Variable Herz estimates for fractional integral operators
UDC 517.5 In this paper, the author study the boundedness of fractional integral operators on a variable Herz-type Hardy space $HK^{\alpha (\cdot )}_{p(\cdot ),q(\cdot )}(\mathbb{R}^n)$ by using the atomic decomposition.
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In this paper, the author study the boundedness of fractional integral operators on a variable Herz-type Hardy space $HK^{\alpha (\cdot )}_{p(\cdot ),q(\cdot )}(\mathbb{R}^n)$ by using the atomic decomposition. |
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DOI: 10.37863/umzh.v72i8.6024
UDC 517.5
R. Heraiz (M’sila Univ., Algeria)
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS
ЗМIННI ОЦIНКИ ГЕРЦА ДЛЯ ДРОБОВИХ IНТЕГРАЛЬНИХ ОПЕРАТОРIВ
In this paper, the author study the boundedness of fractional integral operators on a variable Herz-type Hardy space
H \.K
\alpha (\cdot )
p(\cdot ),q(\cdot ) (\BbbR
n) by using the atomic decomposition.
За допомогою атомарної декомпозицiї вивчається обмеженiсть дробових iнтегральних операторiв у змiнному про-
сторi Гардi H \.K
\alpha (\cdot )
p(\cdot ),q(\cdot ) (\BbbR
n) типу Герца.
1. Introduction. Function spaces with variable exponents have been intensively studied in the recent
years by a significant number of authors. The motivation to study such function spaces comes from
applications to other fields of applied mathematics, such that fluid dynamics and image processing
(see [2, 13]).
Herz spaces K\alpha ,p
q(\cdot ) and \.K\alpha ,p
q(\cdot ) with variable exponent q but fixed \alpha \in \BbbR and p \in (0,\infty ] were
recently studied by Izuki [6, 8]. These spaces with variable exponents \alpha (\cdot ) and q(\cdot ) were studied in
[1], where they gave the boundedness results for a wide class of classical operators on these function
spaces. The spaces K
\alpha (\cdot ),p(\cdot )
q(\cdot ) (\BbbR n) and \.K
\alpha (\cdot ),p(\cdot )
q(\cdot ) (\BbbR n) , were first introduced by Izuki and Noi in
[9]. In [5], the authors gave a new equivalent norms of these function spaces. See [14], where new
variable Herz spaces are given. For more details, we refer the reader to the reference [4].
H. Wang, L. Zongguang and F. Zunwei [16] considered variable Herz-type Hardy spaces H \.K\alpha ,p
q(\cdot )
with variable q, were the boundedness of fractional integral operators and their commutators on these
spaces are obtained.
H. B. Wang and Z. G. Liu [15] studied Herz-type Hardy spaces H \.K
\alpha (\cdot ),p
q(\cdot ) with variables \alpha
and q, but fixed p, where the authors introduced the anisotropic Herz spaces and established their
block decomposition, also they obtain some boundedness on the anisotropic Herz spaces with two
variable exponents for a class of sublinear operators. D. Drihem and F. Seghiri in [5] introduce a
new Herz-type Hardy spaces with variable exponent, where all the three parameters are variables.
This paper is organized as follows. In Section 2, we give some preliminaries where we fix some
notations and recall some basic facts on function spaces with variable integrability. In Section 3, we
give some key technical lemmas needed in the proofs of the main statements. Finally, in Section 4,
we present main results. In particular we will prove the boundedness of fractional integral operators
and their commutators on Herz-type Hardy spaces, where all the three parameters are variables.
2. Preliminaries. As usual, we denote by \BbbR n the n-dimensional real Euclidean space, \BbbN the
collection of all natural numbers and \BbbN 0 = \BbbN \cup \{ 0\} . The letter \BbbZ stands for the set of all integer
numbers. For a multiindex \alpha = (\alpha 1, . . . , \alpha n) \in \BbbN n
0 , we write | \alpha | = \alpha 1 + . . . + \alpha n. The Euclidean
scalar product of x = (x1, . . . , xn) and y = (y1, . . . , yn) is given by x \cdot y = x1y1 + . . .+ xnyn. The
expression f \lesssim g means that f \leq c g for some independent constant c (and nonnegative functions
f and g), and f \approx g means f \lesssim g \lesssim f. As usual for any x \in \BbbR , [x] stands for the largest integer
smaller than or equal to x.
c\bigcirc R. HERAIZ, 2020
1034 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS 1035
For x \in \BbbR n and r > 0, we denote by B(x, r) the open ball in \BbbR n with center x and radius r.
By supp f we denote the support of the function f, i.e., the closure of its non-zero set. 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.
The symbol \scrS (\BbbR n) is used in place of the set of all Schwartz functions on \BbbR n and 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 on \BbbR n with range in
[c,\infty [ for some c > 0. We denote the set of such functions by \scrP 0(\BbbR n). The subset of variable
exponents with range [1,\infty ) is denoted by \scrP . For p \in \scrP 0(\BbbR n), we use the notation
p - = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}
x\in \BbbR n
p(x), p+ = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n
p(x).
