The Existence of Primitives for Continuous Functions in a Quasi-banach Space

We show that if X is a quasi-Banach space with trivial dual then every continuous function f : [0, 1] → X has a primitive, answering a question of M.M. Popov. Let X be a quasi-Banach space and let f : [0, 1] → X be a continuous function. We say that f has a primitive if there is a differentiable function F : [0, 1] → X so that F (t) = f(t) for 0 ≤ t ≤ 1. M.M. Popov has asked where every continuous function f : [0, 1] → Lp where 0 < p < 1 has a primitive; more generally, he asks the same question for any space with trivial dual [4]. We show here that the answer to this question is positive. We remark that by an old result of Mazur and Orlicz [3], [6], every continuous f is Riemann-integrable if and only if X is a Banach space. Let us suppose for convenience that X is p-normed where 0 < p < 1, and let I = [0, 1]. Let C(I;X) be the usual quasi-Banach space of continuous functions f : I → X with the quasi-norm ‖f‖∞ = max0≤t≤1 ‖f(t)‖. We also introduce the space C(I : X) of all functions f ∈ C(I;X) which are differentiable at each t and such that the function g : I → X is continuous where g(t, t) = f (t) for 0 ≤ t ≤ 1 and g(s, t) = f(s)− f(t) s− t when s 6= t. It is easily verified that C(I;X) is a quasi-Banach space under the quasi-norm ‖f‖C1 = ‖f(0)‖+ sup 0≤s 0 we show the existence of f ∈ C 0 (I;X) with ‖Df−g‖∞ < ǫ and ‖f‖C1 < 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 ‖g − h‖∞ < ǫ and ‖h‖∞ < 1. Since h has finite-dimensional range there exists H ∈ C 0 (I;X) with DH = h. Now let n be a natural number, and let xkn = H(k/n)−H((k − 1)/n). For k = 1, 2, . . . n define fk,n ∈ C 1 0 (I;X) so that Df = 0, ‖fk,n‖C1 0 ≤ M‖xkn‖ and fk,n(1) = xkn. Then we define Fn ∈ C 1 0 (I;X) by Fn(t) = H(t)−H( k − 1 n )− fkn(nt− k + 1) for (k − 1)/n ≤ t ≤ k/n. Clearly DFn = DH = h. It remains to estimate ‖Fn‖C1 0 . Let η(ǫ) = sup |t−s|≤ǫ ‖H(t)−H(s)‖ |t− s| . It is easy to see that limǫ→0 η(ǫ) = ‖h‖∞ < 1. Now suppose k−1 n ≤ s < t ≤ k n for some 1 ≤ k ≤ n. Then ‖Fn(t)− Fn(s)‖ ≤ (η( 1 n ) + n‖fkn‖ p C1) (t− s) ≤ (η( 1 n ) +Mn‖xkn‖ )(t− s) ≤ (M + 1)η( 1 n )(t− s). Since Fn( k n ) = 0 for 0 ≤ k ≤ n we obtain that for any 0 ≤ s < t ≤ 1, ‖Fn(t)− Fn(s)‖ ≤ 2 (M + 1)η( 1 n )min(t− s, 1 n ). By taking n large enough we have ‖Fn‖C1 0 < 4M. Thus the theorem follows. We close with a few remarks on the general problem of classifying those quasiBanach spaces X so that the map D : C 0 (I;X) → C(I;X) is surjective; let us say that such a space is a D−space. The following facts are clear: