Continuous characterization of the Besov spaces of variable smoothness and integrability
UDC 517.9 We obtain new equivalent quasinorms of the Besov spaces of variable smoothness and integrability.  Our main tools are the continuous version of the Calderón reproducing formula, maximal inequalities, and the variable-exponent technique; however, allow...
Gespeichert in:
| Datum: | 2023 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2023
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6578 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512452881416192 |
|---|---|
| author | Benmahmoud, S. Drihem, D. Benmahmoud, S. Drihem, D. |
| author_facet | Benmahmoud, S. Drihem, D. Benmahmoud, S. Drihem, D. |
| author_sort | Benmahmoud, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-01-23T14:02:44Z |
| description | UDC 517.9
We obtain new equivalent quasinorms of the Besov spaces of variable smoothness and integrability.  Our main tools are the continuous version of the Calderón reproducing formula, maximal inequalities, and the variable-exponent technique; however, allowing the parameters to vary from point to point leads to additional difficulties which, in general, can be overcome by imposing regularity assumptions on these exponents. |
| doi_str_mv | 10.37863/umzh.v74i12.6578 |
| first_indexed | 2026-03-24T03:29:01Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i12.6578
UDC 517.9
S. Benmahmoud, D. Drihem1 (M’sila Univ., Laboratory Funct. Analysis and Geometry Spaces, Algeria)
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES
OF VARIABLE SMOOTHNESS AND INTEGRABILITY
НЕПЕРЕРВНА ХАРАКТЕРИЗАЦIЯ ПРОСТОРIВ БЄСОВА
ЗМIННОЇ ГЛАДКОСТI ТА IНТЕГРОВНОСТI
We obtain new equivalent quasinorms of the Besov spaces of variable smoothness and integrability. Our main tools are
the continuous version of the Calderón reproducing formula, maximal inequalities, and the variable-exponent technique;
however, allowing the parameters to vary from point to point leads to additional difficulties which, in general, can be
overcome by imposing regularity assumptions on these exponents.
Отримано новi еквiвалентнi квазiнорми просторiв Бєсова змiнної гладкостi та iнтегровностi. Нашi основнi iн-
струменти — це неперервна версiя формули вiдтворення Калдерона, максимальнi нерiвностi та технiка змiнної
експоненти. Зазначимо, що дозвiл для параметрiв змiнюватися вiд точки до точки викликає додатковi труднощi, якi,
як правило, можна подолати шляхом накладення припущень регулярностi на вiдповiднi експоненти.
1. Introduction. Besov spaces of variable smoothness and integrability initially appeared in the
paper of A. Almeida and P. Hästö [3], where several basic properties were shown, such as the
Fourier analytical characterization. Later the author [9] characterized these spaces by local means
and established the atomic characterization. After that, Kempka and Vybı́ral [14] characterized these
spaces by ball means of differences and also by local means. The duality of these function spaces is
given in [12, 16].
The interest in these spaces comes not only from theoretical reasons but also from their applica-
tions to several classical problems in analysis. For further considerations of PDEs, we refer to [8]
and references therein.
The main aim of this paper is to present new equivalent quasinorm of these function spaces, which
based on the continuous version of Calderón reproducing formula. Firstly, we define new family of
function spaces and prove their basic properties. Secondly, under some suitable assumptions on the
parameters we prove that these function spaces are just the Besov spaces of variable smoothness
and integrability of Almeida and Hästö. Finally we characterize these function spaces in terms of
continuous local means.
This paper needs some notation. As usual, we denote by \BbbN 0 the set of all nonnegative integers.
The notation f \lesssim g means that f \leq cg for some independent positive constant c (and nonnegative
functions f and g), and f \approx g means that f \lesssim g \lesssim f. For x \in \BbbR , \lfloor x\rfloor stands for the largest integer
smaller than or equal to x.
If E \subset \BbbR n is a measurable set, then | E| stands for the Lebesgue measure of E and \chi E denotes
its characteristic function. By c we denote generic positive constants, which may have different
values at different occurrences. Although the exact values of the constants are usually irrelevant for
our purposes, sometimes we emphasize their dependence on certain parameters (e.g., c(p) means that
c depends on p, etc.).
1 Corresponding author, e-mails: douadidr@yahoo.fr, salahmath2016@gmail.com.
c\bigcirc S. BENMAHMOUD, D. DRIHEM, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12 1601
1602 S. BENMAHMOUD, D. DRIHEM
The symbol \scrS (\BbbR n) is used in place of the set of all Schwartz functions on \BbbR n. We define the
Fourier transform of a function f \in \scrS (\BbbR n) by
\scrF (f)(\xi ) := (2\pi ) - n/2
\int
\BbbR n
e - ix\cdot \xi f(x)dx, \xi \in \BbbR n.
We denote by \scrS \prime (\BbbR n) the dual space of all tempered distributions on \BbbR n. The variable exponents
that we consider are always measurable functions p on \BbbR n with range in (0,\infty ]. We denote by
\scrP 0(\BbbR n) the set of such functions bounded away from the origin (i.e., p - > 0). The subset of
variable exponents with range in [1,\infty ] is denoted by \scrP (\BbbR n). We use the standard notation
p - := \mathrm{e}\mathrm{s}\mathrm{s}-\mathrm{i}\mathrm{n}\mathrm{f}
x\in \BbbR n
p(x) and p+ := \mathrm{e}\mathrm{s}\mathrm{s}-\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n
p(x).
We put
\omega p(t) =
\left\{
tp, if p \in (0,\infty ) and t > 0,
0, if p = \infty and 0 < t \leq 1,
\infty , if p = \infty and t > 1.
The variable exponent modular is defined by
\varrho p(\cdot )(f) :=
\int
\BbbR n
\omega p(x)(| f(x)| ) dx.
The variable exponent Lebesgue space Lp(\cdot ) consists of measurable functions f on \BbbR n such that
\varrho p(\cdot )(\lambda f) <\infty for some \lambda > 0. We define the Luxemburg (quasi)norm on this space by the formula
\| f\| p(\cdot ) := \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\lambda > 0 : \varrho p(\cdot )
\biggl(
f
\lambda
\biggr)
\leq 1
\biggr\}
.
A useful property is that \| f\| p(\cdot ) \leq 1 if and only if \varrho p(\cdot )(f) \leq 1 (see Lemma 3.2.4 from [8]).
Let p, q \in \scrP 0(\BbbR n). The mixed Lebesgue-sequence space \ell q(\cdot )(Lp(\cdot )) is defined on sequences of
Lp(\cdot )-functions by the modular
\varrho \ell q(\cdot )(Lp(\cdot ))((fv)v) :=
\infty \sum
v=0
\mathrm{i}\mathrm{n}\mathrm{f}
\Biggl\{
\lambda v > 0 : \varrho p(\cdot )
\Biggl(
fv
\lambda
1/q(\cdot )
v
\Biggr)
\leq 1
\Biggr\}
.
The (quasi)norm is defined from this as usual:\bigm\| \bigm\| (fv)v\bigm\| \bigm\| \ell q(\cdot )(Lp(\cdot ))
:= \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\mu > 0 : \varrho \ell q(\cdot )(Lp(\cdot ))
\biggl(
1
\mu
(fv)v
\biggr)
\leq 1
\biggr\}
. (1)
If q+ <\infty , then we can replace (1) by a simpler expression
\varrho \ell q(\cdot )(Lp(\cdot ))((fv)v) =
\infty \sum
v=0
\bigm\| \bigm\| | fv| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
.
We use this notation even when q+ = \infty . Let (ft)0<t\leq 1 be a sequence of measurable functions when
t is a continuous variable. We set
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1603
\varrho \widetilde \ell q(\cdot )(Lp(\cdot ))
((ft)0<t\leq 1) :=
1\int
0
\mathrm{i}\mathrm{n}\mathrm{f}
\Biggl\{
\lambda t : \varrho p(\cdot )
\Biggl(
ft
\lambda
1/q(\cdot )
t
\Biggr)
\leq 1
\Biggr\}
dt
t
.
The (quasi)norm is defined by\bigm\| \bigm\| (ft)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
:= \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\mu > 0 : \varrho \widetilde \ell q(\cdot )(Lp(\cdot ))
\biggl(
1
\mu
(ft)0<t\leq 1
\biggr)
\leq 1
\biggr\}
.
We say that a real valued-function g on \BbbR n is locally log-Hölder continuous on \BbbR n, abbreviated
g \in C \mathrm{l}\mathrm{o}\mathrm{g}
loc (\BbbR
n), if there exists a constant c\mathrm{l}\mathrm{o}\mathrm{g}(g) > 0 such that\bigm| \bigm| g(x) - g(y)
\bigm| \bigm| \leq c\mathrm{l}\mathrm{o}\mathrm{g}(g)
\mathrm{l}\mathrm{o}\mathrm{g}(e+ 1/| x - y| )
(2)
for all x, y \in \BbbR n.
We say that g satisfies the log-Hölder decay condition, if there exist two constants g\infty \in \BbbR and
c\mathrm{l}\mathrm{o}\mathrm{g} > 0 such that \bigm| \bigm| g(x) - g\infty
\bigm| \bigm| \leq c\mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{g}(e+ | x| )
for all x \in \BbbR n. We say that g is globally log-Hölder continuous on \BbbR n, abbreviated g \in C \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n),
if it is locally log-Hölder continuous on \BbbR n and satisfies the log-Hölder decay condition. The con-
stants c\mathrm{l}\mathrm{o}\mathrm{g}(g) and c\mathrm{l}\mathrm{o}\mathrm{g} are called the locally log-Hölder constant and the log-Hölder decay constant,
respectively. We note that any function g \in C \mathrm{l}\mathrm{o}\mathrm{g}
loc (\BbbR
n) always belongs to L\infty .
