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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?

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  • $\begingroup$ Do you have a response to the answer below? $\endgroup$ May 3, 2022 at 2:43
  • $\begingroup$ Oh yeah so sorry that I forgot to respond! Thank you so much for your help! I really appreciate it! $\endgroup$
    – mathisfun
    May 21, 2022 at 6:42

1 Answer 1

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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.

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