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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 329571, 27 pages
doi:10.1155/2010/329571
Research Article
Potential Operators in Variable Exponent Lebesgue
Spaces: Two-Weight Estimates
Vakhtang Kokilashvili,
1, 2
Alexander Meskhi,
1, 3
and Muhammad Sarwar
4
1
Department of Mathematical Analysis, A. Razmadze Mathematical Institute, 1. M. Aleksidze Street,
0193 Tbilisi, Georgia
2
Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, 2 University Street,
0143 Tbilisi, Georgia
3
Department of Mathematics, Faculty of Informatics and Control Systems, Georgian Technical University,
77 Kostava Street, 0175 Tbilisi, Georgia
4
Abdus Salam School of Mathematical Sciences, GC University, 68-B New Muslim Town,
Lahore 54600, Pakistan
Correspondence should be addressed to Alexander Meskhi,
Received 17 June 2010; Accepted 24 November 2010
Academic Editor: M. Vuorinen
Copyright q 2010 Vakhtang Kokilashvili et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Two-weighted norm estimates with general weights for Hardy-type transforms and potentials
in variable exponent Lebesgue spaces defined on quasimetric measure spaces X, d, μ are
established. In particular, we derive integral-type easily verifiable sufficient conditions governing
two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then
most of the derived conditions are simultaneously necessary and sufficient for corresponding
inequalities. Appropriate examples of weights are also given.
1. Introduction
We study the two-weight problem for Hardy-type and potential operators in Lebesgue spaces
with nonstandard growth defined on quasimetric measure spaces X, d, μ. In particular, our
aim is to derive easily verifiable sufficient conditions for the boundedness of the operators
T
α·
f
x
X
f
y
μ
B
x, d
x, y
1−αx
dμ
y
,
I
α·
f
x
X
f
y
d
x, y
1−αx
dμ
y
1.1
in weighted L
p·
X spaces which enable us to effectively construct examples of appropriate
weights. The conditions are simultaneously necessary and sufficient for corresponding
2 Journal of Inequalities and Applications
inequalities when the weights are of special type and the exponent p of the space is constant.
We assume that the exponent p satisfies the local log-H
¨
older continuity condition, and if
the diameter of X is infinite, then we suppose that p is constant outside some ball. In the
framework of variable exponent analysis such a condition first appeared in the paper 1,
where the author established the boundedness of the Hardy-Littlewood maximal operator in
L
p·
R
n
. As f ar as we know, unfortunately, an analog of the log-H
¨
older decay condition at
infinity for p : X → 1, ∞ is not known even in the unweighted case, which is well-known
and natural for the Euclidean spaces see 2–5. Local log-H
¨
older continuity condition for
the exponent p, together with the log-H
¨
older decay condition, guarantees the boundedness
of operators of harmonic analysis in L
p·
R
n
spaces see, e.g., 6. The technique developed
here enables us to expect that results similar to those of this paper can be obtained also
for other integral operators, for instance, for maximal and Calder
´
on-Zygmund singular
operators defined on X.
Considerable interest of researchers is focused on the study of mapping properties
of integral operators defined on quasimetric measure spaces. Such spaces with doubling
measure and all their generalities naturally arise when studying boundary value problems
for partial differential equations with variable coefficients, for instance, when the quasimetric
might be induced by a differential operator or tailored to fit kernels of integral operators.
The problem of t he boundedness of integral operators naturally arises also in the Lebesgue
spaces with nonstandard growth. Historically the boundedness of the maximal and f ractional
integral operators in L
p·
X spaces was derived in the papers 7–14. Weighted inequalities
for classical operators in L
p·
w
spaces, where w is a power-type weight, were established in the
papers 10–12, 15–19, while the same problems with general weights for Hardy, maximal,
and fractional integral operators were studied in 10, 20–25. Moreover, in the latter paper,
a complete solution of the one-weight problem for maximal functions defined on Euclidean
spaces is given in terms of Muckenhoupt-type conditions.
It should be emphasized that in the classical Lebesgue spaces the two-weight problem
for fractional integrals is already solved see 26, 27, but it is often useful to construct
concrete examples of weights from transparent and easily verifiable conditions.
To derive two-weight estimates for potential operators, we use the appropriate
inequalities for Hardy-type transforms on X which are also derived i n this paper and
Hardy-Littlewood-Sobolev-type inequalities for T
α·
and I
α·
in L
p·
X spaces.
The paper is organized as follows: in Section 1, we give some definitions and prove
auxiliary results regarding quasimetric measure spaces and the variable exponent Lebesgue
spaces; Section 2 is devoted to the sufficient governing two-weight inequalities for Hardy-
type operators defined on quasimetric measure spaces, while in Section 3 we study the two-
weight problem for potentials defined on X.
Finally we point out that constants often different constants in the same series of
inequalities will generally be denoted by c or C. The symbol fx ≈ gx means that
there are positive constants c
1
and c
2
independent of x such that the inequality fx ≤
c
1
gx ≤ c
2
fx holds. Throughout the paper is denoted the function px/px − 1 by the
symbol p
x.
2. Preliminaries
Let X :X, d, μ be a topological space with a complete measure μ such that the space of
compactly supported continuous functions is dense in L
1
X, μ and there exists a nonnegative
Journal of Inequalities and Applications 3
real-valued function quasimetric d on X × X satisfying the conditions:
i dx, y0 if and only if x y;
ii there exists a constant a
1
> 0, such that dx, y ≤ a
1
dx, zdz, y for all x, y, z ∈
X;
iii there exists a constant a
0
> 0, such that dx, y ≤ a
0
dy, x for all x, y, ∈ X.
We assume that the balls Bx, r : {y ∈ X : dx, y <r} are measurable and 0 ≤
μBx, r < ∞ for all x ∈ X and r>0; for every neighborhood V of x ∈ X, there exists r>0,
such that Bx, r ⊂ V . Throughout the paper we also suppose that μ{x} 0andthat
B
x, R
\ B
x, r
/
∅, 2.1
for all x ∈ X, positive r and R with 0 <r<R<L, where
L : diam
X
sup
d
x, y
: x, y ∈ X
. 2.2
We call the triple X, d, μ a quasimetric measure space. If μ satisfies the doubling
condition μBx, 2r ≤ cμBx, r, where t he positive constant c does not depend on x ∈ X
and r>0, then X, d, μ is called a space of homogeneous type SHT. For the definition,
examples, and some properties of an SHT see, for example, monographs 28 –30.
A quasimetric measure space, where the doubling condition is not assumed, is called
a nonhomogeneous space.
Notice that the condition L<∞ implies that μX < ∞ because we assumed that every
ball in X has a finite measure.
We say that the measure μ is upper Ahlfors Q-regular if there is a positive constant c
1
such that μBx, r ≤ c
1
r
Q
for for all x ∈ X and r>0. Further, μ is lower Ahlfors Q-regular
if there is a positive constant c
2
such that μBx, r ≥ c
2
r
q
for all x ∈ X and r>0. It is easy
to check that if X, d, μ is a quasimetric measure space and L<∞, then μ is lower Ahlfors
regular see also, e.g., 8 for the case when d is a metric.
For the boundedness of potential operators in weighted Lebesgue spaces with constant
exponents on nonhomogeneous spaces we refer, for example, to the monograph 31, Chapter
6 and references cited therein.
Let p be a nonnegative μ-measurable function on X. Suppose that E is a μ-measurable
set in X. We use the following notation:
p
−
E
: inf
E
p; p
E
: sup
E
p; p
−
: p
−
X
; p
: p
X
;
B
x, r
:
y ∈ X : d
x, y
≤ r
,kB
x, r
: B
x, kr
; B
xy
: B
x, d
x, y
;
B
xy
: B
x, d
x, y
; g
B
:
1
μ
B
B
g
x
dμ
x
.