We define the following class of variable exponents:
\scrP \mathrm{l}\mathrm{o}\mathrm{g}
0 (\BbbR n) :=
\biggl\{
p \in \scrP 0(\BbbR n) :
1
p
\in C \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n)
\biggr\}
,
which is introduced in [6] (Section 2). The class \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) is defined analogously. We define
1
p\infty
:= \mathrm{l}\mathrm{i}\mathrm{m}
| x| \rightarrow \infty
1
p(x)
,
and we use the convention
1
\infty
= 0. Note that although
1
p
is bounded, the variable exponent p itself
can be unbounded. We put
\Psi (x) := \mathrm{s}\mathrm{u}\mathrm{p}
| y| \geq | x|
\bigm| \bigm| \varphi (y)\bigm| \bigm|
for \varphi \in L1. We suppose that \Psi \in L1. Then it was proved in [8] (Lemma 4.6.3) that if p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n),
then
\| \varphi \varepsilon \ast f\| p(\cdot ) \leq c\| \Psi \| 1\| f\| p(\cdot )
for all f \in Lp(\cdot ), where
\varphi \varepsilon :=
1
\varepsilon n
\varphi
\Bigl( \cdot
\varepsilon
\Bigr)
, \varepsilon > 0.
We put
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1604 S. BENMAHMOUD, D. DRIHEM
\eta t,m(x) := t - n(1 + t - 1| x| ) - m
for any x \in \BbbR n, t > 0 and m > 0. Note that \eta t,m \in L1, when m > n, and
\bigm\| \bigm\| \eta t,m\bigm\| \bigm\| 1 = c(m) is
independent of t. If t = 2 - v, v \in \BbbN 0, then we put
\eta v,m := \eta 2 - v ,m.
We refer to the recent monograph [5] for further properties, historical remarks and references on
variable exponent spaces.
2. Basic tools. In this section, we present some useful results. The following lemma is proved
in [7] (Lemma 6.1) (see also [14], Lemma 19).
Lemma 1. Let \alpha \in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n), m \in \BbbN 0 and R \geq c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ), where c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) is the constant from
(2), for g = \alpha . Then there exists a constant c > 0 such that
t - \alpha (x)\eta t,m+R(x - y) \leq ct - \alpha (y)\eta t,m(x - y)
for any 0 < t \leq 1 and x, y \in \BbbR n.
The previous lemma allows us to treat the variable smoothness in many cases as if it were not
variable at all. Namely, we can move the factor t - \alpha (x) inside the convolution as follows:
t - \alpha (x)\eta t,m+R \ast f(x) \leq c\eta t,m \ast (t - \alpha (\cdot )f)(x).
The following lemma is from [22] (Lemma 3.14).
Lemma 2. Let p, q \in \scrP 0(\BbbR n) and f be a measurable function on \BbbR n. If\bigm\| \bigm\| | f | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
\geq 1,
then
\| f\| q
-
p(\cdot ) \leq
\bigm\| \bigm\| | f | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
.
The next lemma is a Hardy-type inequality, see [15].
Lemma 3. Let s > 0 and (\varepsilon t)0<t\leq 1 be a sequence of positive measurable functions, when t is
a continuous variable. Let
\eta t = ts
1\int
t
\tau - s\varepsilon \tau
d\tau
\tau
and \delta t = t - s
t\int
0
\tau s\varepsilon \tau
d\tau
\tau
.
Then there exists a constant c > 0 depending only on s such that
1\int
0
\eta t
dt
t
+
1\int
0
\delta t
dt
t
\leq c
1\int
0
\varepsilon t
dt
t
.
Lemma 4. Let r,N > 0, m > n and \theta , \omega \in \scrS (\BbbR n) with \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \omega \subset B(0, 1). Then there exists
a constant c = c(r,m, n) > 0 such that, for all g \in \scrS \prime (\BbbR n), we have\bigm| \bigm| \theta N \ast \omega N \ast g(x)
\bigm| \bigm| \leq c
\bigl(
\eta N,m \ast | \omega N \ast g| r(x)
\bigr) 1/r
, x \in \BbbR n,
where \theta N (\cdot ) := Nn\theta (N \cdot ), \omega N (\cdot ) := Nn\omega (N \cdot ) and \eta N,m := Nn(1 +N | \cdot | ) - m.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1605
The proof of this lemma is given in [10] (Lemma 2.2). The following lemma is from A. Almeida
and P. Hästö [3] (Lemma 4.7) (we use it, since the maximal operator is in general not bounded on
\ell q(\cdot )(Lp(\cdot )), see [3], Example 4.1).
Lemma 5. Let p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) and q \in \scrP 0(\BbbR n) with
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n). For m > n + c\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
q
\biggr)
,
there exists c > 0 such that\bigm\| \bigm\| (\eta v,m \ast fv)v
\bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
\leq c\| (fv)v\| \ell q(\cdot )(Lp(\cdot )).
Lemma 6. Let 0 < \alpha < \beta <\infty , p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) and q \in \scrP (\BbbR n) with
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n). Let
gt(x) :=
\beta t\int
\alpha t
\eta \tau ,m \ast f\tau (x)
d\tau
\tau
, t \in (0, 1], x \in \BbbR n.
(i) Assume that 0 < \beta t \leq 1. The inequality
\bigm\| \bigm\| | cgt| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
\leq
\beta t\int
\alpha t
\bigm\| \bigm\| | f\tau | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
d\tau
\tau
+ t, t \in (0, 1],
holds for every sequence of functions (ft)0<t\leq 1 and constant m > n + c\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
q
\biggr)
such that the first
term on right-hand side is at most one, where the constant c independent of t.
(ii) The inequality \bigm\| \bigm\| (gt)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
\leq c
\bigm\| \bigm\| (ft)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
holds for every sequence of functions (ft)0<t\leq 1 and constant m > n + c\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
q
\biggr)
such that the
right-hand side is finite.
Proof. First let us prove (i). The claim can be reformulated as showing that
J :=
\bigm\| \bigm\| \bigm\| c1\delta - 1
q(\cdot ) gt
\bigm\| \bigm\| \bigm\|
p(\cdot )
\leq 2
1 - 1
q - + \mathrm{l}\mathrm{n}
\beta
\alpha
, t \in (0, 1],
where c1 > 0 and \delta :=
\int \beta t
\alpha t
\bigm\| \bigm\| | f\tau | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
d\tau
\tau
+ t. Applying Lemma 1, with an appropriate choice of
c1, we get
J \leq
\beta t\int
\alpha t
\bigm\| \bigm\| c1\delta - 1
q(\cdot ) (\eta \tau ,m \ast f\tau )
\bigm\| \bigm\|
p(\cdot )
d\tau
\tau
\leq
\leq
\beta t\int
\alpha t
\bigm\| \bigm\| \eta
\tau ,m - c\mathrm{l}\mathrm{o}\mathrm{g}
\bigl(
1
q
\bigr) \ast c1\delta - 1
q(\cdot ) | f\tau |
\bigm\| \bigm\|
p(\cdot )
d\tau
\tau
\leq
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1606 S. BENMAHMOUD, D. DRIHEM
\leq
\beta t\int
\alpha t
\bigm\| \bigm\| \delta - 1
q(\cdot ) f\tau
\bigm\| \bigm\|
p(\cdot )
d\tau
\tau
,
since \delta \in (t, 1 + t] and the convolution with a radially decreasing L1-function is bounded in Lp(\cdot ),
since m > n+ c\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
q
\biggr)
. Write
\beta t\int
\alpha t
\bigm\| \bigm\| \bigm\| \delta - 1
q(\cdot ) f\tau
\bigm\| \bigm\| \bigm\|
p(\cdot )
d\tau
\tau
=
\int
(\alpha t,\beta t]\cap B
. . .
d\tau
\tau
+
\int
(\alpha t,\beta t]\cap Bc
\cdot \cdot \cdot d\tau
\tau
= J1,t + J2,t,
where
B :=
\Biggl\{
\tau > 0 :
\bigm\| \bigm\| \bigm\| | \delta - 1
q(\cdot ) f\tau | q(\cdot )
\bigm\| \bigm\| \bigm\|
p(\cdot )
q(\cdot )
\geq 1
\Biggr\}
.
By Lemma 2, we have
J1,t \leq
\int
(\alpha t,\beta t]\cap B
\bigm\| \bigm\| \bigm\| | \delta - 1
q(\cdot ) f\tau | q(\cdot )
\bigm\| \bigm\| \bigm\| 1
q -
p(\cdot )
q(\cdot )
d\tau
\tau
\leq 2
1 - 1
q - \delta - 1
\beta t\int
\alpha t
\bigm\| \bigm\| \bigm\| | f\tau | q(\cdot )\bigm\| \bigm\| \bigm\| p(\cdot )
q(\cdot )
d\tau
\tau
\leq 2
1 - 1
q -
and
J2,t \leq
\beta t\int
\alpha t
\bigm\| \bigm\| \bigm\| \delta - 1
q(\cdot ) f\tau
\bigm\| \bigm\| \bigm\|
p(\cdot )
d\tau
\tau
\leq
\beta t\int
\alpha t
d\tau
\tau
= \mathrm{l}\mathrm{n}
\beta
\alpha
.
Now we prove (ii). By the scaling argument, it suffices to consider the case\bigm\| \bigm\| (ft)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
= 1
and show that the modular of f on the left-hand side is bounded. In particular, we show that
1\int
0
\bigm\| \bigm\| \bigm\| | cgt| q(\cdot )\bigm\| \bigm\| \bigm\| p(\cdot )
q(\cdot )
dt
t
\leq 2
for some positive constant c. Applying Hardy inequality (see Lemma 3 and the property (i)), we
obtain the desired result.
