Some s numbers of an integral operator of Hardy type on L p(.) spaces

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1 Some s numbers of an integral operator of Hard tpe on L p(. spaces D. E. Edmunds a,, J. Lang b,, A. Nekvinda c a Department of mathematics, Mantell Building, Universit of Sussex, Brighton BN1 9RF, UK b Department of mathematics, The Ohio State Universit, 100 Math Tower, 231 West 18th Avenue, Columbus, OH , USA c Department of Mathematics, Facult of Civil Engineereng, Czech Technical Universit, Thákurova 7, Prague 6, Czech Republic Abstract Let = [a, b] R, let p : (1, be either a step function or strong log-hölder continuous on, let L p(. ( be the usual space of Lebesgue tpe with variable exponent p, and let T : L p(. ( L p(. ( be the operator of Hard tpe defined b T f(x = x a f(tdt. For an n N, let s n denote the n th approximation, Gelfand, Kolmogorov or Bernstein number of T. We show that lim ns n = 1 n 2π { p (tp(t p(t 1 1/p(t sin ( π/p(t dt where p (t = p(t/(p(t 1. The proof hinges on estimates of the norm of the embedding id of L q( ( in L r( (, where q, r : (1, are measurable, bounded awa from 1 and, and such that, for some ε (0, 1, r(x q(x r(x + ε for all x. t is shown that min(1, ε id ε + ε ε, a result that has independent interest. Ke words: Hard-tpe operator, compactness, s-numbers, L p( MSC: 47G10, 47B10 The third author was supported b grant no. MSM 201/08/0383 of the Grant Agenc of the Czech Republic. Corresponding author addresses: davideedmunds@aol.com (D. E. Edmunds, lang@math.ohio-state.edu ( J. Lang, nales@mat.fsv.cvut.cz ( A. Nekvinda URL: lang/ ( J. Lang Preprint submitted to Elsevier December 11, 2008

2 1. ntroduction Let = [a, b] be a compact interval in the real line and let T be the operator of Hard tpe given b T f(x := x a f(tdt (x. (1.1 t is well known that if p (1,, then T is a compact map from L p ( to L p ( (see, for example, [2], Chapter 2, 3; moreover, if s n (T stands for the n th approximation, Bernstein, Gelfand or Kolmogorov number of T, then lim ns n(t = γ p (b a/2, (1.2 n where γ p = π 1 p 1/p (p 1/p sin(π/p and p = p/(p 1. We refer to [2] and [7] for details of this and similar results for more general operators of Hard tpe. The position when T is viewed as a map from L p ( to L q ( and p q is less simple, but nevertheless genuine asmptotic results similar to (1.2 have been obtained for various s numbers of T in particular circumstances: see [3] and [4]. The focus of the present paper is on the behavior of s numbers of the map T when it acts from the variable exponent space L p( ( to L p( (. Here p : (1, and b L p( ( is meant the space of all real-valued functions f on such that for some λ > 0, f(x/λ p(x < ; endowed with the norm f p( := inf { λ > 0 : f(x/λ p(x 1 (1.3 it is a Banach space. Because of their natural occurrence in various significant phsical contexts (see [12], these spaces (which are particular cases of Musielak- Orlicz spaces have been intensivel studied in recent ears, considerable emphasis being placed on the properties on them of such classical operators in harmonic analsis as the Hard-Littlewood maximal operator. Our main result is a direct analogue of (1.2: if p is either a step function or a strong log-hölder continuous function (see Definition 4.10 and note that an Lipschitz or Hölder function p( is a strong log-hölder continuous function, then lim ns n(t = 1 n 2π ( p (xp(x p(x 1 1/p(x sin (π/p(x, (1.4 where s n (T is the n th approximation, Gelfand, Kolmogorov or Bernstein number of T : L p( ( L p( (. So far as we are aware, this is the first result concerning the s numbers of operators acting on spaces with variable exponent, despite the clear importance 2

