ON THE APPROXIMATION NUMBERS OF CERTAIN VOLTERRA INTEGRAL OPERATORS BETWEEN LEBESGUE SPACES

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1 STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LII, Number, Mrch 27 ON THE APPROXIMATION NUMBERS OF CERTAIN VOLTERRA INTEGRAL OPERATORS BETWEEN LEBESGUE SPACES Abstrct. The im of the pper is to study certin Volterr integrl opertors of the form T f(x) = v(x) x u(t)f(t)dt which defines liner mps between the Lebesgue spces L p (R + ) nd L (R + ) where < p +. Under some conditions of the integrbility on its kernel, we show some importnt properties for T such s boundedness, compctness, the mesure of non compctness nd estimting, upper nd lower bounds for its pproximtion numbers. These estimtes hve n ppliction to the spectrl properties of Sturm-Liouville differentil opertors.. Introduction In this pper we study certin liner integrl opertors of the form T f(x) = v(x) x u(t)f(t)dt (x R + := [, + [). (.) These opertors pper nturlly in the theory of differentil eutions nd it is importnt to estblish when opertors of this kind hve properties (under some conditions of the integrbility on the kernel) such s boundedness, compctness nd to estimte their eigenvlues, or their singulr numbers (pproximtion numbers) if these exist. Our concern in this pper lies with the problem which rises when integrbility conditions on the kernel re wekened to locl integrbility reuirements. Here, we consider in () tht u nd v re functions stisfying the locl integrbility nd integrbility conditions respectively nd our objective is to give precise estimtes for the pproximtion numbers of T. This opertor hs been studied extensively during Received by the editors: Mthemtics Subject Clssifiction. 4B7. Key words nd phrses. Volterr integrl, Mesure of non-compctness, Approximtion numbers. 3

2 the lst severl decdes [2, 7, ] in L 2 (R + ) nd L p (R + ) spces, respectively. Lstly Edmunds et l [4] generlized their results to the sitting in which T mps L p (R + ) to L (R + ) with p + by considering the lower nd upper bounds for the pproximtion numbers nd hence for the mesure of non compctness of T. This pper extends their results to the cse < p +. The pper is orgnized s follows. In the next section we stte our results concerning some properties such tht the boundedness, compctness nd of course the mesure of non compctness of the opertor T. In section 3 we give precise estimtes for the pproximtion numbers of T. In section 4 we give n exmple to illustrte our ides. 2. Properties of T Throughout the pper, we use the sme nottion s in [4], nd we shll ssume tht p, [, + [, tht p = p, nd u nd v re prescribed rel-vlued functions p such tht nd u L p loc ( R + ) (2.) v L ( R +). (2.2) Given ny mesure µ on R +, ny µ mesurble subset S of R + nd ny function f in L p (S, µ), we shll write: ( f p,s,µ = S f p dµ) p ( p < + ), f,s,µ = µ ess sup f(t) if µ is Lebesgue mesure we shll omit the suffix µ nd simply write f p,s if no mbiguity will result. For ny R + we put 4 J = + p p ( x ) ( + u(t) p dt v(z) )dz u(x) p dx x (2.3)

3 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS which define continuous nonnegtive function on R Boundedness. Theorem. Let < p +. Then the opertor T defined in () is bounded liner mp of L p (R + ) to L (R + ) if, nd only if, J < < p +, while where J is defined in (4) with =. <, nd in this cse, when ( p ) J T (p ) J (2.4) T = J if = nd < p + (2.5) T = J for < < p = + (2.6) Proof. For ll the cses < p +, we hve + T f(x) = x v(x) u(t)f(t)dt dx nd so the result follows from Mz j [9, theorem.3.2/] Compctness nd mesure of non-compctness. Let K p, denotes the set of ll compct liner mppings from L p (R + ) to L (R + ), nd stnd by F p, for ll those elements of K p, which re of finite rnk, put K p,p = K p, F p,p = F p nd write α(t ) = inf { T P ; P F p, }. (2.7) Since L (R + ) hs the pproximtion property, it follows tht α(t ) is the distnce of T from K p, ( [4]). Theorem 2. Let < p +. If J <, then nd ( p ) lim J α(t ) (p ) lim J, if > (2.8) + + α(t ) = lim J + for =. (2.9) 5

