I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem:


Recall that the Holder norm of a function is defined as follows:
$$[f]\_{C^{\alpha}} \mathrel{:=} \lVert f\rVert\_{L^{\infty}} + \sup\_{x,y}\frac{\lvert f(y)-f(x)\rvert}{\lvert y-x\rvert^{\alpha}}$$
Prove that for any $\alpha \in (0,1)$ and any two functions $f,g \in C^{\alpha}$, we have the following estimate:
$$[fg]\_{C^{\alpha}} \leq c(\alpha,d)([f]\_{C^{\alpha}}\lVert g\rVert \_{L^{\infty}}+[g]\_{C^{\alpha}}\lVert f\rVert\_{L^{\infty}})$$
where $c=c(\alpha,d) > 0$ is some positive constant depending only on $\alpha$, $d$. I have tried applying the Littlewood–Paley projection operators $P\_{k}$ ($k \in \mathbb{Z}$) to $f$ to obtain some estimate of the Hölder norms, but I have made only limited progress. Any hint/idea?



Estimate of Hölder Norms (Littlewood–Paley theory)
Let $a:=\alpha$, $[h]\_a:=[h]\_{C^a}$, and $\|h\|\_\infty:=\|h\|\_{L^\infty}$. For any distinct $x$ and $y$,
\begin{align\*}
 |f(x)g(x)-f(y)g(y)|&=|f(x)g(x)-f(y)g(x)+f(y)g(x)-f(y)g(y)| \\ 
 &\le|f(x)g(x)-f(y)g(x)|+|f(y)g(x)-f(y)g(y)| \\ 
 &=|f(x)-f(y)|\,|g(x)|+|g(x)-g(y)|\,|f(y)| \\ 
 &\le[f]\_a|x-y|^a\,\|g\|\_\infty+[g]\_a|x-y|^a\,\|f\|\_\infty, 
\end{align\*}
whence
\begin{align\*}
 \frac{|f(x)g(x)-f(y)g(y)|}{|x-y|^a}&\le[f]\_a\,\|g\|\_\infty+[g]\_a\,\|f\|\_\infty. 
\end{align\*}
Also,
\begin{equation}
\|fg\|\_\infty\le\|f\|\_\infty\|g\|\_\infty\le[f]\_a\,\|g\|\_\infty\le[f]\_a\,\|g\|\_\infty+[g]\_a\,\|f\|\_\infty. 
\end{equation}
So,
\begin{equation}
 [fg]\_a\le2([f]\_a\,\|g\|\_\infty+[g]\_a\,\|f\|\_\infty),
\end{equation}
as desired.