Everywhere below we shall consider bounded exponents.
Let p belongs to \scrP 0(\BbbR n). The variable exponent Lebesgue space Lp(\cdot )(\BbbR n) is the class of all
measurable functions f on \BbbR n such that the modular
\varrho p(\cdot )(f) :=
\int
\BbbR n
| f(x)| p(x) dx
is finite. This is a quasi-Banach function space equipped with the norm
\| f\| p(\cdot ) := \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\mu > 0 : \varrho p(\cdot )
\biggl(
1
\mu
f
\biggr)
\leq 1
\biggr\}
.
If p(x) \equiv p is constant, then Lp(\cdot )(\BbbR n) = Lp(\BbbR n) is the classical Lebesgue space.
A useful property is that \varrho p(\cdot )(f) \leq 1 if and only if \| f\| p(\cdot ) \leq 1 (unit ball property). This property
is clear for constant exponents due to the obvious relation between the norm and the modular in that
case.
We say that a function g : \BbbR n \rightarrow \BbbR is locally \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous, if there exists a constant
clog > 0 such that
| g(x) - g(y)| \leq
clog
\mathrm{l}\mathrm{n}(e+ 1/| x - y| )
for all x, y \in \BbbR n. If
| g(x) - g(0)| \leq
clog
\mathrm{l}\mathrm{n}(e+ 1/| x| )
for all x \in \BbbR n, then we say that g is \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous at the origin (or has a \mathrm{l}\mathrm{o}\mathrm{g} decay at the
origin). If, for some g\infty \in \BbbR and clog > 0, there holds
| g(x) - g\infty | \leq
clog
\mathrm{l}\mathrm{n}(e+ | x| )
for all x \in \BbbR n, then we say that g is \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous at infinity (or has a \mathrm{l}\mathrm{o}\mathrm{g} decay at infinity).
By \scrP log
0 (\BbbR n) and \scrP log
\infty (\BbbR n) we denote the class of all exponents p \in \scrP (\BbbR n) which have a
log decay at the origin and at infinity, respectively. The notation \scrP log(\BbbR n) is used for all those
exponents p \in \scrP (\BbbR n) which are locally \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous and have a \mathrm{l}\mathrm{o}\mathrm{g} decay at infinity,
with p\infty := \mathrm{l}\mathrm{i}\mathrm{m}| x| \rightarrow \infty p(x). Obviously, we get \scrP log(\BbbR n) \subset \scrP log
0 (\BbbR n) \cap \scrP log
\infty (\BbbR n). Here, p\prime denotes
the conjugate exponent of p given by 1/p(\cdot ) + 1/p\prime (\cdot ) = 1. Note that p \in \scrP log(\BbbR n) if and only if
p\prime \in \scrP log(\BbbR n), and since (p\prime )\infty = (p\infty )\prime we write only p\prime \infty for any of these quantities.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1036 R. HERAIZ
Let p, q \in \scrP 0. 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) =
\sum
v
\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:
\| (fv)v\| \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\}
.
Since q+ < \infty , then we can replace by the simpler expression
\varrho \ell q(\cdot )(Lp(\cdot ))((fv)v) =
\sum
v
\bigm\| \bigm\| \bigm\| | fv| q(\cdot )\bigm\| \bigm\| \bigm\| p(\cdot )
q(\cdot )
.
Furthermore, if p and q are constants, then \ell q(\cdot )(Lp(\cdot )) = \ell q(Lp). It is known that \ell q(\cdot )(Lp(\cdot )) is a
norm if q(\cdot ) \geq 1 is constant almost everywhere (a.e.) on \BbbR n and p(\cdot ) \geq 1, or if
1
p(x)
+
1
q(x)
\leq 1
a.e. on \BbbR n, or if 1 \leq q(x) \leq p(x) < \infty a.e. on \BbbR n.
Very often we have to deal with the norm of characteristic functions on balls (or cubes) when
studying the behavior of various operators in Harmonic Analysis. In classical Lp spaces the norm
of such functions is easily calculated, but this is not the case when we consider variable exponents.
Nevertheless, it is known that for p \in \scrP log we obtain
\| \chi B\| p(\cdot )\| \chi B\| p\prime (\cdot ) \approx | B| . (2.1)
Also,
\| \chi B\| p(\cdot ) \approx | B|
1
p(x) , x \in B, (2.2)
for small balls B \subset \BbbR n (| B| \leq 2n), and
\| \chi B\| p(\cdot ) \approx | B|
1
p\infty (2.3)
for large balls (| B| \geq 1) with constants only depending on the \mathrm{l}\mathrm{o}\mathrm{g}-Hölder constant of p (see, for
example, [3], Section 4.5). Let L1
Loc (\BbbR n) be the collection of all locally integrable functions on \BbbR n.
Recall that the space BMO(\BbbR n) consists of all locally integrable functions f such that
\| f\| BMO := \mathrm{s}\mathrm{u}\mathrm{p}
Q
1
| Q|
\int
Q
| f(x) - fQ| dx < \infty ,
where fQ =
1
| Q|
\int
Q
f(y)dy, the supremum is taken over all cubes Q \subset \BbbR n with sides parallel to
the coordinate axes.
