Distinctness of lip_loc^α(ℝ) and qC^{1,α}(ℝ) for 0<α<1

Determine whether the inclusion lip^α_loc(ℝ) ⊆ qC^{1,α}(ℝ) is strict for exponents α in (0,1); equivalently, ascertain whether lip^α_loc(ℝ) and qC^{1,α}(ℝ) coincide or are distinct for α∈(0,1).

Background

The paper introduces two function classes on ℝ: (i) the locally little Hölder class lip^α_loc(ℝ), defined by a vanishing Hölder modulus on bounded sets, and (ii) the quasi continuously differentiable class qC^{1,α}(ℝ), characterized by uniform continuity of a certain quotient map Φ_α on punctured squares. The authors prove the inclusion lip^α_loc(ℝ) ⊆ qC^{1,α}(ℝ).

They also show that for α=1 the classes diverge strongly: lip^1_loc(ℝ) contains only constants while qC^{1,1}(ℝ)=C^1(ℝ), so the inclusion is strict. For intermediate exponents α∈(0,1), it is unresolved whether the inclusion is strict or equality holds.