In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p((.)) -> q((.))-theorems were proved for the Riesz potential operator I-alpha in the weighted Lebesgue generalized spaces L-p(.)(R-n, p) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x(0) and to infinity, under an additional condition relating the weight exponents at x(0) and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L-p(.) (S-n, p) on the unit sphere S-n in Rn+1 are also improved in the same way. (c) 2006 Elsevier Inc. All rights reserved.info:eu-repo/semantics/publishedVersio

B. Vakulov

E. Shargorodsky

Kováciˇk

S. Samko

Samko

Journal of Mathematical Analysis and Applications

English

AbstractIn [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229–246], Sobolev-type p(⋅)→q(⋅)-theorems were proved for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(⋅)(Rn,ρ) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x0 and to infinity, under an additional condition relating the weight exponents at x0 and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces Lp(⋅)(Sn,ρ) on the unit sphere Sn in Rn+1 are also improved in the same way

Samko, S.

Shargorodsky, E.

Vakulov, B.

Elsevier - Publisher Connector 

J. Math. Anal. Appl. 325 (2007) 745–751
www.elsevier.com/locate/jmaa
Note
Weighted Sobolev theorem with variable exponent
for spatial and spherical potential operators, II
S. Samko a,∗, E. Shargorodsky b, B. Vakulov c
a University of Algarve, Portugal
b King’s College London, UK
c Rostov State University, Russia
Received 9 October 2005
Available online 20 February 2006
Submitted by L. Grafakos
Abstract
In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spher-
ical potential operators, J. Math. Anal. Appl. 310 (2005) 229–246], Sobolev-type p(·) → q(·)-theorems
were proved for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(·)(Rn,ρ)
with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x0 and
to infinity, under an additional condition relating the weight exponents at x0 and at infinity. We show in this
note that those theorems are valid without this additional condition. Similar theorems for a spherical ana-
logue of the Riesz potential operator in the corresponding weighted spaces Lp(·)(Sn,ρ) on the unit sphere
S
n in Rn+1 are also improved in the same way.
© 2006 Elsevier Inc. All rights reserved.
Keywords: Weighted Lebesgue spaces; Variable exponent; Riesz potentials; Spherical potentials; Stereographical
projection
* Corresponding author.
E-mail addresses: ssamko@ualg.pt (S. Samko), eugene.shargorodsky@kcl.ac.uk (E. Shargorodsky),
vakulov@math.rsu.ru (B. Vakulov).0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2006.01.069
746 S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–7511. Introduction
We consider the Riesz potential operator
Iαf (x) =
∫
Rn
f (y)
|x − y|n−α dy, 0 < α < n, (1.1)
in the weighted Lebesgue generalized spaces Lp(·)(Rn, ρ) with a variable exponent p(x) defined