3 of these numbers and the considerable literature devoted to them in the context of classical Lebesgue spaces. A ke step in the proof is the following twosided estimate of the norm of the embedding id of L q( ( in L p( ( when, for ε (0, 1, p(x q(x p(x + ε (x : min(1, ε id ε + ε ε, This has intrinsic interest, being a sharp improvement of the classical embedding theorem for L p( spaces due to Ková cík and Rákosník [6]. 2. Preliminaries Throughout the paper will stand for a compact interval [a, b] in the real line R, and given an measurable subset E of, the Lebesgue measure of E will be denoted b E and the characteristic function of E b χ E. B M( is meant the famil of all extended scalar-valued (real or complex measurable functions on, and P( will stand for the subset of M( consisting of all those functions p(, with values in (1,, such that 1 < p := ess inf x p(x p + := ess sup x p(x <. For all f M(, define and ρ p( (f = f(x p(x f p(, = f p( = inf { λ > 0 : ρ p( (f/λ 1. The generalised Lebesgue space L p( ( (or space with variable exponent is the set L p( ( := {f : f p( <, equipped with the norm p( ; it is routine to verif that it is a Banach space; indeed, it is a Banach function space. We refer to [6] for an account of the fundamental properties of these spaces and in particular for the following basic embedding theorem, in which b X Y we mean that the Banach space X is continuousl embedded in the Banach space Y. Theorem 2.1. Let p(, q( P( be such that for all x, p(x q(x. Then L q( ( L p( (. n what follows we consider the Hard operator T acting on a space L p( ( with variable exponent. To establish its compactness we use the following wellknown result concerning its behaviour on classical Lebesgue spaces. Theorem 2.2. Let r, s (1,. Then T maps L r ( compactl into L s (. 3

4 This follows from [2], Theorems and 2.3.4, for example. Now the compactness of T on spaces with variable exponent follows quickl. Lemma 2.3. Let 1 < c < d < and suppose that p(, q( P( are such that for all x we have p(x, q(x (c, d. Then T maps L p( ( compactl into L q( (. Proof. B Theorem 2.1, L p( ( and L d ( are continuousl embedded in L c ( and L q( (, respectivel. Moreover, b Theorem 2.2, T maps L c ( compactl into L d (. The result now follows b composition of these maps. More detailed information about the compactness properties of T is provided b the approximation, Bernstein, Gelfand and Kolmogorov numbers, and we next recall the definition of these quantities. Let X and Y be Banach spaces and let S : X Y be compact and linear. Then given an n N, the n th approximation number of S is defined to be a n (S = inf S F, where the infimum is taken over all bounded linear maps F : X Y with rank less than n; the n th Bernstein number of S is b n (S = sup inf Sx Y / x X, x X n\{0 where the supremum is taken over all n dimensional subspaces X n of X; the n th Gelfand number of S is c n (S = inf{ SJ M X : M is a linear subspace of X, codim X < n, where JM X is the embedding map from M to X; and the nth Kolmogorov number of S is d n (S = inf sup inf Sf g X n g X Y / f X, n 0< f X 1 where the outer infimum is taken over all n dimensional subspaces X n of X. Further details of these numbers and their basic properties will be found in [1], [9] and [11]; for the moment we simpl note that the approximation numbers are the largest of them. We recall that not all these s numbers have the multiplicative propert detailed in [1], p. 72: the Bernstein numbers fail to have it (see [10]. However, ever s number s n satisfies the following inequalit s n (R T S R s n (T S, (2.1 for arbitrar and appropriatel composed bounded linear maps R, S and T. To determine the properties of T we introduce certain functions that will pla a ke role in our analsis. 4

5 Definition 2.4. Let p(, q( P(, let J = (c, d, and let ε > 0. We define { A p(,q( (J = inf sup J f : f p(,j 1, q(,j { B p( (J = inf sup J f : f p p + J,J J,J 1, { C p(,q( (J = sup T f q(,j : f p(,j 1, (T f(c = (T f(d = 0, { D p( (J = sup T f p J,J : f p + J,J 1, (T f(c = (T f(d = 0 where p J = inf{p(x; x J and p+ J = sup{p(x; x J. Corresponding to these functions we define N Ap(,q( (ε to be the minimum of all those n N such that can be written as = n j=1 j, where each j is a closed sub-interval of, i j = 0 (i j and A p(,q( ( j ε for ever i. The quantities N Bp( (J, N Cp(,q( (ε, N Dp( (J are defined in an exactl similar wa. We shall write A p( (J = A p(,p( (J and C p( (J = C p(,p( (J, denoting these quantities b A p, C p respectivel if p(x = p is a constant function. When p(x = p and q(x = q are constant functions then we will write A p,q (J = A p(,q( (J and C p,q (J = C p(,q( (J. Functions of this kind were introduced in previous work on the s numbers of Hard-tpe operators in the context of classical Lebesgue spaces (see, for example, [2], [3], [4] and [7], and in fact for that situation we have the following result. Lemma 2.5. When J = (c, d, and p is a constant function, so that p(x = p (1, for all x, where π p = 2π p sin(π/p. A p (J = B p (J = (p p p 1 1/p J, 2π p The following lemma was proved in [13]. Lemma 2.6. Let J = (c, d, and p, q (1,. Then sup f T f q,j = (p + q 1 1 p + 1 q (p 1/q q 1/p J 1 1 p + 1 q f p,j B(1/p,, 1/q and the extremals are the non-zero multiples of cos p,q (π p,q x/2. This leads us to the following result. 5

