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 $\alpha>0$?
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2$\begingroup$ My intuition is that such an $\alpha$-Holder-free space would be the Lipschitz-free space of the snowflaking $(X,0,d^{\alpha})$; granted that $\alpha \in (0,1]$ $\endgroup$– ABIMCommented Sep 5, 2019 at 11:12
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2 Answers
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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.
answered Sep 5, 2019 at 12:24
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2$\begingroup$ Thanks very much Mikhail. I'll take a look now $\endgroup$– ABIMCommented Sep 5, 2019 at 12:42
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As AIM_BLB says in the comments, every Holder space of exponent $\alpha < 1$ is a Lipschitz space with respect to the metric $d^\alpha$. So the answer is an immediate "yes". May I add that I discuss Holder spaces at length in my book Lipschitz Algebras (2nd edition).
answered Sep 5, 2019 at 13:58
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$\begingroup$ Thank you very much. I will definitely look that up asap. $\endgroup$– ABIMCommented Sep 5, 2019 at 14:14
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