Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying $$ \|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty. $$ Then the following inequality holds

$$ \|(fg)*\varphi_{\ell} - (f*\varphi_{\ell})(g*\varphi_{\ell})\|_{C^r(\bar\Omega)}\leq C \ell^{2 a-r}\|f\|_{C^{0,a}(\bar\Omega)}\|g\|_{C^{0,a}(\bar\Omega)} $$

where $\varphi \in C^{\infty}_c(\mathbb{R}^n)$ is standard mollifier and

$$ \varphi_{\ell}(x)=\frac{1}{\ell^n}\varphi(x/\ell). $$ The same estimate holds true on a compact smooth manifold. Apparently the proof should be straightforward using a partition of unity argument, but the details are not clear to me. Is this explained somewhere in detail?