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 Asif1 and Alexander Meskhi2
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, alex72meskhi@yahoo.com
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
Rn 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 6–9 see 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 17 for 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 23 see 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 25 see
also 10, 26 for related topics.
Earlier, the trace inequality for the Riesz potentials boundedness of Riesz potentials
q
from Lp to Lv was established in 27, 28. The two-weight criteria for fractional maximal
functions were obtained in 22, 29, 30 see also 25 for more general case.
Necessary and sufficient conditions guaranteeing the compactness of the Riesz
potentials have been derived in 31 see 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 δt 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 o δt o exp−1 , t > 0, are automorphisms in G, which will be again denoted by
δt . The number Q tr A is the homogeneous dimension of G. The 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 rx rx−1 for every x ∈ G;
ii rδt x trx for every x ∈ G and t > 0;
iii rx 0 if and only if x e;
iv there exists co > 0 such that
rxy ≤ co rx ry ,
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, that is,
Ba, ρ : y ∈ G; r ay−1 ≤ ρ .
Ba, ρ : y ∈ G; r ay−1 < ρ ,
2.2
It can be observed that δρ Be, 1 Be, ρ.
Let us fix a Haar measure |·| in G such that |Be, 1| 1. Then, |δt E| tQ |E|. In
particular, |Bx, t| tQ for x ∈ G, t > 0.
Examples of homogeneous groups are the Euclidean n-dimensional space Rn , 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 : rx 1}. There is a (unique)
Radon measure σ on S such that for all u ∈ L1 G,
∞
uxdx u δt y tQ−1 dσydt.
2.3
G
0
S
For the details see, for example, 33, page 14.
We call a weight a locally integrable almost everywhere positive function on G. Denote
p
by Lw G 1 < p < ∞ the weighted Lebesgue space, which is the space of all measurable
functions f : G → C with the norm
1/p
p
p
fx wxdx
< ∞.
2.4
fLw G G
p
L1 G
If w ≡ 1, then we denote
by Lp G.
p
Let X Lw 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 ≤ Ff Fg 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 → Lq G, then, the norm of the operator T is defined
as follows:
2.5
T sup T fLq G : fX ≤ 1 .
Moreover, T is order preserving if T fx ≥ T gx 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, T fx ≥ 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 . If X 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:
k−1
2 bi εUY for some b1 , . . . , b2k−1 ∈ Y ,
2.7
ek T inf ε > 0 : T UX ⊂
j1
where UX and UY 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
limn → ∞ en T .
In the sequel, we assume that X is a Banach space which is a certain subset of all Haarmeasurable functions on G. We denote by SX the class of all bounded sublinear functionals
defined on X, that is,
2.8
SX F : X → R, F-sublinear and F sup Fx < ∞ .
x≤1
4
Journal of Inequalities and Applications
Let M be the set of all bounded functionals F defined on X with the following
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 Lp G:
FL X, L G :
p
T : T fx m
αj fuj , m ∈ N, uj ≥ 0, uj ∈ Lp G,
j1
uj are linearly independent and αj ∈ X
FS X, L G :
p
T : T fx m
∗
M ,
2.10
βj fuj , m ∈ N, uj ≥ 0, uj ∈ L G,
p
j1
uj are linearly independent and βj ∈ SX
M .
If X Lp G, we will denote these classes by FL Lp G and FS Lp G, respectively. It is clear
that if P ∈ FL X, Lp G resp., P ∈ FS X, Lp G, then P is compact linear resp., compact
sublinear from X to Lp G.
We will use the symbol αT for the distance between the operator T : X → Lp G and
the class FS X, Lp G, that is,
αT : dist T, FS X, Lp G .
2.11
For any bounded subset A of Lp G 1 < p < ∞, let
ΦA : inf δ > 0 : A can be covered by finitely many open balls in Lp G of radius δ ,
sup f − P fLp G : f ∈ A .
ΨA :
infp
P ∈FL L G
2.12
We will need a statement similar to Theorem V.5.1 of Chapter V of 35 for Euclidean
spaces see 2.
Theorem A. For any bounded subset K ⊂ Lp G 1 ≤ p < ∞, the inequality
2ΦK ≥ ΨK
2.13
holds.
