# Boundedness in $L_{p}$ spaces for the Hartley-Fourier
convolutions operator and their applications

Trinh Tuan

(Date: April $27^{th}$, 2023 )

###### Abstract.

The paper deals with $L_{p}$-boundedness of the Hartley-Fourier convolutions
operator and their applied aspects. We establish various new Young-type
inequalities and obtain the structure of a normed ring in Banach space when
equipping it with such convolutional multiplication. Weighted $L_{p}$-norm
inequalities of these convolutions are also considered. As applications, we
investigate the solvability and the bounded $L_{1}$-solution of a class of
Fredholm-type integral equations and linear Barbashin’s equations with the
help of factorization identities of such convolutions. Several examples are
provided to illustrate the obtained results to ensure their validity and
applicability. Keywords. $L_{p}$-Boundedness. Convolutions. Hartley–Fourier
transforms. Young’s inequalities. Wiener-Tauberian’s theorem. Barbashin’s
equations. Fredholm’s integral equations.

Mathematics Subject Classification. 42A38. 44A35. 45E10. 45J05. 26D10

E-mail<EMAIL_ADDRESS>
Department of Mathematics, Electric Power University,

235-Hoang Quoc Viet Road, Hanoi, Vietnam.

E-mail<EMAIL_ADDRESS>
##  1\. Introduction

Integral inequalities are fundamental tools in the study of qualitative as
well as quantitative properties of integral transformations and solutions of
differential equations. In particular, the convolution inequality is
indispensable because there are so many integral transformations and solutions
of differential equations that are represented as convolutions (see [8]).
Certainly among all the convolutional transformations, the best known is the
Fourier convolution. In 1912, Young. W. H introduced Young’s inequality for
the bounded estimation of the Fourier convolution (refer [29]), which is
$\lVert(f\underset{F}{*}g)\lVert_{L_{r}(\mathbb{R})}\ \leq\lVert
f\lVert_{L_{p}(\mathbb{R})}\ \lVert g\lVert_{L_{q}(\mathbb{R})},$ where
$p,q,r>1$ such that $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$. R.A. Adams and
J.F. Fournier generalized Young’s inequality for the Fourier convolution
(refer to Theorem 2.24, page 33, in [1]) to include a weight

$\bigg{|}\int\limits_{\mathbb{R}^{n}}(f\underset{F}{*}g)(x).\
\omega(x)dx\bigg{|}\leq\lVert f\lVert_{L_{p}(\mathbb{R}^{n})}\lVert
g\lVert_{L_{q}(\mathbb{R}^{n})}\lVert\omega\lVert_{L_{r}(\mathbb{R}^{n})},$

where $p,q,r$ are real numbers in $(1,\infty)$ such that
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$ and $f\in L_{p}(\mathbb{R}^{n}),g\in
L_{q}(\mathbb{R}^{n}),\omega\in L_{r}(\mathbb{R}^{n})$ with the Fourier
convolution defined by
$(f\underset{F}{*}g)(x):=\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}f(y)\
g(x-y)dy,\ x\in\mathbb{R}$, then Young’s inequality is seen as a consequence
of Young’s theorem [16]. Nevertheless, Young’s inequality does not hold for
the important case $f,g\in L_{2}(\mathbb{R})$. Based on the technique in [1,
10, 23] and derived from the previous results for Hartley–Fourier convolutions
[9, 21, 26], we study the $L_{p}$ norm estimates of these convolutions and
applicabilities, which are also the main contributions of this paper. The
obtained results are new Young type inequalities with their important
applications in solving classes of integro-differential equations. The
applications aspect will be detailed here, for a large class of integral
operators, with the results that will exemplify three different topics:

(i) Non-commutative normed ring structure on Banach space $L_{1}(\mathbb{R})$;

(ii) Solvability and boundedness of $L_{1}$-solutions of Fredholm’s integral
equation of the second kind;

(iii) Solvability and boundedness of $L_{1}$-solutions for Barbashin’s
equations & Cauchy-type problem.
In addition, it should be noted that the factorization property of Hartley-
Fourier convolutions is crucial in solving corresponding convolution-type
equations [5, 21, 26]. It is also clear that convolution-type equations are
very often used in the modeling of a broad range of different problems [6, 7,
25], so additional knowledge about their solvability is very welcome.

This paper is divided into five sections including the introduction and the
remaining four parts. Section 2 is devoted to the presentation of the notions
of Fourier and Hartley’s transforms, and the relationship between them. We
also recall the definitions of Hartley, Hartley-Fourier convolutions and some
results for factorization properties of these convolutions. We also prove that
the assertion of Wiener-Lévy’s theorem holds for the Hartley transform (see
Lemma 2.1). Section 3 consists of three subsections. In Subsection 3.1, we
prove Young’s Theorem of the Hartley-Fourier generalized convolution (2.4) and
corollaries for estimation in $L_{1}(\mathbb{R})$ space. The general
formulation of Young-type inequality for the Hartley convolutions (2.5) is
stated in Subsection 3.2. In Subsection 3.3, using Hölder’s inequality,
Fubini’s theorem, and integral transforms, we establish norm inequalities in
weighted $L_{p}(\mathbb{R},\rho_{j})$ spaces of each above convolutions.
Section 4 deals with applications of the constructed convolutions. In
Subsection 4.1, the linear space $L_{1}(\mathbb{R})$, equipped with the above-
mentioned convolutions, becomes a normed ring, having no unit, which is non-
commutative, and could be used in the theories of Banach algebra. Subsections
4.2 and 4.3 contain the most important results of this section where
Fredholm’s integral equations of the second kind and Barbashin’s equation are
considered simultaneously. By using the constructed convolutions together with
the help of Wiener-Lévy theorem and Schwartz’s function classes, we provide
necessary and sufficient conditions for the solvability of the integral
equations of convolution type and obtain explicit $L_{1}$-solutions. The most
obvious differences when solving the problems of subsections 4.2 and 4.3 in
$L_{1}(\mathbb{R})$ and the previous results in [24, 26] are that the Fourier,
Hartley transforms $L_{2}(\mathbb{R})\longleftrightarrow L_{2}(\mathbb{R})$
are an isometric (unitary) isomorphism [17], meanwhile it is no longer true
for $L_{1}(\mathbb{R})$. Furthermore, in the space $L_{1}(\mathbb{R})$,
Theorem (68, page 92, in [22]) is no longer true. To overcome that, we use
simultaneously the density in $L_{1}(\mathbb{R})$ of Schwartz’s function
classes [14], Wiener-Lévy Lemma 2.1, and factorization properties. Subsection
4.4 is concerned with the Cauchy-type problem. Techniques used here to check
if the solution of the problem satisfies the initial value theorem or not come
from [11] and [28]. Namely, we follow closely the strategy of Wiener-
Tauberian’s theorem. Some computational examples can be found in the final
section of the paper to illustrate the obtained results to ensure their
validity and applicability.

##  2\. Preliminaries

We recall some notions and results coming from [4, 5, 9, 11, 12, 21, 22, 25,
26]. We denote by (F) the Fourier integral transform ([4, 22]) of a function
$f$ defined by
$(Ff)(y)=\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-ixy}f(x)dx$ with
$y\in\mathbb{R},$ where
$e^{-ixy}=\frac{1+i}{2}\operatorname{Cas}(xy)+\frac{1-i}{2}\operatorname{Cas}(-xy).$
The Hartley transform was proposed as an alternative to the Fourier transform
by R. V. L. Hartley in 1942 (see [5]). The Hartley transform is an alternate
means of analyzing a given function in terms of its sinusoids. This transform
is its own inverse and an efficient computational tool in case of real-value
functions. The Hartley transform of a function is a spectral transform and can
be obtained from the Fourier transform by replacing the exponential kernel
$exp(-ixy)$ by $\operatorname{Cas}(-xy)$. The Hartley transform of a function
$f$ can be expressed as either

$\big{(}H_{\left\\{\begin{smallmatrix}1\\\
2\end{smallmatrix}\right\\}}f\big{)}(y)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)\operatorname{Cas}(\pm
xy)dx,\ y\in\mathbb{R},$ (2.1)

where $\operatorname{Cas}(\pm xy)=\cos(xy)\pm\sin(xy)$. Thus, it is obvious
that we have the following correlation between the Hartley transform and the
Fourier transform as follows

$\big{(}H_{\left\\{\begin{smallmatrix}1\\\
2\end{smallmatrix}\right\\}}f\big{)}(y)=F\left(\dfrac{1\pm
i}{2}f(x)+\dfrac{1\mp i}{2}f(-x)\right)(y),$ (2.2)

and

$(Ff)(y)=H_{\left\\{\begin{smallmatrix}1\\\
2\end{smallmatrix}\right\\}}\left(\dfrac{1-i}{2}f(\pm x)+\dfrac{1+i}{2}f(\mp
x)\right)(y).$ (2.3)

Actually, the Hartley transform can be applied to approximate the continuous
transformation of a non-periodic signal of finite duration. Being a real-
valued function, it has some computational advantages such as memory
efficiency and faster harmonic solutions over the Fourier transform which is a
complex tool, readers refer to details in [5, 12]. Consequently, the method
based on the Hartley transform is helpful to estimate harmonic effects for
inrush current by giving a unified iterative solution that exhibits good
convergence.

######  Definition 1.

The following generalized convolution related to Hartley-Fourier $(H_{1},F)$
transforms was introduced recently (see [21, 26]) defined by

$\big{(}f\underset{H_{1},F}{\ast}g\big{)}(x):=\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}}g(y)[f(x+y)+f(x-y)+if(-x-y)-if(-x+y)]dy,\quad\forall
x\in\mathbb{R}.$ (2.4)

For brevity, we call (2.4) it the Hartley-Fourier generalized convolution.

######  Definition 2.

The convolution of two functions $f$ and $g$ for the Hartley
$H_{\left\\{\begin{smallmatrix}1\\\ 2\end{smallmatrix}\right\\}}$ transform
(see [9, 26]) is defined by

$\big{(}f\underset{H_{\left\\{\begin{smallmatrix}1\\\
2\end{smallmatrix}\right\\}}}{\ast}g\big{)}(x):=\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}}f(y)[g(x+y)+g(x-y)+g(-x+y)-g(-x-y)]dy,\quad\forall
x\in\mathbb{R}.$ (2.5)

For brevity, we call (2.5) it the Hartley convolution.

######  Proposition 2.1.

i) [21]. Let $f,g$ be the functions belonging to $L_{1}(\mathbb{R})$, then
Hartley-Fourier generalized convolution is
$\big{(}f\underset{H_{1},F}{\ast}g\big{)}(x)\in L_{1}(\mathbb{R})$.
Furthermore, the factorization equality (2.6) is valid.

$H_{1}\big{(}f\underset{H_{1},F}{\ast}g\big{)}(y)=(H_{1}f)(y)(Fg)(y),\quad
y\in\mathbb{R}.$ (2.6)

ii) [9, 26]. Suppose that $f,g\in L_{1}(\mathbb{R})$, then Hartley convolution
(2.5) is well-defined, which means that
$\big{(}f\underset{H_{\left\\{\begin{smallmatrix}1\\\
2\end{smallmatrix}\right\\}}}{\ast}g\big{)}$ belongs to $L_{1}(\mathbb{R})$.
And the following factorization equality

$H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(y)=\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}g\big{)}(y),\quad y\in\mathbb{R},$ (2.7)

holds true for any $f,g\in L_{1}(\mathbb{R})$.

We need the following auxiliary lemmas. The Wiener-Levy’s Theorem (refer [11])
for the Fourier transform stated that if $\xi\in L_{1}(\mathbb{R})$, then
$1+(F\xi)(y)\neq 0$ for any $y\in\mathbb{R}$ is a necessary and sufficient
condition for the existence of a function $k$ belonging to $L_{1}(\mathbb{R})$
such that $(Fk)(y)=\frac{(F\xi)(y)}{1+(F\xi)(y)}$, where $\xi\in
L_{1}(\mathbb{R})$. This theorem still holds true for the Hartley transform
and we will detail it below.

