# An existence and uniqueness result to evolution equations with sign-changing
pseudo-differential operators and its applications to logarithmic Laplacian
operators and second-order differential operators without ellipticity

Jae-Hwan Choi  Department of Mathematical Sciences, Korea Advanced Institute
of Science and Technology, 291, Daehak-ro, Yuseong-gu, Daejeon 34141, Republic
of Korea<EMAIL_ADDRESS>and  Ildoo Kim  Department of Mathematics,
Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

We broaden the domain of the Fourier transform to contain all distributions
without using the Paley-Wiener theorem and devise a new weak formulation built
upon this extension. This formulation is applicable to evolution equations
involving pseudo-differential operators, even when the signs of their symbols
may vary over time. Notably, our main operator includes the logarithmic
Laplacian operator $\log(-\Delta)$ and a second-order differential operator
whose leading coefficients are not positive semi-definite.

###### Key words and phrases:

Pseudo-differential operators with sign-changing symbols, Fourier transform
beyond tempered distributions, logarithmic Laplacian, Cauchy problem without
ellipticity, Weighted Bessel potential spaces

###### 2020 Mathematics Subject Classification:

35S05, 35S10, 35A01, 35D30, 47G30

This research was supported by the National Research Foundation of Korea(NRF)
grant funded by the Korea government(MSIT) (No.2020R1A2C1A01003959)

##  1\. Introduction

Pseudo-differential operators emerge as a mathematical generalization of
differential operators. An illustrative form of a pseudo-differential operator
is expressed as

$P(x,-i\nabla)u(x)=\mathcal{F}^{-1}\left[\sigma(x,\xi)\mathcal{F}[u](\xi)\right](x),$

where $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier and the inverse
Fourier transforms on $\mathbb{R}^{d}$, respectively. In general, pseudo-
differential operators lack local properties, which make conventional
perturbation methods from partial differential equation theories inapplicable.
Consequently, studying the well-posedness of the equation

$\displaystyle P(x,-i\nabla)u(x)=f(x)$ (1.1)

within a specific framework becomes highly challenging without imposing very
strong assumptions on both the symbol $\sigma(x,\xi)$ and the data $f$.
However, if one considers pseudo-differential operators which are independent
of the space variable $x$, there might be intriguing theories for addressing a
general symbol $\sigma(\xi)$ and data $f$ within a weak formulation.
Specifically, when considering a polynomial symbol $\sigma(\xi)$, there exist
exceptionally elegant theories regarding the existence and uniqueness of
solutions to (elliptic) equations like (1.1) in the weak sense. Remarkably,
these theories hold even in the absence of strong mathematical conditions on
$\sigma$ and $f$. In particular, a solution $u$ exists in the sense of
distributions if the symbol $\sigma$ is a polynomial, $\sigma(\xi)\not\equiv
0$, and $f$ is a distribution with a compact support on $\mathbb{R}^{d}$. Even
more surprisingly, there always exists a distribution solution $u$ to (1.1)
for any distribution $f$ if the symbol $\sigma$ is a non-zero polynomial,
which is shown based on the Paley-Wiener theorem and duality arguments. This
existence result holds even though a nonzero polynomial symbol $\sigma(\xi)$
does not satisfy an ellipticity condition, since the set of all zeros of a
non-zero polynomial has a lower dimension. However, the uniqueness of a weak
solution $u$ is not guaranteed if there is no ellipticity condition. Thus
ellipticity is necessary and sufficient to guarantee both existence and
uniqueness of a weak solution to (1.1) even for a polynomial symbol which is
independent of $x$. Moreover, this well-posedness theory can be generalized to
accommodate non-polynomial symbols $\sigma$ and tempered distributions $f$ by
imposing supplementary conditions involving growth, regularity, and
ellipticity on the symbol $\sigma$. For more in-depth information, we
recommend that the reader consult the books [10, 11, 24, 25, 27]. It is
noteworthy that for a polynomial symbol $\sigma(\xi)$, the sufficient
condition to ensure the weak existence of (1.1) with $f$ having a distribution
with compact support is simply that $\sigma(\xi)$ is not identically zero.
This condition is feasible through a translation of the polynomial
$\sigma(\xi)$ and using properties of convolution and analytic functions (see,
for instance, [10, Theorem 7.3.10] and [11, Theorem 10.3.1]). However, this
approach may not be applicable for non-polynomial symbols, particularly in
scenarios where the symbols lack regularity. In such scenarios, an ellipticity
condition on $\sigma$ becomes crucial, as it prompts the consideration of the
reciprocal $\frac{1}{\sigma}$ to derive a fundamental solution (Green’s
function).

On the other hand, when we consider the evolutionary (or parabolic)
counterparts of these theories, the reciprocal $\frac{1}{\sigma}$ could not
emerge as a factor in acquiring a fundamental solution if the Fourier
transforms are taken solely with respect to the space variable. Moreover, this
approach to take the Fourier transforms only with respect to the spatial
variable facilitates examinations of time-dependent symbols. Certainly, at
least in a formal sense, a fundamental solution to the following simple
second-order evolution equation

$\displaystyle\partial_{t}u(t,x)=a^{ij}(t)u_{x^{i}x^{j}}(t,x),\quad
u(0)=\delta_{0}$

is given by

$\displaystyle\mathcal{F}_{\xi}^{-1}\left[\exp\left(-\int_{0}^{t}a^{ij}(s)\xi^{i}\xi^{j}\mathrm{d}s\right)\right],$
(1.2)

where $\delta_{0}$ is the Dirac delta measure centered at the origin and
$\mathcal{F}_{\xi}^{-1}$ denotes the inverse Fourier transform with respect to
the variable $\xi$. Similarly, we could claim that a solution $u$ to

$\displaystyle\partial_{t}u(t,x)=\mathcal{F}^{-1}\left[\sigma(t,\xi)\mathcal{F}[u(t,\cdot)](\xi)\right]$
$\displaystyle u(0,x)=\delta_{0}$ (1.3)

is given by

$\displaystyle
u(t,x)=\mathcal{F}_{\xi}^{-1}\left[\exp\left(\int_{0}^{t}\sigma(s,\xi)\mathrm{d}s\right)\right](x).$

This leads us to ponder whether ellipticity can be eliminated in a weak
formulation of evolution equations for general symbols. We found that
providing a positive answer to this question is indeed valid when
investigating evolution equations through a novel weak formulation. This paper
is an outcome of developing a suitable weak formulation to eliminate all
elliptic conditions from main operators in evolution equations even though
symbols are not regular (thus definitely are not polynomials). This new weak
formulation relies entirely on an extension of the Fourier and inverse Fourier
transforms. Traditionally, the class of all tempered distributions has been
recognized as an extensive class suitable for these transforms since both
transforms become automorphisms on this class. However, there exists a
remarkable method to extend the domain of the Fourier transform to include all
distributions. This is possible due to the Paley-Wiener theorem. More
precisely, the Fourier transform of a distribution on $\mathbb{R}^{d}$ becomes
an element of the dual space of the subspace of entire functions $F$ so that
for any multi-index $\alpha$ and $N>0$,

$\displaystyle|D^{\alpha}F(z)|\leq
C(1+|z|)^{-N}\mathrm{e}^{B|\Im(z)|}\quad\forall z\in\mathbb{C}^{d},$ (1.4)

where $C$ and $A$ are positive constants depending on $\alpha$ and $N$, and
$\Im(z)$ denotes the imaginary part of the complex vector $z$ (cf. [24,
Proposition 4.1]). Nonetheless, this extension of the Fourier transform is not
appropriate to deal with initial and inhomogeneous data on the Euclidean
spaces without having analytic continuations. Thus we propose a new approach
which enables the Fourier and inverse Fourier transforms to apply for all
distributions without considering analytic continuations. We will provide a
straightforward explanation of these extensions later in the introduction,
following the presentation of the main equations. Alternatively, you can
examine the comprehensive formulations with all the specifics in Definitions
2.1, 2.2, and 2.3 in the upcoming section.

Now, we succinctly present our equations, which naturally arise as a
generalization of (1.3) :

$\begin{cases}\partial_{t}u(t,x)=\psi(t,-i\nabla)u(t,x)+f(t,x),\quad&(t,x)\in(0,T)\times\mathbb{R}^{d},\\\
u(0,x)=u_{0},\quad&x\in\mathbb{R}^{d}.\end{cases}$ (1.5)

Here the main operator $\psi(t,-i\nabla)$ is given by

$\displaystyle\psi(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[\psi(t,\cdot)\mathcal{F}[u(t,\cdot)]\right](x)$
(1.6)

and called a time-measurable pseudo-differential operator, where $\psi(t,\xi)$
represents a complex-valued (Borel) measurable function defined on
$(0,T)\times\mathbb{R}^{d}$ and is commonly referred to as a symbol. To
maintain consistency with the notations used in our previous findings, we
substitute $\psi(t,\xi)$ for $\sigma(t,\xi)$. This adjustment is made since
$\sigma$ conventionally represents the coefficients associated with random
noise terms in the realm of stochastic partial differential equations (cf.
[26]).

Numerous attempts have been undertaken to ease the conditions on symbols while
still ensuring the well-posedness of partial differential equations that
involve these time-measurable pseudo-differential operators. It is reasonable
to anticipate that the symbol may not require a regularity condition
concerning the time variable $t$, as evidenced by many results pertaining to
second-order parabolic equations and diffusion processes (_cf_. [27, 31]).
However, a regularity condition on $\psi(t,\xi)$ with respect to $\xi$ appears
to be essential for defining the operator $\psi(t,-i\nabla)$ in a strong
space, such as an $L_{p}$-space or a Hölder space, as indicated by multiplier
theories in Fourier analysis (_cf_. [8, 9]). One of the most famous examples
is Mihlin’s multiplier and its evolutionary counterpart is given by

$\displaystyle|D_{\xi}^{\alpha}\psi(t,\xi)|\lesssim|\xi|^{\nu-|\alpha|}\quad\forall(t,\xi)\in(0,T)\times\mathbb{R}_{0}^{d}$
(1.7)

for all multi-index $\alpha$ so that
$|\alpha|\leq\lfloor\frac{d}{2}\rfloor+1$, where
$\mathbb{R}_{0}^{d}=\mathbb{R}^{d}\setminus\\{0\\}$, $\nu$ is a positive
constant representing an order of the operator, and
$\lfloor\frac{d}{2}\rfloor$ is the largest integer which is less than or equal
to $\frac{d}{2}$.

Furthermore, a regularity condition on $\psi(t,\xi)$ alone is insufficient to
ensure a strong solution to (1.5), as observed in the case of a fundamental
solution for the second-order equation in (1.2). In essence, the term
$\exp\left(-\int_{0}^{t}a^{ij}(s)\xi^{i}\xi^{j}\mathrm{d}s\right)$ in (1.2)
may exhibit exponential growth if there is no elliptic condition on the
coefficients $a^{ij}$. Thus, we introduce the following strong elliptic
condition: there exists a positive constant $\nu>0$ such that

$\displaystyle-\Re[\psi(t,\xi)]\geq\nu|\xi|^{\gamma}\quad\forall\xi\in\mathbb{R}^{d},$
(1.8)

where $\Re[\psi(t,\xi)]$ represents the real part of the complex number
$\psi(t,\xi)$.

Drawing upon the criteria outlined in (1.7) and (1.8), extensive researches
conducted by the authors and collaborators have consistently shown robust
well-posedness for the equation (1.5). Examples of such research can be found
in [1, 2, 5, 15, 16, 17, 18]. These operators exhibit a close connection to
non-local operators and serve as generators of stochastic processes.
Additionally, for a discussion on the strong well-posedness of these equations
with non-local operators and generators of stochastic processes in
$L_{p}$-spaces, we direct readers to the references [3, 4, 6, 12, 13, 14, 20,
22, 29, 30, 32].

Historically, our attention has been primarily directed towards investigating
the strong well-posedness in $L_{p}$ or Hölder spaces. This emphasis was
rooted in a longstanding belief that the uniform ellipticity, as expressed in
(1.8), was indispensable and could not be eliminated, even when considering
weak solutions to equations like (1.5). The reasoning seems clear and
uncomplicated. If we examine the quantity $\exp[\psi(t,\xi)]$ without an
ellipticity, then it becomes apparent that this function could have
exponential growth in general, hindering it from being a tempered distribution
on $\mathbb{R}^{d}$. Thus, the inverse Fourier transform of the function
$\exp(\psi(t,\xi))$ loses its meaningfulness, particularly when considered as
a fundamental solution to (1.5), even in a weak sense if we do not consider
extensions of equations to $\mathbb{C}^{d}$ after taking the Fourier
transform. Moreover, it is impossible to find a nice analytic continuation of
$\psi(t,\xi)$ if there is no regularity condition on $\psi(t,\xi)$ at all.
Hence, it becomes crucial to ensure that the real part of the symbol
$\psi(t,\xi)$ remains non-positive in order to guarantee the existence of a
fundamental solution even as a tempered distribution-valued function. In other
words, the strong elliptic condition outlined in (1.8) (or non-positivity of
the symbol) seems to be a suitable requirement, even in the context of a weak
formulation. Therefore, initially we were inclined to believe that a weak
well-posedness theory for (1.5) with a strong elliptic condition (1.8) might
not be appealing for research, even if regularity conditions such as (1.7)
could be eliminated in the weak formulation. However, we recently discovered a
method to extend the domain of the Fourier and the inverse Fourier transforms
to all distributions on $\mathbb{R}^{d}$ without using the Paley-Wiener
theorem. In particular, it becomes possible to assign a certain meaning to the
term $\mathcal{F}^{-1}\left[\exp(\psi(t,\xi))\right]$ instead of an element in
the dual space of entire functions $F$ on $\mathbb{C}^{d}$ so that (1.4)
holds, even when the sign of the real part of the symbol $\psi(t,\xi)$ is
strictly positive. With these extensions, we ultimately achieve a satisfactory
weak well-posedness result for (1.5) without imposing any ellipticity and
regularity conditions. A certain local integrability on the symbol
$\psi(t,\xi)$ is sufficient to establish weak well-posedness of (1.5). We
emphasize that not only the existence, but also the uniqueness of a weak
solution is obtained even though a weak solution to equation (1.1) is not
unique even for a polynomial symbol $\sigma(\xi)$ without ellipticity. For
further details, see Theorem 2.21 below.

Next, we offer a brief overview of a fundamental idea behind our weak
formulation. It is essential to recognize that trying to create a weak
formulation across the entire domain $(0,T)\times\mathbb{R}^{d}$ is unfeasible
due to the fact that a symbol $\psi(t,\xi)$ varies with respect to the time.
Additionally, $(0,T)$ could be a finite interval where Schwartz’s functions do
not work effectively. As a result, our testing actions for the equations are
limited to only the spatial variables. The core aspect of our approach lies in
utilizing a different class of test functions. Specifically, we use
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ as a class of test functions
instead of $\mathcal{D}(\mathbb{R}^{d})$. Here,
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ represents a subset of Schwartz
functions whose Fourier transforms are belonging to the class
$\mathcal{D}(\mathbb{R}^{d})$ and $\mathcal{D}(\mathbb{R}^{d})$ is the class
of all functions that are infinitely differentiable and have compact supports.
We establish the Fourier transform and its inverse for a distribution $v$
defined on $\mathbb{R}^{d}$ as linear functionals acting on
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$. This is achieved by applying
straightforward operations on $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$,
which are motivated by Plancherel’s theorem. In other words, we define

$\displaystyle\langle\mathcal{F}^{-1}[v],\varphi\rangle=\langle
v,\mathcal{F}[\varphi]\rangle\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$
(1.9)

and

$\displaystyle\langle\mathcal{F}[v],\varphi\rangle=\langle
v,\mathcal{F}^{-1}[\varphi]\rangle\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$

Specifically, we have the option to consider the Fourier transform and its
inverse for any function which is locally integrable.

Now, suppose that we have a solution $u$ to equation (1.5) such that the
Fourier transform of $u$ has a realization on $\mathbb{R}^{d}$. Furthermore,
we also assume that the product of $\psi(t,\xi)$ and the Fourier transform of
$u$ with respect to $\xi$ is locally integrable. Then, the function referred
to in (1.6) becomes meaningful due to the revised definition of the inverse
Fourier transform presented in (1.9).

To conclude, we can define $u$ as a weak solution to (1.5) if, for every
$\varphi$ belonging to the space
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$, the following equation holds:

$\displaystyle\left\langle u(t,\cdot),\varphi\right\rangle=\langle
u_{0},\varphi\rangle+\int_{0}^{t}\langle\psi(s,-i\nabla)u(s,\cdot),\varphi\rangle\mathrm{d}s+\int_{0}^{t}\left\langle
f(s,\cdot),\varphi\right\rangle\mathrm{d}s\quad(a.e.)~{}t\in(0,T).$

This straightforward concept enables us to naturally manage the exponential
growth that may arise when the real part of the symbol $\psi(t,\xi)$ is sign-
changing.

