Whitney extension theorem for Hölder spaces

Question

The usual Whitney extension theorem says that Whitney data with remainders like $R_\alpha=o(x-y)^{k-|\alpha|}$ extends to a $C^k$ function. If we also have $R_\alpha=o(x-y)^{k+\lambda-|\alpha|}$ for some $\lambda\in(0,1]$, do we get a $C^{k+\lambda}$ function? That is, a function whose $k$th derivative is $\lambda$-Hölder continuous?

Answer 1

Yes, see Stein's Singular Integrals and Differentiability Properties of Functions Chapter VI Theorem 4: https://archive.org/details/singularintegral0000stei