ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version)
doi:http://dx.doi.org/10.12732/ijam.v33i4.11
HARDY OPERATORS IN THE LOCAL “COMPLEMENTARY” GENERALIZED VARIABLE EXPONENT WEIGHTED
MORREY SPACES
Zuleyxa O. Azizova
Azerbaijan State Oil and Industry University AZ 1010, Baku, 20 Azadlig Ave., AZERBAIJAN
Abstract: In this paper we consider local “complementary” generalized weighted Morrey spacescMp(·),ω,ϕ
{x0} (Ω) with variable exponentp(x) and a general
function ω(r) defining the weighted Morrey-type norm. We prove the bound-edness of the Hardy operators in the spacescMp(·),ω,ϕ
{x0} (Ω) in case of unbounded
sets Ω⊂Rn.
AMS Subject Classification: 42B20, 42B25, 42B35
Key Words: local “complementary” generalized variable exponent weighted Morrey spaces; weighted Hardy operator
1. Introduction
Nowadays there is an evident increase of investigations, last two decades re-lated to both the theory of variable exponent function spaces and operator theory in these spaces. We refer for instance to the surveying papers [3], [4], [24], on the progress in this field, including topics of Harmonic Analysis and Operator Theory, see also references therein. Variable exponent Morrey spaces
Lp,λ(Rn), were introduced and studied in [1] in the Euclidean setting. In [1] the
boundedness of the maximal operator was proved in variable exponent Morrey spacesLp,λ(Rn) under the log-condition onp(·) andλ(·), and for potential
op-erators a Sobolev typeLp(·),λ(·)→ Lq(·),λ(·)–theorem was proved under the same
log-condition in the case of bounded sets.
The generalized variable exponent Morrey spaces were introduced and stud-ied in [11] in the case of bounded sets. In [11] the boundedness of the maximal operator, potential operators and singular integral operators in variable expo-nent Morrey spaces under the certain conditions were proved.
We consider the following Hardy operators
Aαβf(x) =|x|α−n+β
Z
|y|≤|x|
f(y)
|y|β dy,
Aαβf(x) =|x|α+β
Z
|y|>|x|
f(y)dy
|y|n+β,
whereα≥0.
We use the following notation: Rn is the n-dimensional Euclidean space, Ω ⊂Rn is an open set, χE(x) is the characteristic function of a set E ⊆ Rn,
B(x, r) = {y ∈ Rn : |x−y|< r}), B(x, r) =e B(x, r)∩Ω, by c, C, c1, c2 etc.,
we denote various absolute positive constants, which may have different values even in the same line.
2. Variable exponent local ”complementary” generalized Morrey spaces
We refer to the book [5] for variable exponent Lebesgue spaces but give some basic definitions and facts. Let p(·) be a measurable function on Ω with values in [1,∞). An open set Ω is assumed to be bounded throughout the whole paper. We mainly suppose that
1< p−≤p(x)≤p+<∞, (2.1)
where p−:= ess inf
x∈Ω p(x) >1, p+:= ess supx∈Ω p(x)<∞.By L
p(·)(Ω) we denote
the space of all measurable functionsf(x) on Ω such that
Ip(·)(f) =
Z
Ω
|f(x)|p(x)dx <∞.
Equipped with the norm
kfkp(·) = inf
η >0 : Ip(·)
f η
≤1
,
For the basics on variable exponent Lebesgue spaces we refer to [26], [16].
P(Ω) is the set of bounded measurable functionsp: Ω→[1,∞);
Plog(Ω) is the set of exponents p∈ P(Ω) satisfying the local log-condition |p(x)−p(y)| ≤ A
−ln|x−y|, |x−y| ≤
1
2 x, y∈Ω, (2.2)
whereA=A(p)>0 does not depend on x, y. We will use also the following decay conditions:
|p(x)−p(0)| ≤ A0
|ln|x||, |x| ≤
1
2, (2.3)
|p(x)−p(∞)| ≤ A∞
|ln|x||, |x| ≥2, (2.4)
wherep∞= lim
x→∞p(x)>1.
Alog(Ω) is the set of bounded exponents p: Ω→Rsatisfying the condition (2.2);
tPlog(Ω) is the set of exponentsp∈ Plog(Ω) with 1< p
−≤p(x)≤p+<∞;
for Ω which may be unbounded, by P∞(Ω), P∞log(Ω), P∞log(Ω), Alog∞(Ω) we
denote the subsets of the above sets of exponents satisfying the decay condition (2.4) (when Ω is unbounded).
