I am going through Terence Tao's "*Nonlinear Dispersive Equations (Local & Global Analysis)*" and trying to work through some of his exercises. However, I find myself being stumped by Exercise A.21. The question goes as follows:



> 
> Let $0<\alpha<1$ and $1\le p\le\infty$. If $f\in S\_x(\mathbb{R}^d)$, we define the *Hölder norm* $\|f\|\_{\Lambda\_\alpha^p(\mathbb{R}^d)}$by the formula
> $$
> \|f\|\_{\Lambda\_\alpha^p(\mathbb{R}^d)}:=\|f\|\_{L^p\_x(\mathbb{R}^d)} + \sup\_{h\in\mathbb{R}^d; 0<|h|\le1}\frac{\|f^h-f\|\_{L^p\_x(\mathbb{R}^d)}}{|h|^\alpha}
> $$
> where $f^h(x)=f(x+h)$ is the translate of $f$ by $h$. Show that
> $$
> \|f\|\_{\Lambda\_\alpha^p(\mathbb{R}^d)} \sim\_{p,\alpha,d} \|f\|\_{L^p\_x(\mathbb{R}^d)} + \sup\_{N\ge 1}N^\alpha\|P\_{N}f\|\_{L^p\_x(\mathbb{R}^d)}
> $$
> (Hint: to control the former by the latter, obtain two bounds for the $L^p\_x(\mathbb{R}^d)$ norm of $P\_Nf^h-P\_Nf$, using triangle inequality for the high frequency case $N\gtrsim |h|^{-1}$ and the fundamental theorem of calculus in the low frequency case $N\lesssim |h|^{-1}$.)
> 
> 
> 


So when attempting to control the former with the latter in the high frequency case, I took his hint and applied triangle inequality, which landed me at
$$
\|P\_Nf^h-P\_Nf\|\_{L^p\_x(\mathbb{R}^d)} \le 2\|P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}
$$


Thereafter,
$$
\frac{\|P\_Nf^h-P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}}{|h|^\alpha} \lesssim N^\alpha\|P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}
$$


From here, I found myself in a rather unpleasant dead-end since moving forward will require me to take the infinite sum over $N$'s, which in this present form will not converge.


May I know how should I side-step this problem? Or am I misunderstanding his hint? Thanks in advance!



Littlewood-Paley characterisation of Hölder regularity
You need to remember that $N$ is a **dyadic number**, so that $\sum\_{N \geq c} N^{-\alpha}$ converges for positive $\alpha$.


So you have


$$ \sum\_{N \geq |h|^{-1}} \frac{\|P\_Nf^h-P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}}{|h|^\alpha} = \sum\_{N \geq |h|^{-1}} N^\alpha\|P\_Nf^h-P\_Nf\|\_{L^p\_x(\mathbb{R}^d)}\cdot \frac{1}{N^\alpha|h|^\alpha} $$


Split the sum using Holder and put the first factor in $\ell^\infty$ and sum the second factor to get a constant (which is $\approx 1$, independently of $h$.)


