††2010 Mathematics Subject Classification. 42A75, 43A60, 47D99.
Key words and phrases. Besicovitch almost periodic functions in
${\mathbb{R}}^{n}$, Doss almost periodic functions in ${\mathbb{R}}^{n}$,
abstract Volterra integro-differential equations.
The author is partially supported by grant 451-03-68/2020/14/200156 of
Ministry of Science and Technological Development, Republic of Serbia.

# Multi-dimensional Besicovitch almost periodic type functions and
applications

M. Kostić  Faculty of Technical Sciences, University of Novi Sad, Trg D.
Obradovića 6, 21125 Novi Sad, Serbia<EMAIL_ADDRESS>
###### Abstract.

In this paper, we analyze multi-dimensional Besicovitch almost periodic type
functions. We clarify the main structural properties for the introduced
classes of Besicovitch almost periodic type functions, explore the notion of
Besicovitch-Doss almost periodicity in the multi-dimensional setting, and
provide some applications of our results to the abstract Volterra integro-
differential equations and the partial differential equations.

##  1\. Introduction and preliminaries

As is well known, the notion of almost periodicity was introduced by the
Danish mathematician H. Bohr around 1924-1926 and later generalized by many
other authors (see the research monographs [8], [20], [30], [33], [40], [41],
[50], [55] and [61] for further information concerning almost periodic
functions and their applications). Suppose that $(X,\|\cdot\|)$ is a complex
Banach space and $F:{\mathbb{R}}^{n}\rightarrow X$ is a continuous function
($n\in{\mathbb{N}}$). Then it is said that the function $F(\cdot)$ is almost
periodic if and only if for each $\epsilon>0$ there exists $l>0$ such that for
each ${\bf t}_{0}\in{\mathbb{R}}^{n}$ there exists ${\bf\tau}\in B({\bf
t}_{0},l)\equiv\\{{\bf t}\in{\mathbb{R}}^{n}:|{\bf t}-{\bf t}_{0}|\leq l\\}$
with

$\displaystyle\bigl{\|}F({\bf t}+{\bf\tau})-F({\bf
t})\bigr{\|}\leq\epsilon,\quad{\bf t}\in{\mathbb{R}}^{n};$

here, $|\cdot-\cdot|$ denotes the Euclidean distance in ${\mathbb{R}}^{n}$ and
$\tau$ is usually called an $\epsilon$-almost period of $F(\cdot)$. Any
trigonometric polynomial in ${\mathbb{R}}^{n}$ is almost periodic, and we know
that a continuous function $F(\cdot)$ is almost periodic if and only if there
exists a sequence of trigonometric polynomials in ${\mathbb{R}}^{n}$ which
converges uniformly to $F(\cdot).$

If the function $F:{\mathbb{R}}^{n}\rightarrow X$ is locally $p$-integrable,
where $1\leq p<\infty,$ then we say that $F(\cdot)$ is Stepanov-$p$-almost
periodic if and only if for every $\epsilon>0$ there exists $l>0$ such that
for each ${\bf t}_{0}\in{\mathbb{R}}^{n}$ there exists ${\bf\tau}\in B({\bf
t}_{0},l)\cap{\mathbb{R}}^{n}$ with

$\displaystyle\bigl{\|}F({\bf t}+{\bf\tau}+{\bf u})-F({\bf t}+{\bf
u})\bigr{\|}_{L^{p}([0,1]^{n}:X)}\leq\epsilon,\quad{\bf
t}\in{\mathbb{R}}^{n}.$

Further on, we say that a locally $p$-integrable function
$F:{\mathbb{R}}^{n}\rightarrow X$ is:

  * (i)

equi-Weyl-$p$-almost periodic if and only if, for every $\epsilon>0,$ there
exist two finite real numbers $l>0$ and $L>0$ such that for each ${\bf
t}_{0}\in{\mathbb{R}}^{n}$ there exists $\tau\in B({\bf
t}_{0},L)\cap{\mathbb{R}}^{n}$ with

$\displaystyle\sup_{{\bf
t}\in{\mathbb{R}}^{n}}\Biggl{[}l^{-\frac{n}{p}}\bigl{\|}F({\bf\tau}+\cdot)-F(\cdot)\bigr{\|}_{L^{p}({\bf
t}+l[0,1]^{n}:X)}\Biggr{]}<\epsilon.$

  * (ii)

Weyl-$p$-almost periodic if and only if, for every $\epsilon>0,$ there exists
a finite real number $L>0$ such that for each ${\bf t}_{0}\in{\mathbb{R}}^{n}$
there exists $\tau\in B({\bf t}_{0},L)\cap{\mathbb{R}}^{n}$ with

$\displaystyle\limsup_{l\rightarrow+\infty}\sup_{{\bf
t}\in{\mathbb{R}}^{n}}\Biggl{[}l^{-\frac{n}{p}}\bigl{\|}F({\bf\tau}+\cdot)-F(\cdot)\bigr{\|}_{L^{p}({\bf
t}+l[0,1]^{n}:X)}\Biggr{]}<\epsilon.$

It is well known that any Bohr almost periodic function is Stepanov-$p$-almost
periodic, as well as that any Stepanov-$p$-almost periodic function is equi-
Weyl-$p$-almost periodic. The most general class is the class of
Weyl-$p$-almost periodic functions which contains all others. Furthermore, we
know that any equi-Weyl-$p$-almost periodic function is Besicovitch-$p$-almost
periodic, and that there exists a Weyl-$p$-almost periodic function
$f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ which is not Besicovitch-$p$-almost
periodic ([41]).

The notion of Besicovitch-$p$-almost periodicity for a function
$F:{\mathbb{R}}^{n}\rightarrow X$ can be introduced in many equivalent ways;
traditionally, if $F\in L_{loc}^{p}({\mathbb{R}}^{n}:X),$ then we first define

$\|F\|_{{\mathcal{M}}^{p}}:=\limsup_{t\rightarrow+\infty}\Biggl{[}\frac{1}{(2t)^{n}}\int_{[-t,t]^{n}}\|F({\bf
s})\|^{p}\,d{\bf s}\Biggr{]}^{1/p}.$

It can be simply proved that $\|\cdot\|_{{\mathcal{M}}^{p}}$ is a seminorm on
the space ${\mathcal{M}}^{p}({\mathbb{R}}^{n}:X)$ consisting of those
$L_{loc}^{p}({\mathbb{R}}^{n}:X)$-functions $F(\cdot)$ for which
$\|F\|_{{\mathcal{M}}^{p}}<\infty.$ Denote
$K_{p}({\mathbb{R}}^{n}:X):=\\{f\in{\mathcal{M}}^{p}({\mathbb{R}}^{n}:X)\,;\,\|F\|_{{\mathcal{M}}^{p}}=0\\}$
and

$M_{p}({\mathbb{R}}^{n}:X):={\mathcal{M}}^{p}({\mathbb{R}}^{n}:X)/K_{p}({\mathbb{R}}^{n}:X).$

The seminorm $\|\cdot\|_{{\mathcal{M}}^{p}}$ on
${\mathcal{M}}^{p}({\mathbb{R}}^{n}:X)$ induces the norm $\|\cdot\|_{M^{p}}$
on $M^{p}({\mathbb{R}}^{n}:X)$ under which $M^{p}({\mathbb{R}}^{n}:X)$ is
complete; hence, $(M^{p}({\mathbb{R}}^{n}:X),\|\cdot\|_{M^{p}})$ is a Banach
space. It is said that a function $F\in L_{loc}^{p}({\mathbb{R}}^{n}:X)$ is
Besicovitch-$p$-almost periodic if and only if there exists a sequence of
trigonometric polynomials (almost periodic functions, equivalently) converging
to $F(\cdot)$ in $(M^{p}({\mathbb{R}}^{n}:X),\|\cdot\|_{M^{p}}).$ The vector
space consisting of all Besicovitch-$p$-almost periodic functions is denoted
by $B^{p}({\mathbb{R}}^{n}:X).$ Clearly, $B^{p}({\mathbb{R}}^{n}:X)$ is a
closed subspace of $M^{p}({\mathbb{R}}^{n}:X)$ and therefore a Banach space
itself. Concerning the Banach space $M^{p}({\mathbb{R}}^{n}:X)$, we would like
to recall that this space is not separable for any finite exponent $p\geq 1;$
see, e.g., [58, Theorem 18] which concerns the one-dimensional case.

For further information about Besicovitch almost periodic functions,
Besicovitch almost automorphic functions and their applications, we refer the
reader to [1, 2, 4, 5, 6, 7, 9, 11, 12], [13, 14, 15, 16, 29, 31, 32, 35],
[36, 49, 51, 53, 54, 56, 57, 58, 60] and references cited therein; we would
like to specially emphasize here the important research monograph [55] by A.
A. Pankov. The spatially Besicovitch almost periodic solutions for certain
classes of nonlinear second-order elliptic equations, single higher-order
hyperbolic equations and nonlinear Schrödinger equations have been
investigated in the fifth chapter of this monograph. For the basic source of
information about homogenization in algebras with mean value and generalized
Besicovitch spaces (the work of J. L. Woukeng and his coauthors), we refer the
reader to [41, Part II, Chapter 9, pp. 619–621].

On the other hand, in [17], A. Chávez et al. have analyzed various classes of
almost periodic type functions of the form $F:\Lambda\times X\rightarrow Y,$
where $(Y,\|\cdot\|_{Y})$ is a complex Banach space and
$\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n}$. This research has been
continued in [18] and [28], where we have analyzed the Stepanov classes and
the Weyl classes of multi-dimensional almost periodic functions
$F:\Lambda\times X\rightarrow Y.$ Here it worth noticing that the concept
Besicovitch-$p$-almost periodicity has not been well explored for the
functions of the form $F:\Lambda\rightarrow X,$ where
$\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n}$ and
$\Lambda\neq{\mathbb{R}}^{n}$ (some particular results in the one-dimensional
setting are given in the monograph [40], with $\Lambda=[0,\infty)$). This fact
has strongly influenced us to write this paper, in which we continue the
research studies [17, 18, 28, 39, 46] by investigating the multi-dimensional
Besicovitch almost periodic type functions $F:\Lambda\times X\rightarrow Y,$
where $(Y,\|\cdot\|_{Y})$ is a complex Banach space and
$\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n}.$ It is worth noting that this
is probably the first research article which examines the existence and
uniqueness of Besicovitch-$p$-almost periodic solutions for certain classes of
PDEs on some proper subdomains of ${\mathbb{R}}^{n};$ even the most simplest
examples of quasi-linear partial differential equations of first order
considered here vividly exhibit the necessity of further analyses of
Besicovitch-$p$-almost periodic functions which are not defined on the whole
Euclidean space ${\mathbb{R}}^{n}.$

The organization and main ideas of this paper can be briefly described as
follows. In Subsection 1.1, we recall the basic definitions and results from
the theory of Lebesgue spaces with variable exponents $L^{p(x)}$. The main aim
of Section 2 is to introduce and analyze various classes of multi-dimensional
Besicovitch almost periodic type functions. We start this section by
introducing the class $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y),$ where $\Lambda$ is a general non-empty
subset of ${\mathbb{R}}^{n},$ $p\in{\mathcal{P}}(\Lambda),$
$\phi:[0,\infty)\rightarrow[0,\infty)$ is Lebesgue measurable and ${\rm
F}:(0,\infty)\rightarrow(0,\infty);$ see Definition 2.1, which is crucial for
our further work. The class $PAP_{0,p}(\Lambda,{\mathcal{B}},{\rm F},\phi)$ of
weighted ergodic components, introduced recently in [41, Definition 6.4.13],
makes a proper subclass of the class $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$ since its definition is obtained by
plugging the trivial sequence $(P_{k}\equiv 0)$ of trigonometric polynomials
in Definition 2.1 (we omit the term “${\mathcal{B}}$” from the notation for
the functions of the form $F:\Lambda\rightarrow Y$). Before proceeding any
further, we would like to note that there is a large class of PDEs of first
order (second order) whose solutions are not Besicovitch almost periodic in
${\mathbb{R}}^{n}$ and which belong to the class
$PAP_{0,p}(\Lambda,{\mathcal{B}},{\rm F},\phi),$ where
$\emptyset\neq\Lambda\subsetneq{\mathbb{R}}^{n}.$ For instance, we have the
following:

######  Example 1.1.

  * (i)

Let $a$ and $c$ be two non-zero real numbers; then the classical
$C^{1}$-solution of the equation $au_{x}+cu=0$ is given by
$u(x,y)=g(y)e^{-(c/a)x},$ $(x,y)\in{\mathbb{R}}^{2}.$ Keeping in mind the
notion introduced after Definition 2.3, it can be simply shown that any non-
trivial solution of this equation cannot be Besicovitch almost periodic in
${\mathbb{R}}^{2}$ as well as that the solution always belongs to the class
$PAP_{0,1}([0,\infty)\times{\mathbb{R}},{\rm F},x)$ provided that the function
$g(\cdot)$ is Besicovitch bounded and $\lim_{t\rightarrow+\infty}{\rm F}(t)=0$
(see also Example 2.2 and Example 2.6 below).

  * (ii)

The solutions of second-order PDEs on rectangular domains, obtained by the
well known method of separation of variables, can belong to the space
$PAP_{0,p}(\Lambda,{\mathcal{B}},{\rm F},\phi)$. Consider, for the
illustration purposes, the heat equation $u_{t}(x,t)=u_{xx}(x,t)$ on the
domain $\Lambda=[0,2\pi]\times[0,\infty),$ equipped with the initial
conditions $u(x,0)=f(x)\in L^{1}[0,2\pi]$ and $u(0,t)=u(2\pi,t)=0.$ A unique
solution of this problem is given by

$u(x,t)=\sum_{k=1}^{\infty}b_{k}\sin(kx/2)e^{-\frac{k^{2}}{4}t},\quad(x,t)\in\Lambda,$

where $f(x)=\sum_{k=1}b_{k}\sin(kx/2),$ $x\in[0,2\pi].$ Since there exists a
finite real constant $M>0$ such that $|b_{k}|\leq M$ for all
$k\in{\mathbb{N}},$ a simple computation shows that $u(x,t)\in
PAP_{0,1}(\Lambda,t^{-\zeta},x)$ for any real number $\zeta>0.$

We provide some structural characterizations of class
$e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$ in
Proposition 2.4 and Theorem 2.5 and Proposition 2.7 (a special attention is
paid to the case in which $\phi(x)\equiv x^{\alpha}$ for some $\alpha>0$). A
composition principle for multi-dimensional Besicovitch almost periodic
functions is clarified in Theorem 2.10. In Subsection 2.1, we investigate the
notion of multi-dimensional Besicovitch normality. The class of
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal
functions is introduced in Definition 2.11 and characterized after that in
Proposition 2.12, Proposition 2.13 and Proposition 2.14.

In Section 3, we consider the multi-dimensional analogues of the important
research results established by R. Doss in [24]-[25]. We pay special attention
to the analysis of conditions (A), (A)∞, and (AS); see Theorem 3.1,
Proposition 3.3 and Proposition 3.5 for some results obtained in this
direction. Subsection 3.1 is focused on the analysis of condition (B); the
main results obtained in this part are Proposition 3.6 and Proposition 3.7. We
feel it is our duty to emphasize that the above-mentioned results of R. Doss
are primarily intended for the analysis of one-dimensional Besicovitch almost
periodic type functions as well as that we have faced ourselves with many
serious problems concerning the multi-dimensional extensions of these results.

Some applications to the abstract Volterra integro-differential equations are
furnished in Section 4; it is worth noting that we establish here some new
results about the convolution invariance of Besicovitch-$p$-almost periodicity
under the actions of infinite convolution products, and a new result
concerning the usually considered convolution invariance of
Besicovitch-$p$-almost periodicity. We provide some new applications in the
one-dimensional setting, a new application in the analysis of the existence
and uniqueness of Besicovitch-$p$-almost periodic solutions of the abstract
semilinear fractional Cauchy inclusions and the abstract nonautonomous
differential equations of first order, some new applications to the
inhomogeneous heat equation in ${\mathbb{R}}^{n}$ and evolution systems
generated by the bounded perturbations of the Dirichlet Laplacian.

The final section of paper is reserved for conclusions and final comments
about the considered notion. We explain here that the notion of
Besicovitch-$(p,c)$-almost periodicity, where
$c\in{\mathbb{C}}\setminus\\{0\\},$ cannot be so simply introduced and
analyzed. Concerning some open problems proposed, we would like to notice that
we introduce the notion of admissibility with respect to the class
${\mathcal{C}}_{\Lambda}$ of Besicovitch almost periodic type functions in
Definition 2.16, and propose after that an open problem concerning the
extensions of Besicovitch almost periodic type functions to the whole
Euclidean space ${\mathbb{R}}^{n}$; a similar question has recently been posed
for the class of (equi-)Weyl almost periodic type functions in [41]. In
addition to the above, we provide numerous illustrative examples and remarks
about the multi-dimensional Besicovitch almost periodic type functions under
our consideration.

We use the standard notation throughout the paper. We assume that
$(X,\|\cdot\|)$, $(Y,\|\cdot\|_{Y})$ and $(Z,\|\cdot\|_{Z})$ are complex
Banach spaces, $n\in{\mathbb{N}},$
$\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n},$ ${\mathcal{B}}$ is a non-
empty collection of non-empty subsets of $X,$ and ${\mathrm{R}}$ is a non-
empty collection of sequences in ${\mathbb{R}}^{n}.$ We assume that for each
$x\in X$ there exists $B\in{\mathcal{B}}$ such that $x\in B;$
$(e_{1},e_{2},...,e_{n})$ denotes the standard basis of ${\mathbb{R}}^{n}.$ By
$L(X,Y)$ we denote the Banach algebra of all bounded linear operators from $X$
into $Y;$ $L(X,X)\equiv L(X)$. If $A:D(A)\subseteq X\mapsto X$ is a closed
linear operator, then its range will be denoted by $R(A)$; ${\rm I}$ stands
for the identity operator on $Y.$ Set ${\mathbb{N}}_{n}:=\\{1,...,n\\}$ and
$\Delta_{n}:=\\{(t,t,...,t)\in{\mathbb{R}}^{n}:t\in{\mathbb{R}}\\}$. By
$C_{b}({\mathbb{R}}^{n}:X)$ we denote the Banach space of all bounded
continuous functions $F:{\mathbb{R}}^{n}\rightarrow X$ equipped with the sup-
norm. By a convex polyhedral in ${\mathbb{R}}^{n},$ we mean any set $\Lambda$
of the form

$\Lambda=\bigl{\\{}\alpha_{1}{\bf v}_{1}+\cdot\cdot\cdot+\alpha_{n}{\bf
v}_{n}:\alpha_{i}\geq 0\mbox{ for all }i\in{\mathbb{N}}_{n}\bigr{\\}},$

where $(v_{1},v_{2},...,v_{n})$ is a basis of ${\mathbb{R}}^{n}.$

For the sequel, we need the following result which can be deduced in almost
the same way as in the proof of [24, Proposition 2]:

######  Lemma 1.2.

Suppose that the function $F:{\mathbb{R}}^{n}\rightarrow X$ is Bohr almost
periodic. If for every $a_{1}\neq 0,...,a_{n}\neq 0$ we have
($a=(a_{1},...,a_{n})$):

$\lim_{k\rightarrow+\infty}\frac{1}{k}\sum_{j=0}^{k-1}F({\bf t}+(k-1)a)=0,$

uniformly in ${\bf t}\in{\mathbb{R}}^{n},$ then $F\equiv 0.$

Although it could be of some importance, we will not discuss here the question
whether the statement of Lemma 1.2 can be extended to the almost automorphic
functions (the uniformly recurrent functions).

We also need the following notion. Suppose that
$\emptyset\neq\Lambda^{\prime}\subseteq{\mathbb{R}}^{n},$
$\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n}$, $F:\Lambda\times X\rightarrow
Y$ is a continuous function and $\Lambda+\Lambda^{\prime}\subseteq\Lambda.$
Then we say that $F(\cdot;\cdot)$ is Bohr
$({\mathcal{B}},\Lambda^{\prime})$-almost periodic (Bohr
${\mathcal{B}}$-almost periodic, if $\Lambda^{\prime}=\Lambda$) if and only if
for every $\epsilon>0$ and $B\in{\mathcal{B}}$ there exists $l>0$ such that
for each ${\bf t}_{0}\in\Lambda^{\prime}$ there exists ${\bf\tau}\in B({\bf
t}_{0},l)\cap\Lambda^{\prime}$ with

$\displaystyle\bigl{\|}F({\bf t}+{\bf\tau};x)-F({\bf
t};x)\bigr{\|}_{Y}\leq\epsilon,\quad{\bf t}\in\Lambda,\ x\in B.$

Furthermore, we say that the function $F(\cdot;\cdot)$ is
$({\mathcal{B}},\Lambda^{\prime})$-uniformly recurrent
(${\mathcal{B}}$-uniformly recurrent, if $\Lambda^{\prime}=\Lambda$) if and
only if, for every $B\in{\mathcal{B}},$ there exists a sequence
$({\bf\tau}_{n})$ in $\Lambda^{\prime}$ such that
$\lim_{k\rightarrow+\infty}|{\bf\tau}_{k}|=+\infty$ and

$\lim_{k\rightarrow+\infty}\sup_{{\bf t}\in\Lambda,x\in B}\bigl{\|}F({\bf
t}+{\bf\tau}_{k};x)-F({\bf t};x)\bigr{\|}_{Y}=0.$

If we consider the functions of the form $F:\Lambda\rightarrow Y,$ then we
omit the term “${\mathcal{B}}$” from the notation.