It is worth noting that we do not rely on any arguments related to weak pre-
compactness to derive our main theorems. Instead, our results are solely
grounded in elementary techniques, such as Hölder and Minkowski inequalities,
combined with well-known properties of the Fourier and inverse Fourier
transforms.

This paper comprises seven sections. Section 1 serves as an introductory
section, presenting an overview of the results. Our primary theorems are
introduced in Section 2. In Section 3, we revisit the classical Bessel
potential spaces and extend them to incorporate weighted variations. Section 4
is dedicated to establishing the uniqueness of a solution $u$ for (1.5) under
a slightly relaxed condition. The proof of the main theorem is provided in
Section 5, along with noteworthy corollaries for practical applications.
Section 6 focuses on handling logarithmic operators as a specific application
of our theorem, while Section 7 addresses second-order differential operators
without ellipticity.

We finish this section with the notations used in the article.

  * •

Let $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, $\mathbb{C}$ denote the natural
number system, the integer number system, the real number system, and the
complex number system, respectively. For $d\in\mathbb{N}$, $\mathbb{R}^{d}$
denotes the $d$-dimensional Euclidean space.

  * •

For $i=1,...,d$, a multi-index $\alpha=(\alpha_{1},...,\alpha_{d})$ with
$\alpha_{i}\in\\{0,1,2,...\\}$, and a function $g$, we set

$\frac{\partial g}{\partial x^{i}}=D_{x^{i}}g,\quad
D^{\alpha}g=D_{x^{1}}^{\alpha_{1}}\cdot...\cdot
D^{\alpha_{d}}_{x^{d}}g,\quad|\alpha|:=\sum_{i=1}^{d}\alpha_{i}.$

For $\alpha_{i}=0$, we define $D^{\alpha_{i}}_{x^{i}}g=g$. If
$g=g(t,x):\mathbb{R}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$, then we denote

$D_{x}^{\alpha}g=D_{x^{1}}^{\alpha_{1}}\cdot...\cdot D^{\alpha_{d}}_{x^{d}}g,$

where $\alpha=(\alpha_{1},\cdots,\alpha_{d})$.

  * •

We employ the identical symbol “$\sup$” to represent both the supremum and the
essential supremum by slightly abusing the notation $\sup$. On occasion,
“$\operatorname*{ess\,sup}$” may be utilized to emphasize its role as the
essential supremum.

  * •

We use $\mathcal{D}(\mathbb{R}^{d})$ to denote the space of all infinitely
differentiable functions with compact supports. $\mathcal{S}(\mathbb{R}^{d})$
represents the Schwartz space on $\mathbb{R}^{d}$ and the topology on
$\mathcal{S}(\mathbb{R}^{d})$ is generated by the Schwartz semi-norms
$\sup_{x\in\mathbb{R}^{d}}|x^{\alpha}(D^{\beta}f)(x)|$ for all multi-indexes
$\alpha$ and $\beta$. $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ is used to denote
the dual space of $\mathcal{S}(\mathbb{R}^{d})$, i.e.
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ is the space of all tempered
distributions on $\mathbb{R}^{d}$. Additionally, we assume that
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ is a topological space equipped with
the weak*-topology if there is no special remark about the topology on
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$.

  * •

Let $F$ be a normed space and $(X,\mathcal{M},\mu)$ be a measure space.

    * –

$\mathcal{M}^{\mu}$ denotes the completion of $\mathcal{M}$ with respect to
the measure $\mu$.

    * –

For $p\in[1,\infty)$, the space of all $\mathcal{M}^{\mu}$-measurable
functions $f:X\to F$ with the norm

$\left\|f\right\|_{L_{p}(X,\mathcal{M},\mu;F)}:=\left(\int_{X}\left\|f(x)\right\|_{F}^{p}\mu(\mathrm{d}x)\right)^{1/p}<\infty$

is denoted by $L_{p}(X,\mathcal{M},\mu;F)$. We also denote by
$L_{\infty}(X,\mathcal{M},\mu;F)$ the space of all
$\mathcal{M}^{\mu}$-measurable functions $f:X\to F$ with the norm

$\|f\|_{L_{\infty}(X,\mathcal{M},\mu;F)}:=\inf\left\\{r\geq 0:\mu(\\{x\in
X:\|f(x)\|_{F}\geq r\\})=0\right\\}<\infty.$

We usually omit the given measure or $\sigma$-algebra in the notations of
$L_{p}$-spaces if there is no confusion (e.g. Lebesgue (or Borel) measure and
$\sigma$-algebra). We similarly leave out the representation of $F$ when it
takes on the scalar values such as $\mathbb{R}$ or $\mathbb{C}$.

  * •

Let $f(t)$ and $g(t)$ be Lebesgue measurable (or Borel measurable) functions
on $\mathcal{U}$. We write

$\displaystyle f(t)=g(t)~{}(a.e.)~{}t\in\mathcal{U}$

if and only if there exists a measurable subset
$\mathcal{T}\subset\mathcal{U}$ such that the Lebesgue measure
$|\mathcal{U}\setminus\mathcal{T}|$ is zero and $f(t)=g(t)$ for all
$t\in\mathcal{T}$. Moreover, we say that a function $f(t)$ is defined $(a.e.)$
on a Lebesgue measurable (or Borel measurable) set $\mathcal{U}$ if there
exists a measurable subset $\mathcal{T}\subset\mathcal{U}$ such that the
Lebesgue measure $|\mathcal{U}\setminus\mathcal{T}|=0$ and $f(t)$ is defined
for all $t\in\mathcal{T}$.

  * •

For $R>0$,

$B_{R}(x):=\\{y\in\mathbb{R}^{d}:|x-y|<R\\},\quad\overline{B_{R}(x)}:=\\{y\in\mathbb{R}^{d}:|x-y|\leq
R\\}$

  * •

For a measurable function $f$ on $\mathbb{R}^{d}$, we denote the
$d$-dimensional Fourier transform of $f$ by

$\mathcal{F}[f](\xi):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{-i\xi\cdot
x}f(x)\mathrm{d}x$

and the $d$-dimensional inverse Fourier transform of $f$ by

$\mathcal{F}^{-1}[f](x):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{ix\cdot\xi}f(\xi)\mathrm{d}\xi.$

Moreover, for a function $f(t,x)$ defined on $(0,T)\times\mathbb{R}^{d}$, we
use the notation

$\displaystyle\mathcal{F}_{x}[f](\xi):=\mathcal{F}[f(t,x)](\xi):=\mathcal{F}[f(t,\cdot)](\xi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{-i\xi\cdot
x}f(t,x)\mathrm{d}x$

and it is called the Fourier transform of $f$ with respect to the space
variable. On the other hand, for the inverse Fourier transform of $f(t,\xi)$
with respect to the space variable, we use the notation

$\displaystyle\mathcal{F}_{\xi}^{-1}[f](x):=\mathcal{F}^{-1}[f(t,\xi)](x):=\mathcal{F}^{-1}[f(t,\cdot)](x)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{ix\cdot\xi}f(t,\xi)\mathrm{d}\xi.$

For the sake of simplicity, we often omit the subscripts denoted by $x$ and
$\xi$. It is well-known that both the Fourier transform and the inverse
Fourier transform have extensions on
$L_{1}(\mathbb{R}^{d})+L_{2}(\mathbb{R}^{d})$. By slightly abusing the
notation, we use the same notation $\mathcal{F}$ and $\mathcal{F}^{-1}$ to
denote these extensions. We discuss more details some properties of these
extensions and corresponding inversion theorems in Section 3.

  * •

We write $\alpha\lesssim\beta$ if there is a positive constant $N$ such that
$\alpha\leq N\beta$. We use $\alpha\approx\beta$ if $\alpha\lesssim\beta$ and
$\beta\lesssim\alpha$. In particular, we use the notation
$\alpha\lesssim_{a,b,\cdots}\beta$ if the constant $N$ so that $\alpha\leq
N\beta$ depends only on $a,b,\cdots$. Moreover, if we write $N=N(a,b,\cdots)$,
this means that the constant $N$ depends only on $a,b,\cdots$. A generic
constant $N$ may change from a location to a location, even within a line. The
dependence of generic constants is usually specified in each statement of
theorems, propositions, lemmas, and corollaries.

  * •

For $z\in\mathbb{C}$, $\Re[z]$ denotes the real part of $z$, $\Im[z]$ is the
imaginary part of $z$ and $\bar{z}$ is the complex conjugate of $z$.

##  2\. Settings and main results

We fix $T\in(0,\infty]$ and $d\in\mathbb{N}$ throughout the paper. Here $T$
and $d$ denote the terminal time of the evolution equation and the dimension
of the space-variable, respectively. Note that $T=\infty$ is possible in our
theory. All functions are complex-valued if there is no special remark about
the range of a function. In particular, $\psi(t,\xi)$ denotes a complex-valued
measurable function defined $(a.e.)$ on $(0,T)\times\mathbb{R}^{d}$ and denote

$\displaystyle\psi(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[\psi(t,\cdot)\mathcal{F}[u(t,\cdot)]\right](x)$
(2.1)

in the whole paper. All the Fourier and the inverse Fourier transforms in the
paper are taken only with respect to the space variable and thus they are
considered only on $\mathbb{R}^{d}$ even in a weak sense.

Our main goal of this paper is to show the existence and uniqueness of a weak
solution $u$ to the Cauchy problem (1.5). Note that it is not easy to recover
a strong solution to (1.5) even in $L_{p}$-spaces if there are no strong
mathematical conditions on the symbol and data. A naive condition is that for
each $t\in(0,T)$, $\psi(t,\cdot)\mathcal{F}[u](t,\cdot)\in
L_{p}(\mathbb{R}^{d})$ with $p\in[1,2]$. Then the function
$t\mapsto\psi(t,-i\nabla)u(t,\cdot)$ is well-defined as an
$L_{p^{\prime}}(\mathbb{R}^{d})$-valued function defined on $(0,T)$ due to the
Riesz-Thorin Theorem, where $p^{\prime}$ denotes the Hölder conjugate of $p$,
i.e.

$\displaystyle p^{\prime}=\begin{cases}&\frac{p}{p-1}\quad\text{if
$p\in(1,2]$}\\\ &\infty\quad\text{if $p=1$}.\end{cases}$

We revisit Riesz-Thorin’s Theorem in Section 3, providing more specific
details about the constants involved.

On the other hand, it appears that there is no need for numerous mathematical
conditions when approaching the operator described in (2.1) in a weak sense.
In particular, if
$\psi(t,\cdot)\mathcal{F}[u](t,\cdot)\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$,
then $t\mapsto\psi(t,-i\nabla)u(t,\cdot)$ is well-defined as an
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued function defined on $(0,T)$,
where $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ denotes the class of all tempered
distributions on $\mathbb{R}^{d}$. Nevertheless, if the function
$\psi(t,\cdot)\mathcal{F}[u]$ is only locally integrable, it does not qualify
as a tempered distribution on $\mathbb{R}^{d}$. Consequently, the operator in
(2.1) is not well-defined in this scenario, as it may have exponential growth.
However, local integrability of $\psi(t,\cdot)\mathcal{F}[u]$ is sufficient to
establish it as a distribution on $\mathbb{R}^{d}$. Hence, if we suggest a new
technique for expanding the domain of the inverse Fourier transform to
encompass all distributions, then the operator defined in (2.1) could be
meaningful even with only local integrability on
$\psi(t,\cdot)\mathcal{F}[u]$. This extension is especially important when
dealing with functions that exhibit exponential growth. Additionally, it is
well-known that there exists an elegant method to define the Fourier transform
of a distribution on $\mathbb{R}^{d}$ as an element in the dual space of a
subspace of entire functions $F$ on $\mathbb{C}^{d}$ so that (1.4) holds.
However, this extension is not appropriate to fulfill our well-posedness
theory since our symbol $\sigma(t,\xi)$ is not a polynomial type in general.
In other words, we cannot consider an analytic continuation of $\sigma(t,\xi)$
on $\mathbb{C}^{d}$ for each $t$. Thus, we suggest a new extension of the
Fourier transform to include all distributions below in Definitions 2.2 and
2.3. Remarkably, we have not come across a suitable reference discussing this
type extension of the Fourier transform beyond tempered distributions without
using analytic continuations to the best of our knowledge.

Utilization of distribution theories has significantly advanced our
understanding of partial differential equations on a broader scale, as noted
in [7, 10, 11]. Nevertheless, a conventional weak formulation utilizing
distributions appears ineffective in the absence of strong ellipticity on the
symbols for our evolution equations due to weak mathematical conditions with
respect to the time variable $t$ and possibility having exponential growth as
highlighted in the introduction. More specifically, if we allow the symbol
$\psi(t,\xi)$ to change its sign, then our equation (1.5) includes second-
order evolution equations without the parabolicity, which clearly shows that
classical weak formulations taken for both time and space variables
simultaneously, cannot be applicable since the coefficients are depending on
the time, $(0,T)$ could be finite interval, and data $u_{0}$ and $f$ might
have not compact supports in general. Moreover, exponential growth could
emerge naturally as a result of the influence of exponential functions
generated by time evolution when examining evolution equations instead of
stationary equations if there is no ellipticity. This occurrence is
particularly evident when Fourier transforms are exclusively applied to the
spatial variable to accommodate variations in symbols with respect to the time
variable.

Thus, it is reasonable to anticipate that the standard framework of weak
formulations typically tested with $\mathcal{D}(\mathbb{R}^{d})$ or
$\mathcal{S}(\mathbb{R}^{d})$ is insufficient to address well-posedness of our
equation (1.5). A typical form of a solution to (1.5) clarifies these reasons.
Indeed, at least in a formal sense, a solution $u$ to (1.5) can be expressed
as follows:

$\displaystyle
u(t,x)=\mathcal{F}^{-1}\left[\exp\left(\int_{0}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[u_{0}]\right](x)+\mathcal{F}^{-1}\left[\int_{0}^{t}\exp\left(\int_{s}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[f(s,\cdot)]\mathrm{d}s\right](x).$
(2.2)

Hence, if our symbol $\psi(t,\xi)$ shows a sign-changing behavior, it can lead
to exponential growth in the frequency of our solution $u$. Nonetheless, it is
widely recognized that tempered distributions do not offer adequate control
over the exponential growth, as has been emphasized multiple times previously.
Thus, it becomes necessary to find a more extensive class than
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ to ensure well-posedness of our
evolution equation (1.5), particularly when we permit symbols to change signs.
However, using the class of all distributions as a direct replacement is not
suitable because the Fourier transforms of distributions are not universally
defined if we do not consider them as elements in a dual space of some entire
functions. Therefore, we propose a new class referred to as _Fourier
transforms of distributions_.

######  Definition 2.1 (Fourier transforms of distributions).

We use $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ to denote the subclass of
the Schwartz class whose Fourier transform is in
$\mathcal{D}(\mathbb{R}^{d})$, i.e.

$\displaystyle\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}):=\\{\varphi\in\mathcal{S}(\mathbb{R}^{d}):\mathcal{F}[\varphi]\in\mathcal{D}(\mathbb{R}^{d})\\},$

where $\mathcal{S}(\mathbb{R}^{d})$ denotes the (complex-valued) Schwartz
class on $\mathbb{R}^{d}$ and $\mathcal{D}(\mathbb{R}^{d})$ denotes the class
of all complex-valued infinitely differentiable functions with compact
supports defined on $\mathbb{R}^{d}$. Recall that there is a topology on
$\mathcal{S}(\mathbb{R}^{d})$ generated by the Schwartz semi-norms. Since
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\subset\mathcal{S}(\mathbb{R}^{d})$,
there is a subspace topology on $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$.
Therefore, one can consider its dual space. We use the notation
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ to denote
the dual space of $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$, i.e.
$u\in\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ if and
only if $u$ is a continuous linear functional defined on
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$. For
$u\in\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ and
$\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$, we write

$\displaystyle\langle u,\varphi\rangle:=u(\varphi).$

In other words, $\langle u,\varphi\rangle$ denotes the image of $\varphi$
under $u$. Defining a topology for
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ is a
straightforward task. Specifically, one can adopt the weak*-topology for
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ and work
with its associated Borel sets to mention measurability of functions whose
range are in
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$.

######  Definition 2.2 (Fourier and inverse Fourier transforms of
$u\in\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$).

