Báo cáo hóa học: "Research Article Weighted Estimates of a Measure of Noncompactness for Maximal and Potential Operators" pot

Volume 2008, Article ID 697407, 19 pagesdoi:10.1155/2008/697407 Research Article Weighted Estimates of a Measure of Noncompactness for Maximal and Potential Operators Muhammad Asif 1 and

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

Received 5 April 2008; Accepted 19 June 2008

Recommended by Siegfried Carl

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

Copyrightq 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 papers1 3, the measure of noncompactness essential norm of maximal functions, singular integrals, and identity operators acting in weighted Lebesgue spaces defined on

Rnwith 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 of2 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 in6 9 see also 10 For two-sided estimates of the essential norm for the Cauchy integrals see11–14 The same problem in the one-weighted setting has been studied

in15,16

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

see, e.g., 18 and for fractional maximal functions and Riesz potentials by Muckenhoupt and Wheeden19 Two-weight criteria for the Hardy-Littlewood maximal functions have been obtained in20 Necessary and sufficient conditions guaranteeing the boundedness of the Riesz potentials from one weighted Lebesgue space into another one were derived by Sawyer21,22 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 also10,26 for related topics

Earlier, the trace inequality for the Riesz potentialsboundedness 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 in22,29,30 see also 25 for more general case

Necessary and sufficient conditions guaranteeing the compactness of the Riesz potentials have been derived in31 see also 10, Section 5.2 The one-weight problem for the Hilbert transform and partial sums of the Fourier series was solved in32

The paper is organized as follows InSection 2, we give basic concepts and prove some lemmas.Section 3is divided into 4 parts.Section 3.1concerns maximal functions; potential operators are discussed in Sections3.2and3.3.Section 3.4is devoted to the partial sums of Fourier series

Constantsoften 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 δ 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 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

r xy ≤ cor x 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,

B a, ρ :y ∈ G; ray−1

< ρ

, B a, ρ :y ∈ G; ray−1

≤ ρ. 2.2

It can be observed that δ ρ B e, 1  Be, ρ.

Let us fix a Haar measure |·| in G such that |Be, 1|  1 Then, |δt 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 Rn, the Heisenberg group, upper triangular groups, and so forth For the definition and basic properties of the homogeneous group, we refer to33, page 12 and 25

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 ∈ L1G,



G

u xdx 

∞

0



S

u

δ t y

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 wG 1 < 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

f xp

w xdx 1/p < ∞. 2.4

If w ≡ 1, then we denote L p

1G by L p G.

Let X  L p wG1 < 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 → L q G, then, the norm of the operator T is defined

as follows:

T  supTfL q G:fX≤ 1. 2.5

Moreover, T is order preserving if Tfx ≥ Tgx almost everywhere for all nonnegative f and g with f x ≥ gx almost everywhere Further, if T is sublinear and order preserving, then obviously it is nonnegative, that is, Tf x ≥ 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 Tκ X,Y or simplyTκ and is defined as

whereKX, Y is the class of all compact sublinear operators from X to Y If X  Y, then we

use the symbolKX 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 kT  inf ε > 0 : T

U X

⊂2

k−1

j1



b i εUY for some b1, , b2k−1 ∈ Y

, 2.7

where UX and UY are the closed unit balls in X and Y, respectively It is well knownsee, e.g., 34, page 8 that the measure of noncompactness of T is greater than or equal to limn→ ∞e nT.

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, that is,

S X  F : X → R, F-sublinear and F  sup

x≤1

F x<∞. 2.8

Let M be the set of all bounded functionals F defined on X with the following

property:

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 : Tf x m

j1

α jfuj , m ∈ N, uj ≥ 0, uj ∈ L p G,

u j are linearly independent and α j ∈ X∗

M

,

F S



X, L p G: T : Tf x m

j1

β jfuj , m ∈ N, uj ≥ 0, uj ∈ L p G,

u j are linearly independent and β j ∈ SXM

.

2.10

If X  L p G, we will denote these classes by FLL p G and FSL p G, respectively It is clear that if P ∈ FLX, L p G resp., P ∈ FSX, 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 FSX, L p G, that is,

α T : distT, F S



For any bounded subset A of L p G 1 < p < ∞, let

ΦA : infδ > 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 − PfL p G : f ∈ A.

2.12

We will need a statement similar to Theorem V.5.1 of Chapter V of35 for Euclidean spaces see2

Theorem A For any bounded subset K ⊂ L p G 1 ≤ p < ∞, the inequality

holds.