For further information concerning multi-dimensional almost periodic type
functions, multi-dimensional almost automorphic type functions and their
applications, we refer the reader to our newly published research monograph
[41]. We have been forced to quote this monograph multiple times henceforth
since some recent results of ours concerning multi-dimensional almost periodic
type functions and their generalizations are still not accepted or published
in the final form.

###  1.1. Lebesgue spaces with variable exponents $L^{p(x)}$

Let $\emptyset\neq\Omega\subseteq{\mathbb{R}}^{n}$ be a nonempty Lebesgue
measurable subset and let $M(\Omega:X)$ denote the collection of all
measurable functions $f:\Omega\rightarrow X;$
$M(\Omega):=M(\Omega:{\mathbb{R}}).$ Further on, by ${\mathcal{P}}(\Omega)$ we
denote the vector space of all Lebesgue measurable functions
$p:\Omega\rightarrow[1,\infty].$ For any $p\in{\mathcal{P}}(\Omega)$ and $f\in
M(\Omega:X),$ we set

$\varphi_{p(x)}(t):=\left\\{\begin{array}[]{l}t^{p(x)},\quad t\geq 0,\ \ 1\leq
p(x)<\infty,\\\ \\\ 0,\quad 0\leq t\leq 1,\ \ p(x)=\infty,\\\ \\\ \infty,\quad
t>1,\ \ p(x)=\infty\end{array}\right.$

and

$\rho(f):=\int_{\Omega}\varphi_{p(x)}(\|f(x)\|)\,dx.$

We define the Lebesgue space $L^{p(x)}(\Omega:X)$ with variable exponent by

$L^{p(x)}(\Omega:X):=\Bigl{\\{}f\in M(\Omega:X):\lim_{\lambda\rightarrow
0+}\rho(\lambda f)=0\Bigr{\\}}.$

It is well known that

$\displaystyle L^{p(x)}(\Omega:X)=\Bigl{\\{}f\in M(\Omega:X):\mbox{ there
exists }\lambda>0\mbox{ such that }\rho(\lambda f)<\infty\Bigr{\\}};$

see, e.g., [21, p. 73]. For every $u\in L^{p(x)}(\Omega:X),$ we introduce the
Luxemburg norm of $u(\cdot)$ by

$\|u\|_{p(x)}:=\|u\|_{L^{p(x)}(\Omega:X)}:=\inf\Bigl{\\{}\lambda>0:\rho(u/\lambda)\leq
1\Bigr{\\}}.$

Equipped with this norm, $L^{p(x)}(\Omega:X)$ becomes a Banach space (see e.g.
[21, Theorem 3.2.7] for the scalar-valued case), coinciding with the usual
Lebesgue space $L^{p}(\Omega:X)$ in the case that $p(x)=p\geq 1$ is a constant
function. Further on, for any $p\in M(\Omega),$ we define

$p^{-}:=\text{essinf}_{x\in\Omega}p(x)\ \ \mbox{ and }\ \
p^{+}:=\text{esssup}_{x\in\Omega}p(x).$

Set

$D_{+}(\Omega):=\bigl{\\{}p\in M(\Omega):1\leq p^{-}\leq p(x)\leq
p^{+}<\infty\mbox{ for a.e. }x\in\Omega\bigr{\\}}.$

If $p\in D_{+}(\Omega),$ then we know

$L^{p(x)}(\Omega:X)=\Bigl{\\{}f\in M(\Omega:X)\,;\,\mbox{ for all
}\lambda>0\mbox{ we have }\rho(\lambda f)<\infty\Bigr{\\}}.$

We will use the following lemma (cf. [21] for the scalar-valued case):

######  Lemma 1.3.

  * (i)

(The Hölder inequality) Let $p,\ q,\ r\in{\mathcal{P}}(\Omega)$ such that

$\frac{1}{q(x)}=\frac{1}{p(x)}+\frac{1}{r(x)},\quad x\in\Omega.$

Then, for every $u\in L^{p(x)}(\Omega:X)$ and $v\in L^{r(x)}(\Omega),$ we have
$uv\in L^{q(x)}(\Omega:X)$ and

$\displaystyle\|uv\|_{q(x)}\leq 2\|u\|_{p(x)}\|v\|_{r(x)}.$

  * (ii)

Let $\Omega$ be of a finite Lebesgue’s measure and let $p,\
q\in{\mathcal{P}}(\Omega)$ such $q\leq p$ a.e. on $\Omega.$ Then
$L^{p(x)}(\Omega:X)$ is continuously embedded in $L^{q(x)}(\Omega:X),$ and the
constant of embedding is less than or equal to $2(1+m(\Omega)).$

  * (iii)

Let $f\in L^{p(x)}(\Omega:X),$ $g\in M(\Omega:X)$ and $0\leq\|g\|\leq\|f\|$
a.e. on $\Omega.$ Then $g\in L^{p(x)}(\Omega:X)$ and
$\|g\|_{p(x)}\leq\|f\|_{p(x)}.$

For further information concerning the Lebesgue spaces with variable exponents
$L^{p(x)},$ we refer the reader to the monograph [21] by L. Diening et al.,
and the list of references quoted therein.

##  2\. Multi-dimensional Besicovitch almost periodic type functions

Suppose that $\Lambda$ is a general non-empty subset of ${\mathbb{R}}^{n}$ as
well as that $p\in{\mathcal{P}}(\Lambda)$, the function
$\phi:[0,\infty)\rightarrow[0,\infty)$ is Lebesgue measurable and ${\rm
F}:(0,\infty)\rightarrow(0,\infty).$ Set

$\Lambda^{\prime\prime}:=\bigl{\\{}\tau\in{\mathbb{R}}^{n}:\tau+\Lambda\subseteq\Lambda\bigr{\\}}.$

Unless stated otherwise, we assume henceforth that
$\emptyset\neq\Omega\subseteq{\mathbb{R}}^{n}$ is a compact set with positive
Lebesgue measure, as well as that $\Lambda+l\Omega\subseteq\Lambda$ for all
$l>0,$ and $\emptyset\neq\Lambda^{\prime}\subseteq\Lambda^{\prime\prime},$
i.e., $\Lambda+\Lambda^{\prime}\subseteq\Lambda.$ In this section, we
investigate the multi-dimensional Besicovitch almost periodic type functions,
paying a special attention to the class $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$ and the class of
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal
functions.

Recall, a trigonometric polynomial $P:\Lambda\times X\rightarrow Y$ is any
linear combination of functions like $({\bf t};x)\mapsto
e^{i\langle\lambda,{\bf t}\rangle}c(x),$ where $c:X\rightarrow Y$ is a
continuous function; a continuous function $F:\Lambda\times X\rightarrow Y$ is
said to be strongly ${\mathcal{B}}$-almost periodic if and only if for every
$B\in{\mathcal{B}}$ we can find a sequence
$(P_{k}^{B}(\cdot;\cdot))_{k\in{\mathbb{N}}}$ of trigonometric polynomials
which converges to $F(\cdot;\cdot)$, uniformly on $\Lambda\times B.$ We omit
the term “${\mathcal{B}}$” from the notation if $X=\\{0\\}.$

We are ready to introduce the following notion:

######  Definition 2.1.

Suppose that $F:\Lambda\times X\rightarrow Y,$
$\phi:[0,\infty)\rightarrow[0,\infty)$ and ${\rm
F}:(0,\infty)\rightarrow(0,\infty).$ Then we say that the function
$F(\cdot;\cdot)$ belongs to the class $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$ if and only if for each set
$B\in{\mathcal{B}}$ there exists a sequence $(P_{k}(\cdot;\cdot))$ of
trigonometric polynomials such that

(2.1)
$\displaystyle\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}{\rm
F}(t)\sup_{x\in B}\Bigl{[}\phi\Bigl{(}\bigl{\|}F({\bf t};x)-P_{k}({\bf
t};x)\bigr{\|}_{Y}\Bigr{)}\Bigr{]}_{L^{p({\bf t})}(\Lambda_{t})}=0.$

If $\phi(x)\equiv x,$ then we omit the term “$\phi$” from the notation; if
$X=\\{0\\},$ then we omit the term “${\mathcal{B}}$” from the notation.

Immediately from definition, it follows that, for every $F\in
e-({\mathcal{B}},{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$ and
$\lambda\in{\mathbb{R}}^{n},$ we have $e^{i\langle\lambda,\cdot\rangle}F\in
e-({\mathcal{B}},{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y).$ The Weyl class
$e-{\mathcal{B}}-W^{p}_{\Omega}(\Lambda\times X:Y),$ introduced in [41,
Definition 6.3.18] with $p(\cdot)\equiv p\in[1,\infty)$ and ${\rm F}(t)\equiv
t^{-n/p},$ makes a subclass of the class $e-({\mathcal{B}},{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y),$ provided some reasonable choices of
compact set $\Omega;$ for example, we have that
$e-{\mathcal{B}}-W^{p}_{\Omega}(\Lambda\times X:Y)\subseteq
e-({\mathcal{B}},{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$ if
$\Omega=[0,1]^{n},$ and $\Lambda=[0,\infty)^{n}$ or
$\Lambda={\mathbb{R}}^{n}.$

Moreover, we have the following:

  * (i)

Equipped with the usual operations, the set $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$ forms a vector space provided that the
function $\phi(\cdot)$ is monotonically increasing and there exists a finite
real constant $c>0$ such that $\phi(x+y)\leq c[\phi(x)+\phi(y)]$ for all $x,\
y\geq 0.$

  * (ii)

For every $\tau\in\Lambda^{\prime\prime},$ $x_{0}\in X$ and $F\in
e-({\mathcal{B}},{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$, we have
$F(\cdot+\tau;\cdot+x_{0})\in e-({\mathcal{B}}_{x_{0}},{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$ with
${\mathcal{B}}_{x_{0}}\equiv\\{-x_{0}+B:B\in{\mathcal{B}}\\},$ provided that
$p(\cdot)\equiv p\in[1,\infty)$ and there exist two finite real constants
$c_{\tau}>0$ and $t_{\tau}>0$ such that ${\rm F}(t)\leq c_{\tau}{\rm
F}(t+|\tau|),$ $t\geq t_{\tau}.$

  * (iii)

Suppose that the function $\phi(\cdot)$ is monotonically increasing,
continuous at the point zero, there exists a finite real constant $c>0$ such
that $\phi(x+y)\leq c[\phi(x)+\phi(y)]$ for all $x,\ y\geq 0,$ and the mapping
$t\mapsto{\rm F}(t)[1]_{L^{p(\cdot)}(\Lambda_{t})},$ $t>0$ is bounded at plus
infinity. Then $F\in e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times
X:Y)$ if and only if for each set $B\in{\mathcal{B}}$ there exists a sequence
$(F_{k}(\cdot;\cdot))$ of strongly ${\mathcal{B}}$-almost periodic functions
such that (2.1) holds with the polynomial $P_{k}(\cdot;\cdot)$ replaced
therein with the function $F_{k}(\cdot;\cdot).$

  * (iv)

Let the assumptions of (iii) hold and let there exist a function
$\varphi:[0,\infty)\rightarrow[0,\infty)$ such that
$\phi(xy)\leq\varphi(x)\phi(y)$ for all $x,\ y\geq 0.$ Suppose that
$h:Y\rightarrow Z$ is a Lipschitz continuous function and $F\in e-(\phi,{\rm
F})-B^{p(\cdot)}(\Lambda:Y).$ Using [41, Proposition 6.1.11] and the fact that
any uniformly continuous Bohr almost periodic function $F:\Lambda\rightarrow
Y,$ where $\Lambda$ is a convex polyhedral in ${\mathbb{R}}^{n},$ is strongly
almost periodic, we can prove that $h\circ F\in e-(\phi,{\rm
F})-B^{p(\cdot)}(\Lambda:Z).$

Concerning the notion introduced in Definition 2.1, it is clear that the use
of constant coefficients $p(\cdot)\equiv p\in[1,\infty)$ is unquestionably the
best. On the other hand, in the introductory part of [41], we have emphasized
that, from the theoretical point of view, the use of constant coefficients is
not adequately enough because many structural results from the theory of
generalized almost periodic functions can be further extended using some
results from the theory of Lebesgue spaces with variable exponents $L^{p(x)}.$
Further on, it is clear that Definition 2.1 covers some cases that can be
freely called patological; for example, case in which $p\notin D_{+}(\Lambda)$
can be considered:

######  Example 2.2.

Suppose that $\Lambda=[0,\infty),$ and $F:\Lambda\rightarrow{\mathbb{R}}$ is
given by $F(t):=1,$ if there exists $j\in{\mathbb{N}}\setminus\\{1\\}$ such
that $t\in[j^{2}-1,j^{2}],$ and $F(t):=0,$ otherwise. Let $p(x)\equiv
1+x^{2},$ $\phi(x)\equiv x$ and

$\limsup_{t\rightarrow+\infty}{\rm
F}(t)\cdot\inf\Biggl{\\{}\lambda>0:\sum_{2\leq
j\leq\sqrt{t}}\lambda^{-j^{4}+2j^{2}-2}\leq 1\Biggr{\\}}=0.$

Then a simple computation with the Luxemburg norm shows that (2.1) is
satisfied with the trivial sequence $(P_{k}\equiv 0)$ of trigonometric
polynomials, so that $F\in PAP_{0,p}(\Lambda,{\rm F},\phi)\subseteq
e-(\phi,{\rm F})-B^{p(\cdot)}(\Lambda:{\mathbb{C}}).$

Let $\rho$ be a binary relation on $Y.$ For the sequel, we need the following
notion:

######  Definition 2.3.

(see [46, Definition 1])

  * (i)

Suppose that the function $F:\Lambda\times X\rightarrow Y$ satisfies that
$\phi(\|F(\cdot;x)\|_{Y})\in L^{p(\cdot)}(\Lambda_{t})$ for all $t>0$ and
$x\in X.$ Then we say that the function $F(\cdot;\cdot)$ is
Besicovitch-$(p,\phi,{\rm F},{\mathcal{B}})$-bounded if and only if, for every
$B\in{\mathcal{B}},$ there exists a finite real number $M_{B}>0$ such that

$\displaystyle\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|F(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}\leq
M_{B}.$

  * (ii)

Suppose that the function $F:\Lambda\times X\rightarrow Y$ satisfies that
$\phi(\|F(\cdot+\tau;x)-y_{\cdot;x}\|_{Y})\in L^{p(\cdot)}(\Lambda_{t})$ for
all $t>0,$ $x\in X,$ $\tau\in\Lambda^{\prime}$ and
$y_{\cdot;x}\in\rho(F(\cdot;x)).$

    * (a)

We say that the function $F:\Lambda\times X\rightarrow Y$ is
Besicovitch-$(p,\phi,{\rm F},{\mathcal{B}},\Lambda^{\prime},\rho)$-continuous
if and only if, for every $B\in{\mathcal{B}}$ as well as for every $t>0,$
$x\in B$ and $\cdot\in\Lambda_{t},$ we have the existence of an element
$y_{\cdot;x}\in\rho(F(\cdot;x))$ such that

$\displaystyle\lim_{\tau\rightarrow
0,\tau\in\Lambda^{\prime}}\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|F(\cdot+\tau;x)-y_{\cdot;x}\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}=0.$

    * (b)

We say that the function $F(\cdot;\cdot)$ is Doss-$(p,\phi,{\rm
F},{\mathcal{B}},\Lambda^{\prime},\rho)$-almost periodic if and only if, for
every $B\in{\mathcal{B}}$ and $\epsilon>0,$ there exists $l>0$ such that for
each ${\bf t}_{0}\in\Lambda^{\prime}$ there exists a point $\tau\in B({\bf
t}_{0},l)\cap\Lambda^{\prime}$ such that, for every $t>0,$ $x\in B$ and
$\cdot\in\Lambda_{t},$ we have the existence of an element
$y_{\cdot;x}\in\rho(F(\cdot;x))$ such that

$\displaystyle\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|F(\cdot+\tau;x)-y_{\cdot;x}\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}<\epsilon.$

    * (c)

We say that the function $F(\cdot;\cdot)$ is Doss-$(p,\phi,{\rm
F},{\mathcal{B}},\Lambda^{\prime},\rho)$-uniformly recurrent if and only if,
for every $B\in{\mathcal{B}},$ there exists a sequence
$(\tau_{k})\in\Lambda^{\prime}$ such that, for every $t>0,$ $x\in B$ and
$\cdot\in\Lambda_{t},$ we have the existence of an element
$y_{\cdot;x}\in\rho(F(\cdot;x))$ such that

$\displaystyle\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}{\rm
F}(t)\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|F(\cdot+\tau_{k};x)-y_{\cdot;x}\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}=0.$

As before, we omit the term “${\mathcal{B}}$” if $X=\\{0\\},$ the term
“$\Lambda^{\prime}$” if $\Lambda^{\prime}=\Lambda,$ and the term “$\rho$” if
$\rho={\mathrm{I}}.$ The usual notion of Besicovitch-$p$-almost periodicity
(Doss-$p$-almost periodicity) for the function $F:\Lambda\rightarrow Y,$ where
$1\leq p<+\infty,$ is obtained by plugging $\phi(x)\equiv x$ and ${\rm
F}(t)\equiv t^{-n/p}$ ($\phi(x)\equiv x$, ${\rm F}(t)\equiv t^{-n/p}$ and
$\Lambda^{\prime}=\Lambda,$ $\rho={\mathrm{I}}$). Further on, we say that a
function $F:\Lambda\rightarrow Y$ is Besicovitch almost periodic (Doss almost
periodic) if and only if $F(\cdot)$ is Besicovitch-$1$-almost periodic
(Doss-$1$-almost periodic). Let us recall that, in the usual setting, a Doss
almost periodic function $f:{\mathbb{R}}\rightarrow{\mathbb{C}}$ is not
generally Besicovitch almost periodic; for example, A. N. Dabboucy and H. W.
Davies have constructed an example of such a function which has the mean value
equal to zero (cf. [19, pp. 352-354] for more details).

Now we will state and prove the following result:

######  Proposition 2.4.

Suppose that ${\mathcal{B}}$ consists of bounded subsets of $X$,
$F:\Lambda\times X\rightarrow Y$ and, for every fixed element $x\in X,$ the
function $F(\cdot;x)$ is Lebesgue measurable.

  * (i)

Suppose that the function $\phi(\cdot)$ is monotonically increasing. If there
exists a finite real constant $t_{0}>0$ such that ${\rm
F}(t)\leq[\|1\|_{L^{p(\cdot)}(\Lambda_{t})}]^{-1},$ $t\geq t_{0},$ then any
function $F\in e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$
is Besicovitch-$(p,\phi,{\rm F},{\mathcal{B}})$-bounded.