4 To prove theorem 2.2 we use the following lemm (decomposition lemm). Lemm 3. ([4, theorem 2]). Suppose T is bounded liner opertor from L p (R + ) to L (R + ). Then for every X (, ) there exist integrl opertors P X nd T X both of the sme type of T such tht:)p X is compct liner opertor from L p (R + ) to L (R + ), 2) T X is bounded liner opertor from L p (R + ) to L (R + ) with T X p JX for > nd T X = J X for = 3) T f = T X f + P X f for ll f L p (R + ). Proof. of the theorem 2.2 Choose X [, ) nd let T = T X + P X be the decomposition in the lemm bove. By reclling some properties of the mesure of the non-compctness (cf. [3, 2] ) for more detils) nd theorem 2., we see tht T X is bounded liner opertor from L p (R + ) to L (R + ) with α(t ) = α(t X ) T X p JX for >. Hence we hve estblished tht since J X < lim X J X then nd α(t ) T X p lim J X for > X + α(t ) lim X J X for =. To estblish the lower bound for α(t ), we use the method employed by Evns nd Hrris [5, 2] to prove similr results for embedding mps. Let < p nd λ > α(t ). Then there exists P F p, (L p (R + ), L (R + )) with rnkp N, such tht for ll f L p (R + ), T f P f λ f p. By the rgument of the [5, lemm 2.2 ], 6

5 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS we my nd shll suppose tht there exists Y R + such tht for ll f L p (R + ), SuppP f [, Y ]. Hence + T f(x) dx = + Y Y v(x) X u(t)f(t)dt dx. (2.) Now observe tht + Y T f(x) dx = + Y + Y v(x) v(x) X X Y u(t)f(t)dt dx u(t)f(t)dt dx. We my ssume tht uf nd by theorem 2. ensure tht there exists f in L p (R + ) such tht + Y From () we hve v(x) X Y u(t)f(t)dt dx ( p ) JY f p. λ f p ( p ) JY f p for < p nd s λ my be chosen rbitrry close to α(t ), we finlly obtin: This completes the proof. ( p ) lim J Y α(t ). Y Corollry 4. The liner integrl opertor T from L p (R + ) to L (R + ) is compct if, nd only if, lim J =. + The proof of the corollry follows immeditely from theorem Approximtion numbers Our objective in this section is to estimte the pproximtion numbers n (T ) of T. This section follows closely the rgument developed in [4] for the cse p nd severl new fetures emerge becuse of the interchnge of the order of p nd. To chieve this gol the following lemms re crucil. Some nottion 7

6 will be useful. Given ny intervl I with end points (, b) with ( b) nd ny fin L p (R + ), set l(i, f) = l(, b, f) = b b v(x)u(x) y x u(t)f(t)dt dxdy (3.) nd define L(I) = [ µ(i) sup { l(, b) : f p,i } ] if µ(i) otherwise (3.2) where I is ny intervl in R + with end points nd b, nd µ is the finite mesure defined by dµ(x) = v(x) dx, so tht µ(i) = I v(x) dx. In fct the two untities stisfy the following lemms. Lemm 5. For ll bounded intervls I R + with end points nd b, the untity { } sup l(, b, f) : f p,i,µ depends continuously upon nd b. Lemm 6. The second untity L(, b) is monotonic decresing s increses nd monotonic incresing s b increses. Remrk. To del with infinite intervl we set L(, ) = possible tht L(, ) my be infinite. Another piece of nottion will be useful. We shll write F (x) = x lim L(, b) nd it is b u(t)f(t)dt ( f in L p (R + ), x ) (3.3) F I (x) = F (x)dµ(x) if µ(i). (3.4) µ(i) I Given ny, b with b > nd ny c in I = ], b[, we put A c = c ( c ) u(t) p dt x x p p v(z) dz u(x) p dx (3.5) 8

