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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 697407, 19 pages
doi:10.1155/2008/697407
Research Article
Weighted Estimates of a Measure
of Noncompactness for Maximal and
Potential Operators
Muhammad Asif
1
and Alexander Meskhi
2
1
Abdus Salam School of Mathematical Sciences, GC University, c-II, M. M. Alam Road, Gulberg III,
Lahore 54660, Pakistan
2
A. Razmadze Mathematical Institute, Georgian Academy of Sciences, 1, M. aleksidze Street,
0193 Tbilisi, Georgia
Correspondence should be addressed to Alexander Meskhi,
Received 5 April 2008; Accepted 19 June 2008
Recommended by Siegfried Carl
A measure of noncompactness essential norm for maximal functions and potential operators
defined on homogeneous groups is estimated in terms of weights. Similar problem for partial
sums of the Fourier series is studied. In some cases, we conclude that there is no weight pair for
which these operators acting between two weighted Lebesgue spaces are compact.
Copyright q 2008 M. Asif and A. Meskhi. 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.
1. Introduction
In the papers 1–3, the measure of noncompactness essential norm of maximal functions,
singular integrals, and identity operators acting in weighted Lebesgue spaces defined on
R
n
with different weights was estimated from below. In this paper, we investigate the same
problem for maximal functions and potentials defined on homogeneous groups. Analogous
estimates for the partial sums of Fourier series are also derived. For truncated potentials, we
have two-sided estimates of the essential norm.
A result analogous to that of 2 has been obtained in 4, 5 for the Hardy-Littlewood
maximal operator with more general differentiation basis on symmetric spaces. The essential
norm for Hardy-type transforms and one-sided potentials in weighted Lebesgue spaces has
been estimated in 6–9see also 10. For two-sided estimates of the essential norm for the
Cauchy integrals see 11–14. The same problem in the one-weighted setting has been studied
in 15, 16.
The one-weight problem for the Hardy-Littlewood maximal functions was solved
by Muckenhoupt 17for maximal functions defined on the spaces of homogeneous type
2 Journal of Inequalities and Applications
see, e.g., 18 and for fractional maximal functions and Riesz potentials by Muckenhoupt
and Wheeden 19. Two-weight criteria for the Hardy-Littlewood maximal functions have
been obtained in 20. Necessary and sufficient conditions guaranteeing the boundedness of
the Riesz potentials from one weighted Lebesgue space into another one were derived by
Sawyer 21, 22 and Gabidzashvili and Kokilashvili 23see also 24. However, conditions
derived in 23 aremore transparent than those of 21. For the solution of the two-weight
problem for operators with positive kernels on spaces of homogeneous type see 25see
also 10, 26 for related topics.
Earlier, the trace inequality for the Riesz potentials boundedness of Riesz potentials
from L
p
to L
q
v
was established in 27, 28. The two-weight criteria for fractional maximal
functions were obtained in 22, 29, 30see also 25 for more general case.
Necessary and sufficient conditions guaranteeing the compactness of the Riesz
potentials have been derived in 31see also 10, Section 5.2. The one-weight problem
for the Hilbert transform and partial sums of the Fourier series was solved in 32.
The paper is organized as follows. In Section 2, we give basic concepts and prove some
lemmas. Section 3 is divided into 4 parts. Section 3.1 concerns maximal functions; potential
operators are discussed in Sections 3.2 and 3.3. Section 3.4 is devoted to the partial sums of
Fourier series.
Constants often different constants in the same series of inequalities will generally
be denoted by c or C.
2. Preliminaries
A homogeneous group is a simply connected nilpotent Lie group G on a Lie algebra g
with the one-parameter group of transformations δ
t
expA log t, t>0, where A is a
diagonalized linear operator in G with positive eigenvalues. In the homogeneous group G,
the mappings exp oδ
t
o exp
−1
, t>0, are automorphisms in G, which will be again denoted by
δ
t
. The number Q tr A is the homogeneous dimension of G. T he symbol e will stand for the
neutral element in G.
It is possible to equip G with a homogeneous norm r : G → 0, ∞ which is continuous
on G, smooth on G \{e}, and satisfies the conditions
i rxrx
−1
for every x ∈ G;
ii rδ
t
xtrx for every x ∈ G and t>0;
iii rx0 if and only if x e;
iv there exists c
o
> 0 such that
rxy ≤ c
o
rxry
,x,y∈ G. 2.1
In the sequel, we denote by Ba, ρ and
Ba, ρ open and closed balls, respectively,
with the center a and radius ρ,thatis,
Ba, ρ :
y ∈ G; r
ay
−1
<ρ
,
Ba, ρ :
y ∈ G; r
ay
−1
≤ ρ
. 2.2
It can be observed that δ
ρ
Be, 1Be, ρ.
Let us fix a Haar measure |·| in G such that |Be, 1| 1. Then, |δ
t
E| t
Q
|E|.In
particular, |Bx, t| t
Q
for x ∈ G, t > 0.
Examples of homogeneous groups are the Euclidean n-dimensional space R
n
,the
Heisenberg group, upper triangular groups, and so forth. For the definition and basic
properties of the homogeneous group, we refer to 33, page 12 and 25.
