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 © 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" 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

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 from Lp to Lqv) 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 St = exp(A log t), t > 0, where A is a diagonalized linear operator in G with positive eigenvalues. In the homogeneous group G, the mappings exp oSto exp-1, t> 0, are automorphisms in G, which will be again denoted by 6t. 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, to) which is continuous on G, smooth on G \ {e}, and satisfies the conditions

(i) r(x) = r(x-1) for every x e G;

(ii) r(Stx) = tr(x) for every x e G and t> 0;

(iii) r(x) = 0 if and only if x = e;

(iv) there exists co> 0 such that

r(xy) < co(r(x)+r(y)), x,y e G. (2.1)

In the sequel, we denote by B(a,p) and B(a,p) open and closed balls, respectively, with the center a and radius p, that is,

B(a,p) := {y e G; r(ay^) < p}, B(a,p) := {y e G; r(ay^) < p}. (2.2)

It can be observed that 8pB(e, 1) = B(e,p).

Let us fix a Haar measure |-| in G such that |B(e, 1)| = 1. Then, |6tE| = tQ|E|. In particular, |B(x,t)| = tQ for x e G, t > 0.

Examples of homogeneous groups are the Euclidean n-dimensional space r", 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].

Proposition A. Let G be a homogeneous group and let S = {x e G : r(x) = 1}. There is a (unique) Radon measure a on S such that for all u e L1 (G),

f u(x)dx = f f u{6ty)tQ-1da(y)dt. (2.3)

Jg Jo Js

For the details see, for example, [33, page 14].

We call a weight a locally integrable almost everywhere positive function on G. Denote by Lvw (G) (1 < p < to) the weighted Lebesgue space, which is the space of all measurable functions f : G ^ c with the norm

f IIlW(G) = (]" If (x)|pw(x)dx)1/p < to. (2.4)

If w = 1, then we denote LP(G) by LP(G).

Let X = Lvw (G)(1 < p < to) 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) F(f + g) < F(f) + F(g) for all nonnegative f,g e X;

(ii) F(af) = |a|F(f) for all f e X and a e C.

Let T be a sublinear operator T : X ^ Lq(G), then, the norm of the operator T is defined as follows:

||T y = sup {\\Tf\\LHG) : I If IIx < 1}. (2.5)

Moreover, T is order preserving if Tf(x) > Tg(x) almost everywhere for all nonnegative f and g with f (x) > g(x) almost everywhere. Further, if T is sublinear and order preserving, then obviously it is nonnegative, that is, Tf(x) > 0 almost everywhere if f (x) > 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\K(x,Y) (or simply ||T||K) and is defined as

11T ||K(X,Y) = dist {T, K(X,Y)} ^ inf {||T - KW : K e K(X,Y)}, (2.6)

where K(X,Y) is the class of all compact sublinear operators from X to Y .If X = Y, then we use the symbol K(X) forK(X,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:

ek(T) = inf je > 0 : T(Ux) c jj (h + sUy) for some b\,..,b2k-1 e (2.7)

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 lim„ ^to en(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 S(X) the class of all bounded sublinear functionals defined on X, that is,

S(X) = f F : X^r, F-sublinear and ||F|| = sup|F(x)| < ^. (2.8)

I hxk! J

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 e X with 0 < f (x) < g(x) almost every. We also need the following classes of operators acting from X to LP(G):

T : Tf (x) = ^ a(f )Uj, m e n, Uj > 0, uj e LP(G), j=1

Uj are linearly independent and aj e X* Q M

(2.10)

Fs(X,Lp(G)) := J T : Tf (x) = ^ fr (f )Uj, m e N, Uj > 0, Uj e LP(G),

Uj are linearly independent and fr e S(X) Q mJ.

If X = LP(G),wewill denote these classes by Fl(Lp(G)) and FS(Lp(G)), respectively. It is clear that if P e FL(X,Lp(G)) (resp., P e FS(X,Lp(G))), then P is compact linear (resp., compact sublinear) from X to LP (G).

We will use the symbol a(T) for the distance between the operator T : X ^ Lp (G) and the class Fs(X,LP(G)), that is,

a(T) := dist{T,Fs(X,LP(G))}. (2.11)

For any bounded subset A of Lp (G) (1 <p < to), let

®(A) := inf {6 > 0 : A can be covered by finitely many open balls in LP(G) of radius ¥(A) := inf sup {||f - Pf\\LP(G) : f e A}.

P eFL (LP(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 c LP ( G) (1 < p < to), the inequality

2®(K) > ¥(K) (2.13)

holds.

Proof. Let s > O(K). Then, there are gi ,g2, ...,gN e Lp (G) such that for all f e K and some i e {1,2,...,N},

llf - gi^LP(G) <S

(2.14)

Further, given 6 > 0, let B be the closed ball in G with center e such that for all i e (1,2,...,N},

\1/p 1

\gi(x)\pdx) < - 6. (2.15)

' G\B / 2

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 = |Jn=1Ei and Ei c Bi (we can assume that E1 = B1 n B, E2 = (B2 \ B-) n B,...,Ek = (Bk \ Uk=-11Bi) n B,...). We define

P/(x) = XfElXEl(x), fz = E\-1 f(x)dx. (2.16)

i=1 JEi

n i - i p

\\gi - Pgi^[p(B) ^ W\\ gi(x) - dy dx y' j=-JEj \EnjEi

^ i^f \gi(x) - gi(y)\pdydx (2.17)

i=ljEj Iej\je,

- sup _\gi(x) - gi(zx)\pdx —> 0

r(z)<2cohJ 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

IIgi - Pgi \\lp(G) <6, i = 1,2,3.....N, (2.18)

when h is sufficiently small. Further,

IIPfllLp(G) = ¿1" Ef'f f(y)dy

j=\j Ej Je,

< tfijE(y)|,dydx (2.19)

< If 11 Lp (G).

It is also clear that the functionals f ^ fEi belong to (Lp(G))* n M. Hence, P e Fl(Lp(G)). Finally, (2.14)-(2.15) and (2.18) yield

If - P/IIlp (G) < If - gi\\ LP(G) + \\gi - Pgi\\ LP (G) + \\P(gi - f) \

(2.20)

< £ + 6 + ||gi - f \\ LP(G) < 2g + 6.

Since 6 is arbitrarily small, we have the desired result. □

Lemma A. Let 1 < p < to and assume that a set K c Lp(G) is compact. Then for any given e > 0, there exist an operator Pe e FL(Lp (G)) such that for all f e K,

\\f- pef\\LP(G) < e. (2.21)

Proof. Let K be a compact set in Lp(G). Using Theorem A, we see that ¥(K) = 0. Hence for e> 0, there exists Pe e Fl(Lp(G)) such that

sup {\\f - Pef\\LP(G) : f e K} < e. (2.22)

Lemma B. Let T : X ^ Lp(G) be compact, order-preserving, and sublinear operator, where 1 < p < to. Then, a(T) = 0.

Proof. Let UX = {f : \\f ||X < 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 e > 0 there exists an operator Pe e Fl(Lp(G)) such that for all f e Ux,

\\Tf - PeTf \\lp(G) < e. (2.23)

Let Pe = Pe ◦ T. Then, Pe e Fs(X,Lp(G)). Indeed, there exist functionals aj e X* n M, j e {1,2,...,m}, and linearly independent functions Uj e Lp (G), j e {1,2,...,m}, such that

Pef (x) = Pe (Tf)(x) = £ aj (Tf)Uj (x) ^ (f)Uj (x), (2.24)

j=1 j=1

where ^j = aj o T belongs to S(X) n M. Since by (2.23),

\\Tf - Pef\\Lp(G) < e (2.25)

for all f e UX, it follows immediately that a(T) = 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 < to. Then,

\\T\\K = a(T). (2.26)

Proof. Let S> 0. Then, there exists an operator K eK(X,Lq(G)), such that \\T-K\\ < \\T\\K + 6. By Lemma B there is P e FS(X,Lq(G)) for which the inequality \\K - P\\ <6 holds. This gives

