THE APPROXIMATION PROPERTY OF SOME VECTOR VALUED SOBOLEV-SLOBODECKIJ SPACES

In this paper we consider the Sobolev-Slobodeckij spaces Wm,p(ℜn,E) where E is a strict (LF)-space, m∈(0,∞)\ℕ and p∈[1,∞). We prove that Wm,p(ℜn,E) has the approximation property provided E has it, furthermore if E is a Banach space with the strict approximation property then Wm,p(ℜn,E) has this property.

t ,(E) e spa of linear ntuo om E wi e 1o of ifo nvergence on precompa sub of E. We y that E has Se (s) approximation prony, e [5, p. 2] or 10], if e ntuous linear oram of finite me (sly) den L,(E). (A subtA of a locally convex spa F is quasi-clo ff it nins all e acculation in in F of i unded sub.e quasi-closure ofA, denotA e intemeion of all qsilod of F at coninA.We of coupe have Cbut general CX. DB enA strictly den B [9, pp.[91][92].e [10] for nher inoation of Se strict approximation pro.) Now it is well own Sat Se bolev spas '(), 1 < p < ,m are omohic to Se be spas L(0,1).Pelcns d nator [8] prov at Sis is al e e aniopic case.
is r we prove at e bolevlj spas '(E), 1 p < ,m (0, ), E a strict (L-space ve e approximation o ovid E s i eore '(E) has e sct approximation prony E nach spa d s e sct approxation progy.e prf is bad on hw ent ove Sat me 1 spas of disbutio ve e sct approximation progy [10, .9 -10].tion I we ove t '(E) Se relig approximation property 10, p. 8].tion 2 we shall prove Sat '(E) s Se tg appxiation prony [10, p. .Finally e last ion we prove at E) scy de '(E) d Sat '(E) s Se (sict) approximation pro ovid E nach spa) s Se (s) approximation proy.

W't'(E) has the Regularizing Appreximatiou Preperty
Let gT(ffr') be the standard space of test functions in fit'.We say that a sequence {l]}s C (ffP) is regularizing ff 1) vl(x) 0 for all x $P and j ( , 2) Vl(X) 0 if Ix 1> e and e 0% 3) f rb(x)dx-1 for all j II.For rl 0() and f t L"(E) we define the convolution * [(x)-[ n(z)(x-z. Since the space of continuous functions with compact support of ffP into E is dense in L'(E) and using arguments similar to the scalar case we have: a) Vl * f (E) @pace of all E-valued infinitely differentiable functions on ffP), D'(Vl * f) (Dq) * f if a $r' and for any cs(E), n */11 nil ,.,., ' , b) v * f _q).,(E)-{f .(E):Df _Le(E) for all ct _l'}, c) ( * f)a converges to f in L " (E).
PROOF.We prove it for {@'};e,t, the other part is similar.First observe that the operator defined by g -%g is continuous in L'(E), the proof is the same as in the case E C. We also have: 1%(z)-%(z)l -I ,Y-/ll,-IIs -sll 1 -II 0'-s)ll, + f-sll, 211f-sll, Hence the mapping L'(E) Cb(ffP), f-.-, Of, is uniformly continuous and the lemma follows.
PROOF.Since E is a strict (LF)-spaee, E -/rid limEt and f _Le(E) then using the same argument as in [2, p. 255] there exists a k E !I such that f(x)GEt almost everywhere.So we can assume that fis a Bochner-measurable function from 9V to Et and using Pettis measurability theorem, valid for Frchet spaces, we have thatF is Boehner measurable from 9P 9P inEt so F is Boelmer-measurable from ffP 9P into E.
PROOF. e prf a &reet application of ows's equaliW.
W " (E) has the Truncating Approximation Property Let (g') be the space of all bounded C complex functions fdefined in R" such that sup IOaf(x) l< kCE We provide B(R") with the topology defined by the family of seminorms {P,}k i-In this section we prove that W"''(E) has the truncating approximation property, that is, given a sequence {0i}i e C D (ffP)   such that lim 0 1 in e(ffP) the space of all C= complex functions in R", and such that {0/}i 1 is bounded in B(R"), then lira 0if-f uniformly in precompaet subsets of W"(E).
Taking e limit as j rends m infinity we ta, by Fatou's lena, L 3.2.t   cs(E).en ere exism c > 0 dending on n, p and only such at for eve () and [ 't(E).
L. Schwartz in [10] uses a weaker definition of the approximation property, strict approximation property, than Cn'othendieck's definition since it is built on the bomology of all the absolutely convex compact sets.In the next theorem, we shall use the following proposition proved in [10, p. 7].PROPOSITION 4.1 Let E be a locally convex space, F is a linear subspace of E with a locally convex topology finer than the topology inherited from E. If the identity I is an accumulation point (strict) ofL(E,F) in Lc(E) and ifFhas the (strict) approximation property thenE also has the (strict) approximation property.THEOREM 4.2 1.If E is a strict (LF)-space and has the approximation property then W"'(E) has this property.
2. If E is a Banach space with the strict approximation property then W"'(E) has this property.PROOF. 1.Let (l)i e a and (0i)i be regularizing and truncating sequences in g(P) respectively.
Consider the following diagram: T 1 * 0T k w -.L() '() O(), where h is the canonical injection, and (E) has the inductive limit topology.Note that all functions of the diagram are continuous.This implies that ({h} [0j]: j,k, E/If} C Z,(W"''(E), and by Theorems 2.3 and 3.4 the identity I belongs to the quasiclosure of Z,(Mm''(E), ggE)) in the space Z,,(W'"(E)).Now (E) is topologically isomorphic to a numerable locally convex direct sum of copies of s ).E 1, Theorem 4] (where s denotes the space of all rapidly decreasing sequences).Hence e) has the approximation property since s has it [5, ('T), p. 284 and (2), p. 245].