2.3
4 Journal of Inequalities and Applications
Assume that 1 ≤ p
−
≤ p
< ∞. The variable exponent Lebesgue space L
p·
X
sometimes it is denoted by L
px
X is the class of all μ-measurable functions f on X for
which S
p
f :
X
|fx|
px
dμx < ∞. The norm in L
p·
X is defined as follows:
f
L
p·
X
inf
λ>0:S
p
f
λ
≤ 1
. 2.4
It is known see, e.g., 8, 15, 32, 33 that L
p·
is a Banach space. For other properties of
L
p·
spaces we refer, for example, to 32–34.
We need some definitions for the exponent p which will be useful to derive the main
results of the paper.
Definition 2.1. Let X, d, μ be a quasimetric measure space and let N ≥ 1 be a constant.
Suppose that p satisfies the condition 0 <p
−
≤ p
< ∞. We say that p belongs to the class
PN, x, where x ∈ X, if there are positive constants b and c which might be depended on
x such that
μ
B
x, Nr
p
−
Bx,r−p
Bx,r
≤ c 2.5
holds for all r,0<r≤ b. Further, p ∈PN if there are positive constants b and c such that
2.5 holds for all x ∈ X and all r satisfying the condition 0 <r≤ b.
Definition 2.2. Let X, d, μ be an SHT. Suppose that 0 <p
−
≤ p
< ∞. We say that p ∈ LHX, x
p satisfies the log-H
¨
older-type condition at a point x ∈ X if there are positive constants b
and c which might be depended on x such that
p
x
− p
y
≤
c
− ln
μ
B
xy
2.6
holds for all y satisfying the condition dx, y ≤ b. Further, p ∈ LHXp satisfies the log-
H
¨
older type condition on X if there are positive constants b and c such that 2.6 holds for
all x, y with dx, y ≤ b.
We will also need another form of the log-H
¨
older continuity condition given by the
following definition.
Definition 2.3. Let X, d, μ be a quasimetric measure space, and let 0 <p
−
≤ p
< ∞.Wesay
that p ∈
LHX, x if there are positive constants b and c which might be depended on x
such that
p
x
− p
y
≤
c
− ln d
x, y
2.7
for all y with dx, y ≤ b. Further, p ∈
LHX if 2.7 holds f or all x, y with dx, y ≤ b.
Journal of Inequalities and Applications 5
It is easy to see that if a measure μ is upper Ahlfors Q-regular and p ∈ LHXresp.,
p ∈ LHX, x, then p ∈
LHXresp., p ∈ LHX, x. Further, if μ is lower Ahlfors Q-regular
and p ∈
LHXresp., p ∈ LHX, x, then p ∈ LHXresp., p ∈ LHX, x.
Remark 2.4. It can be checked easily that if X, d, μ is an SHT, then μB
x
0
x
≈ μB
xx
0
.
Remark 2.5. Let X, d, μ be an SHT with L<∞. It is known see, e.g., 8, 35 that if p ∈
LHX, then p ∈P1. Further, if μ is upper Ahlfors Q-regular, then the condition p ∈P1
implies that p ∈
LHX.
Proposition 2.6. Let c be positive and let 1 <p
−
X ≤ p
X < ∞ and p ∈ LHX (resp., p ∈
LHX, then the functions cp·, 1/p·, and p
· belong to LHXresp., LHX. Further if
p ∈ LHX, xresp., p ∈
LHX, x then cp·, 1/p·, and p
· belong to LHX, xresp., p ∈
LHX, x.
The proof of the latter statement can be checked immediately using the definitions of
the classes LHX, x,LHX,
LHX, x,andLHX.
Proposition 2.7. Let X, d, μ be an SHT and let p ∈P1.ThenμB
xy
px
≤ cμB
yx
py
for all
x, y ∈ X with μBx, dx, y ≤ b,whereb is a small constant, and the constant c does not depend
on x, y ∈ X.
Proof. Due to the doubling condition for μ, Remark 1.1, the condition p ∈P1 and
the fact that x ∈ By, a
1
a
0
1dy, x we have the following estimates: μB
xy
px
≤
μBy, a
1
a
0
1dx, y
px
≤ cμBy, a
1
a
0
1dx, y
py
≤ cμB
yx
py
, which proves the
statement.
The proof of the next statement is trivial and follows directly from the definition of the
classes PN, x and PN. Details are omitted.
Proposition 2.8. Let X, d, μ be a quasimetric measure space and let x
0
∈ X. Suppose that N ≥ 1
be a constant. Then the following statements hold:
i if p ∈PN, x
0
(resp., p ∈PN, then there are positive constants r
0
, c
1
, and c
2
such
that for all 0 <r≤ r
0
and all y ∈ Bx
0
,r (resp., for all x
0
,ywith dx
0
,y <r≤ r
0
), one
has that μBx
0
,Nr
px
0
≤ c
1
μBx
0
,Nr
py
≤ c
2
μBx
0
,Nr
px
0
.
ii Let p ∈PN, x
0
, then there are positive constants r
0
, c
1
, and c
2
(in general, depending
on x
0
) such that for all r (r ≤ r
0
) and all x, y ∈ Bx
0
,r one has μBx
0
,Nr
px
≤
c
1
μBx
0
,Nr
py
≤ c
2
μBx
0
,Nr
px
.
iii Let p ∈PN, then there are positive constants r
0
, c
1
, and c
2
such that for all balls B with
radius r (r ≤ r
0
) and all x, y ∈ B, one has that μNB
px
≤ c
1
μNB
py
≤ c
2
μNB
px
.
It is known that see, e.g., 32, 33 if f is a measurable function on X and E is a
measurable subset of X, then the following inequalities hold:
f
p
E
L
p·
E
≤ S
p
fχ
E
≤
f
p
−
E
L
p·
E
,
f
L
p·
E
≤ 1;
f
p
−
E
L
p·
E
≤ S
p
fχ
E
≤
f
p
E
L
p·
E
,
f
L
p·
E
> 1.
2.8
6 Journal of Inequalities and Applications
Further, H
¨
older’s inequality in the variable exponent Lebesgue spaces has the
following form:
E
fgdμ ≤
1
p
−
E
1
p
−
E
f
L
p·
E
g
L
p
·
E
. 2.9
Lemma 2.9. Let X, d, μ be an SHT.
i If β is a measurable function on X such that β
< −1 and if r is a small positive number,
then there exists a positive constant c independent of r and x such that
X\B
x
0
,r
μB
x
0
y
βx
dμ
y
≤ c
β
x
1
β
x
μ
B
x
0
,r
βx1
. 2.10
ii Suppose that p and α are measurable functions on X satisfying the conditions 1 <p
−
≤
p
< ∞ and α
−
> 1/p
−
. Then there exists a positive constant c such that for all x ∈ X the
inequality
B
x
0
,2d
x
0
,x
μB
x, d
x, y
αx−1p
x
dμ
y
≤ c
μB
x
0
,d
x
0
,x
αx−1p
x1
2.11
holds.
Proof. Part i was proved in 35see also 31, page 372, for constant β. The proof of Part ii
is given in 31, Lemma 6.5.2, page 348 for constant α and p, but repeating those arguments
we can see that it is also true for variable α and p. Details are omitted.
Lemma 2.10. Let X, d, μ be an SHT. Suppose that 0 <p
−
≤ p
< ∞,thenp satisfies the condition
p ∈P1 (resp., p ∈P1,x) if and only if p ∈ LHXresp., p ∈ LHX, x.
Proof. We follow 1.
Necessity. Let p ∈P1,andletx, y ∈ X with dx, y <c
0
for some positive constant c
0
.
Observe that x, y ∈ B, where B : Bx, 2dx, y. By the doubling condition for μ, we have
that μB
xy
−|px−py|
≤ cμB
−|px−py|
≤ cμB
p
−
B−p
B
≤ C, where C is a positive constant
which is greater than 1. Taking now the logarithm in the last inequality, we have that p ∈
LHX.Ifp ∈P1,x, then by the same arguments we find that p ∈ LHX, x.
Sufficiency. Let B : Bx
0
,r. First observe that If x, y ∈ B, then μB
xy
≤ cμBx
0
,r.
Consequently, this inequality and the condition p ∈ LHX yield |p
−
B − p
B|≤
C/ − lnc
0
μBx
0
,r. Further, there exists r
0
such that 0 <r
0
< 1/2andc
1
≤
lnμB/− lnc
0
μB ≤ c
2
, 0 <r≤ r
0
, where c
1
and c
2
are positive constants. Hence
μB
p
−
B−p
B
≤ μB
C/ lnc
0
μB
expC lnμB/ lnc
0
μB ≤ C.
Journal of Inequalities and Applications 7
Let, now, p ∈ LHX, x and let B
x
: Bx, r where r is a small number. We have that
p
B
x
−px ≤ c/−lnc
0
μBx, r and px−p
−
B
x
≤ c/−lnc
0
μBx, r for some positive
constant c
0
. Consequently,
μ
B
x
p
−
B
x
−p
B
x
μ
B
x
px−p
B
x
μ
B
x
p
−
B
x
−px
≤ c
μ
B
x
−2c/−lnc
0
μB
x
≤ C.
2.12
Definition 2.11. Ameasureμ on X is said to satisfy the reverse doubling condition μ ∈
RDCX if there exist constants A>1andB>1 such that the inequality μBa, Ar ≥
BμBa, r holds.
Remark 2.12. It is known that if all annulus in X are not empty i.e., condition 2.1 holds,
then μ ∈ DCX implies that μ ∈ RDCXsee, e.g., 28, page 11, Lemma 20.
Lemma 2.13. Let X, d, μ be an SHT. Suppose that there is a point x
0
∈ X such that p ∈
LHX, x
0
.LetA be the constant defined in Definition 2.11. Then there exist positive constants r
0
and C (which might be depended on x
0
) such that for all r, 0 <r≤ r
0
, the inequality
μB
A
p
−
B
A
−p
B
A
≤ C 2.13
holds, where B
A
: Bx
0
,Ar \ Bx
0
,r and the constant C is independent of r.
Proof. Taking into account condition 2.1 and Remark 2.12, we have that μ ∈ RDCX.
Let B : Bx
0
,r. By the doubling and reverse doubling conditions, we have that μB
A
μBx
0
,Ar − μBx
0
,r ≥ B − 1μBx
0
,r ≥ cμAB. Suppose that 0 <r<c
0
, where c
0
is a sufficiently small constant. Then by using Lemma 2.10 we find that μB
A
p
−
B
A
−p
B
A
≤
cμAB
p
−
B
A
−p
B
A
≤ cμAB
p
−
AB−p
AB
≤ c.
In the sequel we will use the notation:
I
1,k
:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
B
x
0
,
A
k−1
L
a
1
if L<∞,
B
x
0
,
A
k−1
a
1
if L ∞,
I
2,k
:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
B
x
0
, A
k2
a
1
L
\ B
x
0
,
A
k−1
L
a
1
if L<∞,
B
x
0
,A
k2
a
1
\ B
x
0
,
A
k−1
a
1
if L ∞,
I
3,k
:
⎧
⎨
⎩
X \ B
x
0
,A
k2
La
1
if L<∞,
X \ B
x
0
,A
k2
a
1
if L ∞,
8 Journal of Inequalities and Applications
E
k
:
⎧
⎨
⎩
B
x
0
,A
k1
L
\ B
x
0
,A
k
L
if L<∞,
B
x
0
,A
k1
\ B
x
0
,A
k
if L ∞,
2.14
where the constants A and a
1
are taken, respectively, from Definition 2.11 and the triangle
inequality for the quasimetric d,andL is a diameter of X.
Lemma 2.14. Let X, d, μ be an SHT and let 1 <p
−
x ≤ px ≤ qx ≤ q
X < ∞. Suppose
that there is a point x
0
∈ X such that p, q ∈ LHX, x
0
. Assume that if L ∞,thenpx ≡ p
c
≡ const
and qx ≡ q
c
≡ const outside some ball Bx
0
, a. Then there exists a positive constant C such that
k
fχ
I
2,k
L
p·
X
gχ
I
2,k
L
q
·
X
≤ C
f
L
p·
X
g
L
q
·
X
, 2.15
for all f ∈ L
p·
X and g ∈ L
q
·
X.
Proof. Suppose that L ∞. To prove the lemma, first observe that μE
k
≈ μBx
0
,A
k
and
μI
2,k
≈ μBx
0
,A
k−1
. This holds because μ satisfies the reverse doubling condition and,
consequently,
μE
k
μ
B
x
0
,A
k1
\ B
x
0
,A
k
μ
B
x
0
,A
k1
− μB
x
0
,A
k
μ
B
x
0
,AA
k
− μB
x
0
,A
k
≥ BμB
x
0
,A
k
− μB
x
0
,A
k
B − 1
μB
x
0
,A
k
.
2.16
Moreover, the doubling condition yields μE
k
≤ μBx
0
,AA
k
≤ cμBx
0
,A
k
, where c>1.
Hence, μE
k
≈ μBx
0
,A
k
.
Further, since we can assume that a
1
≥ 1, we find that
μI
2,k
μ
B
x
0
,A
k2
a
1
\ B
x
0
,
A
k−1
a
1
μ
B
x
0
,A
k2
a
1
− μB
x
0
,
A
k−1
a
1
μ
B
x
0
,AA
k1
a
1
− μB
x
0
,
A
k−1
a
1
≥ BμB
x
0
,A
k1
a
1
− μB
x
0
,
A
k−1
a
1
≥ B
2
μB
x
0
,
A
k
a
1
− μB
x
0
,
A
k−1
a
1
≥ B
3
μB
x
0
,
A
k−1
a
1
− μB
x
0
,
A
k−1
a
1
B
3
− 1
μB
x
0
,
A
k−1
a
1
.
2.17
Moreover, using the doubling condition for μ we have that μI
2,k
≤ μBx
0
,A
k2
r ≤
cμBx
0
,A
k1
r ≤ c
2
μBx
0
,A
k
/a
1
≤ c
3
μBx
0
,A
k−1
/a
1
. This gives the estimates B
3
−
1μBx
0
,A
k−1
/a
1
≤ μI
2,k
≤ c
3
μBx
0
,A
k−1
/a
1
.
Journal of Inequalities and Applications 9
For simplicity, assume that a 1. Suppose that m
0
is an integer such that A
m
0
−1
/a
1
> 1.
Let us split the sum as follows:
i
fχ
I
2,i
L
p·
X
·
gχ
I
2,i
L
q
·
X
i≤m
0
···
i>m
0
···
: J
1
J
2
. 2.18
Since px ≡ p
c
const,qxq
c
const outside the ball Bx
0
, 1,byusingH
¨
older’s
inequality and the fact that p
c
≤ q
c
, we have
J
2
i>m
0
fχ
I
2,i
L
p
c
X
·
gχ
I
2,i
L
q
c
X
≤ c
f
L
p·
X
·
g
L
q
·
X
. 2.19
Let us estimate J
1
. Suppose that f
L
p·
X
≤ 1andg
L
q
·
X
≤ 1. Also, by
Proposition 2.6, we have that 1/q
∈ LHX, x
0
. Therefore, by Lemma 2.13 and the fact
that 1/q
∈ LHX, x
0
,weobtainthatμI
2,k
1/q
I
2,k
≈χ
I
2,k
L
q·
X
≈ μI
2,k
1/q
−
I
2,k
and
μI
2,k
1/q
I
2,k
≈χ
I
2,k
L
q
·
X
≈ μI
2,k
1/q
−
I
k
, where k ≤ m
0
. Further, observe that these
estimates and H
¨
older’s inequality yield the following chain of inequalities:
J
1
≤ c
k≤m
0
Bx
0
,A
m
0
1
fχ
I
2,k
L
p·
X
·
gχ
I
2,k
L
q
·
X
χ
I
2,k
L
q·
X
·
χ
I
2,k
L
q
·
X
χ
E
k
x
dμ
x
c
B
x
0
,A
m
0
1
k≤m
0
fχ
I
2,k
L
p·
X
·
gχ
I
2,k
L
q
·
X
χ
I
2,k
L
q·
X
·
χ
I
2,k
L
q
·
X
χ
E
k
x
dμ
x
≤ c
k≤m
0
fχ
I
2,k
L
p·
X
χ
I
2,k
L
q·
X
χ
E
k
x
L
q·
Bx
0
,A
m
0
1
×
k≤m
0
gχ
I
2,k
L
q
·
X
χ
I
2,k
L
q
·
X
χ
E
k
x
L
q
·
Bx
0
,A
m
0
1
: cS
1
f
· S
2
g
.