Proof. Let ε > ΦK. Then, there are g1 , g2 , . . . , gN ∈ Lp G such that for all f ∈ K and some
i ∈ {1, 2, . . . , N},
f − gi p < ε.
L G
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},
1/p
1
gi xp dx
2.15
< δ.
2
G\B
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
{B1 , B2 , . . . , Bn }. Further, let us assume that {E1 , E2 , . . . , En } is a family of pairwise disjoint
sets of positive measure such that B ni1 Ei and Ei ⊂ Bi we can assume that E1 B1 ∩ B,
E2 B2 \ B1 ∩ B, . . . , Ek Bk \ k−1
i1 Bi ∩ B, . . .. We define
n
−1
fEi χEi x, fEi Ei fxdx.
2.16
P fx Ei
i1
Then,
gi − P gi p
p
L B
n 1
p
gi x − gi y dy dx
Ej
j1 Ej Ej
m ≤
j1
1
Ej Ej
≤
gi x − gi yp dy dx
2.17
Ej
gi x − gi zxp dx −→ 0
sup
rz≤2co h
B
as h → 0. The latter fact follows from the continuity of the norm Lp G see, e.g., 33, page
19.
From this and 2.14, we find that
gi − P gi p < δ, i 1, 2, 3, . . . , N,
2.18
L G
when h is sufficiently small. Further,
p
P fLp G p
n −1
dx
Ej fydy
Ej
j1
Ej
n Ej −1 fyp dy dx
≤
j1 Ej
Ej
2.19
p
≤ f
Lp B
p
≤ fLp G .
It is also clear that the functionals f → fEi belong to Lp G∗ ∩ M. Hence, P ∈
FL L G. Finally, 2.14–2.15 and 2.18 yield
f − P fLp G ≤ f − gi Lp G gi − P gi Lp G P gi − f Lp G
2.20
< ε δ gi − f Lp G ≤ 2ε δ.
p
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 ⊂ Lp G is compact. Then for any given ε > 0,
there exist an operator Pε ∈ FL Lp G such that for all f ∈ K,
f − Pε f p ≤ ε .
L G
2.21
Proof. Let K be a compact set in Lp G. Using Theorem A, we see that ΨK 0. Hence for
ε > 0, there exists Pε ∈ FL Lp G such that
sup f − Pε f Lp G : f ∈ K ≤ ε.
2.22
Lemma B. Let T : X → Lp G be compact, order-preserving, and sublinear operator, where 1 ≤ p <
∞. Then, αT 0.
Proof. Let UX {f : fX ≤ 1}. From the compactness of T , it follows that T UX is relatively
compact in Lp G. Using Lemma A, we have that for any given ε > 0 there exists an operator
Pε ∈ FL Lp G such that for all f ∈ UX ,
T f − Pε T f p ≤ ε.
L G
2.23
Let Pε Pε ◦ T. Then, Pε ∈ FS X, Lp G. Indeed, there exist functionals αj ∈ X ∗ ∩ M, j ∈
{1, 2, . . . , m}, and linearly independent functions uj ∈ Lp G, j ∈ {1, 2, . . . , m}, such that
Pε fx Pε T fx m
αj T fuj x j1
m
βj fuj x,
2.24
j1
where βj αj ◦ T belongs to SX ∩ M. Since by 2.23,
T f − Pε f p ≤ ε
L G
2.25
for all f ∈ UX , it follows immediately that αT 0.
We will also need the following lemma.
Lemma C. Let T be a bounded, order-preserving, and sublinear operator from X to Lq G, where
1 ≤ q < ∞. Then,
T κ αT .
2.26
Proof. Let δ > 0. Then, there exists an operator K ∈ KX, Lq G, such that T − K ≤ T κ δ.
By Lemma B there is P ∈ FS X, Lq G for which the inequality K − P < δ holds. 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 ∈ FS X, Lq G. Then for every a ∈ G and ε > 0, there exist an
operator R ∈ FS X, Lq G and positive numbers α, α such that for all f ∈ X, the inequality
P − Rf q ≤ εfX
L G
2.29
holds and supp Rf ⊂ Ba, α \ Ba, α.
