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?