7 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS nd B c = b c ( x ) u(t) p dt c b x p p v(z) dz u(x) p dx (3.6) nd we write W (I) = mx(a c, B c ) where c is the minimum point in I such tht : c v(y) dy = 2 b v(y) dy. Remrk 2. W (I) is right continuous function of b since A c nd B c re both right continuous s functions of b nd if for lmost ll x in I, v(x), then c is uniue nd W (I) depends continuously on b. Finlly we write { } sup F F I,I,µ / f p,i in L p (I), f if µ(i) K(I) = if µ(i) =. (3.7) We refer this section to the [4, section 4] for full discussion of the significnce of these definitions nd for the proofs of the previous lemms. Lemm 7. Let < p +. Then given ny, b with < b, 2p + (p ) W (I) if nd p < K(I) 3 22 p (p ) 2 W (I) if = 2 nd p < nd K(I) > ( 2 )( p ) W (I) if < < p <. Proof. First consider the cse 2 nd p < +. To estblish the upper bound for K(I), let c be ny point in ], b[. We hve F F (c),[,c],µ = = c c F (x) F (c) dµ(x) in [, c] x u(t)f(t)dt dµ(x) = T f,[,c],µ. 9

8 By theorem 2. nd simple trnsformtion tht for every f in L p ([, c]), we hve F F (c),[,c],µ (p ) A c f p,[,c],µ. (3.8) Similrly in [c, b] we hve F F (c),[c,b],µ = T f,[c,b],µ nd lso by theorem 2. shows tht for every f in L p ([c, b]) Hence from (8) nd (9) we hve F F (c) p,[,b],µ = F F (c),[c,b],µ (p ) B c f p,[c,b],µ. (3.9) { } p/ F F (c),[,c],µ + F F (c),[c,b],µ p p (p ) { } p/ A c f p,[,c],µ + B c f p,[c,b],µ p p ( ) f (p ) p p,[,c],µ + f p p,[c,b],µ ( ) (p )/p A p/p c + Bc p/p p p (p ) 2 (p )/ (mx(a c, B c )) p f p,[,b],µ, p/ therefore F F (c),[,b],µ (p ) 2 p W (I) f p,[,b],µ, since F F I,I,µ F F (c),i,µ + (F F (c)) I,I,µ = 2 F F (c),i,µ (p ) 2 p + W (I) f p,i,µ, which shows the first ineulity (upper bound for K(I) in the lemm).

9 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS However, if = 2, then { } F F I 2,I,µ = inf F F (c) 2,I,µ : c constnt F F (c) 2,I,µ, 3p 2 2 2p (p ) 2 W (I) f p,i,µ. To estblish the lower bounds for K(I). First consider the cse nd let c be ny point in ], b[. Assume first tht x [, c] then F,[,c],µ = T f,[,c],µ since F (x) = x c u(t)f (t)dt, it follows by theorem 2., tht there exists f in L p [, c] such tht F,[,c],µ ( ) Ac f p p,[,c],µ. (3.) Now define f in [, c] g(x) = otherwise nd G(x) = x u(t)g(t)dt. We wnt to estimte G G I,I,µ G,I,µ G I,I,µ with I = [, b]. Since µ [, c] G [,b] = µ [, b] F [, c], by pplying Holder s ineulity to the bove G[,b] µ [, b] F,[,c],µ (µ [, c]), conseuently G I ( ) µ [, c] F µ [, b].,[,c],µ

10 Therefore G G I,I,µ G,I,µ G I,I,µ ( ) µ [, c] µ [, b] F,[,c],µ ( µ [, c] µ [, b] ( µ [, c] µ [, b] ) ) ( ) Ac f p p,[,c],µ ( ) Ac g p p,i. (3.) Similrly for x [c, b] F 2,[c,b],µ = T f,[c,b],µ ( ) Bc f 2 p p,[c,b],µ (3.2) nd lso by theorem there exists n f 2 L P [c, b] corresponding to c such tht (23) holds. We define lso otherwise h(x) = f 2 in [c, b] nd H(x) = x u(t)h(t)dt, then 2 otherwise H(x) = F 2 in [c, b],

11 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS therefore H H I,[c,b],µ H,[c,b],µ H I,[c,b],µ ( ) µ [c, b] µ [, b] F 2,[c,b],µ (3.3) ( µ [c, b] µ [, b] ( µ [c, b] µ [, b] ) ) ( ) Bc f p p,[c,b],µ ( ) Bc h p p,i,µ. If we tke G(x) if A c < B c g(x) Φ(x) = nd θ(x) = H(x) if A c B c h(x) if A c < B c if A c B c We get ( µ [, c] Φ Φ I,I,µ µ [, b] ) ( ) W (I) θ p,i,µ. p If c is the minimum point in ], b[ for which µ [, c] = µ [c, b] = 2 µ [, b] then Φ Φ I,I,µ 2 ( ) W (I) θ p,i,µ p which the lst ineulity for K(I). The proof of the lemm for < < p + is complete. Remrk 3. Inspection of the proof shows tht the lemm holds with W (I) replced by inf {mx(a α, B α ) : α ], b[}. Lemm 8. [4, lemm 6] Let < p + nd given ny, b with < b then K(I) = 2 L(I) if = 2 3