Proof Let ε > ΦK Then, there are g1, g2, , g N ∈ L p G such that for all f ∈ K and some

i ∈ {1, 2, , N},

f − gi

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 ixp

dx

1/p

< 1

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, , B n} Further, let us assume that {E1, E2, , E n} is a family of pairwise disjoint sets of positive measure such that B n

i1E i and E i ⊂ Bi we can assume that E1  B1∩ B,

E2 B2\ B1 ∩ B, , Ek  Bk\k−1

i1B i ∩ B,  We define

P f x n

i1

f E i χ E i x, fE i E i−1

E i

Then,

g i − Pgip

L p Bn

j1



E j



E1

j



E j



g ix − giydy

p dx

≤m

j1



E j

1

E j



E j

g ix − giyp

dy dx

≤ sup

r z≤2c o h



B

g ix − gizxp dx−→ 0

2.17

as h → 0 The latter fact follows from the continuity of the norm LpG see, e.g., 33, page 19

From this and2.14, we find that

g i − Pgi

L p G < δ, i  1, 2, 3, , N, 2.18

when h is sufficiently small Further,

Pf p

L p Gn

j1



E j



E j−1

E j

f ydy

p dx

≤n

j1



E j

E j−1

E j

f yp

dy dx

≤ f p

L p B

≤ f p

L p G

2.19

It is also clear that the functionals f → fE i belong to L p G∗ ∩ M Hence, P ∈

F LL p G Finally, 2.14–2.15 and 2.18 yield

f − PfL p G≤f − gi

L p G g i − Pgi

L p G P

g i − f

L p G

< ε δ g i − f

Since δ is arbitrarily small, we have the desired result.

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 ∈ FLL p G such that for all f ∈ K,

f − Pε f

Proof Let K be a compact set in L p G Using Theorem A, we see that ΨK  0 Hence for

ε > 0, there exists P ε ∈ FLL p G such that

supf − Pε 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, αT  0.

Proof Let U X  {f : fX ≤ 1} From the compactness of T, it follows that TUX is relatively compact in L p G Using Lemma A, we have that for any given ε > 0 there exists an operator

P ε ∈ FLL p G such that for all f ∈ UX,

Tf − Pε Tf

Let P ε  Pε ◦ T Then,  P ε ∈ FSX, L p G Indeed, there exist functionals αj ∈ X∗∩ M, j ∈ {1, 2, , m}, and linearly independent functions uj ∈ L p G, j ∈ {1, 2, , m}, such that



P ε f x  PεTfx m

j1

α jTfujx m

j1

β jfujx, 2.24

where βj  αj ◦ T belongs to SX ∩ M Since by 2.23,

Tf− P ε f

for all f ∈ UX , it follows immediately that α T  0.

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,

Proof Let δ > 0 Then, there exists an operator K ∈ KX, L q G, such that T −K ≤ Tκ δ.

By Lemma B there is P ∈ FSX, L q G for which the inequality K − P < δ holds This gives

T − P ≤ T − K K − P ≤ Tκ 2δ. 2.27

Hence, αT ≤ Tκ Moreover, it is obvious that

Lemma D Let 1 ≤ q < ∞ and let P ∈ FSX, L q G Then for every a ∈ G and ε > 0, there exist an operator R ∈ FSX, L q G and positive numbers α, α such that for all f ∈ X, the inequality

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

P f x N

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



j1

Let us choose linearly independentΦj ∈ L q G and positive real numbers αj , α jsuch that

u j− Φj

L q G < ε, j ∈ {1, 2, , N} 2.32 and suppΦj⊂ Ba, αj \ Ba, αj If

Rf x N

j1

then it is obvious that R ∈ FSX, L q G and moreover,

Pf − RfL q G≤N

j1

β jfu j− Φj

L q G ≤ cεfX 2.34

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

in35 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 wG to L q vG Suppose that λ > T κ L p

w G,L q

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:

Tf L q

Ba,τ ≤ λf L p

w G ,

Tf L q

Ba,r c≤ λf L p

w G ,

2.35

where f ∈ L p wG.

Proof Let T be bounded from L p wG to L q vG Let T vbe the operator given by

Then, it is easy to see that

T v

κ L p

w G → L q G  T κ L p

By Lemma C, we have that

λ > α

T v

Consequently, there exists P ∈ FSL p wG, L q G such that

Fix a ∈ G According to Lemma D, there are positive constants β1 and β2, β1 < β2, and R ∈

F SL p wG, L q vG for which

P − R ≤ λ−T v − P

and supp Rf ⊂ Ba, β2 \ Ba, β1 for all f ∈ L p wG Hence,

From the last inequality, it follows that if 0 < τ < β1and r > β2, then2.35 holds for f,

f ∈ L p wG.

The following lemmas are taken from2 for 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, where 1 < r, p < ∞, and w is a weight function on Ω Then,

Tκ L r

Lemma G Let Ω be a domain in R n and let P ∈ FSX, L p Ω, where X  L r

w Ω and 1 < r, p < ∞ Then for every a ∈ Ω and ε > 0, there exist an operator R ∈ FSX, L p Ω and positive numbers β1

and β2, β1< β2such that for all f ∈ X, the inequality

holds and supp Rf ⊂ Da, β2 \ Da, β1, where Da, s : ΩB a, 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 wΩ to L q vΩ Assume that λ > T κ L p

w Ω,L qΩ 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:

Tf L q

Ba,τ ≤ λf L p

wΩ; Tf L q

Ω\Ba,r ≤ λf L p

wΩ, 2.44

wheref ∈ L p wΩ.

Lemma I see 36, Chapter IX Let 1 < p, q < ∞, and let X, μ and Y, ν be σ-finite measure

spaces If



k x, y

L ν Y

L q X < ∞, p  p

then the operator

Kf x 



Y

k x, yfydνy, x ∈ X, 2.46

is compact from L p νY into L q μX.

3 Main results

3.1 Maximal functions

Let G be a homogeneous group and let

M α f x  sup

B x

1

|B|1−α/Q



B

f ydy, x ∈ G, 0 ≤ α < Q, 3.1

where the supremum is taken over all balls B containing x If α  0, then Mαbecomes the

Hardy-Littlewood maximal function which will be denoted by M.

It is knownsee, e.g., 17,18 for α  0, and 19, 33, Chapter 6, for α > 0 that if

1 < p < ∞ and 0 ≤ α < Q/p, then the operator Mα is bounded from L p ρ p G to L q

ρ q G, where

q  Qp/Q − αp, if and only if ρ ∈ Ap,qG, that is,

sup

B

 1

|B|



B

ρ q

1/q 1

|B|



B

ρ −p

1/p

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 vG Then, there is no weight pair v, w such that M is compact from L p wG to L p vG Moreover, the inequality

M κ L p

w G,L p G ≥ sup

a ∈G

lim

τ→ 0

1

B a, τB a,τ v xdx

1/p

B a,τ w1−pxdx

1/p

3.3

holds.

Proof Suppose that λ > M κ L p

w → L p

and a ∈ G By Lemma E, we have that



B a,τ v x

 sup

B  x

1

B a, τ



B a,τ

f ydy p

dx ≤ λ p



B a,τ

f xp

w xdx 3.4

for all τ τ ≤ β and all f supported in Ba, τ Substituting fy  χB a,r y w1−py in the

latter inequality and taking into account that

B a,τ w1−pxdx < ∞ see, e.g., 17,18, 25, Chapter 4 for all τ > 0 we find that

1

B a, τp



B a,τ v xdx



B a,τ w1−pxdx

p−1

This inequality and Lebesgue differentiation theorem see 33, page 67 yield the desired result

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 from L p wG to L q vG Then, there is no weight pair v, w such that Mα is compact from L p wG to

L q vG Moreover, the inequality

M α

κ≥ sup

a ∈G lim

τ→ 0

1

B a, τα/Q−1



B a,τ v xdx

1/q

B a,τ w1−pxdx

1/p

3.6

holds.

The proof of this statement is similar to that ofTheorem 3.1; therefore the proof is omitted

Example 3.3 Let 1 < p < ∞, vx  wx  rx γ, where−Q < γ < p − 1Q Then,

M κ L p

w G ≥ Q

γ Q1/p

γ

1− p Q1/p−1

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 ByTheorem 3.1and Proposition A,

we have

M κ L p

w G≥ lim

τ→ 0

1

B e, τB e,τ w xdx

1/p

B e,τ w1−pxdx

1/p

 σSlim

τ→ 0τ −Q

τ

0

t γ Q−1 dt

1/pτ

0

t γ 1−p  Q−1 dt

1/p

 Qγ Q 1/p

γ

1− p Q1/p−1.

3.8

3.2 Riesz potentials

Let G be a homogeneous group and let

I α f x 



G

f y

r

xy−1Q−α dy, 0 < α < Q, 3.9

be the Riesz potential operator It is well knownsee 33, Chapter 6 that Iαis 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 Let Iα be bounded from L p wG to L q vG Then, the following inequality holds

I α

κ ≥ Cα,Qmax

A1, A2, A3

Lemma A Let ≤ p < ∞ and assume... ∞e nT.

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... that