6 Lemma 2.7. Let J = (c, d, and p, q (1, then A p,q (J = C p,q (J = (p + q 1 1 p + 1 q (p 1/q q 1/p J 1 1 p + 1 q 2B(1/p, 1/q (2.2 := B(p, q J 1 1 p + 1 q From Lemma 2.5, b using techniques from [5], [7], [8] and with the help of the well known inequalit b n (T min{c n (T, d n (T max{c n (T, d n (T a n (T, (2.3 we obtain the following theorem. Theorem 2.8. Let p be as in Lemma 2.5 and let T be viewed as a map from L p ( to itself. Then for all n N, a n (T = c n (T = d n (T = b n (T = (p p p 1 1/p π p n, where π p = 2π p sin(π/p. t is known that under the conditions of the last lemma, A(J depends continuousl on the right-hand endpoint of J; that is, with a slight abuse of notation, the function A(c, is continuous. A similar result holds for non-constant p : this is formulated in the next lemma together with the corresponding results for B, C and D. Lemma 2.9. Let p(, q( P(. Then the functions A p(,q( (c, t, B p( (c, t, C p(,q( (c, t and D p( (c, t of the variable t are nondecreasing and continuous. Analogousl the functions A p(,q( (t, d, B p( (t, d, C p(,q( (t, d and D p( (t, d are nondecreasing and continuous. Proof. We start with A := A p(,q(. First we prove that A(c, d A(c, d + h when h 0. Clearl { A(c, d + h = inf sup f(tdt ; f p(,(c,d+h 1 (c,d+h q(,(c,d+h { { = min inf sup f(tdt ; f p(,(c,d+h 1, (c,d q(,(c,d+h { inf sup f(tdt ; f p(,(c,d+h 1 := min{x, Y. (d,d+h q(,(c,d+h Now X { inf sup (c,d f(tdt ; f p(,(c,d 1 = A(c, d q(,(c,d 6

7 and { Y inf sup f(tdt ; f p(,(c,d 1 (d,d+h q(,(c,d { sup f(tdt ; f p(,(c,d 1 q(,(c,d d { inf sup (c,d f(tdt ; f p(,(c,d = A(c, d, q(,(c,d which gives A(c, d + h A(c, d. Let us prove the continuit of A. B Hölder s inequalit (see [6] we have, for some α 1 (independent of f, x and, x f(tdt α 1 p (,(,x f p(,(,x and considering 1 p (,(,x as a function of x we obtain which gives 1 p (,(,x q(,(d,d+h 1 p (,(c,d+h 1 q(,(d,d+h 7

8 A(c, d A(c, d + h { = inf sup (c,d+h inf sup (c,d+h inf sup (c,d+h f(tdt ; f p(,(c,d+h 1 q(,(c,d+h { f(tdt { inf sup (c,d+h q(,(c,d + f(tdt ; f p(,(c,d+h 1 q(,(d,d+h { f(tdt q(,(c,d +α 1 p (,(,x f p(,(,x ; f p(,(c,d+h 1 q(,(d,d+h +α { inf sup (c,d+h f(tdt 1 p (,(,x q(,(c,d ; f p(,(c,d+h 1 q(,(d,d+h f(tdt ; f p(,(c,d+h 1 q(,(c,d + α 1 p (,(c,d+h 1 q(,(d,d+h { inf sup f(tdt ; f p(,(c,d+h 1 (c,d q(,(c,d + α 1 p (,(c,d+h 1 q(,(d,d+h { = inf sup f(tdt ; f p(,(c,d 1 (c,d q(,(c,d + α 1 p (,(c,d+h 1 q(,(d,d+h = A(c, d + α 1 p (,(c,d+h 1 q(,(d,d+h. Since q(x P( we know that 1 q(,(d,d+h 0 as h 0 and so, A(c, is right-continuous. Left-continuit is proved in a corresponding manner, and the continuit of A(c,. follows. The arguments for B, C and D are similar. As an immediate consequence of this and Lemma 2.3 we have Lemma Let p( P(. Then T : L p( ( L p( ( is compact and for all ε > 0 the quantities N Ap( (ε, N Bp( (ε, N Cp( (ε and N Dp( (ε are finite. We now have Lemma Let p( be as in the last lemma. Then given an N N, there exists a unique ε > 0 such that N A (ε = N, and there is a non-overlapping 8