\T - P\\ < \T - K\\ + \K - P\\ < \T\\K + 26. (2.27)

Hence, a(T) < \\T\\K. Moreover, it is obvious that

\T\K < a(T). (2.28)

Lemma D. Let 1 < q < to and let P e FS(X,Lq(G)). Then for every a e G and e> 0, there exist an operator R e Fs (X, Lq(G)) and positive numbers a, a such that for all f e X, the inequality

ii (P - R)f iilq(g) < *Hf I|x (2.29) holds and supp Rf c B(a,a) \ B(a,a).

Proof. There exist linearly independent nonnegative functions Uj e Lq(G), j e {1,2,...,N), such that

Pf(x) = £ P(f )Uj(x), f e X, (2.30)

where Pj are bounded, order-preserving, sublinear functionals Pj : X ^ r. On the other hand, there is a positive constant c for which

Z\Pj (f )| < cyf yX. (2.31)

Let us choose linearly independent Oj e Lq(G) and positive real numbers aj, aj such

II Uj - Oj II Lq(G) <£, j e {1,2.....N) (2.32)

and supp oj c B(a,aj) \ B(a,aj). If

Rf (x) = X Pj (f )oj (x), (2.33)

then it is obvious that R e FS(X,Lq(G)) and moreover,

l|Pf - Rf llLq(G) < £\Pj(f)\Iuj - oj IILq(G) < ceyf yX (2.34)

for all f e X. Besides this, supp Rf c B(a,a) \ B(a,a), where a = min{aj) and a = max{aj).

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 < to, and let T be a bounded, order-preserving, and sublinear operator from Lvw(G) to Lqv(G). Suppose that 1 > yT||K(Lp (G),Lq(G)), and a is a point of G. Then, there exist constants p1, p2, 0 < p1 < p2 < to, such that for all t and r with r > p2, t < p1, the following inequalities hold:

yTfyLqv(B(a,T)) < 1yf yLW(G),

(2.35)

yTf yLV(B(a,r)c) < 1yf yLW(G),

where f e Lpw(G).

Proof. Let T be bounded from LW(G) to Lqv(G). Let T(v) be the operator given by

T (v)f = v1/qTf. (2.36)

Then, it is easy to see that

\\T (v)\\ K(LPw(G) ^ Lq (G)) = ||T ||k(lw(g) ^ Ll(G)). (2.37) By Lemma C, we have that

\>a(T(v)). (2.38)

Consequently, there exists P e FS(LW(G),Lq(G)) such that

\\T(v) - P\\ <X. (2.39)

Fix a e G. According to Lemma D, there are positive constants f1 and f2, f1 < f2, and R e Fs(LPw (G),LV(G)) for which

1 - \\T(v) - P\\

IIP - R||<----(2.40)

and supp Rf c B(a,p2) \ B(a,fc) for all f e LW(G). Hence,

\\T(v) - R\\ <X. (2.41)

From the last inequality, it follows that if 0 <t < f1 and r > f2, then (2.35) holds for f, f e LW(G). □

The following lemmas are taken from [2] (for the linear case see [35]).

Lemma F. Let q be a domain in rn, and let T be a bounded, order-preserving, and sublinear operator from Lrw (q) to LP (q), where 1 <r,p< to, and w is a weight function on Q. Then,

IT ||k(lw (Q),LP(Q)) = a(T). (2.42)

Lemma G. Let q be a domain in rn and let P e FS(X,LP(Q)),whereX = Lrw(q) and 1 <r,p < to. Then for every a e q and e > 0, there exist an operator R e FS(X,LP (q)) and positive numbers f1 and ¡2, f1 < ¡2 such that for all f e X, the inequality

\\(P - R)f \\lp(Q) < e||f iix (2.43)

holds and supp Rf c D(a,f2) \ D(a,f1), where D(a,s) := B(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 q be a domain in rn. Suppose that 1 < p, q < to, and that T is bounded, order-preserving, and sublinear operator from Lw (q) to Lqv (q). Assume that 1 > ||T||K(Lp (Q),Lq(Q)) and a e Q. Then, there exist constants f1, f2, 0 < f1 < f2 < to such that for all t and r with r > f2, t < f1, the following inequalities hold:

ITfILl(B(a,T)) < 1IIf IILW(Q); IIT/IIli{q\B(a,r)) < MIf IIlw(Q), (2.44)

wheref e LWW (q).