We refer the reader to the recent monograph [3] (Section 4.5) for further details, historical remarks
and more references on variable exponent spaces.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS 1037
3. Basic tools. In this section, we present some results which are useful for us. The following
lemma plays an important role in the proof of the main results of this paper, is given in [1], where is
a generalization of (2.1), (2.2), and (2.3) to the case of dyadic annuli.
Lemma 3.1. Let p \in \scrP log
\infty (\BbbR n) and R = B(0, r) \setminus B
\Bigl(
0,
r
2
\Bigr)
. If | R| \geq 2 - n, then
\| \chi R\| p(\cdot ) \approx | R|
1
p(x) \approx | R|
1
p\infty
with the implicit constants independent of r and x \in R.
The left-hand side equivalence remains true for every | R| > 0 if we assume, additionally, that p
belongs to \scrP log
0 (\BbbR n) \cap \scrP log
\infty (\BbbR n).
The next lemma is a Hardy-type inequality which is easy to prove.
Lemma 3.2. Let 0 < a < 1 and 0 < q \leq \infty . Let \{ \varepsilon k\} k\in \BbbZ be a sequence of positive real
numbers such that \bigm\| \bigm\| \{ \varepsilon k\} k\in \BbbZ \bigm\| \bigm\| \ell q = I < \infty .
Then the sequences
\Bigl\{
\delta k : \delta k =
\sum
j\leq k a
k - j\varepsilon j
\Bigr\}
k\in \BbbZ
and
\Bigl\{
\eta k : \eta k =
\sum
j\geq k a
j - k\varepsilon j
\Bigr\}
k\in \BbbZ
belong to \ell q,
and \bigm\| \bigm\| \{ \delta k\} k\in \BbbZ \bigm\| \bigm\| \ell q + \bigm\| \bigm\| \{ \eta k\} k\in \BbbZ \bigm\| \bigm\| \ell q \leq c I
with c > 0 only depending on a and q.
For convenience, we set
Bk := B(0, 2k), Rk := Bk \setminus Bk - 1 and \chi k = \chi Rk
, k \in \BbbZ .
Definition 3.1. Let p, q belong to \scrP 0(\BbbR n) and \alpha : \BbbR n \rightarrow \BbbR with \alpha \in L\infty (\BbbR n). The inhomoge-
neous Herz space K
\alpha (\cdot )
p(\cdot ),q(\cdot ) (\BbbR
n) consists of all f \in L
p(\cdot )
Loc (\BbbR
n) such that
\| f\|
K
\alpha (\cdot )
p(\cdot ),q(\cdot )
:= \| f \chi B0\| p(\cdot ) +
\bigm\| \bigm\| \bigm\| \bigm\| \Bigl( 2k\alpha (\cdot )f \chi k
\Bigr)
k\geq 1
\bigm\| \bigm\| \bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
< \infty .
Similarly, the homogeneous Herz space \.K
\alpha (\cdot )
p(\cdot ),q(\cdot ) (\BbbR
n) is defined as the set of all f \in
\in L
p(\cdot )
Loc (\BbbR
n \setminus \{ 0\} ) such that
\| f\| \.K
\alpha (\cdot )
p(\cdot ),q(\cdot )
:=
\bigm\| \bigm\| \bigm\| \Bigl( 2k\alpha (\cdot )f \chi k
\Bigr)
k\in \BbbZ
\bigm\| \bigm\| \bigm\|
\ell q(\cdot )(Lp(\cdot ))
< \infty .
If \alpha and p, q are constant, then \.K
\alpha (\cdot ),p(\cdot )
q(\cdot ) (\BbbR n) is the classical Herz spaces \.K\alpha ,p
q (\BbbR n) .
The following proposition is very important for the proof of the main results it is from D. Drihem
and F. Seghiri in [5].
Proposition 3.1. Let \alpha \in L\infty (\BbbR n), p, q \in \scrP 0(\BbbR n). If \alpha and q are \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous at
infinity, then
K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) = K\alpha \infty ,q\infty
p(\cdot ) (\BbbR n) .
Additionally, if \alpha and q have a \mathrm{l}\mathrm{o}\mathrm{g} decay at the origin, then
\| f\| \.K
\alpha (\cdot ),q(\cdot )
p(\cdot )
\approx
\Biggl( - 1\sum
k= - \infty
\| 2k\alpha (0)f \chi k\|
q(0)
p(\cdot )
\Biggr) 1/q(0)
+
\Biggl( \infty \sum
k=0
\| 2k\alpha \infty f \chi k\| q\infty p(\cdot )
\Biggr) 1/q\infty
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1038 R. HERAIZ
Let \varphi belongs to C\infty
0 (\BbbR n) with supp \varphi \subseteq B0,
\int
\BbbR n
\varphi (x)dx \not = 0 and \varphi t (\cdot ) = t - n\varphi
\bigl( \cdot
t
\bigr)
for any
t > 0. Let M\varphi (f) be the grand maximal function of f defined by
M\varphi (f)(x) := \mathrm{s}\mathrm{u}\mathrm{p}
t>0
| \varphi t \ast f(x)| .