For $u\in\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$,
one can define the Fourier and inverse Fourier transforms of $u$ in the
following way:

$\displaystyle\langle\mathcal{F}[u],\varphi\rangle:=\langle
u,\mathcal{F}^{-1}[\varphi]\rangle$

and

$\displaystyle\langle\mathcal{F}^{-1}[u],\varphi\rangle:=\langle
u,\mathcal{F}[\varphi]\rangle$

for all $\varphi\in\mathcal{D}(\mathbb{R}^{d})$. Thus for any
$u\in\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$, we
have $\mathcal{F}[u]\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$ and
$\mathcal{F}^{-1}[u]\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$ since it is
obvious that both $\mathcal{F}^{-1}[\varphi]$ and $\mathcal{F}[\varphi]$ are
in $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ for all
$\varphi\in\mathcal{D}(\mathbb{R}^{d})$ due to well-known properties of the
Fourier and inverse Fourier transforms of a Schwartz function. The continuity
of $\mathcal{F}[u]$ and $\mathcal{F}^{-1}[u]$ on $\mathcal{D}(\mathbb{R}^{d})$
to be distributions is easily obtained from properties of the Fourier
transform and the Schwartz functions as well.

It is essential to reiterate that, for any
$u\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$, the conventional definition of its
Fourier transform or inverse Fourier transform is not applicable without
considering analytic continuations as discussed previously. Nonetheless, we
can ultimately establish a meaningful definition for the Fourier and inverse
Fourier transforms of all distributions, treating them as elements in
$(\mathcal{F}^{-1}\mathcal{D})^{\prime}(\mathbb{R}^{d})$ without the Paley-
Wiener theorem.

######  Definition 2.3 (Fourier and inverse Fourier transforms of
$u\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$).

Let $u\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$. We define the Fourier and
inverse Fourier transforms of $u$ as follows:

$\displaystyle\langle\mathcal{F}[u],\varphi\rangle:=\langle
u,\mathcal{F}^{-1}[\varphi]\rangle\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$

and

$\displaystyle\langle\mathcal{F}^{-1}[u],\varphi\rangle:=\langle
u,\mathcal{F}[\varphi]\rangle\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$

It is obvious that both transforms above are well-defined since
$\mathcal{F}[\varphi]$ and $\mathcal{F}^{-1}[\varphi]$ are in
$\mathcal{D}(\mathbb{R}^{d})$ for all
$\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$.

Due to Definitions 2.2 and 2.3, any distribution $u$ on $\mathbb{R}^{d}$
becomes the Fourier transform (or inverse Fourier transform) of an element in
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ and any
element in $\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$
is the Fourier transform (or inverse Fourier transform) of a distribution on
$\mathbb{R}^{d}$. In this sense, we could say that
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$ is the
space of all Fourier transforms or all inverse Fourier transforms of
distributions on $\mathbb{R}^{d}$. Thus we use the notation
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ or
$\mathcal{F}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ instead of
$\left(\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\right)^{\prime}$.

Besides, recall that both Fourier and inverse Fourier transforms are
homeomorphisms on the class of all tempered distributions with respect to
weak*-topologies. Since
$\mathcal{D}(\mathbb{R}^{d})\subset\mathcal{S}(\mathbb{R}^{d})$ and
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})\subset\mathcal{S}(\mathbb{R}^{d})$,
it is obvious that both spaces $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ and
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ are lager than
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$. We state below that both transforms
still become homeomorphisms with respect to weak*-topologies. We do not give a
detailed proof since the proof is almost identical with that of the tempered
distributions based on properties of transforms for functions in the Schwartz
class (cf. [8, Section 2.2.3] and [10, Theorem 7.1.10]).

######  Theorem 2.4.

  1. (i)

Both Fourier and inverse Fourier transforms are homeomorphisms from
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ onto
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ with respect to
weak*-topologies.

  2. (ii)

Both Fourier and inverse Fourier transforms are homeomorphisms from
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ onto
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ with respect to weak*-topologies.

In total, both transforms $\mathcal{F}$ and $\mathcal{F}^{-1}$ are mappings
from
$\mathcal{D}^{\prime}(\mathbb{R}^{d})\cup\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$
onto
$\mathcal{D}^{\prime}(\mathbb{R}^{d})\cup\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$.

Therefore, we could freely apply the Fourier inversion formula to all
distributions and Fourier transforms of distributions on $\mathbb{R}^{d}$ due
to the above theorem. In particular, $\mathcal{F}[u]=\mathcal{F}[v]$ implies
$u=v$ for all distributions $u$ and $v$ on $\mathbb{R}^{d}$.

It is crucial to grasp how distributions are manifested in practice. In our
main theorem, the mathematical conditions are given by realizations of Fourier
transforms of distributions. Recall that any locally integrable function $v$
on $\mathbb{R}^{d}$ is a distribution on $\mathbb{R}^{d}$ due to the
identification $v$ with the mapping

$\varphi\in\mathcal{D}(\mathbb{R}^{d})\mapsto\int_{\mathbb{R}^{d}}v(x)\overline{\varphi(x)}\mathrm{d}x.$

Here $v$ is locally integrable on $\mathbb{R}^{d}$ if and only if

$\displaystyle\int_{|\xi|<R}v(\xi)\mathrm{d}\xi<\infty\quad\forall
R\in(0,\infty),$

where $B_{R}$ denotes the Euclidean ball whose center is zero and radius is
$R$, i.e.

$\displaystyle B_{R}:=\\{y\in\mathbb{R}^{d}:|y|<R\\}.$

We write $v\in L_{1,\ell oc}(\mathbb{R}^{d})$ if $v$ is locally integrable on
$\mathbb{R}^{d}$. Especially, if there exists a $v\in L_{1,\ell
oc}(\mathbb{R}^{d})$ such that

$\displaystyle\langle\mathcal{F}[u],\varphi\rangle=\left(v,\varphi\right)_{L_{2}(\mathbb{R}^{d})}:=\int_{\mathbb{R}^{d}}v(\xi)\overline{\varphi(\xi)}\mathrm{d}\xi\quad\forall\varphi\in\mathcal{D}(\mathbb{R}^{d}),$

then we identify $\mathcal{F}[u]=v$ and consider $\mathcal{F}[u]$ as a
function defined $(a.e.)$ on $\mathbb{R}^{d}$, where $\overline{\varphi(\xi)}$
denotes the complex conjugate of $\varphi(\xi)$. In this case, we usually say
that the distribution $\mathcal{F}[u]$ has the realization $v$ on
$\mathbb{R}^{d}$. This identification even works for any element
$u\in\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$. In other words, we
say that $u\in\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ is locally
integrable if and only if there is a $v\in L_{1,\ell oc}(\mathbb{R}^{d})$

$\displaystyle\langle
u,\varphi\rangle:=\int_{\mathbb{R}^{d}}v(\xi)\overline{\varphi(\xi)}\mathrm{d}\xi\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$

This function $v$ is uniquely determined $(a.e.)$ on $\mathbb{R}^{d}$ since
both $\mathcal{D}(\mathbb{R}^{d})$ and
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ are dense in $L_{\infty}(B_{R})$
for all $R\in(0,\infty)$ (by considering restrictions of
$\mathcal{D}(\mathbb{R}^{d})$ or $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$
on each $B_{R}$). Due to this identification, we could introduce an important
subclass of $\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ consisting
of all functions whose Fourier transform is locally integrable. Here is the
precise definition. We use $\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})$ to
denote the subspace of $\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$
whose Fourier transform is in $L_{1,\ell oc}(\mathbb{R}^{d})$, i.e.
$u\in\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})$ if and only if

$\displaystyle\left\langle
u,\varphi\right\rangle=\left\langle\mathcal{F}[u],\mathcal{F}[\varphi]\right\rangle=\int_{\mathbb{R}^{d}}\mathcal{F}[u](\xi)\overline{\mathcal{F}[\varphi](\xi)}\mathrm{d}\xi\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$

Since $\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\subset\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$,
there is the subspace topology on $\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})$ and the Borel sets generated by this topology.
Additionally, this identification with a locally integrable function could be
easily performed in any domain of $\mathbb{R}^{d}$ due to the localizations of
distributions. More precisely, if there exist a open set
$\mathcal{U}\subset\mathbb{R}^{d}$ and a locally integrable function $v$ on
$\mathcal{U}$ so that

$\displaystyle\langle\mathcal{F}[u],\varphi\rangle=\int_{\mathbb{R}^{d}}v(\xi)\overline{\varphi(\xi)}\mathrm{d}\xi\quad\forall\varphi\in\mathcal{D}(\mathcal{U}),$
(2.3)

then we identify $\mathcal{F}[u]$ with $v$ on $\mathcal{U}$, where
$\mathcal{D}\left(\mathcal{U}\right)$ denotes the set of all infinitely
differential functions on $\mathbb{R}^{d}$ with compact supports in
$\mathcal{U}$. In such a case, we say that $\mathcal{F}[u]$ has a realization
$v$ on $\mathcal{U}$ and consider $\mathcal{F}[u]$ as a locally integrable
function on $\mathcal{U}$. In other words, for any $u$ in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$, we say that
$\mathcal{F}[u]$ has a realization $v$ on $\mathcal{U}$ or simply
$\mathcal{F}[u]$ is locally integrable on $\mathcal{U}$ if (2.3) holds.

According to these realizations of elements in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$, we are ready to define
the operator $\psi(t,-i\nabla)$. To define the operator universally for all
$t$, we slightly abuse the notation by considering $\psi(-i\nabla)$ with a
function $\psi(\xi)$ on $\mathbb{R}^{d}$ below.

######  Definition 2.5.

Let $g\in\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ so that
$\mathcal{F}[g]$ has a realization on an open set $\mathcal{U}$ covering the
support of $\psi$ and
$\xi\in\mathbb{R}^{d}\mapsto\psi(\xi)\mathcal{F}[g](\xi)\in\mathbb{C}$ is a
locally integrable function on $\mathbb{R}^{d}$. Then we define
$\psi(-i\nabla)g$ as an element in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ so that

$\displaystyle\langle\psi(-i\nabla)g,\varphi\rangle=\int_{\mathbb{R}^{d}}\psi(\xi)\mathcal{F}[g](\xi)\mathcal{F}[\varphi](\xi)\mathrm{d}\xi\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d}).$

Due to the definition provided above, equation (1.5) can be considered valid
even without imposing any regularity requirement on the symbol. This will be
further elaborated upon in the subsequent discussion regarding the definition
of a solution, as outlined with more depth in Definition 2.10.

Next we consider classes comprising Borel measurable functions with values in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$, defined almost
everywhere on the interval $(0,T)$.

######  Definition 2.6.

We use the notation
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
to represent the set of all Borel measurable functions $f$ with values in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ defined almost
everywhere on the interval $(0,T)$. In particular, $f\in
L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
implies that

$\displaystyle\langle f(t,\cdot),\varphi\rangle:=\langle
f(t),\varphi\rangle<\infty\quad(a.e.)~{}t\in(0,T)\quad\text{and}\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$

######  Definition 2.7.

$L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
is used to denote the class of all Borel measurable functions $f$ with values
in $\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ defined $(a.e.)$ on
$(0,T)$ such that

$\displaystyle\int_{0}^{t}\langle
f(s,\cdot),\varphi\rangle\mathrm{d}s:=\int_{0}^{t}\langle
f(s),\varphi\rangle\mathrm{d}s<\infty\quad\forall
t\in(0,T)\quad\text{and}\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$

It is obvious that
$L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)\subset
L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$.
Naturally, we could consider subspaces of
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
and
$L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
with values in $\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})$.

######  Definition 2.8.

We define $L_{0}\left((0,T);\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\right)$ as the subspace of
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
consisting of all Borel measurable functions $f$ with values in
$\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})$ defined $(a.e.)$ on $(0,T)$ so
that

$\displaystyle\int_{B_{R}}|\mathcal{F}[f(t,\cdot)](\xi)|\mathrm{d}\xi:=\int_{B_{R}}|\mathcal{F}[f(t)](\xi)|\mathrm{d}\xi<\infty\quad(a.e.)~{}t\in(0,T)\quad\text{and}\quad\forall
R\in(0,\infty).$

######  Definition 2.9.

We write $f\in L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\right)$ if $f$ is a $\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})$-valued Borel measurable function defined $(a.e.)$ on
$(0,T)$ such that

$\displaystyle\int_{0}^{t}\int_{B_{R}}|\mathcal{F}[f(s,\cdot)](\xi)|\mathrm{d}\xi\mathrm{d}s:=\int_{0}^{t}\int_{B_{R}}|\mathcal{F}[f(s)](\xi)|\mathrm{d}\xi\mathrm{d}s<\infty\quad\forall(t,R)\in(0,T)\times(0,\infty).$

Generally, $f\in L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\right)$ cannot be identified with a complex-valued
function defined on $(0,T)\times\mathbb{R}^{d}$ in the typical almost
everywhere sense. Thus the terms $\mathcal{F}[f(s,\cdot)]$ and
$\mathcal{F}[f(s,\cdot)](\xi)$ should be understood as the notation to denote
$\mathcal{F}[f(s)]$ and $\mathcal{F}[f(s)](\xi)$, respectively. On the other
hand, if $f$ is a nice function defined on $(0,T)\times\mathbb{R}^{d}$, then
one can understand $\mathcal{F}[f(s)](\xi)$ as

$\displaystyle\mathcal{F}[f(s,\cdot)](\xi):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{-ix\cdot\xi}f(s,x)\mathrm{d}x.$

We use both notations $\mathcal{F}[f(s)](\xi)$ and
$\mathcal{F}[f(s,\cdot)](\xi)$ to give the same meaning in either way.

Now assume that $u_{0}\in L_{1,\ell oc}(\mathbb{R}^{d})$ and $f\in
L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\right)$. In such cases, all the components on the right
side of equation (2.2) are well-defined and meaningful, provided that the
symbol $\psi(t,\xi)$ is locally bounded with the help of the extension of the
inverse Fourier transform.

Finally, we suggest a definition of a weak solution to equation (1.5). We call
it a _Fourier-space weak solution_ to (1.5).

######  Definition 2.10 (Fourier-space weak solution).

Let $u_{0}\in\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$, $f\in
L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$,
and $u\in
L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$.
We say that $u$ is a _Fourier-space weak solution_ to equation (1.5) if for
any $\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$,

$\displaystyle\left\langle u(t,\cdot),\varphi\right\rangle=\langle
u_{0},\varphi\rangle+\int_{0}^{t}\left\langle\psi(s,-i\nabla)u(s,\cdot),\varphi\right\rangle\mathrm{d}s+\int_{0}^{t}\left\langle
f(s,\cdot),\varphi\right\rangle\mathrm{d}s\quad(a.e.)~{}t\in(0,T).$ (2.4)

The term “Fourier-space weak solution” as described above is evidently derived
from the fact that the Fourier transform is applied to the spatial variable of
a solution $u$.

######  Remark 2.11.

Note that the term
$\int_{0}^{t}\left\langle\psi(s,-i\nabla)u(s,\cdot),\varphi\right\rangle\mathrm{d}s$
is not well-defined in general even though we consider the extended Fourier
and inverse Fourier transforms. It is because the multiplication action

$\langle\psi(t,\cdot)\mathcal{F}[u](t,\cdot),\varphi\rangle:=\langle\mathcal{F}[u](t,\cdot),\psi(t,\cdot)\varphi\rangle$

is not well-defined since the symbol $\psi(t,\xi)$ does not satisfy a
regularity condition in general. The easiest way to make the term well-defined
is to give a sufficient smooth assumption on the symbol $\psi(t,\xi)$.
However, it is not the case we are interested in our weak formulation by
recalling the goal that we want to remove all regularity conditions on the
symbol $\psi(t,\xi)$. Fortunately, there is a method to make the term well-
defined without any regularity condition on $\psi(t,\xi)$ by using a
realization of $\mathcal{F}[u(t,\cdot)]$ as discussed in Definition 2.5, which
will be revisited in the next remark.

######  Remark 2.12.

Let $u$ be a Fourier-space weak solution to (1.5). Assume that for each
$t\in(0,T)$, the function $\psi(t,\xi)\mathcal{F}[u(t,\cdot)](\xi)$ is locally
integrable with respect to $\xi$. In this case, it is clear that
$\psi(t,\xi)\mathcal{F}[u(t,\cdot)](\xi)$ belongs to the space of
distributions $\mathcal{D}^{\prime}(\mathbb{R}^{d})$. Consequently, we can
proceed to consider the inverse Fourier transform of
$\psi(t,\xi)\mathcal{F}[u(t,\cdot)](\xi)$ due to Definition 2.3. This implies
that the main operator component, $\psi(t,i\nabla)u(t,\cdot)$, becomes
meaningful as a function with values in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$. Additionally according
to definitions of actions on
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$, we have

$\displaystyle\int_{0}^{t}\langle\psi(s,-i\nabla)u(s,\cdot),\varphi\rangle\mathrm{d}s=\int_{0}^{t}\left(\psi(s,\cdot)\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi](\cdot)\right)_{L_{2}(\mathbb{R}^{d})}\mathrm{d}s$
(2.5)

for all $\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$, where
$(\cdot,\cdot)_{L_{2}(\mathbb{R}^{d})}$ denotes the
$L_{2}(\mathbb{R}^{d})$-inner product, i.e.,

$\displaystyle\left(\psi(s,\cdot)\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi](\cdot)\right)_{L_{2}(\mathbb{R}^{d})}=\int_{\mathbb{R}^{d}}\psi(s,\xi)\mathcal{F}[u(s,\cdot)](\xi)\overline{\mathcal{F}[\varphi](\xi)}\mathrm{d}\xi.$

Therefore, we can use the identity in (2.5) to define
$\langle\psi(s,-i\nabla)u(s,\cdot),\varphi\rangle$ as
$\left(\psi(s,\cdot)\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi]\right)_{L_{2}(\mathbb{R}^{d})}$.
In other words, the term $\psi(s,-i\nabla)g$ could be defined for any $g$
whose Fourier transform has a realization so that
$\psi(s,\xi)\mathcal{F}[g](\xi)$ is locally integrable even though the symbol
$\psi(t,\xi)$ is not regular at all. It seems obvious that this realization of
$g$ is sufficient to exist within the support of the symbol $\psi$ which will
be discussed further in the subsequent remark.