  * (ii)

Suppose that $\phi(\cdot)$ is monotonically increasing and continuous at the
point $t=0$. Let $p(\cdot)\equiv p\in[1,\infty)$, and let there exist finite
real constants $c>0$ and $t_{0}>0$ such that, for every $t\geq t_{0},$ we have
${\rm F}(t+1)\geq c{\rm F}(t)$ and ${\rm F}(t)\leq[m(\Lambda_{t})]^{-(1/p)}.$
Then any function $F\in e-({\mathcal{B}},\phi,{\rm F})-B^{p}(\Lambda\times
X:Y)$ is Besicovitch-$(p,\phi,{\rm F},{\mathcal{B}},\Lambda^{\prime},{\rm
I})$-continuous for any set $\Lambda^{\prime}\subseteq\Lambda^{\prime\prime}.$

  * (iii)

Suppose that $p(\cdot)\equiv p\in[1,\infty),$ $\phi(\cdot)$ has the same
properties as in _(ii)_ , as well as that for every real number $a>0$ there
exist finite real constants $c_{a}>0$ and $t_{a}>0$ such that, for every
$t\geq t_{a},$ we have ${\rm F}(t+a)\geq c_{a}{\rm F}(t)$ and ${\rm
F}(t)\leq[m(\Lambda_{t})]^{-(1/p)}.$ Let $F\in e-({\mathcal{B}},\phi,{\rm
F})-B^{p}(\Lambda\times X:Y).$ Then the following holds:

    * (a)

The function $F(\cdot;\cdot)$ is Doss-$(p,\phi,{\rm
F},{\mathcal{B}},\Lambda,{\rm I})$-almost periodic, provided that
$\Lambda+\Lambda\subseteq\Lambda$ and, for every points
$(t_{1},...,t_{n})\in\Lambda$ and $(\tau_{1},...,\tau_{n})\in\Lambda,$ the
points $(t_{1},t_{2}+\tau_{2},...,t_{n}+\tau_{n}),$
$(t_{1},t_{2},t_{3}+\tau_{3},...,t_{n}+\tau_{n}),...,$
$(t_{1},t_{2},...,t_{n-1},t_{n}+\tau_{n}),$ also belong to $\Lambda.$

    * (b)

The function $F(\cdot;\cdot)$ is Doss-$(p,\phi,{\rm
F},{\mathcal{B}},\Lambda\cap\Delta_{n},{\rm I})$-almost periodic, provided
that $\Lambda\cap\Delta_{n}\neq\emptyset,$
$\Lambda+(\Lambda\cap\Delta_{n})\subseteq\Lambda$ and that, for every points
$(t_{1},...,t_{n})\in\Lambda$ and $(\tau,...,\tau)\in\Lambda\cap\Delta_{n},$
the points $(t_{1},t_{2}+\tau,...,t_{n}+\tau),$
$(t_{1},t_{2},t_{3}+\tau,...,t_{n}+\tau),...,$
$(t_{1},t_{2},...,t_{n-1},t_{n}+\tau),$ also belong to
$\Lambda\cap\Delta_{n}.$

###### Proof.

In order to prove (i), fix a set $B\in{\mathcal{B}}.$ Then $B$ is bounded and
we have the existence of a trigonometric polynomial $P_{k}(\cdot;\cdot)$ such
that

$\displaystyle\sup_{x\in B}$
$\displaystyle\Bigl{[}\phi\bigl{(}\|F(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}$
$\displaystyle\leq\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|F(\cdot;x)-P_{k}(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}+\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|P_{k}(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}$
$\displaystyle\leq(\epsilon/2{\rm F}(t))+\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|P_{k}(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}.$

Let
$P_{k}(\cdot;x)=\sum_{j=0}^{l}e^{i\langle\lambda_{l},\cdot\rangle}c_{l}(x)$
for some integer $l\in{\mathbb{N}},$ points $\lambda_{1},...,\lambda_{l}$ from
${\mathbb{R}}^{n}$ and continuous functions $c_{1}(\cdot),...,c_{l}(\cdot)$
from $X$ into $Y.$ Then we have the existence of a finite real constant
$c_{B}>0$ such that (see also Lemma 1.3(ii)):

$\displaystyle\sup_{x\in B}$
$\displaystyle\Bigl{[}\phi\bigl{(}\|P_{k}(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}\leq\sup_{x\in
B}\Biggl{[}\phi\Biggl{(}\Biggl{\|}\sum_{j=0}^{l}e^{i\langle\lambda_{l},\cdot\rangle}c_{l}(x)\Biggr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p(\cdot)}(\Lambda_{t})}$
$\displaystyle\leq\sup_{x\in
B}\Biggl{[}\phi\Biggl{(}\sum_{j=0}^{l}\bigl{\|}c_{l}(x)\bigr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p(\cdot)}(\Lambda_{t})}\leq\Bigl{[}\phi\bigl{(}c_{B}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}\leq\phi\bigl{(}c_{B}\bigr{)}{\rm
F}(t)^{-1},\quad t\geq t_{0}.$

The proof of (ii) is quite similar and follows from the decomposition:

$\displaystyle\sup_{x\in B}$
$\displaystyle\Bigl{[}\phi\bigl{(}\|F(\cdot+\tau;x)-F(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}$
$\displaystyle\leq\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|F(\cdot+\tau;x)-P_{k}(\cdot+\tau;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}$
$\displaystyle+\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|P_{k}(\cdot+\tau;x)-P_{k}(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}$
$\displaystyle+\sup_{x\in
B}\Bigl{[}\phi\bigl{(}\|P_{k}(\cdot;x)-F(\cdot;x)\|_{Y}\bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})};$

let us only note that we need the continuity of $\phi(\cdot)$ at the point
$t=0$ because, in the final steps of computation, we get a term of form
$\phi(c_{B}\sum_{j=0}^{l}|e^{i\langle\lambda_{j},\tau\rangle}-1|),$ which
tends to zero as $\tau\rightarrow 0+.$ The proof of part (a) in (iii) follows
from a relatively simple argumentation involving the decomposition used for
proving (ii), the given assumptions and the fact that the trigonometric
polynomial $P_{k}(\cdot;\cdot)$ is Bohr ${\mathcal{B}}$-almost periodic due to
[41, Proposition 6.1.25(iv)]; the proof of part (b) in (iii) is quite similar
because the prescribed assumptions imply that the trigonometric polynomial
$P_{k}(\cdot;\cdot)$ is Bohr $({\mathcal{B}},\Lambda\cap\Delta_{n})$-almost
periodic due to [41, Proposition 6.1.25(v)] (cf. also [41, Definition 6.1.9,
Definition 6.1.14] for the notion). ∎

Suppose that ${\mathcal{B}}$ consists of bounded subsets of $X$, $\Lambda$ is
unbounded, $F,\ G\in e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times
X:Y)$, for every fixed element $x\in X,$ the function $\phi(\cdot)$ is
monotonically increasing, $\phi(x+y)\leq\phi(x)+\phi(y)$ for all $x,\ y\geq
0,$ $\limsup_{t\rightarrow+\infty}{\rm F}(t)=0,$ and there exists a finite
real constant $t_{0}>0$ such that ${\rm
F}(t)\leq[\|1\|_{L^{p(\cdot)}(\Lambda_{t})}]^{-1},$ $t\geq t_{0}.$ Due to
Proposition 2.4(i), we have that the function $F(\cdot;\cdot)$ is
Besicovitch-$(p,\phi,{\rm F},{\mathcal{B}})$-bounded. Let a set
$B\in{\mathcal{B}}$ be fixed. Then

$d_{B}(F,G):=\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Bigl{[}\phi\Bigl{(}\bigl{\|}F({\cdot};x)-G({\cdot};x)\bigr{\|}_{Y}\Bigr{)}\Bigr{]}_{L^{p(\cdot)}(\Lambda_{t})}$

defines a pseudometric on the set $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y).$ Using the idea from the original proof
of J. Marcinkiewicz [53], we can prove the following theorem (see also [50,
pp. 249–252]):

######  Theorem 2.5.

Let the requirements stated in the previous paragraph hold. Then, for every
set $B\in{\mathcal{B}}$, the pseudometric space $(e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y),d_{B})$ is complete.

The classes with $\phi(x)\equiv x^{\alpha}$, $\alpha>0.$ Without any doubt,
the most important case in Definition 2.1 and Definition 2.3 is that one in
which we have $\phi(x)\equiv x^{\alpha}$ for some real number $\alpha>0.$ If
so, then all requirements necessary for applying Proposition 2.4 and the
statements stated preceding it hold. The assumptions of Theorem 2.5 hold in
case $\alpha\in(0,1],$ when we can provide some proper generalizations of the
usual notion of Besicovitch-$P$-almost periodicity. For example, if $1\leq
P<+\infty,$ $1\leq p<+\infty,$ $\alpha p\in(0,1)$ and the function
$F:{\mathbb{R}}\rightarrow Y$ is Besicovitch-$P$-almost periodic, then the
Hölder inequality implies that, for every trigonometric polynomial $P(\cdot)$,
we have:

$\displaystyle\Biggl{(}\frac{1}{2t}\int^{t}_{-t}\bigl{\|}F(t)-P(t)\bigr{\|}_{Y}^{\alpha
p}\,dt\Biggr{)}^{1/p}\leq\Biggl{(}\frac{1}{2t}\int^{t}_{-t}\bigl{\|}F(t)-P(t)\bigr{\|}_{Y}^{P}\,dt\Biggr{)}^{\alpha/P},\quad
t>0,$

so that $F\in e-(x^{\alpha},t^{-(1/p)})-B^{p}({\mathbb{R}}:Y).$ The converse
statement does not hold in general, as the following illustrative example
shows:

######  Example 2.6.

(cf. also Example 2.8 below) Let $\zeta>1/2$ and $\alpha\zeta\in(0,1/2).$
Define the function $F:{\mathbb{R}}\rightarrow{\mathbb{R}}$ by
$F(t):=m^{\zeta}$ if $t\in[m^{2},m^{2}+\sqrt{|m|})$ for some
$m\in{\mathbb{Z}},$ and $F(t):=0,$ otherwise. Then it can be simply shown that
the function $F(\cdot)$ is not Besicovitch bounded and therefore not
Besicovitch almost periodic. On the other hand, we have $F\in
PAP_{0,p}({\mathbb{R}},t^{-1},x^{\alpha})\subseteq
e-(x^{\alpha},t^{-1})-B^{1}({\mathbb{R}}:{\mathbb{C}}).$

Let the numbers $\alpha>0$ and $\beta>0$ be arbitrary. Using the functions
$\phi(x)\equiv x^{\alpha/p}$ and ${\rm F}(t)\equiv t^{-\beta/p}$ in our
approach, we can consider the generalized Besicovitch class
$B_{\alpha,\beta}({\mathbb{R}}:Y)$ consisting of those Lebesgue measurable
functions $F:{\mathbb{R}}\rightarrow Y$ such that, for every $\epsilon>0,$
there exist a trigonometric polynomial $P(\cdot)$ and a real number $t_{0}>0$
such that

$\int_{-t}^{t}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}^{\alpha}\,ds\leq\epsilon
t^{\beta},\quad t\geq t_{0};$

a multi-dimensional generalization can be introduced analogously. Fairly
complete analysis of the generalized Besicovitch class
$B_{\alpha,\beta}({\mathbb{R}}:Y)$ and its multi-dimensional analogues is
without scope of this paper (let us only observe here that the space
$W_{\alpha}$, considered by M. A. Picardello [58] in the usual one-dimensional
setting with $0<\alpha\leq 1,$ is nothing else but the space
$B_{1,\alpha}({\mathbb{R}}:{\mathbb{C}})$).

We will provide the main details of the proof of the following proposition for
the sake of completeness:

######  Proposition 2.7.

Suppose that $p,\ q,\ r\in[1,\infty),$ $1/r=1/p+1/q$, ${\rm F}_{1}(t)\equiv
t^{-n/p}$, ${\rm F}_{2}(t)\equiv t^{-n/q},$ ${\rm F}(t)\equiv t^{-n/r}$,
$\phi(x)\equiv x^{\alpha}$ for some real number $\alpha>0,$ and any set $B$ of
collection ${\mathcal{B}}$ is bounded in $X.$ If $F_{1}\in
e-({\mathcal{B}},\phi,{\rm F}_{1})-B^{p}(\Lambda\times X:{\mathbb{C}})$ and
$F_{2}\in e-({\mathcal{B}},\phi,{\rm F}_{2})-B^{q}(\Lambda\times X:Y),$ then
the function $F:\Lambda\times X\rightarrow Y,$ given by $F({\bf
t};x):=F_{1}({\bf t};x)F_{2}({\bf t};x),$ ${\bf t}\in\Lambda,$ $x\in X,$
belongs to the class $e-({\mathcal{B}},\phi,{\rm F})-B^{r}(\Lambda\times
X:Y).$

###### Proof.

Let $\epsilon>0$ and $B\in{\mathcal{B}}$ be given. Then there exist a finite
real number $t_{0}>0,$ a scalar-valued trigonometric polynomial
$P_{1}(\cdot;\cdot)$ and a $Y$-valued trigonometric polynomial
$P_{2}(\cdot;\cdot)$ such that, for every $x\in B,$ we have

(2.2)
$\displaystyle\Bigl{[}\phi\bigl{(}\bigl{\|}F_{1}(\cdot;x)-P_{1}(\cdot;x)\bigr{\|}_{Y}\bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}\leq\epsilon
t^{n/p},\quad t\geq t_{0},$

and

(2.3)
$\displaystyle\Bigl{[}\phi\bigl{(}\bigl{\|}F_{2}(\cdot;x)-P_{2}(\cdot;x)\bigr{\|}_{Y}\bigr{)}\Bigr{]}_{L^{q}(\Lambda_{t})}\leq\epsilon
t^{n/q},\quad t\geq t_{0}.$

Clearly, $P_{1}(\cdot;\cdot)P_{2}(\cdot;\cdot)$ is a $Y$-valued trigonometric
polynomial. Applying Proposition 2.4(i), we get that the function
$P_{1}(\cdot;\cdot)$ is Besicovitch-$(p,\phi,{\rm
F}_{1},{\mathcal{B}})$-bounded and the function $F_{2}(\cdot;\cdot)$ is
Besicovitch-$(q,\phi,{\rm F}_{2},{\mathcal{B}})$-bounded. Keeping in mind that
$1/r=1/p+1/q$, the final conclusion simply follows using this fact,
(2.2)-(2.3), the existence of a finite real number $c_{\alpha}>0$ such that

$\displaystyle\phi\Bigl{(}\bigl{\|}$ $\displaystyle
F_{1}(\cdot;x)F_{2}(\cdot;x)-P_{1}(\cdot;x)P_{2}(\cdot;x)\bigr{\|}_{Y}\Bigr{)}$
$\displaystyle\leq
c_{\alpha}\Biggl{[}\phi\Bigl{(}\bigl{|}F_{1}(\cdot;x)-P_{1}(\cdot;x)\bigr{|}\Bigr{)}\cdot\phi\Bigl{(}\bigl{\|}F_{2}(\cdot;x)\bigr{\|}_{Y}\Bigr{)}$
$\displaystyle+\phi\Bigl{(}\bigl{|}P_{1}(\cdot;x)\bigr{|}\Bigr{)}\cdot\phi\Bigl{(}\bigl{\|}F_{2}(\cdot;x)-P_{2}(\cdot;x)\bigr{\|}_{Y}\Bigr{)}\Biggr{]},$

and the Hölder inequality. ∎

We continue by providing the following illustrative application of Proposition
2.7:

######  Example 2.8.

Suppose that $1\leq p_{1},...,p_{n},p<+\infty$ and
$1/p=1/p_{1}+1/p_{2}+...+1/p_{n}.$ Define the function
$F_{j}:{\mathbb{R}}\rightarrow{\mathbb{R}}$ by $F_{j}(t):=m^{1/2p_{j}}$ if
$t\in[m^{2},m^{2}+\sqrt{|m|})$ for some $m\in{\mathbb{Z}},$ and $F_{j}(t):=0,$
otherwise ($1\leq j\leq n$). Then we know that the function $F_{j}(\cdot)$ is
Besicovitch-$p_{j}$-almost periodic but not Besicovitch-$q$-almost periodic if
$q>p_{j};$ see [7, p. 42] and [3, Example 6.24]. Define $F({\bf
t}):=F_{1}(t_{1})\cdot F_{2}(t_{2})\cdot...\cdot F_{n}(t_{n}),$ ${\bf
t}\in{\mathbb{R}}^{n}.$ Applying Proposition 2.7 and a simple argumentation,
it follows that the function $F(\cdot)$ is Besicovitch-$p$-almost periodic but
not Besicovitch-$q$-almost periodic if $q>p.$

Sometimes we need the value of coefficient $p=+\infty$ in Proposition 2.7 and
sometimes the usual choice ${\rm F}(t)\equiv t^{-n/p}$ is wrong if the region
$\Lambda$ is bounded in direction of some real axes:

######  Example 2.9.

Suppose that $1\leq p<+\infty,$ the function
$f:[0,2\pi]\rightarrow{\mathbb{R}}$ is absolutely continuous and the function
$g:[0,\infty)\rightarrow Y$ is Besicovitch-$p$-almost periodic. Since the
Fourier series of function $f(\cdot)$ converges uniformly to this function,
arguing as in the proof of Proposition 2.7 we may conclude that the function
$F(x,y):=f(x)g(y),$ $(x,y)\in\Lambda\equiv[0,2\pi]\times[0,\infty)\rightarrow
Y$ belongs to the class $e-(x,t^{-1/p})-B^{p}(\Lambda:Y).$

Further on, the composition principles for one-dimensional
Besicovitch-$p$-almost periodic functions have been analyzed for the first
time by M. Ayachi and J. Blot in [5, Lemma 4.1]. In the following theorem, we
consider the Besicovitch-$p$-almost periodicity of the multi-dimensional
Nemytskii operator $W:{\mathbb{R}}^{n}\times X\rightarrow Z,$ given by

(2.4) $\displaystyle W({\bf t};x):=G({\bf t};F({\bf t};x)),\quad{\bf
t}\in{\mathbb{R}}^{n},\ x\in X,$

where $F:{\mathbb{R}}^{n}\times X\rightarrow Y$ and $G:{\mathbb{R}}^{n}\times
Y\rightarrow Z.$ We follow the ideas from [5] in (i):

######  Theorem 2.10.

Suppose that $1\leq p,\ q<+\infty,$ $\alpha>0,$ $p=\alpha q,$ ${\rm
F}(t)\equiv t^{-n/p},$ $\phi(x)\equiv x^{\zeta}$ for some real number
$\zeta>0,$ $F\in e-({\mathcal{B}},\phi,{\rm F})-B^{p}({\mathbb{R}}^{n}\times
X:Y),$ and ${\mathcal{B}}$ is the collection consisting of all bounded subsets
of $X.$

  * (i)

Suppose that $G:{\mathbb{R}}^{n}\times Y\rightarrow Z$ is Bohr
${\mathcal{B}}$-almost periodic and there exists a finite real constant $a>0$
such that

(2.5) $\displaystyle\Bigl{\|}G({\bf t};y)-G({\bf
t};y^{\prime})\Bigr{\|}_{Z}\leq
a\bigl{\|}y-y^{\prime}\bigr{\|}_{Y}^{\alpha},\quad{\bf t}\in{\mathbb{R}}^{n},\
y,\ y^{\prime}\in Y.$

Then the function $W(\cdot;\cdot),$ given by (2.4), belongs to the class
$e-({\mathcal{B}},\phi,t^{-n/q})-B^{q}({\mathbb{R}}^{n}\times X:Z).$

  * (ii)

Define

${\mathcal{B}}^{\prime}:=\Biggl{\\{}\bigcup_{{\bf t}\in{\mathbb{R}}^{n}}F({\bf
t};B)\,;\,B\in{\mathcal{B}}\Biggr{\\}}.$

By
$e-({\mathcal{B}}^{\prime},\phi,t^{-n/q})-B^{q}_{a,\alpha}({\mathbb{R}}^{n}\times
Y:Z)$ we denote the class of all functions $G_{1}(\cdot;\cdot)$ such that for
each set $B^{\prime}\equiv\cup_{{\bf t}\in{\mathbb{R}}^{n}}F({\bf
t};B)\in{\mathcal{B}}^{\prime}$ there exists a sequence of Bohr
${\mathcal{B}}$-almost periodic functions $(G_{1}^{k}(\cdot;\cdot))$ such that
(2.5) holds with the function $G(\cdot;\cdot)$ replaced therein by the
function $G_{1}^{k}(\cdot;\cdot)$ for all $k\in{\mathbb{N}},$ the equation
(2.1) holds with the function $F(\cdot;\cdot)$ replaced therein by the
function $G(\cdot;\cdot)$, the polynomial $P_{k}(\cdot;\cdot)$ replaced
therein by the function $G_{1}^{k}(\cdot;\cdot),$ the set $B$ replaced therein
with the set $B^{\prime},$ and the exponent $p(\cdot)$ replaced therein by the
constant exponent $q$. If $G\in
e-({\mathcal{B}}^{\prime},\phi,t^{-n/q})-B^{q}_{a,\alpha}({\mathbb{R}}^{n}\times
Y:Z),$ then $W\in e-({\mathcal{B}},\phi,t^{-n/q})-B^{q}({\mathbb{R}}^{n}\times
X:Z).$

###### Proof.

Let $\epsilon>0$ and $B\in{\mathcal{B}}$ be given. Then there exist a
trigonometric polynomial $P(\cdot;\cdot)$ and a finite real number $t_{0}>0$
such that

$\sup_{x\in B}\int_{[-t,t]^{n}}\bigl{\|}F({\bf t};x)-P({\bf
t};x)\bigr{\|}_{Y}^{\zeta p}\,d{\bf t}<\epsilon t^{n},\quad t\geq t_{0}.$

We will first prove (i). Since we have assumed that (2.5) holds and
${\mathcal{B}}$ is the collection consisting of all bounded subsets of $X$,
the argumentation contained in the proof of [41, Theorem 6.1.47, Corollary
6.1.48] shows that the function $W_{1}:{\mathbb{R}}^{n}\times X\rightarrow Z,$
given by $W_{1}({\bf t};x):=G({\bf t};P({\bf t};x)),$ ${\bf
t}\in{\mathbb{R}}^{n},$ $x\in X,$ is Bohr ${\mathcal{B}}$-almost periodic.
Then the final conclusion simply follows by observing that $p=\alpha q,$ using
the next estimate which holds for any $t>0$ and $x\in B;$ see (2.5):

$\displaystyle\int_{[-t,t]^{n}}\bigl{\|}W({\bf t};x)-W_{1}({\bf
t};x)\bigr{\|}_{Z}^{\zeta q}\,d{\bf t}\leq a^{\zeta
q}\int_{[-t,t]^{n}}\bigl{\|}F({\bf t};x)-P({\bf t};x)\bigr{\|}_{Y}^{\zeta
p}\,d{\bf t}.$