We will give the definition of the homogeneous Herz-type Hardy spaces H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) .
Definition 3.2. Let p, q belong to \scrP 0(\BbbR n) and \alpha : \BbbR n \rightarrow \BbbR with \alpha \in L\infty (\BbbR n). The homo-
geneous Herz-type Hardy space H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) is defined as the set of all f \in \scrS \prime (\BbbR n) such that
M\varphi (f) \in \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) and we define
\| f\|
H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot )
:= \| M\varphi (f)\| \.K
\alpha (\cdot ),q(\cdot )
p(\cdot )
.
It can be shown that, if p, q, and \alpha satisfy the conditions of definition, then the quasinorm
\| f\|
H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot )
does not depend, up to the equivalence of quasinorms, on the choice of the function
\varphi and, hence, the space H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) is defined independently of the choice \varphi . If p belongs to
\scrP log
0 (\BbbR n) \cap \scrP log
\infty (\BbbR n) with
- n
p+
< \alpha - \leq \alpha + < n - n
p -
and q \in \scrP 0(\BbbR n), then
H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) = \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n).
If \alpha (\cdot ) = 0, p (\cdot ) = q (\cdot ) , then H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) and \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) coicide with Lp(\cdot )(\BbbR n).
One recognizes immediately that if \alpha , p and q are constants, then the spaces H \.K\alpha ,q
p are just the
usual Herz-type Hardy spaces were recently studied in [11, 12].
Now, we introduce the basic notation of atomic decomposition.
Definition 3.3. Let \alpha \in L\infty (\BbbR n), p \in \scrP (\BbbR n), q \in \scrP 0(\BbbR n) and s \in \BbbN 0. A function a is said to
be a central (\alpha (\cdot ), p(\cdot ))-atom, if
(i) \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} a \subset B(0, r) = \{ x \in \BbbR n : | x| \leq r\} , r > 0,
(ii) \| a\| p(\cdot ) \leq | B(0, r)| - \alpha (0)/n, 0 < r < 1,
(iii) \| a\| p(\cdot ) \leq | B(0, r)| - \alpha \infty /n, r \geq 1,
(iv)
\int
\BbbR n
x\beta a(x)dx = 0, | \beta | \leq s.
A function a on \BbbR n is said to be a central (\alpha (\cdot ), p(\cdot ))-atom of restricted type, if it satisfies the
conditions (iii), (vi) above and \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}a \subset B(0, r), r \geq 1.
If r = 2k for some k \in \BbbZ in Definition 3.3, then the corresponding central (\alpha (\cdot ), p(\cdot ))-atom is
called a dyadic central (\alpha (\cdot ), p(\cdot ))-atom.
Now we establish characterizations of the spaces H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) in terms of central atomic
decompositions, which make it convenient to study the boundedness of operators on these spaces. In
[5], we have the following the atomic decomposition characterization of spaces H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS 1039
Theorem 3.1. Let \alpha and q are be \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous, both at the origin and at infinity and
p \in \scrP log(\BbbR n) with 1 < p - \leq p+ < \infty . For any f \in H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) , we have
f =
\infty \sum
k= - \infty
\lambda kak,
where the series converges in the sense of distributions, \lambda k \geq 0, each ak is a central (\alpha (\cdot ), p(\cdot ))-
atom with \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} a \subset Bk and\Biggl( - 1\sum
k= - \infty
| \lambda k| q(0)
\Biggr) 1/q(0)
+
\Biggl( \infty \sum
k=0
| \lambda k| q\infty
\Biggr) 1/q\infty
\leq c\| f\|
H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot )
.
Conversely, if \alpha (\cdot ) \geq n
\biggl(
1 - 1
p -
\biggr)
and s \geq
\biggl[
\alpha + + n
\biggl(
1
p -
- 1
\biggr) \biggr]
, and if holds, then f belongs to
H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot ) (\BbbR n) , and
\| f\|
H \.K
\alpha (\cdot ),q(\cdot )
p(\cdot )
\approx \mathrm{i}\mathrm{n}\mathrm{f}
\left\{
\Biggl( - 1\sum
k= - \infty
| \lambda k| q(0)
\Biggr) 1/q(0)
+
\Biggl( \infty \sum
k=0
| \lambda k| q\infty
\Biggr) 1/q\infty
\right\} ,
where the infimum is taken over all the decompositions of f as above.
The following lemma is from [9] (Lemma 2.9), see also [16] (Lemma 0.5).
Lemma 3.3. Let p belongs to \scrP log (\BbbR n) , k be a positive integer and B be a ball in \BbbR n. Then,
for all b \in BMO(\BbbR n)and all i, j \in \BbbZ with j > i, we have
1
c
\| b\| kBMO \leq \mathrm{s}\mathrm{u}\mathrm{p}
B
1
\| \chi B\| p(\cdot )
\| (b - bB)
k\chi B\| p(\cdot ) \leq c \| b\| kBMO ,
\| (b - bBi)
k\chi Bj\| p(\cdot ) \leq c(j - i)k \| b\| kBMO \| \chi Bj\| p(\cdot ) .