For brevity, byP0,∞log(Ω) we denote the set of bounded measurable functions (not necessarily with values in [1,∞)), which satisfy the decay conditions (2.3) and (2.4).
By ϕwe always denote a weight, i.e. a positive, locally integrable function with Rn. The weighted Lebesgue space Lp(·)
ϕ (Rn) is defined as the set of all
measurable functions for which
kfkLp(·)
ϕ (Rn)
=kf ϕkLp(·)(Rn).
Let us define the classAp(·)(Rn) (see [6]) to consist of those weights ϕ for
which
[ϕ]Ap(·) ≡sup B
|B|−1kϕkLp(·)(B(x,r))e kϕ
−1k
Lp′(·)(B(x,r))e <∞.
A weight function ϕbelongs to the classAp(·),q(·)(Rn) if
[ϕ]Ap(·),q(·) ≡sup
B
rθp(x,r)−θq(x,r)−nkϕk
Lq(·)(B(x,r))e kϕ
−1k
Lp′(·)(B(x,r))e <∞.
Everywhere in the sequel the functions ω(r), ω1(r) and ω2(r) used in the
Definition 2.1. Let x0 ∈ Ω, 1 ≤ p− ≤ p(x) ≤ p+ < ∞. The local “complementary” generalized Morrey spacecMp(·),ω
{x0} (Ω) is defined by the norms
kfkcMp(·),ω {x0}
= sup
r>0
rθp′(x0,r)
ω(r) kfkLp(·)(Ω\B(x0,r))e ,
kfkcMp(·),ω,ϕ {x0}
= sup
r>0
kϕkLp′(·)(B(x0,r))e
ω(r) kfkLp(·)
ϕ (Ω\B(xe 0,r))
.
Everywhere in the sequel we assume that
sup
r>0
kϕkLp′(·)(B(xe 0,r))
ω(r) <∞, (2.5)
which makes the spacecMp(·),ω
{x0} (Ω) non-trivial, since it containsLp(·)(Ω) in this
case.
If also inf
r>0 1
ω(r) >0, then cM p(·),ω
{x0} (Ω) = Lp(·)(Ω). Therefore, to guarantee
that the “complementary” spacecMp(·),ω
{x0} (Ω) is strictly larger thanLp(Ω),one
should be interested in the cases where
lim
r→0
rθp′(x0,r)
ω(r) = 0. (2.6)
Clearly, the spacecMp(·),ω
{x0} (Ω) may contain functions with a non-integrable
singularity at the pointx0, if no additional assumptions are introduced.
3. Weighted Hardy operator in the spaces cMp(·),ω,ϕ{0} (Rn)
The proof of the main result of this section presented in Theorems 3.1 and 3.2 is based on the estimate given in the following preliminary theorems.
Theorem 3.1. Let p ∈ P0,∞log(Rn), ϕ ∈ A
p(·),q(·)(Rn). Suppose also that
β ≥0 and the functions(ω1, ω2) satisfy the conditions
Z t
0
sα−θp(0,s)+θq(0,s)ω
1(s) kϕkLp′(·)(B(0,s))
ds s ≤C
ω2(t) kϕkLq′(·)(B(0,s))
, (3.1)
Z t
0
ω1(s) sβkϕk
Lp′(·)(B(0,s))
ds s ≤C
ω1(t) tβkϕk
Lp′(·)(B(0,t))
, (3.2)
Then the weighted Hardy operatorAα
β is bounded from the spacecM p(·),ω1,ϕ {0} (Rn)
to the space cMq(·),ω2,ϕ{0} (Rn).
Proof. Letf ∈cMp(·),ω1,ϕ{0} (Rn), δ >0. We have
Z
|z|<r |f(z)|
|z|β dz=
Z
B(0,r) |f(z)|
|z|β dz=δ
Z
B(0,r) |f(z)| |z|β+δ
Z |z|
0
sδ−1ds
! dz =δ Z r 0 sδ−1 Z
{z∈Rn:s<|z|<r}
|f(z)| |z|β+δdz
!
ds.
Applying H¨older inequality, we get
Z
|z|<r |f(z)|
|z|β dz≤C
Z r
0
1
sβ kfkLp(·),ϕ(Rn\B(0,s))kϕ
−1k
Lp′(·)(B(0,r))
ds s .