Lemma 7. Let 0 < r <\infty and m > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
n,
n
r
\biggr)
. Let \{ \scrF \Phi ,\scrF \varphi \} be a resolution of unity (see
Section 3)
\scrF \Phi (\xi ) +
1\int
0
\scrF \varphi (t\xi )dt
t
= 1, \xi \in \BbbR n.
(i) Let \theta \in \scrS (\BbbR n) be such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \theta \subset
\bigl\{
\xi \in \BbbR n : | \xi | \leq 2
\bigr\}
. There exists a constant c > 0
such that
| \theta \ast f | r \leq c\eta 1,mr \ast | \Phi \ast f | r + c
1\int
1/4
\eta 1,mr \ast | \varphi \tau \ast f | r
d\tau
\tau
for any f \in \scrS \prime (\BbbR n), where \varphi \tau = \tau - n\varphi
\biggl(
\cdot
\tau
\biggr)
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1607
(ii) Let \omega \in \scrS (\BbbR n) be such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \omega \subset
\biggl\{
\xi \in \BbbR n :
1
2
\leq | \xi | \leq 2
\biggr\}
. There exists a constant
c > 0 such that
| \omega t \ast f | r \leq c\eta 1,mr \ast | \Phi \ast f | r + c
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\eta \tau ,mr \ast | \varphi \tau \ast f | r
d\tau
\tau
for any f \in \scrS \prime (\BbbR n) and any 0 < t \leq 1, where \omega t = t - n\omega
\biggl(
\cdot
t
\biggr)
.
Proof. We split the proof into two steps. First the case 1 \leq r < \infty follows by the Hölder
inequality.
Step 1. Proof of (i). Since \{ \scrF \Phi ,\scrF \varphi \} is a resolutions of unity, it follows that
\theta \ast f = \Phi \ast \theta \ast f +
1\int
1/4
\theta \ast \varphi \tau \ast f
d\tau
\tau
.
First recall the elementary inequality
dn\eta d,m(y - z) \leq d2n\eta d, - m(y - x)\eta d,m(x - z), d > 0, x, y, z \in \BbbR n,
which together with Lemma 4 implies that
| \Phi \ast \theta \ast f(y)| r \lesssim \eta 1,mr \ast | \Phi \ast f | r(y) =
= c
\int
\BbbR n
\eta 1,mr(y - z)| \Phi \ast f(z)| rdz \lesssim
\lesssim \eta 1, - mr(y - x)\eta 1,mr \ast | \Phi \ast f | r(x)
for any x \in \BbbR n and any m >
n
r
. Furthermore,
| \Phi \ast \theta \ast f(y)| \leq
\int
\BbbR n
\eta 1,N (y - z)| \theta \ast f(z)| dz \leq
\leq \eta 1, - m(y - x)\theta \ast ,m1 f(x)
\int
\BbbR n
\eta 1,N - m(y - z)dz \lesssim
\lesssim \eta 1, - m(y - x)\theta \ast ,m1 f(x)
for any N > m+ n, where
\theta \ast ,m1 f(x) = \mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
| \theta \ast f(y)|
(1 + | y - x| )m
, x \in \BbbR n.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1608 S. BENMAHMOUD, D. DRIHEM
Therefore,
| \Phi \ast \theta \ast f(y)| \lesssim \eta 1, - m(y - x)(\theta \ast ,m1 f(x))1 - r\eta 1,mr \ast | \Phi \ast f | r(x)
for any x \in \BbbR n and any m > n. Again from Lemma 4 we conclude that
| \theta \ast \varphi \tau \ast f(y)| r \lesssim \eta 1,mr \ast | \varphi \tau \ast f | r(y) \lesssim (1 + | y - x| )mr\eta 1,mr \ast | \varphi \tau \ast f | r(x)
and
| \theta \ast \varphi \tau \ast f(y)| \lesssim
\int
\BbbR n
\eta \tau ,N (y - z)| \theta \ast f(z)| dz \lesssim
\lesssim (1 + | y - x| )m\theta \ast ,m1 f(x),
1
4
\leq \tau \leq 1,
for any x \in \BbbR n, any m > n and any N > m+ n. Consequently,
\theta \ast ,m1 f(x) \leq c(\theta \ast ,m1 f(x))1 - r
\left( \eta 1,mr \ast | \Phi \ast f | r(x) +
1\int
1/4
\eta 1,mr \ast | \varphi \tau \ast f | r(x)
d\tau
\tau
\right) , (3)
which implies that
| \theta \ast f(x)| r \leq c\eta 1,mr \ast | \Phi \ast f | r(x) + c
1\int
1/4
\eta 1,mr \ast | \varphi \tau \ast f | r(x)
d\tau
\tau
, (4)
when \theta \ast ,m1 f(x) <\infty , which is true if m \geq n
r
+N0 (order of distribution). We will use the Strömberg
and Torchinsky idea [18]. Observe that the right-hand side of (4) decreases as m increases. Therefore,
we have (4) for all m >
n
r
but with c = c(f) depending on f. We can easily check that if the right-
hand side of (4), with c = c(f), is finite imply that \theta \ast ,m1 f(x) < \infty , otherwise, there is nothing
to prove. Returning to (3) and having in mind that now \theta \ast ,m1 f(x) < \infty , we obtain the desired
estimate (4).
Step 2. Proof of (ii). We have
\omega t \ast f =
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\omega t \ast \varphi \tau \ast f
d\tau
\tau
+
\left\{
0, if 0 < t <
1
4
,
\omega t \ast \Phi \ast f, if
1
4
\leq t \leq 1.
Let
gt(y) :=
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\omega t \ast \varphi \tau \ast f(y)
d\tau
\tau
, y \in \BbbR n, 0 < t \leq 1.
It follows from Lemma 4 that
| \omega t \ast \varphi \tau \ast f(y)| r \lesssim \eta t,mr \ast | \varphi \tau \ast f | r(y) \lesssim \eta \tau ,mr \ast | \varphi \tau \ast f | r(y) =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1609
= c
\int
\BbbR n
\eta \tau ,mr(y - z)| \varphi \tau \ast f(z)| rdz \lesssim
\lesssim (1 + \tau - 1| y - x| )mr\eta \tau ,mr \ast | \varphi \tau \ast f | r(x)
and
| \omega t \ast \varphi \tau \ast f(y)| \lesssim
\int
\BbbR n
\eta \tau ,N (y - z)| \omega t \ast f(z)| dz \lesssim
\lesssim \omega \ast
t,mf(y)
\int
\BbbR n
\eta \tau ,N (y - z)(1 + t - 1| y - z| )mdz \lesssim
\lesssim \omega \ast ,m
t f(y) \lesssim (1 + t - 1| y - x| )m\omega \ast ,m
t f(x)
for any x, y \in \BbbR n, any
t
4
\leq \tau \leq \mathrm{m}\mathrm{i}\mathrm{n}(1, 4t), 0 < t \leq 1, and any N > m+ n, where
\omega \ast ,m
t f(x) = \mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
| \omega t \ast f(y)|
(1 + t - 1| y - x| )m
, x, y \in \BbbR n, 0 < t \leq 1.
Therefore, | gt(y)| can be estimated from above by
c(\omega \ast ,m
t f(x))1 - r(1 + t - 1| y - x| )m(1 - r)\times
\times
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
(1 + \tau - 1| y - x| )mr\eta \tau ,mr \ast | \varphi \tau \ast f | r(x)
d\tau
\tau
\lesssim
\lesssim (1 + t - 1| y - x| )m(\omega \ast ,m
t f(x))1 - r
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\eta \tau ,mr \ast | \varphi \tau \ast f | r(x)
d\tau
\tau
,
if 0 < t \leq 1. Now if
1
4
\leq t \leq 1, we easily obtain
| \omega t \ast \Phi \ast f(y)| = | \omega t \ast \Phi \ast f(y)| 1 - r| \omega t \ast \Phi \ast f(y)| r \lesssim
\lesssim (1 + t - 1| y - x| )m(1 - r)(\omega \ast ,m
t f(x))1 - r\eta 1,mr \ast | \Phi \ast f | r(y) \lesssim
\lesssim (1 + t - 1| y - x| )m(\omega \ast ,m
t f(x))1 - r\eta 1,mr \ast | \Phi \ast f | r(x),
which yields that
\mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
| \omega t \ast \Phi \ast f(y)|
(1 + t - 1| y - x| )m
\lesssim (\omega \ast ,m
t f(x))1 - r\eta 1,mr \ast | \Phi \ast f | r(x).
Consequently,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1610 S. BENMAHMOUD, D. DRIHEM
| \omega t \ast f(x)| r \lesssim
\bigl(
\omega \ast ,m
t f(x)
\bigr) r
\lesssim \eta 1,mr \ast | \Phi \ast f | r(x) +
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\eta \tau ,mr \ast | \varphi \tau \ast f | r(x)
d\tau
\tau
,
when \omega \ast ,m
t f(x) <\infty , 0 < t \leq 1 and x \in \BbbR n . Using a combination of the arguments used in (i), we
arrive at the desired estimate.
Lemma 7 is proved.
The following lemma is from [17] (Lemma 1).
Lemma 8. Let \varrho , \mu \in \scrS (\BbbR n), and M \geq - 1 an integer such that\int
\BbbR n
x\alpha \mu (x)dx = 0
for all | \alpha | \leq M. Then, for any N > 0, there exists a constant c(N) > 0 such that
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbR n
| t - n\mu (t - 1\cdot ) \ast \varrho (z)| (1 + | z| )N \leq c(N)tM+1, 0 < t \leq 1.