2.20
Now we claim that S
1
f ≤ cIf, where
I
f
:
k≤m
0
fχ
I
2,k
L
p·
X
χ
I
2k
L
p·
X
χ
E
k
·
L
p·
Bx
0
,A
m
0
1
,
2.21
and the positive constant c does not depend on f. Indeed, suppose that If ≤ 1. Then taking
into account Lemma 2.13 we have that
k≤m
0
1
μ
I
2,k
E
k
fχ
I
2,k
px
L
p·
X
dμ
x
≤ c
Bx
0
,A
m
0
1
k≤m
0
fχ
I
2,k
L
p·
X
χ
I
2,k
L
p·
X
χ
E
k
x
px
dμ
x
≤ c.
2.22
10 Journal of Inequalities and Applications
Consequently, since px ≤ qx,E
k
⊆ I
2,k
and f
L
p·
X
≤ 1, we find that
k≤m
0
1
μ
I
2,k
E
k
fχ
I
2,k
qx
L
p·
X
dμ
x
≤
k≤m
0
1
μ
I
2,k
E
k
fχ
I
2,k
px
L
p·
X
dμ
x
≤ c. 2.23
This implies that S
1
f ≤ c. Thus, the desired inequality is proved. Further, let us introduce
the following function:
P
y
:
k≤2
p
I
2,k
χ
E
k
y
. 2.24
It is clear that py ≤ Py because E
k
⊂ I
2,k
. Hence
I
f
≤ c
k≤m
0
fχ
I
2,k
L
p·
X
χ
I
2k
L
p·
X
χ
E
k
·
L
P·
Bx
0
,A
m
0
1
2.25
for some positive constant c. Then, by using this inequality, the definition of the function P,
the condition p ∈ LHX, and the obvious estimate χ
I
2,k
p
I
2,k
L
p·
X
≥ cμI
2,k
,wefindthat
B
x
0
,A
m
0
1
k≤m
0
fχ
I
2,k
L
p·
X
χ
I
2,k
L
p·
X
χ
E
k
x
Px
dμ
x
B
x
0
,A
m
0
1
⎛
⎝
k≤m
0
fχ
I
2,k
p
I
2,k
L
p·
X
χ
I
2,k
p
I
2,k
L
p·
X
χ
E
k
x
⎞
⎠
dμ
x
≤ c
B
x
0
,A
m
0
1
⎛
⎝
k≤m
0
fχ
I
2,k
p
I
2,k
L
p·
X
μ
I
2,k
χ
E
k
x
⎞
⎠
dμ
x
≤ c
k≤m
0
fχ
I
2,k
p
I
2,k
L
p·
X
≤ c
k≤m
0
I
2,k
f
x
px
dμ
x
≤ c
X
f
x
px
dμ
x
≤ c.
2.26
Consequently, If ≤ cf
L
p·
X
. Hence, S
1
f ≤ cf
L
p·
X
. Analogously taking into
account the fact that q
∈ DLX and arguing as above, we find that S
2
g ≤ cg
L
q
·
X
.Thus,
summarizing these estimates we conclude that
i≤m
0
fχ
I
i
L
p·
X
gχ
I
i
L
q
·
X
≤ c
f
L
p·
X
g
L
q
·
X
. 2.27
Lemma 2.14 for L
p·
0, 1 spaces defined with respect to the Lebesgue measure was
derived in 24see also 22 for X R
n
, dx, y|x − y|,anddμxdx.
Journal of Inequalities and Applications 11
3. Hardy-Type Transforms
In this section, we derive two-weight estimates for the operators:
T
v,w
f
x
v
x
B
x
0
x
f
y
w
y
dμ
y
,T
v,w
f
x
v
x
X\B
x
0
x
f
y
w
y
dμ
y
.
3.1
Let a be a positive constant, and let p be a measurable function defined on X.Letus
introduce the notation:
p
0
x
: p
−
B
x
0
x
; p
0
x
:
⎧
⎨
⎩
p
0
x
if d
x
0
,x
≤ a;
p
c
const if d
x
0
,x
>a.
p
1
x
: p
−
B
x
0
,a
\ B
x
0
x
; p
1
x
:
⎧
⎨
⎩
p
1
x
if d
x
0
,x
≤ a;
p
c
const if d
x
0
,x
>a.
3.2
Remark 3.1. If we deal with a quasimetric measure space with L<∞, then we will assume
that a L. Obviously, p
0
≡ p
0
and p
1
≡ p
1
in this case.
Theorem 3.2. Let X, d, μ be a quasimetric measure space. Assume that p and q are measurable
functions on X satisfying the condition 1 <p
−
≤ p
0
x ≤ qx ≤ q
< ∞. In the case when L ∞,
suppose that p ≡ p
c
≡ const, q ≡ q
c
≡ const, outside some ball Bx
0
,a. If the condition
A
1
: sup
0≤t≤L
t<d
x
0
,x
≤L
v
x
qx
d
x
0
,x
≤t
w
p
0
x
y
dμ
y
qx/ p
0
x
dμ
x
< ∞, 3.3
holds, then T
v,w
is bounded from L
p·
X to L
q·
X.
Proof. Here we use the arguments of the proofs of Theorem 1.1.4 in 31, see page 7 and of
Theorem 2.1 in 21.First,wenoticethatp
−
≤ p
0
x ≤ px for all x ∈ X.Letf ≥ 0andlet
S
p
f ≤ 1. First, assume that L<∞. We denote
I
s
:
d
x
0
,y
<s
f
y
w
y
dμ
y
for s ∈
0,L
. 3.4
Suppose that IL < ∞, then IL ∈ 2
m
, 2
m1
for some m ∈ Z. Let us denote s
j
: sup{s :
Is ≤ 2
j
},j≤ m,ands
m1
: L. Then {s
j
}
m1
j−∞
is a nondecreasing sequence. It is easy to check
12 Journal of Inequalities and Applications
that Is
j
≤ 2
j
,Is > 2
j
for s>s
j
,and2
j
≤
s
j
≤dx
0
,y≤s
j1
fywydμy.Ifβ : lim
j →−∞
s
j
,
then dx
0
,x <Lif and only if dx
0
,x ∈ 0,β ∪
m
j−∞
s
j
,s
j1
.IfIL∞, then we take
m ∞. Since 0 ≤ Iβ ≤ Is
j
≤ 2
j
for every j, we have that Iβ0. It is obvious that
X
j≤m
{x : s
j
<dx
0
,x ≤ s
j1
}. Further, we have that
S
q
T
v,w
f
X
T
v,w
f
x
qx
dμ
x
X
v
x
B
x
0
,d
x
0
,x
f
y
w
y
dμ
y
qx
dμ
x
X
v
x
qx
B
x
0
,d
x
0
,x
f
y
w
y
dμ
y
qx
dμ
x
≤
m
j−∞
s
j
<dx
0
,x≤s
j1
v
x
qx
d
x
0
,y
<s
j1
f
y
w
y
dμ
y
qx
dμ
x
.