######  Remark 2.13.

Let $u\in
L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$.
Then even the term

$\displaystyle\int_{0}^{t}\left(\psi(s,\cdot)\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi](\cdot)\right)_{L_{2}(\mathbb{R}^{d})}\mathrm{d}s=\int_{0}^{t}\left(\mathcal{F}[u(s,\cdot)],\overline{\psi}(s,\cdot)\mathcal{F}[\varphi](\cdot)\right)_{L_{2}(\mathbb{R}^{d})}\mathrm{d}s$
(2.6)

is not well-defined in general. Moreover, there exists the possibility of
finding a function $u$ that does not belong to the space
$L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\right)$, yet under the influence of $\psi(s,\cdot)$, the
expression (2.6) can still be made meaningful. To address this issue, we aim
to find appropriate functions, which will be presented in our main theorem and
belong to spaces whose Fourier transforms are in weighted local
$L_{p,q}$-spaces which are subspaces of
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$.
In particular, if $u$ is a Fourier-space weak solution to (1.5) so that the
term in (2.6) is well-defined, then at least, it implies that for each
$s\in(0,T)$, there exists a realization of $\mathcal{F}[u(s,\cdot)](\xi)$ on
the support of $\psi(s,\xi)$. In other words, for each $s$, there exists a
locally integrable function $v$ on $\mathbb{R}^{d}$ and a neighborhood
$\mathcal{U}$ of $\operatorname{supp}\psi(s,\cdot)$ so that

$\displaystyle\langle\mathcal{F}[u(s,\cdot)],\varphi\rangle=\int_{\mathbb{R}^{d}}v(\xi)\varphi(\xi)\mathrm{d}\xi\quad\forall\varphi\in\mathcal{D}\left(\mathcal{U}\right),$

where
$\operatorname{supp}\psi(s,\cdot)=\overline{\\{x\in\mathbb{R}^{d}:\psi(s,x)\neq
0\\}}$, $\overline{\\{x\in\mathbb{R}^{d}:\psi(s,x)\neq 0\\}}$ is the closure
of ${\\{x\in\mathbb{R}^{d}:\psi(s,x)\neq 0\\}}$. However, we can give any
value to $\mathcal{F}[u(s,\cdot)]$ outside of
$\operatorname{supp}\psi(s,\cdot)$ since it does not influence on the terms in
(2.6). Thus it might suggest that the space
$L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
is not appropriate to guarantee the uniqueness of a solution $u$. By the way,
there is a typical action to extend by putting

$\displaystyle\langle\mathcal{F}[u(s,\cdot)],\varphi\rangle=0\quad\forall\varphi\in\mathcal{D}\left(\mathbb{R}^{d}\setminus\operatorname{supp}\psi(s,\cdot)\right).$

Then the support of the distribution $\mathcal{F}[u(s,\cdot)]$ becomes a
subset of $supp\psi(s,\cdot)$ and the realization $v$ satisfies $v=v\cdot
1_{supp\psi(s,\cdot)}$, where $1_{supp\psi(s,\cdot)}$ denotes the indicator
function on the set $supp\psi(s,\cdot)$, i.e.

$1_{supp\psi(s,\cdot)}(x)=\begin{cases}&1\quad x\in supp\psi(s,\cdot)\\\
&0\quad x\in\mathbb{R}^{d}\setminus supp\psi(s,\cdot).\end{cases}$

######  Remark 2.14.

In the classical weak formulation of partial differential equations, the
commonly preferred class of test functions is $\mathcal{D}(\mathbb{R}^{d})$.
This preference is shaped by several factors. One of the primary reasons is
that $\mathcal{D}(\mathbb{R}^{d})$ serves as a well-behaved dense subset.
Specifically, $\mathcal{D}(\mathbb{R}^{d})$ is dense in
$L_{p}(\mathbb{R}^{d})$ for any $p\in[1,\infty)$, and it is also dense in
$\mathcal{S}(\mathbb{R}^{d})$. These properties of denseness are significant
considerations in favor of utilizing $\mathcal{D}(\mathbb{R}^{d})$ as a space
of test functions. They play a crucial role in facilitating a connection
between strong and weak solutions.

It is worth noting that the class
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ serves as an excellent
alternative for a class of test functions, as it possesses the same dense
properties. These properties can be easily shown based on well-known
properties of $\mathcal{D}(\mathbb{R}^{d})$ and $\mathcal{S}(\mathbb{R}^{d})$.
Here is a brief description of a proof. Recall that
$\mathcal{D}(\mathbb{R}^{d})$ is dense in $\mathcal{S}(\mathbb{R}^{d})$
concerning the topology generated by the Schwartz semi-norms. Additionally,
the Fourier transform acts as a homeomorphism from
$\mathcal{S}(\mathbb{R}^{d})$ to itself. Consequently, it is almost
straightforward to prove that $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ is
dense in $\mathcal{S}(\mathbb{R}^{d})$. An interesting observation is that
both $\mathcal{D}(\mathbb{R}^{d})$ and
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ are dense in
$\mathcal{S}(\mathbb{R}^{d})$, but their intersection contains only the zero
element.

We classify our solutions and data by examining their Fourier transforms.
Specifically, we focus on an initial data $u_{0}$ and an inhomogeneous data
$f$ in equation (1.5) whose Fourier transforms belong to weighted
$L_{p}$-spaces. To establish clear definitions for these data spaces, we need
to review notations associated with $L_{p}$-spaces related to weighted
Lebesgue measures. It is important to note that our weighted $L_{p}$-spaces
include the special case of $p=\infty$. Nevertheless, we temporarily exclude
this extreme case in order to simplify the comprehension of functions solely
in terms of integrations.

Let $p,q\in[1,\infty)$, $w$, and $W$ be non-negative (Borel) measurable
functions defined $(a.e.)$ on $(0,T)$ and $(0,T)\times\mathbb{R}^{d}$,
respectively. We write $u\in
L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)$
if and only if

$\displaystyle\|u(t,x)\|_{L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)}:=\left(\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|u(t,x)|^{q}W^{q}(t,x)\mathrm{d}x\right)^{p/q}w^{p}(t)\mathrm{d}t\right)^{1/p}<\infty.$

It is obvious that

$L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)=L_{p,q}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}tw^{q}(t)W^{q}(t,x)\mathrm{d}x\right).$

In particular,

$L_{p,p}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{p}(t,x)\mathrm{d}x\right)=L_{p,p}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}tw^{p}(t)W^{p}(t,x)\mathrm{d}x\right).$

In this case, our space becomes identical to the $L_{p}$-space with the
measure $w^{p}(t)W^{p}(t,x)\mathrm{d}t\mathrm{d}x$, i.e.

$\displaystyle
L_{p}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)W^{p}(t,x)\mathrm{d}t\mathrm{d}x\right)$
$\displaystyle=L_{p,p}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{p}(t,x)\mathrm{d}x\right)$
$\displaystyle=L_{p,p}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}tw^{p}(t)W^{p}(t,x)\mathrm{d}x\right).$

In addition, observe that

$\displaystyle\|u\|_{L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)}$
$\displaystyle=\|u\|_{L_{p}\left((0,T),w^{p}(t)\mathrm{d}t;L_{q}\left(\mathbb{R}^{d},W^{q}(t,x)\mathrm{d}x\right)\right)}$
$\displaystyle=\left(\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|u(t,x)|^{q}W^{q}(t,x)\mathrm{d}x\right)^{p/q}w^{p}(t)\mathrm{d}t\right)^{1/p}$
$\displaystyle=\left(\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|w(t)W(t,x)u(t,x)|^{q}\mathrm{d}x\right)^{p/q}\mathrm{d}t\right)^{1/p}$
$\displaystyle=\|wWu\|_{L_{p}\left((0,T);L_{q}\left(\mathbb{R}^{d}\right)\right)}.$

To include the extreme case that $p=\infty$ or $q=\infty$, we define

$\displaystyle\|u(t,x)\|_{L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)}:=\|wWu\|_{L_{p}\left((0,T);L_{q}\left(\mathbb{R}^{d}\right)\right)}.$

For $p,q\in[1,\infty)$, we remark that the simpler notations

$L_{p,q}\left((0,T)\times\mathbb{R}^{d},w(t)\mathrm{d}tW(t,x)\mathrm{d}x\right):=L_{p,q}\left((0,T)\times\mathbb{R}^{d},(w^{1/p})^{p}(t)\mathrm{d}t(W^{1/q})^{q}(t,x)\mathrm{d}x\right).$

are used on occasion for the sake of consistency with usual notations of
weighted $L_{p,q}$-spaces.

We usually identify
$L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)$
and
$L_{p}\left((0,T),w^{p}(t)\mathrm{d}t;L_{q}\left(\mathbb{R}^{d},W^{q}(t,x)\mathrm{d}x\right)\right)$
even though the iterated measurability does not imply the joint measurability.
We also would like to point out that we would not consider the intricate
matters between Borel and Lebesgue measurabilities by opting for the
completion of a measure space. Furthermore, we adopt a conventional method to
establish the equivalence in the almost everywhere sense across all
$L_{p,q}$-spaces without delving into significant details. In particular, we
do not make a strict distinction between functions defined throughout
$(0,T)\times\mathbb{R}^{d}$ and those defined almost everywhere within
$(0,T)\times\mathbb{R}^{d}$. Similarly, this relaxed approach extends to
functions defined on $\mathbb{R}^{d}$ as well. With these prerequisites in
place, we can now introduce a concept of our weighted local $L_{p,q}$-spaces.
It is worth noting that this local characteristic is presented in a slightly
different manner with respect to the time variable, which will be further
explored in Remark 2.20.

######  Definition 2.15 (Weighted local $L_{p,q}$-spaces).

Let $p,q\in[1,\infty]$.

  1. (i)

We define $L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)$
as the class of all measurable functions $u$ on $(0,T)\times\mathbb{R}^{d}$
such that

$\displaystyle\|u\|_{L_{p,q}\left((0,t_{1})\times
B_{R},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)}:=\|1_{(0,t_{1})}(t)1_{B_{R}}(x)u(t,x)\|_{L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)}<\infty$

for all $0<t_{1}<T$ and $R\in(0,\infty)$, where
$B_{R}:=\\{x\in\mathbb{R}^{d}:|x|<R\\}$ and $1_{B_{R}}$ denotes the indicator
function of $B_{R}$, i.e.

$1_{B_{R}}(x)=\begin{cases}&1\quad x\in B_{R},\\\ &0\quad
x\in\mathbb{R}^{d}\setminus B_{R}.\end{cases}$

  2. (ii)

We define
$L_{p,q,t\text{-}loc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)$
as the class of all measurable functions $u$ on $(0,T)\times\mathbb{R}^{d}$
such that

$\displaystyle\|u(t,x)\|_{L_{p,q}\left((0,t_{1})\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)}<\infty$

for all $0<t_{1}<T$.

We now turn our attention to a significant subset within
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$,
which comprises functions $u$ whose Fourier transform is in an weighted local
$L_{p,q}$-space. This subset can be formally expressed as:

$\displaystyle\left\\{u:\mathcal{F}[u(t,\cdot)](\xi)\in
L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,x)\mathrm{d}x\right)\right\\}.$
(2.7)

However, precisely defining the expression above (2.7) poses challenges due to
the potential degeneracy of our weight function $W(t,\xi)$ for certain
$\xi\in\mathbb{R}^{d}$. It requires employing sophisticated techniques
involving identifications with complex-valued functions, as illustrated in
(2.3).

Let $u\in
L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$.
Then for each $t\in(0,T)$, we have
$\mathcal{F}[u(t)]\in\mathcal{D}^{\prime}(\mathbb{R}^{d})$. Additionally,
assume that for each $t\in(0,T)$, $\mathcal{F}[u(t)]$ has a realization on an
open set $\mathcal{U}_{t}$ covering the support of $W(t,\cdot)$, i.e.
$\overline{\\{\xi\in\mathbb{R}^{d}:W(t,\xi)\neq 0\\}}\subset\mathcal{U}_{t}$,
where $\overline{\\{\xi\in\mathbb{R}^{d}:W(t,\xi)\neq 0\\}}$ denotes the
closure of $\\{\xi\in\mathbb{R}^{d}:W(t,\xi)\neq 0\\}$. Thus by identifying
$\mathcal{F}[u(t)]$ with the realization on $\mathcal{U}$, we may assume that
$(t,\xi)\mapsto\mathcal{F}[u(t,\cdot)](\xi):=\mathcal{F}[u](t,\xi):=\mathcal{F}[u(t)](\xi)$
is a complex-valued function defined on
$\\{(t,\xi)\in(0,T)\times\mathbb{R}^{d}:(t,\xi)\in(0,T)\times\mathcal{U}_{t}\\}$.
In particular, the term
$\|wW\mathcal{F}[u]\|_{L_{p}\left((0,T);L_{q}\left(\mathbb{R}^{d}\right)\right)}$
perfectly makes sense for all $p,q\in[1,\infty]$ by letting the product
$wW\mathcal{F}[u]$ be zero outside the set
$\\{(t,\xi)\in(0,T)\times\mathbb{R}^{d}:(t,\xi)\in(0,T)\times\mathcal{U}_{t}\\}$
since $W(t,\xi)=0$ for all $(t,\xi)\notin(0,T)\times\mathcal{U}_{t}$.
Therefore simply we may say that

$\mathcal{F}[u(t,\cdot)](\xi)\in
L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right),$

if and only if $u$ is an element in
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
so that for each $t\in(0,T)$, $\mathcal{F}[u(t)]$ has a realization on an open
set $\mathcal{U}_{t}$ covering the support of $W(t,\cdot)$ and
$\|wW\mathcal{F}[u]\|_{L_{p}\left((0,T);L_{q}\left(\mathbb{R}^{d}\right)\right)}<\infty$.
Finally, we define subspaces of
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
consisting of $u$ whose Fourier transform lies in a weighted (local)
$L_{p,q}$-spaces in the following definition based on the aforementioned
criteria.

######  Definition 2.16.

  1. (i)

We define subclasses of
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$
consisting of elements whose Fourier transforms with respect to the space
variable are in weighted (local) $L_{p,q}$-spaces. We write

$u\in\mathcal{F}^{-1}L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right),$
$u\in\mathcal{F}^{-1}L_{p,q,t\text{-}loc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right),$

and

$u\in\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)$

if

$\mathcal{F}[u(t,\cdot)](\xi)\in
L_{p,q}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(\xi)\mathrm{d}\xi\right),$
$\mathcal{F}[u(t,\cdot)](\xi)\in
L_{p,q,t\text{-}loc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right),$

and

$\mathcal{F}[u(t,\cdot)](\xi)\in L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right),$

respectively. In other words,

$u\in\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}x\right)$

if and only if

$\displaystyle\|\mathcal{F}[u]\|_{L_{p,q}\left((0,t_{1})\times
B_{R},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)}:=\|1_{B_{R}}(\xi)\mathcal{F}[u(t,\cdot)](\xi)\|_{L_{p,q}\left((0,t_{1})\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)}<\infty$

for all $t_{1}\in(0,T)$ and $R\in(0,\infty)$.

The weights $w^{p}(t)$, $W^{q}(t,x)$, and $W^{q}(t,\xi)$ in the notations are
skipped if they are merely $1$. If all weights are $1$, then we also omit
$\mathrm{d}t\mathrm{d}x$ and $\mathrm{d}t\mathrm{d}\xi$ in the notations.

These local spaces are employed to give weaker conditions on symbols and find
a class of solutions to (1.5) where the existence and uniqueness are
guaranteed. Recall Definition 2.9. Then it is obvious that

$\displaystyle L_{1,t\text{-loc}}\left((0,T);\mathcal{F}^{-1}L_{1,\ell
oc}(\mathbb{R}^{d})\right)=\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right).$

In addition, by Definition 2.6 and Hölder’s inequality, we have the inclusion
that

$\displaystyle\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)\subset\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)\subset
L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$

for all $p,q\in[1,\infty]$. However,

$\displaystyle\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)$
$\displaystyle\not\subset\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)$ (2.8)

in general, since even there is no strict positivity assumption on the weights
$w$ and $W$.