12 nd L(I) K(I) L(I) if 2 2 where I = [, b]. Now we re redy to stte the result concerning the pproximtion numbers of T. Lemm 9. Let < p. Let ε > nd suppose tht there exists N N(N denotes the set of ll nturl numbers) nd numbers c k (k =,,..., N + ) with = c < c < < c N+ = such tht L(I k ) ε for k =,,..., N where I k = ]c k, c k+ [. Then with σ = if 2 nd σ = 2 if = 2, we hve N+2 σ ε(n + ) p. Proof. Let f L p (R + ) be such tht f p,r + = nd we write P f = N P Ik f where P I = χ I v(x)f I. Then P is bounded liner mp from L P (R + ) to L (R + ) with rnk P N +. Also we hve k= T f P f,r = + = N T f P Ik f,i K k= N F F Ik,I k,µ, k= { N F FIk },I sup k,µ f p,i k k= f p,ik N K (I k ) f p,i k. k= By the hypothesis of the lemm nd the lemm 3.3 we hve 4 K(I k ) L(I k )σ εσ

13 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS then T f P f,r + < N ε σ f p,i k k= Applying Holder s ineulity to the N therefore N k= f,i k N = ε σ f,i k. f,i k k= k= k=, we hve ( N ) p ( ) N f p,i k k= ( < (N + ) N ) p f p,i k k= ( T f P f,r < ε σ (N + ) N ) p f p p + p,i k. k= Hence T f P f,r + εσ (N + ) p f p,ik nd since then f p,ik f p,r + =, T f P f,r + εσ (N + ) p which shows tht N+2 σ ε(n + ) p. Lemm. Let < p. Let ε > nd suppose tht there exists N N (N denotes the set of ll nturl numbers) nd numbers c k (k =,,..., N + ) with = c < c < < c N <, such tht L(I k ) ε for k =,,..., N where I k = ]c k, c k+ [. Then N (T ) ν ε where ν 2 = 2 nd ν = 4 if 2. 5

14 Proof. Let λ ], [. By lemm 3.3 nd the hypothesis tht L(I k ) ε for k =,,..., N, there exists θ k L p (I k ) such tht Φ k (Φ k ) Ik,I,µ / θ k p,ik > λη ε, where Φ k (x) = u(t)θ k (t)dt, nd η = /2 if 2 nd η 2 = / 2; set θ k (x) = for I k ll x R + I k. Let P F(L p (R + ), L (R + )) with rnk N, then there re constnts λ, λ 2,..., λ N not ll zero, such tht P ( N λ k θ k ) = with N λ k θ k =. Put k= k= p,r + θ = n x λ k θ k nd Φ(x) = u(t)θ(t)dt (x R + ). For ll x I k, Φ(x) = λ k θ k + µ k for k= some constnt µ k, for ll constnt C, we hve F F I,I,µ F C,I,µ + (C F ) I,I,µ 2 F C,I,µ when 2, while for = 2 we hve { } F F I 2,I,µ = inf F C 2, I, µ : C constnt, where the infinimum is tken over ll constnt C. Hence } F F I,I,µ δ inf { F C, I, µ where δ = 2 if 2 nd δ = if = 2. Thus T θ P θ,r + = T θ,r + N k= λ k Φ k + µ k,i k,µ 6 = N k= Φ,I k,µ δ N λ k Φ k (λ k Φ k ) Ik,I k,µ k= N = δ λ k Φ k (λ k Φ k ) Ik,I k,µ k= N = δ λ k Φ k (Φ k ) Ik,I k,µ > δ k= N (λη ε) k= λ k θ k,i k,µ