######  Lemma 2.1.

(Wiener-Lévy’s Theorem for Hartley transform) For the Hartley transform is
defined by (2.1) and suppose that $\varphi\in L_{1}(\mathbb{R})$, then
$1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)\neq 0$, $\forall y\in\mathbb{R}$ is
a necessary and sufficient condition for the existence of a function $\ell\in
L_{1}(\mathbb{R})$ such that $\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\ell\big{)}(y)=\frac{\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)}{1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)},$ for any $y\in\mathbb{R}$.

###### Proof.

We put $\xi(x):=\frac{1\pm i}{2}\varphi(x)+\frac{1\mp i}{2}\varphi(-x).$ This
implies that $\xi\in L_{1}(\mathbb{R})$ if and only if $\varphi$ belongs to
$L_{1}(\mathbb{R})$. Hence, from the correlation between Hartley and Fourier
transforms (2.2), we obtain
$(F\xi)(y)=\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y),\forall y\in\mathbb{R}.$ Applying
Wiener–Levy’s theorem to the Fourier (F) transform (see[11]), we have
$1+(F\xi)(y)\neq 0$, which implies that
$1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)\neq 0$, $\forall y\in\mathbb{R}$ is
a necessary and sufficient condition for the existence of a function $k\in
L_{1}(\mathbb{R})$ such that

$(Fk)(y)=\frac{(F\xi)(y)}{1+(F\xi)(y)}=\frac{\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)}{1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)},\forall y\in\mathbb{R}.$ (2.8)

Conversely, if set $\ell(x):=\frac{1-i}{2}k(\pm x)+\frac{1+i}{2}k(\mp x),$ by
the same above argument, we deduce $\ell$ belongs to the $L_{1}(\mathbb{R})$
if and only if $k\in L_{1}(\mathbb{R})$ and by formula (2.3), we obtain

$\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\ell\big{)}(y)=(Fk)(y),\forall y\in\mathbb{R}.$ (2.9)

Coupling (2.8) with (2.9), we have $\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\ell\big{)}(y)=(Fk)(y)=\frac{(F\xi)(y)}{1+(F\xi)(y)}=\frac{\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)}{1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\varphi\big{)}(y)},\forall y\in\mathbb{R}.$ ∎

######  Lemma 2.2.

Suppose that $g\in C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$ such that
$g^{\prime},g^{\prime\prime}\in L_{1}(\mathbb{R})$ and
$\lim\limits_{|x|\to\infty}g(x)=\lim\limits_{|x|\to\infty}g^{\prime}(x)=0$.
For any function $f$ belongs to $L_{1}(\mathbb{R})$, we obtain

$H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\bigg{\\{}\left(1-\frac{d^{2}}{dx^{2}}\right)\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(x)\bigg{\\}}(y)=(1+y^{2})H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(y),$ (2.10)

###### Proof.

By (2.1) and assumption of $g(x)$, we have

$\displaystyle\left(H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}g^{\prime\prime}(x)\right)(y)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\pi}}\int_{\mathbb{R}}g^{\prime\prime}(x)\mathrm{Cas}(\pm
xy)dx,\quad y\in\mathbb{R}$ (2.11) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\bigg{\\{}g^{\prime}(x)\mathrm{Cas}(\pm
xy)\Big{|}_{-\infty}^{\infty}\pm y\mathrm{Cas}(\pm
xy)g(x)\Big{|}_{-\infty}^{\infty}-y^{2}\int_{\mathbb{R}}g(x)\mathrm{Cas}(\pm
xy)dx\bigg{\\}}$ $\displaystyle=$
$\displaystyle-y^{2}\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}g(x)\big{)}(y).$

Since $f,g,g^{\prime},$ and $g^{\prime\prime}$ are functions belonging to
$L_{1}(\mathbb{R})$, we infer that the following convolutions
$\Big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\Big{)}$;
$\Big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g^{\prime}\Big{)}$; and
$\Big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g^{\prime\prime}\Big{)}$ both belong to
$L_{1}(\mathbb{R})$ (see [21, 26]). Notation $\mathcal{S}$ is the space of
functions $g(x)\in C^{\infty}(\mathbb{R})$ whose derivative decreases rapidly
when $|x|$ tends to $\infty$, it is also known as Schwartz space. Then, the
closure of $\mathcal{S}=L_{1}(\mathbb{R})$, which means that $\mathcal{S}$ is
the dense set in the $L_{1}(\mathbb{R})$ space. Thus, for any $g,g^{\prime},$
and $g^{\prime\prime}$ belong to the $L_{1}(\mathbb{R})$ there exists a
sequence of functions $\\{g_{n}\\}\in S$ such that $\\{g_{n}\\}\rightarrow g,\
\\{g^{\prime}_{n}\\}\rightarrow g^{\prime}$ and
$\\{g^{\prime\prime}_{n}\\}\rightarrow g^{\prime\prime}$ when n tends to
$\infty$. With the above function classes, together with the assumption that
$g$ is a function belonging to the $L_{1}(\mathbb{R})$ space, the integral in
the formula (2.5) is convergent. We continue to change the order of the
integration and the differentiation as follow

$\displaystyle\frac{d^{2}}{dx^{2}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g_{n}\big{)}(x)$
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\int_{-\infty}^{\infty}f(y)\frac{d^{2}}{dx^{2}}\left\\{g_{n}(x+y)+g_{n}(-x+y)g_{n}(x-y)-g_{n}(-x-y)\right\\}dy$
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\int_{-\infty}^{\infty}f(y)\left\\{g^{\prime\prime}_{n}(x+y)+g^{\prime\prime}_{n}(-x+y)+g^{\prime\prime}_{n}(x-y)-g^{\prime\prime}_{n}(-x-y)\right\\}dy$
$\displaystyle=\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g^{\prime\prime}_{n}\big{)}(x)\in
L_{1}(\mathbb{R}).$

Therefore
$\lim\limits_{n\to\infty}\frac{d^{2}}{dx^{2}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g_{n}\big{)}(x)=\lim\limits_{n\to\infty}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g^{\prime\prime}_{n}\big{)}(x).$ This implies
that $\frac{d^{2}}{dx^{2}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(x)=\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g^{\prime\prime}\big{)}(x)$ belongs to
$L_{1}(\mathbb{R}).$ Coupling factorization equality (2.7) with (2.11), we
have

$\displaystyle H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\bigg{\\{}\frac{d^{2}}{dx^{2}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(x)\bigg{\\}}(y)$
$\displaystyle=H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g^{\prime\prime}\big{)}(y)=\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}g^{\prime\prime}\big{)}(y)$
$\displaystyle=-y^{2}\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}g\big{)}(y)$
$\displaystyle=-y^{2}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(y).$

From the above equality, we infer the desired conclusion of the lemma. ∎

Throughout the article, we shall make frequent use of the weighted Lebesgue
spaces $L_{p}(\mathbb{R},w(x)dx)$, $1\leq p<\infty$ with respect to a positive
measure $w(x)dx$ equipped with the norm for which $\lVert
f\lVert_{L_{p}(\mathbb{R},w)}=\bigg{(}\int\limits_{\mathbb{R}}|f(x)|^{p}w(x)dx\bigg{)}^{1/p}<+\infty.$
If the weighted function $w=1$, then $L_{p}(\mathbb{R},w)\equiv
L_{p}(\mathbb{R})$. In case $p=\infty$, then the norm of functions is defined
by $\lVert
f\lVert_{L_{\infty}(\mathbb{R})}=\sup\limits_{x\in\mathbb{R}}|f(x)|<+\infty$.

##  3\. $L_{p}$-boundedness for the Hartley-Fourier convolutions

###  3.1. Young’s theorem for Hartley-Fourier generalized convolution

######  Theorem 3.1 (Young-type Theorem for Hartley-Fourier generalized
convolution).

Let $p,q$, and $r$ be real numbers in open interval $(1,\infty)$ such that
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$. For any functions $g\in
L_{p}(\mathbb{R})$, $f\in L_{q}(\mathbb{R})$, and $h\in L_{r}(\mathbb{R})$, we
obtain the following inequality

$\bigg{|}\int_{\mathbb{R}}\big{(}f\underset{H_{1},F}{*}g\big{)}(x)\cdot
h(x)\,dx\bigg{|}\leq\sqrt{\frac{2}{\pi}}\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}\|h\|_{L_{r}(\mathbb{R})},$
(3.1)

where $\big{(}f\underset{H_{1},F}{*}g\big{)}$ is defined by (2.4).

###### Proof.

Let $p_{1},q_{1},r_{1}$ be the conjugate exponentials of $p,q,r$,
respectively. This means that
$\frac{1}{p}+\frac{1}{p_{1}}=\frac{1}{q}+\frac{1}{q_{1}}=\frac{1}{r}+\frac{1}{r_{1}}=1$,
together with the assumption of theorem, we get the correlation between
exponential numbers as follows

$\left\\{\begin{array}[]{l}\frac{1}{p_{1}}+\frac{1}{q_{1}}+\frac{1}{r_{1}}=1,\\\
p\left(\frac{1}{q_{1}}+\frac{1}{r_{1}}\right)=q\left(\frac{1}{p_{1}}+\frac{1}{r_{1}}\right)=r\left(\frac{1}{p_{1}}+\frac{1}{q_{1}}\right)=1.\end{array}\right.$
(3.2)

For simplicity, we set

$\displaystyle A_{1}(x,y)$ $\displaystyle=$
$\displaystyle\left|f(x+y)+f(x-y)+if(-x-y)-if(-x+y)\right|^{\frac{q}{p_{1}}}\cdot|h(x)|^{\frac{r}{p_{1}}}\in
L_{p_{1}}(\mathbb{R}^{2}),$ $\displaystyle A_{2}(x,y)$ $\displaystyle=$
$\displaystyle|f(x+y)+f(x-y)+if(-x-y)-if(-x+y)|^{\frac{q}{r_{1}}}|g(x)|^{\frac{p}{r_{1}}}\in
L_{r_{1}}(\mathbb{R}^{2}),$ $\displaystyle A_{3}(x,y)$ $\displaystyle=$
$\displaystyle|g(y)|^{\frac{p}{q_{1}}}|h(x)|^{\frac{r}{q_{1}}}\in
L_{q_{1}}(\mathbb{R}^{2}).$

From (3.2), we deduce that

$\prod_{i=1}^{3}A_{i}(x,y)=|f(x+y)+f(x-y)+if(-x-y)-if(-x+y)|\
|g(x)||h(x)|,\quad\forall(x,y)\in\mathbb{R}^{2}.$ (3.3)

On the other hand, since $q>1$, then $t^{q}$ is a convex function, using the
change of variables theorem, we obtain

$\displaystyle\int_{\mathbb{R}}|f(x+y)+f(x-y)+if(-x-y)-if(-x+y)|^{q}dy$
$\displaystyle\leq
4^{q-1}\int_{\mathbb{R}}\left(|f(x+y)|^{q}+|f(x-y)|^{q}+|if(-x-y)|^{q}+|if(-x+y)|^{q}\right)dy=4^{q}\int_{\mathbb{R}}|f(t)|^{q}dt.$

Based on the assumption of $f\in L_{q}(\mathbb{R}),h\in L_{r}(\mathbb{R})$,
using Fubini’s theorem, we obtain $L_{p_{1}}(\mathbb{R}^{2})$-norm estimation
for the operator $A_{1}(x,y)$ as follows

$\displaystyle\|A_{1}\|^{p_{1}}_{L_{p_{1}}(\mathbb{R}^{2})}$
$\displaystyle=\int_{\mathbb{R}^{2}}|f(x+y)+f(x-y)+if(-x-y)-if(-x+y)|^{q}|h(x)|^{r}dxdy$
$\displaystyle\leq
4^{q}\int_{\mathbb{R}}\biggl{\\{}\int_{\mathbb{R}}|f(t)|^{q}dt\biggr{\\}}|h(x)|^{r}dx=4^{q}\bigg{(}\int_{\mathbb{R}}|f(t)|^{q}dt\bigg{)}\bigg{(}\int_{\mathbb{R}}|h(x)|^{r}dx\bigg{)}=4^{q}\|f\|^{q}_{L_{q}(\mathbb{R})}\|h\|^{r}_{L_{r}(\mathbb{R})}.$

This means that

$\|A_{1}\|_{L_{p_{1}}(\mathbb{R}^{2})}=4^{\frac{q}{p_{1}}}\|f\|^{\frac{q}{p_{1}}}_{L_{q}(\mathbb{R})}\|h\|^{\frac{r}{p_{1}}}_{L_{r}(\mathbb{R})}.$
(3.4)