M. Asif and A. Meskhi 3
Proposition A. Let G be a homogeneous group and let S {x ∈ G : rx1}. There is a (unique)
Radon measure σ on S such that for all u ∈ L
1
G,
G
uxdx
∞
0
S
u
δ
t
y
t
Q−1
dσydt. 2.3
For the details see, for example, 33, page 14.
We call a weight a locally integrable almost everywhere positive function on G. Denote
by L
p
w
G1 <p<∞ the weighted Lebesgue space, which is the space of all measurable
functions f : G → C with the norm
f
L
p
w
G
G
fx
p
wxdx
1/p
< ∞. 2.4
If w ≡ 1, then we denote L
p
1
G by L
p
G.
Let X L
p
w
G1 <p<∞ and denote by X
∗
the space of all bounded linear
functionals on X. We say that a real-valued functional F on X is sublinear if
i Ff g ≤ FfFg for all nonnegative f, g ∈ X;
ii Fαf|α|Ff for all f ∈ X and α ∈ C.
Let T be a sublinear operator T : X → L
q
G, then, the norm of the operator T is defined
as follows:
T sup
Tf
L
q
G
: f
X
≤ 1
. 2.5
Moreover, T is order preserving if Tfx ≥ Tgx almost everywhere for all nonnegative f
and g with fx ≥ gx almost everywhere. Further, if T is sublinear and order preserving,
then obviously it is nonnegative, that is, Tfx ≥ 0 almost everywhere if fx ≥ 0.
The measure of noncompactness for an operator T which is bounded, order
preserving, and sublinear from X into a Banach space Y will be denoted by T
κX,Y
or
simply T
κ
and is defined as
T
κX,Y
dist
T, KX, Y
≡ inf
T − K : K ∈KX, Y
, 2.6
where KX, Y is the class of all compact sublinear operators from X to Y.IfX Y , then we
use the symbol KX for KX, Y .
Let X and Y be Banach spaces and let T be a continuous linear operator from X to Y .
The entropy numbers of the operator T are defined as follows:
e
k
Tinf
ε>0:T
U
X
⊂
2
k−1
j1
b
i
εU
Y
for some b
1
, ,b
2
k−1
∈ Y
, 2.7
where U
X
and U
Y
are the closed unit balls in X and Y, respectively. It is well known see,
e.g., 34, page 8 that the measure of noncompactness of T is greater than or equal to
lim
n →∞
e
n
T.
In the sequel, we assume that X is a Banach space which is a certain subset of all Haar-
measurable functions on G. We denote by SX the class of all bounded sublinear functionals
defined on X,thatis,
SX
F : X → R,F-sublinear and F sup
x≤1
Fx
< ∞
. 2.8
4 Journal of Inequalities and Applications
Let M be the set of all bounded functionals F defined on X with the f ollowing
property:
0 ≤ Ff ≤ Fg, 2.9
for any f, g ∈ X with 0 ≤ fx ≤ gx almost every. We also need the following classes of
operators acting from X to L
p
G:
F
L
X, L
p
G
:
T : Tfx
m
j1
α
j
fu
j
,m∈ N,u
j
≥ 0,u
j
∈ L
p
G,
u
j
are linearly independent and α
j
∈ X
∗
M
,
F
S
X, L
p
G
:
T : Tfx
m
j1
β
j
fu
j
,m∈ N,u
j
≥ 0,u
j
∈ L
p
G,
u
j
are linearly independent and β
j
∈ SX
M
.
2.10
If X L
p
G, we will denote these classes by F
L
L
p
G and F
S
L
p
G, respectively. It is clear
that if P ∈ F
L
X, L
p
G resp., P ∈ F
S
X, L
p
G, then P is compact linear resp., compact
sublinear from X to L
p
G.
We will use the symbol αT for the distance between the operator T : X → L
p
G and
the class F
S
X, L
p
G, that is,
αT : dist
T, F
S
X, L
p
G
. 2.11
For any bounded subset A of L
p
G1 <p<∞,let
ΦA : inf
δ>0:A can be covered by finitely many open balls in L
p
G of radius δ
,
ΨA : inf
P∈F
L
L
p
G
sup
f − Pf
L
p
G
: f ∈ A
.
2.12
We will need a statement similar to Theorem V.5.1 of Chapter V of 35for Euclidean
spaces see 2.
Theorem A. For any bounded subset K ⊂ L
p
G1 ≤ p<∞, the inequality
2ΦK ≥ ΨK2.13
holds.
