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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, 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 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 [...]... 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 References 1 D E Edmunds, A Fiorenza, and A Meskhi, “On a measure of non-compactness for some classical operators,” Acta Mathematica Sinica, vol 22, no 6, pp 1847–1862, 2006 2 D E Edmunds and A Meskhi, “On a measure of. .. non-compactness of maximal operators,” Real Analysis Exchange, vol 28, no 2, pp 439–446, 2002 6 D E Edmunds, W D Evans, and D J Harris, “Two-sided estimates of the approximation numbers of certain Volterra integral operators,” Studia Mathematica, vol 124, no 1, pp 59–80, 1997 7 D E Edmunds and V D Stepanov, “The measure of non-compactness and approximation numbers of certain Volterra integral operators,”... non-compactness for maximal operators,” Mathematische Nachrichten, vol 254-255, no 1, pp 97–106, 2003 3 A Meskhi, “On a measure of non-compactness for singular integrals,” Journal of Function Spaces and Applications, vol 1, no 1, pp 35–43, 2003 4 G G Oniani, “On the measure of non-compactness of maximal operators,” Journal of Function Spaces and Applications, vol 2, no 2, pp 217–225, 2004 5 G G Oniani,... “Capacitary inequalities for fractional integrals, with applications to partial differential equations and Sobolev multipliers,” Arkiv f¨ r Matematik, vol 33, no 1, pp 81–115, o 1995 29 E T Sawyer and R L Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces,” American Journal of Mathematics, vol 114, no 4, pp 813–874, 1992 30 R L Wheeden, A characterization of. .. of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, UK, 1998 26 D E Edmunds, V Kokilashvili, and A Meskhi, “On Fourier multipliers in weighted Triebel-Lizorkin spaces,” Journal of Inequalities and Applications, vol 7, no 4, pp 555–591, 2002 27 D R Adams, A trace inequality for generalized potentials,” Studia Mathematica, vol 48, pp 99–105, 1973 28 V G Maz’ya and I E... 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Alam Road, Gulberg III, Lahore. 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. 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
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