Similar to what we did with the evaluation (3.4) of $A_{1}(x,y)$, we also get
the norm estimation of $A_{2}$ on $L_{r_{1}}(\mathbb{R}^{2})$ as follows

$\|A_{2}\|_{L_{r_{1}}(\mathbb{R}^{2})}=4^{\frac{q}{r_{1}}}\|f\|_{L_{q}(\mathbb{R})}^{\frac{q}{r_{1}}}\|g\|_{L_{p}(\mathbb{R})}^{\frac{p}{r_{1}}}.$
(3.5)

And $L_{q_{1}}(\mathbb{R}^{2})$-norm estimation for the operator $A_{3}$ has
the following form

$\|A_{3}\|_{L_{q_{1}}(\mathbb{R}^{2})}^{q_{1}}=\int_{\mathbb{R}^{2}}|g(y)|^{p}|h(x)|^{r}dxdy=\bigg{(}\int_{\mathbb{R}}|g(y)|^{p}dy\bigg{)}\bigg{(}\int_{\mathbb{R}}|h(x)|^{r}dx\bigg{)}=\|g\|^{p}_{L_{p}(\mathbb{R})}\|h\|^{r}_{L_{r}(\mathbb{R})}.$

Therefore, we have

$\|A_{3}\|_{L_{q_{1}}(\mathbb{R}^{2})}=\|g\|^{\frac{p}{q_{1}}}_{L_{p}(\mathbb{R})}\|h\|^{\frac{r}{q_{1}}}_{L_{r}(\mathbb{R})}.$
(3.6)

Combining (3.4)(3.5) and (3.6), under condition (3.2), we obtain

$\|A_{1}\|_{L_{p_{1}}(\mathbb{R}^{2})}\|A_{2}\|_{L_{r_{1}}(\mathbb{R}^{2})}\|A_{3}\|_{L_{q_{1}}(\mathbb{R}^{2})}\leq
4\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}\|h\|_{L_{r}(\mathbb{R})}.$
(3.7)

Furthermore, from (2.4) and (3.3) we have

$\displaystyle\bigg{|}\int_{\mathbb{R}}\big{(}f\underset{H_{1},F}{\ast}g\big{)}(x)\cdot
h(x)dx\bigg{|}$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}^{2}}|g(y)||f(x+y)+f(x-y)+if(-x-y)-if(-x+y)||h(x)|dydx$
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}^{2}}\
\prod_{i=1}^{3}A_{i}(x,y)dxdy$

Thus $\frac{1}{p_{1}}+\frac{1}{q_{1}}+\frac{1}{r_{1}}=1$. Using Hölder
inequality and (3.7), then $\forall g\in L_{p}(\mathbb{R}),~{}f\in
L_{q}(\mathbb{R}),~{}h\in L_{r}(\mathbb{R})$ we have

$\displaystyle\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}^{2}}\
\prod_{i=1}^{3}A_{i}(x,y)dxdy$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\left(\int_{\mathbb{R}^{2}}|A_{1}(x,y)|^{p_{1}}dxdy\right)^{1\textfractionsolidus
p_{1}}\left(\int_{\mathbb{R}^{2}}|A_{2}(x,y)|^{r_{1}}dxdy\right)^{1\textfractionsolidus
r_{1}}\left(\int_{\mathbb{R}^{2}}|A_{3}(x,y)|^{q_{1}}dxdy\right)^{1\textfractionsolidus
q_{1}}$
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\|A_{1}\|_{L_{p_{1}}(\mathbb{R}^{2})}\|A_{2}\|_{L_{r_{1}}(\mathbb{R}^{2})}\|A_{3}\|_{L_{q_{1}}(\mathbb{R}^{2})}\leq\sqrt{\frac{2}{\pi}}\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}\|h\|_{L_{r}(\mathbb{R})}$

∎

######  Remark 1.

Readers can find another result similar to this theorem through Theorem 2.4 in
[25].

In case the given function $h(x)$ becomes Hartley-Fourier generalized
convolution operator (2.4), the following Young-type inequality is a direct
consequence of Theorem 3.1.

######  Corollary 3.1 (Young’s inequality for Hartley-Fourier generalized
convolution).

Let $p,q,r>1$ be real numbers, satisfying
$\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$. If $g\in L_{p}(\mathbb{R})$, $f\in
L_{q}(\mathbb{R})$, then the convolution (2.4) is well-defined and belongs to
$L_{r}(\mathbb{R})$. Moreover, the following inequality holds

$\big{\|}f\underset{H_{1},F}{*}g\big{\|}_{L_{r}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}.$
(3.8)

###### Proof.

Let $r_{1}$ be the conjugate exponent of $r$, i.e
$\frac{1}{r}+\frac{1}{r_{1}}=1$. From the assumptions of Corollary 3.1, we
have $\frac{1}{p}+\frac{1}{q}+\frac{1}{r_{1}}=2$, which shows the numbers $p$,
$q$, and $r_{1}$ satisfy the conditions of Theorem 3.1 (with $r$ being
replaced by $r_{1}$). Therefore, if $g\in L_{p}(\mathbb{R})$, $f\in
L_{q}(\mathbb{R})$, then the linear operator

$Lh:=\int_{\mathbb{R}}\big{(}f\underset{H_{1},F}{*}g\big{)}(x)\cdot h(x)dx$

is bounded in $L_{r_{1}}(\mathbb{R})$. Consequently, by the Riesz’s
representation theorem [19], then generalized convolution
$\big{(}f\underset{H_{1},F}{*}g\big{)}$ belongs to $L_{r}(\mathbb{R})$. To
prove the inequality (3.8), we choose the function

$h(x):=\mathrm{sign}\bigg{\\{}\big{(}f\underset{H_{1},F}{*}g\big{)}(x)\bigg{\\}}^{r}\times\bigg{\\{}\big{(}f\underset{H_{1},F}{*}g\big{)}(x)\bigg{\\}}^{\frac{r}{r_{1}}}.$

Then $h\in L_{r_{1}}(\mathbb{R})$, with the norm
$\|h\|_{L_{r_{1}}(\mathbb{R})}=\big{\|}f\underset{H_{1},F}{*}g\big{\|}^{\frac{r}{r_{1}}}_{L_{r}(\mathbb{R})}.$
Applying inequality (3.1) to such function $h(x)$, we get

$\displaystyle\big{\|}f\underset{H_{1},F}{*}g\big{\|}_{L_{r}(\mathbb{R})}^{r}$
$\displaystyle=\int_{\mathbb{R}}\big{|}\big{(}f\underset{H_{1},F}{*}g\big{)}(x)\big{|}^{r}dx=\bigg{|}\int_{\mathbb{R}}\big{(}f\underset{H_{1},F}{*}g\big{)}(x)\cdot
h(x)\bigg{|}$
$\displaystyle\leq\sqrt{\frac{2}{\pi}}\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}\|h\|_{L_{r_{1}}(\mathbb{R})}=\sqrt{\frac{2}{\pi}}\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}\big{\|}f\underset{H_{1},F}{*}g\big{\|}^{\frac{r}{r_{1}}}_{L_{r}(\mathbb{R})}.$

Since $r-\frac{r}{r_{1}}=1$, from the above equality, we arrive at the
conclusion of the corollary. ∎

Notice that, inequalities (3.1) and (3.8) do not hold for the important case
$f,g\in L_{2}(\mathbb{R})$. We know that the problem is determining the duals
of the space $L_{p}$ with $p\in$ $[1,\infty)$. There are basically two cases
of this problem that are $p=1$, and the case $1<p<\infty$. The major
difference between these two cases is the fact that for $1<p<\infty$, there is
a “nice”characterization of the dual of $L_{p}$ which identifies it with
$L_{q}$ (with $1/p+1/q=1$), and this identification holds without any
restriction on the underlying space. Otherwise, the duality of $L_{1}$ will be
defined by $L_{\infty}$ locally, but this identification will work only if the
underlying measure space is decomposable. Now let us consider the case
$p=q=r=1$, this means $f$ and $g$ are functions belonging to the
$L_{1}(\mathbb{R})$. We deduce that $\big{(}f\underset{H_{1},F}{*}g\big{)}$
belongs to $L_{1}(\mathbb{R})$ (see [21, 26]) and get the estimate
$L_{1}$-norm as follows.

######  Theorem 3.2.

For any $f,g\in L_{1}(\mathbb{R})$, then

$\big{\|}f\underset{H_{1},F}{*}g\big{\|}_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|f\|_{L_{1}(\mathbb{R})}\|g\|_{L_{1}(\mathbb{R})}.$
(3.9)

###### Proof.

Indeed, by the Definition 2, we infer that

$\displaystyle\big{\|}f\underset{H_{1},F}{*}g\big{\|}_{L_{1}(\mathbb{R})}$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}^{2}}|g(y)|\
|f(x+y)+f(x-y)+if(-x-y)-if(-x+y)|dydx$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\bigg{\\{}\int_{\mathbb{R}^{2}}|g(y)||f(x+y)|dydx+\int_{\mathbb{R}^{2}}|g(y)|\
|f(x-y)|dydx$ $\displaystyle+\int_{\mathbb{R}^{2}}|g(y)|\
|if(-x-y)|dydx+\int_{\mathbb{R}^{2}}|g(y)|\ |if(-x+y)|dydx\bigg{\\}}.$

Since $f,g\in L_{1}(\mathbb{R})$, through the change of variables combined
with Fubini’s theorem, we obtain

$\displaystyle\big{\|}f\underset{H_{1},F}{*}g\big{\|}_{L_{1}(\mathbb{R})}$
$\displaystyle\leq\frac{4}{2\sqrt{2\pi}}\bigg{(}\int_{\mathbb{R}}|g(y)|dy\bigg{)}\bigg{(}\int_{\mathbb{R}}|f(t)|dt\bigg{)}=\sqrt{\frac{2}{\pi}}\|f\|_{L_{1}(\mathbb{R})}\|g\|_{L_{1}(\mathbb{R})}.$

∎

What about the case $r=\infty$?

######  Theorem 3.3.

Suppose that $p,q>1$ and satisfy $\frac{1}{p}+\frac{1}{q}=1$. If $g\in
L_{p}(\mathbb{R}),f\in L_{q}(\mathbb{R})$, then convolution operator (2.4) is
a bounded function for any $x\in\mathbb{R}$. Moreover, the following
inequality holds

$\lVert
f\underset{H_{1},F}{*}\lVert_{L_{\infty}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}.$
(3.10)

###### Proof.

From (2.4), we have

$|(f\underset{H_{1},F}{*})|\leq\frac{1}{2\sqrt{2\pi}}\int\limits_{\mathbb{R}}|g(y)|\
(|f(x+y)|+|f(x-y)|+|if(-x-y)|+|if(-x+y)|)dy.$

Using Hölder’s inequalities for the pair of conjugate exponents $p$ and $q$,
we deduce that

$|(f\underset{H_{1},F}{*})|\leq\frac{1}{2\sqrt{2\pi}}\left(\int_{\mathbb{R}}|g(\xi)|^{p}d\xi\right)^{\frac{1}{p}}\left(\int_{\mathbb{R}}4^{q}|f(t)|^{q}dt\right)^{\frac{1}{q}}=\frac{4}{2\sqrt{2\pi}}\\\
|g\|_{L_{p}(\mathbb{R})}\|f\|_{L_{q}(\mathbb{R})}<\infty.$

This implies that the convolution operator $(f\underset{H_{1},F}{*})$ is
bounded function $\forall x\in\mathbb{R}$ and the desired conclusion
inequality (3.10). ∎

Even in the case of the classical Fourier convolution, one has only the
Young’s convolution (3.9) for the case $p=q=r=1$. Therefore, the Young-type
inequality (3.9) is a specific characteristic of the $\underset{H_{1},F}{*}$
convolution introduced here. Indeed, suppose that $f,g$ are functions
belonging to $L_{1}(\mathbb{R})$ space, then the Fourier convolution
$(f\underset{F}{*}g)\in L_{1}(\mathbb{R})$ and $L_{1}$-norm estimation
inequality are as follows
$\lVert(f\underset{F}{*}g)\lVert_{L_{1}(\mathbb{R})}\leq\lVert
f\lVert_{L_{1}(\mathbb{R})}\lVert g\lVert_{L_{1}(\mathbb{R})},$ (see [16]). It
shows, on the $L_{1}(\mathbb{R})$ space, when being equipped with the
multiplication defined as Fourier convolution multiplication, we get
$\left(L_{1}(\mathbb{R}),\underset{F}{*}\right)$ as a commutative normed ring
[16] for the aforementioned convolution multiplication. However, the ring
structure is no longer commutative if the convolution multiplication is
replaced by the Hartley-Fouier generalized convolution (2.4), this will be
demonstrated in detail in Subsection 4.1 by applying inequality (3.9).