######  Remark 2.17.

There are numerous mathematical conveniences when we exclusively consider
positive weights. Nevertheless, it is crucial not to disregard the possibility
of weights becoming zero, as this allows us to include the interesting cases
in which the symbol $\psi(t,\xi)$ may become degenerate. Furthermore, it is
true that, for all values of $p$ and $q$ within the range of $[1,\infty]$,

$\displaystyle\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)\subset
L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$

due to the definition of $\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)$.
If both weights $w$ and $W$ are strictly positive, then

$\displaystyle\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)\subset
L_{0}\left((0,T);\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})\right)$

for all $p,q\in[1,\infty]$ since $|\mathcal{F}[u(t,\cdot)](\xi)|<\infty$
$(a.e.)$ $(t,\xi)\in(0,T)\times\mathbb{R}^{d}$ based on realizations on
$\mathbb{R}^{d}$ for any

$u\in\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right).$

However, the space $\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}tW^{q}(t,\xi)\mathrm{d}\xi\right)$
is not a metric space in general if one of the weights is degenerate since
$\mathcal{F}[u(t,\cdot)]$ could have any values outside of the supports of the
weights as discussed in Remark 2.13.

######  Remark 2.18.

We skip specifying the exact definitions of corresponding weighted
$L_{q}$-spaces treating data given on $\mathbb{R}^{d}$ since they are easily
induced from corresponding weighted $L_{p,q}$-spaces on
$(0,T)\times\mathbb{R}^{d}$. For instance, $\mathcal{F}^{-1}L_{q,\ell
oc}(\mathbb{R}^{d})$ denotes the subspace of
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ so that

$\displaystyle\|\mathcal{F}[u]\|_{L_{q}(B_{R})}<\infty\quad\forall
R\in(0,\infty).$

######  Remark 2.19.

Let $\mathscr{E}^{\prime}(\mathbb{R}^{d})$ represent the dual space of
$C^{\infty}(\mathbb{R}^{d})$. Then, for any element $u$ belonging to
$\mathscr{E}^{\prime}(\mathbb{R}^{d})$, its Fourier transform, denoted by
$\mathcal{F}[u]$, can be interpreted as a function that is locally bounded. To
clarify, for every $\xi$ within $\mathbb{R}^{d}$, the function
$\mathrm{e}^{-ix\cdot\xi}$ is a smooth function with respect to the variable
$x$, and the mapping from $\xi$ to $\langle u,\mathrm{e}^{-ix\cdot\xi}\rangle$
is a continuous function. Moreover, it is widely recognized that this space
$\mathscr{E}^{\prime}(\mathbb{R}^{d})$ corresponds to a subset of
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ with compact supports, as discussed in
[10, Section 2.3]. As a result, we can conclude that
$\mathscr{E}^{\prime}(\mathbb{R}^{d})$ is contained within
$\mathcal{F}^{-1}L_{q,\ell oc}(\mathbb{R}^{d})$ for all $q\in[1,\infty]$.

######  Remark 2.20.

We can describe a function $u$ belonging to the space
$L_{p,q,t\text{-}loc,x\text{-}\ell oc}\left((0,T)\times\mathbb{R}^{d}\right)$
as having a strong local integrability with respect to the variable $t$ and a
weaker form of local integrability with respect to the variable $x$. To
clarify further, consider a topological space denoted by $X$, and recall that
when we say a property holds locally, it means that the property is satisfied
for all compact subsets $K\subset X$. In this context, our function $u\in
L_{p,q,t\text{-}loc,x\text{-}\ell oc}\left((0,T)\times\mathbb{R}^{d}\right)$
can be described as locally integrable with respect to $x$, but it satisfies a
somewhat stronger condition compared to the conventional notion of local
integrability concerning the time variable $t$. This distinction is why we
employed different notations, namely, “$loc$” and ‘$\ell oc$”, to indicate the
local integrability properties of $t$ and $x$, respectively.

We finally reach our main theorem.

######  Theorem 2.21.

Let $u_{0}\in\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})$ and
$f\in\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)$. Suppose that for all $t\in(0,T)$
and $R\in(0,\infty)$,

$\displaystyle\int_{B_{R}}\exp\left(\int_{0}^{t}\Re[\psi(r,\xi)]\mathrm{d}r\right)\mathcal{F}[u_{0}](\xi)\mathrm{d}\xi+\int_{B_{R}}\int_{0}^{t}\exp\left(\int_{s}^{t}\Re[\psi(r,\xi)]\mathrm{d}r\right)\left|\mathcal{F}[f(s,\cdot)](\xi)\right|\mathrm{d}s\mathrm{d}\xi$
$\displaystyle<\infty,$ (2.9)

where $B_{R}:=\\{\xi\in\mathbb{R}^{d}:|\xi|<R\\}$. Additionally, assume that
for all $t\in(0,T)$ and $R\in(0,\infty)$,

$\displaystyle\int_{B_{R}}\int_{0}^{t}|\psi(\rho,\xi)|\exp\left(\int_{0}^{\rho}\Re[\psi(r,\xi)]\mathrm{d}r\right)\mathcal{F}[u_{0}](\xi)\mathrm{d}\rho\mathrm{d}\xi$
$\displaystyle+\int_{B_{R}}\int_{0}^{t}|\psi(\rho,\xi)|\int_{0}^{\rho}\exp\left(\int_{s}^{\rho}\Re[\psi(r,\xi)]\mathrm{d}r\right)\left|\mathcal{F}[f(s,\cdot)](\xi)\right|\mathrm{d}s\mathrm{d}\rho\mathrm{d}\xi<\infty.$
(2.10)

Then there exists a unique Fourier-space weak solution

$u\in\mathcal{F}^{-1}L_{\infty,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)\cap\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}t|\psi(t,\xi)|\mathrm{d}\xi\right)$

to (1.5), which is given by

$\displaystyle\langle u(t,\cdot),\varphi\rangle$
$\displaystyle=\int_{\mathbb{R}^{d}}\exp\left(\int_{0}^{t}\psi(r,\xi)\mathrm{d}r\right)\mathcal{F}[u_{0}](\xi)\mathcal{F}[\varphi](\xi)\mathrm{d}\xi$
$\displaystyle\quad+\int_{\mathbb{R}^{d}}\left(\int_{0}^{t}\exp\left(\int_{s}^{t}\psi(r,\xi)\mathrm{d}r\right)\mathcal{F}[f(s,\cdot)](\xi)\mathrm{d}s\right)\mathcal{F}[\varphi](\xi)\mathrm{d}\xi\quad\forall\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}).$
(2.11)

The proof of Theorem 2.21 is given in Section 5.

######  Remark 2.22.

The solution $u$ to (1.5) in Theorem 2.21 is not defined as a complex-valued
function on $[0,T)\times\mathbb{R}^{d}$. It is a function in the class
$L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$,
which is clarified by the action in (2.11). Therefore, formally, we may say
that

$\displaystyle
u(t,x)=\mathcal{F}^{-1}\left[\exp\left(\int_{0}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[u_{0}]\right](x)+\mathcal{F}^{-1}\left[\int_{0}^{t}\exp\left(\int_{s}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[f(s,\cdot)]\mathrm{d}s\right](x)$
(2.12)

is a solution to (1.5). It is obvious that this representation for the
solution $u$ is identical to the classical one if $\psi$, $u_{0}$, and $f$
satisfy nice mathematical conditions by considering convolutions with kernels
as follow :

$\displaystyle
u(t,x)=\int_{\mathbb{R}^{d}}K(0,t,x)u_{0}(x-y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{R}^{d}}K(s,t,y)f(s,x-y)\mathrm{d}y\mathrm{d}s,$

where the kernel $K(s,t,x)$ is given by

$\displaystyle
K(s,t,x)=\mathcal{F}^{-1}\left[\exp\left(\int_{s}^{t}\psi(r,\cdot)\mathrm{d}r\right)\right](x).$

This kernel representation shows that it is not hopeful to expect that $u$ has
a realization on $[0,T)\times\mathbb{R}^{d}$ if the symbol $\psi(t,\xi)$ is
not negative.

######  Remark 2.23.

We intentionally split the mathematical assumptions in Theorem 2.21 into two
parts, (2.9) and (2.10), since the condition (2.10) is sufficient to guarantee
the uniqueness of a solution, which is specified in Theorem 4.3. On the other
hand, (2.9) guarantee the function $u(t,x)$ defined as (2.12) is well-defined
as $\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$-valued function. A
naive sufficient condition for both (2.9) and (2.10) is a local bounded
property on the symbol $\psi(t,\xi)$. More precisely, if
$\left\|\psi(\rho,\xi)\right\|_{L_{\infty,\infty}\left((0,t)\times
B_{R},\mathrm{d}\rho\mathrm{d}\xi\right)}<\infty$ for all $t\in(0,T)$ and
$R\in(0,\infty)$, then both (2.9) and (2.10) are satisfied, which will be
relaxed in Corollary 5.1 with positive weights and general exponents. We also
want to mention that both (2.9) and (2.10) are given by combinations of
properties on the symbol $\psi$ and data $u_{0}$ and $f$ in a complicated way
on purpose to emphasize that the condition on the symbol could be weaken if
data $u_{0}$ and $f$ are in better spaces. One of the good examples showing
weakening condition on the symbol $\psi$ by enhancing the condition on data
$u_{0}$ and $f$ is presented in Corollary 5.3. Additionally, these complicated
conditions (2.9) and (2.10) make us handle logarithmic operators instead of
merely considering locally bounded symbols, which will be specified in Theorem
2.27 below.

######  Remark 2.24.

We revisit to mention the importance of the extension of the inverse Fourier
transform to all distributions. Assume that the symbol $\psi(t,\xi)$ is
locally bounded. Then the function
$\exp\left(\int_{0}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[u_{0}]$ is
merely locally integrable with respect to $\xi$. Thus, there is no assurance
that $\exp\left(\int_{0}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[u_{0}]$
qualifies as a tempered distribution on $\mathbb{R}^{d}$. Consequently, the
function $u$ in equation (2.12) cannot be defined because the inverse Fourier
transform is typically applicable only to tempered distributions on
$\mathbb{R}^{d}$ (if we do not consider an extension to $\mathbb{C}^{d}$).
However, it is important to note that we extended the inverse Fourier
transform to all distributions as an element in
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$. Therefore, the
function $u$ is well-defined as a new extension of the inverse Fourier
transform to a distribution even if the functions
$\exp\left(\int_{0}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[u_{0}]$ and
$\int_{0}^{t}\exp\left(\int_{s}^{t}\psi(r,\cdot)\mathrm{d}r\right)\mathcal{F}[f(s,\cdot)]\mathrm{d}s$
are merely locally integrable. Especially, it is easy to check that

$u\in\mathcal{F}^{-1}L_{\infty,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)\cap\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}t|\psi(t,\xi)|\mathrm{d}\xi\right)$

due to (2.9) and (2.10).

######  Remark 2.25.

It is not trivial to understand realizations of data $u_{0}$ and $f$ since all
data $u_{0}$ and $f$ are given based on Fourier transforms acting on
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$. However, it is
important to note that $\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$
is a broader class than $\mathcal{S}^{\prime}(\mathbb{R}^{d})$. Within
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$, there exist various well-known
subspaces that help characterize regularities of elements in
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$. One notable example is the Bessel
potential space. Given that
$\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$ is larger than any
subspace of $\mathcal{S}^{\prime}(\mathbb{R}^{d})$, we could apply our theorem
to data $u_{0}$ and $f$ residing in Bessel potential spaces. We suggest a
weighted Bessel potential space as an important example of realizations of
data $u_{0}$ and $f$ in the next section. The specific results will be
explicitly presented in Corollaries 5.4, 5.6, and 5.7.

It is readily verifiable that the logarithmic Laplacian operator can be
regarded as a specific example of the operator $\psi(t,-i\nabla)$. Recall that
the logarithmic Laplacian operator could be defined as follows

$\displaystyle\mathcal{F}[\log(-\Delta)u]=\log|\xi|^{2}\mathcal{F}[u]$

for a nice function $u$. This means that the logarithmic Laplacian operator
$\log(-\Delta)$ is a pseudo-differential operator characterized by the symbol
$\log|\xi|^{2}$. The logarithmic Laplacian operator is an intriguing example
of $\psi(t,-i\nabla)$ because its symbol is negative for $|\xi|<1$ and
positive for $|\xi|>1$. Additionally, the symbol is not a polynomial. In other
words, the logarithmic Laplacian operator is a simple illustration of a
pseudo-differential operator with a non-polynomial symbol that changes its
sign. As an application of Theorem 2.21, we explore more generalized
logarithmic-type operators with complex-valued coefficients, which naturally
encompass the logarithmic Laplacian operator. We now present an evolution
equation involving a logarithmic-type operator as a specific case of (1.5)
below:

$\begin{cases}\partial_{t}u(t,x)=\beta(t)\log\left(\psi_{exp}(t,-i\nabla)\right)u(t,x)+f(t,x),\quad&(t,x)\in(0,T)\times\mathbb{R}^{d},\\\
u(0,x)=u_{0},\quad&x\in\mathbb{R}^{d},\end{cases}$ (2.13)

where

$\displaystyle\log\left(\psi_{exp}(t,-i\nabla)\right)u(t,x):=\mathcal{F}^{-1}\left[\log\left(\psi_{exp}(t,\xi)\right)\mathcal{F}[u(t,\cdot)]\right](x),$
(2.14)

$\beta(t)$ is a complex-valued measurable function defined on $(0,T)$, and
$\psi_{exp}(t,\xi)$ is a complex-valued measurable function defined on
$(0,T)\times\mathbb{R}^{d}$. In particular, if $\beta(t)=1$ and
$\psi_{exp}(t,\xi)=|\xi|^{2}$ for all $t$, then

$\displaystyle\beta(t)\log\left(\psi_{exp}(t,-i\nabla)\right)u(t,x)=\log(-\Delta)u(t,x).$

A comprehensive understanding of a weak solution $u$ to the equation (2.13)
can be achieved by referring to Definition 2.10. Moreover, we assume

$\displaystyle\psi_{exp}(t,\xi)\neq
0\quad(a.e.)~{}(t,\xi)\in(0,T)\times\mathbb{R}^{d}.$ (2.15)

It is essential for the right-hand side of (2.14) to be meaningful that the
condition stated in (2.15) is satisfied. This requirement is necessary due to
the presence of the logarithm’s domain.

######  Remark 2.26.

Let’s define $\psi(t,\xi)$ as the natural logarithm of $\psi_{exp}(t,\xi)$,
i.e.

$\psi(t,\xi)=\log\left(\psi_{exp}(t,\xi)\right).$

From a heuristic standpoint, we can express this relationship as

$\exp\left(\psi(t,\xi)\right)=\psi_{exp}(t,\xi),$

which explains the reason using the notation $\psi_{exp}(t,\xi)$. In other
words, the subscript in $\psi_{exp}(t,\xi)$ signifies the exponential
operation.

We restate a theorem opting for (2.13), which is one of the major applications
of Theorem 2.21.

######  Theorem 2.27.

Let $u_{0}\in\mathcal{F}^{-1}L_{1,\ell oc}(\mathbb{R}^{d})$ and
$f\in\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)$. Assume that (2.15) holds and

$\displaystyle\int_{0}^{t}|\beta(s)|\mathrm{d}s<\infty\quad\forall t\in(0,T).$
(2.16)

Additionally, suppose that

$\displaystyle\sup_{(s,\xi)\in[0,t)\times
B_{R}}\left(\frac{1}{t-s}\int_{s}^{t}|\psi_{exp}(r,\xi)|^{(t-s)\Re[\beta(r)]}\mathrm{d}r\right)<\infty$
(2.17)

and

$\displaystyle\sup_{(s,\xi)\in[0,t)\times
B_{R}}\left(\int_{s}^{t}|\beta(\rho)|\left(1+\left|\log\left(|\psi_{\exp}(\rho,\xi)|\right)\right|\right)\left(\frac{1}{\rho-s}\int_{s}^{\rho}\left(|\psi_{exp}(r,\xi)|^{(\rho-s)\Re[\beta(r)]}\right)\mathrm{d}r\right)\mathrm{d}\rho\right)<\infty$
(2.18)

for all $t\in(0,T)$ and $R\in(0,\infty)$. Then there exists a unique Fourier-
space weak solution $u$ to (2.13) in

$\mathcal{F}^{-1}L_{\infty,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d}\right)\cap\mathcal{F}^{-1}L_{1,1,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},|\beta(t)|\mathrm{d}t|\log\left(\psi_{exp}(t,\xi)\right)|\mathrm{d}\xi\right).$

The proof of this theorem will be given in Section 6.

