for FIT 1 (left) and FIT 2 (right). Comparison of theoretical results to
the FNAL data from [20] for real Compton scattering ($Q^{2}=0$) and to the
DVCS HERA data is shown. The upper black solid line is for $Q^{2}=0$, the
orange long-dashed-dotted line is for $Q^{2}=1$ GeV2. The remaining lines
correspond to the values $Q^{2}=2.4,3.2,4,6.2,8,10,15.5,25$ GeV2 (from top to
bottom) and should be compared with the HERA data from [23, 21, 24, 22].
Middle panels: Fit results for $Q^{2}=8$ GeV2 together with the H1 data [22].
The interference term of soft and hard pomeron is shown by the green dotted
line. Bottom panels: Comparison of cross-sections as a function of $Q^{2}$ to
the ZEUS data from [24].

In Fig. 3 we show the differential cross-sections $d\sigma/dt$ for different
$\langle W\rangle$ and $\langle Q^{2}\rangle$. We use

$\frac{d\sigma}{dt}=\frac{d\sigma_{\rm T}}{dt}+\varepsilon\frac{d\sigma_{\rm
L}}{dt}$ (15)

with $\varepsilon=1$ ($\varepsilon\approx 1$ for the HERA kinematic region).
In the top panels of Fig. 3 the upper line corresponds to $W=12.7$ GeV and
$Q^{2}=0$ and should be compared to the averaged FNAL data (top data points)
for the $\gamma p\to\gamma p$ reaction [20]. In this kinematic range, for
$Q^{2}=0$ and at intermediate $W$, the reggeon plus soft-$\mathbb{P}$
contributions dominate and the hard-$\mathbb{P}$ exchange gives a negligible
contribution. As expected, there is a significant interference between the
reggeon and soft-pomeron components. The slope parameters $b_{2}$ and $b_{1}$
in (14) are adjusted to the FNAL $d\sigma/dt$ data on the real-photon-proton
scattering. At higher $W$ and $Q^{2}$ measured at HERA the hard pomeron plays
an increasingly important role. The slope parameter $b_{0}$ for the hard-
pomeron exchange is adjusted to the DVCS HERA data. As we noted above,
$d\sigma(\gamma^{*}p\to\gamma p)/dt$ is the sum of two contributions
$d\sigma_{\rm T}/dt$ and $d\sigma_{\rm L}/dt$ with the latter term becoming
very small for $|t|\to 0$. This is understandable since for the
$\gamma^{*}p\to\gamma p$ forward scattering only the double-helicity-flip
amplitudes can contribute to $d\sigma_{\rm L}/dt$. Furthermore, we find that
$d\sigma_{\rm T}/dt$ is dominated by the $b$-type couplings and $d\sigma_{\rm
L}/dt$ is dominated by the $a$-type couplings.

In the middle panels of Fig. 3, we show the complete theoretical result and
individual components contributing to the cross-section $d\sigma/dt$, see
(15), for $W=82$ GeV and $Q^{2}=8$ GeV2 together with the H1 data [22]. The
contributions of soft $\mathbb{P}$ (the blue short-dashed lines), hard
$\mathbb{P}$ (the red long-dashed lines), the interference term (the green
dotted lines), and their sum total (the thin full lines) for ${\rm T}$ and
${\rm L}$ components individually are also shown. The constructive
interference of the soft and hard pomeron terms is a salient feature there.