Proof. Let ε>ΦK. Then, there are g
1
,g
2
, ,g
N
∈ L
p
G such that for all f ∈ K and some
i ∈{1, 2, ,N},
f − g
i
L
p
G
<ε. 2.14
M. Asif and A. Meskhi 5
Further, given δ>0, let
B be the closed ball in G with center e such that for all i ∈
{1, 2, ,N},
G\B
g
i
x
p
dx
1/p
<
1
2
δ. 2.15
It is known see 33, page 8 that every closed ball in G is a compact set. Let us
cover
B by open balls with radius h. Since B is compact, we can choose a finite subcover
{B
1
,B
2
, ,B
n
}. Further, let us assume that {E
1
,E
2
, ,E
n
} is a family of pairwise disjoint
sets of positive measure such that
B
n
i1
E
i
and E
i
⊂ B
i
we can assume that E
1
B
1
∩ B,
E
2
B
2
\ B
1
∩ B, ,E
k
B
k
\
k−1
i1
B
i
∩ B, . We define
Pfx
n
i1
f
E
i
χ
E
i
x,f
E
i
E
i
−1
E
i
fxdx. 2.16
Then,
g
i
− Pg
i
p
L
p
B
n
j1
E
j
1
E
j
E
j
g
i
x − g
i
y
dy
p
dx
≤
m
j1
E
j
1
E
j
E
j
g
i
x − g
i
y
p
dy dx
≤ sup
rz≤2c
o
h
B
g
i
x − g
i
zx
p
dx −→ 0
2.17
as h → 0. The latter fact follows from the continuity of the norm L
p
Gsee, e.g., 33, page
19.
From this and 2.14,wefindthat
g
i
− Pg
i
L
p
G
<δ, i 1, 2, 3, ,N, 2.18
when h is sufficiently small. Further,
Pf
p
L
p
G
n
j1
E
j
E
j
−1
E
j
fydy
p
dx
≤
n
j1
E
j
E
j
−1
E
j
fy
p
dy dx
≤f
p
L
p
B
≤f
p
L
p
G
.
2.19
It is also clear that the functionals f → f
E
i
belong to L
p
G
∗
∩ M. Hence, P ∈
F
L
L
p
G. Finally, 2.14–2.15 and 2.18 yield
f − Pf
L
p
G
≤
f − g
i
L
p
G
g
i
− Pg
i
L
p
G
P
g
i
− f
L
p
G
<ε δ
g
i
− f
L
p
G
≤ 2ε δ.
2.20
Since δ is arbitrarily small, we have the desired result.
6 Journal of Inequalities and Applications
Lemma A. Let 1 ≤ p<∞ and assume that a set K ⊂ L
p
G is compact. Then for any given ε>0,
there exist an operator P
ε
∈ F
L
L
p
G such that for all f ∈ K,
f − P
ε
f
L
p
G
≤ ε. 2.21
Proof. Let K be a compact set in L
p
G. Using Theorem A, we see that ΨK0. Hence for
ε>0, there exists P
ε
∈ F
L
L
p
G such that
sup
f − P
ε
f
L
p
G
: f ∈ K
≤ ε. 2.22
Lemma B. Let T : X → L
p
G be compact, order-preserving, and sublinear operator, where 1 ≤ p<
∞. Then, αT0.
Proof. Let U
X
{f : f
X
≤ 1}. From the compactness of T, it follows that TU
X
is relatively
compact in L
p
G. Using Lemma A, we have that for any given ε>0 there exists an operator
P
ε
∈ F
L
L
p
G such that for all f ∈ U
X
,
Tf − P
ε
Tf
L
p
G
≤ ε. 2.23
Let
P
ε
P
ε
◦ T. Then,
P
ε
∈ F
S
X, L
p
G. Indeed, there exist functionals α
j
∈ X
∗
∩ M, j ∈
{1, 2, ,m}, and linearly independent functions u
j
∈ L
p
G,j∈{1, 2, ,m}, such that
P
ε
fxP
ε
Tfx
m
j1
α
j
Tfu
j
x
m
j1
β
j
fu
j
x, 2.24
where β
j
α
j
◦ T belongs to SX ∩ M. Since by 2.23,
Tf −
P
ε
f
L
p
G
≤ ε 2.25
for all f ∈ U
X
, it follows immediately that αT0.
We will also need the following lemma.
Lemma C. Let T be a bounded, order-preserving, and sublinear operator from X to L
q
G,where
1 ≤ q<∞. Then,
T
κ
αT. 2.26
Proof. Let δ>0. Then, there exists an operator K ∈KX, L
q
G, such that T − K≤T
κ
δ.
By Lemma B there is P ∈ F
S
X, L
q
G for which the inequality K − P <δholds. This gives
T − P ≤T − K K − P≤T
κ
2δ. 2.27
Hence, αT ≤T
κ
. Moreover, it is obvious that
T
κ
≤ αT. 2.28
M. Asif and A. Meskhi 7
Lemma D. Let 1 ≤ q<∞ and let P ∈ F
S
X, L
q
G. Then for every a ∈ G and ε>0, there exist an
operator R ∈ F
S
X, L
q
G and positive numbers α, α such that for all f ∈ X, the inequality
P − Rf
L
q
G
≤ εf
X
2.29
holds and supp Rf ⊂ Ba,
α \ Ba, α.