9 covering of b intervals A i (i = 1,..., N such that A(i A = ε for i = 1,..., N. The same result holds when A is replaced b B, C, D. Proof. The existence follows from the continuit properties established in Lemma 2.9. For uniqueness, observe that given two non-overlapping coverings of, {A i N and {JA i N, there are m, j, k, l such that m A J j A and J A k l A. Now, assuming A(A i = ε 1, A(JA i = ε 2 we obtain ε 1 ε 2 ε 1 b the monotonicit of A. 3. The case when p( is a step-function Let {J i m be a disjoint covering of b intervals and let p be the stepfunction defined b m p(x = χ Ji (xp i, (3.1 where each p i belongs to (1,. For simplicit, in this section we shall write A instead of A p( ; B, C, D will have the analogous meaning. Lemma 3.1. Let p( be the step-function given b (3.1. Then T : L p( ( L p( ( is compact and for sufficientl small ε > 0, (i b NC (ε m(t > ε, (ii a NA (ε+2m 1(T < ε. Proof. Let ε > 0. The compactness of T follows from Lemma 2.10, as does the finiteness of N A (ε and N C (ε. (i B the continuit of C(c,, there exists a set of non-overlapping intervals { i : i =,..., N C (ε covering and such that C( i = ε whenever 1 i < N C (ε and C( NC (ε ε. Let η (0, ε. Then corresponding to each i, 1 i < N C (ε, there is a function f i such that suppf i i := (a i, a i+1, f i p( = 1, ε η < T f i p( ε and (T f(a i = (T f(a i+1 = 0. B { ik M k=1 we denote the set of those intervals i, 1 i < N C (ε, each of which is contained in one of the intervals J l from the definition (3.1 of p(. Then Define b N C (ε m M N C (ε. X M = { f = M α ir f ir ; α ir r=1 R an M dimensional subspace of L p( (. Note that since p( is constant on ir, p(x = p ir on ir. Choose 0 f X M. 9

10 With λ 0 := T f p we have Hence p(x 1 T f(x M p(x λ 0 T f(x r=1 ir λ 0 M ( pi 1 r M ( pi = T f(x pir ε η r f(x pir λ r=1 0 ir λ r=1 0 ir M p(x p(x = f(x r=1 i λ r 0 /(ε η = f(x M r=1 λ ir 0 /(ε η p(x = f(x λ 0 /(ε η. f p, T f p, /(ε η, and so b NC (ε m b M (T ε η. (ii This follows a pattern similar to that of (i. This time we let { i N A(ε be a set of non-overlapping intervals covering for which A( i = ε for i = 1,..., N A (ε 1 and A( NA (ε ε. B { + i M we denote the famil of all nonempt intervals for which there exist j and k such that + i = j J k. Clearl N A (ε M N A (ε + 2(m 1. Let η > 0. Then given an i {1, 2,..., M there exists i + i such that { sup : f p, + = 1 ε + η. i Define i f P ε (f = p, + i M ( i a f(x χ +. i Plainl P ε is a linear map from L p( ( to L p( ( with rank M. Let p i be the constant value of p( on + i. Then we have for an λ 0 (0, and f L p( (, (T P ε f λ 0 p(x = M M + i x i f λ 0 p(x λ p i 0 (ε + η pi = Now choose λ 0 = (1 η (T P ε f p,. Then 1 < (T P ε f λ 0 p(x + i M λ p i 0 + i x i f pi f p i = f(x λ 0 /(ε + η f(x λ 0 /(ε + η p(x, p(x. 10

11 from which we see that so that f p(, > (1 η (T P ε f p(, /(ε + η, ε + η 1 η > (T P εf p(, f p(,. The proof is completed on letting η 0. Lemma 3.2. Let p( be the step-function given b (3.1. Then lim εn(ε = 1 ( p (xp(x p(x 1 1/p(x sin(π/p(x, ε 0 2π where N stands for N A, N B, N C or N D. Proof. Simpl use the fact that p( is a step function together with Lemmas 2.5 and Finall we can give the main result of this section. Theorem 3.3. Let p( be the step-function given b (3.1. Then for the compact map T : L p( ( L p( ( we have lim ns n(t = 1 ( p (xp(x p(x 1 1/p(x sin(π/p(x, n 2π where s n denote the n th approximation, Gelfand, Kolmogorov or Bernstein number of T. Proof. Using Lemmas 3.1 together with inequalities (2.3, we have and εn A (ε a NA (ε+2m 1N A (ε b NA (ε+2m 1N A (ε εn C (ε b NC (ε mn C (ε. Now use Lemma 3.2 to obtain the result for the approximation and Bernstein numbers. The rest follows from (2.3 again. 4. The case when p( is strongl log-hölder-continuous To obtain a result in this case similar to that of Theorem 3.3 the idea is to approximate p b step-functions. This requires that control be kept of the changes in the various norms when p is replaced b an approximating function, and we begin b giving such a result, which has independent interest. Let p(, q( P( be such that for some ε (0, 1, 1 < p(x q(x p(x + ε for all x. (4.1 We know from Theorem 2.1 that L q( ( is continuousl embedded in L p( (; denote b id the norm of the corresponding embedding. Our object is to obtain upper and lower bounds for id in terms of ε. 11