Lemma I (see [36, Chapter IX]). Let 1 < p, q < to, and let (X, p) and (Y, v) be a-finite measure spaces. If

IIII^IIlV(Y)L,(X) < P = ^, (2.45)

then the operator

Kf (x) = J k(x,y)f (y)dv(y), x e X, (2.46)

is compact from LV (Y) into L^ (X).

3. Main results 3.1. Maximal functions

Let G be a homogeneous group and let

Maf (x) = sup 1 f If (y) |dy, x e G, 0 < a<Q,

B3x \B\ /Q JB

where the supremum is taken over all balls B containing x. If a = 0, then Ma becomes the Hardy-Littlewood maximal function which will be denoted by M.

It is known (see, e.g., [17, 18] for a = 0, and [19], [33, Chapter 6], for a > 0) that if 1 <p < to and 0 < a < Q/p, then the operator Ma is bounded from LvpP (G) to Lqpq (G), where q = Qp/(Q - ap), if and only if p e Apq(G), that is,

?(mi/) ""( mis') w< (3.2)

Now, we formulate the main results of this subsection.

Theorem 3.1. Let 1 < p < to. Suppose that the maximal operator M is bounded from Lpw (G) to

rp the,™ hicvc ic mo-in-lif 1-m-iV <r/t) cmli A/T ic rrwnnr^ frn™ tJ^ (G) to Lv (

Lv (G). Then, there is no weight pair (v, w ) such that M is compact from Lpw (G) to Lpv (G). Moreover, the inequality

\\m\\k(lW(G),LV(G)) > supJ™m(*T^ (f v(x)dx) (f w1-p'(x)dx) (3.3)

aeG ^0 IB(a,T)I \JB(a,T) / \J B(a,r) /

holds.

Proof. Suppose that 1 > \\M\\K(Lp^Lp) and a e G. By Lemma E, we have that

v(x)(sup 1 f |f (y)IdyYdx < Xp\ If (x)Ipw(x)dx (3.4)

JB(a,T) \Bs x IB(a,T)^ B(a,T) / J B(a,T)

for all t (t < ¡5) and all f supported in B(a,T). Substituting f (y) = xB(a,r)(y) w1-p'(y) in the latter inequality and taking into account that jb(^at)W1-p'(x)dx < to (see, e.g., [17, 18], [25, Chapter 4]) for all t > 0 we find that

■ p, 1 v(x)dxVf w1-p'(x)dxY 1 < Xp. (3.5)

IB(a,T)f \JB(a,T) / \J B(a,T) /

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 < to, 0 < a< Q/p and let q = Qp/(Q - ap). Suppose that Ma is bounded from LWW (G) to Lqv (G). Then, there is no weight pair (v,w) such that Ma is compact from LWW (G) to Lqv(G). Moreover, the inequality

\\Ma\\K > suplimn--1 ia/Q 1 (f v(x)dx) ({ w1-P'(x)dx) (3.6)

aeGT ^0 \B(a, T )\ Q \ J B(a,r) / \JB(a,T) /

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 < to, v(x) = w(x) = r(x)Y, where -Q < y < (p - 1)Q. Then,

IIMiiklw(G)) > Q[(r + Q)1/P(r(1 -P') + Q)1/P']-1. (3.7)

Indeed, first observe that the fact \B(e, 1)| = 1 and Proposition A implies a(S) = Q, where S is the unit sphere in G and a(S) is its measure. By Theorem 3.1 and Proposition A, we have

k(LW (G)) ^ ,

* sotb^ (Ljw(x)dx) 1/P(LT)wl~pl(x)dx)1/p

= a (S) lim t ~qçjt tr+Q-1df) (f? (1-p')+Q-1d^ (3.8)

= Q[(r + Q)1/p (r (1 - p ) + Q)1/p'1

3.2. Riesz potentials

Let G be a homogeneous group and let

!af(x)= \ , f (y)Q-a dy, 0 < a < Q

JGr(xy-1) Q

be the Riesz potential operator. It is well known (see [33, Chapter 6]) that Ia is bounded from LP(G) to Lq(G), 1 <p,q< to, if and only if q = Qp/(Q - ap).