###  3.2. Young’s inequalities for Hartley convolution

In a similar way, we obtain Young type theorem for the Hartley convolution
(2.5). The results of this part are proved to be similar to those in
Subsection 3.1, respectively.

######  Theorem 3.4.

Let $p,q,r>1$ be real numbers satisfying
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$. Then the following inequalities hold
true for all $f\in L_{p}(\mathbb{R})$, $g\in L_{q}(\mathbb{R})$ and $h\in
L_{r}(\mathbb{R})$.

$\bigg{|}\int_{\mathbb{R}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}g\big{)}(x)\cdot
h(x)dx\bigg{|}\leq\sqrt{\frac{2}{\pi}}\|f\|_{L_{p}(\mathbb{R})}\|g\|_{L_{q}(\mathbb{R})}\|h\|_{L_{r}(\mathbb{R})},$
(3.11)

######  Corollary 3.2.

If $p,q,r>1$ and satisfy $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$, for any
$f\in L_{p}(\mathbb{R})$, $g\in L_{q}(\mathbb{R})$, then convolution
$\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}g\big{)}$ belongs to $L_{r}(\mathbb{R})$. Then we
obtain the following norm estimation

$\big{\|}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}g\big{\|}_{L_{r}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|f\|_{L_{p}(\mathbb{R})}\|g\|_{L_{q}(\mathbb{R})}.$
(3.12)

The proof is similar to Corollary 3.1. And $L_{1}$-norm estimate of Hartley
convolution (2.5) is as follows.

$\big{\|}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}g\big{\|}_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|f\|_{L_{1}(\mathbb{R})}\|g\|_{L_{1}(\mathbb{R})}\quad\forall
f,g\in L_{1}(\mathbb{R}).$ (3.13)

The same thing for the case $r=\infty$ is also obtained with convolution
(2.5).

###  3.3. Weighted $L_{p}$-boundedness for the Hartley convolutions

By considering the $L_{p}$ norms in more naturally determined weighted spaces,
S. Saitoh [15] gave Fourier convolution norm inequalities in the form

$\big{\lVert}\big{(}(F_{1}\rho_{1})\underset{F}{*}(F_{2}\rho_{2})\big{)}\cdot(\rho_{1}\underset{F}{*}\rho_{2})^{\frac{1}{p}-1}\big{\lVert}_{L_{p}(\mathbb{R})}\leq\big{\lVert}F_{1}\big{\lVert}_{L_{p}(\mathbb{R},|\rho_{1}|)}\big{\lVert}F_{2}\big{\lVert}_{L_{p}(\mathbb{R},|\rho_{2}|)},\quad
p>1,$

where $\rho_{j}$ are non-vanishing functions, $F_{j}\in
L_{p}(\mathbb{R},|\rho_{j}|),\,j=1,2$. Here, the norm of $F_{j}$ in the
weighted space $L_{p}(\mathbb{R},\rho_{j})$ is understood as
$\big{\lVert}F_{j}\big{\lVert}_{L_{p}(\mathbb{R},\rho_{j})}=\bigg{\\{}\displaystyle\int\limits_{\mathbb{R}}|F_{j}(x)|^{p}\rho_{j}(x)\mathrm{d}x\bigg{\\}}^{\frac{1}{p}}.$
This type of inequality is very convenient as many applications require the
“same”$L_{p}$ norms. Saitoh’s basic idea in weighted $L_{p}$ norm inequalities
is fairly simple, however, it has been successfully implemented in many
application [15]. Saitoh’s inequality also can be applied to estimating the
solution to a parabolic integro-differential equation [10] modeling a
scattered acoustic field. Based on the above aspects, together with using
Hölder’s inequality, and Fubini’s theorem, we established another result in
weighted space $L_{p}(\mathbb{R},\rho_{j})$ for Hartley convolution (2.5).
Some techniques used in the proof of our theorem come from [23, 27], and we
follow closely the strategy of these results.

######  Theorem 3.5.

Suppose that $\rho_{1},\rho_{2}$ are non-vanishing positive functions such
that convolution $(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\rho_{2})$, given by (2.5), is well-defined. For
any functions $F_{1}\in L_{p}(\mathbb{R},\rho_{1})$ and $F_{2}\in
L_{p}(\mathbb{R},\rho_{2})$ with $p>1$, the following
$L_{p}(\mathbb{R})$-weighted inequality holds true

$\big{\lVert}\big{(}F_{1}\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}F_{2}\rho_{2}\big{)}(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\rho_{2})^{\frac{1}{p}-1}\big{\lVert}_{L_{p}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\big{\lVert}F_{1}\big{\lVert}_{L_{p}(\mathbb{R},\rho_{1})}\big{\lVert}F_{2}\big{\lVert}_{L_{p}(\mathbb{R},\rho_{2})}.$
(3.14)

###### Proof.

From (2.5), we have

$\displaystyle\big{\lVert}\big{(}F_{1}\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}F_{2}\rho_{2}\big{)}(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\rho_{2})^{\frac{1}{p}-1}\big{\lVert}^{p}_{L_{p}(\mathbb{R})}=\int_{\mathbb{R}}\bigg{|}\big{(}F_{1}\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}F_{2}\rho_{2}\big{)}(x)(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\rho_{2})^{\frac{1}{p}-1}(x)\bigg{|}^{p}dx$
(3.15)
$\displaystyle=\frac{1}{2\sqrt{2\pi}}\times\int_{\mathbb{R}}\biggl{\\{}\bigg{|}\int_{\mathbb{R}}(F_{1}\rho_{1})(y)\bigg{(}(F_{2}\rho_{2})(x+y)+(F_{2}\rho_{2})(x-y)+(F_{2}\rho_{2})(-x+y)-(F_{2}\rho_{2})(-x-y)\bigg{)}dy\bigg{|}^{p}$
$\displaystyle\times\bigg{|}\int_{\mathbb{R}}\rho_{1}(y)\bigg{(}\rho_{2}(x+y)+\rho_{2}(x-y)+\rho_{2}(-x+y)-\rho_{2}(-x-y)\bigg{)}dy\bigg{|}^{1-p}\biggr{\\}}dx$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}}\big{(}\mathcal{A}_{1}(x,y)+\mathcal{A}_{2}(x,y)+\mathcal{A}_{3}(x,y)+\mathcal{A}_{4}(x,y)\big{)}\mathscr{T}^{1-p}_{(x,y)}dx,\
p>1,$

where we put

$\displaystyle A_{1}(x,y)=\int_{\mathbb{R}}|(F_{1}\rho_{1})(y)|\
|(F_{2}\rho_{2})(x+y)|dy,$ $\displaystyle
A_{2}(x,y)=\int_{\mathbb{R}}|(F_{1}\rho_{1})(y)|\ |(F_{2}\rho_{2})(x-y)|dy,$
$\displaystyle A_{3}(x,y)=\int_{\mathbb{R}}|(F_{1}\rho_{1})(y)|\
|(F_{2}\rho_{2})(-x+y)|dy,$ $\displaystyle
A_{4}(x,y)=\int_{\mathbb{R}}|(F_{1}\rho_{1})(y)|\ |(F_{2}\rho_{2})(-x-y)|dy.$

And

$\mathscr{T}^{1-p}_{(x,y)}=\biggl{\\{}\int_{\mathbb{R}}\rho_{1}(y)\big{(}\rho_{2}(x+y)+\rho_{2}(x-y)+\rho_{2}(-x+y)+\rho_{2}(-x-y)\big{)}dy\biggr{\\}}^{1-p}.$

Using Hölder’s inequalities for the pair of conjugate exponents $p$ and $q$
towards operators $A_{i},i=\overline{1,4}$, we have

$\displaystyle
A_{1}(x,y)\leq\biggl{\\{}\int_{\mathbb{R}}|F_{1}(y)|^{p}\rho_{1}(y)|F_{2}(x+y)|^{p}\rho_{2}(x+y)dy\biggr{\\}}^{\frac{1}{p}}\times\biggl{\\{}\int_{\mathbb{R}}\rho_{1}(y)\rho_{2}(x+y)dy\biggr{\\}}^{\frac{1}{q}},$
$\displaystyle
A_{2}(x,y)\leq\biggl{\\{}\int_{\mathbb{R}}|F_{1}(y)|^{p}\rho_{1}(y)|F_{2}(x-y)|^{p}\rho_{2}(x-y)dy\biggr{\\}}^{\frac{1}{p}}\times\biggl{\\{}\int_{\mathbb{R}}\rho_{1}(y)\rho_{2}(x-y)dy\biggr{\\}}^{\frac{1}{q}},$
$\displaystyle
A_{3}(x,y)\leq\biggl{\\{}\int_{\mathbb{R}}|F_{1}(y)|^{p}\rho_{1}(y)|F_{2}(-x+y)|^{p}\rho_{2}(-x+y)dy\biggr{\\}}^{\frac{1}{p}}\times\biggl{\\{}\int_{\mathbb{R}}\rho_{1}(y)\rho_{2}(-x+y)dy\biggr{\\}}^{\frac{1}{q}},$
$\displaystyle
A_{4}(x,y)\leq\biggl{\\{}\int_{\mathbb{R}}|F_{1}(y)|^{p}\rho_{1}(y)|F_{2}(-x-y)|^{p}\rho_{2}(-x-y)dy\biggr{\\}}^{\frac{1}{p}}\times\biggl{\\{}\int_{\mathbb{R}}\rho_{1}(y)\rho_{2}(-x-y)dy\biggr{\\}}^{\frac{1}{q}}.$

Therefore, recalling that $t^{\frac{1}{p}},t^{\frac{1}{q}}\ (p,q>1)$ are
concave functions, together with the above-estimates, we obtain

$\displaystyle\sum_{i=1}^{4}A_{i}(x,y)\leq\biggl{\\{}\int_{\mathbb{R}}|(F_{1}\rho_{1})(y)|^{p}\rho_{1}(y)\bigg{(}|F_{2}(x+y)|^{p}\rho_{2}(x+y)+|F_{2}(x-y)|^{p}\rho_{2}(x-y)$
(3.16)
$\displaystyle+|F_{2}(-x+y)|^{p}\rho_{2}(-x+y)+|F_{2}(-x-y)|^{p}\rho_{2}(-x-y)\bigg{)}dy\biggr{\\}}^{\frac{1}{p}}\times\mathscr{T}^{\frac{1}{q}}_{(x,y)}.$

From (3.15) and (3.16), we deduce that

$\displaystyle\big{\lVert}\big{(}F_{1}\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}F_{2}\rho_{2}\big{)}(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\rho_{2})^{\frac{1}{p}-1}\big{\lVert}^{p}_{L_{p}(\mathbb{R})}$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}}\biggl{\\{}\int_{\mathbb{R}}|F_{1}(y)|^{p}\rho_{1}(y)\bigg{(}|F_{2}(x+y)|^{p}\rho_{2}(x+y)+|F_{2}(x-y)|^{p}\rho_{2}(x-y)+|F_{2}(-x+y)|^{p}\rho_{2}(-x+y)$
$\displaystyle+|F_{2}(-x-y)|^{p}\rho_{2}(-x-y)\bigg{)}dy\times\mathscr{T}^{\frac{p}{q}+1-p}_{(x,y)}\biggr{\\}}dx.$

Since $p$ and $q$ are a conjugate pair ($\frac{1}{p}+\frac{1}{q}=1$), it
implies that $\frac{p}{q}+1-p=0$, by using the Fubini theorem, we obtain