3. Variable Besov spaces. In this section, we present the definition of Besov spaces of variable
smoothness and integrability, and prove the basic properties in analogy to the case of fixed exponents.
Select a pair of Schwartz functions \Phi and \varphi satisfying
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \Phi \subset \{ x \in \BbbR n : | x| \leq 2\} , \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \varphi \subset
\biggl\{
x \in \BbbR n :
1
2
\leq | x| \leq 2
\biggr\}
(5)
and
\scrF \Phi (\xi ) +
1\int
0
\scrF \varphi (t\xi )dt
t
= 1, \xi \in \BbbR n. (6)
Such a resolution (5) and (6) of unity can be constructed as follows. Let \mu \in \scrS (\BbbR n) be such that
| \scrF \mu (\xi )| > 0 for 1/2 < | \xi | < 2. There exists \eta \in \scrS (\BbbR n) with
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \eta \subset
\biggl\{
x \in \BbbR n :
1
2
< | x| < 2
\biggr\}
such that
\infty \int
0
\scrF \mu (t\xi )\scrF \eta (t\xi )dt
t
= 1, \xi \not = 0,
see [4, 11, 13]. We set \scrF \varphi = \scrF \mu \scrF \eta and
\scrF \Phi (\xi ) =
\left\{
\int \infty
1
\scrF \varphi (t\xi ) dt
t
, if \xi \not = 0,
1, if \xi = 0.
Then \scrF \Phi \in \scrS (\BbbR n), and as \scrF \eta is supported in
\biggl\{
x \in \BbbR n :
1
2
\leq | x| \leq 2
\biggr\}
, we see that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \Phi \subset
\subset \{ x \in \BbbR n : | x| \leq 2\} .
Now we define the spaces under consideration.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1611
Definition 1. Let \alpha : \BbbR n \rightarrow \BbbR and p, q \in \scrP 0(\BbbR n). Let \{ \scrF \Phi ,\scrF \varphi \} be a resolution of unity and
we put \varphi t = t - n\varphi
\biggl(
\cdot
t
\biggr)
, 0 < t \leq 1. The Besov space \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ) is the collection of all f \in \scrS \prime (\BbbR n)
such that
\| f\| \Phi ,\varphi
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
:=
\bigm\| \bigm\| \Phi \ast f
\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| (t - \alpha (\cdot )\varphi t \ast f)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
<\infty .
When q = \infty , the Besov space \frakB
\alpha (\cdot )
p(\cdot ),\infty consist of all distributions f \in \scrS \prime (\BbbR n) such that
\| f\| \Phi ,\varphi
\frakB
\alpha (\cdot )
p(\cdot ),\infty
:=
\bigm\| \bigm\| \Phi \ast f
\bigm\| \bigm\|
p(\cdot ) + \mathrm{s}\mathrm{u}\mathrm{p}
t\in (0,1]
\bigm\| \bigm\| t - \alpha (\cdot )(\varphi t \ast f)\bigm\| \bigm\| p(\cdot ) <\infty .
One recognizes immediately that \frakB \alpha (\cdot )
p(\cdot ),q(\cdot ) is a quasinormed space and if \alpha , p and q are constants,
then
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ) = B\alpha
p,q,
where B\alpha
p,q is the usual Besov spaces.
Now, we are ready to show that the definition of these function spaces is independent of the
chosen resolution \{ \scrF \Phi ,\scrF \varphi \} of unity. This justifies our omission of the subscript \Phi and \varphi in the
sequel.
Theorem 1. Let \{ \scrF \Phi ,\scrF \varphi \} and \{ \scrF \Psi ,\scrF \psi \} be two resolutions of unity. Let \alpha : \BbbR n \rightarrow \BbbR and
p, q \in \scrP 0(\BbbR n). Assume that p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}
0 (\BbbR n) and \alpha ,
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n). Then
\| f\| \Phi ,\varphi
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
\approx \| f\| \Psi ,\psi
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
.
Proof. It is sufficient to show that there exists a constant c > 0 such that, for all f \in \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ),
we have
\| f\| \Phi ,\varphi
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
\lesssim \| f\| \Psi ,\psi
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
.
In view of Lemma 7, the problem can be reduced to the case of p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) and q \in \scrP (\BbbR n) with
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n). By the scaling argument, it suffices to consider the case \| f\| \Psi ,\psi
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
= 1 and show
that \bigm\| \bigm\| \Phi \ast f
\bigm\| \bigm\|
p(\cdot ) \lesssim 1
and
1\int
0
\bigm\| \bigm\| | ct - \alpha (\cdot )(\varphi t \ast f)| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
dt
t
\leq 1
for some positive constant c. Interchanging the roles of (\Psi , \psi ) and (\Phi , \varphi ), we obtain the desired
result. We have
\scrF \Phi (\xi ) = \scrF \Phi (\xi )\scrF \Psi (\xi ) +
1\int
1/4
\scrF \Phi (\xi )\scrF \psi (\tau \xi )d\tau
\tau
and
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1612 S. BENMAHMOUD, D. DRIHEM
\scrF \varphi (t\xi ) =
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\scrF \varphi (t\xi )\scrF \psi (\tau \xi )d\tau
\tau
+
\left\{
0, if 0 < t <
1
4
,
\scrF \varphi (t\xi )\scrF \Psi (\xi ), if
1
4
\leq t \leq 1,
for any \xi \in \BbbR n. Then we see that
\Phi \ast f = \Phi \ast \Psi \ast f +
1\int
1/4
\Phi \ast \psi \tau \ast f
d\tau
\tau
and
\varphi t \ast f =
\mathrm{m}\mathrm{i}\mathrm{n}(1,4t)\int
t/4
\varphi t \ast \psi \tau \ast f
d\tau
\tau
+
\left\{
0, if 0 < t <
1
4
,
\varphi t \ast \Psi \ast f, if
1
4
\leq t \leq 1.
First observe that
| \Phi \ast \psi \tau \ast f | \lesssim | \eta 1,m \ast \psi \tau \ast f | \lesssim \eta 1,m \ast \tau - \alpha (\cdot )| \psi \tau \ast f | ,
1
4
\leq \tau \leq 1, m > n,
and
| \Phi \ast \Psi \ast f | \lesssim \eta 1,m \ast | \Psi \ast f | , m > n.
Therefore,
| \Phi \ast f | \leq \eta 1,m \ast | \Psi \ast f | +
1\int
1/4
\eta 1,m \ast \tau - \alpha (\cdot )| \psi \tau \ast f |
d\tau
\tau
=
= \eta 1,m \ast | \Psi \ast f | + g.
Since p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) and the convolution with a radially decreasing L1-function is bounded on Lp(\cdot ),
we have \bigm\| \bigm\| \eta 1,m \ast | \Psi \ast f |
\bigm\| \bigm\|
p(\cdot ) \lesssim
\bigm\| \bigm\| \Psi \ast f
\bigm\| \bigm\|
p(\cdot ) \leq 1.
Now, for some suitable positive constant c1,\bigm\| \bigm\| c1g\bigm\| \bigm\| p(\cdot ) \leq 1
if and only if \bigm\| \bigm\| | c1g| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
\leq 1,
which follows by Lemma 6 (i). Therefore,\bigm\| \bigm\| \Phi \ast f
\bigm\| \bigm\|
p(\cdot ) \lesssim 1.
Using the fact that the convolution with a radially decreasing L1-function is bounded in Lp(\cdot ), we
obtain \bigm\| \bigm\| | c\varphi t \ast \Psi \ast f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
\leq 1,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1613
with an appropriate choice of c and any t \in (0, 1]. Observe that
| \varphi t \ast f | \lesssim
4t\int
t/4
\eta \tau ,m \ast | \psi \tau \ast f |
d\tau
\tau
, m > n+ c\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
q
\biggr)
, t \in
\biggl(
0,
1
4
\biggr]
.
Applying again Lemma 6 (ii), we find that
1
4\int
0
\bigm\| \bigm\| | ct - \alpha (\cdot )(\varphi t \ast f)| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
dt
t
\leq 1
for some suitable positive constant c.
Theorem 1 is proved.
Let a > 0, \alpha : \BbbR n \rightarrow \BbbR and f \in \scrS \prime (\BbbR n). Then we define the Peetre maximal function as follows:
\varphi \ast ,a
t t - \alpha (\cdot )f(x) := \mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
t - \alpha (y)| \varphi t \ast f(y)|
(1 + t - 1| x - y| )a
, t > 0,
and
\Phi \ast ,af(x) := \mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
| \Phi \ast f(y)|
(1 + | x - y| )a
.
We now present a fundamental characterization of the spaces under consideration.
Theorem 2. Let \alpha ,
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n), p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}
0 (\BbbR n), q - \geq p - and a >
n+ c\mathrm{l}\mathrm{o}\mathrm{g}(1/q)
p -
+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ).
Then
\| f\| \ast
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
:=
\bigm\| \bigm\| \Phi \ast ,a\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| (\varphi \ast ,a
t t - \alpha (\cdot )f)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
is an equivalent quasinorm in \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ).
Proof. It is easy to see that, for any f \in \scrS \prime (\BbbR n) with \| f\| \ast
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
< \infty and any x \in \BbbR n, we
have
t - \alpha (x)| \varphi t \ast f(x)| \leq \varphi \ast ,a
t t - \alpha (\cdot )f(x).
