In a book 'Elliptic Partial Differential Equations of Second Order' by Gilbarg and Trudinger I stumbled upon the following inequality for two functions $f, g$ and their Holder norms in bounded domains $\Omega$ (eq 4.7):
$$[f g]_{\gamma} \leq \max(1, d^{\alpha + \beta – 2\gamma}) [f]_{\alpha} [g]_{\beta} $$
where $\gamma = \min(\alpha, \beta)$ and $d = \text{diam}\ \Omega$, $[f]_{\alpha} = \sup_{x \neq y} \frac{ f(x) – f(y) }{ \lvert x – y \rvert^{\alpha}}$
It's easy to prove that $fg$ is $\gamma$-Holder (we use Holder embedding in compact spaces):
$$[fg]_{\gamma} = \sup_{x \neq y} \frac{ \lvert f(y) g(y) – f(x) g(x) \rvert }{ \lvert x – y \rvert ^{\gamma} }\leq \sup_{x \neq y} \bigg( | f(x) | \frac{ |g(x) – g(y)| }{|x-y|^{\gamma}} + |g(x)|\frac{|f(x) – f(y)|}{|x-y|^{\gamma}} \bigg) \leq |f|_{\infty} [g]_{\gamma} + |g|_{\infty} [f]_{\gamma} $$
so $[fg]_{\gamma}$ is bounded by finite norms of $f,g$.
Then $|f|_{\infty} = \sup_{x \in \Omega} |f(x)| = \sup_{x \in \Omega} | f(x) – f(x_0) + f(x_0) | \leq |f(x_0)| + [f]_{\gamma} |x – x_0|^{\gamma}$
now if $x_0$ is taken such that $f(x_0) = 0$, then we can bound it by:
$$[fg]_{\gamma} \leq 2 [f]_{\gamma} [g]_{\gamma} d^{\gamma}$$
which is far from satisfying – not to mention that such $x_0$ does not necessarily have to exist.
Best Answer
So I checked the Gilbarg and Trudinger, but the equation (4.7) is actually not for the Hölder seminorms $[f]_\alpha$ but for the norms $$ \|f\|_{C^{\alpha}} = \|f\|_{C^{0,\alpha}} = \|f\|_{L^\infty} + [f]_\alpha. $$ For these norms, the relation indeed holds since (taking to simplify $\alpha =\beta$) $$ \begin{align*} \|fg\|_{C^{0,\alpha}} &= \|fg\|_{L^\infty} + [fg]_\alpha \\ &≤ \|f\|_{L^\infty}\|g\|_{L^\infty} + \|f\|_{L^\infty}[g]_\alpha + [f]_\alpha \|g\|_{L^\infty} \\ &≤ \|f\|_{C^{0,\alpha}} \, \|g\|_{C^{0,\alpha}} \end{align*} $$