15 Since < p nd N It follows tht Therefore which shows tht APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS k= = δ N (λη ε) k= λ k θ k p p,i k,µ =, then T θ P θ,r + λ k θ k p p,i k,µ. λ k θ k p,i k,µ > λ kθ k p p,i k,µ. δ (λη ε) θ p p,r +, nd θ p p,r + =. T θ P θ,r + δ (λη ε) N (T ) (λη ε)δ. Since λ my be chosen rbitrrily close to it follows tht N (T ) (η ε)δ. Provided with these lemms ( lemm 3.5 nd lemm 3.6 ) we my produce our min result concerning the pproximtion numbers of T. Given ny ε >, define numbers c k by the rule tht c =, c k+ = inf {t; L(c k, t) > ε} (3.4) with understnding the inf =. We shll refer to these numbers s forming the (ε, L)-seuence for given ε there re two possibilities :(i) the (ε, L)-seuence is finite. Then there is n integer N such tht c < c <... < c N < c N+ =, nd by the continuity of L we hve {L(c k, c k+ ) = ε for k =,,..., N nd L(c N, c N+ ) ε. By the length of the (ε, L)-seuence we shll men the integer N +. (ii) The (ε, L)-seuence is infinite. Then L(c k, c k+ ) = ε for ll k N. If (i) holds, then by lemm 3.5 nd lemm 3.6 we see tht N+2 σ ε(n + ) p nd N+2 εν. 7

16 If (ii) holds,then by lemm 3.6 shows tht for ll n N ; N (T ) εν ; note lso tht in this cse, c k s k ; for if not nd thus for ll k. c k c <, c k c k+ L(c k, c k+ ) = ε Theorem. Let < p, ν = 4, σ = ( 2) nd ν 2 = ν = 2. Then ) T is bounded if, nd only, if L(, ) <. 2) Let ε (σ, L(, )) (where σ = lim L(x, )) nd let N + be the length x of the (ε, L)-seuence. Then we hve nd N+2 σ ε(n + ) p N (T ) ν ε. Proof. () First suppose L(, ) < nd tke ε = L(, ). In view of the monotonicity of L we see tht the length of the (ε, L) seuence is tht is N =.This is by lemm 3.5, 2 (T ) σ ε, tht is inf { T P } εσ where the infinimum is tken over ll bounded mps P of rnk from L p (R + ) to L (R + ). Since ech such mp is bounded, it follows tht is bounded. Conversely, suppose tht is bounded. Then for ny intervl I, K(I) 2L(I). Hence by the lemm 3.5 is bounded, uniformly in I. Thus L(, ) is bounded. 8

17 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS (2) Let ε (σ, L(R + )). First, suppose ε > σ nd suppose if possible tht the (ε, L)-seuence is infinite, then c k s k. We hve ε = L(c k, c k+ ) L(c k, ) nd hence ε σ which contrdicts ε > σ. Therefore the seuence is finite nd by lemm 3.5, N+2 (N + ) p εσ. Second, suppose if possible tht the seuence is finite of length N +. Then σ = lim x L(x, ) L(c N, ) = L(c N, c N+ ) ε nd we hve contrdiction. Therefore the seuence is infinite. Then by lemm 3.6, it follows tht Therefore Since this holds for rbitrry ε < σ, then This completes the proof. N (T ) ν ε. α(t ) = lim N N (T ) εν. α(t ) Lν. 4. An exmple To illustrte the scope of the theorem 3.7 we del with the sitution in which u(x) = e Ax, v(x) = e Bx (4.) for ll x R + where < A < B. From theorem 2. nd theorem 2.2 (Boundedness nd compctness of T ), we see tht: J = (Ap ) p (B) e (B A) β [ (p ) (p ), (B A ] A ) p (4.2) 9

18 (where β denotes the bet function) which shows tht J < nd lim J =. Thus T is compct liner mp from L p (R + ) to L (R + ) nd (Ap ) p (B( p )) (Ap ) p (B) β [ (p ) β [ (p ) (p ), (B A ] A ) p (p ), (B A ] A ) p T (4.3) Now, we estblish lower nd upper bounds for n (T ). For the lower bound, we obtin it by considering the expression for L(I) directly rther thn the expressions of K(I) or W (I) which re difficult to compute in prctice. First, we stte the following lemm which is useful in developing the previous tsk. Lemm 2. If x < y nd α ], [. Then Proof. We hve so α xα y α. (x y)xα d tα ( dt t ) = αtα + (α )t α + ( t) 2 = φ(t) ( t) 2, tα t The results follow on putting t = y x. t α lim t t = α. Now we hve l(i, f) ( b ( b c 2 e By dy c e By dy ) ( c2 c ) ( c ) ( y e Bx dx x ) e At f(t)dt ) ( c2 e Bx dx e dt) Ap t p c by suitble choice of f with f LP (I) =. Tke c, c 2 to be such ( c ) ( ) ( b ) e By dy = e By dy = b e By dy. c 2 3 2