12 Lemma 4.1. Suppose that p( and q( satisf (4.1 and that f M( is such that f(x q(x 1 Then f(x p(x ε + ε ε. Proof. Set 1 = {x : f(x < ε, 2 = {x : ε f(x 1 and 3 = {x : 1 < f(x.then 3 3 f(x p(x = f(x p(x = A j, sa. j j=1 Evidentl A 1 ε p(x ε ε ( and A 3 f(x q(x. (4.3 3 Since ε f(x 1 on 2 and ε < 1 we have, b (4.1, on 2, and so Hence j=1 ε ε ε q(x p(x f(x q(x p(x 1 1 f(x p(x q(x ε ε. A 2 = f(x q(x f(x p(x q(x ε ε 2 f(x q(x. 2 (4.4 Now (4.2, (4.3 and (4.4 give f(x p(x ε + ε ε f(x q(x + f(x q(x 2 3 ε + ε ( ε f(x q(x + f(x q(x 2 3 ε + ε ε f(x q(x ε + ε ε, as required. Lemma 4.2. Suppose that p( and q( satisf (4.1. Then id ε + ε ε. Proof. Observe that K := ε +ε ε > 1. Given an f such that f(x q(x 1, b Lemma 4.1 we see that f(x/k p(x f(x/k 1/p(x p(x = K 1 f(x p(x Thus id K. ( ε + ε ε /K = 1. 12

13 We now turn to lower bounds. Lemma 4.3. Suppose that p( and q( satisf (4.1 and that 1. Then id 1. Proof. Define a function g b g(x = 1/q(x (x. Then g(x q(x = 1. Since p(x/q(x 1, we have for each λ (0, 1, g(x/λ p(x = p(x/q(x λ p(x Hence id λ for each λ (0, 1, and so id λp(x λ = 1 λ > 1. Lemma 4.4. Suppose that p( and q( satisf (4.1 and that < 1. Then id ε. Proof. Again we consider the function g(x = 1/q(x : g(x q(x = 1. Since p(x 1 q(x = q(x p(x q(x ε/q(x ε, we have g(x p(x = Thus, for each positive λ < ε, p(x q(x = 1 p(x 1 q(x ε. g(x/λ p(x p(x p(x > g(x g(x ε ε/p(x = ε g(x p(x ε ε = 1. t follows that id λ for each λ < ε, which gives the result. Putting these results together we have the following theorem and corollar. Theorem 4.5. Suppose that p( and q( satisf (4.1. Then min(1, ε id ε + ε ε. Corollar 4.6. Let p( P( and suppose that for each n N, q n ( P( and ε n > 0, where lim n ε n = 0, and for all n N and all x, 1 < p(x q(x p(x + ε n. Denote b id n the natural embedding of L qn( ( in L p( (. Then lim id n = 1. n 13

14 n the next, we prove a few technical lemmas. Lemma 4.7. Let δ > 0 and let J be an interval and p(, q( P(J. Assume p(x q(x p(x + δ in J. Then (δ J + δ δ 2 A p( +δ,p( (J A q( (J (δ J + δ δ 2 A p(,p( +δ (J. Proof. Set B 1 = {f; f q( 1, B 2 = {f; f p( δ J + δ δ, where the norms are with respect to the interval J. B Theorem 4.5 we have f p( (δ J + δ δ f q( which gives B 1 B 2 and { { A q( (J = inf sup f ; f q( 1 = inf sup f ; f B 1 J q( J q( { inf sup (δ J + δ δ f ; f B 2 J p( +δ { = (δ J + δ δ 2 inf sup f f p(. J δ J + δ p( +δ δ ; δ J + δ δ 1 { = (δ J + δ δ 2 inf sup g ; g p( 1 J p( +δ = (δ J + δ δ 2 A p(,p( +δ (J The second part of the inequalit can be proved analogousl. Lemma 4.8. Let an interval J, with J 1, and p (1, be given. Then there exists a bounded positive function η defined on (0, 1, with η(δ 0 as δ 0, such that if p(, q( P(J with p p(x p + δ, p q(x p + δ in J, then (1 η(δ J 2δ A p( (J A q( (J (1 + η(δ J 2δ. Proof. t suffices to prove onl the right part of the inequalit. B Lemma 4.7 and (2.2 we have A p( (J A q( (J (δ J + δ δ 4 A p,p+δ(j A p+δ,p (J = (δ J + δ δ 4 B(p, p + δ B(p + δ, p J 1 1 p + 1 p+δ J 1 1 p+δ + 1 p = (δ J + δ δ 4 B(p, p + δ B(p + δ, p J 2δ p(p+δ (δ J + δ δ 4 B(p, p + δ B(p + δ, p J 2δ. 14