Proof. There exist linearly independent nonnegative functions uj ∈ Lq G, j ∈ {1, 2, . . . , N},
such that
P fx N
βj fuj x,
f ∈ X,
2.30
j1
where βj are bounded, order-preserving, sublinear functionals βj : X → R. On the other hand,
there is a positive constant c for which
N βj f ≤ cfX .
2.31
j1
Let us choose linearly independent Φj ∈ Lq G and positive real numbers αj , αj such
that
uj − Φj q < ε,
L G
j ∈ {1, 2, . . . , N}
2.32
and supp Φj ⊂ Ba, αj \ Ba, αj . If
Rfx N
βj f Φj x,
2.33
j1
then it is obvious that R ∈ FS X, Lq G and moreover,
P f − RfLq G ≤
N βj fuj − Φj q ≤ cεfX
L G
2.34
j1
for all f ∈ X. Besides this, supp Rf ⊂ Ba, α \ Ba, α, where α min{αj } and α max{αj }.
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
p
q
Lw G to Lv G. Suppose that λ > T κLpw G,Lqv G , and a is a point of G. Then, there exist constants
β1 , β2 , 0 < β1 < β2 < ∞, such that for all τ and r with r > β2 , τ < β1 , the following inequalities hold:
T fLqv Ba,τ ≤ λfLpw G ,
T fLqv Ba,rc ≤ λfLpw G ,
p
where f ∈ Lw G.
2.35
8
Journal of Inequalities and Applications
p
q
Proof. Let T be bounded from Lw G to Lv G. Let T v be the operator given by
T v f v1/q T f.
2.36
v T p
T κLpw G → Lqv G .
κLw G → Lq G
2.37
Then, it is easy to see that
By Lemma C, we have that
λ > α T v .
2.38
p
Consequently, there exists P ∈ FS Lw G, Lq G such that
v
T − P < λ.
2.39
Fix a ∈ G. According to Lemma D, there are positive constants β1 and β2 , β1 < β2 , and R ∈
p
q
FS Lw G, Lv G for which
λ − T v − P P − R ≤
2.40
2
p
and supp Rf ⊂ Ba, β2 \ Ba, β1 for all f ∈ Lw G. Hence,
v
T − R < λ.
2.41
From the last inequality, it follows that if 0 < τ < β1 and r > β2 , then 2.35 holds for f,
p
f ∈ Lw G.
The following lemmas are taken from 2 for the linear case see 35.
Lemma F. Let Ω be a domain in Rn , and let T be a bounded, order-preserving, and sublinear operator
from Lrw Ω to Lp Ω, where 1 < r, p < ∞, and w is a weight function on Ω. Then,
T κLrw Ω,Lp Ω αT .
2.42
Lemma G. Let Ω be a domain in Rn and let P ∈ FS X, Lp Ω, where X Lrw Ω and 1 < r, p < ∞.
Then for every a ∈ Ω and ε > 0, there exist an operator R ∈ FS X, Lp Ω and positive numbers β1
and β2 , β1 < β2 such that for all f ∈ X, the inequality
P − Rf p
≤ εfX
2.43
L Ω
holds and supp Rf ⊂ Da, β2 \ Da, β1 , where Da, 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 Rn . Suppose that 1 < p, q < ∞, and that T is bounded, orderp
q
preserving, and sublinear operator from Lw Ω to Lv Ω. Assume that λ > T κLpw Ω,Lqv Ω and
a ∈ Ω. Then, there exist constants β1 , β2 , 0 < β1 < β2 < ∞ such that for all τ and r with r > β2 ,
τ < β1 , the following inequalities hold:
T fLqv Ba,τ ≤ λfLpw Ω ;
p
wheref ∈ Lw Ω.
T fLqv Ω\Ba,r ≤ λfLpw Ω ,
2.44
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
p
,
2.45
< ∞, p kx, yLp Y q
ν
Lμ X
p−1
then the operator
Kfx kx, yfydνy,
x ∈ X,
2.46
Y
p
q
is compact from Lν Y into Lμ X.