by the norm
‖f ‖Lp(·)(Rn,ρ) = inf
{
λ > 0:
∫
Rn
ρ(x)
( |f (x)|
λ
)p(x)
dx  1
}
, (1.2)
where
ρ(x) = ργ0,γ∞(x) = |x|γ0
(
1 + |x|)γ∞−γ0 . (1.3)
We refer to [3–6] for the basics of the spaces Lp(·) with variable exponent.
We assume that the exponent p(x) satisfies the standard conditions
1 < p−  p(x) p+ < ∞, x ∈Rn, (1.4)∣∣p(x) − p(y)∣∣ A
ln 1|x−y|
, |x − y| 1
2
, x, y ∈Rn, (1.5)
and also the following condition at infinity
∣∣p∗(x) − p∗(y)∣∣ A∞ln 1|x−y| , |x − y|
1
2
, x, y ∈Rn, (1.6)
where p∗(x) = p( x|x|2 ). Conditions (1.5) and (1.6) taken together are equivalent to the following
global condition:
∣∣p(x) − p(y)∣∣ C
ln
( 2√1+|x|2√1+|y|2
|x−y|
) , x, y ∈Rn. (1.7)
Let
1
q(x)
= 1
p(x)
− α
n
.
The following statement was proved in [8].
Theorem 1.1. Under assumptions (1.4)–(1.6) and the condition
p+ <
n
α
(1.8)
the operator Iα is bounded from the space Lp(·)(Rn, ργ0,γ∞) into the space Lq(·)(Rn, ρμ0,μ∞),
where
μ0 = q(0)
p(0)
γ0 and μ∞ = q(∞)
p(∞)γ∞, (1.9)
if
αp(0) − n < γ0 < n
[
p(0) − 1], αp(∞) − n < γ∞ < n[p(∞) − 1], (1.10)
S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751 747and the exponents γ0 and γ∞ are related to each other by the equality
q(0)
p(0)
γ0 + q(∞)
p(∞)γ∞ =
q(∞)
p(∞)
[
(n + α)p(∞) − 2n]. (1.11)
The goal of this note is to prove that Theorem 1.1 is valid without the additional condi-
tion (1.11). We consider also a similar statement for the spherical potential operators
(
Kαf
)
(x) =
∫
Sn
f (σ )
|x − σ |n−α dσ, x ∈ Sn, 0 < α < n, (1.12)
in the corresponding weighted spaces Lp(·)(Sn, ρ) on the unit sphere Sn in Rn+1.
2. Preliminaries
We need the following theorem for bounded domains proved in [7].
Theorem 2.1. Let Ω be a bounded domain in Rn and x0 ∈ Ω and let p(x) satisfy conditions
(1.4), (1.5) and (1.8) in Ω . Then the following estimate∥∥Iαf ∥∥
Lq(·)(Ω,|x−x0|μ)  C‖f ‖Lp(·)(Ω,|x−x0|γ ) (2.1)
is valid, if
αp(x0) − n < γ < n
[
p(x0) − 1
] (2.2)
and
μ q(x0)
p(x0)
γ. (2.3)
3. The case of the spatial potential operator
We prove the following theorem.
Theorem A. Under assumptions (1.4)–(1.6) and (1.8), the operator Iα is bounded from the space
Lp(·)(Rn, ργ0,γ∞) into the space Lq(·)(Rn, ρμ0,μ∞), where
μ0 = q(0)
p(0)
γ0 and μ∞ = q(∞)
p(∞)γ∞, (3.1)
if
αp(0) − n < γ0 < n
[
p(0) − 1], αp(∞) − n < γ∞ < n[p(∞) − 1]. (3.2)
Proof. Let ‖f ‖Lp(·)(Rn,ρ)  1. To estimate the integral
∫
Rn
ρμ0,μ∞(x)|Iαf (x)|q(x) dx, we split
it, as in [8], in the following way:∫
Rn
ρμ0,μ∞(x)
∣∣Iαf (x)∣∣q(x) dx  c(A++ + A+− + A−+ + A−−),
where
748 S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751A++ =
∫
|x|<1
|x|μ0
∣∣∣∣
∫
|y|<1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx,
A+− =
∫
|x|<1
|x|μ0
∣∣∣∣
∫
|y|>1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx,
and
A−+ =
∫
|x|>1
|x|μ∞
∣∣∣∣
∫
|y|<1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx,
A−− =
∫
|x|>1
|x|μ∞
∣∣∣∣
∫
|y|>1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx.
The boundedness of the terms A++ and A−− was shown in [8] without condition (1.11). So we
only have to treat the terms A+− and A−+.
10. The term A−+. We split A−+ as
A−+ = A1 + A2,
where
A1 =
∫
1<|x|<2
|x|μ∞
∣∣∣∣
∫
|y|<1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx
and
A2 =
∫
|x|>2
|x|μ∞
∣∣∣∣
∫
|y|<1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx.
The term
A1 C
∫
1<|x|<2
|x|μ0
∣∣∣∣
∫
|y|<1
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx  C
∫
|x|<2
|x|μ0
∣∣∣∣
∫
|y|<2
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx
is covered by Theorem 2.1. For the term A2 we have
|x − y| |x| − |y| |x|
2
.
Therefore,
A2 C
∫
|x|>2
|x|μ∞+(α−n)q(x)
( ∫
|y|<1
∣∣f (y)∣∣dy
)q(x)
dx.
It follows from condition (1.6) (see also (1.7)) that
∣∣p(x) − p(∞)∣∣ C , |x| 2,ln |x|
S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751 749and then the same is valid for q(x), so that
A2  C
∫
|x|>2
|x|μ∞+(α−n)q(∞)
( ∫
|y|<1
∣∣f (y)∣∣dy
)q(x)
dx.
Observe that∫
|y|<1
∣∣f (y)∣∣dy  C‖f ‖Lp(·)(Rn,ρ). (3.3)
Indeed, denote g(y) = [ρ(y)]− 1p(y) ; by the Hölder inequality for variable Lp(·)-spaces we get∫
|y|<1
∣∣f (y)∣∣dy =
∫
|y|<1
g(y)
[
ρ(y)
] 1
p(y)
∣∣f (y)∣∣dy
 k‖g‖
Lp
′(·)
∥∥ρ 1p f ∥∥
Lp(·) = k‖g‖Lp′(·)‖f ‖Lp(·)(Rn,ρ). (3.4)
To arrive at (3.3), we have to show that ‖g‖
Lp
′(·) < ∞. Under condition (1.4) one has
‖g‖
Lp
′(·) < ∞ ⇐⇒
∫
|y|<1
∣∣g(y)∣∣p′(y) dy < ∞. (3.5)
As is easily seen, the last integral is finite since γ0 < n[p(0) − 1]. Therefore, from (3.4) there
follows (3.3).
Then A2  C < ∞ if we take into account that μ∞ + (α −n)q(∞) < −n under the condition
γ∞ < n[p(∞) − 1].
20. The term A+− is estimated similarly to A−+: we split A+− as
A+− = A3 + A4,
where
A3 =
∫
|x|<1
|x|μ0
∣∣∣∣
∫
1<|y|<2
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx
and
A4 =
∫
|x|<1
|x|μ0
∣∣∣∣
∫
|y|>2
f (y)dy
|x − y|n−α
∣∣∣∣
q(x)
dx.
The term A3 is covered by Theorem 2.1 similarly to the term A1 in 10. For the term A4, we have
|x − y| |y| − |x| |y|2 . Then∣∣∣∣
∫
|y|>2
f (y)dy
|x − y|n−α
∣∣∣∣ C
∫
|y|>2
|f (y)|dy
|y|n−α = C
∫
|y|>2
|f0(y)|dy
|y|n−α+ γ∞p(∞)
,
where f0(y) = |y|
γ∞
p(∞) f (y). It is easily seen that f0(y) ∈ Lp(·)(Rn\B(0,2)), since [ρ(y)]
1
p(y) ×
f (y) ∈ Lp(·)(Rn) and [ρ(y)] 1p(y) ∼ |y| γ∞p(∞) for |y| 2 under the log-condition at infinity. Hence
by the Hölder inequality and the same log-condition at infinity,
750 S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751
∣∣∣∣
∫
|y|>2
f (y)dy
|x − y|n−α
∣∣∣∣ C1‖f0‖Lp(·)(Rn\B(0,2))
∥∥|y|α−n− γ∞p(∞) ∥∥
Lp
′(·)(Rn\B(0,2))
 C‖f ‖Lp(·)(Rn,ργ0,γ∞ )
∥∥|y|α−n− γ∞p(∞) ∥∥
Lp
′(·)(Rn\B(0,2))
 C
∥∥|y|α−n− γ∞p(∞) ∥∥
Lp
′(·)(Rn\B(0,2)), (3.6)
where the last norm is finite under the condition αp(∞) − n < γ∞ (use the argument given in
(3.5)). 
Corollary 3.1. Let 0 < α < n, p(x) satisfy conditions (1.4)–(1.6) and (1.8). Then the operator
Iα is bounded from the space Lp(·)(Rn) into the space Lq(·)(Rn), 1
q(x)
= 1
p(x)