In the bottom panels of Fig. 3 we show the ratios of the $\gamma^{*}p\to\gamma
p$ cross-sections for longitudinally and transversely polarized virtual
photons,

$\displaystyle R(Q^{2},W^{2})=\frac{\sigma_{\rm L}(Q^{2},W^{2})}{\sigma_{\rm
T}(Q^{2},W^{2})}\,,\quad\tilde{R}(Q^{2},W^{2},t)=\frac{\frac{d\sigma_{\rm
L}}{dt}(Q^{2},W^{2},t)}{\frac{d\sigma_{\rm T}}{dt}(Q^{2},W^{2},t)}\,,$ (16)

as functions of $Q^{2}$ and $|t|$, respectively. The cross-section
$\sigma_{\rm L}$ vanishes proportionally to $Q^{2}$ for $Q^{2}\to 0$. The
ratio $\tilde{R}(Q^{2},W^{2},t)$ strongly grows with $|t|$. We must emphasize
that this behaviour of $\tilde{R}(Q^{2},W^{2},t)$ depends crucially on our
(reasonable) assumption that $a$ and $b$ couplings in the
$\mathbb{P}_{j}\gamma^{*}\gamma^{*}$ vertex functions have the same $t$
dependence for a given $j$.

Figure 3: Top panels: The differential cross-sections $d\sigma/dt$ for FIT 1
(left) and FIT 2 (right) compared to experimental data for different $W$ and
$Q^{2}$. The upper line corresponds to $W=12.7$ GeV and $Q^{2}=0$ and averaged
FNAL data [20]. The bottom lines correspond to theoretical results for
$\gamma^{*}p\to\gamma p$ at higher $Q^{2}$ for transverse (long-dashed lines)
and longitudinal (short-dashed lines) polarization of the $\gamma^{*}$ and
their sum (solid lines). Data for $\langle W\rangle=104$ GeV are from [24] and
for $\langle W\rangle=82$ GeV from [22]. Middle panels: Comparison of the
results for $\gamma^{*}p\to\gamma p$ for transverse (${\rm T}$) and
longitudinal (${\rm L}$) polarization of the $\gamma^{*}$ individually and
their sum ${\rm T+L}$ (see the upper solid lines) to DVCS H1 data [22] for
$W=82$ GeV and $Q^{2}=8$ GeV2. Bottom panels: The ratios (16) for the
$\gamma^{*}p\to\gamma p$ reaction. Note that the meaning of the lines on these
two panels is different.

In [25, 26] also the Regge theory was applied to DVCS. These authors consider
only leading helicity amplitudes (only transversely polarized initial
$\gamma^{*}$’s). In contrast, in our tensor-pomeron approach we present a
complete model (all helicity amplitudes). Therefore, we could make, e.g.,
predictions for $\sigma_{\rm L}/\sigma_{\rm T}$ which can and should be
checked by experiments. The contribution of the interference term found in
[26] is considerable for intermediate values of $Q^{2}$, but smaller than our
findings.

##  4 Conclusions

The two-tensor-pomeron model proposed previously to describe low $x$ DIS data
[2] was applied to deeply virtual Compton scattering (DVCS) for high c.m.
energies $W$ and small Bjorken $x$, say $x\lesssim 0.02$ [1].

In particular, the transition from the small-$Q^{2}$ regime, including the
photoproduction ($Q^{2}=0$) limit, to the large-$Q^{2}$ regime, the DIS limit
is well described. We compared predictions of the two-tensor-pomeron model to
the DVCS data measured at HERA. We onsidered FIT 1 in which a ’minimal’
modification of the $Q^{2}$ dependence of only one
$\gamma^{*}(Q^{2})\gamma\mathbb{P}$ coupling function was made. We considered
also FIT 2 in which the size of the hard-pomeron component was increased,
especially for larger $Q^{2}$, and the soft-pomeron component was reduced
relative to FIT 1. We kept here, on purpose, the same parameters of the form
factors (14) as in FIT 1. The FIT 2 better describes the data at larger $|t|$
for $Q^{2}\gtrsim 8$ GeV2 (see Fig. 3).

The model describes the $W$, $Q^{2}$ and $t$ dependences of
$d\sigma(\gamma^{*}p\to\gamma p)/dt$ measured at HERA and of the elastic
photon-proton cross-section measured at FNAL. A good description of the DVCS
data is achieved due to a sizeable interference of soft and hard pomeron
contributions. The soft component and also the interference of soft and hard
terms are very important up to at least $Q^{2}\simeq 20$ GeV2.