$\displaystyle\big{\lVert}\big{(}F_{1}\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}F_{2}\rho_{2}\big{)}(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\rho_{2})^{\frac{1}{p}-1}\big{\lVert}^{p}_{L_{p}(\mathbb{R})}$
$\displaystyle\leq\frac{4}{2\sqrt{2\pi}}\int_{\mathbb{R}^{2}}|F_{1}(y)|^{p}\rho_{1}(y)\
|F_{2}(t)|^{p}\rho_{2}(t)\ dydt$
$\displaystyle=\sqrt{\frac{2}{\pi}}\bigg{(}\int_{\mathbb{R}}|F_{1}(y)|^{p}\rho_{1}(y)dy\bigg{)}\bigg{(}\int_{\mathbb{R}}|F_{2}(t)|^{p}\rho_{2}(t)dt\bigg{)}.$

From the above equality, we infer the desired conclusion of the theorem. ∎

Clearly, if $\rho_{2}(x)$ is an identical function $1$ for all
$x\in\mathbb{R}$ and $0<\rho_{1}(x)\in L_{1}(\mathbb{R})$, then we have

$\big{|}(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}1)(x)\big{|}=\frac{2}{2\sqrt{2\pi}}\int_{\mathbb{R}}\rho_{1}(x)dx=\frac{1}{\sqrt{2\pi}}\|\rho_{1}\|_{L_{1}(\mathbb{R})}<\infty.$

This means that $(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}1)$ is well-defined and consequently

$\big{|}(\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}1)(x)\big{|}^{1-\frac{1}{p}}\leq\left(\frac{1}{\sqrt{2\pi}}\right)^{1-\frac{1}{p}}\lVert\rho_{1}\lVert^{1-\frac{1}{p}}_{L_{1}(\mathbb{R})}.$

Combining with Theorem 3.5, we arrive at the following corollary

######  Corollary 3.3.

Let $\rho_{1}$ be a positive function belonging to $L_{1}(\mathbb{R})$. If
$F_{1},F_{2}$ are functions belonging to $L_{p}(\mathbb{R},\rho_{1})$ and
$L_{p}(\mathbb{R})$, respectively, with $p>1$, then we have the following
estimate

$\lVert F_{1}\rho_{1}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}F_{2}\lVert_{L_{p}(\mathbb{R})}\leq\frac{1}{\pi}(\sqrt{2\pi})^{\frac{1}{p}}\lVert\rho_{1}\lVert^{1-\frac{1}{p}}_{L_{1}(\mathbb{R})}\lVert
F_{1}\rho_{1}\lVert_{L_{p}(\mathbb{R},\rho_{1})}\lVert
F_{2}\lVert_{L_{p}(\mathbb{R})}.$ (3.17)

Note that, the inequalities obtained in Theorem 3.5 and Corollary 3.3 are
still true for case $p=2$. In a similar way as above, we also obtained
$L_{p}$-boundedness in weighted space $L_{p}(\mathbb{R},\rho_{j})$ for
generalized convolution operator (2.4).

##  4\. Some applications

###  4.1. Structure Nomerd ring

To avoid confusion with the notation of exponential function $(e)$, we denote
$\mathcal{U}$ as the unit element for the multiplicative on vector space $V$.

######  Definition 3.

(See [11]). A vector space $V$ is with a ring structure and a vector norm is
called a normed ring if $\lVert v\ w\lVert\leq\lVert v\lVert\lVert w\lVert,$
for all $v,w\in V$. If $V$ has a multiplicative unit element, then it also
requires that $\lVert\mathcal{U}\lVert=1$.

######  Theorem 4.1.

The Banach space $L_{1}(\mathbb{R})$, when being equipped with the Hartley-
Fourier convolution (2.4) multiplication,
$\big{(}L_{1}(\mathbb{R}),\underset{H_{1},F}{*}\big{)}$ becomes a non-
commutative normed ring and has no unit element.

###### Proof.

For any $f,g$ which belong to $L_{1}(\mathbb{R})$ space, we get
$(f\underset{H_{1},F}{*}g)\in L_{1}(\mathbb{R})$ (refer [21]). Without loss of
generality, we may assume that
$\|f\|_{L_{1}(\mathbb{R})}=\sqrt[4]{\frac{2}{\pi}}\int_{\mathbb{R}}|f(x)|dx$ ,
by inequality (3.9), we get
$\big{\|}f\underset{H_{1},F}{\ast}g\big{\|}_{L_{1}(\mathbb{R})}\leq\|f\|_{L_{1}(\mathbb{R})}\|g\|_{L_{1}(\mathbb{R})},\forall
f,g\in L_{1}(\mathbb{R}).$ This implies that structure
$(L_{1}(\mathbb{R}),\underset{H_{1},F}{*})$ is a normed ring. On the other
hand, by the factorization identity (2.6), we conclude that the normed ring
here is non-commutative for the aforementioned convolutional multiplication.

Next, we need to prove that $(L_{1}(\mathbb{R}),\underset{H_{1},F}{*})$ has no
unit element. Suppose that, there exists an element $\mathcal{U}\in
L_{1}(\mathbb{R})$ such that
$(\mathcal{U}\underset{H_{1},F}{*}f)(x)=(f\underset{H_{1},F}{*}\mathcal{U})(x)=f(x),$
for any $f\in L_{1}(\mathbb{R}).$ The factorization identity (2.6) implies
$H_{1}(f\underset{H_{1},F}{\ast}\mathcal{U})(y)=(H_{1}f)(y),\forall f\in
L_{1}(\mathbb{R}).$ This is equivalent to
$(H_{1}f)(y)(F\mathcal{U})(y)=(H_{1}f)(y),$ so we deduce that

$(H_{1}f)\big{[}(F\mathcal{U})(y)-1\big{]}=0,\forall f\in L_{1}(\mathbb{R}).$
(4.1)

Now, we choose $f(x)=\sqrt{\frac{\pi}{2}}e^{-|x|}\in L_{1}(\mathbb{R})$.
Obviously $f(x)$ belongs to the $L_{1}(\mathbb{R})$ and we obtain
$H_{1}\left(\sqrt{\frac{\pi}{2}}e^{-|x|}\right)(y)=\frac{1}{1+y^{2}}>0.$ Then
the equality (4.1) holds if and only if $(F\mathcal{U})(y)=1$, this means that

$\displaystyle\mathcal{U}(y)=(F1)(y)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-ixy}dx=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}(\cos
xy-i\sin xy)dx\notin L_{1}(\mathbb{R}).$

Therefore, the equality (4.1) is not satisfied $\forall f\in
L_{1}(\mathbb{R})$, implying that the convolution multiplications have no unit
element. ∎

In a similar way, we also obtain a structure non-commutative normed ring, that
has no unit element, when $L_{1}(\mathbb{R})$ space is equipped with the
Hartley convolution (2.5) multiplication. We next turn our attention to the
zero-divisor properties of the normed ring, which is Titchmarsh’s theorem [22,
11]. The Titchmarsh convolution theorem asserts that if $f,g\in
L_{1}(\mathbb{R})$ vanish on $(-\infty,0)$, and $(f*g)(x)=0$ for $x\leq T$,
then there exist real numbers $\alpha$ and $\beta$, such that
$\alpha+\beta=T$, and the functions $f$ and $g$ vanish on $[0,\alpha]$ and
$[0,\beta]$, respectively. Based on this, we give a new version of the
Titchmarsh-type theorem for the Hartley-Fourier generalized convolution
operator (2.4). To get started, we need the following definition.

######  Definition 4.

Denote $L_{1}(\mathbb{R},e^{|x|})$ be the normed space of all measurable
functions $f(x)$ on $\mathbb{R}$ such that $\int\limits_{\mathbb{R}}|f(x)|\
e^{|x|}dx<+\infty,$ with the norm defined by
$||f||_{L_{1}(\mathbb{R},e^{|x|})}:=\int\limits_{\mathbb{R}}|f(x)|\
e^{|x|}dx.$

######  Theorem 4.2 (The Titchmarsh-type Theorem).

Let $f,g$ be continuous functions on $\mathbb{R}$ and such that $f,g\in
L_{1}(\mathbb{R},e^{|x|})$. It follows that
$(f\underset{H_{1},F}{*}g)(x)\equiv 0$ almost everywhere on $\mathbb{R}$ if
and only if $f(x)$ or $g(x)$ is vanish almost everywhere on $\mathbb{R}$.

###### Proof.

If $(f\underset{H_{1},F}{\ast}g)\equiv 0$ almost everywhere on $\mathbb{R}$,
then $H_{1}(f\underset{H_{1},F}{\ast}g)(y)$ is vanish almost everywhere on
$\mathbb{R}$. From (2.6), we obtain

$H_{1}(f\underset{H_{1},F}{\ast}g)(y)=(H_{1}f)(y)(Fg)(y)\equiv 0\ \text{almost
everywhere on }\ \mathbb{R}.$ (4.2)

We prove that $(H_{1}f)(y)$ and $(Fg)(y)$ are analytic functions. Indeed,
using (2.1) together with the Lebesgue’s dominated convergence theorem, we can
exchange orders of the differentiation with the integration, we infer that

$\displaystyle\left|\frac{d^{k}}{dy^{k}}(H_{1}f)(y)\right|$
$\displaystyle\leq\frac{1}{2\sqrt{2\pi}}\int_{\mathbb{R}}\left|f(x)\left(\frac{d^{k}}{dy^{k}}\mathrm{Cas}\,xy\right)\right|dx=\frac{1}{2~{}\sqrt{2\pi}}\int_{\mathbb{R}}\left|f(x)x^{k}\right|\left|\cos\left(xy+\frac{k\pi}{2}\right)+\sin\left(xy+\frac{k\pi}{2}\right)\right|dx$
$\displaystyle\leq\frac{1}{2~{}\sqrt{\pi}}\int_{\mathbb{R}}\left|f(x)x^{k}\right|dx\leq\frac{1}{2~{}\sqrt{\pi}}\int_{\mathbb{R}}e^{-|x|}\frac{|x|^{k}}{k!}k!|f(x)|e^{|x|}dx$
$\displaystyle\leq\frac{1}{2~{}\sqrt{\pi}}\int_{\mathbb{R}}k!|f(x)|e^{|x|}dx\quad\left(\text{since
}e^{-|x|}|x^{k}|=e^{-|x|}\frac{|x^{k}|}{k!}k!\leq k!\right)$
$\displaystyle=\frac{1}{2~{}\sqrt{\pi}}k!\|f\|_{L_{1}(\mathbb{R},e^{|x|})}=k!C,\
\text{where }\ C=\frac{1}{2~{}\sqrt{\pi}}\|f\|_{L_{1}(\mathbb{R},e^{|x|})}.$

On the other hand, we evaluate the remainder of Taylor’s series expansion in
neighborhood of a point $y_{0}$ as follows

$\left|\frac{1}{k!}\frac{d^{k}(H_{1}f)(\theta+y_{0})}{dy^{k}}(y-y_{0})^{k}\right|\leq\frac{1}{k!}k!C|y-y_{0}|^{k}=C|y-y_{0}|^{k},\
\text{with}\ 0<\theta<1.$

This means that at $y_{0}$ and in the neighborhood of $y_{0}$ $(H_{1}f)(y)$
can be expanded to Taylor series and converge on $|y-y_{0}|<1$, which implies
that $(H_{1}f)(y)$ is an analytic function $\forall y\in\mathbb{R}$. In a
similar way, we can also prove $(Fg)(y)$ is an analytic function. Therefore,
from (4.2), we infer that $(H_{1}f)(y)$ or $(Fg)(y)$ is vanish almost
everywhere on $\mathbb{R}$. Furthermore, the uniqueness in $L_{1}(\mathbb{R})$
of Hartley, Fourier transforms (refer [5, 14, 17]), imples that $f(x)$ or
$g(x)$ are functions of vanishing almost everywhere on $\mathbb{R}$. ∎

Theorem 4.2 can also be stated for convolution (2.5) with a similar proof.