Given 0 < \sigma < n, for an appropriate function f, the commutator with m-order of fractional
integral operators Im\sigma ,b, m = 1, 2,. . . , is defined by
Im\sigma ,b (f) (x) :=
\int
\BbbR n
(b(x) - b(y))m
| x - y| n - \sigma f(y)dy.
We denote I1\sigma ,b by [b, I\sigma ] and I0\sigma ,b by the fractional integral operator I\sigma , respectively.
The next two lemmas are from [5] treat the case when m = 0, 1 for Im\sigma ,b.
Lemma 3.4. Suppose that p1, p2 belong to \scrP log(\BbbR n) with p+1 <
n
\sigma
and
1
p1(\cdot )
- 1
p2(\cdot )
=
\sigma
n
.
Then, for all f \in Lp1(\cdot ) (\BbbR n) , we have
\| I\sigma (f)\| p2(\cdot ) \leq c \| f\| p1(\cdot ) .
Lemma 3.5. Suppose that p1, p2 belong to \scrP log(\BbbR n) with p+1 <
n
\sigma
,
1
p1(\cdot )
- 1
p2(\cdot )
=
\sigma
n
and
b \in BMO(\BbbR n). Then, for all f \in Lp1(\cdot ) (\BbbR n) , we have
\| [b, I\sigma ](f)\| p2(\cdot ) \leq C \| b\| BMO \| f\| p1(\cdot ) .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1040 R. HERAIZ
4. Boundedness of fractional integral and their commutators on variable Herz-type Hardy
spaces. In this section, we present the boundedness of fractional integral operators and their com-
mutators on variable Herz-type Hardy spaces.
First, we treat the boundedness of I\sigma on variable Herz-type Hardy spaces.
Theorem 4.1. Suppose that p1, p2 belong to \scrP log(\BbbR n) with p+1 <
n
\sigma
and
1
p1(\cdot )
- 1
p2(\cdot )
=
\sigma
n
,
\alpha \in L\infty (\BbbR n), q1, q2 \in \scrP 0(\BbbR n). If \alpha , q1 and q2 are \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous, both at the origin and
at infinity with \alpha (\cdot ) \geq n
\Bigl(
1 - 1
p - 1
\Bigr)
, q1(0) \leq q2(0) and (q1)\infty \leq (q2)\infty , then I\sigma is bounded from
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot ) (\BbbR n) to \.K
\alpha (\cdot ),q2(\cdot )
p2(\cdot ) (\BbbR n).
Next, we present the boundedness of [b, I\sigma ] on variable Herz-type Hardy spaces.
Theorem 4.2. Suppose that p1, p2 belong to \scrP log(\BbbR n) with p+1 <
n
\sigma
and
1
p1(\cdot )
- 1
p2(\cdot )
=
\sigma
n
,
\alpha \in L\infty (\BbbR n), q1, q2 \in \scrP 0(\BbbR n). If \alpha , q1 and q2 are \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous, both at the origin and at
infinity with \alpha (\cdot ) \geq n(1 - 1
p - 1
), q1(0) \leq q2(0), (q1)\infty \leq (q2)\infty and b \in BMO(\BbbR n), then [b, I\sigma ] is
bounded from H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot ) (\BbbR n) to \.K
\alpha (\cdot ),q2(\cdot )
p2(\cdot ) (\BbbR n).
Remark 4.1. If \alpha , q1 and q2 are constants, then the statements corresponding to Theorems 4.1
and 4.2 can be found in [16] (Theorems 1.1 and 1.2).
Our proofs use partially some techniques already used in [16] where \alpha , q1 and q2 are constants.
Proof of Theorem 4.1. We must show that
\| I\sigma (f)\| \.K
\alpha (\cdot ),q2(\cdot )
p2(\cdot )
(\BbbR n)
\leq c \| f\|
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot )
(\BbbR n)
for all f \in H \.K
\alpha 1(\cdot ),q1(\cdot )
p1(\cdot ) (\BbbR n) . By using Theorem 3.1, we may assume that
f =
+\infty \sum
i= - \infty
\lambda iai
where \lambda i \geq 0 and ai’s are (\alpha (\cdot ) , p1 (\cdot ))-atom with \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} ai \subseteq Bi. By using Proposition 3.1, we
have
\| I\sigma (f)\| \.K
\alpha (\cdot ),q2(\cdot )
p2(\cdot )
(\BbbR n)
\approx
\approx
\Biggl\{ - 1\sum
k= - \infty
2k\alpha (0)q2(0) \| I\sigma (f)\chi k\|
q2(0)
p2(\cdot )
\Biggr\} 1/q2(0)
+ c
\Biggl\{
+\infty \sum
k=0
2k\alpha \infty (q2)\infty \| I\sigma (f)\chi k\|
(q2)\infty
p2(\cdot )
\Biggr\} 1/(q2)\infty
\leq
\leq
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| \| I\sigma (ai)\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
+
+
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl( \infty \sum
i=k - 1
| \lambda i| \| I\sigma (ai)\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS 1041
+
\left\{
+\infty \sum
k=0
2k\alpha \infty (q2)\infty
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| \| I\sigma (ai)\chi k\| p2(\cdot )
\Biggr) (q2)\infty
\right\}
1/(q2)\infty
+
+
\left\{
+\infty \sum
k=0
2k\alpha \infty (q2)\infty
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| I\sigma (ai)\chi k\| p2(\cdot )
\Biggr) (q2)\infty
\right\}
1/(q2)\infty
=:
=: H1 +H2 +H3 +H4.