15 Since we can choose η(δ such that to establish our assertion. lim (δ J + δ 0 δ δ 4 B(p, p + δ B(p + δ, p = 1, η(δ := max{δ, (δ J + δ δ 4 B(p, p + δ B(p + δ, p 1 Lemma 4.9. Let δ > 0, a 1 < b 1 a 2 < b 2 and J i = (a i, b i, i = 1, 2. Assume that f 1, f 2 are functions on such that suppf i J i, i = 1, 2, and T f 1 p(,j1 > δ. Then T (f 1 f 2 p(, > δ. Proof. Since T (f 1 /δ p(,j1 > 1 we have b1 a 1 x a 1 f 1 (t δ dt p(x > 1. Then b T (f 1 f 2 (t p(x = δ a a1 = a b1 a 1 b1 a2 b a x a b a 1 b 1 a 2 x f 1 (t f 2 (t dt p(x b1 = δ a f 1 (t f 2 (t dt p(x δ a 1 b b 2... x a f 1 (t δ dt p(x > 1 and so T (f 1 f 2 p(, > δ. Next we recall the well-known concept of a log-hölder continuous function which is widel used in the theor of variable exponent spaces. Following current terminolog we shall sa that p( is log-hölder continuous if there is a positive constant L such that p(x p( ln x L for all x, with 0 < x < 1 2. n what follows we will require a little stronger condition on the function p( defined on. We remind the reader that = [a, b] is a compact interval. Definition Let p( P(. We sa that p( is strong log-hölder continuous (and write p( SLH( if there is an increasing continuous function ψ(t defined on [0, ] such that lim t 0 + ψ(t = 0 and p(x p( ln x ψ( x for all x, in with 0 < x < 1/2. (4.5 t is eas to see an Lipschitz or Hölder function p( is SLH(. 15

16 Lemma Let p( P( be strong log-hölder continuous on. Then lim εn(ε = 1 ( p (xp(x p(x 1 1/p(x sin(π/p(x, ε 0 2π where N stands for N Ap( or N Cp(. Proof. We prove onl the case N = N Ap(, the case N = N Cp( follows b a simple modification. Let N N. B Lemma 2.11 there exists a constant ε N > 0 and a set of non-overlapping intervals {i N N covering such that A p( (i N = ε N for ever i. Define a step function q N (x b and set q N (x = p + χ i N N i δ N,i = p + p i N i N (x.. Then p(x q N (x p(x + δ N,i for all i = 1, 2,..., N. Claim 1. ε N 0 as N. Proof. Clearl, ε N is non-increasing. Assume for a moment that there exists δ > 0 such that ε N > δ for all N. Fix N and denote i N := i = (a i, a i+1. Since A p(,i > δ there are f i, with suppf i i, such that f i p(,i 1 and a i f i = T f i p(,i > δ for i = 1,..., N. B Lemma 4.9, p(,i T (f i f j p(, > δ for i < j and so, we have found N functions f 1, f 2,..., f N from the unit ball such that T (f i f j p(, > δ. The fact that N can be arbitrar contradicts the compactness of T. Claim 2. lim N max{ N i ; i = 1, 2,..., N = 0. Proof. Assume the contrar. Then there are sequences N k, i k {1, 2,..., N k and an interval J such that J N k i k and so, ε Nk which contradicts the fact that ε N 0. = A p( ( N k i k A p( (J > 0 Claim 3. There is a sequence β N 1 such that holds for all i {1, 2,..., N. β 1 N ε N N i 2δ N,i A qn ( ( N i β N ε N N i 2δ N,i 16

17 Proof. Since p q N (x, p(x p + δ N,i on N i we have b Lemma 4.8, (1 η(δ N,i N i 2δ N,i A p( ( N i A qn ( ( N i (1 + η(δ N,i N i 2δ N,i. Now, using ε N = A p( ( N i we have ε N 1 + η(δ N,i N i 2δ N,i A qn ( (i N ε N 1 η(δ N,i N i 2δ N,i and the assertion follows. Claim 4. The inequalit holds for all N and i {1, 2,..., N. N i δ N,i e ψ( N i Proof. Fix i N. Since p( SLH( we know that p( is a continuous function on. Because p + p = δ i N i N N,i there are points x, i N with p(x p( = δ N,i. Using (4.5 we obtain N i δ N,i x p(x p( e ψ( x e ψ( N i. Claim 5. There is a constant C > 0 such that the inequalit C 1 ε N N i C ε N holds for all N and i {1, 2,..., N. Proof. Remark that q N ( = p i N and so, b Lemma 2.7, + δ N,i := r N,i is a constant function on N i A qn ( ( N i = B(r N,i, r N,i N i. t is eas to see that there is a > 0 such that a 1 B(r N,i, r N,i a holds for all N and i {1, 2,..., N. Using Claim 4 we have and b Claim 3, Hence Since β N 1, the assertion follows. N i 2δ N,i e 2ψ( N i e 2ψ( := K, K 1 β 1 N ε N B(r N,i, r N,i N i Kβ N ε N. a 1 K 1 β 1 N ε N N i akβ N ε N. 17