Our calculation include the contributions of both the transverse and
longitudinal virtual photons. The longitudinal cross-section $d\sigma_{\rm
L}/dt$ is predicted to be very small for $|t|\to 0$ but to be sizeable for
$0.5\;{\rm GeV}^{2}\lesssim|t|\lesssim 1.0\;{\rm GeV}^{2}$. We give also
predictions for $\sigma_{\rm L}/\sigma_{\rm T}$ which can and should be
checked by experiments. The corresponding ratio of ${\rm L/T}$ grows strongly
with $|t|$. We showed the $Q^{2}$ dependence of this ratio for different c.m.
energies of the $\gamma^{*}p$ system.

We presented predictions for low-$x$ DVCS of the two-tensor-pomeron model
which previously was successfully applied to low $x$ DIS in [2]. The model
provides amplitudes for all helicity configurations and, thus, can be checked
by experimentalists in many ways. We are looking forward to further tests of
the non-perturbative QCD dynamics embodied in our tensor-pomeron exchanges in
future electron-hadron collisions in the low-$x$ regime at the EIC [4] and
LHeC [5] colliders.

## Acknowledgments

P.L. thanks the organizers of XXIX Cracow Epiphany Conference for a well-
organized conference with many stimulating discussions. This work was
partially supported by the Polish National Science Centre Grant No.
2018/31/B/ST2/03537.

## References

  * [1] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Lett. B 835, 137497 (2022).
  * [2] D. Britzger, C. Ewerz, S. Glazov, O. Nachtmann, and S. Schmitt, Phys. Rev. D 100 114007 (2019).
  * [3] M. Diehl, Phys. Rept. 388, 41 (2003).
  * [4] R. Abdul Khalek et al., Nucl. Phys. A 1026, 122447 (2022).
  * [5] P. Agostini et al., (LHeC, FCC-he Study Group), J. Phys. G 48, 110501 (2021).
  * [6] C. Ewerz, M. Maniatis, and O. Nachtmann, Annals Phys. 342, 31 (2014).
  * [7] C. Ewerz, P. Lebiedowicz, O. Nachtmann, and A. Szczurek, Phys. Lett. B 763, 382 (2016).
  * [8] A. Donnachie and P. V. Landshoff, Phys. Lett. B 437, 408 (1998).
  * [9] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Annals Phys. 344, 301 (2014).
  * [10] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 91, 074023 (2015).
  * [11] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 93, 054015 (2016).
  * [12] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 97, 094027 (2018).
  * [13] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 99, 094034 (2019).
  * [14] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 101, 094012 (2020).
  * [15] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 102, 114003 (2020).
  * [16] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 105, 014022 (2022).
  * [17] P. Lebiedowicz, O. Nachtmann, A. Szczurek, Phys. Rev. D 106, 034023 (2022).
  * [18] P. Lebiedowicz, O. Nachtmann, A. Szczurek, arXiv: 2302.07192 [hep-ph].
  * [19] T. Arens, O. Nachtmann, M. Diehl, and P. V. Landshoff, Z. Phys. C 74, 651 (1997).
  * [20] A. M. Breakstone et al., Phys. Rev. Lett. 47, 1778 (1981).
  * [21] A. Aktas et al., (H1 Collaboration), Eur. Phys. J. C 44, 1 (2005).
  * [22] F. D. Aaron et al., (H1 Collaboration), Phys. Lett. B 681, 391 (2009).
  * [23] S. Chekanov et al., (ZEUS Collaboration), Phys. Lett. B 573, 46 (2003).
  * [24] S. Chekanov et al., (ZEUS Collaboration), JHEP 05, 108 (2009).
  * [25] M. Capua, S. Fazio, R. Fiore, L. L. Jenkovszky, and F. Paccanoni, Phys. Lett. B 645, 161 (2007).
  * [26] S. Fazio, R. Fiore, L. Jenkovszky, and A. Salii, Phys. Rev. D 90, 016007 (2014).