In order to prove (ii), it suffices to apply (i) and use the decomposition

$\displaystyle\phi\Bigl{(}\bigl{\|}G({\bf t};F({\bf t};x))-G_{k}({\bf
t};P({\bf t};x))\bigr{\|}_{Z}\Bigr{)}$ $\displaystyle\leq
2^{\zeta}\Biggl{[}\phi\Bigl{(}\bigl{\|}G({\bf t};F({\bf t};x))-G_{k}({\bf
t};F({\bf t};x))\bigr{\|}_{Z}\Bigr{)}+\phi\Bigl{(}\bigl{\|}G_{k}({\bf
t};F({\bf t};x))-G_{k}({\bf t};P({\bf t};x))\bigr{\|}_{Z}\Bigr{)}\Biggr{]},$

where $G_{k}(\cdot;\cdot)$ properly approximates $G(\cdot;\cdot)$ in the space
$e-({\mathcal{B}}^{\prime},\phi,t^{-n/q})-B^{q}({\mathbb{R}}^{n}\times Y:Z).$
∎

###  2.1. Multi-dimensional Besicovitch normal type functions

The notion of a Besicovitch $p$-normal function
$f:{\mathbb{R}}\rightarrow{\mathbb{C}}$ was introduced by R. Doss in [23] and
later reconsidered by the same author in [25]; cf. also [41, Subsection 8.3.2,
Definition 8.3.18], where we have recently analyzed the concept Weyl
$p$-almost automorphy (of type $2$) without limit functions. In this
subsection, we will consider the following notion:

######  Definition 2.11.

Suppose that ${\mathrm{R}}$ is any collection of sequences in
$\Lambda^{\prime\prime},$ $F:\Lambda\times X\rightarrow Y,$
$\phi:[0,\infty)\rightarrow[0,\infty)$ and ${\rm
F}:(0,\infty)\rightarrow(0,\infty).$ Then we say that the function
$F(\cdot;\cdot)$ is Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}$-normal if and only if for every set $B\in{\mathcal{B}}$ and
for every sequence $({\bf b}_{k})_{k\in{\mathbb{N}}}$ in ${\mathrm{R}}$ there
exists a subsequence $({\bf b}_{k_{m}})_{m\in{\mathbb{N}}}$ of $({\bf
b}_{k})_{k\in{\mathbb{N}}}$ such that, for every $\epsilon>0,$ there exists an
integer $m_{0}\in{\mathbb{N}}$ such that, for every integers $m,\
m^{\prime}\geq m_{0},$ we have

$\displaystyle\limsup_{t\rightarrow+\infty}{\mathrm{F}}(t)\sup_{x\in
B}\Biggl{[}\phi\Bigl{(}\bigl{\|}F({\bf t}+{\bf b}_{k_{m}};x)-F({\bf t}+{\bf
b}_{k_{m^{\prime}}};x)\bigr{\|}_{Y}\Bigr{)}\Biggr{]}_{L^{p({\bf
t})}(\Lambda_{t})}<\epsilon.$

The usual notion of Besicovitch-$p$-normality for the function
$F:\Lambda\rightarrow Y,$ where $1\leq p<+\infty,$ is obtained by plugging
$\phi(x)\equiv x$ and ${\rm F}(t)\equiv t^{-n/p},$ with ${\mathrm{R}}$ being
the collection of all sequences in $\Lambda^{\prime\prime}.$

In the sequel, we will occasionally use the following conditions:

  * (I)

$\phi(\cdot)$ is monotonically increasing, continuous at the point $t=0,$ and
$p(\cdot)\equiv p\in[1,\infty).$

  * (II)

There exists $c\in(0,1]$ such that $\phi(x+y)\leq c[\phi(x)+\phi(y)]$ for all
$x,\ y\geq 0$, and there exists a function
$\varphi:[0,\infty)\rightarrow[0,\infty)$ such that
$\phi(xy)\leq\phi(x)\varphi(y)$ for all $x,\ y\geq 0$ and
$D:=\sup_{m\in{\mathbb{N}}}[m\varphi(1/m)]<+\infty.$

  * (III)

$\limsup_{t\rightarrow+\infty}[{\rm F}(t)m(\Lambda_{t})^{1/p}]<+\infty$ and,
for every real number $a>0,$ we have $\limsup_{t\rightarrow+\infty}[{\rm
F}(t)/{\rm F}(t+a)]\leq 1.$

It is clear that (II) holds provided that $\phi(x)\equiv x^{\alpha}$ for some
real number $\alpha\geq 1$ as well as that (II) does not hold if
$\phi(x)\equiv x^{\alpha}$ for some real number $\alpha\in(0,1).$

Repeating verbatim the argumentation contained in the proof of [41, Theorem
6.3.19], where we have analyzed the concept Weyl
$({\mathrm{R}},{\mathcal{B}},p)$-normality, the following result can be
deduced without any substantial difficulties:

######  Proposition 2.12.

Suppose that $F:\Lambda\times X\rightarrow Y$, $F\in
e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y),$ and
conditions _(I)-(III)_ hold. Then $F(\cdot;\cdot)$ is
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal.

Even in the usual one-dimensional framework, we know that the converse
statement of Proposition 2.12 is not true in general (see, e.g., [3]) as well
as that the usual Besicovitch-$p$-normality does not imply
Besicovitch-$p$-continuity (see, e.g., [25]). Further on, let
$k\in{\mathbb{N}}$ and $F_{i}:\Lambda\times X\rightarrow Y_{i}$ ($1\leq i\leq
k$). Then we define the function $(F_{1},...,F_{k}):\Lambda\times X\rightarrow
Y_{1}\times...\times Y_{k}$ by

$\bigl{(}F_{1},...,F_{k}\bigr{)}({\bf t};x):=\bigl{(}F_{1}({\bf
t};x),...,F_{k}({\bf t};x)\bigr{)},\quad{\bf t}\in\Lambda,\ x\in X.$

The following result is trivial and its proof is therefore omitted:

######  Proposition 2.13.

Suppose that $k\in{\mathbb{N}}$,
$\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n},$ and we have that, for any
sequence which belongs to ${\mathrm{R}},$ any its subsequence also belongs to
${\mathrm{R}}.$ If the function $F_{i}(\cdot;\cdot)$ is
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal
for $1\leq i\leq k$, then the function $(F_{1},...,F_{k})(\cdot;\cdot)$ is
also Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}$-normal.

The interested reader may try to formulate certain conditions ensuring that
the limit function of a sequence of uniformly convergent
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal
functions (the functions belonging to the class $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$) is also
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal
(belongs to the class $e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda\times X:Y)$); see [41] for many results of this type.

Several structural properties of functions belonging to the class
$e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$ can be simply
reformulated for the class of
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}$-normal
functions. For example, we have the following analogue of Proposition 2.7:

######  Proposition 2.14.

Suppose that $p,\ q,\ r\in[1,\infty),$ $1/r=1/p+1/q$, ${\rm F}_{1}(t)\equiv
t^{-n/p}$, ${\rm F}_{2}(t)\equiv t^{-n/q},$ ${\rm F}(t)\equiv t^{-n/r}$,
$\phi(x)\equiv x^{\alpha}$ for some real number $\alpha>0$ and, for any
sequence which belongs to ${\mathrm{R}},$ any its subsequence also belongs to
${\mathrm{R}}.$ If the function $F_{1}:\Lambda\times X\rightarrow{\mathbb{C}}$
is Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F}_{1})-B^{p}$-normal
and Besicovitch-$(p,\phi,{\rm F}_{1},{\mathcal{B}})$-bounded as well as the
function $F_{2}:\Lambda\times X\rightarrow Y$ is
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F}_{2})-B^{q}$-normal and
Besicovitch-$(q,\phi,{\rm F}_{2},{\mathcal{B}})$-bounded, then the function
$F:\Lambda\times X\rightarrow Y,$ given by $F({\bf t};x):=F_{1}({\bf
t};x)F_{2}({\bf t};x),$ ${\bf t}\in\Lambda,$ $x\in X,$ is
Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm F}_{1})-B^{r}$-normal.

###### Proof.

Let a set $B\in{\mathcal{B}}$ and a sequence $({\bf
b}_{k})_{k\in{\mathbb{N}}}$ in ${\mathrm{R}}$ be given. Keeping in mind the
proof of Proposition 2.7 and our assumption that for any sequence which
belongs to ${\mathrm{R}}$ any its subsequence also belongs to ${\mathrm{R}}$,
it suffices to show that there exist two sufficiently large real numbers
$t_{0}>0$ and $M>0$ such that

(2.6) $\displaystyle\sup_{x\in
B,k\in{\mathbb{N}}}\Bigl{[}\phi\bigl{(}|F_{1}(\cdot+{\bf
b}_{k};x)|\bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}\leq Mt^{n/p},\quad t\geq
t_{0}$

and

(2.7) $\displaystyle\sup_{x\in
B,k\in{\mathbb{N}}}\Bigl{[}\phi\bigl{(}\|F_{2}(\cdot+{\bf
b}_{k};x)\|_{Y}\bigr{)}\Bigr{]}_{L^{q}(\Lambda_{t})}\leq Mt^{n/q},\quad t\geq
t_{0}.$

Since the function $F_{1}:\Lambda\times X\rightarrow{\mathbb{C}}$ is
Besicovitch-$(p,\phi,{\rm F}_{1},{\mathcal{B}})$-bounded, it can be simply
proved that

$\displaystyle\sup_{x\in B}\Bigl{[}\phi\bigl{(}|F_{1}(\cdot+{\bf
b}_{k};x)|\bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}\ \leq Mt^{n/p},\quad t\geq
t_{0}.$

Moreover, we have the existence of a finite real number $t_{0}>0$ and an
integer $k_{0}\in{\mathbb{N}}$ such that, for every integers $k,\
k^{\prime}\geq k_{0},$ we have

$\displaystyle\sup_{x\in B}\Bigl{[}\phi\Bigl{(}\bigl{|}F_{1}(\cdot+{\bf
b}_{k};x)-F_{1}(\cdot+{\bf
b}_{k^{\prime}};x)\bigr{|}\Bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}<Mt^{n/p},\quad
t\geq t_{0}.$

Further on, there exists a finite real constant $c_{\alpha}>0$ such that, for
every integers $k,\ k^{\prime}\geq k_{0},$ we have:

$\displaystyle\Bigl{[}$ $\displaystyle\phi\bigl{(}|F_{1}(\cdot+{\bf
b}_{k};x)|\bigr{)}-\phi\bigl{(}|F_{1}(\cdot+{\bf
b}_{k^{\prime}};x)|\bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}$ $\displaystyle\leq
c_{\alpha}\Bigl{[}\phi\Bigl{(}\Bigr{|}|F_{1}(\cdot+{\bf
b}_{k};x)|-|F_{1}(\cdot+{\bf
b}_{k^{\prime}};x)|\Bigr{|}\Bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}$
$\displaystyle\leq c_{\alpha}\Bigl{[}\phi\Bigl{(}\bigl{|}F_{1}(\cdot+{\bf
b}_{k};x)-F_{1}(\cdot+{\bf
b}_{k^{\prime}};x)\bigr{|}\Bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}\leq
c_{\alpha}Mt^{n/p},\quad t\geq t_{0},\ x\in B.$

This simply implies (2.6) because

$\displaystyle\sup_{x\in B}$
$\displaystyle\Bigl{[}\phi\bigl{(}|F_{1}(\cdot+{\bf
b}_{k};x)|\bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}$ $\displaystyle\leq\sup_{x\in
B}\Bigl{[}\phi\bigl{(}|F_{1}(\cdot+{\bf
b}_{k_{0}};x)|\bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}+c_{\alpha}Mt^{n/p},\quad
t\geq t_{0},\ k\geq k_{0}.$

The estimate (2.7) can be proved analogously, finishing the proof. ∎

Concerning the asumptions on Besicovitch boundedness used in the formulation
of Proposition 2.14, we would like to recall that A. Haraux and P. Souplet
have proved (see [34, Theorem 1.1]) that the function
$f:{\mathbb{R}}\rightarrow{\mathbb{R}},$ given by

(2.8) $\displaystyle
f(t):=\sum_{n=1}^{\infty}\frac{1}{n}\sin^{2}\Bigl{(}\frac{t}{2^{n}}\Bigr{)}\,dt,\quad
t\in{\mathbb{R}},$

is uniformly continuous, uniformly recurrent (the sequence $(\alpha_{k}\equiv
2^{k}\pi)_{k\in{\mathbb{N}}}$) can be chosen in definition of uniform
recurrence) and Besicovitch unbounded; see [41] for the notion. Let ${\rm R}$
denote the collection consisting of the seqeunce
$(\alpha_{k})_{k\in{\mathbb{N}}}$ and all its subsequences. Then the function
$f(\cdot)$ is Besicovitch$-({\mathrm{R}},x,t^{-1/p})-B^{p}$-normal but not
Besicovitch-$(p,x,t^{1/p})$-bounded for any finite exponent $p\geq 1.$
Furthermore, we have the following:

######  Example 2.15.

Suppose that $p\in[1,\infty),$ $\sigma\in(0,1),$ $F(x):=|x|^{\sigma},$
$x\in{\mathbb{R}},$ and $a>1-(1-\sigma)p>0.$ Then we know that the function
$F(\cdot)$ is not Besicovitch-$p$-bounded and that, for every
$t\in{\mathbb{R}}$ and $\omega\in{\mathbb{R}},$ we have:

(2.9)
$\displaystyle\lim_{l\rightarrow+\infty}l^{-a}\int^{l}_{-l}\Bigl{|}\bigl{|}x+t+\omega\bigr{|}^{\sigma}-\bigl{|}x+t\bigr{|}^{\sigma}\Bigr{|}^{p}\,dx=0;$

see [41, Theorem 8.3.8] and its proof. Let ${\mathrm{R}}$ denote the
collection of all sequences in ${\mathbb{R}}$ and let ${\rm F}(t)\equiv
t^{-a/p}.$ Then the limit equality (2.9) simply implies that the function
$F(\cdot)$ is Besicovitch$-({\mathrm{R}},x,{\rm F})-B^{p}$-normal. Hence, the
usual Besicovitch-$p$-normality of a function $F(\cdot)$ does not imply its
Besicovitch-$p$-boundedness as well.

Now we would like to recall that any Doss-$p$-almost periodic function
$F:[0,\infty)\rightarrow Y$, where $p\in[1,\infty),$ can be extended to a
Doss-$p$-almost periodic function $\tilde{F}:{\mathbb{R}}\rightarrow Y$
defined by $\tilde{F}(t):=0,$ $t<0$ (cf. [46] for the notion used in this
paragraph). A similar type of extension can be achieved in a much more general
situation; for example, we know that, under certain reasonable conditions, any
Doss-$(p,\phi,{\rm F},{\mathcal{B}},\Lambda^{\prime},\rho)$-almost periodic
function $F:\Lambda\times X\rightarrow Y$ can be extended to a
Doss-$(p,\phi,{\rm F},{\mathcal{B}},\Lambda^{\prime},\rho_{1})$-almost
periodic function $\tilde{F}:{\mathbb{R}}^{n}\times X\rightarrow Y,$ defined
by $\tilde{F}({\bf t}):=0,$ $t\notin\Lambda,$ $\tilde{F}({\bf t}):=F({\bf
t}),$ $t\in\Lambda,$ with $\rho_{1}:=\rho\cup\\{(0,0)\\}$ (the corresponding
analysis from [46] contains small typographical errors that will be corrected
in our forthcoming monograph [42]).

We would like to emphasize that a similar analysis cannot be carried out for
Besicovitch almost periodic type functions. We close this section by
introducing the following notion and raising the following issue:

######  Definition 2.16.

Let $\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n}$, and let
${\mathcal{C}}_{\Lambda}=e-({\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}(\Lambda:Y)$ or ${\mathcal{C}}_{\Lambda}$ be the class
consisting of all Besicovitch$-({\mathrm{R}},{\mathcal{B}},\phi,{\rm
F})-B^{p(\cdot)}$-normal functions. Then we say that the set $\Lambda$ is
admissible with respect to the class ${\mathcal{C}}_{\Lambda}$ if and only if
for any complex Banach space $Y$ and for any function $F:\Lambda\rightarrow Y$
there exists a function $\tilde{F}\in{\mathcal{C}}_{{\mathbb{R}}^{n}}$ such
that $\tilde{F}({\bf t})=F({\bf t})$ for all ${\bf t}\in\Lambda.$

Problem. It is still not known whether the set
$[0,\infty)\subseteq{\mathbb{R}}$ is admissible with respect to the class of
Besicovitch-$p$-almost periodic functions, i.e., whether a
Besicovitch-$p$-almost periodic function $f:[0,\infty)\rightarrow Y$ can be
extended to a Besicovitch-$p$-almost periodic function
$\tilde{f}:{\mathbb{R}}\rightarrow Y$ defined on the whole real line ($1\leq
p<\infty$). We would like to ask here a more general question: Is it true that
a convex polyhedral $\Lambda$ in ${\mathbb{R}}^{n}$ is admissible with respect
to the class of multi-dimensional Besicovitch-$p$-almost periodic functions
(Besicovitch-$p$-normal functions)?

##  3\. Besicovitch-Doss almost periodicity

In this section, we discuss and reexamine several structural results
established by R. Doss in [24]-[25]. We work in the multi-dimensional setting
here, considering especially the following conditions:

  * (A)

For every $B\in{\mathcal{B}}$ and $a\in\Lambda^{\prime\prime},$ there exists a
function $F^{(a)}_{B}:\Lambda\times X\rightarrow Y$ such that
$F^{(a)}_{B}(\cdot;x)$ is $a$-periodic for every fixed element $x\in B$, i.e.,
$F^{(a)}_{B}({\bf t}+a;x)=F^{(a)}_{B}({\bf t};x)$ for all ${\bf t}\in\Lambda,$
$x\in B,$ $\|F^{(a)}_{B}({\bf t};x)\|_{Y}\in L^{p({\bf t})}(\Lambda_{t})$ for
all $t>0,$ $x\in B$, and

(3.1)
$\displaystyle\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}{\rm
F}(t)\sup_{x\in
B}\Biggl{[}\phi\Biggl{(}\Biggr{\|}\frac{1}{k}\sum_{j=0}^{k-1}F({\bf
t}+ja;x)-F^{(a)}_{B}({\bf t};x)\Biggr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p({\bf
t})}(\Lambda_{t})}=0.$

  * (A)∞

For every $B\in{\mathcal{B}}$ and $a\in\Lambda^{\prime\prime},$ there exists a
function $F^{(a)}_{B}:\Lambda\times X\rightarrow Y$ such that
$F^{(a)}_{B}(\cdot;x)$ is $a$-periodic for every fixed element $x\in B$,
$\|F^{(a)}_{B}(\cdot;x)\|_{Y}\in L^{\infty}({\mathbb{R}}^{n})$ for all $x\in
B$, and (3.1) holds.

  * (AS)

For every $B\in{\mathcal{B}}$ and
$a=(a_{1},a_{2},...,a_{n})\in\Lambda^{\prime\prime}$ such that
$a_{j}e_{j}\in\Lambda^{\prime\prime}$ for all $j\in{\mathbb{N}}_{n},$ there
exists a function $F^{(a)}_{B}:\Lambda\times X\rightarrow Y$ such that
$F^{(a)}_{B}(\cdot;x)$ is $(a_{j})_{j\in{\mathbb{N}}_{n}}$-periodic for every
fixed element $x\in B$, i.e., $F^{(a)}_{B}({\bf
t}+a_{j}e_{j};x)=F^{(a)}_{B}({\bf t};x)$ for all ${\bf t}\in\Lambda,$ $x\in
B,$ $j\in{\mathbb{N}}_{n},$ $\|F^{(a)}_{B}({\bf t};x)\|_{Y}\in L^{p({\bf
t})}(\Lambda_{t})$ for all $t>0,$ $x\in B$, and (3.1) holds.

