arXiv:math/9408206v1 [math.FA] 24 Aug 1994 spoe n[1]. in proved is esythat say We osdrtemap the consider quasi- othat so when function Let hsqeto spstv.W eakta ya l euto M of result old an by her continuous that show every remark We We [6], [4]. positive. dual is trivial question with this space any for question same I f space and uhta h function the that such [0 = : Let upre yNFgatDMS-9201357 grant NSF by Supported 1991 e sspoefrcneinethat convenience for suppose us Let I C Popov. otnosfunction continuous Abstract. H XSEC FPIIIE O CONTINUOUS FOR PRIMITIVES OF EXISTENCE THE → 0 1 s C , X ( ahmtc ujc Classification Subject 1] F 1 I 6= f X ( ; . ′ eaqaiBnc pc n let and space quasi-Banach a be I X ( t [0 : t Let ti aiyvrfidthat verified easily is It . : ihtequasi-norm the with = ) etecoe usaeof subspace closed the be ) X UCIN NAQAIBNC SPACE QUASI-BANACH A IN FUNCTIONS f , 1] falfunctions all of ) C a rmtv fteei ieetal function differentiable a is there if primitive a has f eso htif that show We → ( ( D I t o 0 for ) ; L X k : f p f nvriyo Missouri-Columbia of University C eteuulqaiBnc pc fcniuu functions continuous of space quasi-Banach usual the be ) g f k hr 0 where sReanitgal fadol if only and if Riemann-integrable is 0 1 C ( : [0 : I 1 ≤ I ; = 2 X , t 1] → k ) ≤ X f g → → < p < X ( f (0) 1 saqaiBnc pc ihtiilda hnevery then dual trivial with space quasi-Banach a is ,t s, k . X ..Kalton N.J. ∈ f C scniuu where continuous is ..Ppvhsakdweeeeycontinuous every where asked has Popov M.M. k k = ) ( C a rmtv,aseigaqeto fM.M. of question a answering primitive, a has ∞ sup + 46A16. . I C ( a rmtv;mr eeal,h ssthe asks he generally, more primitive; a has 1 ; I 0 X max = 1 ≤ f ; X C 1 ( X s

f(1) = x and kfkC1 ≤ Mkxk. Now suppose g ∈ C(I; X) with kgk∞ < 1. For any ǫ > 0 we show the existence 1 1/p of f ∈ C0 (I; X) with kDf −gk∞ < ǫ and kfkC1 < 4 M. Once this is achieved the Theorem follows again from a well-known variant of the Open Mapping Theorem. Since g is uniformly continuous, there is a piecewise linear function h so that

kg − hk∞ < ǫ and khk∞ < 1. Since h has finite-dimensional range there exists 1 H ∈ C0 (I; X) with DH = h. Now let n be a natural number, and let xkn = 1 H(k/n) − H((k − 1)/n). For k =1, 2,...n define fk,n ∈ C0 (I; X) so that Df =0, 1 1 kfk,nkC0 ≤ Mkxknk and fk,n(1) = xkn. Then we define Fn ∈ C0 (I; X) by k − 1 F (t) = H(t) − H( ) − f (nt − k + 1) n n kn

1 for (k − 1)/n ≤ t ≤ k/n. Clearly DFn = DH = h. It remains to estimate kFnkC0 . Let kH(t) − H(s)k η(ǫ) = sup . |t−s|≤ǫ |t − s| k−1 k 0 It is easy to see that limǫ→ η(ǫ) = khk∞ < 1. Now suppose n ≤ s

1 p p p 1/p kF (t) − F (s)k ≤ (η( ) + n kf k 1 ) (t − s) n n n kn C 1 ≤ (η( )p + M pnpkx kp)1/p(t − s) n kn 1 ≤ (M p + 1)1/pη( )(t − s). n k Since Fn( n )=0 for 0 ≤ k ≤ n we obtain that for any 0 ≤ s

We close with a few remarks on the general problem of classifying those quasi- 1 Banach spaces X so that the map D : C0 (I; X) → C(I; X) is surjective; let us say that such a space is a D−space. The following facts are clear: 2 Proposition 3. (1) Any quotient of a D-space is a D-space. (2) If X and Y are D-spaces then X ⊕ Y is a D-space. Proof. (1) Let E be a closed subspace of X and let π : X → X/E be the quotient map. Letπ ˜ : C(I : X) → C(I; X/E) be the induced mapπf ˜ = f ◦ π. We start with

the observation thatπ ˜ is surjective. If g ∈ C(I; X/E) with kgk∞ < 1 then we can 1/p−1 find f ∈ C(I; X) with kfk∞ < 2 and kπf˜ −gk∞ < 1. To do thi suppose N is an integer and let fN be a function which is linear on each interval [(k −1)/N, k/N] for

1 ≤ k ≤ N and such that πfN (k/N) = g(k/N) with kfN (k/N)k < 1 for 0 ≤ k ≤ N. For large enough N we have kg − πf˜ N k∞ < 1 and our claim is substantiated. Now if X is a D-space and g ∈ C(I; X/E) then there exists f ∈ C(I; X) with 1 πf˜ = g. Let F ∈ C0 (I; X) with DF = f. Then if G =πF ˜ we have DG = g. (2) is trivial. In [1] the notion of the core is defined: if X is a quasi- then core X is the maximal subspace with trivial dual. Theorem 4. If core X = {0} then X is a D-space if and only if X is a Banach space (i.e. is locally convex). Proof. Suppose core X = {0} and X is a D-space. Suppose DF = 0 where F ∈ 1 C0 (I; X). Let Y be the closed subspace generated by {F (s):0 ≤ s ≤ 1}. We show Y = {0}; if not there exists a nontrivial continuous linear functional y∗ on Y. Then D(y∗ ◦ F ) = 0 so that y∗(F (s)) = 0 for 0 ≤ s ≤ 1. But then y∗ = 0 on Y. We conclude that Y = {0} and so F = 0. Hence D is one-one and surjective and by the Theorem D is an isomorphism. 1 Let M be a constant so that kDF k∞ ≤ 1 implies kF kC1 ≤ M for F ∈ C0 (I; X). Let φ be any C∞− real function on R with φ(t)=0for t ≤ 0 and φ(t)=1for t ≥ 1. ′ Let K = max0≤t≤1 |φ (t)|. For any N and any x1,...xN ∈ X with max kxkk ≤ 1, N 1 we define F (t) = Pk=1 φ(Nt − k + 1)xk. Then F ∈ C0 (I; X) and kDF k∞ ≤ NK. Hence kF (1)k ≤ NMK, i.e. 1 k (x1 + ··· + x )k ≤ MK. N N This implies X is locally convex. Combining Proposition 3 and Theorem 4 gives that if X is a D-space then X/core X is a Banach space. It is, however, possible to construct an example to show that the converse to this statement is false, and there does not seem, therefore to be any nice classification of D-spaces in general. To construct the example we observe the following theorem. First for any quasi-

Banach space X let aN (X) = sup{kx1 +···+xN k : kxik ≤ 1} (so that aN (X) ≥ N). 3 Theorem 5. Suppose X is a D-space; then for some constant C we have aN (X) ≤ CaN (core X).

Proof. Let bN = aN (core X). Suppose x1,...,xN ∈ X with kxik ≤ 1 and define as N in Theorem 4, F (t) = Pk=1 φ(Nt − k + 1)xk. Then kDF k∞ ≤ NK and so by the 1 Open Mapping Theorem, for some constant M = M(X), there exists G ∈ C0 (I; X)

with DG = DF and kGkC1 ≤ MNK. Then kG(k/N) − G((k − 1)/N)k ≤ MK for 1 ≤ k ≤ N. Let H(t) = F (t) − G(t). Since DH = 0 and X/core X is a Banach space H

has range in core X. Now for 1 ≤ k ≤ N, H(k/N) − H((k − 1)/N) = xk − (G(k/N) − G((k − 1)/N) so that kH(k/N) − H((k − 1)/N)k ≤ (M pKp + 1)1/p. p p p Hence if C = M K + 1, we have kH(1)k ≤ CbN or kx1 + ··· + xnk ≤ Cbn. To construct our example we start with the Ribe space Z ([2],[5]) which is a space with a one-dimensional subspace L so that Z/L is isomorphic to ℓ1. A

routine calculation shows aN (Z) ≥ cN log N for some c > 0. Then let Y be any

quasi-Banach space with trivial dual so that aN (Y ) = o(N log N) (for example a Lorentz space L(1, p) where 1 < p < ∞). Let j : L → Y be an isometry and let X be the quotient of Y × Z by the subspace of all (jz,z) for z ∈ L. Then Z embeds

into X so that an(X) ≥ cN log N but core X ∼ Y so that X cannot be a D-space. However X/core X is isomorphic to Z/L which is a Banach space.

References

1. N.J. Kalton, Curves with zero derivative in F-spaces, Glasgow Math. J. 22 (1981), 19-29. 2. N.J. Kalton, N.T. Peck and J. W. Roberts, An F-space sampler, London Math. Soc. Lecture Notes 89, Cambridge University Press, 1985. 3. S. Mazur and W. Orlicz, Sur les espaces lin´eaires m´etriques I, Studia Math. 10 (1948), 184-208. 4. M.M. Popov, On integrability in F -spaces, to appear, Studia Math. 5. M. Ribe, Examples for the nonlocally convex three space problem, Proc. Amer. Math. Soc. 237 (1979), 351-355. 6. S. Rolewicz, Metric linear spaces, PWN Warsaw, 1985.

Department of Mathematics, University of Missouri Columbia, MO 65211, U.S.A. E-mail address: [email protected]

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