######  Remark 2.28.

It is notable that the complex-valued function $\psi_{exp}(t,\xi)$ could have
a negative real value. It is possible because we have not imposed any
regularity conditions on the symbol $\log\left(\psi_{exp}(t,\xi)\right)$,
which means there is no need to consider the analytic continuation of the
natural logarithm.

######  Remark 2.29.

The condition (2.18) could be replaced by

$\displaystyle\sup_{(s,\xi)\in[0,t)\times
B_{R}}\left(\int_{s}^{t}|\beta(\rho)|\left|\log\left(\psi_{\exp}(\rho,\xi)\right)\right|\left(\frac{1}{\rho-s}\int_{s}^{\rho}\left(|\psi_{exp}(r,\xi)|^{(\rho-s)\Re[\beta(r)]}\right)\mathrm{d}r\right)\mathrm{d}\rho\right)<\infty$

since it is obvious that

$\left(1+\left|\log\left(|\psi_{\exp}(\rho,\xi)|\right)\right|\right)\approx\left|\log\left(\psi_{\exp}(\rho,\xi)\right)\right|$

by considering a branch cut.

######  Remark 2.30.

Consider the logarithmic Laplacian operator

$\displaystyle\log(-\Delta)u(t,x)$

as a special case of

$\displaystyle\beta(t)\log\left(\psi_{exp}(t,-i\nabla)\right)u(t,x)$

in Theorem 2.27. It is easy to confirm that (2.17) is valid for this simple
case since $\beta(t)=1$ and $\psi_{exp}(t,\xi)=|\xi|^{2}$ for all values of
$t$ and $\xi$. Nonetheless, it might seem that (2.18) is not straightforward.
However, it can be readily established through elementary calculations, as
demonstrated in the proof of Corollary 6.1. Furthermore, we present a
straightforward extension of this result for dealing with operators that
result from a composition involving the $\log$ function and a general second-
order differential operator, which is detailed in Corollary 6.1.

Next, we consider another important case handling a second-order differential
operator without ellipticity. We investigate a second-order evolution equation
with complex-valued coefficients $a^{ij}(t)$, $b^{i}(t)$, and $c(t)$ as
follows:

$\begin{cases}\partial_{t}u(t,x)=a^{ij}(t)u_{x^{i}x^{j}}(t,x)+b^{j}(t)u_{x^{j}}(t,x)+c(t)u(t,x)+f(t,x),\quad&(t,x)\in(0,T)\times\mathbb{R}^{d},\\\
u(0,x)=u_{0},\end{cases}$ (2.19)

where Einstein’s summation convention is being enforced here. Suppose that all
coefficients $a^{ij}(t)$, $b^{j}(t)$, and $c(t)$ are measurable and defined on
$(0,T)$. Formally, due to some properties of the Fourier transform and the
inverse Fourier transform,

$\displaystyle
a^{ij}(t)u_{x^{i}x^{j}}(t,x)+b^{j}(t)u_{x^{j}}(t,x)+c(t)u(t,x)=\mathcal{F}^{-1}\left[\left(-a^{ij}(t)\xi^{i}\xi^{j}+ib^{j}(t)+c(t)\right)\mathcal{F}[u(t,\cdot)](\xi)\right](x).$

Hence, equation (2.19) can be seen as a specific instance of equation (1.5)
when considering the symbol:

$\displaystyle\psi(t,\xi)=-a^{ij}(t)\xi^{i}\xi^{j}+ib^{j}(t)\xi^{j}+c(t),$
(2.20)

where $i$ beside $b^{j}(t)$ is the imaginary number so that $i^{2}=-1$. To
maintain clarity and facilitate readers’ comprehension, we provide a modified
version of the definition of a Fourier-space weak solution for second-order
cases. This indicates that the structure of our weak formulation closely
parallels that of the classical one.

######  Definition 2.31 (A weak solution tested by
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$).

Let $u\in
L_{0}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$,
$u_{0}\in\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})$, and $f\in
L_{1,t\text{-}loc}\left((0,T);\mathcal{F}^{-1}\mathcal{D}^{\prime}(\mathbb{R}^{d})\right)$.
Then we say that $u$ is a _weak solution_ (tested by
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$) to (2.19) if for any
$\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$,

$\displaystyle\langle u(t,\cdot),\varphi\rangle=\langle
u_{0},\varphi\rangle+\int_{0}^{t}\left(a^{ij}(s)\langle
u(s,\cdot),\varphi_{x^{i}x^{j}}\rangle-b^{j}(s)\langle
u(s,\cdot),\varphi_{x^{j}}\rangle+c(s)\langle
u(s,\cdot),\varphi\rangle\right)\mathrm{d}s+\int_{0}^{t}\langle
f(s,\cdot),\varphi\rangle\mathrm{d}s$ $\displaystyle(a.e.)~{}t\in(0,T).$
(2.21)

######  Remark 2.32.

Let us reiterate that a test function is conventionally given by a smooth
function with compact support in a weak formulation. In simpler terms, we
commonly say that $u$ qualifies as a (classical) weak solution to equation
(2.19) if (2.21) is valid for all $\varphi\in\mathcal{D}(\mathbb{R}^{d})$
instead of $\varphi\in\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$. To ensure
the meaningfulness of this traditional weak formulation, it is essential for
$u$ to be valued in $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ and defined almost
everywhere on $(0,T)$. Sufficiently assume that $u$ is a
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued measurable function on $(0,T)$
and it is a classical weak solution. Then this classical solution $u$ becomes
identical with the weak solution (tested by
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$) in Definition 2.31 since
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$ is dense within
$\mathcal{S}(\mathbb{R}^{d})$ concerning the topology defined by the Schwartz
semi-norms. However, it is crucial to highlight that for a solution $u$ to
equation (2.19), $u(t,\cdot)$ does not belong to
$\mathcal{D}^{\prime}(\mathbb{R}^{d})$ (and thus
$u(t,\cdot)\notin\mathcal{S}^{\prime}(\mathbb{R}^{d})$) if there is no
ellipticity in the leading coefficients $a^{ij}(t)$. This emphasizes the
necessity for a novel concept of a weak solution, such as the Fourier-space
weak solution.

######  Remark 2.33.

The weak solution (tested by $\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d})$) is
equivalent to the Fourier-space weak solution introduced in Definition 2.10
for the particular symbol $\psi$ defined in (2.20) if the Fourier transform of
$u$ is locally integrable (with respect to the space variable) since

$\displaystyle a^{ij}(s)\langle
u(s,\cdot),\varphi_{x^{i}x^{j}}\rangle-b^{j}(s)\langle
u(s,\cdot),\varphi_{x^{j}}\rangle+c(s)\langle u(s,\cdot),\varphi\rangle$
$\displaystyle=a^{ij}(s)\left(\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi_{x^{i}x^{j}}]\right)_{L_{2}(\mathbb{R}^{d})}-b^{j}(s)\left(\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi_{x^{j}}]\right)_{L_{2}(\mathbb{R}^{d})}+c(s)\left(\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi]\right)_{L_{2}(\mathbb{R}^{d})}$
$\displaystyle=\left(\left(-a^{ij}(s)\xi^{i}\xi^{j}+ib^{j}(s)\xi^{j}+c(s)\right)\mathcal{F}[u(s,\cdot)],\mathcal{F}[\varphi]\right)_{L_{2}(\mathbb{R}^{d})}$
$\displaystyle=\left\langle\psi(s,-i\nabla)u(s,\cdot),\varphi\right\rangle,$

which are easily derived from some properties of the Fourier transform.

We are now prepared to present the theorem regarding equation (2.19). It is
important to note that all the conditions specified for the coefficients in
the theorem merely involve local integrability properties. Especially, all
coefficients could be complex-valued, and the matrix consisting of the leading
coefficients $a^{ij}(t)$ does not need to be positive semi-definite.

######  Theorem 2.34.

Let $p,q\in[1,\infty]$, $W_{0}(\xi)$ and $W_{1}(\xi)$ be positive measurable
functions on $\mathbb{R}^{d}$, $W_{2}(t,\xi)$ be a positive measurable
function on $(0,T)\times\mathbb{R}^{d}$, $u_{0}\in\mathcal{F}^{-1}L_{q,\ell
oc}(\mathbb{R}^{d},W_{0}^{q}(\xi)\mathrm{d}\xi)$, and

$f\in\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}tW_{2}^{q}(t,\xi)\mathrm{d}\xi\right).$

Assume that $a^{ij},b^{j},c\in L_{p,loc}\left((0,T)\right)$ for all
$i,j\in\\{1,\ldots,d\\}$, i.e.

$\displaystyle\sum_{i=1}^{d}\sum_{j=1}^{d}\left(\|a^{ij}\|_{L_{p}\left((0,t)\right)}+\|b^{j}\|_{L_{p}\left((0,t)\right)}+\|c\|_{L_{p}\left((0,t)\right)}\right)<\infty\quad\forall
t\in(0,T).$ (2.22)

Additionally, suppose that $W_{1}$ is a lower bound of $W_{2}$ and both
$W_{0}$ and $W_{1}$ have local lower bounds, i.e. for each $t\in(0,T)$ and
$R\in(0,\infty)$, there exist positive constants $\kappa_{0}(R)$ and
$\kappa_{1}(R)$ so that

$\displaystyle W_{0}(\xi)\geq\kappa_{0}(R)\quad\forall\xi\in B_{R}$ (2.23)

and

$\displaystyle W_{2}(s,\xi)\geq
W_{1}(\xi)\geq\kappa_{1}(R)\quad\forall(s,\xi)\in(0,t)\times B_{R}.$ (2.24)

Then there exists a unique weak solution $u$ $($tested by
$\mathcal{F}^{-1}\mathcal{D}(\mathbb{R}^{d}))$ to (2.19) in

$\mathcal{F}^{-1}L_{p,q,t\text{-}loc,x\text{-}\ell
oc}\left((0,T)\times\mathbb{R}^{d},\mathrm{d}t\left(1+|a^{ij}(t)\xi^{i}\xi^{j}-ib^{j}(t)\xi^{j}-c(t)|\right)\mathrm{d}\xi\right)$

and the solution $u$ is given by

$\displaystyle u(t,x)$
$\displaystyle=\mathcal{F}^{-1}\left[\exp\left(\int_{0}^{t}\left(-a^{ij}(r)\xi^{i}\xi^{j}+ib^{j}(r)\xi^{j}+c(r)\right)\mathrm{d}r\right)\mathcal{F}[u_{0}](\xi)\right](x)$
$\displaystyle\quad+\mathcal{F}^{-1}\left[\int_{0}^{t}\exp\left(\int_{s}^{t}\left(-a^{ij}(r)\xi^{i}\xi^{j}+ib^{j}(r)\xi^{j}+c(r)\right)\mathrm{d}r\right)\mathcal{F}[f(s,\cdot)]\mathrm{d}s\right](x).$

Here the solution $u$ is given formally as discussed in Remark 2.22.

We will provide the proof of this theorem in Section 7. It is worth noting
that when dealing with data $u_{0}$ and $f$ in weighted Bessel potentials,
this theorem can be readily applied. We demonstrate these applications as
corollaries of Theorem 2.34 in Section 7.

##  3\. Fourier transforms and weighted Bessel potential spaces

In this section, our objective is to present two types of weighted Bessel
potential spaces. These spaces exhibit numerous intriguing properties that can
be effectively characterized through the use of Fourier and inverse Fourier
transforms. To begin, we recall the definitions and some properties of the
Fourier and inverse Fourier transforms. For a measurable function $f$ on
$\mathbb{R}^{d}$, we denote the $d$-dimensional Fourier transform of $f$ by

$\mathcal{F}[f](\xi):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{-i\xi\cdot
x}f(x)\mathrm{d}x$

and the $d$-dimensional inverse Fourier transform of $f$ by

$\mathcal{F}^{-1}[f](x):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}\mathrm{e}^{ix\cdot\xi}f(\xi)\mathrm{d}\xi.$

Due to the Parseval-Plancherel identity,

$\displaystyle\|f\|_{L_{2}(\mathbb{R}^{d})}=\|\mathcal{F}[f]\|_{L_{2}(\mathbb{R}^{d})}=\|\mathcal{F}^{-1}[f]\|_{L_{2}(\mathbb{R}^{d})}\quad\forall
f\in L_{1}(\mathbb{R}^{d})\cap L_{2}(\mathbb{R}^{d}).$

The above equalities imply that the Fourier and inverse Fourier transforms are
$L_{2}(\mathbb{R}^{d})$-isometries by considering $L_{2}$-extensions based on
completeness of $L_{2}(\mathbb{R}^{d})$. Additionally, due to the Riesz–Thorin
theorem, there exist the extensions of the Fourier and inverse Fourier
transforms which are bounded from
$L_{1}(\mathbb{R}^{d})+L_{2}(\mathbb{R}^{d})$ to
$L_{\infty}(\mathbb{R}^{d})+L_{2}(\mathbb{R}^{d})$. In particular, for any
$\theta\in[0,1]$,

$\displaystyle\|\mathcal{F}[f]\|_{L_{2/\theta}}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(1-\theta)}{2}}\|f\|_{L_{2/(2-\theta)}}\quad\forall
f\in L_{2/(2-\theta)}$

and

$\displaystyle\|\mathcal{F}^{-1}[f]\|_{L_{2/\theta}}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(1-\theta)}{2}}\|f\|_{L_{2/(2-\theta)}}\quad\forall
f\in L_{2/(2-\theta)},$

where $\frac{2}{0}:=\infty$. Therefore, for any $p\in[1,2]$ and $f\in
L_{p}(\mathbb{R}^{d})$, it is possible to regard the Fourier transform and the
inverse Fourier transform of $f$ as functions within
$L_{p^{\prime}}(\mathbb{R}^{d})$, where $p^{\prime}=\frac{p}{p-1}$ and
$\frac{1}{0}:=\infty$. This is a crucial foundation for our theory, as it
allows us to address our solutions and data in a strong sense. To be more
specific, for any $p\in[2,\infty]$ and any function $f$ in
$L_{p^{\prime}}(\mathbb{R}^{d})$, we have the following inequalities:

$\displaystyle\|\mathcal{F}[f]\|_{L_{p}}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(p-2)}{2p}}\|f\|_{L_{p^{\prime}}}$
(3.1)

and

$\displaystyle\|\mathcal{F}^{-1}[f]\|_{L_{p}}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(p-2)}{2p}}\|f\|_{p^{\prime}}.$
(3.2)

We now turn our attention to the Fourier and inverse Fourier transforms
applied to the space of tempered distributions on $\mathbb{R}^{d}$. These
transformations are established in a weak sense, making use of the
advantageous properties of the Schwartz class $\mathcal{S}(\mathbb{R}^{d})$.
Suppose we have a function $f$ belonging to
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$. In that case, we define the Fourier
and inverse Fourier transforms as follows:

$\displaystyle\langle\mathcal{F}[f],\varphi\rangle=\langle
f,\mathcal{F}^{-1}[\varphi]\rangle\quad\forall\varphi\in\mathcal{S}(\mathbb{R}^{d})$

and

$\displaystyle\langle\mathcal{F}^{-1}[f],\varphi\rangle=\langle
f,\mathcal{F}[\varphi]\rangle\quad\forall\varphi\in\mathcal{S}(\mathbb{R}^{d}).$

Here, the notation $\langle f,\mathcal{F}^{-1}[\varphi]\rangle$ represents the
duality pairing, which means that $\langle
f,\mathcal{F}^{-1}[\varphi]\rangle=f\left(\mathcal{F}^{-1}[\varphi]\right)$ is
the value assigned to $\mathcal{F}^{-1}[\varphi]$ under the action of the
linear functional $f$. It is important to note that the Fourier and inverse
Fourier transforms of $f$ once again become tempered distributions on
$\mathbb{R}^{d}$, thanks to the well-established properties of functions
within the Schwartz class. Furthermore, it is worth noting that
$(1+|\cdot|^{2})^{\gamma_{2}/2}f$ remains a tempered distribution as well if
we define the action on test functions as follows:

$\displaystyle\left\langle(1+|\cdot|^{2})^{\gamma_{2}/2}f,\varphi\right\rangle:=\left\langle
f,(1+|\cdot|^{2})^{\gamma_{2}/2}\varphi(\cdot)\right\rangle\quad\text{for
all}\quad\varphi\in\mathcal{S}(\mathbb{R}^{d}).$

This is possible because the function $(1+|x|^{2})^{\gamma_{2}/2}\varphi(x)$
belongs to the Schwartz class $\mathcal{S}(\mathbb{R}^{d})$ for any
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$. As a result, these operations on
tempered distributions enable us to establish the Bessel potential spaces.

