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We say a measure is $C^{1,\alpha}$ $d$-rectifiable if there is a countableunion of $C^{1,\alpha}$ $d$-surfaces whose complement has measure zero. Weprovide sufficient conditions for a Radon measure in $\mathbb{R}^n$ to be$C^{1,\alpha}$ $d$-rectifiable, with $\alpha \in (0,1]$. The conditions involvea Bishop-Jones type square function and all statements are quantitative in thatthe $C^{1,\alpha}$ constants depend on such a function. Along the way we alsogive sufficient conditions for $C^{1,\alpha}$ parametrizations for Reifenbergflat sets in terms of the same square function. Key tools for the proof comefrom David and Toro's Reifenberg parametrizations of sets with holes in theH\"{o}lder and Lipschitz categories.

Sufficient conditions for $C^{1,\alpha}$ parametrization and rectifiability