Then we have Z
|z|<r |f(z)|
|z|β dz
≤C kϕ−1kLp′(·)(B(0,r)) Z r
0
1
sβ kfkLp(·)(Rn\B(0,s))
ds
s . (3.3)
Therefore by (3.3), (3.2) and (3.1) we have
kAαβfkLq(·),ϕ(Rn\B(0,t))
≤C
kϕ−1kLp′(·)(B(0,|·|))| · |α−n+β
×
Z |·|
0
kfkLp(·),ϕ(Rn\B(0,s))
sβ ds s
Lq(·),ϕ(Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
×
| · |α−θp(0,|·|)+θq(0,|·|)+β
kϕkLq(·)(B(0,|·|))
Z |·|
0
ω1(s)
sβkϕk
Lp′(·)(B(0,s))
ds s
Lq(·),ϕ(Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
×
| · |α−θp(0,|·|)+θq(0,|·|)+βω
1(| · |) | · |βkϕk
Lp′(·)(B(0,|·|))kϕkLq(·)(B(0,|·|))
Lq(·),ϕ(Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
×
| · |α−θp(0,|·|)+θq(0,|·|)ω
1(| · |) kϕkLq(·)(B(0,|·|))kϕkLp′(·)(B(0,|·|))
Lq(·),ϕ(Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
Z t
0
sα−θp(0,s)+θq(0,s)ω
1(s) kϕkLp′(·)(B(0,s))
ds s
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
ω2(t)
kϕkLq′(·)(B(0,s))
.
Theorem 3.2. Let p ∈ P0,∞log(Rn), ϕ ∈ A
p(·),q(·)(Rn) Suppose also that
β <−nand the functions(ω1, ω2) satisfy the conditions
Z t
0
sα−θp(0,s)+θq(0,s)ω
1(s) kϕkLp′(·)(B(0,s))
ds s ≤C
ω2(t) kϕkLq′(·)(B(0,t))
, (3.4)
Z ∞
t
s−θp(0,s)+θq(0,s)−β−1
kϕkLq(·)(B(0,s))
ds≤Ct
−θp(0,t)+θq(0,t)−β
kϕkLq(·)(B(0,t))
(3.5)
wheret >0.
Then the weighted Hardy operatorAα
β is bounded from the spacecM p(·),ω1,ϕ {0} (Rn)
to the space cMq(·),ω2,ϕ {0} (Rn).
Proof. Letf ∈ Mp(·),ω1,ϕ{0} (Rn),δ >0,−δ < n+β. We have
Z
|z|>r |f(z)| |z|n+βdz =
Z
Rn\B(0,r)
|f(z)| |z|n+βdz
=δ
Z
Rn\B(0,r)
|z|−n−β−δ|f(z)|
Z |z|
0
sδ−1ds
! dz =δ Z ∞ r sδ−1 Z
{z∈Rn:r<|z|<s}
|z|−n−β−δ|f(z)|dz
!
Hence applying H¨older inequality we get
Z
|z|>r |f(z)| |z|n+βdz
≤C
Z ∞
r
s−n−β−1kfkLp(·)
ϕ (Rn\B(0,r))
kϕ−1kLp′(·)(B(0,s))ds. (3.6)
Therefore by (3.6), we obtain
kAαβfkLq(·)
ϕ (Rn\B(0,t))
≤C
| · |α+βkfkLp(·)
ϕ (Rn\B(0,|·|))
×
Z ∞
|·|
s−n−β−1kϕ−1kLp′(·)(B(0,s))ds
Lq(·)
ϕ (Rn\B(0,t))
≤C | · |
α+βkfk Lp(·)
ϕ (Rn\B(0,|·|))
Z ∞
|·|
s−θp(0,s)+θq(0,s)−β−1
kϕkLq(·)(B(0,s))
ds
Lq(·)
ϕ (Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
×
| · |α+βω1(| · |)
kϕkLp′(·)(B(0,|·|))
Z ∞
|·|
s−θp(0,s)+θq(0,s)−β−1
kϕkLq(·)(B(0,s))
ds
Lq(·)
ϕ (Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
| · |α−θp(0,|·|)+θq(0,|·|)ω
1(| · |) kϕkLq(·)(B(0,|·|))kϕkLp′(·)(B(0,|·|))
Lq(·)
ϕ (Rn\B(0,t))
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
Z t
0
sα−θp(0,s)+θq(0,s)ω
1(s) kϕkLp′(·)(B(0,s))
ds s
≤Ckfkc
Mp(·),ω1,ϕ
{0} (Rn)
ω2(t) kϕkLq′(·)(B(0,t))
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