It is clear that (AS) implies (A) as well as that both conditions are
equivalent in the one-dimensional setting; it is also clear that (A)∞ implies
(A). Further on, for every Lebesgue measurable set
$E\subseteq{\mathbb{R}}^{n}$ and for every Lebesgue measurable function
$F:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}}$, we set

$\nu(E):=\limsup_{t\rightarrow+\infty}\frac{1}{t^{n/p}}\int_{|{\bf t}|\leq
t}\chi_{E}({\bf t})\,d{\bf t}\mbox{ and
}\overline{{\mathcal{M}}}^{E}[F]:=\limsup_{t\rightarrow+\infty}\frac{1}{t^{n/p}}\int_{|{\bf
t}|\leq t}F({\bf t})\chi_{E}({\bf t})\,d{\bf t}.$

If

$\lim_{t\rightarrow+\infty}\frac{1}{t^{n/p}}\int_{|{\bf t}|\leq t}F({\bf
t})\chi_{E}({\bf t})\,d{\bf t}$

exists in ${\mathbb{C}},$ then we denote this quantity by
${\mathcal{M}}^{E}[F].$

Suppose now that the function
$F^{(a)}:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}}$ is
$(a_{1},a_{2},...,a_{n})$-periodic. Then we can find a sequence of infinitely
differentiable functions $(\varphi_{k})_{k\in{\mathbb{N}}}$ with compact
support in $S=[0,|a_{1}|]\times...\times[0,|a_{n}|]$ such that
$\varphi_{k}\rightarrow F^{(a)}$ as $k\rightarrow+\infty,$ in $L^{p}(S).$
After that, we extend $\varphi_{k}(\cdot)$ to an
$(a_{1},a_{2},...,a_{n})$-periodic function $\widetilde{\varphi_{k}}(\cdot)$
defined on the whole space ${\mathbb{R}}^{n}$ in the usual way. Then it is
very simple to prove that

(3.2)
$\displaystyle\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}\Biggl{[}t^{-(n/p)}\bigl{\|}F^{(a)}-\widetilde{\varphi_{k}}\bigr{\|}_{L^{p}(({\mathbb{R}}^{n})_{t})}\Biggr{]}=0;$

cf. also [24, p. 483, l. 7-l. 9]. Keeping this observation in mind, the
following result can be deduced, in a plus-minus technical way, following the
argumentation contained in the proofs of [24, Proposition 1, Proposition 3,
Corollary, Lemma 2] (cf. also Lemma 1.2, which is needed for the proof of
(ii)):

######  Theorem 3.1.

  * (i)

Suppose that $p\in[1,\infty),$ $q\in(1,\infty],$ $1/p+1/q=1,$ the function
$F:\Lambda\rightarrow Y$ satisfies that $\|F(\cdot)\|_{Y}\in
L^{p}(\Lambda_{t})$ for all $t>0$, as well as that
$\|F(\cdot+\tau)-F(\cdot)\|_{Y}\in L^{p}(\Lambda_{t})$ for all $t>0$ and
$\tau\in\Lambda^{\prime\prime}.$ Suppose, further, that the function
$F(\cdot)$ is Besicovitch-$(p,x,{\rm F}_{1})$-bounded and
Besicovitch-$(p,x,{\rm F}_{1},\Lambda^{\prime\prime})$-continuous as well as
that condition _(III)_ holds, and the set
$\Lambda^{\prime\prime}\cap{\mathbb{Q}}^{n}$ is dense in
$\Lambda^{\prime\prime}.$ If the function $G:\Lambda\rightarrow{\mathbb{C}}$
satisfies that $G(\cdot)\in L^{q}(\Lambda_{t})$ for all $t>0,$ and $G(\cdot)$
is Besicovitch-$(q,x,{\rm F}_{2})$-bounded, then for each sequence
$(L_{m})_{m\in{\mathbb{N}}}$ there exists a subsequence
$(T_{m})_{m\in{\mathbb{N}}}$ of $(L_{m})_{m\in{\mathbb{N}}}$ such that the
function

$H(\tau):=\lim_{m\rightarrow+\infty}{\rm F}_{1}\bigl{(}T_{m}\bigr{)}{\rm
F}_{2}\bigl{(}T_{m}\bigr{)}\int_{\Lambda_{T_{m}}}G({\bf s})F({\bf
s}+\tau)\,d{\bf s},\quad\tau\in\Lambda^{\prime\prime}$

is well-defined and bounded. Furthermore, if the function $F(\cdot)$ is
Doss-$(p,x,{\rm F}_{1},\Lambda^{\prime})$-almost periodic (Doss-$(p,x,{\rm
F}_{1},\Lambda^{\prime})$-uniformly recurrent) and
$\Lambda^{\prime}+\Lambda^{\prime\prime}\subseteq\Lambda^{\prime\prime},$ then
the function $H(\cdot)$ is Bohr $\Lambda^{\prime}$-almost periodic
($\Lambda^{\prime}$-uniformly recurrent).

  * (ii)

Suppose that $p\in[1,\infty),$ $q\in(1,\infty],$ $1/p+1/q=1,$ the assumptions
in _(i)_ hold for the function $F:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}},$
with $\Lambda=\Lambda^{\prime}=\Lambda^{\prime\prime}={\mathbb{R}}^{n},$ ${\rm
F}_{1}(t)={\rm F}_{2}(t)=t^{-(n/p)},$ $t\in{\mathbb{R}}^{n},$ and $F(\cdot)$
is Doss-$(p,x,t^{-(n/p)})$-almost periodic. If condition _(A) ∞_ or _(AS)_
holds with $X=\\{0\\},$ $\phi(x)\equiv x$ and $p(\cdot)\equiv p,$ then, for
every real number $\epsilon>0,$ there exists a finite real number $\delta>0$
such that, for every Lebesgue measurable set $E\subseteq{\mathbb{R}}^{n},$ the
assumption $\nu(E)<\delta$ implies
$\overline{{\mathcal{M}}}^{E}[|F|_{Y}^{p}]<\epsilon.$

  * (iii)

Let the assumptions of _(ii)_ hold, and let for each $N>0$ the function
$F_{N}:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}}$ be defined by $F_{N}({\bf
t}):=F({\bf t})$, if $|F({\bf t})|\leq N,$ and $F_{N}({\bf
t}):=Ne^{i\arg(F({\bf t}))}$, if $|F({\bf t})|>N.$ Then
$\lim_{N\rightarrow+\infty}\overline{{\mathcal{M}}}^{{\mathbb{R}}^{n}}[F-F_{N}]=0.$

  * (iv)

Suppose that the assumptions in _(i)_ hold for the function
$F:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}},$ with $p=1,$
$\Lambda=\Lambda^{\prime}=\Lambda^{\prime\prime}={\mathbb{R}}^{n},$ ${\rm
F}_{1}(t)={\rm F}_{2}(t)=t^{-n},$ $t\in{\mathbb{R}}^{n},$ and $F(\cdot)$ is
Doss-$(1,x,t^{-n})$-almost periodic. If condition _(A) ∞_ or _(AS)_ holds with
$X=\\{0\\},$ $\phi(x)\equiv x$ and $p(\cdot)\equiv 1,$ then for each
$a\in{\mathbb{R}}^{n}$ we have that ${\mathcal{M}}^{{\mathbb{R}}^{n}}[F]$
exists in ${\mathbb{C}}$ and
${\mathcal{M}}^{{\mathbb{R}}^{n}}[F]={\mathcal{M}}^{{\mathbb{R}}^{n}}[F^{(a)}].$

Before proceeding further, let us note that it is not clear how we can extend
the statement (ii) to the vector-valued functions. It is also worth noting
that a serious difficulty in our analysis of the multi-dimensional case
presents the fact that it is not clear whether we can further generalize the
above-mentioned statement by using condition (A) in place of (A)∞ or (AS).
Concerning conditions (A), (A)∞, (AS) and the equation (3.2), we would like to
present the following example:

######  Example 3.2.

  * (i)

(see also [41, Example 7.2.2]) Suppose that
$F_{0}:\\{(x,y)\in{\mathbb{R}}^{2}:0\leq x+y\leq 2\\}\rightarrow[0,\infty)$ be
any continuous function such that the following conditions hold:

    * (a)

$F_{0}(x,y)=F_{0}(x+1,y+1)$ for every $(x,y)\in{\mathbb{R}}^{2}$ such that
$x+y=0;$

    * (b)

Let $P_{k}=A_{k}B_{k}C_{k}D_{k}$ be the rectangle in ${\mathbb{R}}^{2}$ with
vertices $A_{k}=(4k-(2/3),(2/3)-4k),$ $B_{k}=(4k-(1/3),(1/3)-4k),$
$C_{k}=(4k+(2/3),(4/3)-4k)$ and $D_{k}=(4k+(1/3),(5/3)-4k),$ for each integer
$k\in{\mathbb{Z}}.$ We have $F_{0}(x,y)\geq 2^{|k|}$ for all integers
$k\in{\mathbb{Z}}$ and $(x,y)\in P_{k}.$

We extend the function $F_{0}(\cdot;\cdot)$ to a continuous $(1,1)$-periodic
function $F:{\mathbb{R}}^{2}\rightarrow[0,\infty)$ in the obvious way. Then
the function $F(\cdot,\cdot)$ is not Besicovitch almost periodic since it is
not Besicovitch bounded; this follows from the following simple estimates:

$\displaystyle\int_{\Lambda_{8k\sqrt{2}}}F(x,y)\,dx\,dy\geq
16k\sum_{l=0}^{4k}\frac{\sqrt{2}}{3}2^{l}\geq\frac{16k\sqrt{2}}{3}\bigl{(}2^{4k+1}-1\bigr{)},\quad
k\in{\mathbb{N}}.$

Furthermore, there do not exist a finite real constant $M>0$ and an
essentially bounded function $G:{\mathbb{R}}^{2}\rightarrow{\mathbb{C}}$ such
that

$\displaystyle\limsup_{t\rightarrow+\infty}\Biggl{[}t^{-2}\int_{|{\bf t}|\leq
t}|F({\bf t})-G({\bf t})|\,d{\bf t}\Biggr{]}\leq M;$

cf. also (3.2).

  * (ii)

Set $P(x,y):=e^{i[\sqrt{2}x+y]}+e^{i[2x+y]},$ $x,\ y\in{\mathbb{R}},$
$a_{1}:=\pi(2+\sqrt{2}),$ $a_{2}:=2\pi(1-\sqrt{2})$ and $a:=(a_{1},a_{2}).$
Then there is no continuous $(a_{j})_{j\in{\mathbb{N}}_{2}}$-periodic function
$F^{(a_{1},a_{2})}(x,y)$ such that (3.1) holds with $\phi(x)\equiv x,$
$p(\cdot)\equiv 1$ and ${\rm F}(t)\equiv t^{-2}$ ($X=\\{0\\}$). In actual
fact, we have $\sqrt{2}a_{1}+a_{2}=2\pi,$ $2a_{1}+a_{2}=2\pi$ and the validity
of (3.1) would imply

(3.3)
$\displaystyle\limsup_{t\rightarrow+\infty}\Biggl{[}t^{-2}\int_{|(x,y)|\leq
t}\Bigl{|}e^{i[\sqrt{2}x+y]}+e^{i[2x+y]}-F^{(a_{1},a_{2})}(x,y)\Bigr{|}\,dx\,dy\Biggr{]}=0.$

To see that (3.3) cannot be true, it suffices to observe that there exists a
real number
$\epsilon_{0}:=\min_{(x,y)\in[0,a_{1}]\times[0,-a_{2}]}|e^{i[\sqrt{2}x+y]}+e^{i[2x+y]}-F^{(a_{1},a_{2})}(x,y)|>0$
such that

$\int_{(x,y)\in[0,t]^{2}}\Bigl{|}e^{i[\sqrt{2}x+y]}+e^{i[2x+y]}-F^{(a_{1},a_{2})}(x,y)\Bigr{|}\,dx\,dy\geq\frac{t^{2}}{\lfloor
a_{1}\rfloor\lfloor|a_{2}|\rfloor}\epsilon_{0}.$

If $F:\Lambda\times X\rightarrow Y$, $G:\Lambda\times X\rightarrow Y$ and
$B\in{\mathcal{B}},$ then we set

$\displaystyle{\mathcal{M}}_{B}^{\phi,{\rm
F}}(F,G):=\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Bigl{[}\phi\Bigl{(}\bigl{\|}F({\bf t};x)-G({\bf
t};x)\bigr{\|}_{Y}\Bigr{)}\Bigr{]}_{L^{p({\bf t})}(\Lambda_{t})}.$

Then the quantity ${\mathcal{M}}_{B}^{\phi,{\rm F}}(F,G)$ always exists in
$[0,+\infty].$ The subsequent result states that, under certain reasonable
assumptions on the functions $\phi(\cdot)$ and ${\rm F}(\cdot),$ any function
$F\in e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y)$
satisfies condition (A):

######  Proposition 3.3.

Suppose that $F:\Lambda\times X\rightarrow Y$, $F\in
e-({\mathcal{B}},\phi,{\rm F})-B^{p(\cdot)}(\Lambda\times X:Y),$ _(I)-(III)_
and the following conditions hold:

  * (IV)

If $B\in{\mathcal{B}},$ $a\in\Lambda^{\prime\prime}$, and
$(F_{k}:\Lambda\times X\rightarrow Y)_{k\in{\mathbb{N}}}$ is any sequence of
functions which satisfies that $F_{k}({\bf t}+a;x)=F_{k}({\bf t};x),$ ${\bf
t}\in\Lambda,$ $x\in B$ and for each $\epsilon>0$ there exists
$k_{0}\in{\mathbb{N}}$ such that ${\mathcal{M}}_{B}^{\phi,{\rm
F}}(F_{k},F_{k^{\prime}})<\epsilon$ for all integers $k,\ k^{\prime}\geq
k_{0},$ then there exists a function $F:\Lambda\times X\rightarrow Y$ such
that $F({\bf t}+a;x)=F({\bf t};x),$ ${\bf t}\in\Lambda,$ $x\in B$ and
$\lim_{k\rightarrow+\infty}{\mathcal{M}}_{B}^{\phi,{\rm F}}(F_{k},F)=0$ (here
we assume that for each element $x\in X$ the functions $F_{k}(\cdot;x)$ and
$F(\cdot;x)$ are Lebesgue measurable ($k\in{\mathbb{N}}$)).

  * (V)

The collection ${\mathcal{B}}$ consists of bounded subsets of $X$.

Then the function $F(\cdot;\cdot)$ satisfies condition _(A)._

###### Proof.

We will slightly modify the original argumentation of R. Doss (see [24, pp.
477-478]). First of all, we will prove that condition (A) holds for any
trigonometric polynomial
$P(\cdot;x)=\sum_{l=0}^{m}e^{i\langle\lambda_{l},\cdot\rangle}c_{l}(x),$ $x\in
X.$ Let $B\in{\mathcal{B}}$ and $a\in\Lambda^{\prime\prime}$ be given; then
${\bf t}+ja\in\Lambda$ for all $j\in{\mathbb{N}}.$ Define
$P^{(a)}_{B}(\cdot;x):=\sum_{l\in
L_{a}}e^{i\langle\lambda_{l},\cdot\rangle}c_{l}(x),$ $x\in X$, where $L_{a}$
denotes the set of all integers $l\in[0,m]$ such that
$e^{i\langle\lambda_{l},a\rangle}=1.$ Clearly, the function
$P^{(a)}_{B}(\cdot;x)$ is $a$-periodic for every fixed element $x\in B$. Then
a simple computation shows that

$\displaystyle{\rm F}(t)\sup_{x\in B}$
$\displaystyle\Biggl{[}\phi\Biggl{(}\Biggr{\|}\frac{1}{k}\sum_{j=0}^{k-1}P({\bf
t}+ja;x)-P^{(a)}_{B}({\bf
t};x)\Biggr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p}(\Lambda_{t})}$
$\displaystyle={\rm F}(t)\sup_{x\in
B}\Biggl{[}\phi\Biggl{(}\Biggr{\|}\frac{1}{k}\sum_{j=0}^{k-1}\sum_{l\notin
L_{a}}e^{i\langle\lambda_{l},{\bf
t}\rangle}e^{i\langle\lambda_{l},aj\rangle}c_{l}(x)\Biggr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p}(\Lambda_{t})}$
$\displaystyle={\rm F}(t)\sup_{x\in
B}\Biggl{[}\phi\Biggl{(}\Biggr{\|}\frac{1}{k}\sum_{l\notin
L_{a}}e^{i\langle\lambda_{l},{\bf
t}\rangle}\frac{e^{i\langle\lambda_{l},ak\rangle}-1}{e^{i\langle\lambda_{l},a\rangle}-1}c_{l}(x)\Biggr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p}(\Lambda_{t})},\quad
t>0.$

Using the facts that the function $\phi(\cdot)$ is monotonically increasing
and the collection ${\mathcal{B}}$ consists of bounded subsets of $X$, the
above computation yields the existence of a finite real constant $c_{B}>0$
such that

$\displaystyle{\rm F}(t)\sup_{x\in B}$
$\displaystyle\Biggl{[}\phi\Biggl{(}\Biggr{\|}\frac{1}{k}\sum_{j=0}^{k-1}P({\bf
t}+ja;x)-P^{(a)}_{B}({\bf t};x)\Biggr{\|}_{Y}\Biggr{)}\Biggr{]}_{L^{p({\bf
t})}(\Lambda_{t})}$
$\displaystyle\leq\phi\Bigl{(}\frac{c_{B}}{k}\Bigr{)}\Bigl{[}{\rm
F}(t)m(\Lambda_{t})^{1/p}\Bigr{]},\quad t>0.$

Then the required conclusion follows from the continuity of function
$\phi(\cdot)$ at zero and the assumption $\limsup_{t\rightarrow+\infty}[{\rm
F}(t)m(\Lambda_{t})^{1/p}]<+\infty$. Before proceeding to the general case,
let us note that our assumptions on the function $\phi(\cdot),$ the assumption
that, for every real number $a>0,$ we have $\limsup_{t\rightarrow+\infty}[{\rm
F}(t)/{\rm F}(t+a)]\leq 1,$ and a relatively simple argumentation shows that:

  * (a)

Define

$\displaystyle{\mathcal{M}}_{B,a}^{\phi,{\rm
F}}(F,G):=\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Bigl{[}\phi\Bigl{(}\bigl{\|}F({\bf t}+a;x)-G({\bf
t}+a;x)\bigr{\|}_{Y}\Bigr{)}\Bigr{]}_{L^{p}(\Lambda_{t})}.$

Then we have

$\displaystyle{\mathcal{M}}_{B,a}^{\phi,{\rm
F}}(F,G)\leq{\mathcal{M}}_{B}^{\phi,{\rm F}}(F,G).$

  * (b)

${\mathcal{M}}_{B}^{\phi,{\rm
F}}(dF,dG)\leq\varphi(d){\mathcal{M}}_{B}^{\phi,{\rm F}}(F,G)$ for all real
numbers $d>0.$

  * (c)

${\mathcal{M}}_{B}^{\phi,{\rm F}}(F,G)\leq c[{\mathcal{M}}_{B}^{\phi,{\rm
F}}(F,H)+{\mathcal{M}}_{B}^{\phi,{\rm F}}(H,G)].$

  * (d)

${\mathcal{M}}_{B}^{\phi,{\rm F}}(F+G,H+W)\leq c[{\mathcal{M}}_{B}^{\phi,{\rm
F}}(F,H)+{\mathcal{M}}_{B}^{\phi,{\rm F}}(G,W)].$

Suppose now that there exists a sequence $(P_{k}(\cdot;\cdot))$ of
trigonometric polynomials such that (2.1) holds, i.e.,
$\lim_{k\rightarrow+\infty}{\mathcal{M}}_{B}^{\phi,{\rm F}}(F,P_{k})=0.$ Let
$\epsilon>0$ be fixed. Using (a) and (d), we get:

$\displaystyle{\mathcal{M}}_{B}^{\phi,{\rm
F}}\Biggl{(}\frac{F(\cdot;\cdot)+F(\cdot+a;\cdot)+...+F(\cdot+(m-1)a;\cdot)}{m},\frac{P_{k}(\cdot;\cdot)+P_{k}(\cdot+a;\cdot)+...+P_{k}(\cdot+(m-1)a;\cdot)}{m}\Biggr{)}$
(3.4)
$\displaystyle\leq\bigl{(}c+...+c^{m-1}\bigr{)}\varphi(1/m){\mathcal{M}}_{B}^{\phi,{\rm
F}}(F,P_{k})\leq m\varphi(1/m){\mathcal{M}}_{B}^{\phi,{\rm F}}(F,P_{k})\leq
D{\mathcal{M}}_{B}^{\phi,{\rm F}}(F,P_{k}),\quad k,\ m\in{\mathbb{N}}.$

On the other hand, using (c) we get the existence of an integer
$k_{0}(\epsilon)\equiv k_{0}$ such that:

$\displaystyle{\mathcal{M}}_{B}^{\phi,{\rm F}}(P_{k},P_{k^{\prime}})\leq
2c\epsilon,\quad k,\ k^{\prime}\geq k_{0}.$

Keeping in mind (3.4) and this estimate, we get

(3.5) $\displaystyle{\mathcal{M}}_{B}^{\phi,{\rm
F}}\Biggl{(}\frac{1}{m}\sum_{j=0}^{m-1}P_{k}(\cdot+ja;\cdot),\frac{1}{m}\sum_{j=0}^{m-1}P_{k^{\prime}}(\cdot+ja;\cdot)\Biggr{)}\leq
2cD\epsilon,\quad k,\ k^{\prime}\geq k_{0},\ m\in{\mathbb{N}}.$

If $k,\ k^{\prime}\geq k_{0},$ then we find the functions $P_{k,B}^{(a)}$ and
$P_{k^{\prime},B}^{(a)}$ from condition (A); then we can use (c) and (3.5) to
get the existence of a finite real constant $d>0$ such that
${\mathcal{M}}_{B}^{\phi,{\rm
F}}(P_{k,B}^{(a)},P_{k^{\prime},B}^{(a)})<d\epsilon$ for all integers $k,\
k^{\prime}\geq k_{0}.$ Let $F^{(a)}_{B}$ be any function such that
$F^{(a)}_{B}({\bf t}+a;x)=F^{(a)}_{B}({\bf t};x)$ for all ${\bf t}\in\Lambda,$
$x\in B$ and $\lim_{k\rightarrow+\infty}{\mathcal{M}}_{B}^{\phi,{\rm
F}}(P_{k,B}^{(a)},F^{(a)}_{B})=0;$ see (iv). Then the final conclusion follows
using (3.4) and condition (A) for $P_{k}(\cdot;\cdot).$ ∎

######  Remark 3.4.

In connection with condition (IV), we would like to note that the
argumentation contained on [24, p. 478; l. 3-l. 6] is a bit incorrect because
the completeness of $L^{p}$-spaces has been mistakenly used. It is also far
from being immediately clear why the function $f^{(a)}(x)$ appearing here must
be of period $a.$

If the assumptions (I)-(III) hold, then we have the validity of statements
(a)-(d) given in the proof of the afore-mentioned proposition. Keeping in mind
this observation, we can repeat verbatim the argumentation contained in the
proof of [24, Lemma 1] to deduce the following:

######  Proposition 3.5.

Suppose that the function $F:\Lambda\times X\rightarrow Y$ satisfies that
$\phi(\|F(\cdot;x)\|_{Y})\in L^{p(\cdot)}(\Lambda_{t})$ for all $t>0$ and
$x\in X.$ If the assumptions _(I)-(III)_ hold and the function
$F(\cdot;\cdot)$ satisfies condition _(A)_ , then $F(\cdot;\cdot)$ is
Besicovitch-$(p,\phi,{\rm F},{\mathcal{B}})$-bounded.