18 Now, since b Claim 1, ε N 0 we know b Claim 5 that max{ N i ; i = 1, 2,..., N 0 as N and b Claim 4, N i 2δ N,i e 2ψ( N i e 2ψ(max{ N i ;,2,...,N := γ N 1. Setting α N = β N γ N we obtain α N 1, and b Claim 2 we have Moreover, b Claim 5 we have which gives, b 4.6, Consequentl, Nε N (α 1 N α 1 N ε N A qn ( ( N i α N ε N. (4.6 Nε N = C C 1 ε N C i N = C 1 N (α 1 N ε N ε N A qn ( ( N i (α N ε N ε N Nε N (α N 1. A qn ( (i N Nε N 0 as N. Nε N On the other hand we have b Lemma 2.5 (recall again that q N ( is constant on i N, A qn ( (i N = 1 (q N ( q N ( q N ( 1 1/ q N ( sin(π/q N ( i N 2π and so, 1 2π ( p (xp(x p(x 1 1/p(x sin(π/p(x, lim Nε N = 1 ( p (xp(x p(x 1 1/p(x sin(π/p(x. N 2π Since ε N is monotone it is not difficult to see lim N Nε N = lim ε 0 εn(ε and consequentl lim εn(ε = 1 ( p (xp(x p(x 1 1/p(x sin(π/p(x. ε 0 2π 18

19 Given p( SLH(, we construct step-functions that are approximations to p(. Let N N and use Lemma 2.11, applied to the function D := D p( : there exists ε > 0 such that N D (ε = N and there are non-overlapping intervals i D (i = 1,..., N that cover and are such that D(i D = ε for i = 1,..., N. Define N p + D,N (x = N χ D i (x, p D,N (x = (x; p + D i step-functions p + B,N ( and p B,N ( are defined in an exactl similar wa, with the function B in place of D and with intervals i B arising from the use of that part of Lemma 2.11 related to B. Lemma Let p( P( and N N. Let ε > 0 correspond to N in the sense of Lemma 2.11, applied to B, so that N B (ε = N, and write p D i χ D i p (x = p B,N (x, p+ (x = p + B,N (x, where p B,N ( and p+ B,N ( are defined as indicated above. Then a N+1 (T : L p ( ( L p+ ( ( ε. Proof. n the notation of Lemma 2.11, there are intervals i B such that B(i B = ε for i = 1,..., N. For each i there exists i i B such that B( B i = sup { i f p +, B i : f p, B i 1. Define P N f(x = i a f(d χ B i plainl P N has rank N. Let f L p ( ( and set (x; Then 1 = f(x λ 0 /ε p (x λ 0 = ε f p,. (4.7 = B i f(x λ 0 /ε p (x Recall that on i B the functions p ( and p + ( have constant values p i, p+ i, sa, respectivel, with p + i /p i 1. Thus. ( 1 i B f(x λ 0 /ε p i p + i /p i = (ε/λ 0 p+ i ( B i f(x p i p + i /p i. 19

20 Use of the fact that ε = sup f ( B i x i p + 1/p + i i / ( f(d B i f( p i d 1/p i now gives 1 (1/λ 0 p+ i B i x i = (T P N (f(x λ 0 f(d p + (x, p + i N = i B x i f(d λ 0 p + i from which it follows that (T P N f p +, λ 0. Using the definition (4.7 of λ 0 we see that (T P N f p +, ε f p,, and so a N+1 (T : L p ( ( L p+ ( ( ε, as claimed. We next obtain a lower estimate for the Bernstein numbers. Lemma Let p( P( and N N. Let ε > 0 correspond to N in the sense of Lemma 2.11, applied to D, so that N D (ε = N, and write p (x = p D,N (x, p+ (x = p + D,N (x, where p D,N ( and p+ D,N ( are defined as indicated above. Then b N (T : L p+ ( ( L p ( ( ε. Proof. n the notation of Lemma 2.11, there are intervals i D such that D(i D = ε for i = 1,..., N. Since T is compact, for each i there exists f i L p+ (i D with supp f i i D, T f i p, / f i D i p +, = ε, (4.8 i D and T f i (c i = T f i (c i+1 = 0, where c i and c i+1 are the endpoints of i D. On each i D the functions p ( and p + ( are constant; denote these constant values b p i and p + i, respectivel and note that p i /p+ i 1. Set X N = { f = α i f i ; α i R. 20