Theorem 3.4. Let 1 <p < q < to, 0 < a < Q. Let Ia be bounded from LW (G) to Lqv (G). Then, the following inequality holds

||Ia||K > Ca,Q max {AuA2,A3},

(3.10)

Ca,Q =

(2co)Q

A1 = sup limra-Q{( v(x)dx\ ([ w1-(x)dx\ ,

aeGr ^0 \jB(a,r) ' \J B(a,r) '

A2 = sup lim ([ v(x)dx~\ ([ r (ay_1)(" Q)P w1-p'(y)dy

aeGr^0\JB(a,r) / \J (B(a,r))c

A3 = sup lim ([ w1^P' (x)dx\ ([ r(ay~1)^a Q)qv(y)dy)

aeGr ^0\ JB(a,r) ' \J (B(a,r))c '

(3.11)

v1/p' /t . \1/q

aeG 0\ JB(a,r) / \J (B(a,r))c

(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:

¡q,af (x)=f f (y)|x - y\a-ndy, x e Q.

(3.12)

Theorem 3.5. Let q c rn be a domain in Rn. Let 1 < p < q < to. If JQa is bounded from Lw(q) to Lqv(q), then one has

IIJqAk > 2a-nb1, (3.13)

B1 = sup limra-n ( f v)/q({ w1-A /. (3.14)

aeQ r^0 \JB(a,r) ' \J B(a,r) '

In particular, if q = rn, then

IIJqAk > 2a-n max {B2,Bs}, (3.15)

1/q /t , , \ 1/p'

I ( I '

aeRnr ^ 0\ JB(a,r) / \J Rn\B(a,r)

B2 = sup lim (f v(x)dx) q(f \a - y\(a-n)p'w1-p'(y)dy) V,

aeRnr ^ 0\ JB(a,r) ' \JRn\B(a,r) '

B3 = sup lim (\ w1-p' (x)dx\ ^ (( \a - y\(a-n)qv(y)dy) ^.

aeR»r ^ 0\ JB(a,r) ' \JR"\B(a,r) '

(3.16)

Corollary 3.6. Let 1 <p < to, 1 <p <Q/a, q = pQ/(Q - ap), then there is no weight pair (v,w) for which Ia is compact from LWW (G) to Lqv (G). Moreover, if Ia is bounded from LWW (G) to Lqv(G), then

IIIaIIK > Ca,QA1, (3.17)

where Ca,Q and A1 are defined in Theorem 3.4.

Proof of Theorem 3.4. By Lemma E, we have that for X > 11^« (G),4 (G)) and a e G, there are positive constants ß1 and ß2 (ß1 < ß2) such that for all t,s (t < ß1, s > ß2),

v(x)\Iaf (x)\qdx < Xq(( \f(x)\pw(x)dxY p (3.18)

J B(a,T) \Jg /

for f e Lp (G), and

v(x)\I«f (x)\qdx < Xq(( |f(x)|pW(x)d^q/P (3.19)

JB(a,s)c \Jß(a,s) '

for supp f c B(a, s).