###  4.2. $L_{1}$-solutions of the Fredholm integral equation second kind

Now, we consider the Fredholm integral equation of the second kind as follows
(see [18])

$f(x)-\lambda\int_{\textbf{S}}K(x,y)f(y)d(y)=g(x),$ (4.3)

Here the right-hand side $g(x)$ and the kernel $K(x,y)$ are some known
functions, $\lambda$ is a known parameter, $S$ is a piecewise-smooth surface
(or line), and $f$ is an unknown function. Up to now, during the last two
decades, the theory of abstract Volterra and Fredholm integral equation has
undergone rapid development. To a large extent, this was due to the
applications of this theory to problems in mathematical physics, such as
viscoelasticity, heat conduction in materials with memory, electrodynamics
with memory, and to the need for tools to tackle the problems arising in these
fields. Many interesting phenomena are not found with differential equations
but observed in specific examples of integral equations (refer [13, 20]). For
general kernels $K(x,y)$, an explicit solution to Eq. (4.3) is not known, and
approximate solutions have been sought instead. Nevertheless, some authors
tried to get explicit analytic solutions to particular cases, for example, in
[18] have found analytic solutions to Eq. (4.3) for the kernels form
$K(x,\tau)=K(x-\tau)=(x-\tau)^{\alpha};e^{-a|x-\tau|};\sinh(a(x-\tau))$, and
$aJ_{1}(a(x-\tau)$ with $J_{1}$ being the Bessel functions. We will consider
solvability Eq. (4.3) in $L_{1}(\mathbb{R})$ for case $\lambda=-1$,
$S=\mathbb{R}$ with kernel
$K(x,y)=\frac{1}{2\sqrt{2\pi}}[\varphi(x+y)+\varphi(x-y)+\varphi(-x+y)-\varphi(-x-y)],$
and $g(x)=(\varphi\underset{H_{1},F}{\ast}\xi)(x).$ We can be rewrite (4.3) in
the convolution form as follows

$f(x)+\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\varphi\big{)}(x)=(\varphi\underset{H_{1},F}{\ast}\xi)(x),\quad
x\in\mathbb{R}.$ (4.4)

######  Theorem 4.3.

Suppose that $\varphi$ and $\xi$ are given functions belonging to
$L_{1}(\mathbb{R})$ and satisfy the condition $1+(H_{1}\varphi)(y)\neq 0$ for
any $y\in\mathbb{R}$. Then (4.4) has the unique solution in
$L_{1}(\mathbb{R})$ which can be represented in the form
$f(x)=(\ell\underset{H_{1},F}{\ast}\xi)(x).$ Here $\ell\in L_{1}(\mathbb{R})$
is defined by $(H_{1}\ell)(y)=\frac{(H_{1}\varphi)(y)}{1+(H_{1}\varphi)(y)}.$
Furthermore, the following $L_{1}$-norm estimate holds
$\|f\|_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|\ell\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})}.$

###### Proof.

Applying the $(H_{1})$ transform to both sides of (4.4), we deduce that

$(H_{1}f)(y)+H_{1}(f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\varphi)(y)=H_{1}\big{(}\varphi\underset{H_{1},F}{\ast}\xi\big{)}(y).$

Using the factorization properties (2.7) and (2.6), together with the
condition $1+(H_{1}\varphi)(y)\neq 0$ for any $y\in\mathbb{R}$, we obtain
$(H_{1}f)(y)[1+(H_{1}\varphi)(y)]=(H_{1}\varphi)(y)(F\xi)(y),\quad
y\in\mathbb{R}$ This is equivalent to
$(H_{1}f)(y)=\frac{(H_{1}\varphi)(y)}{1+(H_{1}\varphi)(y)}(F\xi)(y).$ By Lemma
(2.1), the existence of a function $\ell\in L_{1}(\mathbb{R})$ such that
$(H_{1}\ell)(y)=\frac{(H_{1}\varphi)(y)}{1+(H_{1}\varphi)(y)}.$ This means
that

$(H_{1}f)(y)=(H_{1}\ell)(y)(F\xi)(y)=H_{1}(\ell\underset{H_{1},F}{\ast}\xi)(y).$

Therefore, $f(x)=\big{(}\ell\underset{H_{1},F}{\ast}\xi\big{)}(x)$ almost
everywhere for any $x\in\mathbb{R}$. Moreover, since $\ell,\xi$ are functions
belonging $L_{1}(\mathbb{R})$, we deduce that $f\in L_{1}(\mathbb{R})$ (see
[21, 26]). Applying the inequality (3.9) we obtain norm estimation of the
solution on $L_{1}$ space as follows
$\|f\|_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|\ell\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})}.$
∎

######  Remark 2.

If $p,q,r>1$ satisfy the condition $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$,
applying inequality (3.8), for any $f\in L_{r}(\mathbb{R})$, $\ell\in
L_{p}(\mathbb{R})$, and $\xi\in L_{q}(\mathbb{R})$, we get the following
solution estimate
$\|f\|_{L_{r}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|\ell\|_{L_{p}(\mathbb{R})}\|\xi\|_{L_{q}(\mathbb{R})}.$

Furthermore, in case $r=\infty$, by applying (3.10), we obtain
$L_{\infty}$-boundedness of solution for Eq.(4.4) as follows $\lVert
f\lVert_{L_{\infty}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\lVert\ell\lVert_{L_{p}(\mathbb{R})}\lVert\xi\lVert_{L_{q}(\mathbb{R})}.$

###  4.3. $L_{1}$-solutions for Barbashin’s equations

We are concerned with integro-differential equation of the form

$\frac{\partial f(x,y)}{\partial
y}=c(x,y)f(x,y)+\int_{a}^{b}K(x,y,t)f(x,t)dt+\varphi(x,y).$ (4.5)

Here
$c:J\times[a,b]\rightarrow\mathbb{R},K:J\times[a,b]\times[a,b]\rightarrow\mathbb{R}$,
and mapping $\varphi:J\times[a,b]\rightarrow\mathbb{R}$ are given functions,
where $J$ is a bounded or unbounded interval, the function $f$ is unknown. The
equation (4.5) was first studied by E.A. Barbashin [3] and his pupils. For
this reason, this is nowadays called integro-differential equation of
Barbashin type or simply the Barbashin equation. The (4.5) has been applied to
many fields such as mathematical physics, radiation propagation, mathematical
biology and transport problems, e.g., for more details refer [2]. One of the
characteristics of Barbashin equation is that studying solvability of the
equation is heavily dependent on the kernel $K(x,s,\rho)$ of the equation. In
many cases, we can reduce equation (4.5) to the form of an ordinary
differential equation and use the Cauchy integral operator or evolution
operator to study it when the kernel does not depend on $x$. In some other
cases, we need to use the partial integral operator to study this equation
(see [2]). However, in the general case of $K(x,s,\rho)$ as an arbitrary
kernel, the problem of finding a solution for Barbashin equation remains open.
In some other cases, we need to use the partial integral operator to study
this equation (see [2]). On the other hand, if we view ${A}$ as the operator
defined by ${A}:=\partial\textfractionsolidus\partial y-c(x,y)\mathcal{I}$,
where $\mathcal{I}$ is the identity operator, then Eq. (4.5) is written in the
following form
$\mathcal{A}f(x,y)=\int\limits_{a}^{b}K(x,y,t)f(x,t)dt+\varphi(x,y).$ We will
consider solvability Eq. (4.5) in $L_{1}(\mathbb{R})$ for case operator
$\mathcal{A}f(x,y)=\left(1-\frac{d^{2}}{dx^{2}}\right)\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(x)$ with kernel
$K(x,y,t)=-\frac{1}{2\sqrt{2\pi}}[h(x+y)+h(x-y)+h(-x+y)-h(-x-y)]$, domain
$(a,b)=\mathbb{R}$, and
$\varphi(x,y)=\Big{(}h\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\xi\Big{)}(x)$. Then original equation (4.5)
becomes the linear equation and can be rewritten in the convolution form

$\left(1-\frac{d^{2}}{dx^{2}}\right)\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}g\big{)}(x)+\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}h\big{)}(x)=\big{(}h\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\xi\big{)}(x),$ (4.6)

where $g$, $h$, $\xi$ are given functions, convolution
$(\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast})$ is defined by (2.5), and $f$ is the
function to find.

######  Theorem 4.4.

Let $\xi\in L_{1}(\mathbb{R})$, $g(x)=\sqrt{\frac{\pi}{2}}e^{-|x|}$ , and $h$
be functions belonging to $L_{1}(\mathbb{R})$ such that
$1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)\neq 0$, $\forall y\in\mathbb{R}$. Then
Eq. (4.6) has the unique solution
$f(x)=\Big{(}\eta\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\xi\Big{)}(x)\in L_{1}(\mathbb{R})$, where
$\eta\in L_{1}(\mathbb{R})$ defined by
$\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\eta\big{)}(y)=\frac{\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)}{1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)}$ for all $y\in\mathbb{R}.$ Moreover, the
estimation of $L_{1}$-norm is as follows
$\|f\|_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|\eta\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})}.$

###### Proof.

It is easy to check $g(x)=\sqrt{\frac{\pi}{2}}e^{-|x|}$ actually satisfies the
assumptions of Lemma 2.2. By using (2.10), we obtain
$H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\bigg{\\{}\big{(}1-\frac{d^{2}}{dx^{2}}\big{)}\bigg{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\bigg{)}(x)\bigg{\\}}(y)=(1+y^{2})H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\big{)}(y).$
Applying the $H_{\left\\{\begin{subarray}{c}1\\\ 2\end{subarray}\right\\}}$
transform to both sides of (4.6) combining the factorization equality (2.7),
we get

$(1+y^{2})H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\left(f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(y)+H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}f\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}h\big{)}(y)=H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}h\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\xi\big{)}(y),\forall y>0,$

equivalent to $(1+y^{2})\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\left(\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(y)+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)=\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\xi\big{)}(y).$ Since
$H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\Big{(}\sqrt{\frac{\pi}{2}}e^{-|t|}\Big{)}(y)=\frac{1}{1+y^{2}}$
is finite (see [4], 1.4.1, page 23), under the condition
$1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)\neq 0$, $\forall y\in\mathbb{R}$, we
deduce that $\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)=\frac{\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)}{1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)}\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\xi\big{)}(y)$. Following Lemma 2.1, there exists a
function $\eta$ which belongs to $L_{1}(\mathbb{R}_{+})$ such that

$\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\eta\big{)}(y)=\frac{\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)}{1+\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h\big{)}(y)},\quad\forall y\in\mathbb{R}.$

Thus, we infer that $\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}f\big{)}(y)=\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\eta\big{)}(y)\big{(}H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\xi\big{)}(y)=H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\big{(}\eta\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\xi\big{)}(y)$, implies that
$f(x)=\big{(}\eta\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{\ast}\xi\big{)}(x)$ almost everywhere on
$\mathbb{R}$. Since $\eta,\xi\in L_{1}(\mathbb{R})$, then $f$ belongs to
$L_{1}(\mathbb{R})$ (see [21, 26]). Applying the inequality (3.13), we obtain
the boundedness in $L_{1}$ of
solution$\|f\|_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|\eta\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})}.$
∎

######  Remark 3.

Let $p,q,r>1$ such that $\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$ and if $f\in
L_{r}(\mathbb{R})$, $\eta\in L_{p}(\mathbb{R})$, and $\xi\in
L_{q}(\mathbb{R})$. Applying the inequality (3.12), we get the following
solution estimate
$\|f\|_{L_{r}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\|\eta\|_{L_{p}(\mathbb{R})}\|\xi\|_{L_{q}(\mathbb{R})}.$

######  Remark 4.

If there is a case where $\eta$ is the product of two functions
($\eta=\varphi.\rho$), $\rho$ is a positive function belonging to
$L_{1}(\mathbb{R})$, and $\varphi\in L_{1}(\mathbb{R},\rho)\cap
L_{p}(\mathbb{R},\rho)$, with $p>1$. For any function $\xi\in
L_{1}(\mathbb{R})\cap L_{p}(\mathbb{R})$, apply (3.17), then we have the
following estimate $\lVert
f\lVert_{L_{p}(\mathbb{R})}\leq\frac{1}{\pi}(\sqrt{2\pi})^{\frac{1}{p}}\lVert\rho\lVert^{1-\frac{1}{p}}_{L_{1}(\mathbb{R})}\lVert\varphi.\xi\lVert_{L_{p}(\mathbb{R},\rho)}\lVert\xi\lVert_{L_{p}(\mathbb{R})}$.