Proof. There exist linearly independent nonnegative functions u
j
∈ L
q
G,j ∈{1, 2, ,N},
such that
Pfx
N
j1
β
j
fu
j
x,f∈ X, 2.30
where β
j
are bounded, order-preserving, sublinear functionals β
j
: X → R. On t he other hand,
there is a positive constant c for which
N
j1
β
j
f
≤ cf
X
. 2.31
Let us choose linearly independent Φ
j
∈ L
q
G and positive real numbers α
j
, α
j
such
that
u
j
− Φ
j
L
q
G
<ε, j∈{1, 2, ,N} 2.32
and supp Φ
j
⊂ Ba, α
j
\ Ba, α
j
. If
Rfx
N
j1
β
j
fΦ
j
x, 2.33
then it is obvious that R ∈ F
S
X, L
q
G and moreover,
Pf − Rf
L
q
G
≤
N
j1
β
j
f
u
j
− Φ
j
L
q
G
≤ cεf
X
2.34
for all f ∈ X. Besides this, supp Rf ⊂ Ba, α \ Ba, α, where α min{α
j
} and α max{α
j
}.
Lemmas C and D for Lebesgue spaces defined on Euclidean spaces have been proved
in 35 for the linear case and in 2 for sublinear operators.
Lemma E. Let 1 <p,q<∞, and let T be a bounded, order-preserving, and sublinear operator from
L
p
w
G to L
q
v
G. Suppose that λ>T
κL
p
w
G,L
q
v
G
, and a is a point of G. Then, there exist constants
β
1
,β
2
, 0 <β
1
<β
2
< ∞, such that for all τ and r with r>β
2
, τ<β
1
, the following inequalities hold:
Tf
L
q
v
Ba,τ
≤ λf
L
p
w
G
,
Tf
L
q
v
Ba,r
c
≤ λf
L
p
w
G
,
2.35
where f ∈ L
p
w
G.
8 Journal of Inequalities and Applications
Proof. Let T be bounded from L
p
w
G to L
q
v
G.LetT
v
be the operator given by
T
v
f v
1/q
Tf. 2.36
Then, it is easy to see that
T
v
κL
p
w
G → L
q
G
T
κL
p
w
G → L
q
v
G
. 2.37
By Lemma C, we have that
λ>α
T
v
. 2.38
Consequently, there exists P ∈ F
S
L
p
w
G,L
q
G such that
T
v
− P
<λ. 2.39
Fix a ∈ G. According to Lemma D, there are positive constants β
1
and β
2
,β
1
<β
2
, and R ∈
F
S
L
p
w
G,L
q
v
G for which
P − R≤
λ −
T
v
− P
2
2.40
and supp Rf ⊂ Ba, β
2
\ Ba, β
1
for all f ∈ L
p
w
G. Hence,
T
v
− R
<λ. 2.41
From the last inequality, it follows that if 0 <τ<β
1
and r>β
2
, then 2.35 holds for f,
f ∈ L
p
w
G.
The following lemmas are taken from 2for the linear case see 35.
Lemma F. Let Ω be a domain in R
n
, and let T be a bounded, order-preserving, and sublinear operator
from L
r
w
Ω to L
p
Ω,where1 <r,p<∞, and w is a weight function on Ω. Then,
T
κL
r
w
Ω,L
p
Ω
αT. 2.42
Lemma G. Let Ω be a domain in R
n
and let P ∈ F
S
X, L
p
Ω,whereX L
r
w
Ω and 1 <r,p<∞.
Then for every a ∈ Ω and ε>0, there exist an operator R ∈ F
S
X, L
p
Ω and positive numbers β
1
and β
2
, β
1
<β
2
such that for all f ∈ X, the inequality
P − Rf
L
p
Ω
≤ εf
X
2.43
holds and supp Rf ⊂ Da, β
2
\ Da, β
1
,whereDa, s :Ω
Ba, s.
Lemmas F and G yield the next statement which follows in the same manner as Lemma
E was proved; therefore we give it without proof.
Lemma H. Let Ω be a domain in R
n
. Suppose that 1 <p,q<∞, and that T is bounded, order-
preserving, and sublinear operator from L
p
w
Ω to L
q
v
Ω. Assume that λ>T
κL
p
w
Ω,L
q
v
Ω
and
a ∈ Ω. Then, there exist constants β
1
,β
2
, 0 <β
1
<β
2
< ∞ such that for all τ and r with r>β
2
,
τ<β
1
, the following inequalities hold:
Tf
L
q
v
Ba,τ
≤ λf
L
p
w
Ω
; Tf
L
q
v
Ω\Ba,r
≤ λf
L
p
w
Ω
, 2.44
wheref ∈ L
p
w
Ω.
M. Asif and A. Meskhi 9
Lemma I see 36, Chapter IX. Let 1 <p,q<∞, and let X, μ and Y, ν be σ-finite measure
spaces. If
kx, y
L
p
ν
Y
L
q
μ
X
< ∞,p
p
p − 1
, 2.45
then the operator
Kfx
Y
kx, yfydνy,x∈ X, 2.46
is compact from L
p
ν
Y into L
q
μ
X.
3. Main results
3.1. Maximal functions
Let G be a homogeneous group and let
M
α
fxsup
Bx
1
|B|
1−α/Q
B
fy
dy, x ∈ G, 0 ≤ α<Q, 3.1
where the supremum is taken over all balls B containing x.Ifα 0, then M
α
becomes the
Hardy-Littlewood maximal function which will be denoted by M.