Now taking f (x) = Xß(a,r)(x)w1^p (x) in (3.18) and observing that JB(ar)W1-p (x)dx < go for all r > 0 (see also [25, Chapter 3]), we find that

v(x)(f W p (yQ dyXdx < »(( w1-p(x)dx\q// < g. (3.20)

JB(a,r) \Jß(a,r) r(xy-1)Q « ' \J B(a,r) /

Further if x,y e B(a,T),then

r(xy^1) < Co(r(xa-1) + r(ay^1)) < 2coT. (3.21)

Hence,

||I«||K > C«,qAx. (3.22)

If f (x) = XB(a,T )c (x)(w1-p' (x)/r (ay-1)(Q-")(p-1)), then

i v(x)(i -Q-I'p'(y)dy(Q )(l1))qdx<Wf p(xQdx})q

JB(a,T) \JB(a,T)Cr (xy-1)Q-«r (ay-1 )(Q-»)(p'-1)/ VJB(a,T)c r(ay-1)(q-«)p /

(3.23)

Ibm s VJB(a,T)c^xy-Oß-"r(ay-'/«-«""V \lBarf r (.ay-!)0-''"

Let r(xa-1) < T and r(ya-1) > T. Then,

r(xy~l) < co(r(xa~l) + r(ay~1)) < co(t + r(ay~1)) < 2cor(ay~l). (3.24)

Hence, by (3.18) we have

1 /f / w \/f (y)dy /f w1-p'(x)dx

ft v(x)dxVf ^^ V < ^f w1-p' f.

\JB(a,T) / \JB(aTYr(ay-1)(Q-a)p J V J B(a,T)c r (ay-1)(Q-a)p/

B(a,T) J \JB(a,T)cr(ay-1) ^'J V J B(a,T)c r(ay-1) (Q-")p'

(3.25)

The latter inequality implies

L >-Q—A-2. (3.26)

lK (2co)Q-a

Further, observe that (3.19) means that the norm of the operator

af (X) = f

f (y)dy

B(a,s) r(y-1a)Q-a

Iaf (x) = (3.27)

can be estimated as follows:

\\Ia\\LPW(B(a,s)) ^Lqv(B(a,s)c) < M (3.28)

Now by duality, we find that

La\\Lpw (B(a,s)) ^ Ll(B(a,s)c) II^IIl' , (B(a,s)c) ^ if. , (B(a,s))'

v1 ' w1 p

'B(a,s)cr{xy-!) Q " Indeed, by Fubini's theorem and Holder's inequality, we have

IIvII Lv (B(a s)c) < sup \g(x)(Iaf (x)) \dx

llgM i <1 J B(a,s)c

suP \f (y)\ ia(|gl)(y) dy

1 B(a s)

gii ' <u B(a,s)

Lq I (B(a,s)c) v1 '

< sup (f If fw) 1/p(f (Ia(lgl))P' w'-A1/P

Hell , <1 \ JB(a ,s) / \JB(a,s) /

<1 \ JB(a,s) / \JB(a,s)

Lyq' (B(as)C)

INI^ If IPW

B(a s)

Hence, \\Ia\\ < \\Ia\\. Analogously, \\Ia\\ < \\Ia\\. Further, (3.19) implies

(3.29)

i*g(y>=j; -g^. (3.30)

(3.31)

w1-p'(x) f g(y)dy dxP < \p'({ \g(x)\q'v1-q(x)dxY \ (3.32)

JB(a,s) J(B(a,s))cr(xy-1)\J (B(a,s))c J

Now, taking g(x) = xB(a,sy (x)r(xa ')(Q a)(1 q)v(x) in the last inequality we conclude

that \\Ia\\K > (1/(2c0)Q-a)Ä3. ' □

Theorem 3.5 follows in the same manner as Theorem 3.4 was obtained. We only need to use Lemma H.

3.3. Truncated potentials

This subsection is devoted to the two-sided estimates of the essential norm for the operator:

Taf (x)=f / f {y]Q_a, X e G. (3.33)

JB(e,2r(x)) r(xy )Q

A necessary and sufficient condition guaranteeing the trace inequality for Ta 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 < œ and let a> Q/p. Then,

(i) Ta is bounded from Lp(G) to Lqv(G) if and only if

B := supB(t) := sup( f v(x)r(x)(a-Q)qdx) ^fi1'?' < œ; (3.34)

t>0 t>0 \Jr(x)>t '