This shows that \| f\|
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
\leq \| f\| \ast
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
. We will prove that there exists a constant C > 0 such
that, for every f \in \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ),
\| f\| \ast
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
\leq C\| f\|
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
. (7)
By Lemmas 1 and 4, the estimate
t - \alpha (y)| \varphi t \ast f(y)| \leq C1t
- \alpha (y)\bigl( \eta t,\sigma p - \ast | \varphi t \ast f | p
-
(y)
\bigr) 1/p - \leq
\leq C2
\bigl(
\eta t,(\sigma - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ))p - \ast (t - \alpha (\cdot )| \varphi t \ast f | )p
-
(y)
\bigr) 1/p -
(8)
is true for any y \in \BbbR n, \sigma >
n+ c\mathrm{l}\mathrm{o}\mathrm{g}(1/q)
p -
+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) and t > 0. Now dividing both sides of (8) by\bigl(
1 + t - 1| x - y|
\bigr) a
, in the right-hand side we use the inequality
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1614 S. BENMAHMOUD, D. DRIHEM\bigl(
1 + t - 1| x - y|
\bigr) - a \leq \bigl( 1 + t - 1| x - z|
\bigr) - a\bigl(
1 + t - 1| y - z|
\bigr) a
, x, y, z \in \BbbR n,
while in the left-hand side we take the supremum over y \in \BbbR n, we find that, for all f \in \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
any t > 0 and any \sigma > \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
n+ c\mathrm{l}\mathrm{o}\mathrm{g}(1/q)
p -
+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ), a+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )
\biggr)
,
\varphi \ast ,a
t t - \alpha (\cdot )f(x) \leq C2
\bigl(
\eta t,ap - \ast (t - \alpha (\cdot )p - | \varphi t \ast f | p
-
)(x)
\bigr) 1/p -
,
where C2 > 0 is independent of x, t and f. Assume that the right-hand side of (7) is less than or
equal 1. We will prove that\bigm\| \bigm\| \Phi \ast ,a\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| \bigm\| \Bigl( \bigl( \eta t,ap - \ast (t - \alpha (\cdot )p - | \varphi t \ast f | p
-
)
\bigr) 1/p - \Bigr)
0<t\leq 1
\bigm\| \bigm\| \bigm\| \widetilde \ell q(\cdot )(Lp(\cdot ))
\lesssim 1. (9)
Observe that the second quasinorm of the left-hand side of (9) can be rewritten as\bigm\| \bigm\| \bigm\| \Bigl( \eta t,ap - \ast (t - \alpha (\cdot )p - | \varphi t \ast f | p
-
)
\Bigr)
0<t\leq 1
\bigm\| \bigm\| \bigm\| 1
p -
\widetilde
\ell
q(\cdot )
p - (L
p(\cdot )
p - )
. (10)
Let 0 < t <
1
4
. In view the proof of Lemma 7 (ii), we obtain
t - \alpha (\cdot )p
- | \varphi t \ast f | p
-
\lesssim
4t\int
t/4
\tau - \alpha (\cdot )p
-
\eta \tau ,ap - \ast | \varphi \tau \ast f | p
- d\tau
\tau
\lesssim
\lesssim
4t\int
t/4
\eta \tau ,ap - - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )p - \ast \tau - \alpha (\cdot )p - | \varphi \tau \ast f | p
- d\tau
\tau
,
by Lemma 1. Therefore,
\eta t,ap - \ast (t - \alpha (\cdot )p - | \varphi t \ast f | p
-
) \lesssim
4t\int
t/4
\eta t,ap - \ast \eta \tau ,ap - - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )p - \ast \tau - \alpha (\cdot )p - | \varphi \tau \ast f | p
- d\tau
\tau
\lesssim
\lesssim
4t\int
t/4
\eta \tau ,ap - - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )p - \ast \tau - \alpha (\cdot )p - | \varphi \tau \ast f | p
- d\tau
\tau
by [7] (Lemma A.3). Applying Lemma 6, we deduce that (10), with 0 < t <
1
4
, is bounded by
\bigm\| \bigm\| \bigm\| \bigl( t - \alpha (\cdot )p - | \varphi t \ast f | p - \bigr) 0<t\leq 1
\bigm\| \bigm\| \bigm\| 1
p -
\widetilde
\ell
q(\cdot )
p - (L
p(\cdot )
p - )
\lesssim 1.
Now let
1
4
\leq t \leq 1. Again, by Lemma 7 (ii), we get
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1615
| \varphi t \ast f | p
- \leq c\eta 1,ap - \ast | \Phi \ast f | p - + c
1\int
t/4
\eta \tau ,ap - \ast | \varphi \tau \ast f | p
- d\tau
\tau
.
As above, we obtain
\eta t,ap - \ast
\bigl(
t - \alpha (\cdot )p
- | \varphi t \ast f | p
- \bigr) \leq
\leq c\eta 1,ap - \ast | \Phi \ast f | p - + c
1\int
t/4
\eta \tau ,ap - - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )p - \ast \tau - \alpha (\cdot )p - | \varphi \tau \ast f | p
- d\tau
\tau
=
= c\eta 1,ap - \ast | \Phi \ast f | p - + ht(x).
We need to prove that\bigm\| \bigm\| \bigm\| \bigl( \eta 1,ap - \ast | \Phi \ast f | p -
\bigr)
1
4
\leq t\leq 1
\bigm\| \bigm\| \bigm\| \widetilde
\ell
q(\cdot )
p - (L
p(\cdot )
p - )
\lesssim 1 and
\bigm\| \bigm\| (ht) 1
4
\leq t\leq 1
\bigm\| \bigm\|
\widetilde
\ell
q(\cdot )
p - (L
p(\cdot )
p - )
\lesssim 1. (11)
Applying Lemma 6, we obtain the second estimate of (11). Let us prove the first one. This is
equivalent to \bigm\| \bigm\| | \eta 1,ap - \ast | \Phi \ast f | p - |
q(\cdot )
p -
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
\lesssim 1,
which is equivalent to \bigm\| \bigm\| \eta 1,ap - \ast | \Phi \ast f | p -
\bigm\| \bigm\|
p(\cdot )
p -
\lesssim 1.
Since
p(\cdot )
p -
\in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) and the convolution with a radially decreasing L1-function is bounded in
Lp(\cdot ), we have \bigm\| \bigm\| \eta 1,ap - \ast | \Phi \ast f | p -
\bigm\| \bigm\|
p(\cdot )
p -
\lesssim
\bigm\| \bigm\| | \Phi \ast f | p -
\bigm\| \bigm\|
p(\cdot )
p -
= c
\bigm\| \bigm\| \Phi \ast f
\bigm\| \bigm\| p -
p(\cdot ) \lesssim 1.
The estimate of
\bigm\| \bigm\| \Phi \ast ,a\bigm\| \bigm\|
p(\cdot ) follows easily from the fact that
\bigm\| \bigm\| \Phi \ast ,a\bigm\| \bigm\|
p(\cdot ) \lesssim
\bigm\| \bigm\| \eta 1,ap - \ast | \Phi \ast f | p -
\bigm\| \bigm\| 1
p -
p(\cdot )
p -
\lesssim 1.
Theorem 2 is proved.
4. Relation between \bffrakB
\bfitalpha (\cdot )
\bfitp (\cdot ),\bfitq (\cdot ) and \bfitB
\bfitalpha (\cdot )
\bfitp (\cdot ),\bfitq (\cdot ) . In this section, we present the coincidence
between the above function spaces and the variable Besov spaces of Almeida and Hästö, where to
define these function spaces we first need the concept of a smooth dyadic resolution of unity. Let \Psi
be a function in \scrS (\BbbR n) satisfying \Psi (x) = 1 for | x| \leq 1 and \Psi (x) = 0 for | x| \geq 2. We define \psi 0
and \psi 1 by \scrF \psi 0(x) = \Psi (x), \scrF \psi 1(x) = \Psi
\biggl(
x
2
\biggr)
- \Psi (x) and
\scrF \psi v(x) = \scrF \psi 1(2
1 - vx) for v = 2, 3, . . . .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1616 S. BENMAHMOUD, D. DRIHEM
Then \{ \scrF \psi v\} v\in \BbbN 0 is a smooth dyadic resolution of unity,
\sum \infty
v=0
\scrF \psi v(x) = 1 for all x \in \BbbR n. Thus
we obtain the Littlewood – Paley decomposition
f =
\infty \sum
v=0
\psi v \ast f
for all f \in \scrS \prime (\BbbR n) (convergence in \scrS \prime (\BbbR n)).
We state the definition of the spaces Bs(\cdot )
p(\cdot ),q(\cdot ), which introduced and investigated in [3].
Definition 2. Let \{ \scrF \psi v\} v\in \BbbN 0
be a resolution of unity, s : \BbbR n \rightarrow \BbbR and p, q \in \scrP 0(\BbbR n). The
Besov space Bs(\cdot )
p(\cdot ),q(\cdot ) consists of all distributions f \in \scrS \prime (\BbbR n) such that
\| f\|
B
s(\cdot )
p(\cdot ),q(\cdot )
:=
\bigm\| \bigm\| (2vs(\cdot )\psi v \ast f)v\bigm\| \bigm\| \ell q(\cdot )(Lp(\cdot ))
<\infty .
Taking s \in \BbbR and q \in (0,\infty ] as constants we derive the spaces Bs
p(\cdot ),q studied by Xu in [23].
We refer the reader to the recent papers [1, 2, 9, 14] for further details, historical remarks and more
references on these function spaces. For any p, q \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}
0 (\BbbR )n and s \in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} , the space Bs(\cdot )
p(\cdot ),q(\cdot ) does
not depend on the chosen smooth dyadic resolution of unity \{ \scrF \psi v\} v\in \BbbN 0 (in the sense of equivalent
quasinorms) and
\scrS (\BbbR n) \lhook \rightarrow B
s(\cdot )
p(\cdot ),q(\cdot ) \lhook \rightarrow \scrS \prime (\BbbR n).
Moreover, if p, q, s are constants, we reobtain the usual Besov spaces Bs
p,q, studied in detail in
[20, 21], see also [19].
