For $\alpha \in \mathbb{R}_+ \setminus \mathbb{N}$, I say that $f \in \mathcal{D}'(\mathbb{R}^d)$ is $\alpha$-Holder (and write $f \in C^\alpha(\mathbb{R}^d)$) if for every $x \in \mathbb{R}^d$ there is a polynomial $P_x$ such that \begin{align*} |\langle f - P_x, \varphi_x^\lambda\rangle| \lesssim \lambda^\alpha \end{align*} locally uniformly in $x$, uniformly in $\lambda \in (0,1]$ and uniformly over test functions $\varphi$ supported in the unit ball such that the derivatives of $\varphi$ up to order $\lceil \alpha \rceil$ are bounded above by $1$.
For $\alpha \in (0,1)$, I can prove that this definition coincides with the usual definition of locally $\alpha$-Holder functions. For $\alpha > 1$, I would like to show that this space is $C_{\mathrm{loc}}^{\lfloor \alpha \rfloor, [\alpha]}(\mathbb{R}^d)$ where $[\alpha]$ is the fractional part of $\alpha$.
To do this, it seems natural to me to note that by the case I can prove, if $|k| = \lfloor \alpha \rfloor$, then $g_k := D^k f$ is a $[\alpha]$-Holder function; where $D^k$ is the distributional derivative. It seems like this should imply that $f$ is a $\lfloor \alpha \rfloor$ times continuously differentiable function and that $g_k$ is its $k$-th derivative in the usual sense. However I am unable to prove this.
Is my claim true and if so how can I prove it?