In particular, Proposition 3.5 implies that a uniformly recurrent function
$f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ need not satisfy condition (A); see
(2.8) with $\phi(x)\equiv x,$ $p=1$ and ${\rm F}(t)\equiv t^{-1}.$

In the multi-dimensional setting, it seems very plausible that a trigonometric
polynomial $P(\cdot)$ does not satisfy condition (AS) in general (see Example
3.2(ii)), so that we can freely say that the results established in [24] are
primarily intended for the analysis of Besicovitch almost periodic functions
of one real variable. We want also to emphasize here that the proof of [24,
Proposition 4] contains a small gap since the author has not proved that, for
any function $f:{\mathbb{R}}\rightarrow{\mathbb{C}}$ of class $(D)$ and for
any real number $\lambda\in{\mathbb{R}},$ the function
$e^{-i\lambda\cdot}f(\cdot)$ is Doss almost periodic, i.e., satisfies
condition [24, 2., p. 477] with $p=1$ (this seems to be true, but not
specifically proved in the paper). Finally, we would like to mention that it
is very likely that the statements of [24, Proposition 4-Proposition 6] admit
extensions to the multi-dimensional setting; hence, it seems very reasonable
that a Doss-$p$-almost periodic function
$F:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}}$ which is
Besicovitch-$p$-continuous and satisfies condition (AS) is
Besicovitch-$p$-almost periodic ($1\leq p<+\infty$). This is a very
unsatisfactory result in the multi-dimensional setting and we will skip all
details with regards to this question here.

###  3.1. On Condition (B)

In this subsection, we will consider the following condition:

  * (B)

Let $\Omega=[0,1]^{n},$ $l\Omega\subseteq\Lambda$ and
$\Lambda+l\Omega\subseteq\Lambda$ for all $l>0.$ If ${\rm
F}_{1}:(0,\infty)\rightarrow(0,\infty),$ ${\rm
F}:(0,\infty)\rightarrow(0,\infty)$ and $p\in{\mathcal{P}}(\Lambda),$ then
condition (B) means that, for every $\lambda\in{\mathbb{R}}^{n},$ we have:

(3.6) $\displaystyle\lim_{l\rightarrow+\infty}{\rm
F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}e^{i\lambda{\bf
t}}F({\bf t};x)\,d{\bf t}\Biggr{\|}_{L^{p(y)}(\Lambda_{t}:Y)}=0.$

Let us note that, in the original analysis of R. Doss [25], we have
$p(\cdot)\equiv 1,$ ${\rm F}_{1}(l)\equiv 1/l,$ ${\rm F}(t)\equiv 1/t$,
$\Lambda={\mathbb{R}}$ and $X=\\{0\\}.$ The situation in which the equation
(3.6) holds for all values of
$\lambda\in{\mathbb{R}}^{n}\setminus\\{\lambda_{0}\\}$ but not for the exactly
one value $\lambda=\lambda_{0}\in{\mathbb{R}}^{n}$ is possible; for example,
in the one-dimensional setting, we know that the function
$F:{\mathbb{R}}\rightarrow{\mathbb{C}}$, given by $F(t):=e^{-i\lambda_{0}t},$
$t\geq 0$ and $F(t):=-e^{-i\lambda_{0}t},$ $t<0,$ satisfies (3.6) for all
values $\lambda\in{\mathbb{R}}\setminus\\{\lambda_{0}\\}$ but not for
$\lambda_{0}$ ([25]).

Concerning condition (B), we will first clarify the following result for the
multivariate trigonometric polynomials:

######  Proposition 3.6.

Suppose that $\Omega=[0,1]^{n},$ $\Lambda={\mathbb{R}}^{n}$ and
$\lim_{l\rightarrow+\infty}[l^{n-1}{\rm F}_{1}(l)]=0.$ If the collection
${\mathcal{B}}$ consists solely of bounded subsets of $X$ and conditions
_(I)-(III)_ hold, then condition _(B)_ holds for any trigonometric polynomial
$P(\cdot;\cdot).$

###### Proof.

Let $P({\bf t};x)=\sum_{s=0}^{m}e^{i\langle\lambda_{s},{\bf
t}\rangle}c_{s}(x)$ for some continuous functions $c_{s}(\cdot).$ It suffices
to show that

$\lim_{l\rightarrow+\infty}{\rm F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm
F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}P({\bf
t};x)\,d{\bf t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}=0.$

Let $k_{s}$ denote the number of all non-zero components of vector
$\lambda_{s}=(\lambda_{s}^{1},...,\lambda_{s}^{n}).$ If $k_{s}=n,$ then the
term $e^{i\langle\lambda_{s},{\bf t}\rangle}c_{s}(x)$ is meaningless after the
integration over the cubes $y+l\Omega$ and $l\Omega.$ Because of that, we may
assume without loss of generality that $k_{s}\leq n-1$ for all
$s\in\\{0,1,...,m\\}$ as well as that $\lambda_{s}^{j}=0$ for all
$s\in\\{0,1,...,m\\}$ and $j\in\\{1,...,k\\},$ where $k\leq n-1.$ Therefore,
we need to prove that

$\displaystyle\lim_{l\rightarrow+\infty}$ $\displaystyle{\rm
F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}$
$\displaystyle\times\sum_{s=0}^{m}\sum_{j=k+1}^{n}e^{i[\lambda_{s}^{k+1}t_{k+1}+...+\lambda_{s}^{n}t_{n}]}c_{s}(x)\,dt_{1}\,dt_{2}\,...\,dt_{n}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}=0,$

i.e., that

$\displaystyle\lim_{l\rightarrow+\infty}$ $\displaystyle{\rm
F}_{1}(l)l^{k}\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}$
$\displaystyle\times\sum_{s=0}^{m}\sum_{j=k+1}^{n}e^{i[\lambda_{s}^{k+1}t_{k+1}+...+\lambda_{s}^{n}t_{n}]}c_{s}(x)\,dt_{k+1}\,dt_{k+2}\,...\,dt_{n}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}=0.$

It is clear that we have the existence of a finite real constant $c>0$ such
that, for every $y\in\Lambda_{t}$ and $l>0,$ we have:

$\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}\sum_{s=0}^{m}\sum_{j=k+1}^{n}e^{i[\lambda_{s}^{k+1}t_{k+1}+...+\lambda_{s}^{n}t_{n}]}c_{s}(x)\,dt_{k+1}\,dt_{k+2}\,...\,dt_{n}\Biggr{\|}_{Y}\leq
c.$

Since $\lim_{l\rightarrow+\infty}[l^{n-1}{\rm F}_{1}(l)]=0$ and
$\limsup_{t\rightarrow+\infty}{\rm F}(t)[m(\Lambda_{t})]^{1/p}<+\infty,$ this
simply completes the proof. ∎

In the continuation of subsection, we will first consider the one-dimensional
setting, show that the necessity in [25, Theorem 1] can be extended to the
vector-valued Besicovitch-$p$-almost periodic functions, where $1\leq
p<\infty,$ and emphasize some difficulties in proving the sufficiency in this
theorem in the case that $p>1$ (in the existing literature, we have found many
open problems regarding Besicovitch-$p$-almost periodic functions with the
exponent $p>1$). Keeping in mind the proof of the above-mentioned theorem,
which works in the vector-valued case, as well as the statements of
Proposition 2.4(ii) and Proposition 2.12, it suffices to show that condition
(B) holds for Besicovitch-$p$-almost periodic functions
$F:{\mathbb{R}}\rightarrow Y,$ with $1<p<\infty,$ $\lambda=0,$ ${\rm
F}_{1}(l)\equiv 1/l$ and ${\rm F}_{1}(t)\equiv 1/t^{1/p}.$ Further on, keeping
in mind Proposition 3.7 below and its proof (condition (3.7) holds on account
of the Hölder inequality), it suffices to show that there exists a finite real
constant $c>0$ such that, for every real number $\epsilon\in(0,1),$ there
exists a real number $l_{0}>0$ such that the assumptions $l\geq l_{0}$ and
${\mathcal{M}}^{x,{\rm F}}(F,P)<\epsilon$ imply the existence of a
sufficiently large number $t_{l}>0$ such that, for every $t\geq t_{l},$ we
have

$\displaystyle
l^{-1}t^{-(1/p)}\Biggl{(}\int_{-t}^{t}\Biggl{[}\int_{y}^{y+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}\,ds\Biggr{]}^{p}\,dy\Biggr{)}^{1/p}<c\epsilon.$

Let $\epsilon>0$ be given, let $l_{0}=1$ and $l\geq 1.$ We will show that we
can take $c=5$ in the above requirement. First of all, we have the existence
of a finite real number $t_{0}>0$ such that

$\displaystyle\int_{-t}^{t}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}^{p}\,ds\leq\epsilon
t^{p},\quad t\geq t_{0}.$

Then there exists a sufficiently large number $t_{l}^{1}>0$ such that

$\displaystyle
l^{-1}t^{-(1/p)}\Biggl{(}\int_{-(t_{0}+l)}^{t_{0}+l}\Biggl{[}\int_{y}^{y+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}\,ds\Biggr{]}^{p}\,dy\Biggr{)}^{1/p}<\epsilon,\quad
t\geq t_{l}^{1}.$

There exists a great similarity in the analysis of estimates for the intervals
$[-t,-(t_{0}+l)]$ and $[t_{0}+l,t]$ and, because of that, we will only prove
that there exists a sufficiently large number $t_{l}^{2}>t_{0}$ such that

$\displaystyle
l^{-1}t^{-(1/p)}\Biggl{(}\int_{t_{0}+l}^{t}\Biggl{[}\int_{y}^{y+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}\,ds\Biggr{]}^{p}\,dy\Biggr{)}^{1/p}<2\epsilon,\quad
t\geq t_{l}^{2}.$

This follows from the next computation (at the fourth line, we can use the
inequality appearing on l. 12, p. 134 of [25], which is a consequence of a
simple computation with double integrals):

$\displaystyle l^{-1}$ $\displaystyle
t^{-(1/p)}\Biggl{(}\int_{t_{0}+l}^{t}\Biggl{[}\int_{y}^{y+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}\,ds\Biggr{]}^{p}\,dy\Biggr{)}^{1/p}$
$\displaystyle\leq
l^{-1}t^{-(1/p)}\Biggl{(}\int_{t_{0}+l}^{t}\Biggl{[}l^{1-\frac{1}{p}}\Biggl{(}\int_{y}^{y+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}^{p}\,ds\Biggr{)}^{1/p}\Biggr{]}^{p}\,dy\Biggr{)}^{1/p}$
$\displaystyle=l^{-(1/p)}t^{-(1/p)}\Biggl{(}\int_{t_{0}+l}^{t}\int_{y}^{y+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}^{p}\,ds\,dy\Biggr{)}^{1/p}$
$\displaystyle\leq
l^{-(1/p)}t^{-(1/p)}\Biggl{(}l\int_{-t}^{t+l}\bigl{\|}F(s)-P(s)\bigr{\|}_{Y}^{p}\,ds\Biggr{)}^{1/p}$
$\displaystyle\leq\epsilon
t^{-(1/p)}[2\bigl{(}t+l\bigr{)}]^{1/p}<2\epsilon,\quad t\geq t_{l}^{2}.$

If $p>1,$ then it is very difficult to show that the validity of condition (B)
with ${\rm F}_{1}(l)\equiv l^{-1}$ and ${\rm F}(t)\equiv t^{-(1/p)}$ for a
Besicovitch-$p$-continuous function $F:{\mathbb{R}}\rightarrow Y$ implies the
validity of condition (A) for $F(\cdot),$ even in the scalar-valued case. In
actual fact, it is very simple to prove that (B) implies the validity of
equation obtained by replacing the term $|\cdot|$ in the equation [25, (4)]
with the term $|\cdot|^{p}.$ But, if we replace the term (we will consider the
scalar-valued case, only)

$\Biggl{|}c^{-1}\int^{c}_{0}\Biggl{[}n^{-1}\sum_{k=0}^{n-1}f(t+x+kc)-n^{-1}\sum_{k=0}^{n-1}f(x+kc)\Biggr{]}K_{m}(t)\,dt\Biggr{|}$

in the equation [25, (*), p. 136] with the term

$\displaystyle\Biggl{|}c^{-1}\int^{c}_{0}\Biggl{[}n^{-1}\sum_{k=0}^{n-1}f(t+x+kc)-n^{-1}\sum_{k=0}^{n-1}f(x+kc)\Biggr{]}K_{m}(t)\,dt\Biggr{|}^{p},$

which can be majorized by

$\displaystyle\leq
c^{-p}\int^{c}_{0}\Biggl{|}n^{-1}\sum_{k=0}^{n-1}f(t+x+kc)-n^{-1}\sum_{k=0}^{n-1}f(x+kc)\Biggr{|}^{p}\,dt\cdot\Biggl{(}\int^{c}_{0}K_{m}^{q}(t)\,dt\Biggr{)}^{p/q}$

with the help of the Hölder inequality, then it is impossible to control the
term $(\int^{c}_{0}K_{m}^{q}(t)\,dt)^{p/q}$ as $m\rightarrow+\infty.$ This can
be done only in the case that $p=1$ because the Fejér kernels

$K_{m}(t)=m^{-1}\sin^{2}(m\pi t/c)/\sin^{2}(\pi t/c),\quad t\in{\mathbb{R}}\ \
(m\in{\mathbb{N}})$

are uniformly integrable in a neighborhood of zero with respect to
$m\in{\mathbb{N}}$ but not uniformly $q$-integrable in a neighborhood of zero
with respect to $m\in{\mathbb{N}},$ if $q>1.$

Concerning the multi-dimensional setting, we will prove the following result:

######  Proposition 3.7.

Let $\Omega=[0,1]^{n},$ $\Lambda={\mathbb{R}}^{n}$ or
$\Lambda=[0,\infty)^{n},$ $\lim_{l\rightarrow+\infty}[l^{n-1}{\rm
F}_{1}(l)]=0,$ and let

(3.7) $\displaystyle\lim_{l\rightarrow+\infty}{\rm F}_{1}(l)\sup_{x\in
B}\int_{l\Omega}\Bigl{\|}F({\bf t};x)-P({\bf t};x)\Bigr{\|}_{Y}\,d{\bf
t}=0,\quad B\in{\mathcal{B}}.$

If there exists a finite real constant $c>0$ such that, for every real number
$\epsilon\in(0,1)$ and for every set $B\in{\mathcal{B}},$ there exists a real
number $l_{0}>0$ such that the assumptions $l\geq l_{0}$ and
${\mathcal{M}}_{B}^{x,{\rm F}}(F,P)<\epsilon$ imply the existence of a
sufficiently large number $t_{l}>0$ such that, for every $t\geq t_{l},$ we
have

(3.8) $\displaystyle{\rm F}_{1}(l){\rm F}(t)\sup_{x\in
B}\Biggl{\|}\int_{y+l\Omega}\Bigl{(}F({\bf t};x)-P({\bf t};x)\Bigr{)}\,d{\bf
t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}<c\epsilon,$

the collection ${\mathcal{B}}$ consists solely of bounded subsets of $X$ and
conditions _(I)-(III)_ hold, then condition _(B)_ holds for any function $F\in
e-({\mathcal{B}},{\rm F})-B^{p}(\Lambda\times X:Y).$

###### Proof.

Let $\epsilon>0$ and $B\in{\mathcal{B}}$ be given; we will consider the value
$\lambda=0$ in (B), only. It is clear that there exist a trigonometric
polynomial $P(\cdot;\cdot)$ and a finite real number $t_{0}>0$ such that, for
every real number $t\geq t_{0},$ we have

$\displaystyle{\rm F}(t)\sup_{x\in B}\bigl{\|}F({\bf t};x)-P({\bf
t};x)\bigr{\|}_{L^{p}(\Lambda_{t}:Y)}<\epsilon.$

Then we have ($l>0$):

$\displaystyle{\rm F}_{1}(l)$ $\displaystyle\limsup_{t\rightarrow+\infty}{\rm
F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}F({\bf
t};x)\,d{\bf t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}$ $\displaystyle\leq{\rm
F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}\Bigl{(}F({\bf
t};x)-P({\bf t};x)\Bigr{)}\,d{\bf t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}$
$\displaystyle+{\rm F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}P({\bf
t};x)\,d{\bf t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}$ $\displaystyle\leq{\rm
F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\int_{y+l\Omega}\Bigl{(}F({\bf t};x)-P({\bf t};x)\Bigr{)}\,d{\bf
t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}$ $\displaystyle+{\rm
F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\int_{l\Omega}\Bigl{(}F({\bf t};x)-P({\bf t};x)\Bigr{)}\,d{\bf
t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}$ $\displaystyle+{\rm
F}_{1}(l)\limsup_{t\rightarrow+\infty}{\rm F}(t)\sup_{x\in
B}\Biggl{\|}\Biggl{[}\int_{y+l\Omega}-\int_{l\Omega}\Biggr{]}P({\bf
t};x)\,d{\bf t}\Biggr{\|}_{L^{p}(\Lambda_{t}:Y)}.$

The third term in the last estimate can be majorized as in Proposition 3.6.
For the second term, we can use the assumptions (3.7) and
$\limsup_{t\rightarrow+\infty}{\rm F}(t)[m(\Lambda_{t})]^{1/p}<+\infty.$ For
the first term, we can use our assumption (3.8). ∎

It is not difficult to prove that, for every locally integrable function
$F:[0,\infty)^{n}\rightarrow Y,$ $t>0$ and $t_{0}\in(0,t),$ we have the
estimate

$\int_{\Omega_{0}}\int_{y+l\Omega}\|F({\bf s})\|_{Y}\,d{\bf
s}\,dy\leq\int_{(t+l)\Omega}\|F({\bf s})\|_{Y}\,d{\bf s},$

where $\Omega_{0}:=t\Omega\setminus[0,t_{0}]^{n}.$ Using the Hölder inequality
and repeating verbatim the argumentation given in the one-dimensional setting,
we can prove that the requirements of Proposition 3.7 hold with ${\rm
F}_{1}(l)\equiv l^{-n}$ and ${\rm F}(t)\equiv t^{-(n/p)}.$

Finally, it seems reasonable to ask whether the validity of condition (B) with
$p(\cdot)\equiv 1$ implies, along with the corresponding Besicovitch
continuity assumption, the validity of condition (A) in the multi-dimensional
setting. We will not consider this question here as well as certain
possibilities to extend the results established by A. S. Kovanko [47]-[48] to
the multi-dimensional setting.

##  4\. Applications to the abstract Volterra integro-differential equations

In this section, we will provide several applications of our results to the
various classes of abstract Volterra integro-differential equations and the
partial differential equations.

1\. In this part, we will first prove a new result about the invariance of
Besicovitch-$p$-almost periodicity under the actions of infinite convolution
product

(4.1) $\displaystyle t\mapsto F(t):=\int^{t}_{-\infty}R(t-s)f(s)\,ds,\quad
t\in{\mathbb{R}};$

we will only note here, without going into full details, that this result can
be formulated in the multi-dimensional setting as well ([41]). We assume that
the operator family $(R(t))_{t>0}\subseteq L(X,Y)$ satisfies that there exist
finite real constants $M>0$, $\beta\in(0,1]$ and $\gamma>1$ such that

(4.2) $\displaystyle\bigl{\|}R(t)\bigr{\|}_{L(X,Y)}\leq
M\frac{t^{\beta-1}}{1+t^{\gamma}},\quad t>0,$

and $f(\cdot)$ is Besicovitch-$p$-almost periodic.

