Let $U \subseteq \mathbb{R}^n$ be the unit ball, let $k \in \mathbb{Z}_{\ge 0}$, and let $0 < \alpha < 1$. Let $f, g \in C^{k, \alpha}(U)$ be Hölder functions, i.e. $f$ and $g$ are of class $C^k$, $$ |f|_{k, \alpha} = \sum_{|I| \le k} \sup_{x \in U} |D^If(x)| + \sum_{|I| = k} \sup_{x \neq y \in U}\frac{|D^If(x) - D^If(y)|}{|x - y|^\alpha} < \infty $$ and $|g|_{k, \alpha} < \infty$.
Question. Is there a way to estimate $|fg|_{k,\alpha}$ in terms of some Hölder norms of $f$ and $g$? For instance, is it true that there exists $C > 0$ such that $$ |fg|_{k, \alpha} \le C(|f|_{k, \alpha} |g|_k + |f|_k |g|_{k, \alpha}) $$ or something of that sort?
I read in a book that the above inequality is true when $k = 0$ and I'm wondering what is the right generalization.