Let $\gamma\in\mathbb{R}$ and $p\in[1,\infty]$. Recall that we define the
Bessel potential space with order $\gamma$ and exponent $p$, denoted by
$H_{p}^{\gamma}(\mathbb{R}^{d})$, as the class of all tempered distributions
$f$ on $\mathbb{R}^{d}$ satisfying the condition:

$\displaystyle(I-\Delta)^{\gamma/2}f:=\mathcal{F}^{-1}\left[\left(1+|\cdot|^{2}\right)\mathcal{F}[f]\right]\in
L_{p}(\mathbb{R}^{d}).$

For a more comprehensive understanding, further details on these concepts can
be found in references such as [10] and [8]. Now, we proceed to introduce two
generalizations of the Bessel potential spaces with additional weighting
factors below. We call these spaces an inner weighted Bessel potential space
and an outer weighted Bessel potential space, respectively. It is worth noting
that we have not found any previous literature that discusses these types of
weighted Bessel potential space in general. However, certain outer weighted
Bessel potential spaces can be regarded as particular cases of weighted Bessel
potential spaces with Muckenhoupt’s weight. We will provide further
clarification of this relationship before presenting Proposition 3.5 later on.
We are now ready to present the definitions for both inner and outer weighted
Bessel potential spaces.

######  Definition 3.1 (Inner weighted Bessel potential space).

Let $\gamma_{1},\gamma_{2}\in\mathbb{R}$ and $p\in[1,\infty]$. We use the
notation $H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ to denote the
space of all tempered distributions $f$ on $\mathbb{R}^{d}$ such that

$\displaystyle(I-\Delta)^{\gamma_{1}/2}\left((1+|\cdot|^{2})^{\gamma_{2}/2}f\right):=\mathcal{F}^{-1}\left[\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\mathcal{F}\left[(1+|\cdot|^{2})^{\gamma_{2}/2}f\right]\right]\in
L_{p}(\mathbb{R}^{d}),$

which implies that there exists a $g\in L_{p}(\mathbb{R}^{d})$ such that

$\displaystyle\left\langle\mathcal{F}^{-1}\left[\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\mathcal{F}\left[\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}f\right]\right],\varphi\right\rangle$
$\displaystyle:=\left\langle
f,\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}\varphi\right\rangle$
$\displaystyle=\int_{\mathbb{R}^{d}}g(x)\varphi(x)\mathrm{d}x\qquad\forall\varphi\in\mathcal{S}(\mathbb{R}^{d}).$

Then $H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ becomes a Banach space
with the norm

$\displaystyle\|f\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}:=\left\|(I-\Delta)^{\gamma_{1}/2}\left((1+|\cdot|^{2})^{\gamma_{2}/2}f\right)\right\|_{L_{p}(\mathbb{R}^{d})}.$

We say that $\gamma_{1}$, $\gamma_{2}$, and $p$ are regularity exponent,
weight exponent, and integrability exponent of
$H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$, respectively.

######  Definition 3.2 (Outer weighted Bessel potential space).

Let $\gamma_{1},\gamma_{2}\in\mathbb{R}$ and $p\in[1,\infty]$. We use the
notation $H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ to denote the
space of all tempered distributions $f$ on $\mathbb{R}^{d}$ such that

$\displaystyle(1+|\cdot|^{2})^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}f\in
L_{p}(\mathbb{R}^{d}).$

Then $H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ becomes a Banach
space with the norm

$\displaystyle\|f\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}:=\left\|\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}f\right\|_{L_{p}(\mathbb{R}^{d})}.$

Same terminologies are used for exponents $\gamma_{1}$, $\gamma_{2}$, and $p$.
In other words, $\gamma_{1}$, $\gamma_{2}$, and $p$ are called regularity
exponent, weight exponent, and integrability exponent of
$H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$, respectively.

We would not delve into the specific details demonstrating that these two
spaces are indeed Banach spaces. This can be readily proved by relying on the
completeness of $L_{p}$-spaces. In addition, it is obvious that

$H_{p}^{\gamma_{1}}(\mathbb{R}^{d})=H_{p,in}^{\gamma_{1},0}(\mathbb{R}^{d})=H_{p,out}^{\gamma_{1},0}(\mathbb{R}^{d})$

and

$H_{p,in}^{0,\gamma_{2}}(\mathbb{R}^{d})=H_{p,out}^{0,\gamma_{2}}(\mathbb{R}^{d}).$

######  Remark 3.3.

Let $\gamma_{1}>0$. It is a well-known fact that the Bessel potentials
$(I-\Delta)^{-\gamma_{1}/2}$ have elegant representations in terms of Green
functions (as discussed in [9, Section 1.2.2]). In essence, this means that
there exists a smooth function $G_{\gamma_{1}}(x)$ defined on $\mathbb{R}^{d}$
except for the origin, such that

$\displaystyle(I-\Delta)^{-\gamma_{1}/2}g(x)=\int_{\mathbb{R}^{d}}G_{\gamma_{1}}(y)g(x-y)\mathrm{d}y.$

for any function $g$ in $L_{p}(\mathbb{R}^{d})$. For this particular case, it
is possible to understand realizations of all tempered distributions within
these weighted spaces solely through integrals. In other words, a tempered
distribution $f$ on $\mathbb{R}^{d}$ belongs to
$H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ if and only if there exists
a function $g$ in $L_{p}(\mathbb{R}^{d})$ such that

$\displaystyle
f(x)=(1+|x|^{2})^{-\gamma_{2}/2}\int_{\mathbb{R}^{d}}G_{\gamma_{1}}(y)g(x-y)\mathrm{d}y.$

Similarly, $f\in H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ if and
only if there exist a $g\in L_{p}(\mathbb{R}^{d})$ such that

$\displaystyle
f(x)=\int_{\mathbb{R}^{d}}G_{\gamma_{1}}(y)(1+|x-y|^{2})^{-\gamma_{2}/2}g(x-y)\mathrm{d}y.$

The function $G_{\gamma_{1}}(x)$ is given by

$\displaystyle
G_{\gamma_{1}}(x)=\frac{1}{\Gamma(\frac{\gamma_{1}}{2})}\int_{0}^{\infty}\mathrm{e}^{-t}\mathrm{e}^{-|x|^{2}/(4t)}t^{(\gamma_{1}-d)/2}\frac{\mathrm{d}t}{t},$

where $\Gamma$ denotes the Gamma function. Additionally, the behaviors of $G$
is characterized as follows ([9, Proposition 1.2.5]): For large values of
$|x|$, it satisfies

$\displaystyle G_{\gamma_{1}}(x)\lesssim\mathrm{e}^{-|x|^{2}/2}$

and for small values of $|x|$, the behavior of $G_{\gamma_{1}}(x)$ can be
described by the following cases according to the dimension $d$:

$\displaystyle
G_{\gamma_{1}}(x)\approx\begin{cases}&|x|^{\gamma_{1}-d}+1+O(|x|^{\gamma_{1}-d+2})\quad\text{if
$0<\gamma_{1}<d$}\\\ &\log\frac{2}{|x|}+1+O(|x|^{2})\quad\text{if
$\gamma_{1}=d$}\\\ &1+O(|x|^{\gamma_{1}-d})\quad\text{if
$\gamma_{1}>d$}.\end{cases}$

We establish a few straightforward embedding inequalities for these two
weighted spaces, beginning with the inner spaces. The first embedding
presented below is effective concerning the regularity exponents of inner
spaces.

######  Proposition 3.4.

Let $\gamma_{1},\gamma_{2},\tilde{\gamma}_{1}\in\mathbb{R}$, $p\in(1,\infty]$,
and $f\in H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$. Assume that
$\gamma_{1}\geq\tilde{\gamma}_{1}$. Then

$\displaystyle\left\|f\right\|_{H_{p,in}^{\tilde{\gamma}_{1},\gamma_{2}}(\mathbb{R}^{d})}\leq\left\|f\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

###### Proof.

Let $\varphi\in\mathcal{S}(\mathbb{R}^{d})$. Then by Hölder’s inequality and
an $L_{p}$-boundedness of the Bessel potential (_cf_. [9, Section 1.2.2]),

$\displaystyle\left|\left\langle f,\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle(I-\Delta)^{\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}f\right),(I-\Delta)^{-\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\varphi\right)\right\rangle\right|$
$\displaystyle\leq\left\|f\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}\left\|(I-\Delta)^{-\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\varphi\right)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}$
$\displaystyle=\left\|f\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}\left\|(I-\Delta)^{(\tilde{\gamma}_{1}-\gamma_{1})/2}\left(I-\Delta\right)^{-\tilde{\gamma}_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\varphi\right)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}$
$\displaystyle\leq\left\|f\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}\left\|\left(I-\Delta\right)^{-\tilde{\gamma}_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\varphi\right)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}.$

Thus considering

$\displaystyle\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}\left(I-\Delta\right)^{\tilde{\gamma}_{1}/2}\varphi$

instead of $\varphi$, we have

$\displaystyle\left|\left\langle\left(I-\Delta\right)^{\tilde{\gamma}_{1}/2}\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}f,\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle
f,\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}\left(I-\Delta\right)^{\tilde{\gamma}_{1}/2}\varphi\right\rangle\right|$
$\displaystyle\leq\left\|f\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}\left\|\varphi\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}.$

Taking the supremum over all $\varphi$ such that
$\|\varphi\|_{L_{p^{\prime}}(\mathbb{R}^{d})}=1$, we have

$\displaystyle\left\|f\right\|_{H_{p,in}^{\tilde{\gamma}_{1},\gamma_{2}}(\mathbb{R}^{d})}\leq\left\|f\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

∎

An analogous embedding inequality is valid for outer spaces. This embedding is
provided in relation to the weight exponents of outer spaces.

######  Proposition 3.5.

Let $\gamma_{1},\gamma_{2},\tilde{\gamma}_{2}\in\mathbb{R}$, $p\in(1,\infty]$,
and $f\in H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$. Assume that
$\gamma_{2}\geq\tilde{\gamma}_{2}$. Then

$\displaystyle\left\|f\right\|_{H_{p,out}^{\gamma_{1},\tilde{\gamma}_{2}}(\mathbb{R}^{d})}\leq\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

###### Proof.

Let $\varphi\in\mathcal{S}(\mathbb{R}^{d})$. Then

$\displaystyle\left|\left\langle f,\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}f,\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\varphi\right\rangle\right|$
$\displaystyle\leq\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}}\left\|\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\varphi\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}$
$\displaystyle\leq\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}}\left\|\left(1+|\cdot|^{2}\right)^{(\tilde{\gamma}_{2}-\gamma_{2})/2}\left(1+|\cdot|^{2}\right)^{-\tilde{\gamma}_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\varphi\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}$
$\displaystyle\leq\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}}\left\|\left(1+|\cdot|^{2}\right)^{-\tilde{\gamma}_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\varphi\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}.$

Thus

$\displaystyle\left|\left\langle\left(1+|\cdot|^{2}\right)^{\tilde{\gamma}_{2}/2}(I-\Delta)^{\gamma_{1}/2}f,\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle
f,(I-\Delta)^{\gamma_{1}/2}\left(1+|\cdot|^{2}\right)^{\tilde{\gamma}_{2}/2}\varphi\right\rangle\right|$
$\displaystyle\leq\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}}\left\|\varphi\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}.$

Finally, taking the supremum over all $\varphi$ such that
$\|\varphi\|_{L_{p^{\prime}}(\mathbb{R}^{d})}=1$, we obtain the result. The
proposition is proved. ∎

######  Remark 3.6.

The case $p=1$ is excluded in both Propositions 3.4 and 3.5 since the dual
space of $L_{\infty}(\mathbb{R}^{d})$ is strictly larger than
$L_{1}(\mathbb{R}^{d})$.

The two embedding inequalities mentioned above appear quite straightforward,
as evidenced by their proofs. However, we will now delve into a non-trivial
embedding that relies on a weighted multiplier. To begin, let’s revisit the
definition of Muckenhoupt’s weight. We say that a non-negative measurable
function $w$ on $\mathbb{R}^{d}$ belongs to $A_{p}$ with $p\in(1,\infty)$ if

$\displaystyle\sup\left(\frac{1}{|B|}\int_{B}w(x)\mathrm{d}x\right)\left(\frac{1}{|B|}\int_{B}w(x)^{-1/(p-1)}\mathrm{d}x\right)^{p-1}<\infty,$

where the supremum is taken over all Euclidean balls on $\mathbb{R}^{d}$ and
$|B|$ denotes the Lebesgue measure of $B$. It is well-known that the mapping
$\xi\in\mathbb{R}^{d}\mapsto|\xi|^{\gamma}$ is in $A_{p}$ if and only if
$\gamma\in(-d,d(p-1))$ (cf. [8, Example 7.1.7]). Thus it is easy to verify
that $(1+|\xi|^{2})^{\gamma/2}$ is in $A_{p}$ for all $\gamma\in(-d,d(p-1))$.

It is noteworthy that there is a scarcity of research papers dedicated to the
exploration of weighted Bessel potential spaces with Muckenhoupt’s weights,
despite these spaces representing natural extensions of weighted Sobolev
spaces. Furthermore, handling these spaces becomes relatively straightforward
by applying weighted multiplier theories in conjunction with classical Bessel
potential spaces.

For additional insights and properties of weighted Bessel potential spaces
with Muckenhoupt’s weights, we recommend referring to [1, Appendix] and [23,
Section 3]. The proposition we are about to present can be derived as a
specific case of the results cited above. Nonetheless, we provide a simple
proof to illustrate that it is an easily applicable result within the context
of a weighted multiplier theory.

######  Proposition 3.7.

Let $\gamma_{1},\tilde{\gamma}_{1}\in\mathbb{R}$, $p\in(1,\infty)$,
$\gamma_{2}\in\left(-\frac{d}{p},\frac{d(p-1)}{p}\right)$, and $f\in
H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$. Assume that
$\gamma_{1}\geq\tilde{\gamma}_{1}$. Then

$\displaystyle\left\|f\right\|_{H_{p,out}^{\tilde{\gamma}_{1},\gamma_{2}}(\mathbb{R}^{d})}\lesssim\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

###### Proof.

It is obvious that $(1+|\xi|^{2})^{(p\gamma_{2})/2}$ is in $A_{p}$ since
$-d<p\gamma_{2}<d(p-1)$. Moreover, it is also easy to show that
$(1+|\xi|^{2})^{(\tilde{\gamma}_{1}-\gamma_{1})/2}$ is a weighted
$L_{p}$-multiplier since $\gamma_{1}\geq\tilde{\gamma}_{1}$ (cf. [28]).
Therefore, we have

$\displaystyle\left\|f\right\|_{H_{p,out}^{\tilde{\gamma}_{1},\gamma_{2}}(\mathbb{R}^{d})}$
$\displaystyle=\left\|\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\tilde{\gamma}_{1}/2}f\right\|_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle=\left\|\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{(\tilde{\gamma}_{1}-\gamma_{1})/2}(I-\Delta)^{\gamma_{1}/2}f\right\|_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle=\left\|(I-\Delta)^{(\tilde{\gamma}_{1}-\gamma_{1})/2}(I-\Delta)^{\gamma_{1}/2}f(x)\right\|_{L_{p}(\mathbb{R}^{d},\left(1+|x|^{2}\right)^{(p\gamma_{2})/2}\mathrm{d}x)}$
$\displaystyle\lesssim\left\|(I-\Delta)^{\gamma_{1}/2}f(x)\right\|_{L_{p}(\mathbb{R}^{d},\left(1+|x|^{2}\right)^{(p\gamma_{2})/2}\mathrm{d}x)}$
$\displaystyle=\left\|\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}f\right\|_{L_{p}(\mathbb{R}^{d})}$
$\displaystyle=\left\|f\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

The proposition is proved. ∎

Subsequently, we highlight an evident observation that both Fourier and
inverse Fourier transforms operate similarly within these weighted spaces.
This observation will be employed extensively in the paper without repeated
explicit mention.

######  Proposition 3.8.

Let $\gamma_{1},\gamma_{2}\in\mathbb{R}$, $p\in[1,\infty]$, and $f$ be a
tempered distribution on $\mathbb{R}^{d}$.

  1. (i)

$\displaystyle\mathcal{F}[f]\in
H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})~{}\text{if and only
if}~{}\mathcal{F}^{-1}[f]\in
H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d}).$

Additionally,

$\displaystyle\left\|\mathcal{F}[f]\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}=\left\|\mathcal{F}^{-1}[f]\right\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

  2. (ii)

$\displaystyle\mathcal{F}[f]\in
H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})~{}\text{if and only
if}~{}\mathcal{F}^{-1}[f]\in
H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d}).$

Additionally,

$\displaystyle\left\|\mathcal{F}[f]\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}=\left\|\mathcal{F}^{-1}[f]\right\|_{H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

###### Proof.

It is trivial since

$\displaystyle\mathcal{F}[\varphi](\xi)=\mathcal{F}^{-1}[\varphi](-\xi).$

for all $\varphi\in\mathcal{S}(\mathbb{R}^{d})$ and $\xi\in\mathbb{R}^{d}$. ∎

We also investigate relations between inner and outer spaces using Riesz-
Thorin inequalities (3.1) and (3.2). The Fourier transform operates as a
bridge by interchanging regularity and weight exponents when we connect inner
and outer spaces. It is important to note that the range of $p$ naturally
becomes restrictive in accordance with the constraints imposed by Riesz-
Thorin’s theorem.