3.5
Let us denote
B
j
x
0
:
x ∈ X : s
j−1
≤ d
x
0
,x
≤ s
j
. 3.6
Notice that Is
j1
≤ 2
j1
≤ 4
B
j
x
0
wyfydμy. Consequently, by this estimate and
H
¨
older’s inequality with respect to the exponent p
0
x we find that
S
q
T
v,w
f
≤ c
m
j−∞
s
j
<d
x
0
,x
≤s
j1
v
x
qx
B
j
x
0
f
y
w
y
dμ
y
qx
dμ
x
≤ c
m
j−∞
s
j
<dx
0
,x≤s
j1
v
x
qx
J
k
x
dμ
x
,
3.7
where
J
k
x
:
B
j
x
0
f
y
p
0
x
dμ
y
qx/p
0
x
B
j
x
0
w
y
p
0
x
dμ
y
qx/p
0
x
. 3.8
Journal of Inequalities and Applications 13
Observe now that qx ≥ p
0
x. Hence, this fact and the condition S
p
f ≤ 1 imply that
J
k
x
≤ c
B
j
x
0
∩{y:f
y
≤1}
f
y
p
0
x
dμ
y
B
j
x
0
∩
{
y:f
y
>1
}
f
y
py
dμ
y
qx/p
0
x
×
B
j
x
0
w
y
p
0
x
dμ
y
qx/p
0
x
≤ c
μ
B
j
x
0
B
j
x
0
∩{y:f
y
>1}
f
y
py
dμ
y
×
B
j
x
0
wy
p
0
x
dμ
y
qx/p
0
x
.
3.9
It follows now that
S
q
T
v,w
f
≤ c
⎛
⎝
m
j−∞
μ
B
j
x
0
s
j
<dx
0
,x≤s
j1
v
x
qx
×
B
j
x
0
w
y
p
0
x
dμ
y
qx/p
0
x
dμ
x
m
j−∞
B
j
x
0
∩{y:f
y
>1}
f
y
py
dμ
y
s
j
<d
x
0
,x
≤s
j1
v
x
qx
×
B
j
x
0
w
y
p
0
x
dμ
y
qx/p
0
x
dμ
x
⎞
⎠
: c
N
1
N
2
.
3.10
Since L<∞, it is obvious that
N
1
≤ A
1
m1
j−∞
μ
B
j
x
0
≤ CA
1
,
N
2
≤ A
1
m1
j−∞
B
j
x
0
f
y
py
dμ
y
≤ C
X
f
y
py
dμ
y
A
1
S
p
f
≤ A
1
.
3.11
Finally, S
q
T
v,w
f ≤ cCA
1
A
1
< ∞.Thus,T
v,w
is bounded if A
1
< ∞.
14 Journal of Inequalities and Applications
Let us now suppose that L ∞. We have
T
v,w
f
x
χ
Bx
0
,a
x
v
x
B
x
0
x
f
y
w
y
dμ
y
χ
X\Bx
0
,a
x
v
x
B
x
0
x
f
y
w
y
dμ
y
: T
1
v,w
f
x
T
2
v,w
f
x
.
3.12
By using the already proved result for L<∞ and the fact that diamBx
0
,a < ∞,we
find that T
1
v,w
f
L
q·
Bx
0
,a
≤ cf
L
p·
Bx
0
,a
≤ c because
A
a
1
: sup
0≤t≤a
t<d
x
0
,x
≤a
v
x
qx
d
x
0
,x
≤t
w
p
0
x
y
dμ
y
qx/p
0
x
dμ
x
≤ A
1
< ∞.
3.13
Further, observe that
T
2
v,w
f
x
χ
X\Bx
0
,a
x
v
x
B
x
0
x
f
y
w
y
dμ
y
χ
X\Bx
0
,a
x
v
x
×
d
x
0
,y
≤a
f
y
w
y
dμ
y
χ
X\Bx
0
,a
x
v
x
a≤d
x
0
,y
≤d
x
0
,x
f
y
w
y
dμ
y
: T
2,1
v,w
f
x
T
2,2
v,w
f
x
.
3.14
It is easy to see that see also 31, Theorems 1.1.3 or 1.1.4 the condition
A
a
1
: sup
t≥a
d
x
0
,x
≥t
v
x
q
c
dμ
x
1/q
c
a≤d
x
0
,y
≤t
w
y
p
c
dμ
y
1/p
c
< ∞ 3.15
guarantees the boundedness of the operator
T
v,w
f
x
v
x
a≤d
x
0
,y
<d
x
0
,x
f
y
w
y
dμ
y
3.16
Journal of Inequalities and Applications 15
from L
p
c
X \ Bx
0
,a to L
q
c
X \ Bx
0
,a.Thus,T
2,2
v,w
is bounded. It remains to prove that
T
2,1
v,w
is bounded. We have
T
2,1
v,w
f
L
p·
X
B
x
0
,a
c
v
x
q
c
dμ
x
1/q
c
B
x
0
,a
f
y
w
y
dμ
y
≤
B
x
0
,a
c
v
x
q
c
dμ
x
1/q
c
f
L
p·
Bx
0
,a
w
L
p
·
Bx
0
,a
.
3.17
Observe, now, that the condition A
1
< ∞ guarantees that the integral
B
x
0
,a
c
v
x
q
c
dμ
x
3.18
is finite. Moreover, N : w
L
p
·
Bx
0
,a
< ∞. Indeed, we have that
N ≤
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
B
x
0
,a
w
y
p
y
dμ
y
1/p
−
Bx
0
,a
if
w
L
p
·
Bx
0
,a
≤ 1,
B
x
0
,a
w
y
p
y
dμ
y
1/p
Bx
0
,a
if
w
L
p
·
Bx
0
,a
> 1.
3.19
Further,
B
x
0
,a
w
y
p
y
dμ
y
B
x
0
,a
∩
{
w≤1
}
w
y
p
y
dμ
y
B
x
0
,a
∩
{
w>1
}
w
y
p
y
dμ
y
: I
1
I
2
.
3.20
For I
1
, we have that I
1
≤ μBx
0
,a < ∞. Since L ∞ and condition 2.1 holds, there exists
apointy
0
∈ X such that a<dx
0
,y
0
< 2a. Consequently, Bx
0
,a ⊂ Bx
0
,dx
0
,y
0
and
py ≥ p
−
Bx
0
,dx
0
,y
0
p
0
y
0
, where y ∈ Bx
0
,a. Consequently, the condition A
1
< ∞
yields I
2
≤
Bx
0
,a
wy
p
0
y
0
dy < ∞. Finally, we have that T
2,1
v,w
f
L
p·
X
≤ C. Hence, T
v,w
is bounded from L
p·
X to L
q·
X.
The proof of the following statement is similar to that of Theorem 3.2; therefore, we
omit it see also the proofs of Theorem 1.1.3 in 31 and Theorems 2.6 and 2.7 in 21 for
similar arguments.
16 Journal of Inequalities and Applications
Theorem 3.3. Let X, d, μ be a quasimetric measure space. Assume that p and q are measurable
functions on X satisfying the condition 1 <p
−
≤ p
1
x ≤ qx ≤ q
< ∞.IfL ∞, then, one
assumes that p ≡ p
c
≡ const, q ≡ q
c
≡ const outside some ball Bx
0
,a.If
B
1
sup
0≤t≤L
d
x
0
,x
≤t
v
x
qx
t≤d
x
0
,x
≤L
w
p
1
x
y
dμ
y
qx/ p
1
x
dμ
x
< ∞, 3.21
then T
v,w
is bounded from L
p·
X to L
q·
X.
Remark 3.4. If p ≡ const, then the condition A
1
< ∞ in Theorem 3.2 resp., B
1
< ∞ in
Theorem 3.3 is also necessary for the boundedness of T
v,w
resp., T
v,w
from L
p·
X to
L
q·
X.See31, pages 4-5 for the details.
4. Potentials
In this section, we discuss two-weight estimates for the potential operators T
α·
and I
α·
on
quasimetric measure spaces, where 0 <α
−
≤ α
< 1. If α ≡ const , then we denote T
α·
and
I
α·
by T
α
and I
α
, respectively.