(ii) Ta is compact from Lp(G) to Lqv (G) if and only if

limB(t) = lim B(t)= 0. (3.35)

t^0 t^œ

Theorem 3.7. Let 1 < p < q < to and let 0 < a < Q. Suppose that Ta is bounded from Lvw (G) to Lqv(G). Then, the inequality

llx«llK(C(G) ^Ll(G)) - o b

holds, where

CQ,a = (2co)aQ,

> CqJlim A(a) + lim An,)) (3.36)

\a—> o b —> oo '

A(a) = supff v(x)r(x)(a-Q)qdx\ q([ w1-p'(x)dx)

0<t<a\J B(e,a)\B(e,t) / \J B(e,t) /

A(b) = supff v(x)r(x)(a-Q)qdx) 'Vf w1-p'(x)dx

t>b \J B(e,t)c ' \jB(e,tnB(e,b)

(3.37)

1/q /r , \ i/p'

t>b \ J B(e,t)c / \J B(e,t)\B(e,b)

To prove Theorem 3.7 we need the following lemma.

Lemma 3.8. Let p, q, and a satisfy the conditions of Theorem 3.7 Then from the boundedness of Ta from LWw (G) to Lqv (G), it follows that w1~p is locally integrable on G.

Proof. Let

I (t)=f w1-p' (x)dx = œ (3.38)

B(e t)

for some t > 0. Then, there exists g e Lp(B(e,t)) such that jBet)gw-/p = to. Let us assume that ft(y) = g(y)w'1/P(y)XB(e,t)(y). Then, we haVe

II Taft y lV(G) > IXB(e,t)cTaftILl(C)

ax 1/q , (3.39)

v(x)r (x)(a-Q)qdx) g(y)w~1/p' (y)dy = to.

B(e,t)c ' JB(e,t)

On the other hand,

IIMIiW(G)=f gp(x)dx< to. (3.40)

J B(e,t)

Finally, we conclude that I(t) < to for all t, t > 0. □

Proof of Theorem 3.7. Let X > \\Ta\\K(Lp (G)rLq (G». Then by Lemma E, there exists a positive constant ß such that for all t,, t2, 0 < t, <t2 < ß and f, supp f c B(e,T1), the inequality

IITaf HiV(B(e,T2)\B(e,Tl)) < X\f \\lW(B(e,n)) (3.41)

holds. Observe that if r(x) > t, and r(y) < t,, then r(xy-1) < 2c0r(x). Consequently, taking f = w1-p'xB(e,Tl) and using Lemma 3.8, we find that

-Wf v(x)(r(x^'dx) 1/q(( 1-p ' - ^

(2co )Q \J B(e,T2)\B(e.Ti) / VB(e/n)

w1-p (x)dx ) < X, (3.42)

from which it follows that

(Q-a)q a-> 0

lim A(a) < X. (3.43)

Further, by virtue of Lemma E there exists ß2 such that for all s,, s2 with ß2 < s, < s2 the inequality

ITaf HiV(B(e,s2)c) < X\\f \\lW(B(e,s2)\B(e,s1)) (3.44)

holds, where supp f c B(e,s2) \ B(e,s1). Hence by Lemma 3.8, we find that

-^(f v(x)(r (x))(a-Q)qdx) / ([ w1-p (x)dx)'V < X, (3.45)

(2c0)Q \JB(e,s2)c / \h(e,s2)\B(e,s1) /

which leads us to

Q^^m A(b) < X. (3.46)

Thus, we have the desired result. □

Theorem 3.9. Let 1 <p < q < to and let Q/p < a < Q. Suppose that (3.34) holds. Then, there is a positive constant C such that

ii Ta iik(lp(g) ^LV(G)) < + ^(b)), (3.47)

B(a) = supft_ v(x)r(x)(a-Q]qdx\ \Q/p',

t<a \J B(e,a)\B(e,r) '

B(b) = sup ft v(x)r(x)(a-Q)qdx)1/q(rQ - bQ)1

t>b \JB(et)c '

(3.48)

Proof. Let 0 < a < b < to and represent Taf as follows:

Taf = XB(e,a)Ta(fX~B(e,a)) + XB(e,b)\B(e,a)Ta(fXB(e,b))

+ XG\\B(e,b) Ta(fxB(e,b/2c0)) + XG\B(e,b) Ta (fXG\B(e,b/2c0)) (3.49)

= Pif + P2f + P3f + P4f.