######  Proposition 3.9.

Let $p\in[1,2]$, $\gamma_{1},\gamma_{2}\in\mathbb{R}$, and $f$ be a tempered
distribution on $\mathbb{R}^{d}$.

  1. (i)

If $f\in H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$, then

$\displaystyle\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{p^{\prime},out}(\mathbb{R}^{d})}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(2-p)}{2p}}\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,in}(\mathbb{R}^{d})}.$

  2. (ii)

If $\mathcal{F}[f]\in H_{p,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$, then

$\displaystyle\|f\|_{H^{\gamma_{2},\gamma_{1}}_{p^{\prime},out}(\mathbb{R}^{d})}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(2-p)}{2p}}\|\mathcal{F}[f]\|_{H^{\gamma_{1},\gamma_{2}}_{p,in}(\mathbb{R}^{d})}.$

Here $p^{\prime}$ denotes the Hölder conjugate of $p$.

###### Proof.

First, we prove (i). Let $\varphi\in\mathcal{S}(\mathbb{R}^{d})$. By Hölder’s
inequality,

$\displaystyle\left|\left\langle\mathcal{F}[f],\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle(I-\Delta)^{\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}f\right),(I-\Delta)^{-\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\mathcal{F}^{-1}[\varphi]\right)\right\rangle\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{d}}(I-\Delta)^{\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}f\right)(x)(I-\Delta)^{-\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\mathcal{F}^{-1}[\varphi]\right)(x)\mathrm{d}x\right|$
$\displaystyle\leq\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,in}(\mathbb{R}^{d})}\left\|(I-\Delta)^{-\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\mathcal{F}^{-1}[\varphi]\right)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}.$

Additionally, since $p^{\prime}\in[2,\infty]$, we have

$\displaystyle\left\|(I-\Delta)^{-\gamma_{1}/2}\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}\mathcal{F}^{-1}[\varphi]\right)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d})}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(2-p)}{2p}}\left\|\left(1+|x|^{2}\right)^{-\gamma_{1}/2}\left(I-\Delta\right)^{-\gamma_{2}/2}\varphi\right\|_{L_{p}(\mathbb{R}^{d},\mathrm{d}x)}.$

Considering
$\left(I-\Delta\right)^{\gamma_{2}/2}\left(\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\varphi\right)(x)$
instead of $\varphi(x)$, we have

$\displaystyle\left|\left\langle\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\left(I-\Delta\right)^{\gamma_{2}/2}\mathcal{F}[f],\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle\mathcal{F}[f],\left(I-\Delta\right)^{\gamma_{2}/2}\left(\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\varphi\right)\right\rangle\right|$
$\displaystyle\leq\left(\frac{1}{2\pi}\right)^{\frac{d(2-p)}{2p}}\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,in}(\mathbb{R}^{d})}\|\varphi\|_{L_{p}(\mathbb{R}^{d})}.$

Therefore taking the supremum over all $\varphi\in L_{p}(\mathbb{R}^{d})$ so
that $\|\varphi\|_{L_{p}(\mathbb{R}^{d})}=1$, we have

$\displaystyle\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{p^{\prime},out}(\mathbb{R}^{d})}\leq\left(\frac{1}{2\pi}\right)^{\frac{d(2-p)}{2p}}\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,in}(\mathbb{R}^{d})}.$

Next, we prove (ii). However, it is an easy consequence of $(i)$ by taking
$\mathcal{F}[f]$ instead of $f$ due to Proposition 3.8(i) and the Fourier
inversion theorem. ∎

The Fourier and inverse Fourier transforms exhibit particularly favorable
behaviors as isometries within the $L_{2}(\mathbb{R}^{d})$-space. This
interesting characteristic can also be extended to our weighted Bessel
potential spaces. More specifically, when the exponent of integrability $p$
equals 2, the Fourier transform $\mathcal{F}$ acts as an isometry preserving
the norms between $H_{2,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$ and
$H_{2,out}^{\gamma_{2},\gamma_{1}}(\mathbb{R}^{d})$. To begin, we act the
Fourier transform on elements in
$H_{2,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$.

######  Proposition 3.10.

Let $\gamma_{1},\gamma_{2}\in\mathbb{R}$. Then

$\displaystyle f\in
H_{2,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})~{}\text{if and only
if}~{}\mathcal{F}[f]\in H_{2,out}^{\gamma_{2},\gamma_{1}}(\mathbb{R}^{d}).$

Additionally,

$\displaystyle\left\|f\right\|_{H_{2,in}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}=\left\|\mathcal{F}[f]\right\|_{H_{2,out}^{\gamma_{2},\gamma_{1}}(\mathbb{R}^{d})}.$

###### Proof.

In Proposition 3.9(i), it is already shown that

$\displaystyle\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})}\leq\|f\|_{H^{\gamma_{1},\gamma_{2}}_{2,in}(\mathbb{R}^{d})}\quad\forall
f\in H^{\gamma_{1},\gamma_{2}}_{2,in}(\mathbb{R}^{d}).$

Thus it suffices to show that

$\displaystyle\|f\|_{H^{\gamma_{1},\gamma_{2}}_{2,in}(\mathbb{R}^{d})}\leq\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})}$

for all $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that
$\mathcal{F}[f]\in H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})$. Let
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ so that $\mathcal{F}[f]\in
H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})$. Then

$\displaystyle\left|\left\langle f,\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle\left(\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}(I-\Delta)^{\gamma_{2}/2}\mathcal{F}[f]\right),\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{1}/2}(I-\Delta)^{-\gamma_{2}/2}\mathcal{F}[\varphi]\right)\right\rangle\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{d}}\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}(I-\Delta)^{\gamma_{2}/2}\mathcal{F}[f](x)\left(1+|\cdot|^{2}\right)^{-\gamma_{1}/2}(I-\Delta)^{-\gamma_{2}/2}\mathcal{F}[\varphi](x)\mathrm{d}x\right|$
$\displaystyle\leq\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})}\left\|\left(\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}[\varphi]\right)\right\|_{L_{2}(\mathbb{R}^{d})}$
$\displaystyle=\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})}\left\|(I-\Delta)^{-\gamma_{2}/2}\left(1+|\cdot|^{2}\right)^{-\gamma_{1}/2}\varphi\right\|_{L_{2}(\mathbb{R}^{d})},$

where Plancherel’s theorem is used in the last equality. Thus considering
$\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}\varphi$
instead of $\varphi$, we have

$\displaystyle\left|\left\langle(I-\Delta)^{\gamma_{1}/2}\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}f,\varphi\right\rangle\right|=\left|\left\langle
f,\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}\varphi\right\rangle\right|\leq\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})}\left\|\varphi\right\|_{L_{2}(\mathbb{R}^{d})},$

which implies

$\displaystyle\|f\|_{H^{\gamma_{1},\gamma_{2}}_{2,in}(\mathbb{R}^{d})}\leq\|\mathcal{F}[f]\|_{H^{\gamma_{2},\gamma_{1}}_{2,out}(\mathbb{R}^{d})}.$

The proposition is proved. ∎

Likewise, in the case where the integrability exponent $p$ is equal to 2, the
Fourier transform $\mathcal{F}$ operates from an outer weighted space to an
inner weighted space as an isometry.

######  Proposition 3.11.

Let $\gamma_{1},\gamma_{2}\in\mathbb{R}$. Then

$\displaystyle f\in
H_{2,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})~{}\text{if and only
if}~{}\mathcal{F}[f]\in H_{2,in}^{\gamma_{2},\gamma_{1}}(\mathbb{R}^{d}).$

Additionally,

$\displaystyle\left\|f\right\|_{H_{2,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}=\left\|\mathcal{F}[f]\right\|_{H_{2,in}^{\gamma_{2},\gamma_{1}}(\mathbb{R}^{d})}.$

###### Proof.

It is an easy consequence of Proposition 3.10 by taking $\mathcal{F}[f]$
instead of $f$ with the help of Proposition 3.8 and the Fourier inversion
theorem. ∎

Remarkable connections endure between the inner and outer spaces through the
Fourier transform even if the integrability exponent $p$ surpasses 2.

######  Proposition 3.12.

Let $p\in(2,\infty]$, $\gamma_{1},\gamma_{2}\in\mathbb{R}$, and
$\delta\in\left(\frac{d(p-2)}{2(p-1)},\infty\right)$.

  1. (i)

If $f\in H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$, then

$\displaystyle\mathcal{F}[f]\in
H_{2,in}^{\gamma_{2}-\delta,\gamma_{1}}(\mathbb{R}^{d}).$

Additionally,

$\displaystyle\|\mathcal{F}[f]\|_{H_{2,in}^{\gamma_{2}-\delta,\gamma_{1}}(\mathbb{R}^{d})}\leq\left\|\left(1+|\cdot|^{2}\right)^{-\delta/2}\right\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}\|f\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

  2. (ii)

If $\mathcal{F}[f]\in H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})$, then

$\displaystyle f\in H_{2,in}^{\gamma_{2}-\delta,\gamma_{1}}(\mathbb{R}^{d}).$

Additionally,

$\displaystyle\|f\|_{H_{2,in}^{\gamma_{2}-\delta,\gamma_{1}}(\mathbb{R}^{d})}\leq\left\|\left(1+|\cdot|^{2}\right)^{-\delta/2}\right\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}\|\mathcal{F}[f]\|_{H_{p,out}^{\gamma_{1},\gamma_{2}}(\mathbb{R}^{d})}.$

###### Proof.

It is sufficient to prove (i) since (ii) easily comes from (i) due to
Proposition 3.8 and the Fourier inversion theorem. Let $f\in
H^{\gamma_{1},\gamma_{2}}_{p,out}(\mathbb{R}^{d})$. Then by Hölder’s
inequality,

$\displaystyle\left|\langle\mathcal{F}[f],\varphi\rangle\right|$
$\displaystyle=\left|\left\langle
f,\mathcal{F}^{-1}[\varphi]\right\rangle\right|$
$\displaystyle=\left|\left\langle\left(1+|\cdot|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}f,\left(1+|\cdot|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}^{-1}[\varphi]\right\rangle\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{d}}\left(1+|x|^{2}\right)^{\gamma_{2}/2}(I-\Delta)^{\gamma_{1}/2}f(x)\left(1+|x|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}^{-1}[\varphi](x)\mathrm{d}x\right|$
$\displaystyle\leq\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,out}(\mathbb{R}^{d})}\left\|\left(1+|x|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}^{-1}[\varphi](x)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d},\mathrm{d}x)}.$

Recalling $p^{\prime}\in[1,2)$, we apply Hölder’s inequality again and obtain

$\displaystyle\left\|\left(1+|x|^{2}\right)^{-\gamma_{2}/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}^{-1}[\varphi](x)\right\|_{L_{p^{\prime}}(\mathbb{R}^{d},\mathrm{d}x)}$
$\displaystyle\leq\|(1+|\cdot|^{2})^{-\delta/2}\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}\left\|(1+|\cdot|^{2})^{(\delta-\gamma_{2})/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}^{-1}[\varphi]\right\|_{L_{2}(\mathbb{R}^{d})},$

where $2/0:=\infty$. Since $\delta>\frac{d(p-2)}{2(p-1)}$, we have

$\displaystyle\left\|\left(1+|\cdot|^{2}\right)^{-\delta/2}\right\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}<\infty.$

Thus applying the Plancherel theorem, we have

$\displaystyle\left|\left\langle\mathcal{F}[f],\varphi\right\rangle\right|\leq\left\|\left(1+|\cdot|^{2}\right)^{-\delta/2}\right\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,out}(\mathbb{R}^{d})}\left\|\mathcal{F}\left[(1+|\cdot|^{2})^{(\delta-\gamma_{2})/2}(I-\Delta)^{-\gamma_{1}/2}\mathcal{F}^{-1}[\varphi]\right]\right\|_{L_{2}(\mathbb{R}^{d})}.$

Finally, considering

$\displaystyle\mathcal{F}\left[(I-\Delta)^{\gamma_{1}/2}\left(1+|\cdot|^{2}\right)^{-(\delta-\gamma_{2})/2}\mathcal{F}^{-1}[\varphi]\right]$
$\displaystyle=\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\left(I-\Delta\right)^{-(\delta-\gamma_{2})/2}\varphi$

instead of $\varphi$, we have

$\displaystyle\left|\left\langle\left(I-\Delta\right)^{(\gamma_{2}-\delta)/2}\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\mathcal{F}[f],\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle\mathcal{F}[f],\left(1+|\cdot|^{2}\right)^{\gamma_{1}/2}\left(I-\Delta\right)^{-(\delta-\gamma_{2})/2}\varphi\right\rangle\right|$
$\displaystyle=\left|\left\langle\mathcal{F}[f],\mathcal{F}\left[(I-\Delta)^{\gamma_{1}/2}\left(1+|\cdot|^{2}\right)^{-(\delta-\gamma_{2})/2}\mathcal{F}^{-1}[\varphi]\right]\right\rangle\right|$
$\displaystyle\leq\left\|\left(1+|\cdot|^{2}\right)^{-\delta/2}\right\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,out}(\mathbb{R}^{d})}\left\|\varphi\right\|_{L_{2}(\mathbb{R}^{d})}.$

Finally, taking the supremum with respect to
$\varphi\in\mathcal{S}(\mathbb{R}^{d})$ so that
$\|\varphi\|_{L_{2}(\mathbb{R}^{d})}$, we obtain

$\displaystyle\|\mathcal{F}[f]\|_{H_{2,in}^{\gamma_{2}-\delta,\gamma_{1}}}\leq\left\|\left(1+|\cdot|^{2}\right)^{-\delta/2}\right\|_{L_{2/(2-p^{\prime})}(\mathbb{R}^{d})}\|f\|_{H^{\gamma_{1},\gamma_{2}}_{p,out}(\mathbb{R}^{d})}.$

The proposition is proved. ∎

We are now ready to establish spaces for handling data within our evolutionary
framework, utilizing our newly introduced weighted Bessel potential spaces as
a foundation.

We slightly abuse the notation in the following definition as follows : for
any $\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued (Borel) measurable function
$u$ on $(0,T)$, we put $u(t,\cdot)=u(t)$, i.e.

$\displaystyle\langle u(t,\cdot),\varphi\rangle:=\langle
u(t),\varphi\rangle\quad\forall\varphi\in\mathcal{S}(\mathbb{R}^{d}).$

This notation is useful especially when $u(t,\cdot)$ has a realization for
each $t\in(0,T)$ by putting $u(t,\xi)=u(t)(\xi)$.

######  Definition 3.13 (Weighted Bessel potential-valued spaces).

  1. Let $p,q\in[1,\infty]$, $\gamma_{1},\gamma_{2}\in\mathbb{R}$, and $w(t)$ be a non-negative measurable function on $(0,T)$.

  2. (i)

We define
$\mathbf{H}^{\gamma_{1},\gamma_{2}}_{p,q,in}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}t\right)$
as the class of all $\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued (Borel)
measurable functions $u$ on $(0,T)$ such that

$\displaystyle\|u\|_{\mathbf{H}^{\gamma_{1},\gamma_{2}}_{p,q,in}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}t\right)}$
$\displaystyle:=\|u\|_{L_{p,q}\left((0,T),w^{p}(t)\mathrm{d}t;H^{\gamma_{1},\gamma_{2}}_{q,in}(\mathbb{R}^{d})\right)}$
$\displaystyle:=\int_{0}^{T}\|u(t,\cdot)\|^{p}_{H^{\gamma_{1},\gamma_{2}}_{q,in}(\mathbb{R}^{d})}w^{p}(t)\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\left\|(I-\Delta)^{\gamma_{1}/2}\left((1+|\cdot|^{2})^{\gamma_{2}/2}u(t,\cdot)\right)\right\|^{p}_{L_{q}}w^{p}(t)\mathrm{d}t<\infty.$

  3. (ii)

We define
$\mathbf{H}^{\gamma_{1},\gamma_{2}}_{p,q,out}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}t\right)$
as the class of all $\mathcal{S}^{\prime}(\mathbb{R}^{d})$-valued (Borel)
measurable functions $u$ on $(0,T)$ such that

$\displaystyle\|u\|_{\mathbf{H}^{\gamma_{1},\gamma_{2}}_{p,q,out}\left((0,T)\times\mathbb{R}^{d},w^{p}(t)\mathrm{d}t\right)}$
$\displaystyle:=\|u\|_{L_{p,q}\left((0,T),w^{p}(t)\mathrm{d}t;H^{\gamma_{1},\gamma_{2}}_{q,out}(\mathbb{R}^{d})\right)}$