Let us estimate H1. By the s-order vanishing moments of ai with
s \geq
\biggl[
\alpha + - n
\biggl(
1 - 1
p - 1
\biggr) \biggr]
,
we can subtract the Taylor expansion of | x - y| - n+\sigma at x, we obtain
| I\sigma (ai) (x)| \leq
\int
Bi
| ai (y)| | y| s+1
| x| n - \sigma +s+1 dy \leq
\leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1)
\int
Bi
| ai (y)| dy.
Applying Hölder inequality, we get
| I\sigma (ai) (x)| \leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1) \| ai\| p1(\cdot ) \| \chi Bi\| p\prime 1(\cdot ) . (4.1)
On the other hand (see [7, p. 350]), we have
I\sigma (\chi Bk
) (x) \geq
\int
Bk
dy
| x - y| n - \sigma \chi Bk
(x) \geq c \cdot 2k\sigma \chi Bk
(x) . (4.2)
By (4.1), (4.2) and Lemma 3.4, gives
\| I\sigma (ai)\chi k\| p2(\cdot ) \leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1) \| ai\| p1(\cdot ) \| \chi Bi\| p\prime 1(\cdot ) \| \chi k\| p2(\cdot ) \leq
\leq c \cdot 2 - k(n+s+1)+i(s+1) \| ai\| p1(\cdot ) \| \chi Bi\| p\prime 1(\cdot ) \| I\sigma (\chi Bk
)\| p2(\cdot ) \leq
\leq c \cdot 2 - k(n+s+1)+i(s+1) \| ai\| p1(\cdot ) \| \chi Bi\| p\prime 1(\cdot ) \| \chi Bk
\| p1(\cdot ) .
By (2.2), we have
H1 =
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| \| I\sigma (ai)\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1042 R. HERAIZ
\leq c
\left\{
- 1\sum
k= - \infty
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| 2(i - k)(n+s+1 - (\alpha +n/p1)(0))
\Biggr) q2(0)
\right\}
1/q2(0)
.
Since s+ 1 - \alpha + + n
\biggl(
1 - 1
p - 1
\biggr)
> 0, then, by Lemma 3.2, we obtain
H1 \leq c
\Biggl( - 1\sum
k= - \infty
| \lambda k| q2(0)
\Biggr) 1/q2(0)
\leq c
\Biggl( - 1\sum
k= - \infty
| \lambda k| q1(0)
\Biggr) 1/q1(0)
\leq c\| f\|
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot )
.
Let us estimate H2. By Lemma 3.4, we get
H2 =
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| I\sigma (ai)\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| ai\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl( - 1\sum
i=k - 1
| \lambda i| \| ai\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
+
+c
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=0
| \lambda i| \| ai\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c
\left\{
- 1\sum
k= - \infty
\Biggl( - 1\sum
i=k - 1
| \lambda i| 2(k - i)\alpha (0)
\Biggr) q2(0)
\right\}
1/q2(0)
+
+c
\left\{
- 1\sum
k= - \infty
\Biggl(
+\infty \sum
i=0
| \lambda i| 2(k - i)\alpha - +k(\alpha (0) - \alpha - )+i(\alpha - - \alpha \infty )
\Biggr) q2(0)
\right\}
1/q2(0)
for k < 0 \leq i and since \alpha - \leq \mathrm{m}\mathrm{i}\mathrm{n}(\alpha (0) , \alpha \infty ), we have k(\alpha (0) - \alpha - ) + i(\alpha - - \alpha \infty ) \leq 0. Then,
by Lemma 3.2, we have
H2 \leq c
\Biggl( - 1\sum
k= - \infty
| \lambda k| q2(0)
\Biggr) 1/q2(0)
\leq c
\Biggl( - 1\sum
k= - \infty
| \lambda k| q1(0)
\Biggr) 1/q1(0)
\leq c\| f\|
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot )
.
We can estimate H3 and H4 by the same technique as in the estimation of H1 and H2 when replacing
\alpha (0) and q2(0) by \alpha \infty and (q2)\infty , respectively. A combination of estimations of H1, H2, H3, and
H4 finish the proof ot Theorem 4.1.