Theorem 3. Let \alpha : \BbbR n \rightarrow \BbbR and p, q \in \scrP 0(\BbbR n). Assume that p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}
0 (\BbbR n) and \alpha ,
1
q
\in
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n). Then
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ) = B
\alpha (\cdot )
p(\cdot ),q(\cdot ),
in the sense of equivalent quasinorms.
Proof. Step 1. We will prove that
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ) \lhook \rightarrow B
\alpha (\cdot )
p(\cdot ),q(\cdot ).
From Lemma 7, we only consider the case p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}(\BbbR n) and q \in \scrP (\BbbR n) with
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n). Let
\{ \scrF \Phi ,\scrF \varphi \} and \{ \scrF \psi j\} j\in \BbbN 0
be two resolutions of unity and let f \in \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ) with
\| f\|
\frakB
\alpha (\cdot )
p(\cdot ),q(\cdot )
\leq 1.
We have
\psi v \ast f =
\mathrm{m}\mathrm{i}\mathrm{n}(1,22 - v)\int
2 - v - 2
\psi v \ast \varphi t \ast f
dt
t
+
\left\{ 0, if v \geq 2,
\psi v \ast \Phi \ast f, if v = 0, 1.
Since the convolution with a radially decreasing L1-function is bounded in Lp(\cdot ), we obtain\bigm\| \bigm\| | c\psi v \ast \Phi \ast f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
\leq 1, v = 0, 1,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1617
for some suitable positive constant c. Applying Lemma 6, we get
\bigm\| \bigm\| \bigm\| | c12v\alpha (\cdot )\psi v \ast f | q(\cdot )\bigm\| \bigm\| \bigm\| p(\cdot )
q(\cdot )
\leq
\mathrm{m}\mathrm{i}\mathrm{n}(1,22 - v)\int
2 - v - 2
\bigm\| \bigm\| | t\alpha (\cdot )\varphi t \ast f | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
dt
t
+ 2 - v, v \geq 2,
with an appropriate choice of c1. Taking the sum over v \geq 2, we have \| f\|
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\lesssim 1.
Step 2. We will prove that
B
\alpha (\cdot )
p(\cdot ),q(\cdot ) \lhook \rightarrow \frakB
\alpha (\cdot )
p(\cdot ),q(\cdot ).
Let \{ \scrF \Phi ,\scrF \varphi \} and \{ \scrF \psi v\} v\in \BbbN 0
be two resolutions of unity and let f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ) with
\| f\|
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\leq 1.
We have
\varphi t \ast f =
\infty \sum
v=0
\varphi t \ast \psi v \ast f =
=
\lfloor \mathrm{l}\mathrm{o}\mathrm{g}2( 4
t )\rfloor +1\sum
v=\lfloor \mathrm{l}\mathrm{o}\mathrm{g}2( 1
2t
)\rfloor
\varphi t \ast \psi v \ast f +
\left\{
0, if 0 < t \leq 1
4
,
\psi 0 \ast \Phi \ast f, if t >
1
4
,
and
\Phi \ast f =
2\sum
v=0
\Phi \ast \psi v \ast f.
Notice that if v < 0 then we put \psi v \ast f = 0. Since the convolution with a radially decreasing
L1-function is bounded in Lp(\cdot ), we obtain\bigm\| \bigm\| | c\psi v \ast \Phi \ast f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
\leq 1, v = 0, 1, 2,
which yields \bigm\| \bigm\| c| \Phi \ast f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
\leq 1
for some suitable positive constant c. Let t \in (2 - i, 2 - i+1], i \in \BbbN . We have
t - \alpha (\cdot )| \varphi t \ast f | \lesssim
\lfloor \mathrm{l}\mathrm{o}\mathrm{g}2( 4
t )\rfloor +1\sum
v=\lfloor \mathrm{l}\mathrm{o}\mathrm{g}2( 1
2t)\rfloor
t - \alpha (\cdot )\eta t,m \ast | \psi v \ast f | \lesssim
\lesssim
i - 1\sum
v=i - 3
2(i - v)\alpha
-
\eta v,m - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) \ast 2
v\alpha (\cdot )| \psi v \ast f | \leq
\leq c
- 1\sum
j= - 3
\eta j+i,m - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) \ast 2
(j+i)\alpha (\cdot )| \psi j+i \ast f | ,
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1618 S. BENMAHMOUD, D. DRIHEM
where m > n+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) + c\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1
q
\biggr)
. Now observe that
1\int
0
\bigm\| \bigm\| | ct - \alpha (\cdot )\varphi t \ast f | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
dt
t
=
=
\infty \sum
i=0
21 - i\int
2 - i
\bigm\| \bigm\| | t - \alpha (\cdot )\varphi t \ast f | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
dt
t
\leq
\leq
\infty \sum
i=0
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( c - 1\sum
j= - 3
\eta j+i,m - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) \ast 2
(j+i)\alpha (\cdot )| \psi j+i \ast f |
\right) q(\cdot )
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
p(\cdot )
q(\cdot )
for some suitable positive constant c. The desired estimate follows by Lemma 5.
Theorem 3 is proved.
In order to formulate the main result of this section, let us consider k0, k \in \scrS (\BbbR n) and S \geq - 1
an integer such that for an \varepsilon > 0
| \scrF k0(\xi )| > 0 for | \xi | < 2\varepsilon , (12)
| \scrF k(\xi )| > 0 for
\varepsilon
2
< | \xi | < 2\varepsilon (13)
and \int
\BbbR n
x\alpha k(x)dx = 0 for any | \alpha | \leq S. (14)
Here, (12) and (13) are Tauberian conditions, while (14) states that moment conditions on k. We
recall the notation
kt(x) := t - nk(t - 1x) for t > 0.
For any a > 0, f \in \scrS \prime (\BbbR n) and x \in \BbbR n, we denote
k\ast ,at t - \alpha (\cdot )f(x) := \mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
t - \alpha (y)| kt \ast f(y)|
(1 + t - 1| x - y| )a
, j \in \BbbN 0.
We are now able to state the so called local mean characterization of B\alpha (\cdot )
p(\cdot ),q(\cdot ) spaces, which is a
more general form of Theorem 2.
Theorem 4. Let \alpha ,
1
q
\in C \mathrm{l}\mathrm{o}\mathrm{g}
\mathrm{l}\mathrm{o}\mathrm{c} (\BbbR
n), p \in \scrP \mathrm{l}\mathrm{o}\mathrm{g}
0 (\BbbR n), a >
n
p -
and \alpha + < S + 1. Then
\| f\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
:=
\bigm\| \bigm\| k\ast ,a0 f
\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| (k\ast ,at t - \alpha (\cdot )f)0<t\leq 1
\bigm\| \bigm\|
\widetilde \ell q(\cdot )(Lp(\cdot ))
is an equivalent quasinorm on B\alpha (\cdot )
p(\cdot ),q(\cdot ).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1619
Proof. The idea of the proof is from V. S. Rychkov [17]. The proof is divided into three steps.
Step 1. Let \varepsilon > 0. Take any pair of functions \varphi 0 and \varphi \in \scrS (\BbbR n) such that
| \scrF \varphi 0(\xi )| > 0 for | \xi | < 2\varepsilon ,
| \scrF \varphi (\xi )| > 0 for
\varepsilon
2
< | \xi | < 2\varepsilon .
We prove that there exists a constant c > 0 such that, for any f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ),
\| f\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\leq c
\bigm\| \bigm\| \varphi \ast ,a
0 f
\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| \bigm\| \bigl( \varphi \ast ,a
j 2j\alpha (\cdot )f
\bigr)
j\geq 1
\bigm\| \bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
. (15)
Let \Lambda , \lambda \in \scrS (\BbbR n) such that
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \Lambda \subset \{ \xi \in \BbbR n : | \xi | < 2\varepsilon \} , \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \lambda \subset \{ \xi \in \BbbR n : \varepsilon /2 < | \xi | < 2\varepsilon \}
and
\scrF \Lambda (\xi )\scrF \varphi 0(\xi ) +
\infty \sum
j=1
\scrF \lambda (2 - j\xi )\scrF \varphi (2 - j\xi ) = 1, \xi \in \BbbR n.
In particular, for any f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ), the following identity is true:
f = \Lambda \ast \varphi 0 \ast f +
\infty \sum
j=1
\lambda j \ast \varphi j \ast f,
where
\varphi j := 2jn\varphi (2j \cdot ) and \lambda j := 2jn\lambda (2j \cdot ), j \in \BbbN .
Hence we can write
kt \ast f = kt \ast \Lambda \ast \varphi 0 \ast f +
\infty \sum
j=1
kt \ast \lambda j \ast \varphi j \ast f, t \in (0, 1].
Let 2 - i < t \leq 21 - i, i \in \BbbN . First, let j < i. Writing, for any z \in \BbbR n,
kt \ast \lambda j(z) = 2jnk2jt \ast \lambda (2jz),
we deduce from Lemma 8 that, for any N > 0, there exists a constant c > 0, independent of t and
j, such that
| kt \ast \lambda j(z)| \leq c
\bigl(
2jt
\bigr) S+1
\eta j,N (z), z \in \BbbR n.
This together with Lemma 1 yield that
t - \alpha (y)| kt \ast \lambda j \ast \varphi j \ast f(y)| ,
can be estimated from above by
c2(j - i)(S+1 - \alpha +)\varphi \ast ,a
j 2j\alpha (\cdot )f(y)
\int
\BbbR n
\eta j,N - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) - a(y - z)dz \lesssim 2(j - i)(S+1 - \alpha +)\varphi \ast ,a
j 2j\alpha (\cdot )f(y)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1620 S. BENMAHMOUD, D. DRIHEM
for any N > n+ a+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ), any y \in \BbbR n and any j < i.