For P2, we have

P2f (x) = | k(x, y)dy, (3.50)

where k(x, y) = XB(e/b)\B(e/a)(x)XB(e/2r(x))(y)r(xy-1)tt-Q. Further observe that

m(k(x,y))p dy} v(x)dx = f_ _ ^ f_ (r (xy-1))^ Q)p dy^ v(x)dx

. G ' J B(e,b)\B(e,a) \J B(e,2r(x)) '

< ft _ ft (r(xy-1))ia-Q)p'dy\/P v(x)dx

J B(e,b)\B(e,a)\j B(e,r(x)/2co) '

< ft _ r(x)(a-Q)q+q/p'v(x)dx< TO.

J B(e,b)\B(e,a)

(3.51)

Hence by Lemma I, we conclude that P2 is compact for every a and b. Now we observe that if r(x) > b and r(y) < b/2co, then r(x) < 2cor(xy-1). Due to Proposition A we have that P3 is compact.

Further, we know that (see [38], [10, Chapter 6])

II Pi II < C1B(a), || P41| < C2B(b/c) , (3.52)

where the constants C1 and C2 depend only on p, q , Q, and a. Therefore,

II Ta - P2 - P3II < IIP1II + 11 P4 11 < c(B(a) + B(b)). (3.53)

The last inequality completes the proof. □

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:

^fa B(a) + Si3®) ^ HT«IUp(G),LV (G)) < B(a) + ^J®) (3.54)

for some positive constants c1 and c2 depending only on Q, a, 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:

Snf (x) = - f (t)Dn(t)dt, (3.55)

n J -n

where Dn = 1/2 + En^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 < i. Then, there is no n e n and weight pair (w,v) on T := (-n,n) such that Sn is compact from Lvw (T) to LVv (T). Moreover, if Sn is bounded from Lvw (T) to LVv (T), then

v1/2 _ , 1 Ad+r \ 1/p / 1 Ad+r \ 1/p

(2 + 21/2) 2 -/ 1 fd+r \1/p / i

|Sn || >-2^-suplirn( — I ^ — I w1

(1 fa+r \ 1/p / I fa+r \ 1/p

... 2?LV) (2^J„wlp) , (356)

where I = (a - r,a + r).

ProofTaking X > \\Sn\\K(Lpw(t),lV(T)), by Lemma H we find that

| v(x) | Snf (x) |pdx < Xpj |f (x) ¡pw(x)dx (3.57)

for all intervals I = (a - r,a + r), where r is a small positive number. Let

J1 = J v(x)l[Snf (x)|pdx, J2 = 1^(x)|pw(x)d(x). (3.58)

Suppose that \I| < n/4, and let n be the greatest integer less than or equal to n/4\I|. Then for x e I (see [32]),

1 f |f (d) | sin(3n/8) (3_9)

|Snf (x)| > n \ I—n/n—d°. (3.59)

Using this estimate and taking f := w1-p'(x)jI(x), we find that

J, > (1sin3n(JIv)(/,w-)'. (3.60)

On the other hand, it is easy to see that J2 = jjW1 p < to. Hence, by (3.57) we conclude that

* * 1sinf (i^f (liiM" (3.61)

Now passing r to 0, taking supremum over a e T, and using the fact sin(3n/8) = (2 + 21/2)1/2/2, we find that (3.56) holds. □

Corollary 3.12. Let 1 <p < to and let n e N. Then

(2 + 21/2)

IISn y k(Lp(T)) - 2n . (3.62)

Corollary 3.13. Let 1 <p < to and let n e n. Suppose that w(x) = v(x) = \x\a. Then, one has

(2 + 21/2)1/2 ( 1 )1/P( 1 )1/p'

SnL(Li(T))- 2n \aTTJ U(1 -p) + V . (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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