Proof of Theorem 4.2. We must show that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS 1043
\| [b, I\sigma ]f\| \.K
\alpha (\cdot ),q2(\cdot )
p2(\cdot )
(\BbbR n)
\leq c \| f\|
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot )
(\BbbR n)
for all f \in H \.K
\alpha 1(\cdot ),q1(\cdot )
p1(\cdot ) (\BbbR n) . By using Theorem 3.1, we may assume that
f =
+\infty \sum
i= - \infty
\lambda iai,
where \lambda i \geq 0 and ai’s are (\alpha (\cdot ) , p1 (\cdot ))-atom with \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}ai \subseteq Bi. By using Proposition 3.1, we
obtain
\| [b, I\sigma ]f\| \.K
\alpha (\cdot ),q2(\cdot )
p2(\cdot )
(\BbbR n)
\approx
\approx
\Biggl\{ - 1\sum
k= - \infty
2k\alpha (0)q2(0) \| [b, I\sigma ]f\chi k\|
q2(0)
p2(\cdot )
\Biggr\} 1/q2(0)
+ c
\Biggl\{
+\infty \sum
k=0
2k\alpha \infty (q2)\infty \| [b, I\sigma ]f\chi k\|
(q2)\infty
p2(\cdot )
\Biggr\} 1/(q2)\infty
\leq
\leq
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| \| [b, I\sigma ]f\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
+
+
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| [b, I\sigma ]f\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
+
+
\left\{
+\infty \sum
k=0
2k\alpha \infty (q2)\infty
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| \| [b, I\sigma ]\chi k\| p2(\cdot )
\Biggr) (q2)\infty
\right\}
1/(q2)\infty
+
+
\left\{
+\infty \sum
k=0
2k\alpha \infty (q2)\infty
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| [b, I\sigma ]f\chi k\| p2(\cdot )
\Biggr) (q2)\infty
\right\}
1/(q2)\infty
=:
=: Q1 +Q2 +Q3 +Q4.
Let us estimate Q1. As in H1, we use the Taylor expansion of | x - y| - n+\sigma at x and the s-order
vanishing moments of ai with s \geq
\biggl[
\alpha + - n
\biggl(
1 - 1
p - 1
\biggr) \biggr]
, we get
| [b, I\sigma ] (ai)| \leq
\leq
\int
Bi
| b(x) - b(y)| | ai (y)| | y|
s+1
| x| n - \sigma +s+1 dy \leq
\leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1)
\int
Bi
| ai (y)| | b(x) - b(y)| dy \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1044 R. HERAIZ
\leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1)
\left( | b(x) - bBi |
\int
Bi
| ai (y)| dy +
\int
Bi
| ai (y)| | bBi - b(y)| dy
\right) .
We use the Hölder inequality, the last expression is bounded by
c \cdot 2 - k(n - \sigma +s+1)+i(s+1) \| ai\| p1(\cdot )
\Bigl(
| b(x) - bBi | \| \chi Bi\| p\prime 1(\cdot ) + \| | bBi - b(y)| \chi Bi\| p\prime 1(\cdot )
\Bigr)
.
By (4.2) and Lemma 3.3, we have
\| [b, I\sigma ] (ai)\chi k\| p2(\cdot ) \leq
\leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1) \| ai\| p1(\cdot )
\Bigl(
\| | b(x) - bBi | \chi k\| p2(\cdot ) \| \chi Bi\| p\prime 1(\cdot )+
+ \| | bBi - b(y)| \chi Bi\| p\prime 1(\cdot ) \| \chi k\| p2(\cdot )
\Bigr)
\leq
\leq c \cdot 2 - k(n - \sigma +s+1)+i(s+1) \| ai\| p1(\cdot )
\Bigl(
(k - i) \| b\| BMO \| \chi Bk
\| p2(\cdot ) \| \chi Bi\| p\prime 1(\cdot )+
+ \| b\| BMO \| \chi Bi\| p\prime 1(\cdot ) \| \chi k\| p2(\cdot )
\Bigr)
\leq
\leq c(k - i) \cdot 2 - k(n - \sigma +s+1)+i(s+1) \| ai\| p1(\cdot ) \| b\| BMO \| \chi Bi\| p\prime 1(\cdot ) \| \chi Bk
\| p2(\cdot ) \leq
\leq c(k - i) \cdot 2 - k(n+s+1)+i(s+1) \| ai\| p1(\cdot ) \| b\| BMO \| \chi Bi\| p\prime 1(\cdot ) \| I\sigma (ai)\chi k\| p2(\cdot ) \leq
\leq c(k - i) \cdot 2 - k(n+s+1)+i(s+1) \| ai\| p1(\cdot ) \| b\| BMO \| \chi Bi\| p\prime 1(\cdot ) \| \chi k\| p1(\cdot ) .