3. Main results
3.1. Maximal functions
Let G be a homogeneous group and let
Mα fx sup
Bx
1
|B|1−α/Q
fydy,
x ∈ G, 0 ≤ α < Q,
3.1
B
where the supremum is taken over all balls B containing x. If α 0, then Mα 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, for α > 0 that if
p
q
1 < p < ∞ and 0 ≤ α < Q/p, then the operator Mα is bounded from Lρp G to Lρq G, where
q Qp/Q − αp, if and only if ρ ∈ Ap,q G, that is,
1/q 1/p
1
1
q
−p
sup
ρ
ρ
< ∞.
3.2
|B| B
|B| B
B
Now, we formulate the main results of this subsection.
p
Theorem 3.1. Let 1 < p < ∞. Suppose that the maximal operator M is bounded from Lw G to
p
p
p
Lv G. Then, there is no weight pair v, w such that M is compact from Lw G to Lv G. Moreover,
the inequality
1/p 1/p
1
vxdx
w1−p xdx
3.3
MκLpw G,Lpv G ≥ sup lim Ba,τ
Ba,τ
a∈G τ → 0 Ba, τ
holds.
Proof. Suppose that λ > MκLpw → Lpv and a ∈ G. By Lemma E, we have that
p
1
p
fxp wxdx
fy dy dx ≤ λ
vx sup Ba,τ
Ba,τ
Ba,τ
B x Ba, τ
3.4
for all τ τ ≤ β and all f supported in Ba, τ. Substituting fy χBa,r y w1−p y in the
latter inequality and taking into account that Ba,τ w1−p xdx < ∞ see, e.g., 17, 18, 25,
Chapter 4 for all τ > 0 we find that
p−1
1
vxdx
w1−p xdx
≤ λp .
3.5
p
Ba, τ
Ba,τ
Ba,τ
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/p and let q Qp/Q − αp. Suppose that Mα is bounded
p
q
p
from Lw G to Lv G. Then, there is no weight pair v, w such that Mα is compact from Lw G to
q
Lv G. Moreover, the inequality
1
Mα ≥ sup lim κ
τ
→
0
Ba, τα/Q−1
a∈G
1/q vxdx
w
Ba,τ
1−p
xdx
1/p
3.6
Ba,τ
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 < ∞, vx wx rxγ , where −Q < γ < p − 1Q. Then,
1/p −1
1/p MκLpw G ≥ Q γ Q
.
γ 1 − p Q
3.7
Indeed, first observe that the fact |Be, 1| 1 and Proposition A implies σS Q,
where S is the unit sphere in G and σS is its measure. By Theorem 3.1 and Proposition A,
we have
1
MκLpw G ≥ lim τ → 0 Be, τ
σSlim τ −Q
Be,τ
τ
τ →0
1/p w1−p xdx
wxdx
1/p
Be,τ
1/p τ
tγQ−1 dt
0
1/p
3.8
tγ1−p Q−1 dt
0
1/p −1
Q γ Q1/p γ 1 − p Q
.
3.2. Riesz potentials
Let G be a homogeneous group and let
Iα fx fy
Q−α dy,
G r xy −1
3.9
0 < α < Q,
be the Riesz potential operator. It is well known see 33, Chapter 6 that Iα is bounded from
Lp G to Lq G, 1 < p, q < ∞, if and only if q Qp/Q − αp.
p
q
Theorem 3.4. Let 1 < p ≤ q < ∞, 0 < α < Q. Let Iα be bounded from Lw G to Lv G. Then, the
following inequality holds
Iα ≥ Cα,Q max A1 , A2 , A3 ,
κ
3.10
M. Asif and A. Meskhi
11
where
Cα,Q 1
Q−α ,
2co
1/q A1 sup lim r α−Q
α∈G r → 0
Ba,r
1/q A2 sup lim
vxdx
A3 sup lim
w
a∈G r → 0
1−p
,
xdx
α−Qp 1−p
r ay−1
w ydy
Ba,r
Ba,r
1/p
Ba,r
a∈G r → 0
w1−p xdx
vxdx
c
1/p Ba,rc
Ba,r
α−Qq
r ay−1
vydy
3.11
1/p
,
1/q
.
(co is the constant from the triangle inequality for the homogeneous norms.)
The next statement is formulated for the Riesz potentials defined on domains in Rn :
JΩ,α fx Ω
fy|x − y|α−n dy,
x ∈ Ω.