Next, let j \geq i. Then, again by Lemma 8, we have, for any z \in \BbbR n and any L > 0,
| kt \ast \lambda j(z)| = t - n
\bigm| \bigm| \bigm| \bigm| k \ast \lambda 1
2jt
\biggl(
z
t
\biggr) \bigm| \bigm| \bigm| \bigm| \leq c
\biggl(
1
2jt
\biggr) M+1
\eta t,L(z),
where an integer M \geq - 1 is taken arbitrarily large, since D\beta \scrF \lambda (0) = 0 for all \beta . Hence, again
with Lemma 1,
t - \alpha (y)| kt \ast \lambda j \ast \varphi j \ast f(y)| \leq
\leq t - \alpha (y)
\int
\BbbR n
| kt \ast \lambda j(y - z)| | \varphi j \ast f(z)| dz \lesssim
\lesssim 2(i - j)(M+1+\alpha - ) - jn\varphi \ast ,a
j 2j\alpha (\cdot )f(y)
\int
\BbbR n
\eta j, - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) - a(y - z)\eta i,L(y - z)dz.
We have, for any j \geq i,\bigl(
1 + 2j | z|
\bigr) c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )+a \leq 2(j - i)(c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )+a)
\bigl(
1 + 2i| z|
\bigr) c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha )+a.
Then, by taking L > n+ a+ c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ), we get
t - \alpha (y)| kt \ast \lambda j \ast \varphi j \ast f(y)| \lesssim 2(i - j)(M+1+\alpha - - c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) - a)\varphi \ast ,a
j 2j\alpha (\cdot )f(y).
Let us take M > c\mathrm{l}\mathrm{o}\mathrm{g}(\alpha ) - \alpha - + 2a to estimate the last expression by
c2(i - j)(a+1)\varphi \ast ,a
j 2j\alpha (\cdot )f(y),
where c > 0 is independent of i, j and f. Using the fact that for any z \in \BbbR n and any N > 0
| kt \ast \Lambda (z)| \leq ctS+1\eta 1,N (z),
we obtain by the similar arguments that for any 2 - i \leq t \leq 2 - i+1, i \in \BbbN ,
\mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
t - \alpha (y)| kt \ast \Lambda \ast \varphi 0 \ast f(y)|
(1 + t - 1| x - y| )a
\leq C2 - i(S+1 - \alpha +)\varphi \ast ,a
0 f(x).
Further, note that, for all x, y \in \BbbR n all 2 - i \leq t \leq 21 - i, i \in \BbbN , and any j \in \BbbN 0,
\varphi \ast ,a
j 2j\alpha (\cdot )f(y) \leq \varphi \ast ,a
j 2j\alpha (\cdot )f(x)(1 + 2j | x - y| )a \leq
\leq \varphi \ast ,a
j 2j\alpha (\cdot )f(x)\mathrm{m}\mathrm{a}\mathrm{x}(1, 2(j - i)a)(1 + 2i| x - y| )a.
Hence
\mathrm{s}\mathrm{u}\mathrm{p}
y\in \BbbR n
t - \alpha (y)| kt \ast \lambda j \ast \varphi j \ast f(y)|
(1 + t - 1| x - y| )a
\leq C\varphi \ast ,a
j 2j\alpha (\cdot )f(x)
\Biggl\{
2(j - i)(S+1 - \alpha +), if j < i,
2i - j , if j \geq i.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1621
Therefore, for all f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ), any x \in \BbbR n and any 2 - i \leq t \leq 21 - i, i \in \BbbN 0, we get
k\ast ,at t - \alpha (\cdot )f(x) \lesssim 2 - i(S+1 - \alpha +)\varphi \ast ,a
0 f(x)+
+C
\infty \sum
j=1
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl(
2(j - i)(S+1 - \alpha +), 2i - j
\Bigr)
\varphi \ast ,a
j 2j\alpha (\cdot )f(x) =
= C
\infty \sum
j=0
\mathrm{m}\mathrm{i}\mathrm{n}
\Bigl(
2(j - i)(S+1 - \alpha +), 2i - j
\Bigr)
\varphi \ast ,a
j 2j\alpha (\cdot )f(x) =
= C\Psi i(x).
Assume that the right-hand side of (15) is less than or equal one. Then we have
1\int
0
\bigm\| \bigm\| | k\ast ,at t - \alpha (\cdot )f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
dt
t
=
\infty \sum
i=0
21 - i\int
2 - i
\bigm\| \bigm\| | k\ast ,at t - \alpha (\cdot )f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
dt
t
\leq
\leq
\infty \sum
i=0
\bigm\| \bigm\| | c\Psi i| q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
for some positive constant c. The last term on the right-hand side is less than or equal one if and
only if \bigm\| \bigm\| (c1\Psi i)i
\bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
\leq 1
for some suitable positive constant c1, which follows by Lemma 8 of [14] and the fact that \alpha + <
< S + 1. Also we have, for any z \in \BbbR n, any N > 0 and any integer M \geq - 1,
| k0 \ast \lambda j(z)| \leq c2 - j(M+1)\eta j,N (z) and | k0 \ast \Lambda (z)| \leq c\eta 1,N (z).
As before, we get, for any x \in \BbbR n,
k\ast ,a0 f(x) \leq C\varphi \ast ,a
0 f(x) + C
\infty \sum
j=1
2 - j\varphi \ast ,a
j 2j\alpha (\cdot )f(x). (16)
In (16) taking the Lp(\cdot )-quasinorm and using the embedding \ell q(\cdot )(Lp(\cdot )) \lhook \rightarrow \ell \infty (Lp(\cdot )) we get (15).
Step 2. Let \{ \scrF \varphi j\} j\in \BbbN 0
\subset \scrS (\BbbR n) be such that
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \varphi \subset
\bigl\{
\xi \in \BbbR n : \varepsilon /2 \leq | \xi | \leq 2\varepsilon
\bigr\}
and
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \varphi 0 \subset
\bigl\{
\xi \in \BbbR n : | \xi | \leq 2\varepsilon
\bigr\}
, \varepsilon > 0,
with \varphi j = 2jn\varphi (2j \cdot ), j \in \BbbN . We will prove that\bigm\| \bigm\| \varphi 0 \ast f
\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| \bigm\| \bigl( 2j\alpha (\cdot )(\varphi j \ast f)\bigr) j\geq 1
\bigm\| \bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
\lesssim \| f\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
. (17)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1622 S. BENMAHMOUD, D. DRIHEM
Let \Lambda , \lambda \in \scrS (\BbbR n) such that
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \Lambda \subset \{ \xi \in \BbbR n : | \xi | < 2\varepsilon \} , \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\scrF \lambda \subset \{ \xi \in \BbbR n : \varepsilon /2 < | \xi | < 2\varepsilon \} ,
\scrF \Lambda (\xi )\scrF k0(\xi ) +
1\int
0
\scrF \lambda (\tau \xi )\scrF k(\tau \xi )d\tau
\tau
= 1, \xi \in \BbbR n.
In particular, for any f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ), the following identity is true:
f = \Lambda \ast k0 \ast f +
1\int
0
\lambda \tau \ast k\tau \ast f
d\tau
\tau
.
Hence we can write
\varphi j \ast f =
1\int
0
\varphi j \ast \lambda \tau \ast k\tau \ast f
d\tau
\tau
=
2 - j+2\int
2 - j - 2
\varphi j \ast \lambda \tau \ast k\tau \ast f
d\tau
\tau
, j \geq 2.
Using the fact that
\mathrm{m}\mathrm{a}\mathrm{x}(| k\tau \ast \lambda \tau (z)| , | \varphi j \ast \lambda \tau (z)| ) \lesssim \eta j,N (z), z \in \BbbR n, 2 - j - 2 \leq \tau \leq 2 - j+2, j \in \BbbN ,
and Lemma 1, with N > 0 large enough, we easily obtain
2j\alpha (y)| \varphi j \ast \lambda \tau \ast k\tau \ast f(y)| \lesssim \mathrm{m}\mathrm{i}\mathrm{n}(k\ast ,a\tau \tau - \alpha (\cdot )f(y), \varphi \ast ,a
j 2j\alpha (y)f(y))
for any y \in \BbbR n and any 2 - j+2 \leq \tau \leq 2 - j - 2, j \in \BbbN . Therefore,
2j\alpha (y)| \varphi j \ast f(y)| \lesssim
\bigl(
\varphi \ast ,a
j 2j\alpha (\cdot )f(y)
\bigr) 1 - r 2 - j+2\int
2 - j - 2
\Bigl(
k\ast ,a\tau \tau - \alpha (\cdot )f(y)
\Bigr) r d\tau
\tau
, 0 < r < 1,
which yields that
\varphi \ast ,a
j 2j\alpha (\cdot )f(x) \lesssim
\bigl(
\varphi \ast ,a
j 2j\alpha (\cdot )f(x)
\bigr) 1 - r 2 - j+2\int
2 - j - 2
\Bigl(
k\ast ,a\tau \tau - \alpha (\cdot )f(x)
\Bigr) r d\tau
\tau
.