###  4.4. The Cauchy-type initial value problem

We consider the following Cauchy-type problem

$\left\\{\begin{array}[]{l}f(x)-f^{\prime\prime}(x)+\big{(}1-\frac{d^{2}}{dx^{2}}\big{)}\big{(}f\underset{H_{1},F}{\ast}g\big{)}(x)=\big{(}h\underset{H_{1},F}{\ast}g\big{)}(x),\\\
\lim\limits_{|x|\to\infty}f(x)=\lim\limits_{|x|\to\infty}f^{\prime}(x)=0,\end{array}\right.$
(4.7)

where $g$ is a given function belonging to $g\in L_{1}(\mathbb{R})$, $h\in
C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$ such as
$h^{\prime},h^{\prime\prime}\in L_{1}(\mathbb{R})$, and $f$ is the function to
find.

######  Theorem 4.5.

With the assumption of the function h, g us given as above, so that the
condition $1+(Fg)(y)\neq 0$ for any $y\in\mathbb{R}$. Then problem 4.7 has a
unique solution in $C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$ and is
represented by
$f(x)=\left(\big{(}h\underset{H_{1},F}{\ast}\xi\big{)}\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(x).$
Moreover, we have $\|f\|_{L_{1}(\mathbb{R})}\leq
2\sqrt{\frac{2}{\pi}}\|h\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})},$
where $\xi$ is a function beloning to $L_{1}(\mathbb{R})$ determined by
$(F\xi)(y)=\frac{(Fg)(y)}{1+(Fg)(y)},$ and $(\underset{H_{1},F}{\ast})$
defined by (2.4).

###### Proof.

Without loss of generality, we may assume that $f\in C^{2}(\mathbb{R})\cap
L_{1}(\mathbb{R})$ is a satisfied function condition of the Problem (4.7).
Applying Lemma 2.2 to the function $f$ instead of the role of $g$, by (2.10),
we obtain

$H_{1}\left(\left(1-\frac{d^{2}}{dx^{2}}\right)(f\underset{H_{1},F}{\ast}g)(x)\right)(y)=(1+y^{2})H_{1}(f\underset{H_{1},F}{\ast}g)(y),\quad\forall
g\in L_{1}(\mathbb{R}),~{}y\in\mathbb{R}.$ (4.8)

From (2.11), we deduce that

$\left(H_{1}f^{\prime\prime}(x)\right)(y)=-y^{2}(H_{1}f)(y).$ (4.9)

Applying the $(H_{1})$ transform to both sides of Eq. (4.7), combining (4.8),
(4.9), and using the factorization property (2.6), we infer
$(H_{1}f)(y)+y^{2}(H_{1}f)(y)+(1+y^{2})H_{1}(f\underset{H_{1},F}{\ast}g)(y)=H_{1}(f\underset{H_{1},F}{\ast}g)(y),\forall
y\in\mathbb{R}.$ This equivalent to
$(1+y^{2})(H_{1}f)(y)[1+(Fg)(y)]=(H_{1}h)(y)(Fg)(y),$ implies

$(H_{1}f)(y)=\frac{1}{1+y^{2}}\frac{(Fg)(y)}{1+(Fg)(y)}(H_{1}h)(y).$ (4.10)

By Wiener-Levy’s Theorem [11] for the Fourier transform, we deduce the
existence of a function $\xi$ belonging to $L_{1}(\mathbb{R})$ such that
$(F\xi)(y)=\frac{(Fg)(y)}{1+(Fg)(y)}.$ On the other hand,
$F\left(\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(y)=\frac{1}{1+y^{2}}$ [4] is
finite, and coupling (4.10) with (2.6), we have

$(H_{1}f)(y)=H_{1}\left((h\underset{H_{1},F}{\ast}\xi)\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(y).$

Therefore
$f(x)=\left((h\underset{H_{1},F}{\ast}\xi)\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(x)$
almost everywhere for any $x\in\mathbb{R}.$ Moreover, since $h,\xi,$ and
$\sqrt{\frac{\pi}{2}}e^{-|t|}$ are functions belonging to $L_{1}(\mathbb{R})$,
by [26, 21] lead to $f\in L_{1}(\mathbb{R})$. We putting
$G(x)=(h\underset{H_{1},F}{\ast}\xi)(x)\in L_{1}(\mathbb{R})$. Thus $h\in
C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$, and
$h^{\prime},h^{\prime\prime},\xi\in L_{1}(\mathbb{R})$, by (2.4), we infer
$G\in C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$. This means that $f\in
C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$.

Now, we prove solution $f(x)$ of Problem (4.7) actually satisfies the initial
condition. Indeed, by setting $G(x)$, we obtain

$\displaystyle f(x)$
$\displaystyle=\left(G\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(x)=\frac{\sqrt{\pi}}{2\sqrt{2}}\bigg{\\{}\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-|y|}G(x+y)dy+\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-|y|}G(x-y)dy$
(4.11)
$\displaystyle+\frac{i}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-|y|}G(-x-y)dy-\frac{i}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-|y|}G(-x+y)dy\bigg{\\}}$
$\displaystyle=2\sqrt{\frac{\pi}{2}}\big{\\{}J_{1}(x)+J_{2}(x)+iJ_{3}(x)-iJ_{4}(x)\big{\\}},\quad\forall
x\in\mathbb{R}.$

Observe the integral $J_{1}$ then
$J_{1}(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-|y|}G(x+y)dy=-\frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-|t|}G(x-t)dt=-\big{(}e^{-|t|}\underset{F}{\ast}G\big{)}(x).$
We apply the Wiener-Tauberian’s theorem [28] to the function
$\omega(t)=e^{-|t|}$ with bounded $\forall t\in\mathbb{R}$, and
$\Phi=e^{-|x-y|}\in L_{1}(\mathbb{R})$. It is easy to see
$F\left(e^{-|x-y|}\right)=\sqrt{\frac{2}{\pi}}\frac{1}{1+(x-y)^{2}}\neq
0,\forall x,y\in\mathbb{R}.$ On the other hand,

$\displaystyle\big{(}\omega\underset{F}{\ast}\Phi\big{)}(x)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{(-|y|-|x-y|)}dy=\left\\{\begin{array}[]{ll}\frac{1}{\sqrt{2\pi}}(x+1)e^{-x}&\quad\text{if}\
x>0,\\\ \frac{1}{\sqrt{2\pi}}(1-x)e^{x}&\quad\text{if}\
x<0,\end{array}\right.\quad\text{tends to}\ 0\ \text{when }|x|\to\infty.$

By Wiener-Tauberian’s theorem, we deduce that
$\lim\limits_{|x|\to\infty}J_{1}(x)=-\lim\limits_{|x|\to\infty}\big{(}e^{-|t|}\underset{F}{\ast}G\big{)}(x)=0,\forall
G\in L_{1}(\mathbb{R}).$ By the same argument as above, we obtain
$\lim\limits_{|x|\to\infty}J_{i}(x)=0,$ with $i=2,3,4.$ From (4.11) we get
$\lim_{|x|\to\infty}f(x)=0.$ Furthermore, we also deduce
$f^{\prime}(x)=2\sqrt{\frac{\pi}{2}}\left\\{J^{\prime}_{1}(x)-J^{\prime}_{2}(x)-iJ^{\prime}_{3}(x)+iJ^{\prime}_{4}(x)\right\\},$
where for each integral $J^{\prime}_{i}(x)$, with $i=\overline{1,4}$ can be
represented by the Fourier convolution of the form
$\big{(}e^{-|t|}\underset{F}{\ast}G^{\prime}\big{)}$, $\forall G^{\prime}\in
L_{1}(\mathbb{R})$. Hence, we apply the Wiener-Tauberian’s theorem again
[28](also refer [14]) to the function $G^{\prime}$, with $\omega$, $\Phi$ as
above implies that $\lim\limits_{|x|\to\infty}J^{\prime}_{i}(x)=0,$ with
$i=\overline{1,4}$. This yields $\lim\limits_{|x|\to\infty}f^{\prime}(x)=0$,
and we can conclude that $f(x)$ satisfies the initial conditions of the
Problem (4.7). Applying the equality (3.9), we obtain the boundedness in
$L_{1}$ of solution as follows

$\displaystyle\|f\|_{L_{1}(\mathbb{R})}$
$\displaystyle=\big{\|}(h\underset{H_{1},F}{\ast}\xi)\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\big{\|}_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\big{\|}h\underset{H_{1},F}{\ast}\xi\big{\|}_{L_{1}(\mathbb{R})}\big{\|}\sqrt{\frac{\pi}{2}}e^{-|t|}\big{\|}_{L_{1}(\mathbb{R})}$
$\displaystyle\leq\sqrt{\frac{2}{\pi}}\|h\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})}\|e^{-|t|}\|_{L_{1}(\mathbb{R})}=2\sqrt{\frac{2}{\pi}}\|h\|_{L_{1}(\mathbb{R})}\|\xi\|_{L_{1}(\mathbb{R})}.$

∎

######  Remark 5.

Assume that $p,q,r,s>1$ and satisfy the condition
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2+\frac{1}{s}$. If $f\in
L_{s}(\mathbb{R})$, $h\in L_{p}(\mathbb{R})$, and $\xi\in L_{q}(\mathbb{R})$,
based on (3.8), we get the following solution of Problem (4.7) estimate

$\|f\|_{L_{s}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\left(\frac{2}{r}\right)^{\frac{1}{r}}\|h\|_{L_{p}(\mathbb{R})}\|\xi\|_{L_{q}(\mathbb{R})}.$

##  5\. Examples

In the last section of article, we give numerical examples to illustrate the
calculation to ensure the validity and applicability of the results in Section
4. First, we get started with an illustrative example of Theorem 4.3 and
Remark 2.

######  Example 5.1.

Let $\varphi(x)=\xi(x)=\sqrt{\frac{\pi}{2}}e^{-|x|}$. It is easy to check
$\varphi(x)$ and $\xi$ are functions belonging to $L_{1}(\mathbb{R}).$
According to formula 1.4.1, p. 23 in [4], we have
$H_{1}\left(\sqrt{\frac{\pi}{2}}e^{-|x|}\right)(y)=\frac{1}{1+y^{2}}.$
Therefore $1+(H_{1}\varphi)(y)\neq 0$, $\forall y\in\mathbb{R}$ and
$\frac{(H_{1}\varphi)(y)}{1+(H_{1}\varphi)(y)}=\frac{1}{2+y^{2}}=(H_{1}\ell)(y),$
by Lemma 2.1, we infer there existence of a function $\ell\in
L_{1}(\mathbb{R})$ is satisfied. In this case
$\ell(x)=\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|x|}\in L_{1}(\mathbb{R})$. We obtain
the solution in $L_{1}(\mathbb{R})$ of Eq. (4.4) as follows
$f(x)=\left(\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(x).$
We get the $L_{1}$-norm estimation of $f(x)$ as follows
$\|f\|_{L_{1}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\big{\|}\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\big{\|}_{L_{1}(\mathbb{R})}\big{\|}\sqrt{\frac{\pi}{2}}e^{-|t|}\big{\|}_{L_{1}(\mathbb{R})}=\sqrt{2\pi}.$
Moreover, if there exist real numbers $p,q,r>1$ and satisfy
$\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}$, then

$\|f\|_{L_{r}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\big{\|}\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\big{\|}_{L_{p}(\mathbb{R})}\big{\|}\sqrt{\frac{\pi}{2}}e^{-|t|}\big{\|}_{L_{q}(\mathbb{R})}=\frac{\sqrt{\pi}}{2}\left(\frac{\sqrt{2}}{p}\right)^{\frac{1}{p}}\left(\frac{2}{q}\right)^{\frac{1}{q}}.$

For case $r=\infty$, from inequality (3.10), we obtain
$\|f\|_{L_{\infty}(\mathbb{R})}\leq\frac{\sqrt{\pi}}{2}\left(\frac{\sqrt{2}}{p}\right)^{\frac{1}{p}}\left(\frac{2}{q}\right)^{\frac{1}{q}}.$

Next example illustrates the Theorem 4.4.