3.12
Theorem 3.5. Let Ω ⊆ Rn be a domain in Rn . Let 1 < p ≤ q < ∞. If JΩ,α is bounded from
p
q
Lw Ω to Lv Ω, then one has
JΩ,α κ ≥ 2α−n B1 ,
3.13
where
B1 sup lim r
1/q α−n
a∈Ω r → 0
v
w
Ba,r
1−p
1/p
3.14
.
Ba,r
In particular, if Ω ≡ Rn , then
JΩ,α κ ≥ 2α−n max B2 , B3 ,
3.15
where
1/q B2 sup lim
a∈Rn r → 0
vxdx
B3 sup lim
a∈Rn
r →0
Rn \Ba,r
Ba,r
w1−p xdx
1/p Ba,r
|a − y|α−np w1−p ydy
R \Ba,r
n
1/p
|a − y|α−nq vydy
,
3.16
1/q
.
Corollary 3.6. Let 1 < p < ∞, 1 < p < Q/α, q pQ/Q − αp, then there is no weight pair v, w
p
q
p
q
for which Iα is compact from Lw G to Lv G. Moreover, if Iα is bounded from Lw G to Lv G, then
Iα κ ≥ Cα,Q A1 ,
where Cα,Q and A1 are defined in Theorem 3.4.
3.17
12
Journal of Inequalities and Applications
Proof of Theorem 3.4. By Lemma E, we have that for λ > Iα κLpw G,Lqv G and a ∈ G, there are
positive constants β1 and β2 β1 < β2 such that for all τ, s τ < β1 , s > β2 ,
q
vxIα fx dx ≤ λq
Ba,τ
fxp wxdx
q/p
3.18
G
p
for f ∈ Lw G, and
Ba,sc
q
vxIα fx dx ≤ λq
|fx|p wxdx
q/p
3.19
Ba,s
for supp f ⊂ Ba, s.
Now taking fx χBa,r xw1−p x in 3.18 and observing that Ba,r w1−p xdx <
∞ for all r > 0 see also 25, Chapter 3, we find that
vx
Ba,r
w1−p y
Q−α dy
Ba,r r xy −1
q
dx ≤ λq
w1−p xdx
q/p
< ∞.
3.20
Ba,r
Further if x, y ∈ Ba, τ, then
r xy−1 ≤ co r xa−1 r ay−1 ≤ 2co τ.
3.21
Iα ≥ Cα,Q A1 .
κ
3.22
Hence,
If fx χBa,τc xw1−p x/ray−1 Q−αp −1
w1−p ydy
Q−αp −1
Q−α Ba,τc r xy −1
r ay−1
vx
Ba,τ
q
, then
dx ≤ λ
q
w1−p xdx
Q−αp
Ba,τc r ay −1
q/p
< ∞.
3.23
Let rxa−1 < τ and rya−1 > τ. Then,
r xy−1 ≤ co r xa−1 r ay−1 ≤ co τ r ay−1 ≤ 2co r ay−1 .
3.24
Hence, by 3.18 we have
2co
1
qQ−α
vxdx
Ba,τ
w1−p ydy
Q−αp
Ba,τc r ay −1
q
≤ λq
w1−p xdx
Q−αp
Ba,τc r ay −1
q/p
.
3.25
The latter inequality implies
Iα ≥ κ
1
Q−α A2 .
2co
3.26
M. Asif and A. Meskhi
13
Further, observe that 3.19 means that the norm of the operator
I α fx fydy
3.27
Q−α
Ba,s ry −1 a
can be estimated as follows:
I α p
q
≤ λ.
Lw Ba,s → Lv Ba,sc 3.28
Now by duality, we find that
I α p
q
Iα Lq
Lw Ba,s → Lv Ba,sc v1−q
p
Ba,sc → L
w1−p
Ba,s
3.29
,
where
Iα gy c
Ba,s
gxdx
−1 Q−α .
r xy
3.30
Indeed, by Fubini’s theorem and Hölder’s inequality, we have
I α f q
≤
Lv Ba,sc sup
g
Ba,sc
≤1
q
Lv Ba,sc gx I α fx dx
≤
fy Iα |g| y dy
sup
g
L
q
v1−q
Ba,sc ≤1
Ba,s
≤
|f|p w
sup
g
L
q
v1−q
Ba,sc ≤1
≤ Iα 1/p Ba,s
|f|p w
p 1−p
w
Iα |g|
1/p
3.31
Ba,s
1/p
.