The following result is closely connected with the statements of [40, Theorem
2.11.4, Theorem 2.13.10, Theorem 2.13.12]:

######  Proposition 4.1.

Suppose that the operator family $(R(t))_{t>0}\subseteq L(X,Y)$ satisfies
(4.2), as well as that $a>0,$ $\alpha>0,$ $1\leq p<+\infty,$ $\alpha p\geq 1,$
$ap\geq 1,$ $\alpha p(\beta-1)/(\alpha p-1)>-1$ if $\alpha p>1$, and $\beta=1$
if $\alpha p=1.$ If the function $f:{\mathbb{R}}\rightarrow X$ is
Stepanov-$(\alpha p)$-bounded, i.e.,

$\bigl{\|}f\bigr{\|}_{S^{p}}:=\sup_{t\in{\mathbb{R}}}\int^{t+1}_{t}\bigl{\|}f(s)\bigr{\|}^{\alpha
p}\,ds<+\infty,$

and $f\in e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}:X),$ then the function
$F(\cdot)$, given by (4.1), is bounded, continuous and belongs to the class
$e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}:Y).$

###### Proof.

Arguing as in the proof of [40, Proposition 2.6.11], we may conclude that the
function $F(\cdot)$ is well-defined, bounded and continuous. Let $(P_{k})$ be
a sequence of trigonometric polynomials such that

(4.3)
$\displaystyle\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}\frac{1}{2t^{ap}}\int^{t}_{-t}\bigl{\|}f(s)-P_{k}(s)\bigr{\|}^{\alpha
p}\,ds=0.$

Applying again [40, Proposition 2.6.11], we get that the function $t\mapsto
F_{k}(t)\equiv\int^{t}_{-\infty}R(t-s)P_{k}(s)\,ds,$ $t\in{\mathbb{R}}$ is
almost periodic and we only need to prove that

(4.4)
$\displaystyle\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}\frac{1}{2t^{ap}}\int^{t}_{-t}\bigl{\|}F(s)-F_{k}(s)\bigr{\|}^{\alpha
p}\,ds=0.$

In the remainder of the proof, we will consider case $\alpha p>1$ since the
consideration is quite similar if $\alpha p=1.$ Let $\zeta\in(1/(\alpha
p),(1/(\alpha p))+\gamma-\beta).$ Then it is clear that the function
$s\mapsto|s|^{\beta-1}(1+|s|)^{\zeta}/(1+|s|^{\gamma}),$ $s\in{\mathbb{R}}$
belongs to the space $L^{\alpha p/(\alpha p-1)}((-\infty,0));$ further on,
arguing as in the proof of [40, Theorem 2.11.4], we have that the function
$s\mapsto(1+|s|)^{-\zeta}\|P_{k}(s+z)-f(s+z)\|,$ $s\in{\mathbb{R}}$ belongs to
the space $L^{\alpha p}((-\infty,0))$ for all $k\in{\mathbb{N}}$ and
$z\in{\mathbb{R}}.$ The estimate (4.4) follows from the next computation
($M_{1}>0$ and $c>0$ are finite real constants):

$\displaystyle\frac{1}{2t^{ap}}$
$\displaystyle\int^{t}_{-t}\bigl{\|}F(s)-F_{k}(s)\bigr{\|}^{\alpha p}\,ds$
$\displaystyle\leq\frac{1}{2t^{ap}}\int^{t}_{-t}\Biggl{|}\int^{0}_{-\infty}\|R(-z)\|\cdot\bigl{\|}P_{k}(s+z)-f(s+z)\bigr{\|}\,dz\Biggr{|}^{\alpha
p}\,ds$
$\displaystyle\leq\frac{M}{2t^{ap}}\int^{t}_{-t}\Biggl{|}\int^{0}_{-\infty}\frac{|z|^{\beta-1}(1+|z|)^{\zeta}}{(1+|z|^{\gamma})}\cdot(1+|z|)^{-\zeta}\bigl{\|}P_{k}(s+z)-f(s+z)\bigr{\|}\,dz\Biggr{|}^{\alpha
p}\,ds$
$\displaystyle\leq\frac{M_{1}}{2t^{ap}}\int^{t}_{-t}\int^{0}_{-\infty}\frac{1}{(1+|z|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(s+z)-f(s+z)\bigr{\|}^{\alpha
p}\,dz\,ds$
$\displaystyle=\frac{M_{1}}{2t^{ap}}\int^{t}_{-t}\int^{t}_{z}\frac{1}{(1+|z-s|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,ds\,dz$
$\displaystyle+\frac{M_{1}}{2t^{ap}}\int^{t}_{-\infty}\int^{t}_{-t}\frac{1}{(1+|z-s|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,ds\,dz$
$\displaystyle\leq\frac{M_{1}}{t^{ap}}\int^{t}_{-t}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,dz\cdot\int^{+\infty}_{-\infty}\frac{ds}{(1+|s|^{\zeta})^{\alpha p}}$
$\displaystyle+\frac{M_{1}}{2t^{ap}}\int^{-3t}_{-\infty}\int^{t}_{-t}\frac{1}{(1+|z-s|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,ds\,dz$
$\displaystyle+\frac{M_{1}}{2t^{ap}}\int^{3t}_{-3t}\int^{t}_{-t}\frac{1}{(1+|z-s|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,ds\,dz$
$\displaystyle\leq\frac{M_{1}}{t^{ap}}\int^{t}_{-t}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,dz\cdot\int^{+\infty}_{-\infty}\frac{ds}{(1+|s|^{\zeta})^{\alpha p}}$
$\displaystyle+\frac{M_{1}}{t^{ap}}\int^{3t}_{-3t}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,dz\cdot\int^{+\infty}_{-\infty}\frac{ds}{(1+|s|^{\zeta})^{\alpha p}}$
$\displaystyle+\frac{cM_{1}t}{2t^{ap}}\int^{-3t}_{-\infty}\frac{1}{(1+|z/2|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,dz,$

involving the Hölder inequality, the Fubini theorem and an elementary change
of variables in the double integral; here we use (4.3) and the fact that

$\lim_{t\rightarrow+\infty}\int^{-3t}_{-\infty}\frac{1}{(1+|z/2|^{\alpha\zeta})^{p}}\bigl{\|}P_{k}(z)-f(z)\bigr{\|}^{\alpha
p}\,dz=0.$

∎

In the following result, the inhomogeneity $f(\cdot)$ is not necessarily
Stepanov-$(\alpha p)$-bounded:

######  Proposition 4.2.

Suppose that the operator family $(R(t))_{t>0}\subseteq L(X,Y)$ satisfies
(4.2), as well as that $a>0,$ $\alpha>0,$ $1\leq p<+\infty,$ $\alpha p\geq 1,$
$ap\geq 1,$ $\alpha p(\beta-1)/(\alpha p-1)>-1$ if $\alpha p>1$, and $\beta=1$
if $\alpha p=1.$ If the function $f:{\mathbb{R}}\rightarrow X$ belongs to the
class $e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}:X)$ and there exists a finite
real constant $M>0$ such that $\|f(t)\|\leq M(1+|t|)^{b},$ $t\in{\mathbb{R}}$
for some real constant $b\in[0,\gamma-\beta),$ then the function $F(\cdot)$,
given by (4.1), is continuous, belongs to the class
$e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}:Y),$ and there exists a finite real
constant $M^{\prime}>0$ such that $\|F(t)\|_{Y}\leq M^{\prime}(1+|t|)^{b},$
$t\in{\mathbb{R}}.$

###### Proof.

The proof is very similar to the proof of Proposition 4.1 and we will only
outline the main details. Since $b\in[0,\gamma-\beta),$ it can be simply shown
that the function $F(\cdot)$ is well-defined, measurable as well as that there
exists a finite real constant $M^{\prime}>0$ such that $\|F(t)\|_{Y}\leq
M^{\prime}(1+|t|)^{b},$ $t\in{\mathbb{R}}.$ In order to prove the continuity
of function $F(\cdot)$ at the fixed point $t\in{\mathbb{R}},$ we take first
any integer $k\in{\mathbb{N}}$ such that

(4.5)
$\displaystyle\int_{k}^{+\infty}\frac{s^{\beta-1}(1+s)^{b}}{1+s^{\gamma}}\,ds\leq\frac{\epsilon}{4M(1+|t|^{b})3^{b}},$

where a real number $\epsilon>0$ is given in advance and the real constant
$M>0$ has the same value as in (4.2). Since $\|R(\cdot)\|_{L(X,Y)}\in
L^{\alpha p/(\alpha p-1)}[0,k]$ and $\|f\|\in L^{\alpha p}_{loc}[0,k],$ we can
apply the Hölder inequality in order to see that the function
$F_{k}(\cdot):=\int^{k}_{0}R(s)f(\cdot-s)\,ds,$ $\cdot\in{\mathbb{R}}$ is
continuous. Take any $\delta\in(0,1)$ such that

(4.6)
$\displaystyle\Biggl{\|}\int^{k}_{0}R(s)\bigl{[}f(t-s)-f(t^{\prime}-s)\bigr{]}\,ds\Biggr{\|}_{Y}<\frac{\epsilon}{2}\
\ \mbox{ for }\ \ |t-t^{\prime}|\leq\delta.$

A very simple argumentation involving (4.5) shows that

$\Biggl{\|}\int_{k}^{+\infty}R(s)\bigl{[}f(t-s)-f(t^{\prime}-s)\bigr{]}\,ds\Biggr{\|}_{Y}\leq\epsilon/2,$

which along with (4.6) completes the proof of continuity of function
$F(\cdot)$ at the point $t.$ The belonging of function $F(\cdot)$ to the class
$e-(x^{\alpha},t^{-1/p})-B^{p}({\mathbb{R}}:Y)$ can be proved as above, by
taking any real number $\zeta\in((1/(\alpha p))+b,(1/(\alpha
p))+\gamma-\beta).$ ∎

######  Remark 4.3.

Suppose that there exist finite real constants $M>0,$ $c>0$ and
$\beta\in(0,1]$ such that

$\displaystyle\bigl{\|}R(t)\bigr{\|}_{L(X,Y)}\leq Me^{-ct}t^{\beta-1},\quad
t>0.$

This estimate appears in the analysis of (degenerate) semigroups of operators
satisfying condition [40, (P)] and, in particular, in the analysis of
semigroups of operators generated by almost sectorial operators. In this case,
we do not need the assumption $\alpha<\gamma-\beta.$

It is clear that Proposition 4.1 and Proposition 4.2 can be applied to a large
class of the abstract (degenerate) Volterra integro-differential equations
without initial conditions. For example, we can apply this result in the
analysis of the existence and uniqueness of Besicovitch-$p$-almost periodic
type solutions of the initial value problems with constant coefficients

$\begin{array}[]{l}D_{t,+}^{\gamma}u(t,x)=\sum_{|\alpha|\leq
k}a_{\alpha}f^{(\alpha)}(t,x)+f(t,x),\ t\in{\mathbb{R}},\
x\in\mathbb{R}^{n}\end{array}$

in the space $L^{p}(\mathbb{R}^{n}),$ where $\gamma\in(0,1),$
$D_{t,+}^{\gamma}u(t)$ denotes the Weyl-Liouville fractional derivative of
order $\gamma,$ $1\leq p<\infty$ and some extra assumptions are satisfied. We
can also consider the existence and uniqueness of Besicovitch-$p$-almost
periodic type solutions of the fractional Poisson heat equation in
$L^{p}(\mathbb{R}^{n}),$ and a class of the abstract fractional differential
equations with the higher-order elliptic operators in the Hölder spaces
([40]).

2\. In this part, we analyze the abstract nonautonomous differential equations
of first order. First of all, we will remind the readers of some basic
definitions from the theory of evolution equations, hyperbolic evolution
systems and Green’s functions (see [52] and the references cited in [40,
Section 2.14]).

######  Definition 4.4.

A family $\\{U(t,s):t\geq s,\ t,\ s\in{\mathbb{R}}\\}$ of bounded linear
operators on $X$ is said to be an evolution system if and only if the
following holds:

  * (i)

$U(s,s)=I,$ $U(t,s)=U(t,r)U(r,s)$ for $t\geq r\geq s$ and $t,\ r,\
s\in{\mathbb{R}},$

  * (ii)

$\\{(\tau,s)\in{\mathbb{R}}^{2}:\tau>s\\}\ni(t,s)\mapsto U(t,s)x$ is
continuous for any fixed element $x\in X$.

Here, $I$ denotes the identity operator on $X.$ We assume that the family
$A(\cdot)$ satisfies conditions [40, (H1)-(H2)]; then there exists an
evolution system $\\{U(t,s):t\geq s,\ t,\ s\in{\mathbb{R}}\\}$ generated by
the family $A(t).$ If $\Gamma(\cdot,\cdot)$ denotes the associated Green’s
function, then we know that there exists a finite real constant $M>0$ such
that

(4.7) $\displaystyle\|\Gamma(t,s)\|\leq Me^{-\omega|t-s|},\quad t,\
s\in{\mathbb{R}}.$

The function

(4.8) $\displaystyle u(t):=\int^{+\infty}_{-\infty}\Gamma(t,s)f(s)\,ds,\quad
t\in{\mathbb{R}}$

is said to be a unique mild solution of the abstract Cauchy problem

$\displaystyle u^{\prime}(t)=A(t)u(t)+f(t),\quad t\in{\mathbb{R}}.$

The proof of subsequent theorem can be deduced using the argumentation
employed in the proof of [40, Theorem 3.7.1], where we have assumed that
$p=1,$ the argumentation contained in the proof of Proposition 4.1, the
estimate (4.7), and the following facts:

  * (i)

If $P(\cdot)$ is a trigonometric polynomial, then the functions
$t\mapsto\int_{-\infty}^{t}\Gamma(t,s)P(s)\,ds,$ $t\in{\mathbb{R}}$ and
$t\mapsto\int^{+\infty}_{t}\Gamma(t,s)P(s)\,ds,$ $t\in{\mathbb{R}}$ are almost
periodic; see, e.g., the proof of [22, Theorem 2.3].

  * (ii)

We write
$\int^{+\infty}_{-\infty}\Gamma(t,s)f(s)\,ds=\int^{+\infty}_{t}\Gamma(t,s)f(s)\,ds+\int^{t}_{-\infty}\Gamma(t,s)f(s)\,ds,$
$t\in{\mathbb{R}}.$ The both addends can be considered similarly, while the
second addend is identically equal to
$\int_{-\infty}^{0}\Gamma(t,t+s)f(t+s)\,ds,$ $t\in{\mathbb{R}}.$

######  Theorem 4.5.

Suppose that $a>0,$ $\alpha>0,$ $1\leq p<+\infty,$ $\alpha p\geq 1$ and
$ap\geq 1.$ If the function $f:{\mathbb{R}}\rightarrow X$ is bounded and $f\in
e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}:X)$, then the function $u(\cdot)$,
given by (4.8), is bounded, continuous and belongs to the class
$e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}:X).$

3\. The use of Theorem 2.10 is almost mandatory in the analysis of the
existence and uniqueness of Besicovitch almost periodic type solutions for
some classes of the abstract semilinear Cauchy problems. The first part of
this result has a serious unpleasant drawback because we must impose that the
function $G:{\mathbb{R}}^{n}\times Y\rightarrow Z$ from its formulation is
Bohr ${\mathcal{B}}$-almost periodic, which automatically leads to the
existence and uniqueness of almost periodic solutions of the abstract
semilinear Cauchy problems under our consideration, in a certain sense. Here
we will present the following illustrative application of Theorem 2.10(ii),
with $\zeta=1,$ and Proposition 4.1.

Suppose that the operator family $(R(t))_{t>0}\subseteq L(X)$ satisfies (4.2),
as well as that $1\leq p,\ q<+\infty,$ $1/p+1/q=1,$ $q(\beta-1)>-1$ if $p>1$,
and $\beta=1$ if $p=1.$ If $(R(t))_{t>0}$ is a solution operator family which
governs solutions of the abstract fractional Cauchy inclusion
$D_{t,+}^{\gamma}u(t)\in{\mathcal{A}}u(t)+g(t),$ $t\in{\mathbb{R}},$ where
$\gamma\in(0,1),$ and a closed multivalued linear operator ${\mathcal{A}}$
satisfies condition [40, (P)], then it is usually said that a continuous
function $t\mapsto u(t),$ $t\in{\mathbb{R}}$ is a mild solution of the
abstract semilinear fractional Cauchy inclusion

$\text{(SCP):}\ \ \,D_{t,+}^{\gamma}u(t)\in{\mathcal{A}}u(t)+G(t;u(t)),\quad
t\in{\mathbb{R}}$

if and only if

$\displaystyle u(t)=\int^{t}_{-\infty}R(t-s)G(s;u(s))\,ds,\quad
t\in{\mathbb{R}}.$

Suppose that $G\in e-({\mathcal{B}},x,t^{-n/p})-B^{p}_{a,1}({\mathbb{R}}\times
X:X),$ where ${\mathcal{B}}$ denotes the family of all bounded subsets of $X.$
Suppose, further, that there exists a finite real constant $a>0$ such that
(2.5) holds with $\alpha=1$, $a\int^{\infty}_{0}\|R(s)\|\,ds<1,$ and
$\sup_{t\in{\mathbb{R}},x\in B}\|G(t;x)\|<+\infty$ for every bounded subset
$B$ of $X.$ It can be simply proved that the vector space
$C_{b}({\mathbb{R}}:X)\cap e-(x,t^{-1/p})-B^{p}({\mathbb{R}}:X)$ equipped with
the sup-norm is a Banach space. Applying Proposition 4.1(ii) and Theorem 2.10,
we get that the mapping $\Phi:C_{b}({\mathbb{R}}:X)\cap
e-(x,t^{-1/p})-B^{p}({\mathbb{R}}:X)\rightarrow C_{b}({\mathbb{R}}:X)\cap
e-(x,t^{-1/p})-B^{p}({\mathbb{R}}:X),$ given by $(\Phi
u)(t):=\int^{t}_{-\infty}R(t-s)G(s;u(s))\,ds,$ $t\in{\mathbb{R}},$ is a well-
defined contraction; therefore, there exists a unique mild solution $u(\cdot)$
of the abstract semilinear inclusion (SCP) which belongs to the space
$C_{b}({\mathbb{R}}:X)\cap e-(x,t^{-1/p})-B^{p}({\mathbb{R}}:X).$

4\. Our results about the invariance of Besicovitch-$p$-almost periodicity
under the actions of infinite convolution products can be also formulated for
the usual convolution

(4.9) $\displaystyle f\mapsto
F(x)\equiv\int_{{\mathbb{R}}^{n}}h(x-y)f(y)\,dy,\quad x\in{\mathbb{R}}^{n},$

provided that the function $h\in L^{1}({\mathbb{R}}^{n})$ has a certain growth
order. Before stating a general result in this direction, we will first
consider the inhomogeneous heat equation in ${\mathbb{R}}^{n}$ whose solutions
are governed by the action of Gaussian semigroup

(4.10) $\displaystyle F\mapsto(G(t)F)(x)\equiv\bigl{(}4\pi
t\bigr{)}^{-n/2}\int_{{\mathbb{R}}^{n}}e^{-|y|^{2}/4t}F(x-y)\,dy,\quad t>0,\
x\in{\mathbb{R}}^{n}.$

Without going into full details, we will only note that the formula (4.10)
makes sense even if the function $F(\cdot)$ is polynomially bounded; our basic
assumption will be that there exist two finite real numbers $b\geq 0$ and
$c>0$ such that $|F(x)|\leq c(1+|x|)^{b},$ $x\in{\mathbb{R}}^{n}$ as well as
that $a>0,$ $\alpha>0,$ $1\leq p<+\infty,$ $\alpha p\geq 1,$ $1/(\alpha
p)+1/q=1$ and $F\in
e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}^{n}:{\mathbb{C}}).$ Let us fix a real
number $t_{0}$ in (4.10). Then the mapping $x\mapsto(G(t_{0})F)(x),$
$x\in{\mathbb{R}}^{n}$ is well-defined and has the same growth as the
inhomogeneity $f(\cdot)$. Now we will prove that belongs to the class
$e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}^{n}:{\mathbb{C}})$ as well. Let
$\epsilon>0$, let

$c_{t_{0}}:=\bigl{(}4\pi
t_{0}\bigr{)}^{-n/2}\Bigl{\|}e^{-|\cdot|^{2}/8t_{0}}\Bigr{\|}_{L^{q}({\mathbb{R}}^{n})}^{\alpha
p},$

and let $\epsilon_{0}>0$ be such that

$\epsilon_{0}\cdot
2^{ap}c_{t_{0}}\int_{{\mathbb{R}}^{n}}e^{-|y|^{2}p/8t_{0}}\bigl{(}1+|y|\bigr{)}^{n}\,dy<\epsilon.$

We know that there exist a trigonometric polynomial $P(\cdot)$ and a finite
real number $t_{1}>0$ such that

$\int_{[-t,t]^{n}}\bigl{|}F(x)-P(x)\bigr{|}^{\alpha
p}\,dx<\epsilon_{0}t^{ap},\quad t\geq t_{1}.$

Furthermore, we know that the function $x\mapsto(G(t_{0})P)(x),$
$x\in{\mathbb{R}}^{n}$ is Bohr almost periodic (see [41, Subsection 6.1.7]) so
that the final conclusion simply follows from the next computation (see also
the computation from the proof of Proposition 4.1):

