ON NONSMOOTH SUBSPACES OF SPACES OF CONTINUITY FUNCTIONS

In the geometric theory of Banach spaces of smooth functions the following problem is of considerable interest. Question. Suppose that we are give a Banach space Z of smooth functions. Does there exist an infinite-dimensional closed subspace Y ⊂ Z such that each function y ∈ Y not identically zero is not smoother than the nonsmoothest function from Z? This question was studied by many mathematicians. In the present report we give final answers of this problem in some situations. Let C[0, 1] is Banach space of continuity function on [0, 1]. Suppose we are give a setD ⊂ I = [0, 1] of positive measure. For each f: D → R, we define the modulus of continuity of f on D as follows. For each h> 0, we set Dh = {t ∈ D: t+ h ∈ D}. Then by the modulus of continuity of the function f on D we mean Ω(f, h;D) = sup 0≤δ≤h sup t∈Dδ |f(t+ h) − f(t)|. If D = I, then instead of Ω(f, h; I) we write Ω(f, h). Suppose that ω is a modulus of continuity. As is customary, by Hω(D) we denote the function space ‖f |Hω(D) ‖ = sup t∈D |f(t)|+ sup h>0 Ω(f, h;D) ω(h) Theorem 1. There exists a closed infinite-dimensional subspace E ⊂ C[0, 1] such that each function f ∈ E not identically zero has neither right- or left-hand finite derivative at any point (0, 1). Theorem 2. Choose a Holder space Hω. There exists a closed infinite-dimensional subspace G ⊂ C[0, 1], isomorphic to l1 and such that, for each function f ∈ G not identically zero, its restriction to any set of positive measure does not belong to Hω(D). We denote H0,ω the subset of space Hω, such that for each function f ∈ H0,ω exist tf ∈ (0, 1), into hold one or two equality lim δ→+0 sup h≤δ Ω(f, h;D+δ) ω(h) = 0, lim δ→+0 sup h≤δ Ω(f, h;D−δ) ω(h