It is known see, e.g., 17, 18 for α 0, and 19, 33, Chapter 6,forα>0 that if
1 <p<∞ and 0 ≤ α<Q/p, then the operator M
α
is bounded from L
p
ρ
p
G to L
q
ρ
q
G, where
q Qp/Q − αp, if and only if ρ ∈ A
p,q
G,thatis,
sup
B
1
|B|
B
ρ
q
1/q
1
|B|
B
ρ
−p
1/p
< ∞. 3.2
Now, we formulate the main results of this subsection.
Theorem 3.1. Let 1 <p<∞. Suppose that the maximal operator M is bounded from L
p
w
G to
L
p
v
G. Then, there is no weight pair v, w such that M is compact from L
p
w
G to L
p
v
G. Moreover,
the inequality
M
κL
p
w
G,L
p
v
G
≥ sup
a∈G
lim
τ → 0
1
Ba, τ
Ba,τ
vxdx
1/p
Ba,τ
w
1−p
xdx
1/p
3.3
holds.
Proof. Suppose that λ>M
κL
p
w
→ L
p
v
and a ∈ G. By Lemma E, we have that
Ba,τ
vx
sup
B x
1
Ba, τ
Ba,τ
fy
dy
p
dx ≤ λ
p
Ba,τ
fx
p
wxdx 3.4
for all τ τ ≤ β and all f supported in
Ba, τ. Substituting fyχ
Ba,r
y w
1−p
y in the
latter inequality and taking into account that
Ba,τ
w
1−p
xdx < ∞ see, e.g., 17, 18, 25,
Chapter 4 for all τ>0wefindthat
1
Ba, τ
p
Ba,τ
vxdx
Ba,τ
w
1−p
xdx
p−1
≤ λ
p
. 3.5
This inequality and Lebesgue differentiation theorem see 33, page 67 yield the
desired result.
10 Journal of Inequalities and Applications
For the fractional maximal functions, we have the following theorem.
Theorem 3.2. Let 1 <p<∞, 0 <α<Q/pand let q Qp/Q − αp. Suppose that M
α
is bounded
from L
p
w
G to L
q
v
G. Then, there is no weight pair v, w such that M
α
is compact from L
p
w
G to
L
q
v
G. Moreover, the inequality
M
α
κ
≥ sup
a∈G
lim
τ → 0
1
Ba, τ
α/Q−1
Ba,τ
vxdx
1/q
Ba,τ
w
1−p
xdx
1/p
3.6
holds.
The proof of this statement is similar to that of Theorem 3.1; therefore the proof is
omitted.
Example 3.3. Let 1 <p<∞, vxwxrx
γ
, where −Q<γ<p − 1Q. Then,
M
κL
p
w
G
≥ Q
γ Q
1/p
γ
1 − p
Q
1/p
−1
. 3.7
Indeed, first observe that the fact |Be, 1| 1 and Proposition A implies σSQ,
where S is the unit sphere in G and σS is its measure. By Theorem 3.1 and Proposition A,
we have
M
κL
p
w
G
≥ lim
τ → 0
1
Be, τ
Be,τ
wxdx
1/p
Be,τ
w
1−p
xdx
1/p
σSlim
τ → 0
τ
−Q
τ
0
t
γQ−1
dt
1/p
τ
0
t
γ1−p
Q−1
dt
1/p
Q
γ Q
1/p
γ
1 − p
Q
1/p
−1
.
3.8
3.2. Riesz potentials
Let G be a homogeneous group and let
I
α
fx
G
fy
r
xy
−1
Q−α
dy, 0 <α<Q, 3.9
be the Riesz potential operator. It is well known see 33, Chapter 6 that I
α
is bounded from
L
p
G to L
q
G,1<p,q<∞, if and only if q Qp/Q − αp.
Theorem 3.4. Let 1 <p≤ q<∞, 0 <α<Q.LetI
α
be bounded from L
p
w
G to L
q
v
G. Then, the
following inequality holds
I
α
κ
≥ C
α,Q
max
A
1
,A
2
,A
3
, 3.10
M. Asif and A. Meskhi 11
where
C
α,Q
1
2c
o
Q−α
,
A
1
sup
α∈G
lim
r → 0
r
α−Q
Ba,r
vxdx
1/q
Ba,r
w
1−p
xdx
1/p
,
A
2
sup
a∈G
lim
r → 0
Ba,r
vxdx
1/q
Ba,r
c
r
ay
−1
α−Qp
w
1−p
ydy
1/p
,
A
3
sup
a∈G
lim
r → 0
Ba,r
w
1−p
xdx
1/p
Ba,r
c
r
ay
−1
α−Qq
vydy
1/q
.
3.11
(c
o
is the constant from the triangle inequality for the homogeneous norms.)
The next statement is formulated for the Riesz potentials defined on domains in R
n
:
J
Ω,α
fx
Ω
fy|x − y|
α−n
dy, x ∈ Ω. 3.12
Theorem 3.5. Let Ω ⊆ R
n
be a domain in R
n
.Let1 <p≤ q<∞.IfJ
Ω,α
is bounded from
L
p
w
Ω to L
q
v
Ω, then one has
J
Ω,α
κ
≥ 2
α−n
B
1
, 3.13
where
B
1
sup
a∈Ω
lim
r → 0
r
α−n
Ba,r
v
1/q
Ba,r
w
1−p
1/p
. 3.14
In particular, if Ω ≡ R
n
,then
J
Ω,α
κ
≥ 2
α−n
max
B
2
,B
3
, 3.15
where
B
2
sup
a∈R
n
lim
r → 0
Ba,r
vxdx
1/q
R
n
\Ba,r
|a − y|
α−np
w
1−p
ydy
1/p
,
B
3
sup
a∈R
n
lim
r → 0
Ba,r
w
1−p
xdx
1/p
R
n
\Ba,r
|a − y|
α−nq
vydy
1/q
.
3.16
Corollary 3.6. Let 1 <p<∞, 1 <p<Q/α, q pQ/Q − αp, then there is no weight pair v, w
for which I
α
is compact from L
p
w
G to L
q
v
G. Moreover, if I
α
is bounded from L
p
w
G to L
q
v
G,then
I
α
κ
≥ C
α,Q
A
1
, 3.17
where C
α,Q
and A
1
are defined in Theorem 3.4.
12 Journal of Inequalities and Applications
Proof of Theorem 3.4. By Lemma E, we have that for λ>I
α
κL
p
w
G,L
q
v
G
and a ∈ G, there are
positive constants β
1
and β
2
β
1
<β
2
such that for all τ,s τ<β
1
, s>β
2
,
Ba,τ
vx
I
α
fx
q
dx ≤ λ
q
G
fx
p
wxdx
q/p
3.18
for f ∈ L
p
w
G, and
Ba,s
c
vx
I
α
fx
q
dx ≤ λ
q
Ba,s
|fx|
p
wxdx
q/p
3.19
for supp f ⊂ Ba, s.
Now taking fxχ
Ba,r
xw
1−p
x in 3.18 and observing that
Ba,r
w
1−p
xdx <
∞ for all r>0 see also 25, Chapter 3,wefindthat
Ba,r
vx
Ba,r
w
1−p
y
r
xy
−1
Q−α
dy
q
dx ≤ λ
q
Ba,r
w
1−p
xdx
q/p
< ∞. 3.20
Further if x, y ∈ Ba, τ, then
r
xy
−1
≤ c
o
r
xa
−1
r
ay
−1
≤ 2c
o
τ. 3.21
Hence,
I
α
κ
≥ C
α,Q
A
1
. 3.22
If fxχ
Ba,τ
c
xw
1−p
x/ray
−1
Q−αp
−1
, then
Ba,τ
vx
Ba,τ
c
w
1−p
ydy
r
xy
−1
Q−α
r
ay
−1
Q−αp
−1
q
dx ≤ λ
q
Ba,τ
c
w
1−p
xdx
r
ay
−1
Q−αp
q/p
< ∞.
3.23
Let rxa
−1
<τand rya
−1
>τ.Then,
r
xy
−1
≤ c
o
r
xa
−1
r
ay
−1
≤ c
o
τ r
ay
−1
≤ 2c
o
r
ay
−1
. 3.24
Hence, by 3.18 we have
1
2c
o
qQ−α
Ba,τ
vxdx
Ba,τ
c
w
1−p
ydy
r
ay
−1
Q−αp
q
≤ λ
q
Ba,τ
c
w
1−p
xdx
r
ay
−1
Q−αp
q/p
.
3.25
The latter inequality implies
I
α
κ
≥
1
2c
o
Q−α
A
2
. 3.26
M. Asif and A. Meskhi 13
Further, observe that 3.19 means that the norm of the operator
I
α
fx
Ba,s
fydy
ry
−1
a
Q−α
3.27
can be estimated as follows:
I
α
L
p
w
Ba,s → L
q
v
Ba,s
c
≤ λ. 3.28
Now by duality, we find that
I
α
L
p
w
Ba,s → L
q
v
Ba,s
c
I
α
L
q
v
1−q
Ba,s
c
→ L
p
w
1−p
Ba,s
, 3.29
where
I
α
gy
Ba,s
c
gxdx
r
xy
−1
Q−α
. 3.30
Indeed, by Fubini’s theorem and H
¨
older’s inequality, we have
I
α
f
L
q
v
Ba,s
c
≤ sup
g
L
q
v
Ba,s
c
≤1
Ba,s
c
gx
I
α
fx
dx
≤ sup
g
L
q
v
1−q
Ba,s
c
≤1
Ba,s
fy
I
α
|g|
y dy
≤ sup
g
L
q
v
1−q
Ba,s
c
≤1
Ba,s
|f|
p
w
1/p
Ba,s
I
α
|g|
p
w
1−p
1/p
≤
I
α
Ba,s
|f|
p
w
1/p
.
3.31
Hence,
I
α
≤
I
α
. Analogously,
I
α
≤I
α
.
Further, 3.19 implies
Ba,s
w
1−p
x
Ba,s
c
gydy
r
xy
−1
Q−α
dx
p
≤ λ
p
Ba,s
c
gx
q
v
1−q
xdx
p
/q
. 3.32
Now, taking gxχ
Ba,s
c
xrxa
−1
Q−α1−q
vx in the last inequality we conclude
that I
α
κ
≥ 1/2c
o
Q−α
A
3
.
Theorem 3.5 follows in the same manner as Theorem 3.4 was obtained. We only need
to use Lemma H.
14 Journal of Inequalities and Applications
3.3. Truncated potentials
This subsection is devoted to the two-sided estimates of the essential norm for the operator:
T
α
fx
Be,2rx
fy
rxy
−1
Q−α
,x∈ G. 3.33
A necessary and sufficient condition guaranteeing the trace inequality for T
α
in Euclidean
spaces was established in 37. This result was generalized in 38, 10, Chapter 6,forthe
spaces of homogeneous type. From the latter result as a corollary, we have the following
proposition.
Proposition B. Let 1 <p≤ q<∞ and let α>Q/p. Then,
i T
α
is bounded from L
p
G to L
q
v
G if and only if
B : sup
t>0
Bt : sup
t>0
rx>t
vxrx
α−Qq
dx
1/q
t
Q/p
< ∞; 3.34
ii T
α
is compact from L
p
G to L
q
v
G if and only if
lim
t → 0
Bt lim
t →∞
Bt0. 3.35
Theorem 3.7. Let 1 <p≤ q<∞ and let 0 <α<Q.Suppose that T
α
is bounded from L
p
w
G to
L
q
v
G. Then, the inequality
T
α
κL
p
w
G → L
q
v
G
≥ C
Q,α
lim
a → 0
A
a
lim
b →∞
A
b
3.36
holds, where
C
Q,α
2c
o
α−Q
,
A
a
sup
0<t<a
Be,a\Be,t
vxrx
α−Qq
dx
1/q
Be,t
w
1−p
xdx
1/p
,
A
b
sup
t>b
Be,t
c
vxrx
α−Qq
dx
1/q
Be,t\Be,b
w
1−p
xdx
1/p
.
3.37
To prove Theorem 3.7 we need the following lemma.
Lemma 3.8. Let p, q, and α satisfy the conditions of Theorem 3.7 Then from the boundedness of T
α
from L
p
w
G to L
q
v
G, it follows that w
1−p
is locally integrable on G.
Proof. Let
It
Be,t
w
1−p
xdx ∞ 3.38
M. Asif and A. Meskhi 15
for some t>0. Then, there exists g ∈ L
p
Be, t such that
Be,t
gw
−1/p
∞. Let us assume
that f
t
ygyw
−1/p
yχ
Be,t
y. Then, we have
T
α
f
t
L
q
v
G
≥
χ
Be,t
c
T
α
f
t
L
q
v
G
≥ c
Be,t
c
vxrx
α−Qq
dx
1/q
Be,t
gyw
−1/p
ydy ∞.
3.39
On the other hand,
f
t
L
p
w
G
Be,t
g
p
xdx < ∞. 3.40
Finally, we conclude that It < ∞ for all t, t > 0.
Proof of Theorem 3.7. Let λ>T
α
κL
p
w
G,L
q
v
G
. Then by Lemma E, there exists a positive
constant β such that for all τ
1
,τ
2
, 0 <τ
1
<τ
2
<βand f, supp f ⊂ Be, τ
1
, the inequality
T
α
f
L
q
v
Be,τ
2
\Be,τ
1
≤ λf
L
p
w
Be,τ
1
3.41
holds. Observe that if rx >τ
1
and ry <τ
1
, then rxy
−1
≤ 2c
o
rx. Consequently, taking
f w
1−p
χ
Be,τ
1
and using Lemma 3.8,wefindthat
1
2c
o
Q−α
Be,τ
2
\Be.τ
1
vx
rx
α−Qq
dx
1/q
Be,τ
1
w
1−p
xdx
1/p
≤ λ, 3.42
from which it follows that
1
2c
o
Q−αq
lim
a → 0
A
a
≤ λ. 3.43
Further, by virtue of Lemma E there exists β
2
such that for all s
1
,s
2
with β
2
<s
1
<s
2
the
inequality
T
α
f
L
q
v
Be,s
2
c
≤ λf
L
p
w
Be,s
2
\Be,s
1
3.44
holds, where supp f ⊂ Be, s
2
\ Be, s
1
. Hence by Lemma 3.8,wefindthat
1
2c
o
Q−α
Be,s
2
c
vx
rx
α−Qq
dx
1/q
Be,s
2
\Be,s
1
w
1−p
xdx
1/p
≤ λ, 3.45
which leads us to
1
2c
o
Q−α
lim
b → 0
A
b
≤ λ. 3.46
Thus, we have the desired result.
16 Journal of Inequalities and Applications
Theorem 3.9. Let 1 <p≤ q<∞ and let Q/p < α < Q. Suppose that 3.34 holds. Then, there is a
positive constant C such that
T
α
κL
p
G → L
q
v
G
≤ C
lim
a → 0
B
a
lim
b → 0
B
b
, 3.47
where
B
a
sup
t≤a
Be,a\Be,r
vxrx
α−Qq
dx
1/q
r
Q/p
,
B
b
sup
t≥b
Be,t
c
vxrx
α−Qq
dx
1/q
r
Q
− b
Q
1/p
.
3.48
Proof. Let 0 <a<b<∞ and represent T
α
f as follows:
T
α
f χ
Be,a
T
α
fχ
Be,a
χ
Be,b\Be,a
T
α
fχ
Be,b
χ
G\Be,b
T
α
fχ
Be,b/2c
0
χ
G\Be,b
T
α
fχ
G\Be,b/2c
0
≡ P
1
f P
2
f P
3
f P
4
f.
3.49
For P
2
, we have
P
2
fx
G
kx, ydy, 3.50
where kx, yχ
Be,b\Be,a
xχ
Be,2rx
yrxy
−1
α−Q
.
Further observe that
G
G
kx, y
p
dy
q/p
vxdx
Be,b\Be,a
Be,2rx
r
xy
−1
α−Qp
dy
q/p
vxdx
≤ c
Be,b\Be,a
Be,rx/2c
0
r
xy
−1
α−Qp
dy
q/p
vxdx
≤ c
Be,b\Be,a
rx
α−Qqq/p
vxdx < ∞.
3.51
Hence by Lemma I, we conclude that P
2
is compact for every a and b. Now we observe
that if rx >band ry <b/2c
o
, then rx ≤ 2c
o
rxy
−1
. Due to Proposition A we have that
P
3
is compact.
Further, we know that see 38, 10, Chapter 6
P
1
≤ C
1
B
a
,
P
4
≤ C
2
B
b/2c
o
, 3.52
where the constants C
1
and C
2
depend only on p, q, Q,andα.
Therefore,
T
α
− P
2
− P
3
≤
P
1
P
4
≤ c
B
a
B
b
. 3.53
The last inequality completes the proof.
M. Asif and A. Meskhi 17
Theorem 3.10. Let p and q satisfy the conditions of Theorem 3.9. Suppose that 3.18 holds. Then,
one has the following two-sided estimate:
c
2
lim
a → 0
B
a
lim
b →∞
B
b
≤
T
α
κL
p
G,L
q
v
G
≤ c
1
lim
a → 0
B
a
lim
b →∞
B
b
3.54
for some positive constants c
1
and c
2
depending only on Q, α, p, and q.
Theorem 3.10 follows immediately from Theorems 3.7 and 3.9.
3.4. Partial sums of Fourier series
Here, we investigate the lower estimate of the essential norm for the partial sums of the
Fourier series:
S
n
fx
1
π
π
−π
ftD
n
tdt, 3.55
where D
n
1/2
n
k1
cos kt.
One-weighted inequalities for S
n
were obtained in 32see also 25, Chapter 6. For
basic properties of S
n
in unweighted case; see, for example, 39.
Theorem 3.11. Let 1 <p<∞. Then, there is no n ∈ N and weight pair w, v on T :−π, π such
that S
n
is compact from L
p
w
T to L
p
v
T. Moreover, if S
n
is bounded from L
p
w
T to L
p
v
T,then
S
n
≥
2 2
1/2
1/2
2π
sup
a∈T
lim
r → 0
1
2r
ar
a−r
v
1/p
1
2r
ar
a−r
w
1−p
1/p
, 3.56
where I a − r, a r.
Proof. Taking λ>S
n
κL
p
w
T,L
p
v
T
, by Lemma H we find that
I
vx
S
n
fx
p
dx ≤ λ
p
I
fx
p
wxdx 3.57
for all intervals I a − r, a r, where r is a small positive number.
Let
J
1
I
vx
S
n
fx
p
dx, J
2
I
fx
p
wxdx. 3.58
Suppose that |I|≤π/4, and let n be the greatest integer less than or equal to π/4|I|. Then for
x ∈ I see 32,
S
n
fx
≥
1
π
I
fθ
sin3π/8
π/4n
dθ. 3.59
Using this estimate and taking f : w
1−p
xχ
I
x, we find that
J
1
≥
1
π
sin
3π
8
p
|I|
−p
I
v
I
w
1−p
p
. 3.60
18 Journal of Inequalities and Applications
On the other hand, it is easy to see that J
2
I
w
1−p
< ∞.
Hence, by 3.57 we conclude that
λ ≥
1
π
sin
3π
8
1
|I|
I
v
1/p
1
|I|
I
w
1−p
1/p
. 3.61
Now passing r to 0, taking supremum over a ∈ T, and using the fact sin3π/8
2 2
1/2
1/2
/2, we find that 3.56 holds.
Corollary 3.12. Let 1 <p<∞ and let n ∈ N.Then
S
n
κL
p
T
≥
2 2
1/2
1/2
2π
. 3.62
Corollary 3.13. Let 1 <p<∞ and let n ∈ N. Suppose t hat wxvx|x|
α
. Then, one has
S
n
κL
p
w
T
≥
2 2
1/2
1/2
2π
1
α 1
1/p
1
α
1 − p
1
1/p
. 3.63
Acknowledgments
The authors express their gratitude to the referees for their valuable remarks and suggestions.
The second author was partially supported by the INTAS Grant no. 05-1000008-8157 and the
Georgian National Science Foundation Grant no. GNSF/ST07/3-169.
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