By (2.2) and Lemma 3.1, we obtain
Q1 =
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| \| [b, I\sigma ] (ai)\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c \| b\| BMO
\left\{
- 1\sum
k= - \infty
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| (k - i) \cdot 2(i - k)(s+1+n - (\alpha +n/p1)(0))
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c \| b\| BMO
\left\{
- 1\sum
k= - \infty
\Biggl(
k - 2\sum
i= - \infty
| \lambda i| (k - i) \cdot 2(i - k)(s+1+n - (\alpha +n/p1)(0))
\Biggr) q2(0)
\right\}
1/q2(0)
. (4.3)
Since s+ 1 - \alpha + + n
\biggl(
1 - 1
p - 1
\biggr)
> 0, then, by Lemma 3.2, we get that (4.3) is bounded by
Q1 \leq c \| b\| BMO
\Biggl( - 1\sum
k= - \infty
| \lambda k| q2(0)
\Biggr) 1/q2(0)
\leq c
\Biggl( - 1\sum
k= - \infty
| \lambda k| q1(0)
\Biggr) 1/q1(0)
\leq c\| f\|
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot )
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
VARIABLE HERZ ESTIMATES FOR FRACTIONAL INTEGRAL OPERATORS 1045
Let us estimate Q2. By Lemma 3.3, we get
Q2 =
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| [b, I\sigma ] (ai)\chi k\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c \| b\| BMO
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=k - 1
| \lambda i| \| ai\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c \| b\| BMO
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl( - 1\sum
i=k - 1
| \lambda i| \| ai\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
+
+c \| b\| BMO
\left\{
- 1\sum
k= - \infty
2k\alpha (0)q2(0)
\Biggl(
+\infty \sum
i=0
| \lambda i| \| ai\| p2(\cdot )
\Biggr) q2(0)
\right\}
1/q2(0)
\leq
\leq c \| b\| BMO
\left\{
- 1\sum
k= - \infty
\Biggl( - 1\sum
i=k - 1
| \lambda i| 2(k - i)\alpha (0)
\Biggr) q2(0)
\right\}
1/q2(0)
+
+c \| b\| BMO
\left\{
- 1\sum
k= - \infty
\Biggl(
+\infty \sum
i=0
| \lambda i| 2(k - i)\alpha - +k(\alpha (0) - \alpha - )+i(\alpha - - \alpha \infty )
\Biggr) q2(0)
\right\}
1/q2(0)
for k < 0 \leq i. Since \alpha - \leq \mathrm{m}\mathrm{i}\mathrm{n}(\alpha (0) , \alpha \infty ), we obtain k(\alpha (0) - \alpha - )+ i(\alpha - - \alpha \infty ) \leq 0. Then, by
Lemma 3.2, we have
Q2 \leq c \| b\| BMO
\Biggl( - 1\sum
k= - \infty
| \lambda k| q2(0)
\Biggr) 1/q2(0)
\leq c
\Biggl( - 1\sum
k= - \infty
| \lambda k| q1(0)
\Biggr) 1/q1(0)
\leq c\| f\|
H \.K
\alpha (\cdot ),q1(\cdot )
p1(\cdot )
.
We can estimate Q3 and Q4 by the same technique as in the estimation of Q1 and Q2, when
replacing \alpha (0) and q2(0) by \alpha \infty and (q2)\infty , respectively.
A combination of estimations of Q1, Q2, Q3, and Q4 finish the proof ot Theorem 4.2.
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6. M. Izuki, Herz and amalgam spaces with variable exponent, the Haar wavelets and greediness of the wavelet system,
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Received 11.06.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
|
| id | umjimathkievua-article-6024 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:23Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c2/1223ffeca16e8d10af96d0656d5536c2.pdf |
| spelling | umjimathkievua-article-60242022-03-26T11:02:01Z Variable Herz estimates for fractional integral operators Variable Herz estimates for fractional integral operators Heraiz , R. Heraiz , R. UDC 517.5 In this paper, the author study the boundedness of fractional integral operators on a variable Herz-type Hardy space $HK^{\alpha (\cdot )}_{p(\cdot ),q(\cdot )}(\mathbb{R}^n)$ by using the atomic decomposition. УДК 517.5 Змiннi оцiнки Герца для дробових iнтегральних операторiв За допомогою атомарної декомпозицiї вивчається обмеженiсть дробових iнтегральних операторiв у змiнному просторi Гардi $HK^{\alpha (\cdot )}_{p(\cdot ),q(\cdot )}(\mathbb{R}^n)$ типу Герца. Institute of Mathematics, NAS of Ukraine 2020-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6024 10.37863/umzh.v72i8.6024 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 8 (2020); 1034-1046 Український математичний журнал; Том 72 № 8 (2020); 1034-1046 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6024/8737 |
| spellingShingle | Heraiz , R. Heraiz , R. Variable Herz estimates for fractional integral operators |
| title | Variable Herz estimates for fractional integral operators |
| title_alt | Variable Herz estimates for fractional integral operators |
| title_full | Variable Herz estimates for fractional integral operators |
| title_fullStr | Variable Herz estimates for fractional integral operators |
| title_full_unstemmed | Variable Herz estimates for fractional integral operators |
| title_short | Variable Herz estimates for fractional integral operators |
| title_sort | variable herz estimates for fractional integral operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6024 |
| work_keys_str_mv | AT heraizr variableherzestimatesforfractionalintegraloperators AT heraizr variableherzestimatesforfractionalintegraloperators |