Approximation in Hölder Spaces

We introduce new vanishing subspaces of the homogeneous Hölder space ${\dot{C}}^{0,\omega }(X,Y)$ in the generality of a doubling modulus $\omega$ and normed spaces $X$ and $Y\mathrm{.}$ For many couples $X,Y,$ we show these vanishing subspaces to completely characterize those Hölder functions that admit approximations, in the Hölder seminorm, by smooth, Lipschitz and boundedly supported functions. We present connections to bi-parameter harmonic analysis on the Euclidean space by providing applications to the compactness of the bi-commutator of two Calderón-Zygmund operators