− α
n
.
The statement of the corollary was proved in [1,2] under a weaker than (1.6) version of the
log-condition at infinity.
4. The case of the spherical potential operator
4.1. The space Lp(·)(Sn, ρ)
We consider the weighted space Lp(·)(Sn, ρβa,βb ) with a variable exponent on the unit sphere
S
n = {σ ∈Rn+1: |σ | = 1}, defined by the norm
‖f ‖Lp(·)(Sn,ρβa ,βb ) =
{
λ > 0:
∫
Sn
|σ − a|βa · |σ − b|βb
∣∣∣∣f (σ )λ
∣∣∣∣
p(σ)
dσ  1
}
,
where ρβa,βb (σ ) = |σ − a|βa · |σ − b|βb and a ∈ Sn and b ∈ Sn are arbitrary points on Sn, a = b.
We assume that 0 < α < n and
1 < p−  p(σ) p+ <
n
α
, σ ∈ Sn, (4.1)
∣∣p(σ1) − p(σ2)∣∣ Aln 3|σ1−σ2|
, σ1 ∈ Sn, σ2 ∈ Sn. (4.2)
The following theorem is valid.
Theorem B. Let the function p :Sn → [1,∞) satisfy conditions (4.1) and (4.2). The spherical
potential operator Kα is bounded from the space Lp(·)(Sn, ρβa,βb ) with ρβa,βb (σ ) = |σ − a|βa ·|σ − b|βb , where a ∈ Sn and b ∈ Sn are arbitrary points on the unit sphere Sn, a = b, into the
space Lq(·)(Sn, ρβa,βb ) with ρνa,νb (σ ) = |σ − a|νa · |σ − b|νb , where 1q(σ ) = 1p(σ) − αn , and
αp(a) − n < βa < np(a) − n, αp(b) − n < βb < np(b) − n, (4.3)
νa = q(a)
p(a)
βa, νb = q(b)
p(b)
βb. (4.4)
This theorem was proved in [8] under the additional assumption that the weight exponents βa
and βb are related to each other by the connection
q(a)
βa = q(b) βb. (4.5)p(a) p(b)
S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751 751Now Theorem B without this condition follows from Theorem A by means of the stereo-
graphic projection exactly in the same way as in [8, Section 5].
Corollary 4.1. Under assumptions (4.1) and (4.2), the spherical potential operator Kα is
bounded from Lp(·)(Sn) into Lq(·)(Sn), 1
q(σ )
= 1
p(σ)
− α
n
.
References
[1] C. Capone, D. Cruz-Uribe, A. Fiorenza, The fractional maximal operator on variable Lp spaces, preprint of Istituto
per le Applicazioni del Calcolo “Mauro Picone” – Sezione di Napoli, (RT 281/04):1–22, 2004.
[2] D. Cruz-Uribe, A. Fiorenza, J.M. Martell, C. Perez, The boundedness of classical operators on variable Lp spaces,
preprint, Universidad Autonoma de Madrid, Departamento de Matematicas, www.uam.es/personal_pdi/ciencias/
martell/Investigacion/research.html, 2004, 26 pp.; Ann. Acad. Sci. Fenn., in press.
[3] X. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2) (2001)
749–760.
[4] X. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2) (2001) 424–446.
[5] O. Kovácˇik, J. Rákosnˇik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (116) (1991) 592–618.
[6] S.G. Samko, Convolution and potential type operators in Lp(x) , Integral Transforms Spec. Funct. 7 (3–4) (1998)
261–284.
[7] S.G. Samko, Hardy–Littlewood–Stein–Weiss inequality in the Lebesgue spaces with variable exponent, Fract. Calc.
Appl. Anal. 6 (4) (2003) 421–440.
[8] S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential
operators, J. Math. Anal. Appl. 310 (2005) 229–246.


Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II 

Universidade do Algarve

Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

Crossref

In [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl. 310 (2005) 229-246], Sobolev-type p((.)) -&gt; q((.))-theorems were proved for the Riesz potential operator I-alpha in the weighted Lebesgue generalized spaces L-p(.)(R-n, p) with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x(0) and to infinity, under an additional condition relating the weight exponents at x(0) and at infinity. We show in this note that those theorems are valid without this additional condition. Similar theorems for a spherical analogue of the Riesz potential operator in the corresponding weighted spaces L-p(.) (S-n, p) on the unit sphere S-n in Rn+1 are also improved in the same way. (c) 2006 Elsevier Inc. All rights reserve

Samko, S

Shargorodsky, E

Vakulov, B

King's Research Portal

Sapientia

J. Math. Anal. Appl. 325 (2007) 745–751www.elsevier.com/locate/jmaaNoteWeighted Sobolev theorem with variable exponentfor spatial and spherical potential operators, IIS. Samko a,∗, E. Shargorodsky b, B. Vakulov ca University of Algarve, Portugalb King’s College London, UKc Rostov State University, RussiaReceived 9 October 2005Available online 20 February 2006Submitted by L. GrafakosAbstractIn [S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spher-ical potential operators, J. Math. Anal. Appl. 310 (2005) 229–246], Sobolev-type p(·) → q(·)-theoremswere proved for the Riesz potential operator Iα in the weighted Lebesgue generalized spaces Lp(·)(Rn,ρ)with the variable exponent p(x) and a two-parameter power weight fixed to an arbitrary finite point x0 andto infinity, under an additional condition relating the weight exponents at x0 and at infinity. We show in thisnote that those theorems are valid without this additional condition. Similar theorems for a spherical ana-logue of the Riesz potential operator in the corresponding weighted spaces Lp(·)(Sn,ρ) on the unit sphereSn in Rn+1 are also improved in the same way.© 2006 Elsevier Inc. All rights reserved.Keywords: Weighted Lebesgue spaces; Variable exponent; Riesz potentials; Spherical potentials; Stereographicalprojection* Corresponding author.E-mail addresses: ssamko@ualg.pt (S. Samko), eugene.shargorodsky@kcl.ac.uk (E. Shargorodsky),vakulov@math.rsu.ru (B. Vakulov).0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2006.01.069746 S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–7511. IntroductionWe consider the Riesz potential operatorIαf (x) =∫Rnf (y)|x − y|n−α dy, 0 < α < n, (1.1)in the weighted Lebesgue generalized spaces Lp(·)(Rn, ρ) with a variable exponent p(x) definedby the norm‖f ‖Lp(·)(Rn,ρ) = inf{λ > 