Ba,s
Hence, I α ≤ Iα . Analogously, Iα ≤ I α .
Further, 3.19 implies
w
Ba,s
1−p
x
p p /q
q 1−q
gydy
p
gx v xdx
.
Q−α dx ≤ λ
Ba,sc r xy −1
Ba,sc
Q−α1−q
Now, taking gx χBa,sc xrxa−1 that Iα κ ≥ 1/2co Q−α A3 .
3.32
vx in the last inequality we conclude
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:
Tα fx fy
Q−α
Be,2rx rxy −1 x ∈ G.
,
3.33
A necessary and sufficient condition guaranteeing the trace inequality for Tα in Euclidean
spaces was established in 37. This result was generalized in 38, 10, Chapter 6, for the
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,
q
i Tα is bounded from Lp G to Lv G if and only if
α−Qq
B : supBt : sup
t>0
vxrx
t>0
1/q
dx
tQ/p < ∞;
3.34
rx>t
q
ii Tα is compact from Lp G to Lv G if and only if
limBt lim Bt 0.
3.35
t→∞
t→0
p
Theorem 3.7. Let 1 < p ≤ q < ∞ and let 0 < α < Q. Suppose that Tα is bounded from Lw G to
q
Lv G. Then, the inequality
a
Tα p
q
lim
≥
C
A
lim
A
Q,α
b
κLw G → Lv G
a→0
3.36
b→∞
holds, where
α−Q
CQ,α 2co
,
a
A
sup
vxrx
0<t<a
t>b
1/q dx
w
Be,a\Be,t
Ab sup
α−Qq
Be,tc
1−p
xdx
1/p
,
Be,t
vxrxα−Qq dx
1/q w1−p xdx
3.37
1/p
.
Be,t\Be,b
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 Tα
p
q
from Lw G to Lv G, it follows that w1−p is locally integrable on G.
Proof. Let
w1−p xdx ∞
It Be,t
3.38
M. Asif and A. Meskhi
15
for some t > 0. Then, there exists g ∈ Lp Be, t such that
that ft y gyw−1/p yχBe,t y. Then, we have
Be,t
gw−1/p ∞. Let us assume
Tα ft q
≥ χBe,tc Tα ft Lqv G
Lv G
≥c
α−Qq
Be,tc
vxrx
1/q gyw−1/p ydy ∞.
dx
3.39
Be,t
On the other hand,
ft p
Lw G
g p xdx < ∞.
3.40
Be,t
Finally, we conclude that It < ∞ for all t, t > 0.
Proof of Theorem 3.7. Let λ > Tα κLpw G,Lqv G . Then by Lemma E, there exists a positive
constant β such that for all τ1 , τ2 , 0 < τ1 < τ2 < β and f, supp f ⊂ Be, τ1 , the inequality
Tα f q
≤ λfLpw Be,τ1 Lv Be,τ2 \Be,τ1 3.41
holds. Observe that if rx > τ1 and ry < τ1 , then rxy−1 ≤ 2co rx. Consequently, taking
f w1−p χBe,τ1 and using Lemma 3.8, we find that
1
2co Q−α
Be,τ2 \Be.τ1 α−Qq
vx rx
dx
1/q Be,τ1 w
1−p
xdx
1/p
≤ λ,
3.42
from which it follows that
2co
1
Aa ≤ λ.
Q−αq alim
→0
3.43
Further, by virtue of Lemma E there exists β2 such that for all s1 , s2 with β2 < s1 < s2 the
inequality
Tα f q
≤ λfLpw Be,s2 \Be,s1 Lv Be,s2 c 3.44
holds, where supp f ⊂ Be, s2 \ Be, s1 . Hence by Lemma 3.8, we find that
1
Q−α
2co
Be,s2 c
α−Qq
vx rx
dx
1/q Be,s2 \Be,s1 w
1−p
xdx
1/p
≤ λ,
3.45
which leads us to
1
Ab ≤ λ.