19 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS By holder s ineulity if r = p +, we hve ( c2 Hence c ) ( c2 ) e r(b A)t r dt e By ( c dy e dt) Ap t p. c L(I) = sup f p,i = (l(i, f) I e By dy ( c2 c ) e r(b A)t r dt. Let now v = e B nd v 2 = e Bb, then e Bc = 2 3 v + 3 v 2, e Bc2 = 2 3 v v. We hve X = = ( c2 c ) e r(b A)t r dt = (r(b A)) r (e r(b A)c e r(b A)c2 ), r(b A)) r ( 2 3 v + 3 v 2) B A B B A ( 2 3 v v B ). Then by the bove lemm we hve ( ) B A B X 3 r (v v 2 ) r (r(b A)) r Agin ( dt) b r Y = e r(b A)t = (r(b A)) r ( 2 3 v + 3 v 2 ) B A B r. (4.4) ( e r(b A) e r(b A)b) r. 2

20 Hence Y (r(b A)) B A r (v v 2 ) r B r (v ). (4.5) From (29) nd (3) we see tht X Then from (3) we hve ( 2 3 ) B A B r (r(b A)) r Y. (4.6) ( ) b r L(I) c e r(b A)t dt (4.7) where c is constnt. Let c, c,..., c N, c N+ = be the (ε, L)-seuence defined in (25). Since L(I k ) = ε for k =,,..., N, it follows from (3) tht N N Nε r = L r (I k ) c ck+ k= k= c k e r(b A)x dx. We summrize our results s follows Theorem 3. Let u nd v be defined in (26) nd let < p. Then s n, n (T ) cn r (c is constnt) where r = p +. To derive n upper bound for n (T ) we use the result in [4] tht for the cse p =, n (T ) = ( n ). Let T f(x) = e Bx e At f(t)dt where B > B > A so tht T f(x) = e (B B)x T f(x). Now the mp T : L p (R + ) L (R + ) hs pproximtion numbers n (T ) = O( n ). The mp U : L p (R + ) L (R + ) defined by Uf(x) = e (B B)x f(x) is bounded (by holder s ineulity). It follows from the fct tht n (T ) n (T ) U = O( n ). 22

21 APPROXIMATION NUMBERS OF VOLTERRA INTEGRAL OPERATORS We summrize our results s follows. Theorem 4. Let u( nd) v be defined in (26) nd let < p. Then s n, n (T ) = O. n References [] Bchmn, G., Nrici, L., Functionl Anlysis, Acdemic Press, 978. [2] Chisholm, R.S., Everitt, W.N., On bounded integrl opertors in the spce of integrblesure functions, Proc. Roy. Soc. Edinburgh, A69(97), [3] Edmunds, D.E., Evns, W.D., Spectrl Theory nd Differentil Opertors, University University Press, Oxford, 987. [4] Edmunds, D.E., Evns, W.D., Hrris, D.J., The pproximtion numbers of certin Volterr opertors, J. London. Mth. Society (2), 37(988), [5] Evns, W.D., Hrris, D.J., Sobolev embeddings for generlized ridged domins, Proc. London Mth. Society (3), 54(987), [6] Jorgens, K., Liner integrl opertors (97), (Trnslted by G. F. Roch, 982). [7] Juberg, R.K., The mesure of non-compctness in L p for clss of integrl opertors, Indi Univ. Mth. J23(974), [8] Konig, H., Eigenvlue Distribution of Compct Opertors, Birkhuser, Boston, 986. [9] Mz j, V.G., Sobolev Spces, Springer, Berlin, 985. [] Pietsch, A., Nucler Loclly convex spces, Berlin, Heidelberg, New York, 972. [] Sturt, C.A., The mesure of non-compctness of liner integrl opertors, Proc. Roy. Soc. Edinburgh, A 7 (973), [2] Webb, J.R.L., On seminorms of opertors, J. London Mth. Soc. (2), 7(973), Déprtement de Mthémtiues, Fculté des Sciences Université Ferht Abbs, Sétif 9, Algérie E-mil ddress: Achche m@yhoo.fr 23