The boundedness of Riesz potential operators in L
p·
Ω spaces, where Ω is a domain
in R
n
was established in 5, 6, 36 , 37.
For the following statement we refer to 11.
Theorem A. Let X, d, μ be an SHT . Suppose that 1 <p
−
≤ p
< ∞ and p ∈P1. Assume that
if L ∞,thenp ≡ const outside some ball. Let α be a constant satisfying the condition 0 <α<1/p
.
One sets qxpx/1 − αpx. Then, T
α
is bounded from L
p·
X to L
q·
X.
Theorem B see 9. Let X, d, μ be a nonhomogeneous space with L<∞ and let N be a constant
defined by N a
1
1 2a
0
, where the constants a
0
and a
1
are taken from the definition of the
quasimetric d. Suppose that 1 <p
−
<p
< ∞,p,α∈PN and that μ is upper Ahlfors 1-regular.
One defines qxpx/1 − αxpx,where0 <α
−
≤ α
< 1/p
.ThenI
α·
is bounded from
L
p·
X to L
q·
X.
For the statements and their proofs of this section, we keep the notation of the previous
sections and, in addition, introduce the new notation:
v
1
α
x
: v
x
μB
x
0
x
α−1
,w
1
α
x
: w
−1
x
; v
2
α
x
: v
x
;
w
2
α
x
: w
−1
x
μB
x
0
x
α−1
;
F
x
:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
y ∈ X :
d
x
0
,y
L
A
2
a
1
≤ d
x
0
,y
≤ A
2
La
1
d
x
0
,x
, if L<∞,
y ∈ X :
d
x
0
,y
A
2
a
1
≤ d
x
0
,y
≤ A
2
a
1
d
x
0
,x
, if L ∞,
4.1
where A and a
1
are constants defined in Definition 2.11 and the triangle inequality for d,
respectively. We begin this section with the following general-type statement.
Journal of Inequalities and Applications 17
Theorem 4.1. Let X, d, μ be an SHT without atoms. Suppose that 1 <p
−
≤ p
< ∞ and α is a
constant satisfying the condition 0 <α<1/p
.Letp ∈P1. One sets qxpx/1 − αpx.
Further, if L ∞, then one assumes that p ≡ p
c
≡ const outside some ball Bx
0
,a. Then the
inequality
v
T
α
f
L
q·
X
≤ c
wf
L
p·
X
4.2
holds if the following three conditions are satisfied:
a T
v
1
α
,w
1
α
is bounded from L
p·
X to L
q·
X;
b T
v
2
α
,w
2
α
is bounded from L
p·
X to L
q·
X;
c there is a positive constant b such that one of the following inequalities hold: (1) v
F
x
≤
bwx for μ− a.e. x ∈ X;(2) vx ≤ bw
−
F
x
for μ− a.e. x ∈ X.
Proof. For simplicity, suppose that L<∞. The proof for the case L ∞ is similar to that of
the previous case. Recall that the sets I
i,k
,i 1, 2, 3andE
k
are defined in Section 2.Letf ≥ 0
and let g
L
q
·
X
≤ 1. We have
X
T
α
f
x
g
x
v
x
dμ
x
0
k−∞
E
k
T
α
f
x
g
x
v
x
dμ
x
≤
0
k−∞
E
k
T
α
f
1,k
x
g
x
v
x
dμ
x
0
k−∞
E
k
T
α
f
2,k
x
g
x
v
x
dμ
x
0
k−∞
E
k
T
α
f
3,k
x
g
x
v
x
dμ
x
: S
1
S
2
S
3
,
4.3
where f
1,k
f · χ
I
1,k
,f
2,k
f · χ
I
2,k
,f
3,k
f · χ
I
3,k
.
Observe that if x ∈ E
k
and y ∈ I
1,k
, then dx
0
,y ≤ dx
0
,x/Aa
1
. Consequently, the
triangle inequality for d yields dx
0
,x ≤ A
a
1
a
0
dx, y, where A
A/A − 1. Hence, by
using Remark 2.4,wefindthatμB
x
0
x
≤ cμB
xy
. Applying condition a now, we have that
S
1
≤ c
μB
x
0
x
α−1
v
x
B
x
0
x
f
y
dμ
y
L
qx
X
g
L
q
·
X
≤ c
f
L
p·
X
. 4.4
Further, observe that if x ∈ E
k
and y ∈ I
3,k
, then μB
x
0
y
≤ cμB
xy
. By condition b,
we find that S
3
≤ cf
L
p·
X
.
18 Journal of Inequalities and Applications
Now we estimate S
2
. Suppose that v
F
x
≤ bwx. Theorem A and Lemma 2.14 yield
S
2
≤
k
T
α
f
2,k
·
χ
E
k
·
v
·
L
q·
X
gχ
E
k
·
L
q
·
X
≤
k
v
E
k
T
α
f
2,k
·
L
q·
X
g
·
χ
E
k
·
L
q
·
X
≤ c
k
v
E
k
f
2,k
L
p·
X
g·χ
E
k
·
L
q
·
X
≤ c
k
f
2,k
·
w
·
χ
I
2,k
·
L
p·
X
g
·
χ
E
k
·
L
q
·
X
≤ c
f·w·
L
p·
X
g·
L
q
·
X
≤ c
f
·
w
·
L
p·
X
.
4.5
The estimate of S
2
for the case when vx ≤ bw
−
F
x
is similar to that of the previous
one. Details are omitted.
Theorems 4.1, 3.2,and3.3 imply the following statement.
Theorem 4.2. Let X, d, μ be an SHT. Suppose that 1 <p
−
≤ p
< ∞ and α is a constant satisfying
the condition 0 <α<1/p
.Letp ∈P1. One sets qxpx/1 − αpx.IfL ∞, then, one
supposes that p ≡ p
c
≡ const outside some ball Bx
0
,a. Then inequality 4.2 holds if the following
three conditions are satisfied:
i
P
1
: sup
0<t≤L
t<d
x
0
,x
≤L
v
x
μ
B
x
0
x
1−α
qx
×
d
x
0
,y
≤t
w
−p
0
x
y
dμ
y
qx/ p
0
x
dμ
x
< ∞;
4.6
ii
P
2
: sup
0<t≤L
d
x
0
,x
≤t
v
x
qx
×
t<d
x
0
,y
≤L
w
y
μB
x
0
y
1−α
−p
1
x
dμ
y
qx/ p
1
x
dμ
x
<∞,
4.7
iii condition c of Theorem 4.1 holds.
Remark 4.3. If p p
c
≡ const on X, then the conditions P
i
< ∞, i 1, 2, are necessary for
4.2. Necessity of the condition P
1
< ∞ follows by taking the test function f w
−p
c
χ
Bx
0
,t
in 4.2 and observing that μB
xy
≤ cμB
x
0
x
for those x and y which satisfy the conditions
dx
0
,x ≥ t and dx
0
,y ≤ t see also 31, Theorem 6.6.1, page 418 for the similar arguments
while necessity of the condition P
2
< ∞ can be derived by choosing the test function
Journal of Inequalities and Applications 19
fxw
−p
c
xχ
X\Bx
0
,t
xμB
x
0
x
α−1p
c
−1
and taking into account the estimate μB
xy
≤
μB
x
0
y
for dx
0
,x ≤ t and dx
0
,y ≥ t.
The next statement follows in the same manner as the previous one. In this case,
Theorem B is used instead of Theorem A. The proof is omitted.
Theorem 4.4. Let X, d, μ be a nonhomogeneous space with L<∞.LetN be a constant defined by
N a
1
1 2a
0
. Suppose that 1 <p
−
≤ p
< ∞,p,α∈PN and that μ is upper Ahlfors 1-regular.
We define qxpx/1 − αxpx,where0 <α
−
≤ α
< 1/p
. Then the inequality
v
·
I
α·
f
·
L
q·
X
≤ c
w
·
f
·
L
p·
X
4.8
holds if
i
sup
0≤t≤L
t<d
x
0
,x
≤L
v
x
d
x
0
,x
1−αx
qx
B
x
0
,t
w
−p
0
x
y
dμ
y
qx/p
0
x
dμ
x
< ∞;
4.9
ii
sup
0≤t≤L
B
x
0
,t
v
x
qx
t<d
x
0
,y
≤L
w
y
d
x
0
,y
1−αy
−p
1
x
dμ
y
qx/p
1
x
dμ
x
<∞,
4.10
and iii condition c of Theorem 4.1 is satisfied.
Remark 4.5. It is easy to check that if p and α are constants, then conditions i and ii in
Theorem 4.4 are also necessary for 4.8. This follows easily by choosing appropriate test
functions in 4.8see also Remark 4.3.
Theorem 4.6. Let X, d, μ be an SHT without atoms. Let 1 <p
−
≤ p
< ∞ and let α be a constant
with the condition 0 <α<1/p
. One sets qxpx/1 − αpx. Assume that p has a minimum
at x
0
and that p ∈ LHX. Suppose also that if L ∞,thenp is constant outside some ball Bx
0
,a.
Let v and w be positive increasing functions on 0, 2L. Then the inequality
v
d
x
0
, ·
T
α
f
·
L
q·
X
≤ c
w
d
x
0
, ·
f
·
L
p·
X
4.11
holds if
I
1
: sup
0<t≤L
I
1
t
: sup
0<t≤L
t<d
x
0
,x
≤L
v
d
x
0
,x
μ
B
x
0
x
1−α
qx
×
d
x
0
,y
≤t
w
−p
0
x
d
x
0
,y
dμ
y
qx/ p
0
x
dμ
x
< ∞,
4.12
20 Journal of Inequalities and Applications
for L ∞;
J
1
: sup
0<t≤L
t<d
x
0
,x
≤L
v
d
x
0
,x
μ
B
x
0
x
1−α
qx
×
d
x
0
,y
≤t
w
−p
x
0
d
x
0
,y
dμ
y
qx/p
x
0
dμ
x
< ∞,
4.13
for L<∞.
Proof. We prove the theorem for L ∞. The proof for the case when L<∞ is similar. Observe
that by Lemma 2.10 the condition p ∈ LHX implies p ∈P1. We will show that the
condition I
1
< ∞ implies the inequality vA
2
a
1
t/wt ≤ C for all t>0, where A and a
1
are constants defined in Definition 2.11 and the triangle inequality for d, respectively. Indeed,
let us assume that t ≤ b
1
, where b
1
is a small positive constant. Then, taking into account the
monotonicity of v and w and the facts that p
0
xp
0
xfor small dx
0
,x and μ ∈ RDCX,
we have
I
1
t
≥
A
2
a
1
t≤d
x
0
,x
<A
3
a
1
t
v
A
2
a
1
t
w
t
qx
μB
x
0
,t
α−1/p
0
xqx
dμ
x
≥
v
A
2
a
1
t
w
t
q
−
A
2
a
1
t≤d
x
0
,x
<A
3
a
1
t
μB
x
0
,t
α−1/p
0
xqx
dμ
x
≥ c
v
A
2
a
1
t
w
t
q
−
.
4.14
Hence,
c : lim
t → 0
vA
2
a
1
t/wt < ∞. Further, if t>b
2
, where b
2
is a large number, then
since p and q are constants, for dx
0
,x >t, we have that
I
1
t
≥
A
2
a
1
t≤d
x
0
,x
<A
3
a
1
t
v
d
x
0
,x
q
c
μB
x
0
,t
α−1q
c
dμ
x
×
B
x
0
,t
w
−p
c
x
dμ
x
q
c
/p
c
dμ
x
≥ C
v
A
2
a
1
t
w
t
q
c
A
2
a
1
t≤d
x
0
,x
<A
3
a
1
t
μB
x
0
,t
α−1/p
c
q
c
dμ
x
≥ c
v
A
2
a
1
t
w
t
q
c
.
4.15
In the last inequality we used the fact that μ satisfies the reverse doubling condition.
Journal of Inequalities and Applications 21
Now we show that the condition I
1
< ∞ implies
sup
t>0
I
2
t
: sup
t>0
d
x
0
,x
≤t
v
d
x
0
,x
qx
×
d
x
0
,y
>t
w
−p
1
x
d
x
0
,y
μ
B
x
0
y
α−1p
1
x
dμ
y
qx/ p
1
x
dμ
x
< ∞.
4.16
Due to monotonicity of functions v and w, the condition p ∈ LHX, Proposition 2.6,
Lemmas 2.9,and2.10 and the assumption that p has a minimum at x
0
,wefindthatforall
t>0,
I
2
t
≤
dx
0
,x≤t
v
t
w
t
qx
μ
B
x
0
,t
α−1/px
0
qx
dμ
x
≤ c
d
x
0
,x
≤t
v
t
w
t
qx
μ
B
x
0
,t
α−1/px
0
qx
0
dμ
x
≤ c
⎛
⎝
d
x
0
,x
≤t
v
A
2
a
1
t
w
t
qx
dμ
x
⎞
⎠
μ
B
x
0
,t
−1
≤ C.
4.17
Now, Theorem 4.2 completes the proof.
Theorem 4.7. Let X, d, μ be an SHT with L<∞. Suppose that p, q and α are measurable functions
on X satisfying the conditions: 1 <p
−
≤ px ≤ qx ≤ q
< ∞ and 1/p
−
<α
−
≤ α
< 1. Assume
that α ∈ LHX and there is a point x
0
∈ X such that p, q ∈ LHX, x
0
. Suppose also that w is a
positive increasing function on 0, 2L. Then the inequality
T
α·
f
v
L
q·
X
≤ c
w
d
x
0
, ·
f
·
L
p·
X
4.18
holds if the following two conditions are satisfied:
I
1
: sup
0<t≤L
t≤d
x
0
,x
≤L
v
x
μB
x
0
x
1−αx
qx
×
d
x
0
,x
≤t
w
−p
0
x
d
x
0
,y
dμ
y
qx/p
0
x
dμ
x
< ∞;
I
2
: sup
0<t≤L
d
x
0
,x
≤t
v
x
qx
×
t≤d
x
0
,x
≤L
w
d
x
0
,y
×
μB
x
0
y
1−αx
−p
1
x
dμ
y
qx/p
1
x
dμ
x
< ∞.
4.19
22 Journal of Inequalities and Applications
Proof. For simplicity, assume that L 1. First observe that by Lemma 2.10 we have p, q ∈
P1,x
0
and α ∈P1. Suppose that f ≥ 0andS
p
wdx
0
, ·f· ≤ 1. We will show that
S
q
vT
α·
f ≤ C.
We have
S
q
vT
α·
f
≤ C
q
⎡
⎣
X
v
x
d
x
0
,y
≤d
x
0
,x
/
2a
1
f
y
μB
xy
αx−1
dμ
y
qx
dμ
x
X
v
x
d
x
0
,x
/
2a
1
≤d
x
0
,y
≤2a
1
d
x
0
,x
f
y
μB
xy
αx−1
dμ
y
qx
dμ
x
X
v
x
d
x
0
,y
≥2a
1
d
x
0
,x
f
y
μB
xy
αx−1
dμ
y
qx
dμ
x
⎤
⎦
: C
q
I
1
I
2
I
3
.
4.20
First, observe that by virtue of the doubling condition for μ, Remark 2.4,andsimple
calculation we find that μB
x
0
x
≤ cμB
xy
. Taking into account this estimate and Theorem 3.2
we have that
I
1
≤ c
X
v
x
μB
x
0
x
1−αx
d
x
0
,y
<d
x
0
,x
f
y
dμ
y
qx
dμ
x
≤ C. 4.21
Further, it is easy to see that if dx
0
,y ≥ 2a
1
dx
0
,x, then the triangle inequality for
d and the doubling condition for μ yield that μB
x
0
y
≤ cμB
xy
. Hence, due to Proposition 2.7,
we see that μB
x
0
y
αx−1
≥ cμB
xy
αy−1
for such x and y. Therefore, Theorem 3.3 implies that
I
3
≤ C.
It remains to estimate I
2
. Let us denote:
E
1
x
:
B
x
0
x
\ B
x
0
,
d
x
0
,x
2a
1
; E
2
x
:
B
x
0
, 2a
1
d
x
0
,x
\ B
x
0
x
. 4.22
Then we have that
I
2
≤ C
⎡
⎣
X
v
x
E
1
x
f
y
μB
xy
αx−1
dμ
y
qx
dμ
x
X
v
x
E
2
x
f
y
μB
xy
αx−1
dμ
y
qx
dμ
x
⎤
⎦
: c
I
21
I
22
.
4.23
Journal of Inequalities and Applications 23
Using H
¨
older’s inequality for the classical Lebesgue spaces we find that
I
21
≤
X
v
qx
x
E
1
x
w
p
0
x
d
x
0
,y
f
y
p
0
x
dμ
y
qx/p
0
x
×
E
1
x
w
−p
0
x
d
x
0
,y
μB
xy
αx−1p
0
x
dμ
y
qx/p
0
x
dμ
x
.
4.24
Denote the first inner integral by J
1
and the second one by J
2
.
By using the fact that p
0
x ≤ py, where y ∈ E
1
x,weseethatJ
1
≤ μB
x
0
x
E
1
x
fy
py
wdx
0
,y
py
dμy, while by applying Lemma 2.9,forJ
2
, we have that
J
2
≤ cw
−p
0
x
d
x
0
,x
2a
1
E
1
x
μB
xy
αx−1p
0
x
dμ
y
≤ cw
−p
0
x
d
x
0
,x
2a
1
μB
x
0
x
αx−1p
0
x1
.
4.25
Summarizing these estimates for J
1
and J
2
we conclude that
I
21
≤
X
v
qx
x
μB
x
0
x
qxαx
w
−qx
d
x
0
,x
2a
1
dμ
x
X
v
qx
x
×
E
1
x
w
py
d
x
0
,y
f
y
py
dμ
y
qx/p
0
x
μB
x
0
x
qxαx−1/p
0
x
× w
−qx
d
x
0
,x
2a
1
dμ
x
: I
1
21
I
2
21
.
4.26
By applying monotonicity of w, the reverse doubling property f or μ with the constants
A and B see Remark 2.12, and the condition
I
1
< ∞ we have that
I
1
21
≤ c
0
k−∞
Bx
0
,A
k
\Bx
0
,A
k−1
v
x
qx
Bx
0
,A
k−1
/2a
1
w
−p
0
x
d
x
0
,y
dμ
y
qx/p
0
x
×
μB
x
0
,x
qx/p
0
xαx−1qx
dμ
x
≤ c
0
k−∞
μ
B
x
0
,A
k
q
−
/p
24 Journal of Inequalities and Applications
×
Bx
0
,A
k
\Bx
0
,A
k−1
v
x
qx
Bx
0
,A
k
w
−p
0
x
d
x
0
,y
dμ
y
qx/p
0
x
×
μB
x
0
,x
qxαx−1
dμ
x
≤ c
0
k−∞
μ
B
x
0
,A
k
\ B
x
0
,A
k−1
q
−
/p
≤ c
0
k−∞
μBx
0
,A
k
\Bx
0
,A
k−1
μB
x
0
,x
q
−
/p
−1
dμ
y
≤ c
X
μB
x
0
,x
q
−
/p
−1
dμ
y
< ∞.
4.27
Due to the facts that qx ≥ p
0
x,S
p
wdx
0
, ·f· ≤ 1,
I
1
< ∞ and w is increasing,
for I
2
21
,wefindthat
I
2
21
≤ c
0
k−∞
μB
x
0
,A
k1
a
1
\ B
x
0
,A
k−2
w
py
d
x
0
,y
f
y
py
dμ
y
×
⎛
⎝
μB
x
0
,A
k
\B
x
0
,A
k−1
v
qx
x
B
x
0
,A
k−1
w
−p
0
x
d
x
0
,y
dμ
y
qx/p
0
x
×
μB
x
0
,x
αx−1qx
dμ
x
⎞
⎠
≤ cS
p
f
·
w
d
x
0
, ·
≤ c.
4.28
Analogously, the estimate for I
22
follows. In this case, we use the condition
I
2
< ∞ and
the fact that p
1
x ≤ py when dx
0
,x ≤ dx
0
,y < 2a
1
dx
0
,x. The details are omitted. The
theorem is proved.
Taking into account the proof of Theorem 4.6, we can easily derive the following
statement, proof of which is omitted.
Theorem 4.8. Let X, d, μ be an SHT with L<∞. Suppose that p, q and α are measurable functions
on X satisfying the conditions 1 <p
−
≤ px ≤ qx ≤ q
< ∞ and 1/p
−
<α
−
≤ α
< 1. Assume
that α ∈ LHX. Suppose also that there is a point x
0
such that p, q ∈ LHX, x
0
and p has a
minimum at x
0
.Letv and w be a positive increasing function on 0, 2L satisfying the condition
J
1
< ∞ (see Theorem 4.6). Then inequality 4.11 is fulfilled.
Theorem 4.9. Let X, d, μ be an SHT with L<∞ and let μ be upper Ahlfors 1-regular. Suppose
that 1 <p
−
≤ p
< ∞ and that p ∈ LHX.Letp have a minimum at x
0
. Assume that α is constant
Journal of Inequalities and Applications 25
satisfying the condition α<1/p
. We set qxpx/1−αpx.Ifv and w are positive increasing
functions on 0, 2L satisfying the condition
E : sup
0≤t≤L
t<d
x
0
,x
≤L
v
d
x
0
,x
d
x
0
,x
1−α
qx
×
d
x
0
,x
≤t
w
−p
0
x
y
dμ
y
qx/p
0
x
dμ
x
< ∞,
4.29
then the inequality
v
d
x
0
, ·
I
α
f
·
L
q·
X
≤ c
w
d
x
0
, ·
f
·
L
p·
X
4.30
holds.
Proof. The proof is similar to that of Theorem 4.6, we only discuss some details. First, observe
that due to Remark 2.5 we have that p ∈PN, where N a
1
1 2a
0
. It is easy to check that
the condition E<∞ implies that vA
2
a
1
t/wt ≤ C for all t, where the constant A is defined
in Definition 2.11 and a
1
is from the triangle inequality for d. Further, Lemmas 2.9 and 2.10,
the fact that p has a minimum at x
0
, and the inequality
d
x
0
,y
>t
d
x
0
,y
α−1p
1
x
dμ
y
≤ ct
α−1p
1
x1
, 4.31
where the constant c does not depend on t and x, yield that
sup
0≤t≤L
d
x
0
,x
≤t
v
d
x
0
,x
qx
×
⎛
⎝
d
x
0
,y
>t
w
d
x
0
,y
d
x
0
,y
1−α
−p
1
x
dμ
y
⎞
⎠
qx/p
1
x
dμ
x
< ∞.
4.32
Theorem 4.4 completes the proof.
Example 4.10. Let vtt
γ
and wtt
β
, where γ and β are constants satisfying the condition
0 ≤ β<1/p
−
, γ ≥ max{0, 1 − α − 1/q
− q
−
/q
−β 1/p
−
}. Then v, w satisfies the
conditions of Theorem 4.6.
Acknowledgments
The first and second authors were partially supported by the Georgian National Science
Foundation Grant project numbers: GNSF/ST09/23/3-100 and GNSF/ST07/3-169.Apart
of this work was fulfilled in Abdus Salam School of Mathematical sciences, GC University,
Lahore. The second and third authors are grateful to the Higher Educational Commission of