Q−α blim
→0
2co
Thus, we have the desired result.
3.46
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
a
Tα p
q
,
3.47
≤
C
lim
B
lim
B
b
κL G → Lv G
a→0
where
Ba sup
t≤a
vxrxα−Qq dx
1/q
r Q/p ,
Be,a\Be,r
Bb sup
t≥b
b→0
c
vxrxα−Qq dx
1/q
rQ − b
Q 1/p
3.48
.
Be,t
Proof. Let 0 < a < b < ∞ and represent Tα f as follows:
Tα f χBe,a Tα fχBe,a χBe,b\Be,a Tα fχBe,b
χG\Be,b Tα fχBe,b/2c0 χG\Be,b Tα fχG\Be,b/2c0 3.49
≡ P1 f P2 f P3 f P4 f.
For P2 , we have
P2 fx 3.50
kx, ydy,
G
α−Q
where kx, y χBe,b\Be,a xχBe,2rx yrxy−1 Further observe that
q/p
p vxdx kx, y dy
G
G
Be,b\Be,a
≤c
Be,b\Be,a
≤c
.
−1 α−Qp
dy
r xy
q/p
vxdx
Be,2rx
Be,rx/2c0 −1 α−Qp
dy
r xy
q/p
vxdx
rxα−Qqq/p vxdx < ∞.
Be,b\Be,a
3.51
Hence by Lemma I, we conclude that P2 is compact for every a and b. Now we observe
that if rx > b and ry < b/2co , then rx ≤ 2co rxy−1 . Due to Proposition A we have that
P3 is compact.
Further, we know that see 38, 10, Chapter 6
P4 ≤ C2 Bb/2c ,
P1 ≤ C1 Ba ,
3.52
o
where the constants C1 and C2 depend only on p, q, Q, and α.
Therefore,
Tα − P2 − P3 ≤ P1 P4 ≤ c Ba Bb .
The last inequality completes the proof.
3.53
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:
c2 lim Ba lim Bb ≤ Tα κLp G,Lqv G ≤ c1 lim Ba lim Bb
a→0
a→0
b→∞
b→∞
3.54
for some positive constants c1 and c2 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:
1 π
ftDn tdt,
3.55
Sn fx π −π
where Dn 1/2 nk1 cos kt.
One-weighted inequalities for Sn were obtained in 32 see also 25, Chapter 6. For
basic properties of Sn 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
p
p
p
p
that Sn is compact from Lw T to Lv T . Moreover, if Sn is bounded from Lw T to Lv T , then
1/2
ar 1/p ar
1/p
2 21/2
1
1
1−p
Sn ≥
sup lim
v
w
,
2π
2r a−r
a∈T r → 0 2r a−r
3.56
where I a − r, a r.
Proof. Taking λ > Sn κLpw T ,Lpv T , by Lemma H we find that
p
vxSn fx dx ≤ λp
I
fxp wxdx
3.57
I
for all intervals I a − r, a r, where r is a small positive number.
Let
p
p
J1 vx Sn fx dx,
J2 fx wxdx.
I
3.58
I
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,
fθ sin3π/8
Sn fx ≥ 1
dθ.
3.59
π I
π/4n
Using this estimate and taking f : w1−p xχI x, we find that
J1 ≥
3π
1
sin
π
8
p
|I|
−p
p
1−p
v
w
.
I
I
3.60
18
Journal of Inequalities and Applications
On the other hand, it is easy to see that J2 Hence, by 3.57 we conclude that
λ≥
3π
1
sin
π
8
1
|I|
I
w1−p < ∞.
1/p 1/p
1
v
w1−p
.
|I| I
I
3.61
Now passing r to 0, taking supremum over a ∈ T, and using the fact sin3π/8 2 2 /2, we find that 3.56 holds.
1/2 1/2
Corollary 3.12. Let 1 < p < ∞ and let n ∈ N. Then
Sn κLp T 2 21/2
≥
2π
1/2
3.62
.
Corollary 3.13. Let 1 < p < ∞ and let n ∈ N. Suppose that wx vx |x|α . Then, one has
Sn p
κLw T 2 21/2
≥
2π
1/2 1
α1
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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