0:∫Rnρ(x)( |f (x)|λ)p(x)dx  1}, (1.2)whereρ(x) = ργ0,γ∞(x) = |x|γ0(1 + |x|)γ∞−γ0 . (1.3)We refer to [3–6] for the basics of the spaces Lp(·) with variable exponent.We assume that the exponent p(x) satisfies the standard conditions1 < p−  p(x)  p+ < ∞, x ∈ Rn, (1.4)∣∣p(x) − p(y)∣∣  Aln 1|x−y|, |x − y|  12, x, y ∈ Rn, (1.5)and also the following condition at infinity∣∣p∗(x) − p∗(y)∣∣  A∞ln 1|x−y|, |x − y|  12, x, y ∈ Rn, (1.6)where p∗(x) = p( x|x|2 ). Conditions (1.5) and (1.6) taken together are equivalent to the followingglobal condition:∣∣p(x) − p(y)∣∣  Cln( 2√1+|x|2√1+|y|2|x−y|) , x, y ∈ Rn. (1.7)Let1q(x)= 1p(x)− αn.The following statement was proved in [8].Theorem 1.1. Under assumptions (1.4)–(1.6) and the conditionp+ <nα(1.8)the operator Iα is bounded from the space Lp(·)(Rn, ργ0,γ∞) into the space Lq(·)(Rn, ρμ0,μ∞),whereμ0 = q(0)p(0)γ0 and μ∞ = q(∞)p(∞)γ∞, (1.9)ifαp(0) − n < γ0 < n[p(0) − 1], αp(∞) − n < γ∞ < n[p(∞) − 1], (1.10)S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751 747and the exponents γ0 and γ∞ are related to each other by the equalityq(0)p(0)γ0 + q(∞)p(∞)γ∞ =q(∞)p(∞)[(n + α)p(∞) − 2n]. (1.11)The goal of this note is to prove that Theorem 1.1 is valid without the additional condi-tion (1.11). We consider also a similar statement for the spherical potential operators(Kαf)(x) =∫Snf (σ )|x − σ |n−α dσ, x ∈ Sn, 0 < α < n, (1.12)in the corresponding weighted spaces Lp(·)(Sn, ρ) on the unit sphere Sn in Rn+1.2. PreliminariesWe need the following theorem for bounded domains proved in [7].Theorem 2.1. Let Ω be a bounded domain in Rn and x0 ∈ Ω and let p(x) satisfy conditions(1.4), (1.5) and (1.8) in Ω . Then the following estimate∥∥Iαf ∥∥Lq(·)(Ω,|x−x0|μ)  C‖f ‖Lp(·)(Ω,|x−x0|γ ) (2.1)is valid, ifαp(x0) − n < γ < n[p(x0) − 1](2.2)andμ  q(x0)p(x0)γ. (2.3)3. The case of the spatial potential operatorWe prove the following theorem.Theorem A. Under assumptions (1.4)–(1.6) and (1.8), the operator Iα is bounded from the spaceLp(·)(Rn, ργ0,γ∞) into the space Lq(·)(Rn, ρμ0,μ∞), whereμ0 = q(0)p(0)γ0 and μ∞ = q(∞)p(∞)γ∞, (3.1)ifαp(0) − n < γ0 < n[p(0) − 1], αp(∞) − n < γ∞ < n[p(∞) − 1]. (3.2)Proof. Let ‖f ‖Lp(·)(Rn,ρ)  1. To estimate the integral∫Rnρμ0,μ∞(x)|Iαf (x)|q(x) dx, we splitit, as in [8], in the following way:∫Rnρμ0,μ∞(x)∣∣Iαf (x)∣∣q(x) dx  c(A++ + A+− + A−+ + A−−),where748 S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751A++ =∫|x|<1|x|μ0∣∣∣∣∫|y|<1f (y)dy|x − y|n−α∣∣∣∣q(x)dx,A+− =∫|x|<1|x|μ0∣∣∣∣∫|y|>1f (y)dy|x − y|n−α∣∣∣∣q(x)dx,andA−+ =∫|x|>1|x|μ∞∣∣∣∣∫|y|<1f (y)dy|x − y|n−α∣∣∣∣q(x)dx,A−− =∫|x|>1|x|μ∞∣∣∣∣∫|y|>1f (y)dy|x − y|n−α∣∣∣∣q(x)dx.The boundedness of the terms A++ and A−− was shown in [8] without condition (1.11). So weonly have to treat the terms A+− and A−+.10. The term A−+. We split A−+ asA−+ = A1 + A2,whereA1 =∫1<|x|<2|x|μ∞∣∣∣∣∫|y|<1f (y)dy|x − y|n−α∣∣∣∣q(x)dxandA2 =∫|x|>2|x|μ∞∣∣∣∣∫|y|<1f (y)dy|x − y|n−α∣∣∣∣q(x)dx.The termA1  C∫1<|x|<2|x|μ0∣∣∣∣∫|y|<1f (y)dy|x − y|n−α∣∣∣∣q(x)dx  C∫|x|<2|x|μ0∣∣∣∣∫|y|<2f (y)dy|x − y|n−α∣∣∣∣q(x)dxis covered by Theorem 2.1. For the term A2 we have|x − y|  |x| − |y|  |x|2.Therefore,A2  C∫|x|>2|x|μ∞+(α−n)q(x)( ∫|y|<1∣∣f (y)∣∣dy)q(x)dx.It follows from condition (1.6) (see also (1.7)) that∣∣p(x) − p(∞)∣∣  C , |x|  2,ln |x|S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751 749and then the same is valid for q(x), so thatA2  C∫|x|>2|x|μ∞+(α−n)q(∞)( ∫|y|<1∣∣f (y)∣∣dy)q(x)dx.Observe that∫|y|<1∣∣f (y)∣∣dy  C‖f ‖Lp(·)(Rn,ρ). (3.3)Indeed, denote g(y) = [ρ(y)]− 1p(y) ; by the Hölder inequality for variable Lp(·)-spaces we get∫|y|<1∣∣f (y)∣∣dy =∫|y|<1g(y)[ρ(y)] 1p(y)∣∣f (y)∣∣dy k‖g‖Lp′(·)∥∥ρ 1p f ∥∥Lp(·) = k‖g‖Lp′(·)‖f ‖Lp(·)(Rn,ρ). (3.4)To arrive at (3.3), we have to show that ‖g‖Lp′(·) < ∞. Under condition (1.4) one has‖g‖Lp′(·) < ∞ ⇐⇒∫|y|<1∣∣g(y)∣∣p′(y) dy < ∞. (3.5)As is easily seen, the last integral is finite since γ0 < n[p(0) − 1]. Therefore, from (3.4) therefollows (3.3).Then A2  C < ∞ if we take into account that μ∞ + (α −n)q(∞) < −n under the conditionγ∞ < n[p(∞) − 1].20. The term A+− is estimated similarly to A−+: we split A+− asA+− = A3 + A4,whereA3 =∫|x|<1|x|μ0∣∣∣∣∫1<|y|<2f (y)dy|x − y|n−α∣∣∣∣q(x)dxandA4 =∫|x|<1|x|μ0∣∣∣∣∫|y|>2f (y)dy|x − y|n−α∣∣∣∣q(x)dx.The term A3 is covered by Theorem 2.1 similarly to the term A1 in 10. For the term A4, we have|x − y|  |y| − |x|  |y|2 . Then∣∣∣∣∫|y|>2f (y)dy|x − y|n−α∣∣∣∣  C∫|y|>2|f (y)|dy|y|n−α = C∫|y|>2|f0(y)|dy|y|n−α+ γ∞p(∞),where f0(y) = |y|γ∞p(∞) f (y). It is easily seen that f0(y) ∈ Lp(·)(Rn\B(0,2)), since [ρ(y)]1p(y) ×f (y) ∈ Lp(·)(Rn) and [ρ(y)] 1p(y) ∼ |y| γ∞p(∞) for |y|  2 under the log-condition at infinity. Henceby the Hölder inequality and the same log-condition at infinity,750 S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751∣∣∣∣∫|y|>2f (y)dy|x − y|n−α∣∣∣∣  C1‖f0‖Lp(·)(Rn\B(0,2))∥∥|y|α−n− γ∞p(∞) ∥∥Lp′(·)(Rn\B(0,2)) C‖f ‖Lp(·)(Rn,ργ0,γ∞ )∥∥|y|α−n− γ∞p(∞) ∥∥Lp′(·)(Rn\B(0,2)) C∥∥|y|α−n− γ∞p(∞) ∥∥Lp′(·)(Rn\B(0,2)), (3.6)where the last norm is finite under the condition αp(∞) − n < γ∞ (use the argument given in(3.5)). Corollary 3.1. Let 0 < α < n, p(x) satisfy conditions (1.4)–(1.6) and (1.8). Then the operatorIα is bounded from the space Lp(·)(Rn) into the space Lq(·)(Rn), 1q(x)= 1p(x)− αn.The statement of the corollary was proved in [1,2] under a weaker than (1.6) version of thelog-condition at infinity.4. The case of the spherical potential operator4.1. The space Lp(·)(Sn, ρ)We consider the weighted space Lp(·)(Sn, ρβa,βb ) with a variable exponent on the unit sphereSn = {σ ∈ Rn+1: |σ | = 1}, defined by the norm‖f ‖Lp(·)(Sn,ρβa ,βb ) ={λ > 0:∫Sn|σ − a|βa · |σ − b|βb∣∣∣∣f (σ )λ∣∣∣∣p(σ)dσ  1},where ρβa,βb (σ ) = |σ − a|βa · |σ − b|βb and a ∈ Sn and b ∈ Sn are arbitrary points on Sn, a = b.We assume that 0 < α < n and1 < p−  p(σ)  p+ <nα, σ ∈ Sn, (4.1)∣∣p(σ1) − p(σ2)∣∣  Aln 3|σ1−σ2|, σ1 ∈ Sn, σ2 ∈ Sn. (4.2)The following theorem is valid.Theorem B. Let the function p : Sn → [1,∞) satisfy conditions (4.1) and (4.2). The sphericalpotential operator Kα is bounded from the space Lp(·)(Sn, ρβa,βb ) with ρβa,βb (σ ) = |σ − a|βa ·|σ − b|βb , where a ∈ Sn and b ∈ Sn are arbitrary points on the unit sphere Sn, a = b, into thespace Lq(·)(Sn, ρβa,βb ) with ρνa,νb (σ ) = |σ − a|νa · |σ − b|νb , where 1q(σ ) = 1p(σ) − αn , andαp(a) − n < βa < np(a) − n, αp(b) − n < βb < np(b) − n, (4.3)νa = q(a)p(a)βa, νb = q(b)p(b)βb. (4.4)This theorem was proved in [8] under the additional assumption that the weight exponents βaand βb are related to each other by the connectionq(a)βa = q(b) βb. (4.5)p(a) p(b)S. Samko et al. / J. Math. Anal. Appl. 325 (2007) 745–751 751Now Theorem B without this condition follows from Theorem A by means of the stereo-graphic projection exactly in the same way as in [8, Section 5].Corollary 4.1. Under assumptions (4.1) and (4.2), the spherical potential operator Kα isbounded from Lp(·)(Sn) into Lq(·)(Sn), 1q(σ )= 1p(σ)− αn.References[1] C. Capone, D. Cruz-Uribe, A. Fiorenza, The fractional maximal operator on variable Lp spaces, preprint of Istitutoper le Applicazioni del Calcolo “Mauro Picone” – Sezione di Napoli, (RT 281/04):1–22, 2004.[2] D. Cruz-Uribe, A. Fiorenza, J.M. Martell, C. Perez, The boundedness of classical operators on variable Lp spaces,preprint, Universidad Autonoma de Madrid, Departamento de Matematicas, www.uam.es/personal_pdi/ciencias/martell/Investigacion/research.html, 2004, 26 pp.; Ann. Acad. Sci. Fenn., in press.[3] X. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces Wk,p(x)(Ω), J. Math. Anal. Appl. 262 (2) (2001)749–760.[4] X. Fan, D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2) (2001) 424–446.[5] O. Kovácǐk, J. Rákosnǐk, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41 (116) (1991) 592–618.[6] S.G. Samko, Convolution and potential type operators in Lp(x) , Integral Transforms Spec. Funct. 7 (3–4) (1998)261–284.[7] S.G. Samko, Hardy–Littlewood–Stein–Weiss inequality in the Lebesgue spaces with variable exponent, Fract. Calc.Appl. Anal. 6 (4) (2003) 421–440.[8] S.G. Samko, B.G. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potentialoperators, J. Math. Anal. 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Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, II

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