21 Then dim X N = N. Choose an non-zero f X N and set λ 0 = ε f p +,. Then 1 = f(x λ 0 /ε p + (x = ( f(x i D λ 0 /ε ( = (ε/λ 0 p i p + i D i D i p i /p+ i f(x λ 0 /ε = α i f i (x p+ i p + (x (ε/λ 0 p i p i /p+ i. ( D i f(x p+ i p i /p+ i Use of (4.8 now shows that 1 (1/λ 0 p i = D i D i T f(x λ 0 p i T (α i f i (x p i = T f(x λ 0 p (x which follows that ε b N (T : L p+ ( ( L p ( (, and the proof is complete. Theorem Let p( P( be continuous on. For all N N denote b ε N numbers satisfing N = N B (ε N. Then there are sequences K N, L N with K N 1, L N 1 as N such that (i a N+1 (T : L p( ( L p( ( K N ε N, (ii b N (T : L p( ( L p( ( L N ε N. Proof. First we prove (i. n view of the propert (2.1 of the approximation numbers, a N+1 (T : L p( ( L p( ( is majorized b id N : Lp( ( L p B,N ( an+1 (T : L p B,N ( L p + B,N ( id + N : Lp+ B,N ( L p( (, where id and id + are the obvious embedding maps. 0 and p( is continuous, it is clear that B i p( p B,N ( 0 and p( p + B,N ( 0. When N, since 21

22 (Here i B, p+ B,N (, p B,N ( are the same as in Lemma Thus b Corollar 4.6, id N : Lp( ( L p B,N id ( 1, + N : Lp+ B,N ( L p( ( 1 as N. The result now follows from Lemma To prove (ii we follow the idea of the proof of (i with the help of Lemma Theorem Let p SLH(. Then lim ns n(t = 1 n 2π { p (tp(t p(t 1 1/p(t sin ( π p(t dt, where s n denote the n th approximation, Gelfand, Kolmogorov or Bernstein number of T. Proof. Use Theorem 4.14, Lemma 4.11 and the inequalit (2.3. t is not difficult to combine proofs of Theorem 3.3 and Theorem 4.15 to obtain the following theorem, which contains both these results. Theorem Let J i, i = 1, 2,..., m be a finite decomposition of. Assume that p( be such that p( SLH( i for each i {1, 2,..., m. Then lim ns n(t = 1 { p (tp(t p(t 1 ( 1/p(t π sin dt, n 2π p(t where s n denote the n th approximation, Gelfand, Kolmogorov or Bernstein number of T. References [1] Edmunds, D.E. and Evans, W.D., Spectral theor and differential operators, Oxford Universit Press, [2] Edmunds, D.E. and Evans, W.D., Hard operators, function spaces and embeddings, Springer-Verlag, Berlin, [3] Edmunds, D.E. and Lang, J., Approximation numbers and Kolmogorov widths of Hard-tpe operators in a non-homogeneous case, Math. Nachr. 279 (2006, [4] Edmunds, D.E. and Lang, J., Bernstein widths of Hard-tpe operators in a non-homogeneous case, J. Math. Anal. Appl. 325 (2007, [5] Edmunds, D.E. and Lang, J., Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case, Journal of Functional Analsis, 206 (2004,

23 [6] Ková cik, O. and Rákosník, J., On spaces L p(x (Ω and W k,p(x (Ω, Czech. Math. J. 41 (1991, [7] Lang, J., mproved estimates for the approximation numbers of Hard-tpe operators, J. Approx. Theor 121 (2006, [8] Lang, J. Estimates for n-widths of the Hard-tpe operators. J. Approx. Theor 140 (2006, no. 2, [9] Lorentz, G.G., Golitschek, V. and Makovoz, Yu., Constructive Approximation: advanced problems Springer Verlag, Berlin, 1996 [10] Pietsch, A., Bad properties of the Bernstein numbers, Studia Math. 184 (2008, [11] Pinkus, A., n widths in approximation theor, Springer-Verlag, Berlin, [12] Růžička, M., Electrorheological fluids: modeling and mathematical theor, Lecture Notes in mathematics 1748, Springer-Verlag, Berlin, [13] Schmidt, E. Über die Ungleichung, welche die ntegrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet. (German Math. Ann. 117, (