The Lipschitz-free or Arens-Eells space over a pointed separable metric space $(X,0,d)$ is a well-studied object. My question is, is an analogos Hölder-free space; for a fixed Hölder constant $\alpha>0$?



Existence of a Hölder-free space
Kalton [Collect. Math. 55 (2004), no. 2, 171–217] studied several versions of such spaces, see the definitions on page 180. This paper was reprinted in Nigel J. Kalton selecta. Vol. 2. Birkhäuser/Springer, 2016. 


