RACSAM Rev. R. Acad. Cien. Serie A. Mat. VOL. 97 (2), 2003, pp Análisis Matemático / Mathematical Analysis

Transcription

1 RACSAM Rev R Acad Cien Serie A Mat VOL 97 (2), 23, pp Análisis Matemático / Mathematical Analysis On Nuclear Maps Between Spaces of Ultradifferentiable Jets of Roumieu Type J Schmets and M Valdivia To the memory of our friend Klaus Floret Abstract If K is a non void compact subset of R r, we give a condition under which the canonical injection from E {M },b (K) into E {M },d (K) is nuclear We then consider the mixed case and obtain the existence of a nuclear extension map from E {M 1 }(F ) A into E {M 2 }(R r ) D where F is a proper closed subset of R r and A and D suitable Banach disks We finally apply this last result to the Borel case, ie when F = {} Sobre aplicaciones nucleares entre espacios de jets ultradiferenciables de tipo Roumieu Resumen Si K es un compacto no vacío en R r, damos una condición suficiente para que la inyección canónica de E {M },b (K) en E {M },d (K) sea nuclear Consideramos el caso mixto y obtenemos la existencia de un operador de extension nuclear de E {M 1 }(F ) A en E {M 2 }(R r ) D donde F es un subconjunto cerrado propio de R r y A y D son discos de Banach adecuados Finalmente aplicamos este último resultado al caso Borel, es decir cuando F = {} 1 Introduction Let us first recall the following facts about the quasi-lb-spaces studied in [7] We endow the set N N of the sequences of positive integers with the order defined by (a n ) n N (b n ) n N a n b n, n N A quasi-lb-representation in a locally convex space E is a set { A a : a N N} of Banach disks in E submitted to the following two requirements : (a) { A a : a N N} = E; (b) (a, b N N ; a b) A a A b A quasi-lb-space is a locally convex space having a quasi-lb-representation Let us the remark that the Proposition 12 of [7] leads to the following property that will be used later on Let { A a : a N N} be a quasi-lb-representation in the locally convex space E and let T be a linear map with closed graph from E onto a Banach space F Then for every compact subset K of F, there are a N N and a compact subset H of E Aa such that H A a and T H = K Presentado por José Bonet Recibido: 3 de Abril de 23 Aceptado: 25 de Julio de 23 Palabras clave / Keywords: nuclear (extension) map, ultradifferentiable jet, Roumieu type Mathematics Subject Classifications: 46E1, 26E1 c 23 Real Academia de Ciencias, España 315

2 J Schmets, M Valdivia Throughout the paper, unless otherwise stated, r designates a positive integer and M = (M n ) n N designates a sequence of positive numbers which is (a) normalized, ie M = 1 and M n 1 for every n N; (b) logarithmically convex, ie M 2 n M n 1 M n+1 for every n N We now introduce the main locally convex spaces we shall deal with The space E m (K) For every non void compact subset K of R r and integer m N, E m (K) is the Banach space introduced by Whitney For the sake of completeness we recall that its elements are the Whitney m-jets ϕ on K, ie ϕ is a family (ϕ α ) α N r, α m of continuous functions on K such that where Its norm E m (K) is defined by where sup sup α m x,y K < x y t (R m ϕ α )(x, y) := ϕ α (y) ϕ E m (K) := ϕ E m (K) + sup ϕ E m (K) (R m ϕ α )(x, y) y x m α if t + β m α α m sup x,y K x y ϕ α+β (x) := sup ϕ α K α m (y x)β β! (R m ϕ α )(x, y) y x m α The space E {M } (R r ) Let us briefly recall its definition as introduced in [3] Its elements are the C -functions f on R r such that, for every compact subset K of R r, there are A > and h > such that D α f(x) Ah α M α, x K, α N r Its topology is defined as follows For every ball b(m) := { x R r : x m} and h >, one introduces the Banach space E {M },m,h (R r ): its elements are the restrictions to b(m) of the C -functions f on R r such that D α f b(m) π {M},m,h (f) := sup < α N r h α M α and endows it with the norm π {M},m,h Finally one sets E {M } (R r ) = proj ind E {M },m,k(r r ); k N m N so E {M } (R r ) is a locally convex space More precisely as a Hausdorff projective limit of a sequence of (LB)-spaces it is a quasi-lb-space (cf [7]) The space E {M } (F ) In this notation, F designates a proper closed subset of R r A jet ϕ on F is a family ϕ = (ϕ α ) α N r of continuous function on F The elements of E {M } (F ) are the Whitney jets of class {M} on F, ie the jets ϕ on F such that for every compact subset K of F, there is h > such that ϕ α ϕ K,h := sup K < α N r h α M α 316

3 Nuclear (extension) maps and ϕ K,h := sup sup sup m N α N r x,y K α m x y (R m ϕ α )(x, y) (m α + 1)! h m+1 < M m α +1 m+1 y x Clearly E {M } (F ) is a vector space To define its topology, one proceeds as follows If F = K is compact, then for every positive integer s, E {M },s (K) denotes the vector subspace of E {M } (K) the elements ϕ of which verify ϕ K,s + ϕ K,s <, endowed with the norm K,s + K,s ; it is a Banach space Then we set E {M } (K) = ind s N E {M },s(k), a Hausdorff (LB)-space indeed hence a quasi-lb-space If F is not compact, we consider a sequence (H s ) s N of compact subsets of R r such that H s = Hs, Hs+1, K s = H s F and R r = s=1h s and set It is a Hausdorff (LF)-space hence a quasi-lb-space E {M } (F ) = proj E {M } (K s ) s N Results We establish that the canonical map from E r+1 (K) into E (K) is nuclear; this extends a result of Komatsu (cf [3]) We then obtain a result establishing the nuclearity of the canonical injection from E {M },b (K) into E {M },d (K) for some d > b Finally we consider the mixed problem In this case two sequences M 1 and M 2 are used; they are submitted to a condition of the type L { (D α f F ) α N r : f E {M 2}(R r ) } where L is a vector subspace of E {M 1}(F ) We obtain a result providing the existence of a nuclear extension map from a subspace of L into E {M 2}(R r ) We finally examine the application of this last result to the Borel case, ie when F reduces to {} Let us mention that the problem of the existence of extension maps in the mixed setting has also been examined in [1], [2], [4] and [5] 2 A nuclearity result about the E {M },s (K) spaces For the sake of completeness, let us mention with proof the following Lemma that was obtained by Komatsu in [3], under the assumption that K is regular, ie K has a finite number of connected components and there is a constant C > such that any two points x, y of any connected component H of K are the endpoints of a rectifiable curve contained in H and of length C x y Of course if K is convex, it is regular Let us recall the following property that will be used later on: if the compact subset K is regular, then, for every m N, the norms Em (K) and E m (K) are equivalent on E m (K) (cf [6] page 76, for instance) Lemma 1 For every non void compact subset K of R r, the continuous linear map is nuclear J : E r+1 (K) E (K); ϕ = (ϕ α ) α r+1 ϕ PROOF Let h be a positive integer such that K is contained in the interior of [ h, h] r We are going to use the following Banach space C r+1 H (πh): its elements are the Cr+1 -functions on πh with support contained in H and its norm is := sup α r+1 D α πh The Whitney extension theorem provides the existence of a continuous linear extension map E from E r+1 (K) into C r+1 H (πh), ie such that (Dα Eϕ)(x) = ϕ α (x) for every ϕ E r+1 (K), x K and α N r 317

4 J Schmets, M Valdivia such that α r + 1 Let us denote by E the norm of this map E For every k Z r, we then designate by v k the continuous linear functional v k : C r+1 H (πh) C; g g(y)e iyk/h dy It is well known that if v k is the norm of v k, there is a constant L > such that v k L(1 + k ) 1 r for every k Z r Finally we set u k := v k E for every k Z r as well as ψ k (x) := (2πh) r e ixk/h for every k Z r and x K Of course for every k Z, ψ k belongs to E (K) and u k to the dual of E r+1 (K) If we designate by u k the norm of u k, we successively get u k ψ k E (K) (2πh) r v k E k Z r k Z r L E (2πh) r Hence the conclusion since for every ϕ E r+1 (K), we have (Jϕ)(x) = (2πh) r k Z r e ixk/h πh 1 (1 + k ) k Z r πh r+1 < (Eϕ)(y)e iyk/h dy = k Z r ϕ, u k ψ k (x), x K Theorem 1 Let K be a non empty convex and compact subset of R r and let the sequence M verify the following condition: there are positive constants P and Q such that M n+1 P Q n M n for every n N Then for every b N, there is an integer d > b such that the continuous linear injection from E {M },b (K) into E {M },d (K) is nuclear PROOF Let the map J : E r+1 (K) E (K) as well as the u k E r+1 (K) and ψ k E (K) be defined as in the Proposition 1 and its proof We then order the family (u k, ψ k ) k Z r as a sequence (w j, φ j ) j N ; this leads to w j < and Jϕ = ϕ = ϕ, w j φ j, ϕ E r+1 (K) j=1 To every ϕ E {M },b (K) and α N r, let us associate the (r + 1)-jet Obviously we have ϕ (α) E r+1 (K) hence Now we choose an integer l such that j=1 ϕ (α) := (ϕ α+β ) β N r, β r+1 Jϕ (α) = ϕ α = j=1 l > b and bqr+1 l ϕ (α), w j φ j (1) < 1 (1 + r) 2r Then for every j N and α N r, we designate by u α,j the continuous linear functional defined on E {M },b (K) by ϕ, u α,j := 3 ϕ (α), w j l α M α, ϕ E {M },b (K), 318

5 Nuclear (extension) maps and we denote its norm by u α,j From the inequality (1), we get ϕ α K 1 3 l α M α j=1 ϕ, u α,j hence ϕ α ϕ K,l = sup K 1 α N r l α M α 3 α N r j=1 ϕ, u α,j (2) As K is convex, it is regular Therefore the norms E r+1 (K) and E r+1 (K) are equivalent on E r+1 (K): there is A > such that E r+1 (K) A E r+1 (K) on E r+1 (K) This successively leads to ϕ (α), w j wj ϕ (α) E A w j ϕ (α) E r+1 (K) r+1 (K) with ϕ (α) E r+1 (K) = sup β r+1 ϕ α+β K b α +r+1 M α +r+1 sup β r+1 ϕ α+β K b α+β M α+β b α +r+1 M α +r+1 ϕ K,b hence ϕ (α), w j A wj b α +r+1 M α +r+1 ( ϕ K,b + ϕ K,b ) As the inequalities M α +1 P Q α M α, M α +2 P Q α +1 M α +1, lead to we finally obtain M α +r+1 P r+1 Q α (r+1) Q r(r+1)/2 M α, ( ) bq r+1 α ϕ, u α,j 3A w j (bp Q r/2 ) r+1 ( ϕ l + ϕ K,b K,b ) (3) So if we set B := 3A(bP Q r/2 ) r+1, we get For every s N, this leads to u α,j B w j (1 + r) 2r α, α Z r, j N u α,j B w j s r (1 + r) 2rs B w j (2 s /2 2s ) r = 2 s B w j α =s hence u α,j u α,j 2B w j < (4) j=1 α N r j=1 s= α =s j=1 Given ϕ E {M },b (K) real and m N, the finite jet (ϕ α ) α m+1 belongs of course to E m+1 (K) and the extension theorem of Whitney provides a real function f C m+1 (R r ) such that D α f(x) = ϕ α (x) for every x K and α N r such that α m + 1 If we fix α N r such that α m as well as two 319

6 J Schmets, M Valdivia points x and y of R r, the limited Taylor formula provides the existence of some θ ], 1[ such that D α f(y) is equal to β m α D α+β (y x)β f(x) β! + β =m+1 α D α+β f(x + θ(y x)) (y x)β β! If x and y belong to K, we have x + θ(y x) K since K is convex and this formula applies as well if we replace D α+β f by ϕ α+β If ϕ is not real, we may split it into its real and imaginary parts and therefore get (R m ϕ α )(x, y) 2 β =m+1 α ϕ α+β K y x m+1 α for every ϕ E {M },b (K), m N and α N r such that α m hence successively and finally (R m ϕ α )(x, y) 2 lm+1 M m+1 (m + 1 α )! β =m+1 α 2 lm+1 M m+1 (m + 1 α )! ϕ K,l y x m+1 α ϕ α+β K y x m+1 α (m + 1 α )! l α+β M α+β β! β =m+1 α 2 lm+1 M m+1 (m + 1 α )! ϕ K,l y x m+1 α r m+1 α 2 (rl)m+1 M m+1 (m + 1 α )! ϕ K,l y x m+1 α (R m ϕ α )(x, y) (rl) m+1 M m+1 β! (m + 1 α )! y x m+1 α 2 ϕ K,l (m + 1 α )! β! Obviously we have ϕ K,rl ϕ K,l for every ϕ E {M },b (K) Therefore if J 1 is the canonical injection from E {M },b (K) into E {M },rl (K), we get Applying the inequality (2) leads then to J 1 ϕ K,rl + J 1 ϕ K,rl = ϕ K,rl + ϕ K,rl 3 ϕ K,l J 1 ϕ K,rl + J 1 ϕ K,rl ϕ, u α,j, α N r j=1 ϕ E {M },b (K) This last relation combined with the inequality (4) imply that the linear map J 1 is quasi-nuclear In the same way we may obtain an integer d > rl such that the canonical injection J 2 from E {M },rl (K) into E {M },d (K) is quasi-nuclear Therefore we know that the canonical injection J := J 2 J 1 from E {M },b (K) into E {M },d (K) is nuclear Hence the conclusion 3 Mixed problem: general case Theorem 2 Let M 1 = (M 1,n ) n N and M 2 = (M 2,n ) n N be two sequences of positive numbers which are normalized and logarithmically convex Let moreover A and B be Banach disks in E {M 1}(F ) such that A B and the canonical injection from E {M 1}(F ) A into E {M 1}(F ) B is nuclear 32

7 Nuclear (extension) maps If E {M 1}(F ) B { (D α f F ) α N r : f E {M 2}(R r ) }, then there are an absolutely convex compact subset D of E {M 2}(R r ) and a nuclear linear extension map from E {M 1}(F ) A into E {M 2}(R r ) D PROOF Let us designate by H the vector subspace of E {M 2}(R r ) the elements of which verify Sf := (D α f F ) α N r E {M 1}(F ) B Of course the map S : H E {M 1}(F ) B so defined is linear and surjective As S 1 {} clearly is a closed vector subspace of E {M 2}(R r ), we may consider H/S 1 {} as a vector subspace of E {M 2}(R r )/S 1 {} If we consider the canonical quotient map this allows to define the injective linear map Q: E {M 2}(R r ) E {M 2}(R r )/S 1 {}, T : E {M 1}(F ) B E {M 2}(R r )/S 1 {} by T (Sf) = Qf for every f H We now prove that this map T has a closed graph Let (ϕ j ) j J be a net in E {M 1}(F ) B converging to and such that the net (T ϕ j ) j J converges to u in E {M 2}(R r )/S 1 {} Let f be an element of E {M 2}(R r ) such that Qf = u As u belongs to the closure of H/S 1 {} in E {M 2}(R r )/S 1 {}, f itself belongs to the closure of H in E {M 2}(R r ) Now let { V i : i I} be a fundamental system of neighbourhoods of f in E {M 2}(R r ) and let L be the subset of the elements (i, j) of I J such that T ϕ j QV i We order L with defined by (i 1, j 1 ) (i 2, j 2 ) ( V i2 V i1 and j 1 j 2 ) For every (i, j) L, we choose an element f i,j V i such that Qf i,j = T ϕ j Of course the net (f i,j ) (i,j) (L, ) converges to f; in particular, for every α N r, the net (D α f i,j ) (i,j) (L, ) converges pointwise to D α f and as Sf i,j = (T 1 Q)f i,j = T 1 (Qf i,j ) = T 1 (T ϕ j ) = ϕ j, we get that the net (D α f i,j ) (i,j) (L, ) converges pointwise to on F, ie D α f F = hence f S 1 {} This implies u = and so T has a closed graph Since E {M 2}(R r )/S 1 {} is a quasi-lb-space, the map T is continuous (cf Corollary 15 of [7]) Therefore T A and T B are Banach disks in H/S 1 {} Let us respectively denote by E and G the Banach spaces generated by T A and T B: we have E G and by use of the hypothesis, the canonical injection W : E G is nuclear Let us denote by the norm in E and its conjugate as well, and by the norm in G So we know there are sequences (u n) n N in E and (v n ) n N in G such that For every n N, if we set u n = 1 for every n N, v n <, W u = u, u n v n for every u E λ n := ( v j ) 1/2 ( j=n j=n+1 v j ) 1/2 and ρn := ( v n /λ n ) 1/2, 321

8 J Schmets, M Valdivia we get ( ) v n 1/2 λ n ρ n = vn = λ n ( ( j=n v j ) 1/2 + ( j=n+1 v j ) 1/2 ) 1/2 hence ρ n = v n /(λ n ρ n ) if n Therefore the closed absolutely convex hull P of the set { v n /(λ n ρ n ) : n N} in G is compact Moreover it is clear that the sequence (v n /λ n = ρ n v n /(λ n ρ n )) n N converges to in G P ; therefore the closed absolutely convex hull M of { v n /λ n : n N} is a compact subset of G P Now let { A a : a N N} be a quasi-lb representation of E {M 2}(R r ) To every a N N, we associate the set B a := Q 1 (a 1 P ) A a { where a denotes the sequence (a n := a n+1 ) n N It is then clear that Ba : a N N} is a quasi-lb representation of the subspace L = a N NB a = Q 1 G P of E {M 2}(R r ) The map R: L G P ; f Qf is linear and surjective and has a closed graph As M is a compact subset of G P, the property of the quasi-lb spaces mentioned in the introduction provides b N N and a compact subset D of L Bb such that RD = QD = M Let us denote by the norm of L D as well as the one of G M From v n /λ n 1, we deduce v n λ n v n 1/2 < m=1 For every n N, we then choose an element g n L D such that Rg n = v n and g n 2 v n Now for every ϕ E {M 1}(F ) A, we consider the series σϕ = T ϕ, u n g n We denote by T : E E {M 1}(F ) A the transposed of T : E {M 1}(F ) A E and set w n = T u n for every n N If denotes the norm of E {M 1}(F ) A and of its conjugate space, we have w n = 1 for every n N hence w n g n = g n 2 v n < Moreover we also have σϕ = T ϕ, u n g n = ϕ, w n g n Therefore σ is a linear and nuclear map from E {M 1}(F ) A into E {M 2}(R r ) D To conclude we then have just to compute successively ( (D α (σϕ)) F ) α N r ( ) = Sσϕ = T 1 Qσϕ = T 1 Q T ϕ, u n g n = T 1 T ϕ, u n v n = T 1 W T ϕ = T 1 T ϕ = ϕ which proves that σ is an extension map 322

9 Nuclear (extension) maps 4 Mixed problem: Borel setting In the case F = {}, the space E {M 1}(F ) has to be replaced by the space Λ {M 1} defined as follows It is the vector space of the families c = (c α ) α N r of complex numbers for which there is h > such that c α c h := sup < α N r h α M 1, α Then a) Λ {M 1},h denotes the Banach space of the elements c of Λ {M 1} for which c h <, endowed with the norm h ; b) Λ {M 1} is the inductive limit ind n N Λ {M 1},n It is a Hausdorff (LB)-space hence a quasi-lb space: in fact if we denote by B n the closed unit ball of Λ {M 1},n, it is clear that { A a = a 2 B a1 : a N N} is a quasi-lb representation of Λ {M 1} The proof of the following Lemma is standard since a multiplication operator is nuclear whenever its symbol is absolutely summable Lemma 2 With the notations just introduced, for every a, b N N such that a 1 injection from (Λ {M 1}) a2b a1 into (Λ {M 1}) b2b b1 is nuclear In particular, the space Λ {M 1} is complete, nuclear and conuclear < b 1, the canonical As a direct consequence of the main theorem, we then get the following result Theorem 3 If the inclusion Λ {M 1},m+1 { } (f (α) ()) α N r : f E {M 2}(R r ) holds then there are an an absolutely convex compact subset D of E {M 2}(R r ) and a linear nuclear extension map from Λ {M 1},m into E {M 2}(R r ) D Finally we obtain the following Corollary as a direct consequence of Grothendieck s factorization theorem Corollary 1 If the inclusion Λ {M 1} { } (f (α) ()) α N r : f E {M 2}(R r ) holds then, for every Banach disk B of Λ {M 1}, there are an absolutely convex compact subset D of E {M 2}(R r ) and a linear nuclear extension map from Λ {M 1},B into E {M 2}(R r ) D Acknowledgement J Schmets, partially supported by the Belgian CGRI M Valdivia, partially supported by the MCYT and FEDER Project BFM References [1] Bonet J, Meise R and Taylor B A (1992) On the Range of the Borel Map for Classes of Non-Quasianalytic Functions Progress in Functional Analysis, K D Bierstedt, J Bonet, J Horváth & M Maestre (Eds), North Holland Mathematics Studies 17, [2] Chaumat J and Chollet A-M (1994) Surjectivité de l application restriction à un compact dans les classes de fonctions ultradifférentiables Math Ann, 298,

10 J Schmets, M Valdivia [3] Komatsu H, Ultradistributions, I : Structure theorems and a characterization (1973) J Fac Sc Univ Tokyo, 2, [4] Langenbruch M (1994) Extension of ultradifferentiable functions, Manuscripta Math, 83, [5] Schmets J and Valdivia M On certain extension theorems in the mixed Borel setting, J Math Anal Appl, to appear [6] Tougeron J-Cl (1972) Idéaux de fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 71 Springer [7] Valdivia M (1987) Quasi-LB-spaces, J London Math Soc, 35, [8] Whitney H, (1934) Analytic extensions of differentiable functions defined in closed sets, Trans Amer Math Soc, 36, J Schmets M Valdivia Institut de Mathématique Facultad de Matemáticas Université de Liège Universidad de Valencia Sart Tilman Bât B 37 Dr Moliner 5 B-4 LIEGE 1 E-461 BURJASOT (Valencia) Belgium Spain jschmets@ulgacbe 324