######  Example 5.2.

Choose functions $h(x)=\xi(x)=\sqrt{\frac{\pi}{2}}e^{-|x|}.$ Obviously, these
functions are belonging to $L_{1}(\mathbb{R})$. According to formula 1.4.1, p.
23 in [4], we deduce that $H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\left(\sqrt{\frac{\pi}{2}}e^{-|x|}\right)(y)=\frac{1}{1+y^{2}}.$
Thus $1+H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\left(\sqrt{\frac{\pi}{2}}e^{-|x|}\right)\neq
0,\forall y\in\mathbb{R}$ and $\frac{(H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h)(y)}{1+(H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}h)(y)}=\frac{1}{2+y^{2}}=(H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}\eta)(y),$ by Lemma 2.1, we infer there existence of
a function $\eta=\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|x|}\in L_{1}(\mathbb{R})$ is
satisfied. Therefore, Eq. (4.6) has the unique solution
$f(x)=\bigg{(}\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\underset{H_{\left\\{\begin{subarray}{c}1\\\
2\end{subarray}\right\\}}}{*}\sqrt{\frac{\pi}{2}}e^{-|t|}\bigg{)}(x)$
belonging to $L_{1}(\mathbb{R}).$ The estimate of $f(x)$ is similar in Example
5.1 means $\|f\|_{L_{1}(\mathbb{R})}\leq\sqrt{2\pi}$ and
$\|f\|_{L_{r}(\mathbb{R})}\leq\frac{\sqrt{\pi}}{2}\left(\frac{\sqrt{2}}{p}\right)^{\frac{1}{p}}\left(\frac{2}{q}\right)^{\frac{1}{q}}.$

Now we rewrite function $\eta$ in the following form:

$\eta=\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}=\frac{\sqrt{\pi}}{2}e^{-\frac{\sqrt{2}}{2}|t|}.e^{-\frac{\sqrt{2}}{2}|t|}.$

We set $\varphi=\frac{\sqrt{\pi}}{2}e^{-\frac{\sqrt{2}}{2}|t|}$ and weighted
function $\rho=e^{-\frac{\sqrt{2}}{2}|t|}$. Clearly, in this case, $\varphi\in
L_{1}(\mathbb{R},\rho)\cap L_{p}(\mathbb{R},\rho)$ and $\rho$ is a positive
function belonging to $L_{1}(\mathbb{R})$. According to Remark 4, we obtain

$\displaystyle\lVert f\lVert_{L_{p}(\mathbb{R})}$
$\displaystyle\leq\frac{1}{\pi}(\sqrt{2\pi})^{\frac{1}{p}}\lVert
e^{-\frac{\sqrt{2}}{2}|t|}\lVert_{L_{1}(\mathbb{R})}^{1-\frac{1}{p}}\lVert\frac{\sqrt{\pi}}{2}e^{-\frac{\sqrt{2}}{2}|t|}\lVert_{L_{p}(\mathbb{R},\rho)}\lVert\frac{\sqrt{\pi}}{2}e^{-\frac{\sqrt{2}}{2}|t|}\lVert_{L_{p}(\mathbb{R})},\
\forall p>1.$
$\displaystyle\leq\sqrt{2}\bigg{(}\frac{2\sqrt{2\pi}}{p(p+1)}\bigg{)}^{\frac{1}{p}}.$

We end the article with an example for the Theorem 4.5.

######  Example 5.3.

A simple example is provided by $g=h=\sqrt{\frac{\pi}{2}}e^{-|x|}$. It is not
difficult to check that $g$ belongs to $L_{1}(\mathbb{R})$, and $h\in
C^{2}(\mathbb{R})\cap L_{1}(\mathbb{R})$. Obviously
$h^{\prime},h^{\prime\prime}\in L_{1}(\mathbb{R})$, following [4], we have
$F\big{(}\sqrt{\frac{\pi}{2}}e^{-|x|}\big{)}(y)=\frac{1}{1+y^{2}}$, therefore
$1+(Fg)(y)\neq 0$, $\forall y\in\mathbb{R}$. Moreover
$(F\xi)(y)=\frac{(Fg)(y)}{1+(Fg)(y)}=\frac{1}{2+y^{2}},$ so there is the
function $\xi(x)=\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|x|}\in L_{1}(\mathbb{R})$.
In this case, the solution of Problem (4.7) is

$f(x)=\left(\left(\sqrt{\frac{\pi}{2}}e^{-|t|}\underset{H_{1},F}{\ast}\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\right)\underset{H_{1},F}{\ast}\sqrt{\frac{\pi}{2}}e^{-|t|}\right)(x),$

and boundedness in $L_{1}$ is as follows $\|f\|_{L_{1}(\mathbb{R})}\leq
2\sqrt{\frac{2}{\pi}}\big{\|}\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\big{\|}_{L_{1}(\mathbb{R})}\big{\|}\sqrt{\frac{\pi}{2}}e^{-|t|}\big{\|}_{L_{1}(\mathbb{R})}=2\sqrt{2\pi}.$
Furthermore, if we assume that $p,q,r,s>1$ satisfy condition
$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2+\frac{1}{s}$, we obtain

$\|f\|_{L_{s}(\mathbb{R})}\leq\sqrt{\frac{2}{\pi}}\left(\frac{2}{r}\right)^{\frac{1}{r}}\left\|\sqrt{\frac{\pi}{2}}e^{-|t|}\right\|_{L_{p}(\mathbb{R})}\left\|\frac{\sqrt{\pi}}{2}e^{-\sqrt{2}|t|}\right\|_{L_{q}(\mathbb{R})}=\frac{\sqrt{\pi}}{2}\left(\frac{2}{r}\right)^{\frac{1}{r}}\left(\frac{2}{p}\right)^{\frac{1}{p}}\left(\frac{\sqrt{2}}{q}\right)^{\frac{1}{q}}.$

Acknowledgements. The author would like to thank the referee for carefully
reading the manuscript and giving valuable comments to improve the article.

DATA AVAILIBILITY
My manuscript has no associated data.

FUNDING STATEMENT
No funding was received for conducting this study.

CONFLICT OF INTEREST
The author has no conflicts of interest to declare that are relevant to the
content of this article.

ORCID
Trinh Tuan https://orcid.org/0000-0002-0376-0238

## References

  * [1] R. A. Adams and J. F. Fournier.  Sobolev Spaces.  Pure and Applied Mathematics. Vol. 140, Elsevier/Academic Press, Amsterdam, 2 edition, 2003.
  * [2] J. M. Appell, A. S. Kalitvin, and P. P. Zabrejko.  Partial Integral Operators and Integro-Differential Equations: pure and applied mathematics.  Marcel Dekker Inc, New York, 2000.  https://doi.org/10.1201/9781482270402.
  * [3] E. A. Barbashin.  Conditions for invariance of stability of solutions of integro-differential equations.  (in Russian). Izv. Vyssh. Uchebn. Zaved. Mat., 1:25–34, 1957.
  * [4] H. Bateman and A. Erdélyi.  Tables of Integral Transforms.  McGraw-Hill Book Company, Inc, New York-London, 1954.
  * [5] R. N Bracewell.  The Hartley Transform.  The Clarendon Press, Oxford University Press, New York, 1986.
  * [6] L. P. Castro, R. C. Guerra, and N. M. Tuan.  On Wiener’s Tauberian theorems and convolution for oscillatory integral operators.  Turk. J. Math., 43(3):1124–1147, 2019.  https://doi.org/10.3906/mat-1801-90.
  * [7] L. P. Castro and D. Kapanadze.  Wave diffraction by a half-plane with an obstacle perpendicular to the boundary.  Journal of Differential Equations, 254(2):493–510, 2013.  https://doi.org/10.1016/j.jde.2012.08.030.
  * [8] L. Debnath and B. Bhatta.  Integral Transforms and Their Applications.  Chapman & Hall/CRC Press, New York, 2 edition, 2006.  https://doi.org/10.1201/9781420010916.
  * [9] B. T. Giang, N. V. Mau, and N. M. Tuan.  Operational properties of two integral transforms of Fourier type and their convolutions.  Integr. Equ. Oper. Theory., 65(3):363–386, 2009.  https://doi.org/10.1007/s00020-009-1722-x.
  * [10] P. V. Hoang, T. Tuan, N. X. Thao, and V. K. Tuan.  Boundedness in weighted ${L_{p}}$ spaces for the Kontorovich–Lebedev–Fourier generalized convolutions and applications.  Integral Transforms Spec. Funct., 28(8):590–604, 2017.  https://doi.org/10.1080/10652469.2017.1330825.
  * [11] M. A. Naimark.  Normed Algebras.  Wolters-Noordhoff Publishing, Groningen, 1972.
  * [12] A. D. Poularikas (Ed.).  The Transform and Applications Handbook.  (The Electrical Engineering Handbook), CRC Press, Boca Raton, 3 edition, 2010.  https://doi.org/10.1201/9781315218915.
  * [13] J. Prüss.  Evolutionary Integral Equations and Applications.  Monographs in Mathematics, vol. 87. Birkhäuser Verlag, Basel, 1993\.
  * [14] W. Rudin.  Real and Complex Analysis.  McGraw-Hill Book Company, New York, 3 edition, 1987.
  * [15] S. Saitoh.  Weighted ${L}_{p}$-norm inequalities in convolutions.  Rassias, T.M. (ed.) Survey on Classical Inequalities, Mathematics and its Applications, vol. 517. Springer, Dordrecht, pages 225–234, 2000.  https://doi.org/10.1007/978-94-011-4339-4_8.
  * [16] I. N. Sneddon.  The Use of Integral Transform.  McGraw-Hill Book Company, Inc, New York (NY), London, 1972.
  * [17] C. D. Sogge.  Fourier Integrals in Classical Analysis.  Cambridge University Press, Cambridge, 1993.
  * [18] H. M. Srivastava and R. G. Buschman.  Theory and Applications of Convolution Integral Equations.  Mathematics and Its Applications, Vol. 79. Kluwer Academic Publishers, Dordrecht, 1992.  https://doi.org/10.1007/978-94-015-8092-2.
  * [19] E. M. Stein and G. Weiss.  Introduction to Fourier Analysis on Euclidean Spaces.  Princeton Mathematical Series (PMS-32), Volume 32, Princeton University Press, NJ, 1971.
  * [20] M. A. Sumbatyan and A. Scalia.  Equations of Mathematical Diffraction Theory.  CRC Press, Boca Raton, 1 edition, 2004.  https://doi.org/10.1201/9780203643488.
  * [21] N. X. Thao and H. T. VanAnh.  On the Hartley–Fourier sine generalized convolutions.  Math. Methods Appl. Sci., 37(15):2308–2319, 2014.  https://doi.org/10.1002/mma.2980.
  * [22] E. C. Titchmarsh.  Introduction to the Theory of Fourier Integrals.  Chelsea Publishing Co, New York, 3 edition, 1986.
  * [23] T. Tuan.  On the Fourier-sine and Kontorovich–Lebedev generalized convolution transforms and their applications.  Ukrainian Math. J., 72(2):302–316, 2020.  https://doi.org/10.1007/s11253-020-01782-1.
  * [24] T. Tuan.  Some classes of integral equations of convolutions-pair generated by the Kontorovich-Lebedev, Laplace and Fourier transforms.  Commun. Korean Math. Soc., 36(3):485–494, 2021.  https://doi.org/10.4134/CKMS.c200183.
  * [25] T. Tuan.  Operational properties of the Hartley convolution and its applications.  Mediterr. J. Math., 19, 266, 2022.  https://doi.org/10.1007/s00009-022-02173-5.
  * [26] T. Tuan.  Some results of Watson and Plancherel–type integral transforms related to the Hartley, Fourier convolutions and applications.  Math. Methods Appl. Sci., 45(17):11158–11180, 2022.  https://doi.org/10.1002/mma.8442.
  * [27] T. Tuan and V. K. Tuan.  Young inequalities for a Fourier cosine and sine polyconvolution and a generalized convolution.  Integral Transforms Spec. Funct., pages 1–13, 2023.  https://doi.org/10.1080/10652469.2023.2182776.
  * [28] N. Wiener.  Tauberian theorems.  Annals of Mathematics., 33(1):1–100, 1932.  https://doi.org/10.2307/1968102.
  * [29] W. H. Young.  On classes of summable functions and their Fourier series.  Proceedings of the Royal Society of London, Series A., 87:225–229, 1912.  https://doi.org/10.1098/rspa.1912.0076.