$\displaystyle\frac{1}{t^{ap}}$
$\displaystyle\int_{[-t,t]^{n}}\bigl{|}(G(t_{0})F)(x)-(G(t_{0})P)(x)\bigr{|}^{\alpha
p}\,dx$ (4.11)
$\displaystyle\leq\frac{c_{t_{0}}}{t^{ap}}\int_{[-t,t]^{n}}\int_{{\mathbb{R}}^{n}}e^{-|y|^{2}\alpha
p/8t_{0}}\bigl{|}F(x-y)-P(x-y)\bigr{|}^{\alpha p}\,dy\,dx$
$\displaystyle=\frac{c_{t_{0}}}{t^{ap}}\int_{{\mathbb{R}}^{n}}e^{-|y|^{2}\alpha
p/8t_{0}}\int_{[-t,t]^{n}}\bigl{|}F(x-y)-P(x-y)\bigr{|}^{\alpha p}\,dx\,dy$
$\displaystyle\leq\frac{c_{t_{0}}}{t^{ap}}\int_{{\mathbb{R}}^{n}}e^{-|y|^{2}\alpha
p/8t_{0}}\int_{[-t+|y|,t+|y|]^{n}}\bigl{|}F(x)-P(x)\bigr{|}^{\alpha
p}\,dx\,dy$
$\displaystyle\leq\frac{c_{t_{0}}}{t^{ap}}\int_{{\mathbb{R}}^{n}}e^{-|y|^{2}\alpha
p/8t_{0}}\epsilon_{0}2^{ap}\bigl{(}t^{ap}+|y|^{ap}\bigr{)}\,dy,\quad t\geq
t_{1}.$

The estimate (4.11) is obtained by writing the term
$e^{-|y|^{2}/4t_{0}}=e^{-|y|^{2}/8t_{0}}e^{-|y|^{2}/8t_{0}}$ and applying the
Hölder inequality after that. This can be also done in the general case;
arguing so, we can prove the following result:

######  Theorem 4.6.

Suppose that $b\geq 0,$ $\alpha>0,$ $a>0,$ $1\leq p<+\infty,$ $\alpha p\geq
1$, $1/(\alpha p)+1/q=1,$ $f\in
e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}^{n}:Y)$ and $\|f(x)\|_{Y}\leq
c(1+|x|)^{b},$ $x\in{\mathbb{R}}^{n}.$ If there exist two functions
$h_{1}:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}}$ and
$h_{2}:{\mathbb{R}}^{n}\rightarrow{\mathbb{C}}$ such that $h=h_{1}h_{2}$,
$h_{1}\in L^{q}({\mathbb{R}}^{n})$ and
$|h_{1}(\cdot)|^{\alpha}[1+|\cdot|]^{\zeta}\in L^{p}({\mathbb{R}}^{n})$ with
$\zeta=\max(b\alpha,a),$ then the function $F(\cdot),$ given by (4.9), belongs
to the class $e-(x^{\alpha},t^{-a})-B^{p}({\mathbb{R}}^{n}:Y)$ and has the
same growth order as $f(\cdot).$

We can simply apply Theorem 2.10 and Theorem 4.6 in the analysis of the
existence and uniqueness of bounded Besicovitch-$p$-almost periodic solutions
for a class of the semilinear Hammerstein integral equations of convolution
type on ${\mathbb{R}}^{n};$ see [41, p. 362] for more details. It would be
very tempting to incorporate Theorem 4.6 in the analysis of the existence and
uniqueness of Besicovitch-$p$-almost periodic solutions of the abstract ill-
posed Cauchy problems whose solutions are governed by integrated solution
operator families or $C$-regularized solution operator families (cf. [41, pp.
543–545] for more details).

In the remaining applications, we will consider the usual case $\phi(x)\equiv
x$ and the class of Besicovitch-$p$-almost periodic functions.

5\. Without going into full details, we would like to note that the
argumentation contained in our analysis of [41, Example 3, p. XXXV] enables
one to consider the existence and uniqueness of Besicovitch-$p$-almost
periodic type solutions of the wave equation in ${\mathbb{R}}^{2}$ whose
solutions are given by the famous d’Alembert formula. For example, if the
functions $f(\cdot)$ and $g^{[1]}(\cdot)$ from this example are
Besicovitch-$p$-almost periodic in ${\mathbb{R}}$, then the solution $u(x,t)$
will be Besicovitch-$p$-almost periodic in ${\mathbb{R}}^{2}.$ A similar
conclusion can be clarified for the solutions of the wave equation given by
the Kirchhoff (Poisson) formula; see [41] for more details.

6\. In this issue, we continue our analysis of the evolution systems
considered in the final application of [41, Section 6.3, pp. 426–428]. Suppose
that $Y:=L^{r}({\mathbb{R}}^{n})$ for some $r\in[1,\infty)$ and
$A(t):=\Delta+a(t){\rm I}$, $t\geq 0$, where $\Delta$ is the Dirichlet
Laplacian on $L^{r}(\mathbb{R}^{n}),$ and $a\in L^{\infty}([0,\infty))$. Then
the evolution system $(U(t,s))_{t\geq s\geq 0}\subseteq L(Y)$ generated by the
family $(A(t))_{t\geq 0}$ exists; this evolution system is given by
$U(t,t):=I$ for all $t\geq 0,$ and

$[U(t,s)F]({\bf u}):=\int_{{\mathbb{R}}^{n}}K(t,s,{\bf u},{\bf v})F({\bf
v})\,d{\bf v},\quad F\in L^{r}({\mathbb{R}}^{n}),\quad t>s\geq 0,$

where

$K(t,s,{\bf u},{\bf
v}):=\bigl{(}4\pi(t-s)\bigr{)}^{-\frac{n}{2}}e^{\int_{s}^{t}a(\tau)\,d\tau}\exp\Biggl{(}-\frac{|x-y|^{2}}{4(t-s)}\Biggr{)},\quad
t>s,\ {\bf u},\ {\bf v}\in\mathbb{R}^{n}.$

We know that, for every $\tau\in\mathbb{R}^{n}$, we have

$K(t,s,{\bf u}+\tau,{\bf v}+\tau)=K(t,s,{\bf u},{\bf v}),\quad t>s\geq 0,\
{\bf u},\ {\bf v}\in\mathbb{R}^{n},$

as well as that, under certain assumptions, a unique mild solution of the
abstract Cauchy problem $(\partial/\partial t)u(t,x)=A(t)u(t,x),$ $t>0;$
$u(0,x)=F(x)$ is given by $u(t,x):=[U(t,0)F](x),$ $t\geq 0,$
$x\in{\mathbb{R}}^{n}.$ Suppose now that $F(\cdot)\in Y$ is
Besicovitch-$p$-almost periodic for some finite exponent $p\geq 1.$ If
$\epsilon>0$ is given in advance, then we can find a finite real number
$t_{0}>0$ and a trigonometric polynomial $P(\cdot)$ such that

$\int_{[-t,t]^{n}}\bigl{|}F({\bf u})-P({\bf u})\bigr{|}^{p}\,d{\bf u}<\epsilon
t^{n},\quad t\geq t_{0}.$

The function $u_{P}(t,x):=[U(t,0)P](x),$ $t\geq 0,$ $x\in{\mathbb{R}}^{n}$ is
well-defined, continuous and satisfies that, if $\tau\in{\mathbb{R}}^{n}$ is
an $\epsilon$-almost period of $P(\cdot)$, then

$\displaystyle|u_{P}(t,{\bf u}+\tau)-u_{P}(t,{\bf
u})|=\Biggl{|}\int_{{\mathbb{R}}^{n}}\bigl{[}K(t,0,{\bf u}+\tau,{\bf
v})-K(t,0,{\bf u},{\bf v})\bigr{]}P({\bf v})\,d{\bf v}\Biggr{|}$
$\displaystyle=\Biggl{|}\int_{{\mathbb{R}}^{n}}K(t,0,{\bf u}+\tau,{\bf
v}+\tau)P({\bf v}+\tau)\,d{\bf v}-\int_{{\mathbb{R}}^{n}}K(t,0,{\bf u},{\bf
v})P({\bf v})\,d{\bf v}\Biggr{|}$
$\displaystyle=\Biggl{|}\int_{{\mathbb{R}}^{n}}K(t,0,{\bf u},{\bf
v})\bigl{[}P({\bf v}+\tau)\,d{\bf v}-P({\bf v})\bigr{]}\,d{\bf v}\Biggr{|}$
$\displaystyle\leq c_{t}\int_{{\mathbb{R}}^{n}}e^{-\frac{|{\bf u}-{\bf
v}|^{2}}{4t}}|P({\bf v}+\tau)-P({\bf v})|\,d{\bf v}$ $\displaystyle\leq
c_{t}\epsilon\int_{{\mathbb{R}}^{n}}e^{-\frac{|{\bf u}-{\bf
v}|^{2}}{4t}}\,d{\bf v}=c_{t}\epsilon\int_{{\mathbb{R}}^{n}}e^{-\frac{|{\bf
v}|^{2}}{4t}}\,d{\bf v},\quad t>0,\ {\bf u}\in{\mathbb{R}}^{n};$

hence, the function $u_{P}(t,\cdot)$ is almost periodic for every fixed real
number $t>0.$ Writing

$\int_{{\mathbb{R}}^{n}}K(t,0,{\bf u},{\bf v})\bigl{[}F({\bf v})-P({\bf
v})\bigr{]}\,d{\bf v}=\int_{{\mathbb{R}}^{n}}K(t,0,{\bf u},{\bf v}-{\bf
u})\bigl{[}F({\bf v}-{\bf u})-P({\bf v}-{\bf u})\bigr{]}\,d{\bf v},$

and the term $e^{-|{\bf v}|^{2}/4t}$ as $e^{-|{\bf v}|^{2}/8t}\cdot e^{-|{\bf
v}|^{2}/8t}$ in the corresponding computation after that, we may conclude as
before that the function $u(t,\cdot)$ is Besicovitch-$p$-almost periodic for
every fixed real number $t>0.$

7\. In this part, we will present certain applications of Proposition 2.7 in
the analysis of the existence and uniqueness of Besicovitch almost periodic
solutions for certain classes of PDEs; see, e.g., [59] and references cited
therein. We will revisit here the classical theories of quasi-linear partial
differential equations of first order and the linear partial differential
equations of second order with constant coefficients, considering solutions
defined on certain proper subsets $\Lambda$ of ${\mathbb{R}}^{n},$ where
$n\geq 2.$

7.1. It is well known that the general solution of equation $u_{x}+u_{y}=u$ is
given by $u(x,y)=g(y-x)e^{x},$ $(x,y)\in{\mathbb{R}}^{2},$ where
$g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a continuously differentiable
function. Suppose that $\Lambda:=(-\infty,0]\times{\mathbb{R}}$ and the
function $g(\cdot)$ is Besicovitch-$q$-almost periodic for some finite
exponent $q>1.$ Then there exists a sequence $(P_{k})$ of trigonometric
polynomials such that

$\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}t^{-1}\int_{-t}^{t}\bigl{|}P_{k}(s)-g(s)\bigr{|}^{q}\,ds=0.$

Since $Q_{k}(x,y):=P_{k}(x-y),$ $(x,y)\in{\mathbb{R}}^{2}$ is a sequence of
trigonometric polynomials of two variables, the above simply implies along
with an elementary argumentation that

$\lim_{k\rightarrow+\infty}\limsup_{t\rightarrow+\infty}t^{-2}\int_{-t}^{t}\int_{-t}^{t}\bigl{|}Q_{k}(x,y)-g(x-y)\bigr{|}^{q}\,dx\,dy=0,$

so that the function $G(x,y):=g(x-y),$ $(x,y)\in{\mathbb{R}}^{2}$ is
Besicovitch-$q$-almost periodic. Since the function $(x,y)\mapsto e^{x},$
$(x,y)\in\Lambda$ is Besicovitch-$p$-almost periodic for any finite exponent
$p\geq 1,$ an application of Proposition 2.7 yields that the solution $u(x,y)$
is Besicovitch-$r$-almost periodic on $\Lambda,$ for any exponent $r\in[1,q).$

7.2. Consider the linear partial differential equation of second order with
constant coefficients:

(4.12) $\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+2Du_{x}+2Eu_{y}+Fu=0,$

where $A,\ B,\ C,\ D,E,\ F$ are real constants such that $B>0,$ $C>0,$
$B^{2}\geq AC,$ $E^{2}\geq CF,$ $B^{2}>E^{2}-CF,$ and

$\bigl{(}BE-CD\bigr{)}^{2}=\bigl{(}B^{2}-AC\bigr{)}\bigl{(}E^{2}-CF\bigr{)}.$

As proposed by J. D. Kečkić in [37] (see also [38]), the general solution
$u(x,y)$ of the equation (4.12) is given by

$\displaystyle u(x,y)$
$\displaystyle=e^{\Bigl{(}-\frac{B}{C}+\frac{\sqrt{E^{2}-CF}}{C}\Bigr{)}y}f\Biggl{(}x+\Bigl{(}-\frac{E}{C}+\frac{1}{C}\sqrt{B^{2}-AC}\Bigr{)}y\Biggr{)}$
$\displaystyle+e^{\Bigl{(}-\frac{B}{C}-\frac{\sqrt{E^{2}-CF}}{C}\Bigr{)}y}g\Biggl{(}x+\Bigl{(}-\frac{E}{C}-\frac{1}{C}\sqrt{B^{2}-AC}\Bigr{)}y\Biggr{)},$

where $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ and
$g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ are arbitrary two times continuously
differentiable functions. Suppose that $\Lambda={\mathbb{R}}\times[0,+\infty)$
and the functions $f(\cdot)$ and $g(\cdot)$ are Besicovitch-$q$-almost
periodic for some finite exponent $q>1.$ Arguing as above, an application of
Proposition 2.7 yields that the solution $u(x,y)$ of (4.12) is
Besicovitch-$r$-almost periodic on $\Lambda,$ for any exponent $r\in[1,q).$

We close this section with the observation that it is not clear how one can
prove that the solutions considered above are Besicovitch-$q$-almost periodic.

##  5\. Conclusions and final remarks

In this paper, we have considered various classes of multi-dimensional
Besicovitch almost periodic type functions. The main structural properties of
multi-dimensional Besicovitch almost periodic type functions are established.
Many structural results known in the one-dimensional setting are reconsidered
and extended to the functions of the form $F:\Lambda\times X\rightarrow Y,$
where $\emptyset\neq\Lambda\subseteq{\mathbb{R}}^{n},$ $X$ and $Y$ are complex
Banach spaces. We have also furnished many illustrative examples, useful
remarks and applications to the abstract Volterra integro-differential
equations.

Concerning certain drawbacks of this work, we would like to note that we have
not considered here the differentiation and integration of multi-dimensional
Besicovitch almost periodic type functions as well as the dual spaces of
multi-dimensional Besicovitch almost periodic type functions. Besides many
other topics, in this paper we have not reconsidered the notions of
${\mathcal{M}}^{p}$-almost periodicity and ${\mathcal{M}}^{p}$-regularity in
the multi-dimensional setting as well (see [3] and [7] for more details about
the subject). The class of metrical multi-dimensional Besicovitch almost
periodic functions will be considered somewhere else following the general
approach obeyed in [45].

The class of multi-dimensional $\rho$-almost periodic type functions,
extending the class of multi-dimensional $c$-almost periodic type functions
when $\rho=c{\rm I},$ have recently been investigated by M. Fečkan et al.
[27]; see also [43]-[44]. The class of Besicovitch-$(p,c)$-almost periodic
functions, where $1\leq p<\infty$ and $c\neq 1$, has not been analyzed in the
existing literature so far, even in the one-dimensional setting. At the end of
paper, we will briefly explain that the method proposed in the proof of [39,
Proposition 2.8], a fundamental result about $c$-almost periodic functions
saying that any $c$-almost periodic function $f:{\mathbb{R}}\rightarrow Y$ is
almost periodic, cannot be essentially employed in the analysis of
corresponding classes of Besicovitch-$(p,c)$-almost periodic type functions:

######  Example 5.1.

Suppose that $p\geq 1,$ $c\in{\mathbb{C}}\setminus\\{0\\},$
$m\in{\mathbb{N}},$ $c^{m}=1,$ and the function
$f:{\mathbb{R}}\rightarrow{\mathbb{C}}$ is Besicovitch-$(p,c)$-almost
periodic, i.e., for every $\epsilon>0$ there exists a satisfactorily uniform
set $A=\\{\tau_{i}:i\in{\mathbb{Z}}\\}\subseteq{\mathbb{R}}$ such that

(5.1)
$\displaystyle\limsup_{t\rightarrow+\infty}\Biggl{(}\frac{1}{2t}\int^{t}_{-t}\bigl{|}f(s+\tau_{i})-cf(s)\bigr{|}^{p}\,ds\Biggr{)}^{1/p}<\epsilon$

and, for every $l>0,$

(5.2)
$\displaystyle\limsup_{t\rightarrow+\infty}\Biggl{(}\frac{1}{2t}\int^{t}_{-t}\Biggl{[}\limsup_{k\rightarrow+\infty}\frac{1}{2k+1}\sum_{i=-k}^{k}l^{-1}\int^{x+l}_{x}\bigl{|}f(s+\tau_{i})-cf(s)\bigr{|}^{p}\,ds\Biggr{]}\,dx\Biggr{)}^{1/p}<\epsilon;$

cf. [8] for the notion of a satisfactorily uniform set in ${\mathbb{R}}.$ Then
the set $mA$ is also satisfactorily uniform and for each $i\in{\mathbb{Z}}$ we
have:

$\displaystyle\int^{t}_{-t}$
$\displaystyle\bigl{|}f(s+m\tau_{i})-f(s)\bigr{|}^{p}\,ds$
$\displaystyle\leq\int_{-t}^{t}\Biggl{(}\sum_{j=0}^{m-1}|c|^{j}\bigl{|}f(s+(m-j)\tau_{i})-cf(s+(m-j-1)\tau_{i})\bigr{|}\Biggr{)}^{p}\,ds$
$\displaystyle\leq
c_{m,p}\sum_{j=0}^{m-1}\int^{t+(m-j-1)\tau_{i}}_{-t+(m-j-1)\tau_{i}}\bigl{|}f(s+\tau_{i})-cf(s)\bigr{|}^{p}\,ds$
$\displaystyle\leq
2mc_{m,p}\epsilon^{p}\Bigl{(}t+\bigl{|}t+\tau_{i}\bigr{|}+...+|t+(m-1)\tau_{i}|\Bigr{)}$
$\displaystyle\leq
2mc_{m,p}\epsilon^{p}\Bigl{(}mt+\frac{m^{2}}{2}|\tau_{i}|\Bigr{)}\leq\mbox{Const.}\cdot
2t\epsilon^{p},\quad t\geq t_{0}(\epsilon,i),$

for some finite real constants $c_{m,p}>0$ and $t_{0}(\epsilon,i)>0.$
Therefore, (5.1) holds with the number $c$ replaced by the number $1$ therein,
which can be simply transferred to the multi-dimensional setting. But, it is
not clear how we can deduce the corresponding conclusion for the equation
(5.2); arguing as above, we can only prove that there exists a finite real
constant $c_{m,p}>0$ such that:

$\displaystyle\int^{t}_{-t}$
$\displaystyle\Biggl{[}\limsup_{k\rightarrow+\infty}\frac{1}{2k+1}\sum_{i=-k}^{k}l^{-1}\int^{x+l}_{x}\bigl{|}f(s+\tau_{i})-f(s)\bigr{|}^{p}\,ds\Biggr{]}\,dx$
$\displaystyle\leq
c_{m,p}\Biggl{\\{}\int^{t}_{-t}\Biggl{[}\limsup_{k\rightarrow+\infty}\frac{1}{2k+1}\sum_{i=-k}^{k}l^{-1}\int^{x+l}_{x}\bigl{|}f(s+\tau_{i})-cf(s)\bigr{|}^{p}\,ds\Biggr{]}\,dx$
$\displaystyle+\int^{t}_{-t}\Biggl{[}\limsup_{k\rightarrow+\infty}\frac{1}{2k+1}\sum_{i=-k}^{k}l^{-1}\int^{x+\tau_{i}+l}_{x+\tau_{i}}\bigl{|}f(s+\tau_{i})-cf(s)\bigr{|}^{p}\,ds\Biggr{]}\,dx$
$\displaystyle+...+\int^{t}_{-t}\Biggl{[}\limsup_{k\rightarrow+\infty}\frac{1}{2k+1}\sum_{i=-k}^{k}l^{-1}\int^{x+(m-1)\tau_{i}+l}_{x+(m-1)\tau_{i}}\bigl{|}f(s+\tau_{i})-cf(s)\bigr{|}^{p}\,ds\Biggr{]}\,dx\Biggr{\\}},$

which seems to be completely inapplicable for our purposes.

To the best knowledge of the author, the class of Besicovitch-$p$-almost
periodic functions in ${\mathbb{R}}^{n}$ has not yet been considered in a Bohr
like manner ($n\geq 2$). For further information concerning Besicovitch almost