This estimate gives \bigl(
\varphi \ast ,a
j 2j\alpha (\cdot )f(x)
\bigr) r
\lesssim
2 - j+2\int
2 - j - 2
\Bigl(
k\ast ,a\tau \tau - \alpha (\cdot )f(x)
\Bigr) r d\tau
\tau
and
2j\alpha (x)r| \varphi j \ast f(x)| r \lesssim
2 - j+2\int
2 - j - 2
\Bigl(
k\ast ,a\tau \tau - \alpha (\cdot )f(x)
\Bigr) r d\tau
\tau
, x \in \BbbR n, (18)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
CONTINUOUS CHARACTERIZATION OF THE BESOV SPACES . . . 1623
but if \varphi \ast ,a
j 2j\alpha (\cdot )f(x) <\infty . Using a combination of the arguments used in Lemma 7, we get (18) for
all 0 < r < 1, a > 0 and all f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ). Similarly, we obtain
| \varphi j \ast f(x)| r \lesssim
\bigl(
k\ast ,a0 f(x)
\bigr) r
+
1\int
1
8
\Bigl(
k\ast ,a\tau \tau - \alpha (\cdot )f(x)
\Bigr) r d\tau
\tau
, j = 0, 1,
for any 0 < r < 1, a > 0 and any f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ).
Let \theta > 0 be such that \mathrm{m}\mathrm{a}\mathrm{x}
\biggl(
1,
(1/p)+
(1/q) -
\biggr)
< \theta <
q -
r
. Hölder’s and Minkowski’s inequalities
yield
\bigm\| \bigm\| | c2j\alpha (\cdot )(\varphi j \ast f)| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
\leq
\left( 2 - j+2\int
2 - j - 2
\bigm\| \bigm\| | k\ast ,a\tau \tau - \alpha (\cdot )f | q(\cdot )
\bigm\| \bigm\| 1
\theta
p(\cdot )
q(\cdot )
d\tau
\tau
\right)
\theta
\leq
\leq
2 - j+2\int
2 - j - 2
\bigm\| \bigm\| | k\ast ,a\tau \tau - \alpha (\cdot )f | q(\cdot )
\bigm\| \bigm\|
p(\cdot )
q(\cdot )
d\tau
\tau
.
We get
\infty \sum
j=2
\bigm\| \bigm\| | c2j\alpha (\cdot )(\varphi j \ast f)| q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
\leq 1,
with an appropriate choice of c > 0 such that the left-hand side of (18) it at most one. Similarly, we
have \bigm\| \bigm\| | c\varphi j \ast f | q(\cdot )\bigm\| \bigm\| p(\cdot )
q(\cdot )
\leq 1, j = 0, 1.
The desired estimate follows by the scaling argument.
Step 3. We will prove that, for all f \in B
\alpha (\cdot )
p(\cdot ),q(\cdot ), the following estimates are true:
\| f\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\lesssim \| f\|
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\lesssim
\bigm\| \bigm\| f\bigm\| \bigm\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
. (19)
Let \{ \scrF \varphi j\} j\in \BbbN 0 be a resolution of unity. The first inequality follows by the chain of the estimates
\| f\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\lesssim
\bigm\| \bigm\| \varphi \ast ,a
0 f
\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| \bigm\| \bigl( \varphi \ast ,a
j 2j\alpha (\cdot )f
\bigr)
j\geq 1
\bigm\| \bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
\lesssim \| f\|
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
,
where the first inequality is (15), and the second inequality is obvious (see [9]). Now the second
inequality in (19) can be obtained by the following chain of the estimates:
\| f\|
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
\lesssim
\bigm\| \bigm\| \varphi 0 \ast f
\bigm\| \bigm\|
p(\cdot ) +
\bigm\| \bigm\| \bigm\| \bigl( 2j\alpha (\cdot )(\varphi j \ast f)\bigr) j\geq 1
\bigm\| \bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
\lesssim \| f\| \prime
B
\alpha (\cdot )
p(\cdot ),q(\cdot )
,
where the first inequality is obvious and the second inequality is (17).
Theorem 4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1624 S. BENMAHMOUD, D. DRIHEM
References
1. A. Almeida, A. Caetano, On 2-microlocal spaces with all exponents variable, Nonlinear Anal., 135, 97 – 119 (2016).
2. A. Almeida, A. Caetano, Atomic and molecular decompositions in variable exponents 2-microlocal spaces and
applications, J. Funct. Anal., 270, 1888 – 1921 (2016).
3. A. Almeida, P. Hästö, Besov spaces with variable smoothness and integrability, J. Funct. Anal., 258, 1628 – 1655
(2010).
4. A. P. Calderón, A. Torchinsky, Parabolic maximal functions associated with a distribution, I, II, Adv. Math., 16,
1 – 64 (1975); 24, 101 – 171 (1977).
5. D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue spaces: foundations and harmonic analysis, Birkhäuser-Verlag, Basel
(2013).
6. L. Diening, P. Harjulehto, P. Hästö, Y. Mizuta, T. Shimomura, Maximal functions in variable exponent spaces: limiting
cases of the exponent, Ann. Acad. Sci. Fenn. Math., 34, № 2, 503 – 522 (2009).
7. L. Diening, P. Hästö, S. Roudenko, Function spaces of variable smoothness and integrability. J. Funct. Anal., 256,
№ 6, 1731 – 1768 (2009).
8. L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture
Notes in Math., 2017, Springer-Verlag, Berlin (2011).
9. D. Drihem, Atomic decomposition of Besov spaces with variable smoothness and integrability, J. Math. Anal. and
Appl., 389, 15 – 31 (2012).
10. D. Drihem, Some characterizations of variable Besov-type spaces, Ann. Funct. Anal., 6, 255 – 288 (2015).
11. N. J. H. Heideman, Duality and fractional integration in Lipschitz spaces, Studia Math., 50, 65 – 85 (1974).
12. M. Izuki, T. Noi, Duality of Besov, Triebel – Lizorkin and Herz spaces with variable exponents, Rend. Circ. Mat.
Palermo., 63, 221 – 245 (2014).
13. S. Janson, M. Taibleson, I teoremi di rappresentazione di Calderón, Rend. Semin. Mat. Univ. Politec. Torino, 39,
27 – 35 (1981).
14. H. Kempka, J. Vybı́ral, Spaces of variable smoothness and integrability: characterizations by local means and ball
means of differences, J. Fourier Anal. and Appl., 18, 852 – 891 (2012).
15. M. Moussai, Continuité de certains opérateurs intégraux singuliers sur les espaces de Besov, PhD thesis, Paris
(1987).
16. T. Noi, Duality of variable exponent Triebel – Lizorkin and Besov spaces, J. Funct. Spaces Appl., 2012, Article ID
361807 (2012).
17. V. S. Rychkov, On a theorem of Bui, Paluszynski and Taibleson, Proc. Steklov Inst. Math., 227, 280 – 292 (1999).
18. J.-O. Strömberg, A. Torchinsky, Weighted Hardy spaces, Lecture Notes in Math., 1381, Springer, Berlin (1989).
19. Y. Sawano, Theory of Besov spaces, Develop. Math., 56, Springer, Singapore (2018).
20. H. Triebel, Theory of function spaces, Birkhäuser-Verlag, Basel (1983).
21. H. Triebel, Theory of function spaces II, Birkhäuser-Verlag, Basel (1992).
22. D. Yang, C. Zhuo, W. Yuan, Besov-type spaces with variable smoothness and integrability, J. Funct. Anal., 269,
1840 – 1898 (2015).
23. J. Xu, Variable Besov and Triebel – Lizorkin spaces, Ann. Acad. Sci. Fenn. Math., 33, 511 – 522 (2008).
Received 10.02.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
|
| id | umjimathkievua-article-6578 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:29:01Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/9e/244815b53e3c688581bee1dae058a59e.pdf |
| spelling | umjimathkievua-article-65782023-01-23T14:02:44Z Continuous characterization of the Besov spaces of variable smoothness and integrability Continuous characterization of the Besov spaces of variable smoothness and integrability Benmahmoud, S. Drihem, D. Benmahmoud, S. Drihem, D. Besov space, variable exponent, Calderón reproducing formula. 46E35 UDC 517.9 We obtain new equivalent quasinorms of the Besov spaces of&nbsp;variable smoothness and integrability.&nbsp;&nbsp;Our main tools are the continuous version of the Calderón reproducing formula, maximal inequalities, and the variable-exponent technique; however, allowing the parameters to vary from point to point leads to additional difficulties which, in general, can be overcome by imposing regularity assumptions on these exponents. УДК 517.9 Неперервна характеризація просторів бєсова змінної гладкості та інтегровності Отримано нові еквівалентні квазінорми просторів Бєсова&nbsp;змінної&nbsp; гладкості та інтегровності.&nbsp;&nbsp;Наші основні інструменти – це неперервна версія формули відтворення Калдерона, максимальні нерівності та техніка змінної експоненти. &nbsp;&nbsp;Зазначимо, що дозвіл для параметрів змінюватися від точки до точки викликає додаткові труднощі, які, як правило, можна подолати шляхом накладення припущень регулярності на відповідні експоненти. Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6578 10.37863/umzh.v74i12.6578 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1601 - 1624 Український математичний журнал; Том 74 № 12 (2022); 1601 - 1624 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6578/9339 Copyright (c) 2023 Douadi |
| spellingShingle | Benmahmoud, S. Drihem, D. Benmahmoud, S. Drihem, D. Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title | Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title_alt | Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title_full | Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title_fullStr | Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title_full_unstemmed | Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title_short | Continuous characterization of the Besov spaces of variable smoothness and integrability |
| title_sort | continuous characterization of the besov spaces of variable smoothness and integrability |
| topic_facet | Besov space variable exponent Calderón reproducing formula. 46E35 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6578 |
| work_keys_str_mv | AT benmahmouds continuouscharacterizationofthebesovspacesofvariablesmoothnessandintegrability AT drihemd continuouscharacterizationofthebesovspacesofvariablesmoothnessandintegrability AT benmahmouds continuouscharacterizationofthebesovspacesofvariablesmoothnessandintegrability AT drihemd continuouscharacterizationofthebesovspacesofvariablesmoothnessandintegrability |